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%\Large
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%\begin{center}
%Christoph Sieg\footnote{\label{mi} csieg@mi.infn.it} 
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%Universit\`a degli Studi di Milano \\
%Via Celoria 16, 
%20133 Milano, 
%Italy \\[2mm]
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\begin{flushright}\footnotesize
\texttt{UUITP-34/11} \\
\texttt{HU-EP-11/62} \\
\texttt{HU-MATH-2011-26}
\vspace{0.8cm}\end{flushright}

\begingroup\LARGE
Four-loop anomalous dimensions in %the cubic 
Leigh-Strassler
deformations
\par\endgroup
\vspace{1.5em}
\begingroup\large
J.\ A.\ Minahan ${}^a$, C.\ Sieg ${}^b$% use subscripts to distinguish multiple authors
\par\endgroup
\vspace{1em}
\begingroup\itshape
$^a$  Department of Physics and Astronomy, Uppsala University\\
SE-751 08 Uppsala, Sweden\\
%\vspace{0.2cm}
$^b$ 
Institut f\"ur Mathematik und Institut f\"ur Physik, Humboldt-Universit\"at zu Berlin\\
Johann von Neumann Haus, Rudower Chaussee 25, 12489 Berlin, Germany
\par\endgroup
\vspace{1em}
\begingroup\ttfamily
joseph.minahan@fysast.uu.se\\
csieg@math.hu-berlin.de
\par\endgroup
\vspace{1.5em}
\endgroup

\paragraph{Abstract.}

%We determine the four-loop dilatation operator that yields the anomalous 
%dimensions of certain operators composed of scalars in the 
%Leigh-Strassler deformed  $\mathcal{N}=4$ SYM theory.
We determine the scalar part of the four-loop  chiral dilatation operator for  Leigh-Strassler deformations of  $\mathcal{N}=4$ super Yang-Mills.
% the closed flavor $SU(2)$ 
% subsector of $\mathcal{N}=4$ SYM
% by disentangling 
% contributions that are identified within this subsector. This allows us to 
% determine the mixing among scalar operators in the Leigh-Strassler 
% deformed theory. 
This is sufficient   to find the four-loop anomalous dimensions for operators in closed scalar subsectors.
%In addition to the contribution of the dressing phase we find 
%further transcendental $\zeta(3)$ contributions.
This includes the $SU(2)$ subsector of the (complex) $\beta$-deformation, where we explicitly compute the anomalous dimension for operators with a single impurity.  It also includes the ``3-string null" operators of the cubic Leigh-Strassler deformation. 
%we obtain from this result the anomalous dimensions of the 
%respective composite operators with a single impurity. 
%In this latter case  
Our four-loop results show
% which was previously alluded to in 1108.1583, 
%shows 
that the rational part of the anomalous dimension
is consistent with a conjecture made in arXiv:1108.1583 based 
on the three-loop result of arXiv:1008.3351 and the $\NN=4$ 
magnon dispersion relation. 
Here we find additional $\zeta(3)$ terms.

%already entered the paper 1108.1583 by one of us, where it supplements 
%the conjecture in the first version by additional transcendental 
%contributions. 

%They occur in the 
%respective dilatation operator, and are delayed by one-loop order, i.e.\ 
%they have transcendentality three at four loops and  
%appear in addition to the ones of the dressing phase.                      
%\paragraph{Mathematics Subject Classification (2000).} 
%MSC Numbers
% http://www.ams.org/msc

\paragraph{Keywords.} 
{\it PACS}: 11.15.-q; 11.30.Pb; 11.25.Tq\\
{\it Keywords}: Super-Yang-Mills; Superspace; Anomalous 
dimensions; Integrability;

\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

\label{sec:introduction}

The spectrum of single trace operators in planar  $\NN=4$ supersymmetric Yang-Mills (SYM) is solvable,  at least  in principle if not always in practice, %.
because of an underlying integrability
%The underlying reason is integrability
%for %in 
%computing the
%spectrum of 
%%composite
% single-trace operators in 
% the planar limit of certain 
%superconformal gauge theories 
%in the planar limit 
(see \cite{Beisert:2010jr} for a comprehensive review).
%The %first and 
%most 
%thoroughly %intensely 
%studied example 
%%in this context 
%is %the  
%$\mathcal{N}=4$ supersymmetric Yang-Mills (SYM) theory. 
%\\
Starting with $\NN=4$ SYM,  there exists a class of deformations  that break
 the superconformal symmetry down to $\NN=1$ but preserve the integrability.  However, there exist other deformations that destroy the %do not appear to preserve 
 integrability and we can ask ourselves to what extent  one can calculate the spectra of these theories.  %can be determined.  
  A possible way forward is to compute the   dilatation operator to high enough loop order such that a pattern emerges.   
% In this paper we find the dilation operator for chiral operators  dimensions to four-loop order for certain operators in two classes of  these nonintegrable theories.
%Other examples can be obtained by deforming 
%the 
%$\mathcal{N}=4$ SYM
%superpotential %theory, 
%%thereby 
%breaking some of the supersymmetries.

%Those %Such 
%deformations that 
%%still 
%preserve

The general class of deformations of $\NN=4$ SYM that preserve an
$\mathcal{N}=1$ superconformal symmetry were first catalogued 
%introduced 
by 
Leigh and Strassler \cite{Leigh:1995ep}. The Leigh-Strassler %respective 
superpotential is given by
\begin{equation}\label{superpot}
W=i\kappa\Big[\tr(XYZ-q XZY)+\frac{h}{3}\tr(X^3+Y^3+Z^3)\Big]\col
\end{equation}
which %and it 
depends on three complex parameters $\kappa$, $q$ and $h$, although
%by a chiral field phase rotation 
the imaginary part of $\kappa$ can %finally 
be eliminated
by a chiral field phase rotation.  The deformation is exactly marginal %, i.e.\ preserves 
%also the conformal symmetry, 
if the Yang-Mills 
coupling $g_\YM$ and  the deformation parameters  satisfy the relation
\begin{equation}\label{couprel}
2g_\YM^2=\kappa\bar\kappa(1+q\bar q+h\bar h)+\mathcal{O}((\kappa\bar\kappa)^4)\col
\end{equation}
where we have allowed for a fourth-order correction, to be discussed in 
more detail below.  
This leaves a  three complex dimensional space of 
$\mathcal{N}=1$ superconformal theories.
%Leigh and Strassler's argument is essentially an existence proof for
%the  marginal directions, but  the relation of the couplings is known,
%at least to the first few loop orders,  and in the planar limit is
%given by \cite{Aharony:2002tp} 
%\begin{equation}\label{couprel}
%2g_\YM^2=\kappa\bar\kappa(1+q\bar q+h\bar h)+\mathcal{O}((\kappa\bar\kappa)^4)\col
%\end{equation}
%where we have allowed for a fourth-order correction to be discussed in 
%more detail below.  
%%%%%
Setting the coefficients  to $\kappa=g_\YM$, $q=1$, $h=0$, we recover
 the  $\mathcal{N}=4$  superpotential
  \begin{equation}\label{N4superpot}
W=ig_\YM\tr\comm{X\,}{\,Y}Z\pnt
\end{equation}
%%%%%


The deformation \eqref{superpot} includes some  interesting
special cases.
%First of all, one recovers the  $\mathcal{N}=4$ SYM theory 
%for the choice $\kappa=g_\YM$, $q=1$, $h=0$, where the 
%superpotential becomes
%\begin{equation}\label{N4superpot}
%W=ig_\YM\tr\comm{X\,}{\,Y}Z\pnt
%\end{equation}
%
The most well known deformation is the so-called $\beta$-deformation
(see \cite{Zoubos:2010kh} for a review)
where $q=\e^{-2i\pi\beta}$, $h=0$ with  $\beta$ real.  Inserting this into 
\eqref{couprel} then yields $\kappa=g_\YM$ which is exact in the planar 
limit \cite{Khoze:2005nd,Mauri:2005pa}.
With a field redefinition, the $\beta$-deformed superpotential  can be 
recast %cast 
into the form
\begin{equation}
W=ig_\YM\tr(e^{ i\pi \beta}XYZ-e^{- i\pi\beta}XZY)\pnt
\end{equation}
Furthermore, the computation 
of  %for
the spectrum 
for %of 
local operators is an integrable problem
\cite{Roiban:2003dw,Berenstein:2004ys,Beisert:2005if,Frolov:2005ty}.
In \cite{Berenstein:2004ys} it was shown 
that at the one-loop level %at the one-loop level that
the corresponding Bethe equations are the same as in 
% the case of
$\mathcal{N}=4$ SYM,
except for a $\beta$-dependent shift.  This was extended to all loops
in \cite{Beisert:2005if}.  The supergravity dual for the
$\beta$-deformed theory is known \cite{Lunin:2005jy} and its
world-sheet theory has been shown to be classically integrable, even
though it is not a coset \cite{Frolov:2005dj,Frolov:2005ty}. 
Other integrable deformations with nonzero values of $h$ can be obtained by acting on the $\beta$-deformed theories with similarity transformations  \cite{Bundzik:2005zg,Mansson:2007sh,Mansson:2008xv}.

If the deformation is generalized 
to complex $\beta$, %by allowing the parameter $\beta$ to be complex, 
the relation of the couplings has to be  altered at four-loop order
\cite{Elmetti:2006gr,Elmetti:2007up},
as 
indicated %anticipated
 in \eqref{couprel}. 
The resulting
spin-chain is a known integrable model only 
in a certain subsector
\cite{Berenstein:2004ys}.
 This %mentioned 
subsector plays 
an %a very 
important role in 
%the 
$\mathcal{N}=4$ SYM
%theory 
and in its (complex) $\beta$-deformation, where it is closed, 
at least perturbatively \cite{Minahan:2005jq}.
It consists of operators %that are built by using only two out of the three
composed of %only 
two flavors of complex scalar fields. %, say $X$ and $Z$, 
%which are the lowest
%components of  chiral superfields, %also denoted by 
%say $X$ and $Z$.
%complexified scalar fields that appear as lowest components in the 
%component 
%expansions of the chiral 
%superfields $X$, $Y$ and $Z$. 
The subsector %It 
is called the $SU(2)$ subsector, since an
$SU(2)$ subgroup of the
$SU(4)$ $R$-symmetry of $\mathcal{N}=4$
transforms the two flavors into 
each other. We will stick with %to 
this name, even if the $SU(2)$ $R$-symmetry is 
broken in the presence of the (complex) 
$\beta$-deformation.
In a formulation with manifest $\mathcal{N}=1$ supersymmetry, these
complex scalars are the lowest components of chiral superfields.
Hence, being composites of two of these three superfield flavors, the operators of the $SU(2)$ subsector are themselves chiral
superfields. 

There exist a bigger subsector that consists of all possible
chiral composite operators. This means,  the latter can contain 
all three types of chiral superfield flavors $X$, $Y$, $Z$, and also the 
chiral superfield strength $W_\alpha$. 
Starting at two loops, three chiral
field flavors can  transform into two  $W_\alpha$ and {\it vice versa}.
If $\kappa\ne0$, $h=0$ in \eqref{superpot}, this mixing 
can only occur if all three interacting chiral field flavors are different, 
while at  $h\ne0$ a mixing occurs even if all these flavors are identical.
This guarantees that the $SU(2)$ subsector is closed whenever $h=0$, but it 
also means that in the case $h\ne0$ the mixing extends to the bigger subsector.
When restricting to the lowest lying components of the superfields, 
the field content of this bigger subsector is reduced to the three 
complexified scalars and the two-component gaugino $\psi_\alpha$. In $\NN=4$ SYM this is the $SU(2|3)$ 
subsector \cite{Beisert:2003ys}.  

%sector is %hence 
%spanned by chiral operators composed %out 
%of  two flavors of chiral superfields.

Let us now consider those deformations that have closed scalar subsectors and restrict ourselves to the planar limit. For  the (complex) $\beta$-deformations, the simplest 
%composite 
single-trace operators 
%of this subsector 
are %contain 
$L$ copies of 
%only
 a single superfield flavor $X$,
%and are for instance given by
\begin{equation}\label{SU2groundstate}
\tr X\dots X \col
\end{equation}
 %defined as the number of elementary fields $Z$
which %they 
are protected from quantum corrections. In the spin chain picture
each of these operators 
is % becomes 
the ferromagnetic ground state 
of %for
 a closed spin chain with fixed length $L$. 
One constructs excited states by changing the %field 
flavor of 
one or more %some 
of the %vaccum 
fields.
The composite operator that contains a single 
changed flavor %different field flavor 
\begin{equation}\label{SU2oneexstate}
\tr YX\dots X 
\end{equation}
corresponds to a spin chain with one excitation
(magnon). % moving within the remaining vacuum fields.
While in the $\mathcal{N}=4$  case %theory also the
this operator is still protected,
%composite operators \eqref{SU2oneexstate} are protected, 
%they acquire anomalous dimensions 
in the  (complex) $\beta$-deformed theory it acquires an anomalous dimension.  We can add more excitations by adding more $Y$ fields.  Adding $Z$ 
fields takes us out of the $SU(2)$ subsector.


%Less attention has been paid to  deformations with nonzero $h$,
%although  interesting results have still been found.   
%For deformations with nonzero $h$  \cite{Bundzik:2005zg,Mansson:2007sh,Mansson:2008xv}, it has been
%shown that for %the choice 
%$q=(1+\rho)e^{\pi i n/3}$, $h=\rho e^{\pi
%  im/3}$ where $\rho$ is real and $n$ and $m$ are integers, the
%$R$-matrix in the $SU(3)$ subsector satisfies the Yang-Baxter equation.
%However, it was also shown that all such theories are related to the
%$\beta$-deformed theories via similarity transformations
%\cite{Bundzik:2005zg}.  Another interesting branch has $q=0$ and
%$|h|=1$, which was shown to be integrable at the one-loop level
%\cite{Mansson:2008xv} and also has been conjectured to require no
%higher order corrections to \eqref{couprel} \cite{Bork:2007bj}. 

The only other way to have a closed scalar subsector is to
 take the 
 limit $\kappa\to0$ with fixed $\kappa|h|=\sqrt{2}g_\YM$, such that
the superpotential becomes 
\begin{equation}
W=i\sqrt{2}g_\YM\tr(X^3+Y^3+Z^3)\pnt
\end{equation}
This special case, called the (Fermat) cubic Leigh-Strassler deformation, was recently  analyzed in \cite{Minahan:2011dd}.
This  deformation has no closed subsector resembling the 
$SU(2)$ subsector of the undeformed theory.  However, because the superpotential does not have the ``hopping" term there is no direct mixing of scalar operators in the planar limit.  Scalar mixing can still occur indirectly through intermediate scalar-fermion mixing, but if the operator does not have a sequence of three or more identical scalar fields, then this mixing will  not occur either.  We will call such operators ``3-string null''. 
In the planar limit these operators do not mix with other operators.

In the cubic Leigh-Strassler theory, the ground states are anti-ferromagnetic in nature, having the general form \cite{Minahan:2011dd}
\begin{equation}\label{cubicLSgroundstate}
\tr XYXZYZ\dots XY \col
\end{equation}
where no two neighboring fields have the same flavor.
These operators are
protected 
in the planar limit
to all orders in perturbation theory.  
Operators of the type
\begin{equation}\label{cubiconeexstate}
\tr  XYXYYZ\dots XY
\col 
\end{equation}
where  there are two neighboring fields with the same flavor
correspond to the first excited states. They are not protected,
but they are  3-string null and hence 
do not undergo operator mixing either.
The lack of mixing makes it possible to formulate an all-loop conjecture about the 
rational part of their anomalous dimensions \cite{Minahan:2011dd}.  The conjecture is applicable to any 3-string null operator \cite{Minahan:2011dd}.

The upshot is that we have two classes of Leigh-Strassler theories where the mixing is closed to scalar operators; the $SU(2)$ subsector of the complex $\beta$-deformed theories and the 3-string null operators in the cubic Leigh-Strassler theory.  For these two cases we only need the scalar part of 
the ``chiral'' dilatation operator, where 
``chiral'' indicates the restriction to the field content of the
$SU(2|3)$ subsector.

%such that their identical anomalous dimensions could be conjectured in

%In this paper we determine to four loops the 
%dilatation operator of the planar $\mathcal{N}=1$ SYM theory with
%the Leigh-Strassler deformed superpotential \eqref{superpot}
%that determines the mixing among 
%single-trace operators built built by choosing 
%out of two different chiral field flavors as building blocks.
%In the special cases where these operators form a closed subsector, 
%i.e.\ in the (complex) $\beta$-deformed $\mathcal{N}=4$ SYM theory, 
%this determines the entire four-loop mixing, while in the generic case 
%we only obtain the partial mixing of some particular operators within the 
%full mixing problem. This allows us 
%to determine in the cubic Leigh-Strassler deformation
%the four-loop anomalous dimension
%of the operators \eqref{cubiconeexstate} that do not undergo any mixing.
%Based on our findings, the conjecture in the 
%first version of \cite{Minahan:2011dd} was supplemented by additional 
%transcendental terms.

In this paper we determine  the scalar part of the chiral
dilatation operator to four-loop order.   
The evaluation of the relevant Feynman diagrams becomes manageable by %just 
determining the deviations from the four-loop dilatation operator in the $SU(2)$ subsector of the $\NN=4$ SYM theory. The latter has been determined \cite{Beisert:2007hz} 
%under the assumption of integrability
 as one of 
the conserved local charges \cite{Beisert:2003tq,Beisert:2005wv} using integrability and the postulated form of the magnon dispersion relation.
Hence, by doing perturbation theory in the general Leigh-Strassler case, we 
determine the scalar part of the chiral dilatation operator. These more general deformations are not believed to be integrable.
But our result is also valid for the mixing between all three 
scalar chiral fields of $\mathcal{N}=4$ SYM.

% stipulating  the form of the magnon dispersion relation in $\NN=4$ SYM.  The dispersion relation is determined by symmetries up to an overall function $h^2(g)$ of the 't Hooft coupling \cite{Beisert:2005tm}.  It is known that $h^2(g)=4g^2$ to three-loop order \cite{Sieg:2010tz} and also for very large values of $\la$ by the AdS/CFT correspondence.  We thus make the seemingly reasonable assumption that $h^2(g)=4g^2$ to any loop order.  The dispersion relation then relates the contributions of the less tractable Feynman diagrams to those that are more straightforward to compute.

 The scalar part we find for the chiral dilatation operator is valid for {\it any} Leigh-Strassler deformation.   It is complete when acting on a closed scalar subsector, hence we can apply it to the above cases to find the anomalous dimensions to four-loop order.  When acting on the 3-string null operators in the cubic Leigh-Strassler theory we find that the rational contributions to the anomalous dimensions are consistent with an all-loop conjecture made in  \cite{Minahan:2011dd}.  Furthermore, we see that the  four-loop  anomalous dimensions contain  additional  transcendental terms.

%The paper is organized as follows.
%In section  \ref{sec:Feynmandiag} we perform the Feynman diagram
%calculation that determines the modifications that yield our result in 
%terms of the four-loop dilatation operator in $\mathcal{N}=4$ SYM theory.
%In section \ref{sec:oneimpstates} we simplify the result and
%also determine the eigenvalues of 
%the composite operators \eqref{SU2oneexstate} and \eqref{cubiconeexstate} 
%respectively in the 
%(complex) $\beta$-deformation and the cubic Leigh-Strassler deformation.
%In section \ref{sec:conclusions} we draw our conclusions. 
%Several technical details concerning the so-called chiral functions, 
%similarity transformations and the occurring loop integrals
%are included in various appendices.

The paper is organized as follows.
In section  \ref{sec:Feynmandiag} we analyze the Feynman diagrams that
modify the scalar part of the chiral four-loop dilatation operator 
of $\mathcal{N}=4$ SYM theory
for  Leigh-Strassler deformations.
In section \ref{sec:oneimpstates} we simplify the results from the previous 
section and
 determine the  anomalous dimensions of
the one impurity  operators in the 
(complex) $\beta$-deformation and the 3-string null operators in the cubic Leigh-Strassler deformation.
In section \ref{sec:conclusions} we draw our conclusions. 
Several technical details concerning the so-called chiral functions, 
similarity transformations and relevant loop integrals
are included in various appendices.


\section{Feynman diagram analysis}
\label{sec:Feynmandiag}

The perturbative spectrum of local composite operators in a conformal field theory is given by the bare dimensions of the operators plus their anomalous dimensions.
%the spectrum of composite operators, i.e.\ their conformal dimensions,
%are given as a sum of the bare scaling dimensions to which the 
%anomalous dimensions are added. 
The latter are generated 
by the renormalization of the composite operators % as
\begin{equation}\label{opren}
\mathcal{O}_{a,\text{ren}}
=\mathcal{Z}_{a}{}^b(\lambda,\varepsilon)\mathcal{O}_{b,\text{bare}}
\col
\end{equation}
which in general imply a mixing among the operators.
%The index $a$ labels the different operators, and
$\mathcal{Z}$ is the matrix-valued renormalization constant
that is given as a power series in the %small
 't Hooft coupling constant 
$\lambda=g_\YM^2N$ and %that 
absorbs the overall UV divergences
which are %as 
generated at each loop order. % by the Feynman diagrams.
In $D=4-2\varepsilon$ dimensions the divergences appear %manifest themselves 
as poles in $\varepsilon$.

We determine the anomalous dimensions as eigenvalues of 
the dilatation operator, which in terms of $\mathcal{Z}$ is defined as
\begin{equation}\label{DinZ}
\mathcal{D}
=\mu\frac{\de}{\de\mu}\ln\mathcal{Z}(\lambda\mu^{2\varepsilon},\varepsilon)
=\lim_{\varepsilon\rightarrow0}\left[2\varepsilon\lambda
\frac{\de}{\de \lambda}\ln\mathcal{Z}(\lambda,\varepsilon)\right]
\pnt
\end{equation}
The logarithm in the above description is understood as a series 
expansion in
the orthonormalized basis of composite operators 
for the unrenormalized theory, such that the leading contribution
to $\mathcal{Z}$ is the identity.
All higher order poles in $\ln\mathcal{Z}$ must %have to
 cancel, leading to %and this is the reason why 
 the second relation in \eqref{DinZ}. % holds.
%A diagonalization of $\mathcal{D}$ then yields the anomalous dimensions as eigenvalues.
The anomalous dimensions are then the eigenvalues of $\mathcal{D}$.

%By computing %employing 
%Feynman diagrams 
%using %in 
%an $\mathcal{N}=1$ superfield formulation,
We first calculate the renormalization constant $\mathcal{Z}$
for operators composed of scalar chiral superfields
by computing the relevant Feynman diagrams 
using an $\mathcal{N}=1$ superfield formulation.
%, and then
We then
use \eqref{DinZ} to derive %from it 
the dilatation operator.
For a more detailed description %of the calculational details 
we refer the reader % respectively to \cite{Sieg:2010jt} and 
%\cite{Sieg:2010tz}.
to \cite{Sieg:2010jt,Sieg:2010tz}. It should be understood that $\DD$ is the scalar part of the chiral dilatation operator, since 
we have restricted ourselves to scalar chiral operators. 
The dilatation operator can be expressed as the series expansion
\begin{eqnarray}
\DD=\sum_{n=1}^\infty g^{2n}\DD_n\col
\qquad
g\equiv\frac{\sqrt{\la}}{4\pi}\pnt
\end{eqnarray}
For $\mathcal{N}=4$ SYM, the first three terms in the expansion were found to be % then determined as
\begin{equation}\label{D1D2D3}
\begin{aligned}
\mathcal{D}_1
&={}-{}2\chi(1)
\col\\
\mathcal{D}_2
&={}-{}2[\chi(1,2)+\chi(2,1)]+4\chi(1)
\col\\
\mathcal{D}_3
&=-4(\chi(1,2,3)+\chi(3,2,1))+4i\epsilon_2[\chi(2,1,3)-\chi(1,3,2)]-4\chi(1,3)\\
&\phantom{{}={}}
+16(\chi(1,2)+\chi(2,1))-16\chi(1)-4(\chi(1,2,1)+\chi(2,1,2))
\col
\end{aligned}
\end{equation}
where $\epsilon_2=-\frac{i}{2}$ is a parameter that can be changed by
similarity transformations of the basis of operators.

The terms in \eqref{D1D2D3} are expressed using the very convenient 
basis of so-called chiral functions. 
They capture the structure of the chiral and
anti-chiral superfields within the Feynman diagrams %in superspace % and
%have been introduced in 
\cite{Fiamberti:2007rj,Fiamberti:2008sh} 
(see \cite{Sieg:2010jt} for a review).
Here they are defined as
\begin{equation}\label{chifuncdef}
\begin{aligned}
\chiop(a_1,\dots,a_n)=\sum_{r=0}^{L-1}\prod_{i=1}^n%(\perm-\unitmatrix)
\Fop_{a_i+r\;a_i+r+1}
\pnt
%\chiop(\vec m)&=\sum_{\vec n\subseteq\vec m}
%(-1)^{\dim(\vec m)-\dim(\vec n)}\pone{\vec n}\col
\end{aligned}
\end{equation}
where the operator $\Fop_{ij}$ acts on the flavors at sites $i$ and $j$.
Its precise form will be given below.  
The product of the $\Fop_{ij}$ is specified
by the argument list of the chiral function. 
Periodicity in $L$ is understood 
when this product acts on adjacent elementary flavors of a single-trace 
operator with the length $L$, i.e. with $L$ elementary fields. 
Shifts of all integer entries $a_i$, 
$i=1,\dots,n$ in the argument lists hence do not produce new chiral functions.
We can therefore choose $\min(a_1,\dots,a_n)=1$.
 
In the $\mathcal{N}=4$ SYM theory, the operator $\Fop_{ij}$
is given as the permutation minus the identity. 
In fact, if we consider the Leigh-Strassler 
deformation, only the chiral functions
\eqref{chifuncdef} are modified since $\Fop_{ij}$ 
as a contraction of the 
superpotential \eqref{superpot} with its Hermitean conjugate
is sensitive to the deformation.
There is no further modification of the expressions in 
\eqref{D1D2D3} since they rely on a full Feynman diagram calculation in 
\cite{Sieg:2010tz} that does not make use of relations
between chiral functions.
With the superpotential \eqref{superpot} the fundamental building block
$\Fop_{ij}$ evaluates to
\begin{equation}\label{buildingblock}
\begin{aligned}
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-13\unitlength}%
\raisebox{\eqoff}{%
\fmfframe(-1.5,3)(-11.5,3){%
\begin{fmfchar*}(20,20)
\chionei[phantom]{plain}{plain}{plain}{plain}{plain}
\fmfiv{label=$\scriptstyle i$,l.a=-90,l.dist=4}{vloc(__v3)}
\fmfiv{label=$\scriptstyle j$,l.a=-90,l.dist=4}{vloc(__v4)}
%\fmfiv{label=$\scriptstyle o_1$,l.a=90,l.dist=4}{vloc(__v1)}
%\fmfiv{label=$\scriptstyle 0_2$,l.a=90,l.dist=4}{vloc(__v2)}
\end{fmfchar*}}}
&=\Fop_{ij}%_{i_1i_2}^{o_1o_2}
=\rho^2(\dperm-\unitmatrix+
\bar h\depsilon \ds-h\ds\bar\depsilon+h\bar h\ds\ds)_{ij}%_{i_1i_2}^{o_1o_2}
\col\quad
\rho=\frac{\kappa}{g_\YM}=\sqrt{\frac{2}{1+q\bar q+h\bar h}}+\mathcal{O}(\kappa^6)
\col
\end{aligned}
\end{equation}
where we have used the relation of the couplings \eqref{couprel} and 
the aforementioned field redefinitions in order to make $\kappa$ real. 
The permutation $\dperm$ and $\depsilon$ are
generalizations of the ordinary permutation and the 
$\epsilon$-tensor, which take into account 
the deformation by $q$. 
The only non-vanishing components of the tensors in the products 
up to cyclic permutations are
\begin{equation}
\begin{gathered}
\depsilon_{XYZ}=1\col\quad\depsilon_{YXZ}=-q\col\quad
\bm{\bar\epsilon}_{XYZ}=-\bar q\col\quad\bm{\bar\epsilon}_{YXZ}=1
\col\quad%\\
d_{XXX}=d_{YYY}=d_{ZZZ}=1\pnt
\end{gathered}
\end{equation}
Using $i_i$, $i_j$ and $o_i$, $o_j$ as the flavor degrees of freedom
that enter and leave the building block at the respective positions $i$ and $j$,
the individual terms in \eqref{buildingblock} read
\begin{equation}
\begin{aligned}[]
[(\dperm-\unitmatrix)_{ij}]_{i_ii_j}^{o_io_j}&=-\bm{\epsilon}^{o_io_jk}\bar{\bm{\epsilon}}_{ki_ii_j}\col\\
[(\depsilon \ds)_{ij}]_{i_ii_j}^{o_io_j}&=\bm{\epsilon}^{o_io_jk}d_{ki_ii_j}\col\\
[(\ds\bar\depsilon)_{ij}]_{i_ii_j}^{o_io_j}&=d^{o_io_jk}\bar{\bm{\epsilon}}_{ki_ii_j}\col\\
[(\ds\bar\ds)_{ij}]_{i_ii_j}^{o_io_j}&=d^{o_io_jk}d_{ki_ii_j}
\pnt
\end{aligned}
\end{equation}
The above combinations can be written in terms of bilinears in the 
Gell-Mann matrices $\lambda^a$, $a=1,\dots,8$, which act on the 
flavors at positions $i$ and $j$.  
We form  combinations
$\lambda^{\pm3}=\frac{1}{2}(\lambda^1\pm i\lambda^2)$,
%$\lambda^{-3}=\frac{1}{2}(\lambda^1-i\lambda^2)$,
$\lambda^{\pm2}=\frac{1}{2}(\lambda^4\pm i\lambda^5)$,
%$\lambda^{-2}=\frac{1}{2}(\lambda^4-i\lambda^5)$,
$\lambda^{\pm1}=\frac{1}{2}(\lambda^6\pm i\lambda^7)$,
%$\lambda^{-1}=\frac{1}{2}(\lambda^6-i\lambda^7)$,
such that $\lambda^{\pm i}$ act as rising and lowering 
operators, leaving unaffected the flavor of type $i$, and also 
introduce the identity $\lambda^0=\sqrt{\frac{2}{3}}\unitmatrix$, 
such that $\tr\lambda^A\lambda^B=2\delta^{AB}$, $A,B=0,1,\dots 8$.
In terms of these matrices, the expressions read
\begin{equation}\label{permdef}
\begin{aligned}
(\dperm-\unitmatrix)_{ij}&=\frac{1}{4}
(1+q\bar q)\big(\lambda_i^3\lambda_j^3+\lambda_i^8\lambda_j^8
-2\lambda_i^0\lambda_j^0\big)
+\frac{\sqrt{3}}{4}(1-q\bar q)\big(\lambda_i^8\lambda_j^3-\lambda_i^3\lambda_j^8\big)\\
&\phantom{{}={}}
+q\big(\lambda_i^{-1}\lambda_j^{+1}+\lambda_i^{+2}\lambda_j^{-2}+\lambda_i^{-3}\lambda_j^{+3}\big)
+\bar q\big(\lambda_i^{+1}\lambda_j^{-1}+\lambda_i^{-2}\lambda_j^{+2}+\lambda_i^{+3}\lambda_j^{-3}\big)
\col\\
(\depsilon \ds)_{ij}&=\lambda_i^{+1}\lambda_j^{-3}+\lambda_i^{-2}\lambda_j^{-1}+\lambda_i^{+3}\lambda_j^{+2}
-\bar q\big(\lambda_i^{-1}\lambda_j^{-2}+\lambda_i^{+2}\lambda_j^{+3}+\lambda_i^{-3}\lambda_j^{+1}\big)\col\\
(\ds \depsilon)_{ij}&=\lambda_i^{-1}\lambda_j^{+3}+\lambda_i^{+2}\lambda_j^{+1}+\lambda_i^{-3}\lambda_j^{-2}
-q\big(\lambda_i^{+1}\lambda_j^{+2}+\lambda_i^{-2}\lambda_j^{-3}+\lambda_i^{+3}\lambda_j^{-1}\big)\col\\
\ds\ds_{ij}&=\frac{1}{2}\big(\lambda_i^3\lambda_j^3+\lambda_i^8\lambda_j^8+\lambda_i^0\lambda_j^0\big)
\pnt
\end{aligned}
\end{equation}

A projection of the above expressions onto the subspace of operators that 
contain only the flavors of type $X$ and $Y$, is realized as 
\begin{equation}\label{lambdaSU2proj}
\lambda^{\pm3}\to\sigma^{\pm}\col\qquad
\lambda^{3}\to\sigma^{3}\col\qquad
\lambda^{8}\to\frac{1}{\sqrt{3}}\unitmatrix
\col\qquad
\lambda^{0}\to\sqrt{\frac{2}{3}}\unitmatrix
\col
\end{equation}
where $\sigma^{\pm}$, $\sigma^3$, $\unitmatrix$ are the
Pauli and identity matrices in the two-dimensional flavor space. 
Moreover, we have to drop all combinations of $\lambda^{\pm i}\lambda^{\pm i}$
with $i=1,2$.
% In terms of the Pauli matrices 
% $\sigma_\pm=\frac{1}{2}(\sigma_1\pm i\sigma_2)$, $\sigma_3$ and the
% identity $\unitmatrix$ that act at sites $i$ and $j$, 
% we can write
% \begin{equation}\label{permdef}
% \begin{aligned}
% (\dperm-\unitmatrix)_{ij}&=\frac{1}{4}
% (1+q\bar q)\big(\sigma_i^3\sigma_j^3-\unitmatrix_i\unitmatrix_j\big)
% +\frac{1}{4}(1-q\bar q)\big(\unitmatrix_i\sigma_j^3-\sigma_i^3\unitmatrix_j\big)
% +\bar q\,\sigma_i^+\sigma_j^-
% +q\,\sigma_i^-\sigma_j^+\col\\
% \ds\ds_{ij}&=\frac{1}{2}\big(\sigma_i^3\sigma_j^3+\unitmatrix_i\unitmatrix_j\big)
% \pnt
% \end{aligned}
% \end{equation}
The combination $\dperm-\unitmatrix$ then reduces to the respective
expression in the $SU(2)$ subsector, and also $\ds\ds$ can be consistently 
truncated, while the terms linear in 
$h$, $\bar h$ in \eqref{buildingblock} do not admit a restriction to 
two types of flavors.
% This means that in the (complex) $\beta$-deformation ($h=0$)
% or the cubic Leigh-Strassler deformation ($\rho\to0$, $\rho|h|=\sqrt{2}$)
% the respective operators form closed subsectors at least to one loop,
% but not in the full deformed case with superpotential \eqref{superpot}.
% In the (complex) $\beta$-deformation, the closure persists even to all loop 
% orders.


\subsection{Self energy diagrams}

In a superconformal theory, the anomalous dimensions of the chiral superfields must vanish.  This imposes the relation on the couplings in  \eqref{couprel}.
%The relation \eqref{couprel} immediately follows as condition for 
%the vanishing of the anomalous dimensions of the chiral superfields.
At one loop, the vanishing of the anomalous dimensions %this statement 
is equivalent to the UV finiteness of the 
theory. %, i.e.\ the absence of UV divergences in the self energies of the chiral superfields. 
%In fact the cancellation of the
%divergent one-loop diagrams 
Examining the one-loop diagrams
\begin{equation}\label{sediag}
\begin{aligned}
\seone{1}{1}{plain}{plain}{plain}{plain}{plain}{plain}
=N\kappa\bar\kappa(1+q\bar q+h\bar h)I_1
\col\qquad
\seone{1}{0}{plain}{plain}{photon}{photon}{plain}{plain}
=-2Ng_\YM^2I_1
\col
\end{aligned}
\end{equation}
one can immediately see that the cancellation of the divergences 
leads to the relation in \eqref{couprel}
at leading order. 
This continues to hold  %also 
at two- and three-loop order, where the one-loop  
relation \eqref{couprel}
is not altered 
\cite{Elmetti:2006gr,Elmetti:2007up,Bork:2007bj}.
The reason is that the chiral field lines in the 
two-point %respective 
Feynman diagrams
only form (nested) bubbles.

%However, 
At four loops,
this is no longer the case %since 
and leads to a modification of the %couplings 
relation in \eqref{couprel}.
% is required for vanishing anomalous dimensions, and this mixes contributions
Divergences no longer cancel at the same loop-level, but the four-loop contribution to the anomalous dimension %divergence 
can be cancelled by a one-loop contribution %divergence 
if \eqref{couprel} is modified by a fourth-order term.
%at different loop orders. 
%For the (complex) $\beta$-deformation this 
%was already worked out in \cite{Elmetti:2006gr,Elmetti:2007up}. 
This was previously worked out for the (complex) $\beta$-deformation in
 \cite{Elmetti:2006gr,Elmetti:2007up} 
and for the entire Leigh-Strassler deformation \eqref{superpot}
in \cite{Bork:2007bj}. Here, we recall these results.

%The four-loop diagram that does not have the (nested)
%bubble structure is chiral, 
%i.e.\ it only involves propagators of the (anti-)chiral superfields and no 
%vector fields. 
The only four-loop diagram without a bubble structure 
%It generates a divergent contribution that 
is given by
\begin{equation}\label{sefour}
\begin{aligned}
%\smash[b]{
\csefour[\fmfi{plain}{p3}\fmfi{plain}{p4}\fmfi{plain}{vm1--vm2}]{plain}{plain}{plain}{plain}{plain}{plain}{plain}{plain}
%}
&\to-N^4\frac{(\kappa\bar\kappa)^4}{2}((1+q\bar q+h\bar h)^4
+\delta H(q,\bar q,h,\bar h))I_{4\text{t}}
\col\\
\end{aligned}
\end{equation}
where the first term is %the one that 
is cancelled by the 
remaining four-loop diagrams. 
%in which the chiral field lines form (nested) bubbles. 
The additional 
contribution is a straightforward generalization of the 
result at $h=0$ in \cite{Elmetti:2006gr,Elmetti:2007up} and 
in accord with \cite{Bork:2007bj} we find
\begin{equation}\label{deltaH}
\begin{aligned}
\delta H(q,\bar q,h,\bar h)&=
(1-q\bar q-h\bar h)^4
+8q\bar qh\bar h(q\bar q(3-q\bar q)+h\bar h(3-h\bar h))\\
&\phantom{{}={}}
+8q\bar q(h^3+\bar h^3)
-8h\bar h(q^3+\bar q^3)
-8(q^3\bar h^3+\bar q^3h^3)
\pnt
\end{aligned}
\end{equation}
Since
this term is not cancelled by the other four-loop contributions
it  %and 
will %would 
generate an anomalous dimension for the chiral field, 
unless one allows for fourth order corrections to %of
 the marginal couplings. 
%in \eqref{superpot}. 
Unlike in 
the complex $\beta$-deformed case, it 
is not sufficient to have a series expansion 
 with constant coefficients
for the correction in \eqref{couprel}
Instead, we should %have to
make the ansatz
\begin{equation}\label{kmod}
\kappa\to\kappa(1+\lambda^3\Delta\kappa(q,\bar q,h,\bar h))\col\qquad
\bar\kappa\to\bar\kappa(1+\lambda^3\Delta\bar\kappa(q,\bar q,h,\bar h))
\pnt
\end{equation}
If we absorb these corrections into the 
original $\kappa$, $\bar\kappa$
the relation \eqref{couprel} is modified to %as
\begin{equation}
2g_\YM^2=\kappa\bar\kappa(1+q\bar q+h\bar h)
(1-\lambda^3(\Delta\kappa+\Delta\bar\kappa))
\col
\end{equation}
where further modifications will occur at higher orders.
With these corrections, the leftover piece \eqref{deltaH} from 
the four-loop diagrams can be cancelled by the chiral 
one-loop diagram \eqref{sediag} with modified couplings \eqref{kmod}.
For conformal invariance we have to cancel the anomalous dimensions
which is achieved by setting
\begin{equation}
\begin{aligned}\label{gamma4cancel}
2%(-N\Delta I_1)
\bigg[\seone[\fmfcmd{fill fullcircle scaled 9 shifted vloc(__vc1) withcolor black ;}
\fmfiv{label=\small$\textcolor{white}{\scriptstyle\Delta}$,l.dist=0}{vloc(__vc1)}]{1}{1}{plain}{plain}{plain}{plain}{plain}{plain}
+
\seone[\fmfcmd{fill fullcircle scaled 9 shifted vloc(__vc2) withcolor black ;}
\fmfiv{label=\small$\textcolor{white}{\scriptstyle\Delta}$,l.dist=0}{vloc(__vc2)}]{1}{1}{plain}{plain}{plain}{plain}{plain}{plain}
\bigg]
+
8\bigg[%\Big(-N^4\frac{(\kappa\bar\kappa)^4}{2}\delta H(q,\bar q,h,\bar h)I_{4\text{t}}\Big)
\csefour[\fmfi{plain}{p3}\fmfi{plain}{p4}\fmfi{plain}{vm1--vm2}]{plain}{plain}{plain}{plain}{plain}{plain}{plain}{plain}+\dots\bigg]
=\text{finite}
\col
\end{aligned}
\end{equation}
where the blobs are the vertex corrections which come with factors of $\lambda^3\Delta\kappa$ or $\lambda^3\Delta\bar\kappa$.
%where a blob means that the vertex in the diagram also carries a respective
%factor $g_\YM^6\Delta\kappa$ or $g_\YM^6\Delta\bar\kappa$. 
The ellipsis denotes the remaining four-loop 
diagrams in which the chiral field lines form (nested) bubbles.
The integer prefactors come from the definition 
of the dilatation operator (the anomalous dimension) in 
\eqref{DinZ}. This leads to a  multiplication of the $K$-loop contribution 
by a factor $2K$. 
From the condition \eqref{gamma4cancel} we find
\begin{equation}
\begin{aligned}
\Delta\kappa+\Delta\bar\kappa
=16\frac{\delta H(q,\bar q,h,\bar h)}{(1+q\bar q+h\bar h)^4}\frac{\mathcal{I}_{4\text{t}}}{\mathcal{I}_1}
\pnt
\end{aligned}
\end{equation}
With this adjustment the theory is conformal but %no longer finite, (it wasn't finite to begin with)
not finite, 
since the diagrams that contribute to the chiral field renormalization do not have  the prefactors in \eqref{gamma4cancel} and hence do not cancel.
%since the wave-function renormalization 
%the contribution to the chiral field wave-function
%in the renormalization constant itself the prefactors 
%that occur in \eqref{gamma4cancel} are all absent.

% However, at four-loop order the cancellation of the anomalous dimension 
% of the chiral fields guarantees that we do not have to consider 
% four-loop self energy contributions.
% Up to three loops, the chiral structures in all superspace Feynman diagram only 
% consist of (nested) bubbles \cite{}. This is no longer the case
% at four loops where a new type of chiral structure appears.

% In order to preserve conformal invariance, we have to cancel 
% its contribution to the anomalous dimension of the chiral field.
% This can be achieved by adding the fourth order correction $\Delta$
% to \eqref{couplre} that is determined by the relation


%The alterations which are induced by the new topology of the four-loop
%chiral self-energy diagram \eqref{sefour} also concern the diagrams
%of operator renormalization. At four-loop order they can only affect the
%diagrams associated with the simplest chiral function $\chi(1)$. 
The alterations induced by \eqref{sefour} also affect the operator renormalization. At four-loop order this only concerns
the diagrams associated with the simplest chiral function $\chi(1)$. 
But %Thereby, 
the relation in \eqref{gamma4cancel} guarantees that %also 
these diagrams  obey %the relation
\begin{equation}\label{chionefour}
\begin{aligned}
2\left[\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-13\unitlength}%
\raisebox{\eqoff}{%
\fmfframe(-1.5,3)(-11.5,3){%
\begin{fmfchar*}(20,20)
\chionei{plain}{plain}{plain}{plain}{plain}
\fmfcmd{fill fullcircle scaled 9 shifted vloc(__vc1) withcolor black ;}
\fmfiv{label=\small$\textcolor{white}{\scriptstyle\Delta}$,l.dist=0}{vloc(__vc1)}
\end{fmfchar*}}}
+
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-13\unitlength}%
\raisebox{\eqoff}{%
\fmfframe(-1.5,3)(-11.5,3){%
\begin{fmfchar*}(20,20)
\chionei{plain}{plain}{plain}{plain}{plain}
\fmfcmd{fill fullcircle scaled 9 shifted vloc(__vc2) withcolor black ;}
\fmfiv{label=\small$\textcolor{white}{\scriptstyle\Delta}$,l.dist=0}{vloc(__vc2)}
\end{fmfchar*}}}
\right]
+8
\left[
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-13\unitlength}%
\raisebox{\eqoff}{%
\fmfframe(-1.5,3)(-11.5,3){%
\begin{fmfchar*}(20,20)
\chionei{plain}{plain}{plain}{plain}{plain}
\fmfi{plain}{vd4--vd5}
\fmfi{plain}{vu4--vu5}
\fmfiset{p6}{vd4--vd5}
\fmfiset{p7}{vu4--vu5}
\svertex{vm6}{p6}
\svertex{vm7}{p7}
\fmfi{plain}{vm6--vm7}
\end{fmfchar*}}}
+\dots
\right]
=\text{finite}
\,, %\pnt
\end{aligned}
\end{equation}
%The validity of the above relation 
%is obvious if we consider that the composite operators
%under consideration are themselves chiral superfields. 
%In the above diagrams they are represented
%by a horizontal bold line from which the interacting field lines originate,
%not indicating additional non-interacting lines. 
%Because of the cancellation of their UV divergences in 
%\eqref{chionefour}, the respective diagrams can be omitted 
%in the calculation of the dilatation operator from the very beginning.
whose validity is obvious if one realizes that the chiral operator (represented by the bold line) can be treated like a chiral vertex.  Hence, the diagrams in \eqref{chionefour} have the same topology as those in \eqref{gamma4cancel} and so have the same cancellation of  UV divergences.  These diagrams can then be omitted from the very beginning since they will not contribute to the dilatation operator.

\subsection{Diagrams with reducible chiral functions}

\label{subsec:nnmrange}

In the $SU(2)$ subsector of the (complex) $\beta$-deformed 
theories,
% $\mathcal{N}=4$ SYM theory,
not all of the chiral functions 
in \eqref{chifuncdef} are independent.  Many chiral functions contain terms that antisymmetrize three neighboring sites.  Such terms are zero when there are only two flavors, leading to relations between different chiral functions.  
% that capture the structure of
%the chiral and anti-chiral field lines in the underlying 
%superspace Feynman diagrams 
%are independent. 
%They are related to 
%the vanishing of the total anti-symmetrization
%of three neighboured elementary fields within the composite operators 
%of the subsector. 
%The simplest examples of such relations read 
The simplest example is 
\begin{equation}\label{chiidentities1}
\chiop(1,2,1)|_{SU(2)}=\chiop(2,1,2)|_{SU(2)}=\rho^4q\bar q\chiop(1)|_{SU(2)}\col
\end{equation}
where $|_{SU(2)}$ indicates the projection onto the subsector.
%\footnote{From now on we will not distinguish between 
%the $SU(2)$ subsector of the $\mathcal{N}=4$ SYM theory and
%the respective closed subsector in its (complex) $\beta$-deformation.
%Even if the latter has no $SU(2)$ symmetry, for simplicity we 
%implicitly include it in our statements concerning the $SU(2)$ subsector.
%This concerns also the projection on the respective closed chiral subsector.
%}
The general set of relations %between different chiral functions 
are worked out in  appendix \ref{app:chifuncrel}. 
%Since to each entry in the argument lists of the chiral functions 
%one chiral and one anti-chiral elementary vertex 
%is associated, the chiral functions $\chiop(1,2,1)$ and $\chiop(2,1,2)$
%require at least six orders of the Yang-Mills coupling constant and
 The chiral functions $\chiop(1,2,1)$ and $\chiop(2,1,2)$ %hence
  first appear in the three-loop dilatation operator $\mathcal{D}_3$. 
The respective terms in \eqref{D1D2D3} arise from the chiral Feynman diagram
\begin{equation}
\begin{aligned}\label{crange3}
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\end{fmfchar*}}}%}
&=\lambda^3I_3\chiop(1,2,1)
\col
\end{aligned}
\end{equation}
and its reflection. 
At four loops, the relation \eqref{chiidentities1} generalizes
to %the 
chiral functions with four arguments. %. 
%For the ones that are related to diagrams with overall UV divergences 
%and that hence contribute to the dilatation operator
%the respective relations read
Here there are two types of relations,  given by
\begin{equation}\label{chiidentities2}
\begin{gathered}
\begin{aligned}
\chiop(1,2,1,2)|_{SU(2)}=\chiop(1,2,3,2)|_{SU(2)}=\chiop(2,1,2,3)|_{SU(2)}&=\rho^4q\bar q\chiop(1,2)\col\\
\chiop(2,1,2,1)|_{SU(2)}=\chiop(3,2,1,2)|_{SU(2)}=\chiop(2,3,2,1)|_{SU(2)}&=\rho^4q\bar q\chiop(2,1)\col\\
\end{aligned}\\
\left.
\begin{aligned}
\chiop(1,2,1,3)|_{SU(2)}=\chiop(1,3,2,3)|_{SU(2)}\\
\chiop(3,2,3,1)|_{SU(2)}=\chiop(3,1,2,1)|_{SU(2)}
\end{aligned}
\right\}
=\rho^4q\bar q\chiop(1,3)\col\\
\end{gathered}
\end{equation}
where a relation and its reflection are displayed in tandem.
We call chiral functions ``reducible'' if they simplify
as in \eqref{chiidentities1} and \eqref{chiidentities2}
when projected onto the $SU(2)$ subsector.
%where the pairs of chiral functions which are obtained 
%by reflecting the underlying diagrams are vertically aligned.

In analogy to the three-loop diagram \eqref{crange3},  four-loop
 diagrams that come with chiral functions having %with
 four arguments are chiral. 
%At four loops, there are also contributions with $\chiop(1,2,1)$
%and $\chiop(2,1,2)$. 
%They are determined by promoting the
%chiral three-loop diagram in \eqref{crange3} to 
%four-loop diagrams by adding a vector field interaction.
% Hence a single respective diagram generates 
%the contribution to each of them. 
Hence, each one is generated by one diagram.
After $\D$-algebra, we find
\begin{equation}
\begin{gathered}\label{crange4}
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-1.5,2)(-6.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoonetwoone
\end{fmfchar*}}}%}
=\lambda^4I_4\chiop(1,2,1,2)
\col\,\,
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-1.5,2)(-1.5,2){%
\begin{fmfchar*}(20,20)
\chionetwothreetwo
\end{fmfchar*}}}%}
=\lambda^4I_{4\text{w}}\chiop(1,2,3,2)
\col\,\,
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-0.5,2){%
\begin{fmfchar*}(20,20)
\chitwoonetwothree
\end{fmfchar*}}}%}
=\lambda^4I_4\chiop(2,1,2,3)
\col\\
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-1.5,2)(-1.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoonethree
\end{fmfchar*}}}%}
=\lambda^4I_{4\text{bb}}\chiop(1,2,1,3)
\col\,\,
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-1.5,2)(-1.5,2){%
\begin{fmfchar*}(20,20)
\chionethreetwothree
\end{fmfchar*}}}%}
=\lambda^4I_{4\text{w}}\chiop(1,3,2,3)
\col\,\,
\end{gathered}
\end{equation}
as well as  analogous results for their reflections. 

There are also four-loop diagrams   having chiral function $\chiop(1,2,1)$
or $\chiop(2,1,2)$. 
These diagrams are constructed  by attaching  a vector propagator to  \eqref{crange3} or its reflection.
%chiral three-loop diagram in \eqref{crange3} to 
%four-loop diagrams by adding a vector field interaction.
%
%In the following, we summarize the calculation of all four-loop diagrams 
%that according to the identities \eqref{chiidentities1} and
%\eqref{chiidentities2} in the $\mathcal{N}=4$ SYM theory and in
%its $\beta$-deformation where $q\bar q=1$ 
%are hidden within the contributions 
%of chiral functions of shorter range. 
%This will allow us to disentangle the respective contributions in the 
%four-loop dilatation operator and hence to determine the result
%in the Leigh-Strassler deformed case. 
%
% 
%\subsubsection{Range three reducible diagrams}
%
%As mentioned before, at four loops the contributions which come 
%with chiral functions $\chiop(1,2,2)$ and $\chiop(2,1,2)$
%are determined by the Feynman diagrams that are built 
%from \eqref{crange3} by adding a vector field line.
Since the one-loop chiral self energy is identically zero, 
%at least three vertices must be involved in each of the loops 
%that is formed with the vector field propagator.
the ends of the vector propagator much attach to different chiral propagators.
Moreover, there are further restrictions coming from the finiteness 
conditions of \cite{Sieg:2010tz}.
First, %in order to obtain a diagram with an overall UV divergence, 
the one  chiral vertex in   \eqref{crange3} which is not part of a loop 
%must not become part of a loop 
must remain out of any loop after adding the vector field interaction.  %when the vector field interaction is added.
Second, 
the vector propagator cannot attach to  the neighboring field lines  of  \eqref{crange3}, keeping the range of the diagram to three sites. 
%the interaction must not involve one of the neighboured field lines 
%that are not yet participating in the interactions and that have 
%been omitted in \eqref{crange3}. This means, the range of the diagram, 
%i.e.\ the number of elementary fields of the composite operator 
%involved in the interaction has to remain $R=3$.
The  diagrams that fulfill these constraints, along with their values,  are % yield
% \begin{equation}
% \begin{aligned}
% \settoheight{\eqoff}{$\times$}%
% \setlength{\eqoff}{0.5\eqoff}%
% \addtolength{\eqoff}{-12\unitlength}%
% %\subfigure[]{%
% \raisebox{\eqoff}{%
% \fmfframe(-0.5,2)(-5.5,2){%
% \begin{fmfchar*}(20,20)
% \chionetwo
% \fmfi{phantom,label=$\scriptstyle p_1$,l.dist=2}{p1}
% \fmfi{phantom,label=$\scriptstyle p_2$,l.dist=2}{p2}
% \fmfi{phantom,label=$\scriptstyle p_3$,l.dist=2}{p3}
% \fmfi{phantom,label=$\scriptstyle p_4$,l.dist=2}{p4}
% \fmfi{phantom,label=$\scriptstyle p_5$,l.dist=0}{p5}
% \fmfi{phantom,label=$\scriptstyle p_6$,l.dist=2,l.side=left}{p6}
% \fmfi{phantom,label=$\scriptstyle p_7$,l.dist=2,l.side=left}{p7}
% \fmfi{phantom,label=$\scriptstyle p_8$,l.dist=1,l.side=left}{p8}
% \fmfi{phantom,label=$\scriptstyle p_9$,l.dist=2}{p9}
% \end{fmfchar*}}}%}
% \end{aligned}
% \end{equation}

% \begin{equation}
% \begin{aligned}
% \begin{tabular}{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}
% &
% \settoheight{\eqoff}{$\times$}%
% \setlength{\eqoff}{0.5\eqoff}%
% \addtolength{\eqoff}{-12\unitlength}%
% %\subfigure[]{%
% \raisebox{\eqoff}{%
% \fmfframe(-0.5,2)(-5.5,2){%
% \begin{fmfchar*}(20,20)
% \chionetwo
% \fmfi{photon}{vm3{dir 180}..{dir -45}vm4}
% \end{fmfchar*}}}%}
% &
% %
% \settoheight{\eqoff}{$\times$}%
% \setlength{\eqoff}{0.5\eqoff}%
% \addtolength{\eqoff}{-12\unitlength}%
% %\subfigure[]{%
% \raisebox{\eqoff}{%
% \fmfframe(-0.5,2)(-5.5,2){%
% \begin{fmfchar*}(20,20)
% \chionetwo
% \fmfi{photon}{vm3{dir 0}..{dir -45}vm5}
% \end{fmfchar*}}}%}
% &
% %
% \settoheight{\eqoff}{$\times$}%
% \setlength{\eqoff}{0.5\eqoff}%
% \addtolength{\eqoff}{-12\unitlength}%
% %\subfigure[]{%
% \raisebox{\eqoff}{%
% \fmfframe(-0.5,2)(-5.5,2){%
% \begin{fmfchar*}(20,20)
% \chionetwo
% \fmfi{photon}{vm3{dir 30}..{dir 15}vm6}
% \end{fmfchar*}}}%}
% &
% %
% \settoheight{\eqoff}{$\times$}%
% \setlength{\eqoff}{0.5\eqoff}%
% \addtolength{\eqoff}{-12\unitlength}%
% %\subfigure[]{%
% \raisebox{\eqoff}{%
% \fmfframe(-0.5,2)(-5.5,2){%
% \begin{fmfchar*}(20,20)
% \chionetwo
% \fmfi{photon}{vm4{dir 0}..{dir 90}vm5}
% \end{fmfchar*}}}%}
% &
% %
% \settoheight{\eqoff}{$\times$}%
% \setlength{\eqoff}{0.5\eqoff}%
% \addtolength{\eqoff}{-12\unitlength}%
% %\subfigure[]{%
% \raisebox{\eqoff}{%
% \fmfframe(-0.5,2)(-5.5,2){%
% \begin{fmfchar*}(20,20)
% \chionetwo
% \fmfi{photon}{vm4{dir 15}..{dir 30}vm7}
% \end{fmfchar*}}}%}
% &
% %
% \settoheight{\eqoff}{$\times$}%
% \setlength{\eqoff}{0.5\eqoff}%
% \addtolength{\eqoff}{-12\unitlength}%
% %\subfigure[]{%
% \raisebox{\eqoff}{%
% \fmfframe(-0.5,2)(-5.5,2){%
% \begin{fmfchar*}(20,20)
% \chionetwo
% \fmfi{photon}{vm4{dir 0}..{dir 0}vm8}
% \end{fmfchar*}}}%}
% &
% \settoheight{\eqoff}{$\times$}%
% \setlength{\eqoff}{0.5\eqoff}%
% \addtolength{\eqoff}{-12\unitlength}%
% %\subfigure[]{%
% \raisebox{\eqoff}{%
% \fmfframe(-0.5,2)(-5.5,2){%
% \begin{fmfchar*}(20,20)
% \chionetwo
% \fmfi{photon}{vm5{dir 120}..{dir 30}vm6}
% \end{fmfchar*}}}%}
% &
% %
% \settoheight{\eqoff}{$\times$}%
% \setlength{\eqoff}{0.5\eqoff}%
% \addtolength{\eqoff}{-12\unitlength}%
% %\subfigure[]{%
% \raisebox{\eqoff}{%
% \fmfframe(-0.5,2)(-5.5,2){%
% \begin{fmfchar*}(20,20)
% \chionetwo
% \fmfi{photon}{vm5{dir -45}..{dir 0}vm7}
% \end{fmfchar*}}}%}
% &
% %
% \settoheight{\eqoff}{$\times$}%
% \setlength{\eqoff}{0.5\eqoff}%
% \addtolength{\eqoff}{-12\unitlength}%
% %\subfigure[]{%
% \raisebox{\eqoff}{%
% \fmfframe(-0.5,2)(-5.5,2){%
% \begin{fmfchar*}(20,20)
% \chionetwo
% \fmfi{photon}{vm5{dir -90}..{dir -90}vm8}
% \end{fmfchar*}}}%}
% &
% %
% \settoheight{\eqoff}{$\times$}%
% \setlength{\eqoff}{0.5\eqoff}%
% \addtolength{\eqoff}{-12\unitlength}%
% %\subfigure[]{%
% \raisebox{\eqoff}{%
% \fmfframe(-0.5,2)(-5.5,2){%
% \begin{fmfchar*}(20,20)
% \chionetwo
% \fmfi{photon}{vm6{dir -45}..{dir 180}vm7}
% \end{fmfchar*}}}%}
% &
% %
% \settoheight{\eqoff}{$\times$}%
% \setlength{\eqoff}{0.5\eqoff}%
% \addtolength{\eqoff}{-12\unitlength}%
% %\subfigure[]{%
% \raisebox{\eqoff}{%
% \fmfframe(-0.5,2)(-5.5,2){%
% \begin{fmfchar*}(20,20)
% \chionetwo
% \fmfi{photon}{vm6{dir -90}..{dir -90}vm9}
% \end{fmfchar*}}}%}
% %
% &
% \settoheight{\eqoff}{$\times$}%
% \setlength{\eqoff}{0.5\eqoff}%
% \addtolength{\eqoff}{-12\unitlength}%
% %\subfigure[]{%
% \raisebox{\eqoff}{%
% \fmfframe(-0.5,2)(-5.5,2){%
% \begin{fmfchar*}(20,20)
% \chionetwo
% \fmfi{photon}{vm7{dir 225}..{dir -45}vm8}
% \end{fmfchar*}}}%}
% &
% \settoheight{\eqoff}{$\times$}%
% \setlength{\eqoff}{0.5\eqoff}%
% \addtolength{\eqoff}{-12\unitlength}%
% %\subfigure[]{%
% \raisebox{\eqoff}{%
% \fmfframe(-0.5,2)(-5.5,2){%
% \begin{fmfchar*}(20,20)
% \chionetwo
% \fmfi{photon}{vm7{dir -45}..{dir -135}vm9}
% \end{fmfchar*}}}%}
% &
% \settoheight{\eqoff}{$\times$}%
% \setlength{\eqoff}{0.5\eqoff}%
% \addtolength{\eqoff}{-12\unitlength}%
% %\subfigure[]{%
% \raisebox{\eqoff}{%
% \fmfframe(-0.5,2)(-5.5,2){%
% \begin{fmfchar*}(20,20)
% \chionetwo
% \fmfi{photon}{vm8--vm9}
% \end{fmfchar*}}}%}
% \end{tabular}
% \pnt
% \end{aligned}
% \end{equation}
\begin{equation}
\begin{aligned}\label{chi121r3}
%
&
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm3{dir 180}..{dir 0}vm4}
\end{fmfchar*}}}%}
\col
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm3{dir 0}..{dir -90}vm5}
\end{fmfchar*}}}%}
\col
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm5{dir 90}..{dir 0}vm6}
\end{fmfchar*}}}%}
\col
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm6{dir -90}..{dir 180}vm7}
\end{fmfchar*}}}%}
\col
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm4{dir 180}..{dir 0}vm10}
\end{fmfchar*}}}%}
\col
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm7{dir 0}..{dir -90}vm9}
\end{fmfchar*}}}%}
\col
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm10{dir 180}..{dir -90}vm11}
\end{fmfchar*}}}%}
\col
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm10{dir 0}..{dir -90}vm12}
\end{fmfchar*}}}%}
\to-I_4
\col\\
&
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm4{dir 0}..{dir -90}vm8}
\end{fmfchar*}}}%}
\col
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm8{dir 90}..{dir 0}vm7}
\end{fmfchar*}}}%}
\to-I_{4\text{w}}
\col\qquad
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm8{dir -90}..{dir 180}vm10}
\end{fmfchar*}}}%}
\col
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm9{dir 180}..{dir 90}vm8}
\end{fmfchar*}}}%}
\to-I_{4\text{bt}}
\col\qquad
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm11{dir 0}..{dir 0}vm12}
\end{fmfchar*}}}%}
\to-I_{43\text{t}}
\col\\
%
&
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm3{dir 0}..{dir 0}vm6}
\end{fmfchar*}}}%}
\col
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm3{dir 180}..{dir 0}vm10}
\end{fmfchar*}}}%}
\col
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm6{dir -90}..{dir -90}vm9}
\end{fmfchar*}}}%}
\col
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm4{dir 0}..{dir 0}vm7}
\end{fmfchar*}}}%}
\col
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm10{dir 0}..{dir 0}vm9}
\end{fmfchar*}}}%}
\to I_4
\col\qquad
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm8{dir 90}..{dir 90}vm5}
\end{fmfchar*}}}%}
\to I_{4\text{bt}}
\col\\
%
&
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm4{dir 180}..{dir -90}vm11}
\end{fmfchar*}}}%}
\to I_4+I_{4\beta}-I_{4\text{tr}1}
\col\qquad
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm8{dir -90}..{dir -90}vm12}
\end{fmfchar*}}}%}
\to I_4+I_{4\text{w}}-I_{4\text{tr}2}
\col\\
&
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm3{dir 180}..{dir -90}vm11}
\end{fmfchar*}}}%}
\to-I_{4\text{w}}
\col\qquad
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm12{dir 45}..{dir 45}vm9}
\end{fmfchar*}}}%}
\to-I_4
%
\col
\end{aligned}
\end{equation}
where we have grouped together the diagrams that
lead to the same integrals  after $\D$-algebra.  The integrals are listed in \eqref{IK}.
The above results contain all signs from color and flavor factors, and from 
the $\D$-algebra manipulations. %, but
Finite contributions and a common factor
$\lambda^4\chiop(1,2,1)$ have been omitted.
Note that most of the diagrams 
are easily evaluated %, following the argumentation and counting as explained 
using the arguments
in \cite{Sieg:2010tz}. %Only 
The diagrams in the fourth line
require a little more work and lead to integrals with momenta
in the numerator of their integrands that are contracted as prescribed
by a trace over products of $\gamma$-matrices.
Using the results of appendix \ref{app:integrals}, 
we find for the sum of the most complicated diagrams
\begin{equation}
\begin{aligned}
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm4{dir 180}..{dir -90}vm11}
\end{fmfchar*}}}%}
+
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\fmfi{photon}{vm8{dir -90}..{dir -90}vm12}
\end{fmfchar*}}}%}
\to
-2I''_{4\text{t}1}
+I_{4\text{w}}
+I_{4\text{bt}}
+I_{43\text{t}}
-I_{4\text{t}}
\pnt\\
\end{aligned}
\end{equation}
Summing up all diagrams of \eqref{chi121r3}, and eliminating 
$I_{4\text{w}}$ by making use of the relation \eqref{I4I4wrel} %,
results in
%the result reads
\begin{equation}\label{range3}
\begin{aligned}
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(-0.5,2)(-5.5,2){%
\begin{fmfchar*}(20,20)
\chionetwoone
\end{fmfchar*}}}%}
+\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-2\unitlength}%
%\subfigure[]{%
\raisebox{\eqoff}{%
\fmfframe(1,0)(1,0){%
\begin{fmfchar*}(7.5,4)
\fmfleft{in}
\fmfright{out}
\fmf{photon}{in,out}
\end{fmfchar*}}}%}
%&=\lambda^4(-8I_4-2I_{4\text{w}}-2I_{4\text{bt}}-I_{43\text{t}}
%+5I_4+I_{4\text{bt}}
%-2I''_{4\text{t}1}+I_{4\text{w}}+I_{4\text{bt}}+I_{43\text{t}}-I_{4\text{t}}
%-I_{4\text{w}}-I_4)\chiop(1,2,1)\\
%&=\lambda^4(-4I_4-2I_{4\text{w}}-2I''_{4\text{t}1}-I_{4\text{t}})\chiop(1,2,1)
=\lambda^4(-6I_4+2I''_{4\text{t}1}+I_{4\text{t}})\chiop(1,2,1)
\pnt
\end{aligned}
\end{equation}
The result for the reflected diagrams is obtained by %simply 
replacing % the 
%chiral function 
$\chiop(1,2,1)$ with $\chiop(2,1,2)$.

%\subsubsection{Range four reducible diagrams}
%As mentioned before, at four-loops the diagrams which come with the 
%chiral functions with four arguments given in \eqref{chiidentities2} 
%are purely chiral. Hence a single respective diagram generates 
%the contribution to each of them. 
%After $\D$-algebra, we find
%\begin{equation}
%\begin{gathered}\label{crange4}
%\settoheight{\eqoff}{$\times$}%
%\setlength{\eqoff}{0.5\eqoff}%
%\addtolength{\eqoff}{-12\unitlength}%
%%\subfigure[]{%
%\raisebox{\eqoff}{%
%\fmfframe(-1.5,2)(-6.5,2){%
%\begin{fmfchar*}(20,20)
%\chionetwoonetwoone
%\end{fmfchar*}}}%}
%=\lambda^4I_4\chiop(1,2,1,2)
%\col\,\,
%\settoheight{\eqoff}{$\times$}%
%\setlength{\eqoff}{0.5\eqoff}%
%\addtolength{\eqoff}{-12\unitlength}%
%%\subfigure[]{%
%\raisebox{\eqoff}{%
%\fmfframe(-1.5,2)(-1.5,2){%
%\begin{fmfchar*}(20,20)
%\chionetwothreetwo
%\end{fmfchar*}}}%}
%=\lambda^4I_{4\text{w}}\chiop(1,2,3,2)
%\col\,\,
%\settoheight{\eqoff}{$\times$}%
%\setlength{\eqoff}{0.5\eqoff}%
%\addtolength{\eqoff}{-12\unitlength}%
%%\subfigure[]{%
%\raisebox{\eqoff}{%
%\fmfframe(-0.5,2)(-0.5,2){%
%\begin{fmfchar*}(20,20)
%\chitwoonetwothree
%\end{fmfchar*}}}%}
%=\lambda^4I_4\chiop(2,1,2,3)
%\col\\
%\settoheight{\eqoff}{$\times$}%
%\setlength{\eqoff}{0.5\eqoff}%
%\addtolength{\eqoff}{-12\unitlength}%
%%\subfigure[]{%
%\raisebox{\eqoff}{%
%\fmfframe(-1.5,2)(-1.5,2){%
%\begin{fmfchar*}(20,20)
%\chionetwoonethree
%\end{fmfchar*}}}%}
%=\lambda^4I_{4\text{bb}}\chiop(1,2,1,3)
%\col\,\,
%\settoheight{\eqoff}{$\times$}%
%\setlength{\eqoff}{0.5\eqoff}%
%\addtolength{\eqoff}{-12\unitlength}%
%%\subfigure[]{%
%\raisebox{\eqoff}{%
%\fmfframe(-1.5,2)(-1.5,2){%
%\begin{fmfchar*}(20,20)
%\chionethreetwothree
%\end{fmfchar*}}}%}
%=\lambda^4I_{4\text{w}}\chiop(1,3,2,3)
%\col\,\,
%\end{gathered}
%\end{equation}
%and we also have to consider their reflections. 


\subsection{Four-loop result}
\label{sec:fourloopresults}

%The negative of the sum of \eqref{range3}, and \eqref{crange4} 
%and of their reflections yields 
%the %respective contribution
%reducible contributions
% to the renormalization constant.
%After inserting the explicit expressions in \eqref{IK} for the 
%pole parts of the loop integrals, the definition 
%\eqref{DinZ} then yields the dilatation operator
%by extracting  the simple $\frac{1}{\varepsilon}$ pole of this result
%and multiplying it by $8$.  We find for the reducible terms
The reducible contribution
 to the renormalization constant is the negative  sum of \eqref{range3},  \eqref{crange4} 
and  their reflections.  Using \eqref{DinZ}, one finds that the reducible part 
contributes to the dilatation operator with the coefficient of the $\frac{1}{\varepsilon}$ pole multiplied by $8$. 
Inserting the explicit expressions in \eqref{IK} then gives
\begin{equation}\label{deltaDred}
\begin{aligned}
\delta\mathcal{D}_{4\text{red}}
&=(60-8\zeta(3))[\chiop(1,2,1)+\chiop(2,1,2)]\\
&\phantom{{}={}}-10[\chiop(1,2,1,2)+\chiop(2,1,2,1)]\\
&
\phantom{{}={}}-(10-8\zeta(3))[\chiop(1,2,3,2)+\chiop(3,2,1,2)]\\
&
\phantom{{}={}}-10[\chiop(2,1,2,3)+\chiop(2,3,2,1)]\\
&
\phantom{{}={}}+\Big(\frac{10}{3}-4\zeta(3)\Big)[\chiop(1,2,1,3)+\chiop(3,2,3,1)]\\
&
\phantom{{}={}}-(10-8\zeta(3))[\chiop(1,3,2,3)+\chiop(3,1,2,1)]
\pnt
\end{aligned}
\end{equation}
Restricting to the $SU(2)$ subsector of  $\mathcal{N}=4$ SYM
%by making use of 
 and using
 the identities in \eqref{chiidentities1} and
\eqref{chiidentities2},
%the above terms reduce to
reduces the above term to
\begin{equation}
\begin{aligned}\label{deltaDredSU2}
\delta\mathcal{D}_{4\text{red}}|_{SU(2)}
&=8(15-2\zeta(3))\chiop(1)
-2(15-4\zeta(3))[\chiop(1,2)+\chiop(2,1)]\\
&\phantom{{}={}}
-8\Big(\frac{5}{3}-\zeta(3)\Big)\chiop(1,3)
\pnt
\end{aligned}
\end{equation}

The four-loop dilatation operator $\DD_{4,\NN=4}$ for the $SU(2)$ subsector of $\NN=4$ SYM was first presented in \cite{Beisert:2007hz}. It is determined 
as one of the commuting charges of the integrable system, 
using information
from the all loop Bethe equations  \cite{Beisert:2005fw} and the magnon 
dispersion relation 
\cite{Santambrogio:2002sb,Beisert:2004hm,Beisert:2005tm}
\begin{equation}\label{Ep}
E(p)=\sqrt{1+4h^2(g)\sin^2\tfrac{p}{2}}-1\col
%\pnt
\end{equation}
where we assume that $h^2(g)=4g^2$.  To find $\DD_4$ for the Leigh-Strassler theories we add  \eqref{deltaDred} to $\DD_{4,\NN=4}$ and then subtract \eqref{deltaDredSU2}.  Using the convention in \cite{Fiamberti:2008sh} for $\DD_{4,\NN=4}$, we find
%The four-loop dilatation operator $\mathcal{D}_4$ for the composite operators
%considered here in the Leigh-Strassler deformed theory
%is then obtained from the one for the $SU(2)$ subsector of $\mathcal{N}=4$
%SYM theory in the scheme of $\mathcal{N}=1$ supergraphs as given e.g.\ in 
%\cite{Fiamberti:2008sh}
%by adding \eqref{deltaDred} and subtracting \eqref{deltaDredSU2}.
%This yields
\begin{equation}
\begin{aligned}\label{D4chi}
\mathcal{D}_4&=
{}+{}16(5+\zeta(3))\chiop(1)\\
&\phantom{{}={}}
-8(15+\zeta(3))[\chiop(1,2)+\chiop(2,1)]%\
%&\phantom{{}={}}
+8\Big(\frac{23}{3}-\zeta(3)\Big)\chiop(1,3)%\\
%&\phantom{{}={}}
-4\chiop(1,4)\\
&\phantom{{}={}}
+4(15-2\zeta(3))[\chiop(1,2,1)+\chiop(2,1,2)]
+60[\chiop(1,2,3)+\chiop(3,2,1)]\\
&\phantom{{}={}}
+2(4+\beta+2\epsilon_{3a}-2i\epsilon_{3b}+i\epsilon_{3c}-2i\epsilon_{3d})
\chiop(1,3,2)\\
&\phantom{{}={}}
+2(4+\beta+2\epsilon_{3a}+2i\epsilon_{3b}-i\epsilon_{3c}+2i\epsilon_{3d})
\chiop(2,1,3)\\
&\phantom{{}={}}
-2(2+2i\epsilon_{3b}+i\epsilon_{3c})[\chiop(1,2,4)+\chiop(1,4,3)]\\
&\phantom{{}={}}
-2(2-2i\epsilon_{3b}-i\epsilon_{3c})[\chiop(1,3,4)+\chiop(2,1,4)]\\
&\phantom{{}={}}
-10[\chiop(1,2,1,2)+\chiop(2,1,2,1)]\\
&\phantom{{}={}}
-(10+i\epsilon_{3e}-i\epsilon_{3f})[\chiop(2,1,2,3)+\chiop(2,3,2,1)]\\
&\phantom{{}={}}
-(10-8\zeta(3)-i\epsilon_{3e}+i\epsilon_{3f})[\chiop(1,2,3,2)+\chiop(3,2,1,2)]\\
&\phantom{{}={}}
+\Big(\frac{14}{3}+8\zeta(3)+2\epsilon_{3a}-4i\epsilon_{3b}+i\epsilon_{3e}\Big)[\chiop(1,3,2,3)+\chiop(3,1,2,1)]\\
&\phantom{{}={}}
+\Big(\frac{14}{3}-4\zeta(3)+2\epsilon_{3a}+4i\epsilon_{3b}-i\epsilon_{3e}\Big)[\chiop(1,2,1,3)+\chiop(3,2,3,1)]\\
&\phantom{{}={}}-2(6+\beta+2\epsilon_{3a})\chiop(2,1,3,2)\\
&\phantom{{}={}}
+2(9+2\epsilon_{3a})[\chiop(1,3,2,4)+\chiop(2,1,4,3)]\\
&\phantom{{}={}}
-2(4+\epsilon_{3a}+i\epsilon_{3b})[\chiop(1,2,4,3)+\chiop(1,4,3,2)]\\
&\phantom{{}={}}
-2(4+\epsilon_{3a}-i\epsilon_{3b})[\chiop(2,1,3,4)+\chiop(3,2,1,4)]\\
&\phantom{{}={}}
-10[\chiop(1,2,3,4)+\chiop(4,3,2,1)]\col
\end{aligned}
\end{equation}
where we have also considered the possibility of more general 
similarity transformations 
as
compared to %the 
$\mathcal{N}=4$ SYM.
%case that 
These are parameterized by two additional parameters 
$\epsilon_{3e}$ and $\epsilon_{3f}$ %. They occur 
%due to the fact that in the ansatz for the similarity transformations
%we have to abandon the identities \eqref{chiidentities1}
%and in the final result we must not make use also of \eqref{chiidentities2}.
and they occur because the identities in \eqref{chiidentities1} and  \eqref{chiidentities2}
are no longer applicable.
The details are worked out in appendix 
\ref{app:strafo}.
The dressing phase $\beta$ and the coefficients 
$\epsilon_{3a},\dots,\epsilon_{3c}$ of similarity transformations
in the scheme of $\mathcal{N}=1$ supergraphs are fixed by comparing
the $SU(2)$ subsector projection of the above result to the 
integrability-based expression.
This is not the case for $\epsilon_{3e}$ and $\epsilon_{3f}$
that drop out in the projection and hence can be set to convenient 
values. With our choice we have made the rational numbers
within the coefficients of $\chiop(1,3,2,3)+\chiop(3,1,2,1)$ and $\chiop(1,2,1,3)+\chiop(3,2,3,1)$
equal.
This yields
\begin{equation}
\beta=4\zeta(3)\col\qquad
\epsilon_{3a}=-4\col\qquad
\epsilon_{3b}=-i\frac{4}{3}\col\qquad
\epsilon_{3c}=i\frac{4}{3}\col\qquad
%\epsilon_{3d}=\col\qquad
\epsilon_{3e}=i\frac{4}{3}\col\qquad
\epsilon_{3f}=i\frac{4}{3}
\col
\end{equation}
%where we could choose convenient values for $\epsilon_{3e}$ and
%$\epsilon_{3f}$ such that the rational terms in lines 11 and 12 
%are identical. 
Note that we have not fixed 
the coefficient $\epsilon_{3d}$ in the 
scheme of Feynman diagrams in $\mathcal{N}=1$ superspace.
The calculation would be tedious and unnecessary here,
since the combination 
$\chiop(1,3,2)-\chiop(1,3,2)$ of chiral functions that are conjugate to each 
other vanishes whenever applied to the states, and hence 
$\epsilon_{3d}$ drops out.

\section{Application to single-impurity states}
\label{sec:oneimpstates} 

%We call chiral functions connected if nearest neighboured entries in 
%their argument lists never differ by more than $\pm1$. 
%These are the only chiral functions that in the $SU(2)$ subsector
%do not vanish on the single-impurity states \eqref{SU2oneexstate}
%and \eqref{cubiconeexstate}.
%Also, when applied to the respective momentum eigenstates of a single 
%excitation (magnon), they determine its dispersion relation.
%With the definition of a connected product of chiral functions given
%in \eqref{connprod}, we can write the dilatation operator
%\eqref{D1D2D3} and \eqref{D4chi} as 
%\begin{equation}\label{D1D2D3D4}
%\begin{aligned}
%\mathcal{D}_1&={}-{}2\chiop(1)\col\\
%\mathcal{D}_2&={}-{}2\big[\chiop(1)^2\big]_{\text{c}}\col\\
%\mathcal{D}_3&={}-{}4\big[\chiop(1)^3\big]_{\text{c}}
%+4i\epsilon_2[\chiop(2,1,3)-\chiop(1,3,2)]-4\chiop(1,3)\col\\
%\mathcal{D}_4&=
%{}-{}10\big[\chiop(1)^4\big]_{\text{c}}\\
%&\phantom{{}={}}
%+8\zeta(3)[{}-{}2\chiop(1)+\chiop(1,2)+\chiop(2,1)\\
%&\phantom{{}={}+8\zeta(3)[}
%-\chiop(1,2,1)-\chiop(2,1,2)
%+\chiop(1,2,3,2)+\chiop(3,2,1,2)]\\
%&\phantom{{}={}}
%-i(\epsilon_{3e}-\epsilon_{3f})[\chiop(2,1,2,3)+\chiop(2,3,2,1)-\chiop(1,2,3,2)-\chiop(3,2,1,2)]\\
%&\phantom{{}={}}
%+8\Big(\frac{23}{3}-\zeta(3)\Big)\chiop(1,3)%\\
%%&\phantom{{}={}}
%-4\chiop(1,4)\\
%&\phantom{{}={}}
%+2(4+\beta+2\epsilon_{3a}-2i\epsilon_{3b}+i\epsilon_{3c}-2i\epsilon_{3d})
%\chiop(1,3,2)\\
%&\phantom{{}={}}
%+2(4+\beta+2\epsilon_{3a}+2i\epsilon_{3b}-i\epsilon_{3c}+2i\epsilon_{3d})
%\chiop(2,1,3)\\
%&\phantom{{}={}}
%-2(2+2i\epsilon_{3b}+i\epsilon_{3c})[\chiop(1,2,4)+\chiop(1,4,3)]\\
%&\phantom{{}={}}
%-2(2-2i\epsilon_{3b}-i\epsilon_{3c})[\chiop(1,3,4)+\chiop(2,1,4)]\\
%&\phantom{{}={}}
%+\Big(\frac{14}{3}+8\zeta(3)+2\epsilon_{3a}-4i\epsilon_{3b}+i\epsilon_{3e}\Big)[\chiop(1,3,2,3)+\chiop(3,1,2,1)]\\
%&\phantom{{}={}}
%+\Big(\frac{14}{3}-4\zeta(3)+2\epsilon_{3a}+4i\epsilon_{3b}-i\epsilon_{3e}\Big)[\chiop(1,2,1,3)+\chiop(3,2,3,1)]\\
%&\phantom{{}={}}-2(6+\beta+2\epsilon_{3a})\chiop(2,1,3,2)\\
%&\phantom{{}={}}
%+2(9+2\epsilon_{3a})[\chiop(1,3,2,4)+\chiop(2,1,4,3)]\\
%&\phantom{{}={}}
%-2(4+\epsilon_{3a}+i\epsilon_{3b})[\chiop(1,2,4,3)+\chiop(1,4,3,2)]\\
%&\phantom{{}={}}
%-2(4+\epsilon_{3a}-i\epsilon_{3b})[\chiop(2,1,3,4)+\chiop(3,2,1,4)]
%\pnt
%\end{aligned}
%\end{equation}
%Note that in the $SU(2)$ subsector this combination of chiral functions
%is expected \cite{Sieg:2010tz} from the all-order 
%form of the magnon dispersion relation \cite{Beisert:2004hm,Santambrogio:2002sb,}.
%\marginpar[]{\tiny Some more references here?}
%\begin{equation}\label{Ep}
%E(p)=\sqrt{1+16g^2\sin^2\frac{p}{2}}-1
%\pnt
%\end{equation}
%It continues to holds at least for the rational terms 
%even without the identities \eqref{chiidentities1}, \eqref{chiidentities2},
%since the cancellation of all higher order poles
%in the logarithm of the renormalization constant that enters 
%\eqref{DinZ} has
%to remain guaranteed \cite{Sieg:2010tz}, and the only occurring 
%integrals in the results \eqref{range3} and \eqref{crange4}
%have identical rational pole terms.
%They only differ by $\zeta(3)$ in their simple pole, and this causes 
%the combination starting in the second line of $\mathcal{D}_4$.

We call chiral functions ``connected'' if adjacent %nearest neighboured 
entries in
their arguments %lists never 
differ %by more than 
by $\pm1$.
These are the only chiral functions that %in the $SU(2)$ subsector
do not vanish 
when acting 
on the one-impurity states in \eqref{SU2oneexstate}
and \eqref{cubiconeexstate}. 
For this reason the magnon dispersion relations can only depend on the connected chiral functions.
%When they are applied to the respective momentum
%eigenstates of a single excitation (magnon), they yield the
%dispersion relation. In the integrable cases of $\mathcal{N}=4$ SYM
%theory and its $\beta$-deformation it reads
%\cite{Beisert:2004hm,Santambrogio:2002sb,}.
%\marginpar[]{\tiny Some more references here?}
%\begin{equation}\label{Ep}
%E(p)=\sqrt{1+16g^2\sin^2\frac{p}{2}}-1
%\pnt
%\end{equation}
Using %With 
the definition of a connected product of chiral functions %given
in \eqref{connprod}, we can reexpress the the first four orders of $\DD$ in
\eqref{D1D2D3} and \eqref{D4chi} as
\begin{equation}\label{D1D2D3D4}
\begin{aligned}
\mathcal{D}_1&={}-{}2\chiop(1)\col\\
\mathcal{D}_2&={}-{}2\big[\chiop(1)^2\big]_{\text{c}}\col\\
\mathcal{D}_3&={}-{}4\big[\chiop(1)^3\big]_{\text{c}}
+4i\epsilon_2[\chiop(2,1,3)-\chiop(1,3,2)]-4\chiop(1,3)\col\\
\mathcal{D}_4&=
{}-{}10\big[\chiop(1)^4\big]_{\text{c}}\\
&\phantom{{}={}}
+8\zeta(3)[{}-{}2\chiop(1)+\chiop(1,2)+\chiop(2,1)\\
&\phantom{{}={}+8\zeta(3)[}
-\chiop(1,2,1)-\chiop(2,1,2)
+\chiop(1,2,3,2)+\chiop(3,2,1,2)]\\
&\phantom{{}={}}
-i(\epsilon_{3e}-\epsilon_{3f})[\chiop(2,1,2,3)+\chiop(2,3,2,1)-\chiop(1,2,3,2)-\chiop(3,2,1,2)]\\
&\phantom{{}={}}
+8\Big(\frac{23}{3}-\zeta(3)\Big)\chiop(1,3)%\\
%&\phantom{{}={}}
-4\chiop(1,4)\\
&\phantom{{}={}}
+2(4+\beta+2\epsilon_{3a}-2i\epsilon_{3b}+i\epsilon_{3c}-2i\epsilon_{3d})
\chiop(1,3,2)\\
&\phantom{{}={}}
+2(4+\beta+2\epsilon_{3a}+2i\epsilon_{3b}-i\epsilon_{3c}+2i\epsilon_{3d})
\chiop(2,1,3)\\
&\phantom{{}={}}
-2(2+2i\epsilon_{3b}+i\epsilon_{3c})[\chiop(1,2,4)+\chiop(1,4,3)]\\
&\phantom{{}={}}
-2(2-2i\epsilon_{3b}-i\epsilon_{3c})[\chiop(1,3,4)+\chiop(2,1,4)]\\
&\phantom{{}={}}
+\Big(\frac{14}{3}+8\zeta(3)+2\epsilon_{3a}-4i\epsilon_{3b}+i\epsilon_{3e}\Big)[\chiop(1,3,2,3)+\chiop(3,1,2,1)]\\
&\phantom{{}={}}
+\Big(\frac{14}{3}-4\zeta(3)+2\epsilon_{3a}+4i\epsilon_{3b}-i\epsilon_{3e}\Big)[\chiop(1,2,1,3)+\chiop(3,2,3,1)]\\
&\phantom{{}={}}-2(6+\beta+2\epsilon_{3a})\chiop(2,1,3,2)\\
&\phantom{{}={}}
+2(9+2\epsilon_{3a})[\chiop(1,3,2,4)+\chiop(2,1,4,3)]\\
&\phantom{{}={}}
-2(4+\epsilon_{3a}+i\epsilon_{3b})[\chiop(1,2,4,3)+\chiop(1,4,3,2)]\\
&\phantom{{}={}}
-2(4+\epsilon_{3a}-i\epsilon_{3b})[\chiop(2,1,3,4)+\chiop(3,2,1,4)]
\pnt
\end{aligned}
\end{equation}
The terms with connected products appear naturally in the expansion of the $\NN=4$  magnon dispersion relation \cite{Sieg:2010tz,Minahan:2011dd}.  In particular, the first term in $\DD_4$ is consistent with a conjecture made in
\cite{Minahan:2011dd} that the rational connected parts of $\DD_m$ are given by
\begin{equation}
\frac{\Gamma(3/2)}{\Gamma(m+1)\Gamma(3/2-m)}(-4)^m\big[\chi(1)^m\big]_{\text{c}}
\pnt
\end{equation} 
The conjecture is nontrivial since it allows us to separate terms that would have been equivalent in the $\beta$-deformed or undeformed theory.

The next term in $\DD_4$ is a sum of connected chiral functions multiplied by $8\zeta(3)$.  It vanishes when we project to the 
$SU(2)$ subsector in $\NN=4$ or the real $\beta$-deformed theory and hence
is consistent with the dispersion relation \eqref{Ep} with 
$h^2(g)=4g^2$. However, it contributes when two distinct types of 
scalar magnons approach each other, and hence is associated with 
the %respective 
scattering matrix for different magnons.
In the complex $\beta$-deformed case it is also non-vanishing 
if a single magnon 
is within its interaction range and hence it starts to appear 
within the respective dispersion relation as a four-loop contribution to
$h^2(g)$ in \eqref{Ep}.
%Note that in the integrable cases, the dispersion relation \eqref{Ep}
%guarantees that at each order a particular linear combination of
%connected chiral functions occurs that can be combined
%into a connected product \cite{Sieg:2010tz}.
%In the generalization to the Leigh-Strassler deformation \eqref{superpot}
%where one cannot make use of
%the identities \eqref{chiidentities1}, \eqref{chiidentities2},
%it continues to holds at least to four-loops for the rational terms
%due to the following argument:
%The cancellation of all higher order poles
%in the logarithm of the renormalization constant that enters
%\eqref{DinZ} has
%to remain guaranteed \cite{Sieg:2010tz}, and the only occurring
%integrals in the results \eqref{range3} and \eqref{crange4}
%have identical rational pole terms.
%They only differ by $\zeta(3)$ in their simple pole, and this causes
%the combination starting in the second line of $\mathcal{D}_4$.
%

We can now use \eqref{D1D2D3D4} to determine the
four-loop anomalous dimensions for particular operators.

\subsection{Complex $\beta$-deformation}


The one-impurity operators in the complex $\beta$-deformations,  where $|q|\neq 1$ and $h=0$, are eigenstates of $\chi(1)$ with eigenvalue $c(q,\bar q)$, where
\begin{equation}
 c(q,\bar q)=\rho^2(1-q)(1-\bar q)=\frac{2(1-q)(1-\bar q)}{(1+q\bar q)}\col
\end{equation}
and we have eliminated $\rho$ by applying  \eqref{buildingblock}.
Furthermore, the connected products of $\chi(1)$ are products of $c(q,\bar  q)$.  Using the relation in \eqref{chifuncrel}, we find 
\begin{equation}
\gamma=\big[\sqrt{1+4g^2c(q,\bar q)}-1\big]-8\zeta(3)(1-q\bar q)^2c^2(q,\bar q)
+\mathcal{O}(g^{10})\pnt
\end{equation}
The part of $\gamma$ inside the square bracket comes from the connected products in  \eqref{D1D2D3D4}.  The transcendental term comes from those connected chiral functions that are not included in the connected products.  In the special case where $\beta$ is real,  the transcendental term drops out and 
we find $c(q,\bar q)=4\sin^2\pi\beta$. The remaining part of $\gamma$ is the energy coming from \eqref{Ep}
for one excitation with   momentum shifted to $p=2\pi\beta$ by the twisted boundary conditions.




%In the case of  complex $\beta$-deformations where $|q|\neq 1$ and $h=0$,
%the anomalous dimension of the %composite 
%one-impurity operator in \eqref{SU2oneexstate}
%is easily determined from \eqref{D1D2D3D4}, using the relation 
%\eqref{chifuncrel} and the result for the action of the simplest 
%chiral function that is given by
%\begin{equation}
%\chi(1)\to c(q,\bar q)=\rho^2(1-q)(1-\bar q)=\frac{2(1-q)(1-\bar q)}{(1+q\bar q)}\pnt
%\end{equation}
%Eliminating $\rho$ by using the relation \eqref{buildingblock}, 
%the four-loop anomalous dimension then reads
%\begin{equation}
%\gamma=\sqrt{1+4g^2c(q,\bar q)}-1-8\zeta(3)(1-q\bar q)^2c^2(q,\bar q)
%+\mathcal{O}(g^{10})\pnt
%\end{equation}
%We have summed the rational parts in the square root 
%to make contact with the magnon dispersion relation \eqref{Ep}.
%In the special case of the real $\beta$-deformation, $q=\e^{2i\pi\beta}$, 
%$\bar q=\e^{-2i\pi\beta}$ the transcendental term drops out, and 
%we find $c(q,\bar q)=4\sin^2\pi\beta$. This is 
%the correct result coming from the dispersion relation
%\eqref{Ep}
%after inserting the momentum of the single excitation $p=2\pi\beta$ 
%as determined by the twisted boundary conditions.

% \begin{equation}
% \begin{aligned}\label{D4chicplxbeta}
% \mathcal{D}_3
% &={}-{}4(\chi(1,2,3)+\chi(3,2,1))+4i\epsilon_2(\chi(2,1,3)-\chi(1,3,2))
% -4\chi(1,3)\\
% &\phantom{{}={}}
% +16(\chi(1,2)+\chi(2,1))-8(2+q\bar q)\chi(1)
% \col\\
% \mathcal{D}_4&={}+{}8(25-(1-q\bar q)(15-2\zeta(3)))\chiop(1)\\
% &\phantom{{}={}}
% -2(75-(1-q\bar q)(15-4\zeta(3)))[\chiop(1,2)+\chiop(2,1)]\\
% &\phantom{{}={}}
% +8\Big(10+\epsilon_{3a}-(1-q\bar q)\Big(\frac{7}{3}+\epsilon_{3a}+\zeta(3)\Big)\Big)\chiop(1,3)%\\
% %&\phantom{{}={}}
% -4\chiop(1,4)\\
% &\phantom{{}={}}
% +60[\chiop(1,2,3)+\chiop(3,2,1)]\\
% &\phantom{{}={}}
% +2(4+\beta+2\epsilon_{3a}-2i\epsilon_{3b}+i\epsilon_{3c}-2i\epsilon_{3d})
% \chiop(1,3,2)\\
% &\phantom{{}={}}
% +2(4+\beta+2\epsilon_{3a}+2i\epsilon_{3b}-i\epsilon_{3c}+2i\epsilon_{3d})
% \chiop(2,1,3)\\
% &\phantom{{}={}}
% -2(2+2i\epsilon_{3b}+i\epsilon_{3c})[\chiop(1,2,4)+\chiop(1,4,3)]\\
% &\phantom{{}={}}
% -2(2-2i\epsilon_{3b}-i\epsilon_{3c})[\chiop(1,3,4)+\chiop(2,1,4)]\\
% &\phantom{{}={}}
% -2(6+\beta+2\epsilon_{3a})\chiop(2,1,3,2)\\
% &\phantom{{}={}}
% +2(9+2\epsilon_{3a})[\chiop(1,3,2,4)+\chiop(2,1,4,3)]\\
% &\phantom{{}={}}
% -2(4+\epsilon_{3a}+i\epsilon_{3b})[\chiop(1,2,4,3)+\chiop(1,4,3,2)]\\
% &\phantom{{}={}}
% -2(4+\epsilon_{3a}-i\epsilon_{3b})[\chiop(2,1,3,4)+\chiop(3,2,1,4)]\\
% &\phantom{{}={}}
% -10[\chiop(1,2,3,4)+\chiop(4,3,2,1)]\col
% \end{aligned}
% \end{equation}



\subsection{Cubic Leigh-Strassler deformation}

In the cubic Leigh-Strassler deformation, where $\rho\to0$, $\rho|h|=\sqrt{2}$,
the 3-string null operators are eigenstates of every chiral function.  For $\chi(1)$ the eigenvalue is $-2M$, where $M$ is the number of pairs of adjacent fields with the same flavor.  For all other chiral functions the eigenvalue is 0.  Hence, the eigenvalue of the connected product $[\chi(1)^m]_c$ is $(-2)^m M$.
Inserting these expressions into  \eqref{D1D2D3D4}, we find that the anomalous dimensions of these operators are $\gamma_M=M\gamma_1$, where
\begin{equation}\label{fourloopgamma}
\gamma_1=\big[\sqrt{1+8g^2}-1\big]-32\,\zeta(3)\,g^8+\mathcal{O}(g^{10})
\pnt
\end{equation}

Note that the four-loop result is consistent with a conjecture in \cite{Minahan:2011dd} for the rational part of $\gamma_1$, where it was proposed that it would have the  form in the square brackets to all orders in $g$, assuming that  cancellations similar to \eqref{chionefour} continue to hold\footnote{In  \cite{Minahan:2011dd} the cancellations were also shown to hold at five-loop order.}. However, there is also a transcendental contribution starting at four-loop order.  A natural way to view this is that instead of $g^2$, the square root depends on a $g$ dependent function $\tilde h^2(g)$, such that
\begin{equation}\label{gamma}
\gamma_1=\sqrt{1+8\tilde h^2(g)}-1\,,
\end{equation}
where
\begin{equation}
\tilde h^2(g)=g^2-8\zeta(3)\,g^8+\dots\,.
\end{equation}
Without this correction the strong coupling behavior of $\gamma_1$ would have been $\gamma_1\sim g$,  but with it $\gamma_1$  likely increases with a smaller power of $g$.

%In the cubic Leigh-Strassler deformation, where $\rho\to0$, $\rho|h|=\sqrt{2}$,
%the dilatation operator \eqref{D1D2D3D4} does not describe the mixing in 
%a closed subsector \cite{Minahan:2011dd}. 
%However, the composite operator \eqref{cubiconeexstate} does not mix with 
%other operators. It is not protected, since the
%chiral function $\chiop(1)$ yields a non-vanishing result 
%whenever the elementary building block \eqref{buildingblock} 
%involves two identical adjacent field flavors.
%In this case, one obtains
%\begin{equation}
%\chiop(1)\to-2
%\pnt
%\end{equation}
%All the chiral functions with more arguments require more identical 
%adjacent field flavors and yield zero.
%The anomalous dimension of the first excited state \eqref{cubiconeexstate} is 
%hence given by extracting the term with $\chiop(1)$ from the 
%dilatation operator \eqref{D1D2D3D4} after expanding the connected products.
%To four-loop order, this yields the anomalous dimension
%\begin{equation}\label{fourloopgamma}
%\gamma=4g^2-8g^4+24g^6-32(5+\zeta(3))g^8+\mathcal{O}(g^{10})
%\pnt
%\end{equation}
%In \cite{Minahan:2011dd} arguments in favor of an all-order summation 
%of the rational parts were considered. However, the possibility of 
%additional transcendental contributions was not considered in a first 
%version. In the light of our result that here we find such  
%an additional transcendental term, we cannot exclude that at higher orders
%also the rational terms may deviate from what was expected.
%However, it should still be possible to 
%rearrange the answer in a square root form as 
%\begin{equation}\label{gamma}
%\gamma=\sqrt{1+2h^2(g)}-1\col\qquad h^2(g)=4g^2-32g^8\zeta(3)+\dots
%\col
%\end{equation}
%where the aforementioned deviations appear as higher order corrections 
%to $h^2(g)$. They start at four loops where they are homogeneous 
%transcendental but non-maximal, i.e. they are proportional to $\zeta(3)$. 

\section{Conclusions}
\label{sec:conclusions}

The main result of this paper is the construction of the scalar part of the four-loop chiral dilatation operator \eqref{D1D2D3D4}.  This construction is valid for any Leigh-Strassler deformation and it is complete for the closed scalar subsectors of these deformations.  Completeness for an arbitrary deformation would require the contributions to $\DD$ involving the gauginos $\psi_\alpha$.

For the closed subsectors we have explicitly found the four-loop anomalous dimensions for the one impurity operators in the complex $\beta$-deformed theory and for {\it every} 3-string null operator in the cubic Leigh-Strassler theory.  
These anomalous dimensions have $\zeta(3)$ terms that first appear at four loops\footnote{The transcendental contributions do come one loop after those found 
in \cite{Pomoni:2011jj} for the $\NN=2$ interpolating theory in 
\cite{Gadde:2010zi,Gadde:2009dj}.}.  
These transcendental terms show up because  they are present in the  
coefficients of connected chiral functions in $\DD_4$.  

%In section \ref{sec:oneimpstates} we have observed that the occurring
%$\zeta(3)$ coefficients in dependence on the type of deformation
%are associated with the dispersion relation (in the complex $\beta$-deformed
%case) or the scattering matrix of two different types of scalar magnons (in 
%the integrable cases). This seems to indicate a connection between the 
%occurrence of transcendental terms in the dispersion relation and
%the scattering matrix, when considering larger subsectors.
%Such a connection would be very important since it might allow for an 
%interpretation or even determination of the function $h^2(g)$
%in terms of the BES dressing phase \cite{Beisert:2006ez}
%as an exact expression in $g$ and with a
%weak coupling expansion that is homogeneous transcendental 
%at each order.

From the perspective of $\NN=4$ SYM or its integrable deformations, the $\zeta(3)$ coefficients of the  connected (or disconnected) chiral functions must trace back to the BES dressing phase  \cite{Beisert:2006ez};
 the only place that transcendental terms appear in the all loop Bethe equations  \cite{Beisert:2005fw} is in the dressing phase and so it must be the source for these terms in the chiral dilatation operator.  However, in the complex $\beta$-deformed theory, these same terms are associated with the dispersion relation, since they contribute to the anomalous dimensions for single impurity operators.   This suggests  that the dispersion relation for the complex $\beta$-deformed theory, or the anomalous dimensions of the 3-string null operators in the cubic deformation,  are in principle derivable from the BES dressing phase. This assumes that the tuning mechanism as pictured in \eqref{gamma4cancel} and \eqref{chionefour} continues to hold at higher loops, otherwise further corrections could spoil this relation.

% Ultimately, the $\zeta(3)$ coefficients in the  chiral functions, connected or otherwise, must trace back to the BES dressing phase  \cite{Beisert:2006ez}, since $\DD_4$ is also valid for $\NN=4$ SYM. It is the scalar part of the four-loop dilatation operator in the $SU(2|3)$ subsector. 
% \mk[I do not agree on this statement! These transcendental terms are not from scattering...have to discuss this]
% The only place that transcendental terms appear in the all loop Bethe equations  \cite{Beisert:2005fw} is in the dressing phase and so it must be the source for these terms in the dilatation operator.  Dropping all transcendental terms would then be equivalent to constructing  a dilatation operator from the BDS Bethe equations \cite{Beisert:2004hm}.  Perhaps this information will be useful for constructing $\tilde h(g)$, which would then give  the all loop expression for the coefficient of $\chi(1)$ in the dilation operator via \eqref{gamma}.

For the immediate future, one can use the disentangling of the chiral functions to compute next to leading order wrapping effects, extending the analysis 
in \cite{Sieg:2005kd,Fiamberti:2007rj,Fiamberti:2008sh,Fiamberti:2008sm,Fiamberti:2008sn}.  It would also be interesting to find the complete four-loop chiral dilatation operator.  For this it would be useful to find an extension of the chiral functions that could also include the gauginos.

%In this paper we have determined the four-loop dilatation operator 
%\eqref{D1D2D3D4} for composite operators built by choosing 
%out of two chiral field flavors in the 
%Leigh-Strassler deformed $\mathcal{N}=4$ SYM theory. 
%We find $\zeta(3)$ contributions at four loops. In the dilatation 
%operator they contribute to the coefficients of the connected chiral 
%functions, and hence
%are associated with the dispersion relation of the single excitation.
%This has to be confronted with the $\zeta(3)$ terms that 
%already occur in the $\mathcal{N}=4$ SYM theory and are 
%associated with the scattering matrix rather than with the 
%dispersion relation of the Bethe ansatz in the integrable 
%special limits.
%
%By applying our result to the single-impurity states \eqref{SU2oneexstate} 
%and \eqref{cubiconeexstate} in the 
%special cases of the complex $\beta$-deformed theory and the 
%cubic Leigh-Strassler deformed one, we obtain the respective anomalous
%dimensions. In both cases, the results depend on the transcendental 
%$\zeta(3)$ contributions, and this made it necessary to modify 
%the original conjecture for the anomalous dimension in the 
%cubic Leigh-Strassler case as formulated in \cite{Minahan:2011dd}. 
%Still, the transcendental contributions are delayed by one 
%loop order, i.e.\ in \eqref{gamma} a $\zeta(3)$ contribution does not occur 
%at three loops where it was found \cite{Pomoni:2011jj} in case of a theory 
%interpolating between a $\mathcal{Z}_4$ orbifold of $\mathcal{N}=4$ SYM 
%theory and $\mathcal{N}=2$ superconformal QCD 
%\cite{Gadde:2010zi,Gadde:2009dj}.


% This is similar to the case of $\mathcal{N}=4$ SYM theory, where 
% such contributions also occur at four loops, but there they come from 
% the coefficient $\beta$ in \eqref{D4chi} that is the lowest order term 
% in the dressing phase of the scattering matrix. However, here they also
% appear in the coefficients of connected chiral functions, in particular of 
% $\chiop(1)$, and hence they show up in \eqref{gamma} that is the counterpart of 
% the magnon dispersion relation of $\mathcal{N}=4$ SYM theory in the 
% cubic Leigh-Strassler model.

% In this paper we have determined the four-loop dilatation operator
% that generalizes the result in the $SU(2)$ subsector of the $\mathcal{N}=4$
% SYM theory. Our result \eqref{D4inchi} contains all the 
% individual chiral functions that appear in the Feynman diagrams with 
% overall UV divergences before applying the identities \eqref{chiidentities1}
% and \eqref{chiidentitites2}
% that make them indistinguishable in the $beta$-deformed case.

%In our result for the dilatation operator we have disentangled
%the contributions from Feynman diagrams of different range 
%that are indistinguishable in an integrability-based construction
%in the $SU(2)$ subsector of the $\mathcal{N}=4$ SYM theory 
%and its $\beta$-deformation. Therefore, our result 
%will be an essential ingredient for the calculation of the
%next-to leading wrapping corrections when proceeding 
%along the lines of 
%\cite{Sieg:2005kd,Fiamberti:2007rj,Fiamberti:2008sh,Fiamberti:2008sm,Fiamberti:2008sn}.

\enlargethispage{\baselineskip}

\section*{Acknowledgements}
J.\ A.\ M.\ and C.\ S.\ thank the Perimeter Institute for kind hospitality.
J.\ A.\ M.\ also thanks the CTP at MIT for kind hospitality during the course of this
work. The work of J.\ A.\ M.\ is supported in part by Vetenskapr\aa{}det.
The work of C.\ S.\ is supported by DFG, SFB 647
\emph{Raum -- Zeit -- Materie. Analytische und Geometrische Strukturen}.


\appendix

\section{Relations between chiral functions}
\label{app:chifuncrel}


In the $SU(2)$ subsector, the
building block \eqref{buildingblock} of the chiral functions
\eqref{chifuncdef} reduces to
\begin{equation}
\Fop_{ij}|_{SU(2)}=\rho^2(\dperm-\unitmatrix)_{ij}\col
\end{equation}
where the deformed permutation and the identity on the r.h.s.\ are the ones
 given in \eqref{permdef} but
projected into the subspace of two flavors by applying \eqref{lambdaSU2proj}. 
Using the resulting expressions, one
can check that the r.h.s.\ obeys the relation
\begin{equation}\label{buildingblockid}
(\dperm-\unitmatrix)_{n\,n+1}(\dperm-\unitmatrix)_{n+1\,n+2}(\dperm-\unitmatrix)_{n\,n+1}
=q\bar q(\dperm-\unitmatrix)_{n\,n+1}
\pnt
\end{equation}
This relation has its origin in the fact that three adjacent elementary fields 
within the single trace of the composite operators of the $SU(2)$ subsector
carry maximally two different field flavors. The permutations
then fulfill the relation
\begin{equation}
\begin{aligned}
0
%\epsilon_{i_ni_{n+1}i_{n+2}}\epsilon^{o_no_{n+1}o_{n+2}}|_{SU(2)}
&=%\big[
\unitmatrix-\dperm_{n\,n+1}-\dperm_{n+1\,n+2}+\dperm_{n\,n+1}\dperm_{n\,n+1}+\dperm_{n+\,n+2}\dperm_{n\,n+1}%\\
%&\phantom{{}=\big[{}}
-\dperm_{n\,n+1}\dperm_{n+1\,n+2}\dperm_{n\,n+1}
%\big]_{i_ni_{n+1}i_{n+2}}^{o_no_{n+1}o_{n+2}}
\col
\end{aligned}
\end{equation}
where the r.h.s.\ at vanishing $\beta$-deformation $q=1$ 
is nothing else than the total antisymmetrizer 
$\epsilon_{i_ni_{n+1}i_{n+2}}\epsilon^{o_no_{n+1}o_{n+2}}$ between three field 
flavors when expressed in terms of permutations. In the undeformed case,
a respective relation was already worked out in  \cite{Beisert:2005wv}.
When inserting \eqref{buildingblockid} into the definition of the chiral 
functions \eqref{chifuncdef}, we find that in the 
$SU(2)$ subsector they are reducible as
\begin{equation}\label{chifuncrel}
\chiop(a_1,\dots,a_k,a,b,a,a_{k+4},\dots,a_n)|_{SU(2)}
=\rho^4q\bar q\chiop(a_1,\dots,a_k,a,a_{k+4},\dots,a_n)|_{SU(2)}
\pnt
\end{equation}

For the general Leigh-Strassler deformation, it is very easy to check
that the chiral functions 
fulfill
%\footnote{This equation is valid in our expansion where we
%strip-off factors of $g_\YM$  if we use the 
%relation \eqref{couprel} in order to replace a factor from the
%contraction of a chiral and an anti-chiral vertex as in
%the first diagram of \eqref{sediag} in terms of $g_\YM$.}
\begin{equation}\label{chifuncred}
\begin{aligned}
\chiop(a_1,\dots,a_k,a,a,a_{k+3},\dots,a_n)
&=-\rho^2(1+q\bar q+h\bar h)\chiop(a_1,\dots,a_k,a,a_{k+3},\dots,a_n)\\
&=-2\chiop(a_1,\dots,a_k,a,a_{k+3},\dots,a_n)+\mathcal{O}(\kappa^6)
\col
\end{aligned}
\end{equation}
where the second equality relies on \eqref{buildingblock}.
The previous relation between chiral functions is required in order 
to simplify the (noncommutative but 
associative) products of chiral functions. 
In order two define such products of chiral functions, we assume from 
now on that the length $L$ is always sufficiently 
large, i.e.\ $L\ge\kappa_a+\kappa_b-1$. Thereby, $\kappa_a$, $\kappa_b$
are the ranges, i.e.\ the numbers
of legs involved in the chiral interactions, 
of the two chiral functions that are to be multiplied.
In terms of the arguments of \eqref{chifuncdef} the range is defined as
\begin{equation}
\kappa_a=\max_{a_1,\dots,a_n}-\min_{a_1,\dots,a_n}+2\pnt
\end{equation}
We first introduce the commutator of two chiral functions that is given by
%Assuming that we always choose the representative in which the smallest 
%entry in the argument lists is $1$, we obtain
\begin{equation}\label{commdef}
\begin{aligned}
\comm{\chiop(a_1\dots,a_n)\,}{\,\chiop(b_1,\dots,b_p)}
&=\sum_{\min\limits_{b_1,\dots,b_p}-\max\limits_{a_1,\dots,a_n}-1}^{\max\limits_{b_1,\dots,b_p}-\min\limits_{a_1,\dots,a_n}+1}
\chiop(a_1+k,\dots,a_n+k,b_1,\dots,b_p)
%\sum_{k=0}^{\displaystyle\max_{b_1,\dots,b_p}}\chiop(a_1+k,\dots,a_n+k,b_1,\dots,%b_p)\\
%&\phantom{{}={}}
%+\sum_{k=1}^{\displaystyle\max_{a_1,\dots,a_n}}\chiop(a_1,\dots,a_n,b_1+k,\dots,b%_p+k)\\
%&\phantom{{}={}}
%-\sum_{k=0}^{\displaystyle\max_{a_1,\dots,a_n}}\chiop(b_1+k,\dots,b_p+k,a_1,\dots%,a_n)\\
%&\phantom{{}={}}
%-\sum_{k=1}^{\displaystyle\max_{b_1,\dots,b_p}}\chiop(b_1,\dots,b_p,a_1+k,\dots,a_n+k)\\
\\
&\phantom{{}={}}-\sum_{\min\limits_{a_1,\dots,a_n}-\max\limits_{b_1,\dots,b_p}-1}^{\max\limits_{a_1,\dots,a_n}-\min\limits_{b_1,\dots,b_p}+1}
\chiop(b_1+k,\dots,b_p+k,a_1,\dots,a_n)
\pnt
\end{aligned}
\end{equation}
We call chiral functions ``connected'' if adjacent entries in their list of 
arguments never differ by more than $\pm 1$. For these chiral 
functions, we define the connected product as 
\begin{equation}\label{connprod}
[\chiop(a_1\dots,a_n)\chiop(b_1,\dots,b_p)]_{\text{c}}
=\sum_{k=b_1-a_n-1}^{b_1-a_n+1}\chiop(a_1+k,\dots,a_n+k,b_1,\dots,b_p)\col
\end{equation}
such that the result is again a connected chiral function.
It is understood that for multiple connected chiral functions 
appearing within $[\dots]_\text{c}$, the connected product is
applied for each pairwise multiplication.



\section{Similarity transformations}
\label{app:strafo}

The representation of the dilatation operator is not unique, but 
it may be transformed by a change of the basis of operators that 
does not alter its eigenvalues. In this appendix, we work out such
transformations.
We include non-unitary cases that allow us to remove the 
anti-Hermitean contributions in the three-loop dilatation 
operator given in \eqref{D1D2D3}.
The similarity transformations can be realized as
\begin{equation}\label{strafo}
\mathcal{D}'=\e^{-\chi}\mathcal{D}\e^{\chi}=\mathcal{D}+\delta\mathcal{D}
\col
\end{equation}
where $\chi$ is a linear combination of chiral functions.
We demand that the transformations preserve the structural constraints
coming from the underlying Feynman diagrams, i.e.\
at each loop order in the weak coupling expansion of 
$\mathcal{D}$ the transformation must not generate contributions
that involve chiral functions that can only appear at higher orders.
This is guaranteed if the weak coupling expansion of $\chi$ only contains 
those chiral functions that can be associated with Feynman diagrams at the 
considered order. In the $SU(2)$ subsector of the $\mathcal{N}=4$ SYM theory
the transformations are parameterized by one and four 
free parameters respectively in the three- and
four-loop contribution to the dilatation operator. 
Here, we have to abandon the relation \eqref{chiidentities1} and 
construct a more general transformation at four loops.
The ansatz is given by
\begin{equation}\label{stansatz}
\begin{aligned}
\chi&=g^2i\delta_{11}\chiop(1)
%&\phantom{{}={}}
+g^4\Big(
i\delta_{21}\chiop(1)
+\frac{i}{2}\delta_{22}[\chiop(1,2)+\chiop(2,1)]\Big)\\
&\phantom{{}={}}
+g^6\Big(
i\delta_{31}\chiop(1)
+\frac{i}{2}\delta_{32}[\chiop(1,2)+\chiop(2,1)]
+i\delta_{33}\chiop(1,3)
+\frac{i}{2}\delta_{34}[\chiop(1,2,1)+\chiop(2,1,2)]\\
&\phantom{{}={}+g^6\Big(}
+\frac{1}{2}(i\delta_{35}+\delta_{36})\chiop(1,3,2)
+\frac{1}{2}(i\delta_{35}-\delta_{36})\chiop(2,1,3)\\
&\phantom{{}={}+g^6\Big(}
+\frac{i}{2}\delta_{37}[\chiop(1,2,3)+\chiop(3,2,1)]
\Big)
\col
\end{aligned}
\end{equation}
where we have considered the fact that the chiral functions of the 
first two contributions 
in the last row are adjoint to each other, but the loop integrals
of the respective diagrams are different.

Inserting this ansatz into \eqref{strafo} and expanding in powers of
$g$, we respectively 
obtain for the non-vanishing transformations at two and three loops
\begin{equation}
\begin{aligned}
\delta\mathcal{D}_3&=
-4i\epsilon_2[\chiop(1,3,2)-\chiop(2,1,3)]
\col\\
\delta\mathcal{D}_4&=
2(2\epsilon_{3a}-2i\epsilon_{3b}+i\epsilon_{3c}-2i\epsilon_{3d})
\chiop(1,3,2)%\\
%&\phantom{{}={}}
+2(2\epsilon_{3a}+2i\epsilon_{3b}-i\epsilon_{3c}+2i\epsilon_{3d})
\chiop(2,1,3)\\
&\phantom{{}={}}
-2i(2\epsilon_{3b}+\epsilon_{3c})[
\chiop(1,2,4)+\chiop(1,4,3)
-\chiop(1,3,4)-\chiop(2,1,4)]\\
&\phantom{{}={}}
-i(\epsilon_{3e}-\epsilon_{3f})[
\chiop(2,1,2,3)+\chiop(2,3,2,1)
-\chiop(1,2,3,2)-\chiop(3,2,1,2)]\\
&\phantom{{}={}}
+(2\epsilon_{3a}-4i\epsilon_{3b}+i\epsilon_{3e})[\chiop(1,3,2,3)+\chiop(3,1,2,1)]\\
&\phantom{{}={}}
+(2\epsilon_{3a}+4i\epsilon_{3b}-i\epsilon_{3e})[\chiop(1,2,1,3)+\chiop(3,2,3,1)]\\
&\phantom{{}={}}-4\epsilon_{3a}[\chiop(2,1,3,2)-\chiop(1,3,2,4)-\chiop(2,1,4,3)]\\
&\phantom{{}={}}
-2(\epsilon_{3a}+i\epsilon_{3b})[\chiop(1,2,4,3)+\chiop(1,4,3,2)]\\
&\phantom{{}={}}
-2(\epsilon_{3a}-i\epsilon_{3b})[\chiop(2,1,3,4)+\chiop(3,2,1,4)]
\col
\end{aligned}
\end{equation}
where the independent parameters read
\begin{equation}
\begin{aligned}
\epsilon_2&=\frac{1}{2}(2\delta_{11}-\delta_{22})\col\\
\epsilon_{3a}&=-\frac{3}{2}\delta_{11}(2\delta_{11}-\delta_{22})
-\frac{1}{2}\delta_{36}
\col\qquad
&\epsilon_{3b}&=2\delta_{11}-\frac{1}{2}(\delta_{35}+\delta_{37})
\col\\
\epsilon_{3c}&=-2\delta_{11}-\delta_{33}+\delta_{35}+\delta_{37}
\col\qquad
&\epsilon_{3d}&=\delta_{21}+\delta_{22}-12\delta_{11}-\frac{1}{2}\delta_{32}+\frac{3}{2}\delta_{35}+\delta_{37}
\col\\
\epsilon_{3e}&=\delta_{34}-3\delta_{35}-\delta_{37}
\col\qquad
&\epsilon_{3f}&=-3\delta_{35}
\pnt
\end{aligned}
\end{equation}

\newpage

\section{Integrals}
\label{app:integrals}
\enlargethispage{\baselineskip}

All four-loop integrals $I$ and their pole parts $\mathcal{I}$ used in the 
text are given by
\begin{equation}\label{IK}
\begin{aligned}
I_{4\text{bb}}&=
\intsixpo{%
\fmf{plain}{v1,v2}
\fmf{plain}{v2,v3}
\fmf{plain}{vc,v1}
\fmf{plain}{vc,v2}
\fmf{plain,left=0.25}{v3,v4}
\fmf{plain,right=0.25}{v3,v4}
\fmf{plain,left=0.25}{vc,v4}
\fmf{plain,right=0.25}{vc,v4}}
\col\qquad
\mathcal{I}_{4\text{bb}}
=\frac{1}{(4\pi)^8}\Big(-\frac{1}{12\varepsilon^4}+\frac{1}{4\varepsilon^3}-\frac{1}{12\varepsilon^2}+\frac{1}{\varepsilon}\Big(-\frac{5}{12}+\frac{1}{2}\zeta(3)\Big)\Big) 
\col\\
I_4&=
\intsixpo{%
\fmf{plain}{v1,v2}
\fmf{plain}{v2,v3}
\fmf{plain}{v3,v4}
\fmf{plain}{vc,v1}
\fmf{plain}{vc,v2}
\fmf{plain}{vc,v3}
\fmf{plain,left=0.25}{vc,v4}
\fmf{plain,right=0.25}{vc,v4}}
\col\qquad
\mathcal{I}_4
=\frac{1}{(4\pi)^8}\Big(-\frac{1}{24\varepsilon^4}+\frac{1}{4\varepsilon^3}-\frac{19}{24\varepsilon^2}+\frac{5}{4\varepsilon}\Big) 
\col\\
I_{4\text{c}}&=
\intsixpo{%
\fmf{plain}{v1,v2}
\fmf{plain}{v3,v4}
\fmf{plain}{vc,v1}
\fmf{plain}{vc,v2}
\fmf{plain}{vc,v3}
\fmf{plain,left=0.25}{v2,v3}
\fmf{plain,right=0.25}{v2,v3}
\fmf{plain}{vc,v4}}
\col\qquad
\mathcal{I}_{4\text{c}}
=\frac{1}{(4\pi)^8}\Big(-\frac{1}{12\varepsilon^4}+\frac{5}{12\varepsilon^3}-\frac{13}{12\varepsilon^2}+\frac{1}{\varepsilon}\Big(\frac{11}{12}-\frac{1}{2}\zeta(3)\Big)\Big)
\col\\
I_{4\text{w}}&=
\intsixp{%
\fmf{plain}{v1,v3}
\fmf{plain}{v3,v4}
\fmf{plain}{v4,v5}
\fmf{plain}{v5,v1}
\fmf{plain}{vc,v3}
\fmf{plain}{vc,v4}
\fmf{plain,left=0.25}{vc,v5}
\fmf{plain,right=0.25}{vc,v5}}
%\fourlint[\fmf{plain,right=0.25}{vc,v4}]{plain}{plain}{plain}{plain}{phantom}{plain}{plain}{plain,left=0.25}
\col\qquad
\mathcal{I}_{4\text{w}}
=\frac{1}{(4\pi)^8}\Big(-\frac{1}{24\varepsilon^4}+\frac{1}{4\varepsilon^3}-\frac{19}{24\varepsilon^2}+\frac{1}{\varepsilon}\Big(\frac{5}{4}-\zeta(3)\Big)\Big) 
\col\\
I_{4\beta}&=
\intsixp{%
\fmf{plain}{v1,v3}
\fmf{plain}{v3,v4}
\fmf{plain}{v4,v5}
\fmf{plain}{v5,v1}
\fmf{plain}{vc,v3}
\fmf{plain}{vc,v5}
\fmf{plain,left=0.25}{vc,v4}
\fmf{plain,right=0.25}{vc,v4}}
%\fourlint[\fmf{plain,right=0.25}{vc,v3}]{plain}{plain}{plain}{plain}{phantom}{plain}{plain,left=0.25}{plain}
\col\qquad
\mathcal{I}_{4\beta}
=\frac{1}{(4\pi)^8}\Big(
-\frac{1}{12\varepsilon^4}+\frac{1}{3\varepsilon^3}
-\frac{5}{12\varepsilon^2}
-\frac{1}{\varepsilon}\Big(\frac{1}{2}-\zeta(3)\Big)\Big)
\col\\
I_{4\text{bt}}&=
\intsixp{%
\fmf{plain}{v1,v3}
\fmf{plain}{v3,v4}
\fmf{plain}{v4,v5}
\fmf{plain,left=0.25}{v5,v1}
\fmf{plain,right=0.25}{v5,v1}
\fmf{plain}{vc,v3}
\fmf{plain}{vc,v4}
\fmf{plain}{vc,v5}}
%\fourlint[\fmf{plain,right=0.25}{v4,v1}]{plain}{plain}{plain}{plain,left=0.25}{phantom}{plain}{plain}{plain}
\col\qquad
\mathcal{I}_{4\text{bt}}
=\frac{1}{(4\pi)^8}\Big(-\frac{1}{2\varepsilon^2}\zeta(3)+\frac{1}{\varepsilon}\Big(\frac{3}{2}\zeta(3)+\frac{\pi^4}{120}\Big)\Big)
\col\\
I_{43\text{t}}&=
\intsixp[29]{%
\fmf{plain}{v1,v3}
\fmf{plain}{v3,v4}
\fmf{plain}{v4,v5}
\fmf{plain}{v5,v1}
\fmf{plain}{v3,v5}
\fmfpoly{phantom}{v3,v5,v7}
\fmf{plain}{v3,v7}
\fmf{plain}{v4,v7}
\fmf{plain}{v5,v7}
%\fmf{plain}{vc,v4}
%\fmf{plain}{vc,v5}
}
%\fourlint[\fmf{plain}{vb,v2}\fmf{plain}{vb,v3}\fmf{plain}{vb,v4}]{plain}{plain}{plain}{plain}{phantom}{plain}{phantom}{plain}
\col\qquad
\mathcal{I}_{43\text{t}}
=\frac{1}{(4\pi)^8}\Big(-\frac{3}{2\varepsilon^2}\zeta(3)+\frac{1}{\varepsilon}\Big(\frac{1}{2}\zeta(3)-\frac{\pi^4}{120}\Big)\Big)
\col\\
I_{4\text{t}}&=
\fourlint{plain}{plain}{plain}{plain}{plain}{plain}{plain}{plain}
\col\qquad
\mathcal{I}_{4\text{t}}
=\frac{1}{(4\pi)^8}\frac{1}{\varepsilon}5\zeta(5)
\col\\
I''_{4\text{t}1}&=
\intsixp[18]{%
\fmf{plain}{v1,v2}
\fmf{plain}{v2,v3}
\fmf{phantom}{v5,v7}
\fmf{phantom}{v7,v6}
\fmffreeze
\fmf{derplain,right=0.5}{v7,v3}
\fmf{derplain}{v7,v1}
\fmf{plain}{v2,vi1}
\fmf{plain}{v7,vi1}
\fmf{plain}{v3,vi2}
\fmf{plain}{v7,vi2}
\fmffreeze
\fmf{plain}{vi1,vi2}}
%\fivelint[\fmf{derplain,right=0.5}{v5,v3}\fmf{plain}{vb1,v2}\fmf{plain}{vb1,v5}\fmf{plain}{vb2,v3}\fmf{plain}{vb2,v5}\fmffreeze\fmf{plain}{vb1,vb2}]{plain}{plain}{phantom}{phantom}{derplain}
\col\qquad
\mathcal{I}''_{4\text{t}1}
=\frac{1}{(4\pi)^8}\frac{1}{2\varepsilon}(\zeta(3)-5\zeta(5))
\col\\
I''_{4\text{t}2}&=
\intsixp[18]{%
\fmf{plain}{v1,v2}
\fmf{plain}{v2,v3}
\fmf{phantom}{v5,v7}
\fmf{phantom}{v7,v6}
\fmffreeze
\fmf{plain,right=0.5}{v7,v3}
\fmf{plain}{v7,v1}
\fmf{derplain}{v2,vi1}
\fmf{plain}{v7,vi1}
\fmf{derplain}{v3,vi2}
\fmf{plain}{v7,vi2}
\fmffreeze
\fmf{plain}{vi1,vi2}}
%\fivelint[\fmf{derplain,right=0.5}{v5,v3}\fmf{plain}{vb1,v2}\fmf{plain}{vb1,v5}\fmf{plain}{vb2,v3}\fmf{plain}{vb2,v5}\fmffreeze\fmf{plain}{vb1,vb2}]{plain}{plain}{phantom}{phantom}{derplain}
=
I''_{4\text{t}1}+\frac{1}{2}(I_4-I_{4\text{w}}-I_{43\text{t}}+I_{4\text{t}})
\col\\
I_{4\text{tr}1}&=
\intsixp{%
\fmf{plain}{v1,v2}
\fmf{plain,left=0.5}{v2,v4}
\fmf{derplain,label=$\scriptscriptstyle \delta$,l.side=left,l.dist=2}{v4,v5}
\fmf{plain}{v5,v6}
\fmf{plain}{v6,v1}
\fmf{derplain,label=$\scriptscriptstyle \alpha$,l.side=left,l.dist=2}{v6,vi1}
\fmf{plain}{vi1,v2}
\fmf{plain}{v5,vi2}
\fmf{phantom}{vi2,v3}
%\fmffreeze
\fmf{derplain,label=$\scriptscriptstyle \beta$,l.side=left,l.dist=2}{vi1,vi2}
\fmffreeze
\fmf{derplain,label=$\scriptscriptstyle \gamma$,l.side=left,l.dist=2}{vi2,v4}
}\tr(\gamma^\alpha\gamma^\beta\gamma^\gamma\gamma^\delta)
\col\\
I_{4\text{tr}2}&=
\intsixp{%
\fmf{plain}{v1,v2}
\fmf{plain,left=0.5}{v2,v4}
\fmf{derplain,label=$\scriptscriptstyle \gamma$,l.side=left,l.dist=2}{v4,v5}
\fmf{derplain,label=$\scriptscriptstyle \beta$,l.side=left,l.dist=2}{v5,v6}
\fmf{plain}{v6,v1}
\fmf{derplain,label=$\scriptscriptstyle \alpha$,l.side=left,l.dist=2}{v6,vi1}
\fmf{plain}{vi1,v2}
\fmf{plain}{v5,vi2}
\fmf{phantom}{vi2,v3}
%\fmffreeze
\fmf{plain}{vi1,vi2}
\fmffreeze
\fmf{derplain,label=$\scriptscriptstyle \delta$,l.side=left,l.dist=2}{vi2,v4}
}\tr(\gamma^\alpha\gamma^\beta\gamma^\gamma\gamma^\delta)
\pnt
\end{aligned}
\end{equation}
We only need the sum of the last two integrals. This is much easier to work out than
the individual integrals due to the symmetrization in pairs of indices that occurs 
after decomposing the numerator momentum with Lorentz index $\beta$ in terms of 
the other two momenta at a vertex. One obtains the relation 
\begin{equation}
\begin{aligned}
I_{4\text{tr}1}+I_{4\text{tr}1}
&=2I''_{4\text{t}2}
-I_{4\text{bt}}
+I_4
+I_{4\text{w}}
+I_{4\beta}
% =2I''_{4\text{t}1}
% -I_{4\text{bt}}
% +2I_4
% +I_{4\beta}
% -I_{43\text{t}}
%+I_{4\text{t}}
\pnt
\end{aligned}
\end{equation}
Note also that $\mathcal{I}_{4\text{w}}$ differs from $\mathcal{I}_4$ 
only by an additional $-\zeta(3)$ in the simple $\frac{1}{\varepsilon}$ pole.
Using then
\begin{equation}\label{zeta3simplepole}
2\mathcal{I}''_{4\text{t}1}
+\mathcal{I}_{4\text{t}}=\frac{1}{(4\pi)^8}\frac{1}{\varepsilon}\zeta(3)
\col
\end{equation}
we can express $\mathcal{I}_{4\text{w}}$ in terms of $\mathcal{I}_4$ as
\begin{equation}\label{I4I4wrel}
\mathcal{I}_{4\text{w}}=\mathcal{I}_4-2\mathcal{I}''_{4\text{t}1}
-\mathcal{I}_{4\text{t}}
\pnt
\end{equation}




\end{fmffile}

%\clearpage

\footnotesize
\bibliographystyle{JHEP}
\bibliography{references}

\end{document}







input3.tex
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% \fmfi{photon}{pi2}
% \fmfi{photon}{pi3}
% }
\newcommand{\wigglywrap}[4]{%
\fmfipath{pi[]}
\fmfiset{pi1}{#1 ..controls (xpart(#1)-0.25w,ypart(#1)) and (xpart(vloc(__#2))-0.25w,-0.15w) .. (xpart(vloc(__#2)),-0.15w)}
\fmfiset{pi2}{(xpart(vloc(__#2)),-0.15w) ..(xpart(vloc(__#3)),-0.15w)}
\fmfiset{pi3}{(xpart(vloc(__#3)),-0.15w) ..controls (xpart(vloc(__#3))+0.25w,-0.15w) and (xpart(#4)+0.25w,ypart(#4)) .. #4}
\fmfi{photon}{pi1}
\fmfi{photon}{pi2}
\fmfi{photon}{pi3}
}

\newcommand{\dasheswrap}[4]{%
\fmfipath{pi[]}
\fmfiset{pi1}{vloc(__#1) ..controls (-0.175w,ypart(vloc(__#1))) and (-0.175w,-0.15w) .. (xpart(vloc(__#2)),-0.15w)}
\fmfiset{pi2}{(xpart(vloc(__#2)),-0.15w) ..(xpart(vloc(__#3)),-0.15w)}
\fmfiset{pi3}{(xpart(vloc(__#3)),-0.15w) ..controls (1.175w,-0.15w) and (1.175w,ypart(vloc(__#4))) .. vloc(__#4)}
\fmfiset{pi4}{pi1 ..pi2 ..pi3}
\fmfi{dashes}{pi1}
\fmfi{dashes}{pi2}
\fmfi{dashes}{pi3}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  chiral parts of one-loop diagrams
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\idonei}[5]{%
\fmftop{v1}
\fmfbottom{v3}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v3}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v3,v4}
%
\fmf{phantom}{v1,v3}
\fmf{#4}{v1,vc1}
\fmf{#1}{v3,vc1}
\fmf{phantom}{v2,v4}
\fmf{#5}{v2,vc2}
\fmf{#2}{v4,vc2}
\fmf{plain,tension=0.5,right=0,width=1mm}{v3,v4}
\fmffreeze
\fmfposition
\fmf{#3}{vc1,vc2}
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__v3)}
\fmfiset{p2}{vpath(__v2,__v4)}
}

\newcommand{\idonegi}[3]{%
\fmftop{v1}
\fmfbottom{v3}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v3}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v3,v4}
%
\fmf{phantom}{v1,v3}
\fmf{phantom}{v2,v4}
\fmf{phantom}{v3,vg}
\fmf{phantom}{vg,v4}
\fmf{plain,tension=0.5,right=0,width=1mm}{v3,v4}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfipair{vc[]}
\fmfiset{p1}{vpath(__v1,__v3)}
\fmfiset{p2}{vpath(__v2,__v4)}
\svertex{vc[-1]}{p1}
\svertex{vc[+1]}{p2}
\fmfi{#2}{p1}
\fmfi{#3}{p2}
\fmfi{photon}{vc[#1]{dir (-90-(#1*45))}..{dir -90}vloc(__vg)}
}

\newcommand{\chionei}[6][plain]{%
\fmftop{v1}
\fmfbottom{v3}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v3}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v3,v4}
%
\fmf{#5,tension=0.5,right=0.25}{v1,vc1}
\fmf{#6,tension=0.5,left=0.25}{v2,vc1}
\fmf{#4,tension=1.25}{vc1,vc2}
\fmf{#2,tension=0.5,left=0.25}{v3,vc2}
\fmf{#3,tension=0.5,right=0.25}{v4,vc2}
\fmf{#1,tension=0.5,right=0,width=1mm}{v3,v4}
%\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfipair{vd[],vm[],vu[]}
\fmfiset{p1}{vpath(__v1,__vc1)}
\fmfiset{p2}{vpath(__v2,__vc1)}
\fmfiset{p3}{vpath(__vc1,__vc2)}
\fmfiset{p4}{vpath(__v3,__vc2)}
\fmfiset{p5}{vpath(__v4,__vc2)}
\svertex{vm1}{p1}
\dvertex{vu1}{vd1}{p1}
\svertex{vm2}{p2}
\dvertex{vu2}{vd2}{p2}
\svertex{vm3}{p3}
\dvertex{vu3}{vd3}{p3}
\svertex{vm4}{p4}
\dvertex{vd4}{vu4}{p4}
\svertex{vm5}{p5}
\dvertex{vd5}{vu5}{p5}
}

\newcommand{\chionevgi}[6]{
\chionei{#2}{#3}{#4}{#5}{#6}
\fmfipair{vg,v[]}
\fmfiset{vg}{vm3}
\fmfiset{v[-1]}{vm4}
\fmfiset{v[+1]}{vm5}
%\fmfi{wiggly}{v[-1]{dir 135}..{dir 0}vg}
%\fmfi{wiggly}{v[+1]{dir 45}..{dir 180}vg}
\fmfi{wiggly}{v[#1]{dir (90-(#1*45))}..{dir (90+(#1*90))}vg}
}

\newcommand{\chionegi}[6]{%
\fmftop{v1}
\fmfbottom{v4}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v4}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v4,v5}
\fmffixed{(0.25w,0)}{v5,v6}
%
\fmf{#4,tension=0.5,right=0.25}{v1,vc1}
\fmf{#5,tension=0.5,left=0.25}{v2,vc1}
  \fmf{#3,tension=1.25}{vc1,vc2}
%\fmf{plain,tension=0.5,left=0.125}{vc3,vc2}
\fmf{#1,tension=0.5,left=0.25}{v4,vc2}
\fmf{#2,tension=0.5,right=0.25}{v5,vc2}
\fmf{#6}{v3,v6}
\fmf{plain,tension=0.5,right=0,width=1mm}{v4,v6}
%\fmffreeze
\fmfposition
\fmfipath{p[],pg}
\fmfipair{vd[],vm[],vu[],vgd[],vgm[],vgu[],vg[]}
\fmfiset{p1}{vpath(__v1,__vc1)}
\fmfiset{p2}{vpath(__v2,__vc1)}
\fmfiset{p3}{vpath(__vc1,__vc2)}
\fmfiset{p5}{vpath(__v5,__vc2)}
\fmfiset{p4}{vpath(__v4,__vc2)}
\fmfiset{pg}{vpath(__v3,__v6)}
\svertex{vm1}{p1}
\dvertex{vu1}{vd1}{p1}
\svertex{vm2}{p2}
\dvertex{vu2}{vd2}{p2}
\svertex{vm3}{p3}
\dvertex{vu3}{vd3}{p3}
\svertex{vm4}{p4}
\dvertex{vd4}{vu4}{p4}
\svertex{vm5}{p5}
\dvertex{vd5}{vu5}{p5}
%
\vvertex{vgu2}{vu2}{pg}
\vvertex{vgm2}{vm2}{pg}
\vvertex{vgd2}{vd2}{pg}
\vvertex{vgu3}{vu3}{pg}
\vvertex{vgm3}{vm3}{pg}
\vvertex{vgd3}{vd3}{pg}
\vvertex{vgu5}{vu5}{pg}
\vvertex{vgm5}{vm5}{pg}
\vvertex{vgd5}{vd5}{pg}
\vvertex{vg1}{vloc(__vc1)}{pg}
\vvertex{vg2}{vloc(__vc2)}{pg}
}

\newcommand{\gchionei}[6]{%
\fmftop{v1}
\fmfbottom{v4}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v4}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v4,v5}
\fmffixed{(0.25w,0)}{v5,v6}
%
\fmf{#5,tension=0.5,right=0.25}{v2,vc1}
\fmf{#6,tension=0.5,left=0.25}{v3,vc1}
  \fmf{#4,tension=1.25}{vc1,vc2}
%\fmf{plain,tension=0.5,left=0.125}{vc3,vc2}
\fmf{#2,tension=0.5,left=0.25}{v5,vc2}
\fmf{#3,tension=0.5,right=0.25}{v6,vc2}
\fmf{#1}{v1,v4}
\fmf{plain,tension=0.5,right=0,width=1mm}{v4,v6}
%\fmffreeze
\fmfposition
\fmfipath{p[],pg}
\fmfipair{vd[],vm[],vu[],vgd[],vgm[],vgu[],vg[]}
\fmfiset{p1}{vpath(__v2,__vc1)}
\fmfiset{p2}{vpath(__v3,__vc1)}
\fmfiset{p3}{vpath(__vc1,__vc2)}
\fmfiset{p5}{vpath(__v6,__vc2)}
\fmfiset{p4}{vpath(__v5,__vc2)}
\fmfiset{pg}{vpath(__v1,__v4)}
\svertex{vm1}{p1}
\dvertex{vu1}{vd1}{p1}
\svertex{vm2}{p2}
\dvertex{vu2}{vd2}{p2}
\svertex{vm3}{p3}
\dvertex{vu3}{vd3}{p3}
\svertex{vm4}{p4}
\dvertex{vd4}{vu4}{p4}
\svertex{vm5}{p5}
\dvertex{vd5}{vu5}{p5}
%
\vvertex{vgu2}{vu2}{pg}
\vvertex{vgm2}{vm2}{pg}
\vvertex{vgd2}{vd2}{pg}
\vvertex{vgu3}{vu3}{pg}
\vvertex{vgm3}{vm3}{pg}
\vvertex{vgd3}{vd3}{pg}
\vvertex{vgu5}{vu5}{pg}
\vvertex{vgm5}{vm5}{pg}
\vvertex{vgd5}{vd5}{pg}
\vvertex{vg1}{vloc(__vc1)}{pg}
\vvertex{vg2}{vloc(__vc2)}{pg}
}




\newcommand{\chionetwoi}[9]{%
\fmftop{v1}
\fmfbottom{v4}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v4}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v4,v5}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0,whatever)}{vc1,vc3}
\fmffixed{(0,whatever)}{vc2,vc4}
\fmf{#7,tension=0.5,right=0.25}{v1,vc1}
\fmf{#8,tension=0.5,left=0.25}{v2,vc1}
\fmf{phantom,tension=0.5,right=0.25}{v2,vc2}
\fmf{#9,tension=0.5,left=0.25}{v3,vc2}
%
\fmf{#1,tension=0.5,left=0.25}{v4,vc3}
\fmf{phantom,tension=0.5,right=0.25}{v5,vc3}
\fmf{#2,tension=0.5,left=0.25}{v5,vc4}
\fmf{#3,tension=0.5,right=0.25}{v6,vc4}
%
\fmf{#4,tension=1.25,left=0}{vc1,vc3}
\fmf{#6,tension=1.25,left=0}{vc2,vc4}
\fmffreeze
\fmf{#5,tension=1,left=0}{vc2,vc3}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v4,v6}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfipair{vd[],vm[],vu[]}
\fmfiset{p1}{vpath(__v1,__vc1)}
\fmfiset{p2}{vpath(__v2,__vc1)}
\fmfiset{p6}{vpath(__v3,__vc2)}
\fmfiset{p4}{vpath(__v4,__vc3)}
\fmfiset{p8}{vpath(__v5,__vc4)}
\fmfiset{p9}{vpath(__v6,__vc4)}
\fmfiset{p3}{vpath(__vc1,__vc3)}
\fmfiset{p7}{vpath(__vc2,__vc4)}
\fmfiset{p5}{vpath(__vc2,__vc3)}
\svertex{vm1}{p1}
\svertex{vm2}{p2}
\svertex{vm3}{p3}
\svertex{vm4}{p4}
\svertex{vm5}{p5}
\svertex{vm6}{p6}
\svertex{vm7}{p7}
\svertex{vm8}{p8}
\svertex{vm9}{p9}
%
}

\newcommand{\chitwoonei}[9]{%
\fmftop{v1}
\fmfbottom{v4}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v4}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v4,v5}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0,whatever)}{vc1,vc3}
\fmffixed{(0,whatever)}{vc2,vc4}
\fmf{#9,tension=0.5,left=0.25}{v3,vc1}
\fmf{#8,tension=0.5,right=0.25}{v2,vc1}
\fmf{phantom,tension=0.5,left=0.25}{v2,vc2}
\fmf{#7,tension=0.5,right=0.25}{v1,vc2}
%
\fmf{#3,tension=0.5,right=0.25}{v6,vc3}
\fmf{phantom,tension=0.5,left=0.25}{v5,vc3}
\fmf{#2,tension=0.5,right=0.25}{v5,vc4}
\fmf{#1,tension=0.5,left=0.25}{v4,vc4}
%
\fmf{#6,tension=1.25,left=0}{vc1,vc3}
\fmf{#4,tension=1.25,left=0}{vc2,vc4}
\fmffreeze
\fmf{#5,tension=1,left=0}{vc2,vc3}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v4,v6}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfipair{vd[],vm[],vu[]}
\fmfiset{p1}{vpath(__v3,__vc1)}
\fmfiset{p2}{vpath(__v2,__vc1)}
\fmfiset{p6}{vpath(__v1,__vc2)}
\fmfiset{p4}{vpath(__v6,__vc3)}
\fmfiset{p8}{vpath(__v5,__vc4)}
\fmfiset{p9}{vpath(__v4,__vc4)}
\fmfiset{p3}{vpath(__vc1,__vc3)}
\fmfiset{p7}{vpath(__vc2,__vc4)}
\fmfiset{p5}{vpath(__vc2,__vc3)}
\svertex{vm1}{p1}
\svertex{vm2}{p2}
\svertex{vm3}{p3}
\svertex{vm4}{p4}
\svertex{vm5}{p5}
\svertex{vm6}{p6}
\svertex{vm7}{p7}
\svertex{vm8}{p8}
\svertex{vm9}{p9}
%
}

\newcommand{\chionetwothreei}[9][plain]{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
\fmffixed{(0,whatever)}{vc1,vc4}
\fmffixed{(0,whatever)}{vc2,vc5}
\fmffixed{(0,whatever)}{vc3,vc6}

\fmf{#6,tension=0.5,right=0.25}{v1,vc1}
\fmf{#7,tension=0.5,left=0.25}{v2,vc1}
\fmf{phantom,tension=0.5,right=0.25}{v2,vc2}
\fmf{#8,tension=0.5,left=0.25}{v3,vc2}
\fmf{phantom,tension=0.5,right=0.25}{v3,vc3}
\fmf{#9,tension=0.5,left=0.25}{v4,vc3}
%
\fmf{#2,tension=0.5,left=0.25}{v5,vc4}
\fmf{phantom,tension=0.5,right=0.25}{v6,vc4}
\fmf{#3,tension=0.5,left=0.25}{v6,vc5}
\fmf{phantom,tension=0.5,right=0.25}{v7,vc5}
\fmf{#4,tension=0.5,left=0.25}{v7,vc6}
\fmf{#5,tension=0.5,right=0.25}{v8,vc6}
%
\fmf{dots,tension=1.25,left=0}{vc1,vc4}
\fmf{dots,tension=1.25,left=0}{vc2,vc5}
\fmf{dots,tension=1.25,left=0}{vc3,vc6}
\fmffreeze
\fmf{dots,tension=1,left=0}{vc4,vc2}
\fmf{dots,tension=1,left=0}{vc5,vc3}
%
\fmf{#1,tension=0.5,right=0,width=1mm}{v5,v8}
\fmffreeze
\fmfposition
%
}

\newcommand{\chithreetwoonei}[9][plain]{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
\fmffixed{(0,whatever)}{vc1,vc4}
\fmffixed{(0,whatever)}{vc2,vc5}
\fmffixed{(0,whatever)}{vc3,vc6}

\fmf{#9,tension=0.5,left=0.25}{v4,vc3}
\fmf{#8,tension=0.5,right=0.25}{v3,vc3}
\fmf{phantom,tension=0.5,left=0.25}{v3,vc2}
\fmf{#7,tension=0.5,right=0.25}{v2,vc2}
\fmf{phantom,tension=0.5,left=0.25}{v2,vc1}
\fmf{#6,tension=0.5,right=0.25}{v1,vc1}
%
\fmf{#5,tension=0.5,right=0.25}{v8,vc6}
\fmf{phantom,tension=0.5,left=0.25}{v7,vc6}
\fmf{#4,tension=0.5,right=0.25}{v7,vc5}
\fmf{phantom,tension=0.5,left=0.25}{v6,vc5}
\fmf{#3,tension=0.5,right=0.25}{v6,vc4}
\fmf{#2,tension=0.5,left=0.25}{v5,vc4}
%
\fmf{dots,tension=1.25,left=0}{vc1,vc4}
\fmf{dots,tension=1.25,left=0}{vc2,vc5}
\fmf{dots,tension=1.25,left=0}{vc3,vc6}
\fmffreeze
\fmf{dots,tension=1,left=0}{vc1,vc5}
\fmf{dots,tension=1,left=0}{vc2,vc6}
%
\fmf{#1,tension=0.5,right=0,width=1mm}{v5,v8}
\fmffreeze
\fmfposition
%
}


\newcommand{\chionetwoonei}{%
\fmftop{v1}
\fmfbottom{v4}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v4}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v4,v5}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0,whatever)}{vc1,vc3}
\fmffixed{(0,whatever)}{vb2,vb4}
\fmffixed{(0,whatever)}{vc1,vb1}
\fmffixed{(0,whatever)}{vc1,vb3}
\fmffixed{(whatever,0)}{vb1,vb2}
\fmffixed{(whatever,0)}{vb3,vb4}
%
\fmf{dots,tension=0.5,right=0.25}{v1,vc1}
\fmf{dots,tension=0.5,left=0.25}{v2,vc1}
\fmf{phantom,tension=0.5,right=0.25}{v2,vb2}
\fmf{dots,tension=0.5,left=0.25}{v3,vb2}
%
\fmf{dots,tension=0.5,left=0.25}{v4,vc3}
\fmf{dots,tension=0.5,right=0.25}{v5,vc3}
\fmf{phantom,tension=0.5,left=0.25}{v5,vb4}
\fmf{dots,tension=0.5,right=0.25}{v6,vb4}
%
\fmf{dots,tension=1.25,left=0}{vc1,vb1}
\fmf{dots,tension=1.25,left=0}{vb1,vb3}
\fmf{dots,tension=1.25,left=0}{vb3,vc3}
%\fmf{plain,tension=1.25,left=0}{vc2,vb2}
\fmf{dots,tension=1.25,left=0}{vb2,vb4}
%\fmf{plain,tension=1.25,left=0}{vb4,vc4}
\fmffreeze
\fmf{dots,tension=1,left=0}{vb1,vb2}
\fmf{dots,tension=1,left=0}{vb3,vb4}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v4,v6}
\fmffreeze
\fmfposition
}

\newcommand{\chitwoonetwoi}{%
\fmftop{v1}
\fmfbottom{v4}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v4}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v4,v5}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0,whatever)}{vc1,vc3}
\fmffixed{(0,whatever)}{vb2,vb4}
\fmffixed{(0,whatever)}{vc1,vb1}
\fmffixed{(0,whatever)}{vc1,vb3}
\fmffixed{(whatever,0)}{vb1,vb2}
\fmffixed{(whatever,0)}{vb3,vb4}
%
\fmf{dots,tension=0.5,left=0.25}{v3,vc1}
\fmf{dots,tension=0.5,right=0.25}{v2,vc1}
\fmf{phantom,tension=0.5,left=0.25}{v2,vb2}
\fmf{dots,tension=0.5,right=0.25}{v1,vb2}
%
\fmf{dots,tension=0.5,right=0.25}{v6,vc3}
\fmf{dots,tension=0.5,left=0.25}{v5,vc3}
\fmf{phantom,tension=0.5,right=0.25}{v5,vb4}
\fmf{dots,tension=0.5,left=0.25}{v4,vb4}
%
\fmf{dots,tension=1.25,left=0}{vc1,vb1}
\fmf{dots,tension=1.25,left=0}{vb1,vb3}
\fmf{dots,tension=1.25,left=0}{vb3,vc3}
%\fmf{plain,tension=1.25,left=0}{vc2,vb2}
\fmf{dots,tension=1.25,left=0}{vb2,vb4}
%\fmf{plain,tension=1.25,left=0}{vb4,vc4}
\fmffreeze
\fmf{dots,tension=1,left=0}{vb1,vb2}
\fmf{dots,tension=1,left=0}{vb3,vb4}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v4,v6}
\fmffreeze
\fmfposition
}


\newcommand{\chionetwoonesymi}[9]{%
\fmftop{v1}
\fmfbottom{v4}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v4}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v4,v5}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0,whatever)}{vc1,vc3}
\fmffixed{(0,whatever)}{vb2,vb4}
\fmffixed{(0,whatever)}{vc1,vb1}
\fmffixed{(0,whatever)}{vc1,vb3}
\fmffixed{(whatever,0)}{vb1,vb2}
\fmffixed{(whatever,0)}{vb3,vb4}
%
\fmf{#7,tension=0.5,right=0.25}{v1,vc1}
\fmf{#8,tension=0.5,left=0.25}{v2,vc1}
\fmf{phantom,tension=0.5,right=0.25}{v2,vb2}
\fmf{#9,tension=0.5,left=0.25}{v3,vb2}
%
\fmf{#1,tension=0.5,left=0.25}{v4,vc3}
\fmf{#2,tension=0.5,right=0.25}{v5,vc3}
\fmf{phantom,tension=0.5,left=0.25}{v5,vb4}
\fmf{#3,tension=0.5,right=0.25}{v6,vb4}
%
\fmf{#4,tension=1.25,left=0}{vc1,vb1}
\fmf{#3,tension=1.25,left=0}{vb1,vb3}
\fmf{#4,tension=1.25,left=0}{vb3,vc3}
%\fmf{plain,tension=1.25,left=0}{vc2,vb2}
\fmf{#6,tension=1.25,left=0}{vb2,vb4}
%\fmf{plain,tension=1.25,left=0}{vb4,vc4}
\fmffreeze
\fmf{#5,tension=1,left=0}{vb1,vb2}
\fmf{#5,tension=1,left=0}{vb3,vb4}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v4,v6}
\fmffreeze
\fmfposition
}

\newcommand{\chionetwoonesymii}[9]{%
\fmftop{v1}
\fmfbottom{v4}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v4}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v4,v5}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0,whatever)}{vc1,vc3}
\fmffixed{(0,whatever)}{vb2,vb4}
\fmffixed{(0,whatever)}{vc1,vb1}
\fmffixed{(0,whatever)}{vc1,vb3}
\fmffixed{(whatever,0)}{vb1,vb2}
\fmffixed{(whatever,0)}{vb3,vb4}
%
\fmf{#7,tension=0.5,right=0.25}{v1,vc1}
\fmf{#8,tension=0.5,left=0.25}{v2,vc1}
\fmf{phantom,tension=0.5,right=0.25}{v2,vb2}
\fmf{#9,tension=0.5,left=0.25}{v3,vb2}
%
\fmf{#1,tension=0.5,left=0.25}{v4,vc3}
\fmf{#2,tension=0.5,right=0.25}{v5,vc3}
\fmf{phantom,tension=0.5,left=0.25}{v5,vb4}
\fmf{#3,tension=0.5,right=0.25}{v6,vb4}
%
\fmf{#4,tension=1.25,left=0}{vc1,vb1}
\fmf{#6,tension=1.25,left=0}{vb1,vb3}
\fmf{#4,tension=1.25,left=0}{vb3,vc3}
%\fmf{plain,tension=1.25,left=0}{vc2,vb2}
\fmf{#6,tension=1.25,left=0}{vb2,vb4}
%\fmf{plain,tension=1.25,left=0}{vb4,vc4}
\fmffreeze
\fmf{#5,tension=1,left=0}{vb1,vb2}
\fmf{#5,tension=1,left=0}{vb3,vb4}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v4,v6}
\fmffreeze
\fmfposition
}

\newcommand{\chionetwoonenonsymi}[9]{%
\fmftop{v1}
\fmfbottom{v4}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v4}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v4,v5}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0,whatever)}{vc1,vc3}
\fmffixed{(0,whatever)}{vb2,vb4}
\fmffixed{(0,whatever)}{vc1,vb1}
\fmffixed{(0,whatever)}{vc1,vb3}
\fmffixed{(whatever,0)}{vb1,vb2}
\fmffixed{(whatever,0)}{vb3,vb4}
%
\fmf{#7,tension=0.5,right=0.25}{v1,vc1}
\fmf{#8,tension=0.5,left=0.25}{v2,vc1}
\fmf{phantom,tension=0.5,right=0.25}{v2,vb2}
\fmf{#9,tension=0.5,left=0.25}{v3,vb2}
%
\fmf{#1,tension=0.5,left=0.25}{v4,vc3}
\fmf{#2,tension=0.5,right=0.25}{v5,vc3}
\fmf{phantom,tension=0.5,left=0.25}{v5,vb4}
\fmf{#3,tension=0.5,right=0.25}{v6,vb4}
%
\fmf{#6,tension=1.25,left=0}{vc1,vb1}
\fmf{#5,tension=1.25,left=0}{vb1,vb3}
\fmf{#4,tension=1.25,left=0}{vb3,vc3}
%\fmf{plain,tension=1.25,left=0}{vc2,vb2}
\fmf{#5,tension=1.25,left=0}{vb2,vb4}
%\fmf{plain,tension=1.25,left=0}{vb4,vc4}
\fmffreeze
\fmf{#3,tension=1,left=0}{vb1,vb2}
\fmf{#9,tension=1,left=0}{vb3,vb4}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v4,v6}
\fmffreeze
\fmfposition
}

\newcommand{\chionethreei}[8]{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
%
\fmf{#5,tension=0.5,right=0.25}{v1,vc1}
\fmf{#6,tension=0.5,left=0.25}{v2,vc1}
  \fmf{dots,tension=1.25}{vc1,vc2}
\fmf{#7,tension=0.5,left=0.25}{v5,vc2}
\fmf{#8,tension=0.5,right=0.25}{v6,vc2}
%
\fmf{#1,tension=0.5,right=0.25}{v3,vc3}
\fmf{#2,tension=0.5,left=0.25}{v4,vc3}
\fmf{dots,tension=1.25}{vc3,vc4}
\fmf{#3,tension=0.5,left=0.25}{v7,vc4}
\fmf{#4,tension=0.5,right=0.25}{v8,vc4}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v8}
%\fmffreeze
\fmfposition
\fmfipath{pl[],pr[]}
\fmfipair{vld[],vlm[],vlu[],vrd[],vrm[],vru[]}
\fmfiset{pl1}{vpath(__v1,__vc1)}
\fmfiset{pl2}{vpath(__v2,__vc1)}
\fmfiset{pl3}{vpath(__vc1,__vc2)}
\fmfiset{pl5}{vpath(__v6,__vc2)}
\fmfiset{pl4}{vpath(__v5,__vc2)}
%
\fmfiset{pr1}{vpath(__v3,__vc3)}
\fmfiset{pr2}{vpath(__v4,__vc3)}
\fmfiset{pr3}{vpath(__vc3,__vc4)}
\fmfiset{pr5}{vpath(__v8,__vc4)}
\fmfiset{pr4}{vpath(__v7,__vc4)}
%
\svertex{vlm1}{pl1}
\dvertex{vlu1}{vld1}{pl1}
\svertex{vlm2}{pl2}
\dvertex{vlu2}{vld2}{pl2}
\svertex{vlm3}{pl3}
\dvertex{vlu3}{vld3}{pl3}
\svertex{vlm4}{pl4}
\dvertex{vld4}{vlu4}{pl4}
\svertex{vlm5}{pl5}
\dvertex{vld5}{vlu5}{pl5}
%
\svertex{vrm1}{pr1}
\dvertex{vru1}{vrd1}{pr1}
\svertex{vrm2}{pr2}
\dvertex{vru2}{vrd2}{pr2}
\svertex{vrm3}{pr3}
\dvertex{vru3}{vrd3}{pr3}
\svertex{vrm4}{pr4}
\dvertex{vrd4}{vru4}{pr4}
\svertex{vrm5}{pr5}
\dvertex{vrd5}{vru5}{pr5}

}


\newcommand{\chitwoonethreei}[8]{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
\fmffixed{(whatever,0.5h)}{v5,vc1}
\fmffixed{(0,whatever)}{vc1,vc4}
\fmffixed{(0,whatever)}{vc2,vc5}
\fmffixed{(0,whatever)}{vc3,vc6}
\fmffixed{(whatever,0)}{vc1,vc3}
\fmffixed{(whatever,0)}{vc3,vc5}

\fmf{#5,tension=0.5,right=0.125}{v1,vc1}
\fmf{phantom,tension=0.5,left=0.25}{v2,vc1}
\fmf{#6,tension=0.5,right=0.25}{v2,vc2}
\fmf{#7,tension=0.5,left=0.25}{v3,vc2}
\fmf{phantom,tension=0.5,right=0.25}{v3,vc3}
\fmf{#8,tension=0.5,left=0.125}{v4,vc3}
%
\fmf{#1,tension=0.5,left=0.25}{v5,vc4}
\fmf{#2,tension=0.5,right=0.25}{v6,vc4}
\fmf{phantom,tension=0.5,left=0.25}{v6,vc5}
\fmf{phantom,tension=0.5,right=0.25}{v7,vc5}
\fmf{#3,tension=0.5,left=0.25}{v7,vc6}
\fmf{#4,tension=0.5,right=0.25}{v8,vc6}
%
\fmf{dots,tension=1.25,left=0}{vc1,vc4}
\fmf{dots,tension=1.25,left=0}{vc2,vc5}
\fmf{dots,tension=1.25,left=0}{vc3,vc6}
\fmffreeze
\fmf{dots,tension=1,left=0}{vc1,vc5}
\fmf{dots,tension=1,left=0}{vc5,vc3}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v8}
\fmffreeze
\fmfposition
%
}

\newcommand{\chionethreetwoi}[8]{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
\fmffixed{(whatever,0.5h)}{v5,vc2}
\fmffixed{(0,whatever)}{vc1,vc4}
\fmffixed{(0,whatever)}{vc2,vc5}
\fmffixed{(0,whatever)}{vc3,vc6}
\fmffixed{(whatever,0)}{vc2,vc4}
\fmffixed{(whatever,0)}{vc4,vc6}
%
\fmf{#5,tension=0.5,right=0.25}{v1,vc1}
\fmf{#6,tension=0.5,left=0.25}{v2,vc1}
\fmf{phantom,tension=0.5,right=0.25}{v2,vc2}
\fmf{phantom,tension=0.5,left=0.25}{v3,vc2}
\fmf{#7,tension=0.5,right=0.25}{v3,vc3}
\fmf{#8,tension=0.5,left=0.25}{v4,vc3}
%
\fmf{#1,tension=0.5,left=0.125}{v5,vc4}
\fmf{phantom,tension=0.5,right=0.25}{v6,vc4}
\fmf{#2,tension=0.5,left=0.25}{v6,vc5}
\fmf{#3,tension=0.5,right=0.25}{v7,vc5}
\fmf{phantom,tension=0.5,left=0.25}{v7,vc6}
\fmf{#4,tension=0.5,right=0.125}{v8,vc6}
%
\fmf{dots,tension=1.25,left=0}{vc1,vc4}
\fmf{dots,tension=1.25,left=0}{vc2,vc5}
\fmf{dots,tension=1.25,left=0}{vc3,vc6}
\fmffreeze
\fmf{dots,tension=1,left=0}{vc2,vc4}
\fmf{dots,tension=1,left=0}{vc2,vc6}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v8}
\fmffreeze
\fmfposition
%
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  chiral parts of the two-loop diagrams
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\chione}[1][black]{%
\fmftop{v1}
\fmfbottom{v3}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v3}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v3,v4}
%
\fmf{plain,tension=0.5,right=0.25,fore=#1}{v1,vc1}
\fmf{plain,tension=0.5,left=0.25,fore=#1}{v2,vc1}
\fmf{plain,tension=1.25,fore=#1}{vc1,vc2}
\fmf{plain,tension=0.5,left=0.125,fore=#1}{vc3,vc2}
\fmf{plain,tension=0.5,left=0.25,fore=#1}{v3,vc2}
\fmf{plain,tension=0.5,right=0.25,fore=#1}{v4,vc2}
\fmf{plain,tension=0.5,right=0,width=1mm,fore=#1}{v3,v4}
%\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfipair{vd[],vm[],vu[]}
\fmfiset{p1}{vpath(__v1,__vc1)}
\fmfiset{p2}{vpath(__v2,__vc1)}
\fmfiset{p3}{vpath(__vc1,__vc2)}
\fmfiset{p4}{vpath(__v3,__vc2)}
\fmfiset{p5}{vpath(__v4,__vc2)}
\svertex{vm1}{p1}
\dvertex{vu1}{vd1}{p1}
\svertex{vm2}{p2}
\dvertex{vu2}{vd2}{p2}
\svertex{vm3}{p3}
\dvertex{vu3}{vd3}{p3}
\svertex{vm4}{p4}
\dvertex{vd4}{vu4}{p4}
\svertex{vm5}{p5}
\dvertex{vd5}{vu5}{p5}
}

\newcommand{\chioneg}[1][black]{%
\fmftop{v1}
\fmfbottom{v4}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v4}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v4,v5}
\fmffixed{(0.25w,0)}{v5,v6}
%
\fmf{plain,tension=0.5,right=0.25,fore=#1}{v1,vc1}
\fmf{plain,tension=0.5,left=0.25,fore=#1}{v2,vc1}
  \fmf{plain,tension=1.25,fore=#1}{vc1,vc2}
%\fmf{plain,tension=0.5,left=0.125,fore=#1}{vc3,vc2}
\fmf{plain,tension=0.5,left=0.25,fore=#1}{v4,vc2}
\fmf{plain,tension=0.5,right=0.25,fore=#1}{v5,vc2}
\fmf{plain,fore=#1}{v3,v6}
\fmf{plain,tension=0.5,right=0,width=1mm,fore=#1}{v4,v6}
%\fmffreeze
\fmfposition
\fmfipath{p[],pg}
\fmfipair{vd[],vm[],vu[],vgd[],vgm[],vgu[],vg[]}
\fmfiset{p1}{vpath(__v1,__vc1)}
\fmfiset{p2}{vpath(__v2,__vc1)}
\fmfiset{p3}{vpath(__vc1,__vc2)}
\fmfiset{p5}{vpath(__v5,__vc2)}
\fmfiset{p4}{vpath(__v4,__vc2)}
\fmfiset{pg}{vpath(__v3,__v6)}
\svertex{vm1}{p1}
\dvertex{vu1}{vd1}{p1}
\svertex{vm2}{p2}
\dvertex{vu2}{vd2}{p2}
\svertex{vm3}{p3}
\dvertex{vu3}{vd3}{p3}
\svertex{vm4}{p4}
\dvertex{vd4}{vu4}{p4}
\svertex{vm5}{p5}
\dvertex{vd5}{vu5}{p5}
%
\vvertex{vgu2}{vu2}{pg}
\vvertex{vgm2}{vm2}{pg}
\vvertex{vgd2}{vd2}{pg}
\vvertex{vgu3}{vu3}{pg}
\vvertex{vgm3}{vm3}{pg}
\vvertex{vgd3}{vd3}{pg}
\vvertex{vgu5}{vu5}{pg}
\vvertex{vgm5}{vm5}{pg}
\vvertex{vgd5}{vd5}{pg}
\vvertex{vg1}{vloc(__vc1)}{pg}
\vvertex{vg2}{vloc(__vc2)}{pg}
}

\newcommand{\gchioneg}[1][black]{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
%
\fmf{plain,fore=#1}{v1,v5}
\fmf{plain,tension=0.5,right=0.25,fore=#1}{v2,vc1}
\fmf{plain,tension=0.5,left=0.25,fore=#1}{v3,vc1}
  \fmf{plain,tension=1.25,fore=#1}{vc1,vc2}
\fmf{plain,tension=0.5,left=0.25,fore=#1}{v6,vc2}
\fmf{plain,tension=0.5,right=0.25,fore=#1}{v7,vc2}
\fmf{plain,fore=#1}{v4,v8}
\fmf{plain,tension=0.5,right=0,width=1mm,fore=#1}{v5,v8}
%\fmffreeze
\fmfposition
\fmfipath{p[],pgl,pgr}
\fmfipair{vd[],vm[],vu[],vgd[],vglm[],vglu[],vgld[],vgl[],vgrm[],vgru[],vgrd[],vgr[]}
\fmfiset{p1}{vpath(__v2,__vc1)}
\fmfiset{p2}{vpath(__v3,__vc1)}
\fmfiset{p3}{vpath(__vc1,__vc2)}
\fmfiset{p5}{vpath(__v7,__vc2)}
\fmfiset{p4}{vpath(__v6,__vc2)}
\fmfiset{pgl}{vpath(__v1,__v5)}
\fmfiset{pgr}{vpath(__v4,__v8)}
\svertex{vm1}{p1}
\dvertex{vu1}{vd1}{p1}
\svertex{vm2}{p2}
\dvertex{vu2}{vd2}{p2}
\svertex{vm3}{p3}
\dvertex{vu3}{vd3}{p3}
\svertex{vm4}{p4}
\dvertex{vd4}{vu4}{p4}
\svertex{vm5}{p5}
\dvertex{vd5}{vu5}{p5}
%
\vvertex{vglu1}{vu1}{pgl}
\vvertex{vglm1}{vm1}{pgl}
\vvertex{vgld1}{vd1}{pgl}
\vvertex{vglu3}{vu3}{pgl}
\vvertex{vglm3}{vm3}{pgl}
\vvertex{vgld3}{vd3}{pgl}
\vvertex{vglu5}{vu5}{pgl}
\vvertex{vglm5}{vm5}{pgl}
\vvertex{vgld5}{vd5}{pgl}
\vvertex{vgl1}{vloc(__vc1)}{pgl}
\vvertex{vgl2}{vloc(__vc2)}{pgl}
%
\vvertex{vgru2}{vu2}{pgr}
\vvertex{vgrm2}{vm2}{pgr}
\vvertex{vgrd2}{vd2}{pgr}
\vvertex{vgru3}{vu3}{pgr}
\vvertex{vgrm3}{vm3}{pgr}
\vvertex{vgrd3}{vd3}{pgr}
\vvertex{vgru5}{vu5}{pgr}
\vvertex{vgrm5}{vm5}{pgr}
\vvertex{vgrd5}{vd5}{pgr}
\vvertex{vgr1}{vloc(__vc1)}{pgr}
\vvertex{vgr2}{vloc(__vc2)}{pgr}
}

\newcommand{\chionetwo}[1][black]{%
\fmftop{v1}
\fmfbottom{v4}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v4}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v4,v5}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0,whatever)}{vc1,vc3}
\fmffixed{(0,whatever)}{vc2,vc4}
\fmf{plain,tension=0.5,right=0.25}{v1,vc1}
\fmf{plain,tension=0.5,left=0.25}{v2,vc1}
\fmf{phantom,tension=0.5,right=0.25}{v2,vc2}
\fmf{plain,tension=0.5,left=0.25}{v3,vc2}
%
\fmf{plain,tension=0.5,left=0.25}{v4,vc3}
\fmf{phantom,tension=0.5,right=0.25}{v5,vc3}
\fmf{plain,tension=0.5,left=0.25}{v5,vc4}
\fmf{plain,tension=0.5,right=0.25}{v6,vc4}
%
\fmf{plain,tension=1.25,left=0}{vc1,vc3}
\fmf{plain,tension=1.25,left=0}{vc2,vc4}
\fmffreeze
\fmf{plain,tension=1,left=0}{vc2,vc3}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v4,v6}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfipair{vd[],vm[],vu[]}
\fmfiset{p1}{vpath(__v1,__vc1)}
\fmfiset{p2}{vpath(__v2,__vc1)}
\fmfiset{p6}{vpath(__v3,__vc2)}
\fmfiset{p4}{vpath(__v4,__vc3)}
\fmfiset{p8}{vpath(__v5,__vc4)}
\fmfiset{p9}{vpath(__v6,__vc4)}
\fmfiset{p3}{vpath(__vc1,__vc3)}
\fmfiset{p7}{vpath(__vc2,__vc4)}
\fmfiset{p5}{vpath(__vc2,__vc3)}
\svertex{vm1}{p1}
\svertex{vm2}{p2}
\svertex{vm3}{p3}
\svertex{vm4}{p4}
\svertex{vm5}{p5}
\svertex{vm6}{p6}
\svertex{vm7}{p7}
\svertex{vm8}{p8}
\svertex{vm9}{p9}
%
}

\newcommand{\chionetwoone}[1][black]{%
\fmftop{v1}
\fmfbottom{v4}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v4}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v4,v5}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0,whatever)}{vc1,vc3}
\fmffixed{(0,whatever)}{vb2,vb4}
\fmffixed{(0,whatever)}{vc1,vb1}
\fmffixed{(0,whatever)}{vc1,vb3}
\fmffixed{(whatever,0)}{vb1,vb2}
\fmffixed{(whatever,0)}{vb3,vb4}
%
\fmf{plain,tension=0.5,right=0.25}{v1,vc1}
\fmf{plain,tension=0.5,left=0.25}{v2,vc1}
\fmf{phantom,tension=0.5,right=0.25}{v2,vb2}
\fmf{plain,tension=0.5,left=0.25}{v3,vb2}
%
\fmf{plain,tension=0.5,left=0.25}{v4,vc3}
\fmf{plain,tension=0.5,right=0.25}{v5,vc3}
\fmf{phantom,tension=0.5,left=0.25}{v5,vb4}
\fmf{plain,tension=0.5,right=0.25}{v6,vb4}
%
\fmf{plain,tension=2,left=0}{vc1,vb1}
\fmf{plain,tension=2,left=0}{vb1,vb3}
\fmf{plain,tension=2,left=0}{vb3,vc3}
%\fmf{plain,tension=2,left=0}{vc2,vb2}
\fmf{plain,tension=2,left=0}{vb2,vb4}
%\fmf{plain,tension=2,left=0}{vb4,vc4}
\fmffreeze
\fmf{plain,tension=2,left=0}{vb1,vb2}
\fmf{plain,tension=2,left=0}{vb3,vb4}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v4,v6}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfipair{vd[],vm[],vu[]}
\fmfiset{p1}{vpath(__v1,__vc1)}
\fmfiset{p2}{vpath(__v2,__vc1)}
\fmfiset{p3}{vpath(__vc1,__vb1)}
\fmfiset{p4}{vpath(__vb1,__vb3)}
\fmfiset{p5}{vpath(__vb1,__vb2)}
\fmfiset{p6}{vpath(__v3,__vb2)}
\fmfiset{p7}{vpath(__vb2,__vb4)}
\fmfiset{p8}{vpath(__vb3,__vb4)}
\fmfiset{p9}{vpath(__v6,__vb4)}
\fmfiset{p10}{vpath(__vb3,__vc3)}
\fmfiset{p11}{vpath(__v4,__vc3)}
\fmfiset{p12}{vpath(__v5,__vc3)}
\svertex{vm1}{p1}
\svertex{vm2}{p2}
\svertex{vm3}{p3}
\svertex{vm4}{p4}
\svertex{vm5}{p5}
\svertex{vm6}{p6}
\svertex{vm7}{p7}
\svertex{vm8}{p8}
\svertex{vm9}{p9}
\svertex{vm10}{p10}
\svertex{vm11}{p11}
\svertex{vm12}{p12}
}

\newcommand{\chionethree}[1][black]{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
%
\fmf{plain,tension=0.5,right=0.25,fore=#1}{v1,vc1}
\fmf{plain,tension=0.5,left=0.25,fore=#1}{v2,vc1}
  \fmf{plain,tension=1.25,fore=#1}{vc1,vc2}
\fmf{plain,tension=0.5,left=0.25,fore=#1}{v5,vc2}
\fmf{plain,tension=0.5,right=0.25,fore=#1}{v6,vc2}
%
\fmf{plain,tension=0.5,right=0.25,fore=#1}{v3,vc3}
\fmf{plain,tension=0.5,left=0.25,fore=#1}{v4,vc3}
  \fmf{plain,tension=1.25,fore=#1}{vc3,vc4}
\fmf{plain,tension=0.5,left=0.25,fore=#1}{v7,vc4}
\fmf{plain,tension=0.5,right=0.25,fore=#1}{v8,vc4}
%
\fmf{plain,tension=0.5,right=0,width=1mm,fore=#1}{v5,v8}
%\fmffreeze
\fmfposition
\fmfipath{pl[],pr[]}
\fmfipair{vld[],vlm[],vlu[],vrd[],vrm[],vru[]}
\fmfiset{pl1}{vpath(__v1,__vc1)}
\fmfiset{pl2}{vpath(__v2,__vc1)}
\fmfiset{pl3}{vpath(__vc1,__vc2)}
\fmfiset{pl5}{vpath(__v6,__vc2)}
\fmfiset{pl4}{vpath(__v5,__vc2)}
%
\fmfiset{pr1}{vpath(__v3,__vc3)}
\fmfiset{pr2}{vpath(__v4,__vc3)}
\fmfiset{pr3}{vpath(__vc3,__vc4)}
\fmfiset{pr5}{vpath(__v8,__vc4)}
\fmfiset{pr4}{vpath(__v7,__vc4)}
%
\svertex{vlm1}{pl1}
\dvertex{vlu1}{vld1}{pl1}
\svertex{vlm2}{pl2}
\dvertex{vlu2}{vld2}{pl2}
\svertex{vlm3}{pl3}
\dvertex{vlu3}{vld3}{pl3}
\svertex{vlm4}{pl4}
\dvertex{vld4}{vlu4}{pl4}
\svertex{vlm5}{pl5}
\dvertex{vld5}{vlu5}{pl5}
%
\svertex{vrm1}{pr1}
\dvertex{vru1}{vrd1}{pr1}
\svertex{vrm2}{pr2}
\dvertex{vru2}{vrd2}{pr2}
\svertex{vrm3}{pr3}
\dvertex{vru3}{vrd3}{pr3}
\svertex{vrm4}{pr4}
\dvertex{vrd4}{vru4}{pr4}
\svertex{vrm5}{pr5}
\dvertex{vrd5}{vru5}{pr5}

}


\newcommand{\chionetwothree}[1][black]{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
\fmffixed{(0,whatever)}{vc1,vc4}
\fmffixed{(0,whatever)}{vc2,vc5}
\fmffixed{(0,whatever)}{vc3,vc6}

\fmf{plain,tension=0.5,right=0.25}{v1,vc1}
\fmf{plain,tension=0.5,left=0.25}{v2,vc1}
\fmf{phantom,tension=0.5,right=0.25}{v2,vc2}
\fmf{plain,tension=0.5,left=0.25}{v3,vc2}
\fmf{phantom,tension=0.5,right=0.25}{v3,vc3}
\fmf{plain,tension=0.5,left=0.25}{v4,vc3}
%
\fmf{plain,tension=0.5,left=0.25}{v5,vc4}
\fmf{phantom,tension=0.5,right=0.25}{v6,vc4}
\fmf{plain,tension=0.5,left=0.25}{v6,vc5}
\fmf{phantom,tension=0.5,right=0.25}{v7,vc5}
\fmf{plain,tension=0.5,left=0.25}{v7,vc6}
\fmf{plain,tension=0.5,right=0.25}{v8,vc6}
%
\fmf{plain,tension=1.25,left=0}{vc1,vc4}
\fmf{plain,tension=1.25,left=0}{vc2,vc5}
\fmf{plain,tension=1.25,left=0}{vc3,vc6}
\fmffreeze
\fmf{plain,tension=1,left=0}{vc4,vc2}
\fmf{plain,tension=1,left=0}{vc5,vc3}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v8}
\fmffreeze
\fmfposition
%
}

\newcommand{\chitwoonethree}[1][black]{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
\fmffixed{(whatever,0.5h)}{v5,vc1}
\fmffixed{(0,whatever)}{vc1,vc4}
\fmffixed{(0,whatever)}{vc2,vc5}
\fmffixed{(0,whatever)}{vc3,vc6}
\fmffixed{(whatever,0)}{vc1,vc3}
\fmffixed{(whatever,0)}{vc3,vc5}

\fmf{plain,tension=0.5,right=0.125}{v1,vc1}
\fmf{phantom,tension=0.5,left=0.25}{v2,vc1}
\fmf{plain,tension=0.5,right=0.25}{v2,vc2}
\fmf{plain,tension=0.5,left=0.25}{v3,vc2}
\fmf{phantom,tension=0.5,right=0.25}{v3,vc3}
\fmf{plain,tension=0.5,left=0.125}{v4,vc3}
%
\fmf{plain,tension=0.5,left=0.25}{v5,vc4}
\fmf{plain,tension=0.5,right=0.25}{v6,vc4}
\fmf{phantom,tension=0.5,left=0.25}{v6,vc5}
\fmf{phantom,tension=0.5,right=0.25}{v7,vc5}
\fmf{plain,tension=0.5,left=0.25}{v7,vc6}
\fmf{plain,tension=0.5,right=0.25}{v8,vc6}
%
\fmf{plain,tension=1.25,left=0}{vc1,vc4}
\fmf{plain,tension=1.25,left=0}{vc2,vc5}
\fmf{plain,tension=1.25,left=0}{vc3,vc6}
\fmffreeze
\fmf{plain,tension=1,left=0}{vc1,vc5}
\fmf{plain,tension=1,left=0}{vc5,vc3}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v8}
\fmffreeze
\fmfposition
%
}

\newcommand{\chionethreetwo}[1][black]{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
\fmffixed{(whatever,0.5h)}{v5,vc2}
\fmffixed{(0,whatever)}{vc1,vc4}
\fmffixed{(0,whatever)}{vc2,vc5}
\fmffixed{(0,whatever)}{vc3,vc6}
\fmffixed{(whatever,0)}{vc2,vc4}
\fmffixed{(whatever,0)}{vc4,vc6}
%
\fmf{plain,tension=0.5,right=0.25}{v1,vc1}
\fmf{plain,tension=0.5,left=0.25}{v2,vc1}
\fmf{phantom,tension=0.5,right=0.25}{v2,vc2}
\fmf{phantom,tension=0.5,left=0.25}{v3,vc2}
\fmf{plain,tension=0.5,right=0.25}{v3,vc3}
\fmf{plain,tension=0.5,left=0.25}{v4,vc3}
%
\fmf{plain,tension=0.5,left=0.125}{v5,vc4}
\fmf{phantom,tension=0.5,right=0.25}{v6,vc4}
\fmf{plain,tension=0.5,left=0.25}{v6,vc5}
\fmf{plain,tension=0.5,right=0.25}{v7,vc5}
\fmf{phantom,tension=0.5,left=0.25}{v7,vc6}
\fmf{plain,tension=0.5,right=0.125}{v8,vc6}
%
\fmf{plain,tension=1.25,left=0}{vc1,vc4}
\fmf{plain,tension=1.25,left=0}{vc2,vc5}
\fmf{plain,tension=1.25,left=0}{vc3,vc6}
\fmffreeze
\fmf{plain,tension=1,left=0}{vc2,vc4}
\fmf{plain,tension=1,left=0}{vc2,vc6}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v8}
\fmffreeze
\fmfposition
%
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% four loop chiral functions that are reduced in the SU(2) subsector 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\chionetwothreetwo}[1][black]{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
\fmffixed{(0,whatever)}{va1,va2}
\fmffixed{(whatever,0)}{va1,vc1}
\fmffixed{(whatever,0)}{va2,vb3}
\fmffixed{(0,whatever)}{vc1,vc3}
\fmffixed{(0,whatever)}{vb2,vb4}
\fmffixed{(0,whatever)}{vc1,vb1}
\fmffixed{(0,whatever)}{vc1,vb3}
\fmffixed{(whatever,0)}{vb1,vb2}
\fmffixed{(whatever,0)}{vb3,vb4}
%
\fmf{phantom,tension=0.5,right=0.25}{v2,vc1}
\fmf{plain,tension=0.5,left=0.25}{v3,vc1}
\fmf{phantom,tension=0.5,right=0.25}{v3,vb2}
\fmf{plain,tension=0.5,left=0.25}{v4,vb2}
%
\fmf{plain,tension=0.5,left=0.25}{v6,vc3}
\fmf{plain,tension=0.5,right=0.25}{v7,vc3}
\fmf{phantom,tension=0.5,left=0.25}{v7,vb4}
\fmf{plain,tension=0.5,right=0.25}{v8,vb4}
%
\fmf{plain,tension=2,left=0}{vc1,vb1}
\fmf{plain,tension=2,left=0}{vb1,vb3}
\fmf{plain,tension=2,left=0}{vb3,vc3}
%\fmf{plain,tension=2,left=0}{vc2,vb2}
\fmf{plain,tension=2,left=0}{vb2,vb4}
%\fmf{plain,tension=2,left=0}{vb4,vc4}
\fmffreeze
%
\fmf{plain,tension=0.5,right=0.25}{v1,va1}
\fmf{plain,tension=0.5,left=0.25}{v2,va1}
\fmf{plain,tension=1}{va1,va2}
\fmf{plain,tension=0}{va2,vc1}
\fmf{plain,tension=0.5,left=0.25}{v5,va2}
\fmf{phantom,tension=0.5,right=0.25}{v6,va2}
%
\fmf{plain,tension=1,left=0}{vb1,vb2}
\fmf{plain,tension=1,left=0}{vb3,vb4}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v8}
}

\newcommand{\chitwothreetwoone}[1][black]{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
\fmffixed{(0,whatever)}{va1,va2}
\fmffixed{(whatever,0)}{va2,vc3}
\fmffixed{(whatever,0)}{va1,vb1}
\fmffixed{(0,whatever)}{vc1,vc3}
\fmffixed{(0,whatever)}{vb2,vb4}
\fmffixed{(0,whatever)}{vc1,vb1}
\fmffixed{(0,whatever)}{vc1,vb3}
\fmffixed{(whatever,0)}{vb1,vb2}
\fmffixed{(whatever,0)}{vb3,vb4}
%
\fmf{plain,tension=0.5,right=0.25}{v2,vc1}
\fmf{plain,tension=0.5,left=0.25}{v3,vc1}
\fmf{phantom,tension=0.5,right=0.25}{v3,vb2}
\fmf{plain,tension=0.5,left=0.25}{v4,vb2}
%
\fmf{phantom,tension=0.5,left=0.25}{v6,vc3}
\fmf{plain,tension=0.5,right=0.25}{v7,vc3}
\fmf{phantom,tension=0.5,left=0.25}{v7,vb4}
\fmf{plain,tension=0.5,right=0.25}{v8,vb4}
%
\fmf{plain,tension=2,left=0}{vc1,vb1}
\fmf{plain,tension=2,left=0}{vb1,vb3}
\fmf{plain,tension=2,left=0}{vb3,vc3}
%\fmf{plain,tension=2,left=0}{vc2,vb2}
\fmf{plain,tension=2,left=0}{vb2,vb4}
%\fmf{plain,tension=2,left=0}{vb4,vc4}
\fmffreeze
%
\fmf{plain,tension=0.5,right=0.25}{v1,va1}
\fmf{phantom,tension=0.5,left=0.25}{v2,va1}
\fmf{plain,tension=1}{va1,va2}
\fmf{plain,tension=0}{va1,vc3}
\fmf{plain,tension=0.5,left=0.25}{v5,va2}
\fmf{plain,tension=0.5,right=0.25}{v6,va2}
%
\fmf{plain,tension=1,left=0}{vb1,vb2}
\fmf{plain,tension=1,left=0}{vb3,vb4}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v8}
}

\newcommand{\chitwoonetwothree}[1][black]{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
\fmffixed{(0,whatever)}{va1,va2}
\fmffixed{(whatever,0)}{va2,vc3}
\fmffixed{(whatever,0)}{va1,vb1}
\fmffixed{(0,whatever)}{vc1,vc3}
\fmffixed{(0,whatever)}{vb2,vb4}
\fmffixed{(0,whatever)}{vc1,vb1}
\fmffixed{(0,whatever)}{vc1,vb3}
\fmffixed{(whatever,0)}{vb1,vb2}
\fmffixed{(whatever,0)}{vb3,vb4}
%
\fmf{plain,tension=0.5,right=0.25}{v2,vc1}
\fmf{plain,tension=0.5,left=0.25}{v3,vc1}
\fmf{plain,tension=0.5,right=0.25}{v1,vb2}
\fmf{phantom,tension=0.5,left=0.25}{v2,vb2}
%
\fmf{plain,tension=0.5,left=0.25}{v6,vc3}
\fmf{phantom,tension=0.5,right=0.25}{v7,vc3}
\fmf{plain,tension=0.5,left=0.25}{v5,vb4}
\fmf{phantom,tension=0.5,right=0.25}{v6,vb4}
%
\fmf{plain,tension=2,left=0}{vc1,vb1}
\fmf{plain,tension=2,left=0}{vb1,vb3}
\fmf{plain,tension=2,left=0}{vb3,vc3}
%\fmf{plain,tension=2,left=0}{vc2,vb2}
\fmf{plain,tension=2,left=0}{vb2,vb4}
%\fmf{plain,tension=2,left=0}{vb4,vc4}
\fmffreeze
%
\fmf{phantom,tension=0.5,right=0.25}{v3,va1}
\fmf{plain,tension=0.5,left=0.25}{v4,va1}
\fmf{plain,tension=1}{va1,va2}
\fmf{plain,tension=0}{va1,vc3}
\fmf{plain,tension=0.5,left=0.25}{v7,va2}
\fmf{plain,tension=0.5,right=0.25}{v8,va2}
%
\fmf{plain,tension=1,left=0}{vb1,vb2}
\fmf{plain,tension=1,left=0}{vb3,vb4}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v8}
}

\newcommand{\chithreeonetwoone}[1][black]{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
\fmffixed{(0,whatever)}{va1,va2}
\fmffixed{(whatever,0)}{va1,vc1}
\fmffixed{(whatever,0)}{va2,vb1}
\fmffixed{(0,whatever)}{vc1,vc3}
\fmffixed{(0,whatever)}{vb2,vb4}
\fmffixed{(0,whatever)}{vc1,vb1}
\fmffixed{(0,whatever)}{vc1,vb3}
\fmffixed{(whatever,0)}{vb1,vb2}
\fmffixed{(whatever,0)}{vb3,vb4}
%
\fmf{plain,tension=0.5,right=0.25}{v1,vc1}
\fmf{plain,tension=0.5,left=0.25}{v2,vc1}
\fmf{phantom,tension=0.5,right=0.25}{v2,vb2}
\fmf{phantom,tension=0.5,left=0.25}{v3,vb2}
%
\fmf{plain,tension=0.5,left=0.25}{v5,vc3}
\fmf{plain,tension=0.5,right=0.25}{v6,vc3}
\fmf{phantom,tension=0.5,left=0.125}{v6,vb4}
\fmf{plain,tension=0.5,right=0.125}{v7,vb4}
%
\fmf{plain,tension=2,left=0}{vc1,vb1}
\fmf{plain,tension=2,left=0}{vb1,vb3}
\fmf{plain,tension=2,left=0}{vb3,vc3}
%\fmf{plain,tension=2,left=0}{vc2,vb2}
\fmf{plain,tension=2,left=0}{vb2,vb4}
%\fmf{plain,tension=2,left=0}{vb4,vc4}
\fmffreeze
%
\fmf{plain,tension=0.5,right=0.25}{v3,va1}
\fmf{plain,tension=0.5,left=0.25}{v4,va1}
\fmf{plain,tension=1}{va1,va2}
\fmf{plain,tension=0}{va2,vb2}
\fmf{phantom,tension=0.5,left=0.125}{v7,va2}
\fmf{plain,tension=0.5,right=0.125}{v8,va2}
%
\fmf{plain,tension=1,left=0}{vb1,vb2}
\fmf{plain,tension=1,left=0}{vb3,vb4}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v8}
}

\newcommand{\chionethreetwothree}[1][black]{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
\fmffixed{(0,whatever)}{va1,va2}
\fmffixed{(whatever,0)}{va1,vc1}
\fmffixed{(whatever,0)}{va2,vb1}
\fmffixed{(0,whatever)}{vc1,vc3}
\fmffixed{(0,whatever)}{vb2,vb4}
\fmffixed{(0,whatever)}{vc1,vb1}
\fmffixed{(0,whatever)}{vc1,vb3}
\fmffixed{(whatever,0)}{vb1,vb2}
\fmffixed{(whatever,0)}{vb3,vb4}
%
\fmf{plain,tension=0.5,right=0.25}{v3,vc1}
\fmf{plain,tension=0.5,left=0.25}{v4,vc1}
\fmf{phantom,tension=0.5,right=0.25}{v2,vb2}
\fmf{phantom,tension=0.5,left=0.25}{v3,vb2}
%
\fmf{plain,tension=0.5,left=0.25}{v7,vc3}
\fmf{plain,tension=0.5,right=0.25}{v8,vc3}
\fmf{plain,tension=0.5,left=0.125}{v6,vb4}
\fmf{phantom,tension=0.5,right=0.125}{v7,vb4}
%
\fmf{plain,tension=2,left=0}{vc1,vb1}
\fmf{plain,tension=2,left=0}{vb1,vb3}
\fmf{plain,tension=2,left=0}{vb3,vc3}
%\fmf{plain,tension=2,left=0}{vc2,vb2}
\fmf{plain,tension=2,left=0}{vb2,vb4}
%\fmf{plain,tension=2,left=0}{vb4,vc4}
\fmffreeze
%
\fmf{plain,tension=0.5,right=0.25}{v1,va1}
\fmf{plain,tension=0.5,left=0.25}{v2,va1}
\fmf{plain,tension=1}{va1,va2}
\fmf{plain,tension=0}{va2,vb2}
\fmf{plain,tension=0.5,left=0.125}{v5,va2}
\fmf{phantom,tension=0.5,right=0.125}{v6,va2}
%
\fmf{plain,tension=1,left=0}{vb1,vb2}
\fmf{plain,tension=1,left=0}{vb3,vb4}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v8}
}


\newcommand{\chionetwoonethree}[1][black]{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
\fmffixed{(0,whatever)}{va1,va2}
\fmffixed{(whatever,0)}{va1,vb3}
\fmffixed{(whatever,0)}{va2,vc3}
\fmffixed{(0,whatever)}{vc1,vc3}
\fmffixed{(0,whatever)}{vb2,vb4}
\fmffixed{(0,whatever)}{vc1,vb1}
\fmffixed{(0,whatever)}{vc1,vb3}
\fmffixed{(whatever,0)}{vb1,vb2}
\fmffixed{(whatever,0)}{vb3,vb4}
%
\fmf{plain,tension=0.5,right=0.25}{v1,vc1}
\fmf{plain,tension=0.5,left=0.25}{v2,vc1}
\fmf{phantom,tension=0.5,right=0.125}{v2,vb2}
\fmf{plain,tension=0.5,left=0.125}{v3,vb2}
%
\fmf{plain,tension=0.5,left=0.25}{v5,vc3}
\fmf{plain,tension=0.5,right=0.25}{v6,vc3}
\fmf{phantom,tension=0.5,left=0.25}{v6,vb4}
\fmf{phantom,tension=0.5,right=0.25}{v7,vb4}
%
\fmf{plain,tension=2,left=0}{vc1,vb1}
\fmf{plain,tension=2,left=0}{vb1,vb3}
\fmf{plain,tension=2,left=0}{vb3,vc3}
%\fmf{plain,tension=2,left=0}{vc2,vb2}
\fmf{plain,tension=2,left=0}{vb2,vb4}
%\fmf{plain,tension=2,left=0}{vb4,vc4}
\fmffreeze
%
\fmf{phantom,tension=0.5,right=0.125}{v3,va1}
\fmf{plain,tension=0.5,left=0.125}{v4,va1}
\fmf{plain,tension=1}{va1,va2}
\fmf{plain,tension=0}{va1,vb4}
\fmf{plain,tension=0.5,left=0.25}{v7,va2}
\fmf{plain,tension=0.5,right=0.25}{v8,va2}
%
\fmf{plain,tension=1,left=0}{vb1,vb2}
\fmf{plain,tension=1,left=0}{vb3,vb4}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v8}
}


\newcommand{\chionetwoonetwoone}[1][black]{%
\fmftop{v1}
\fmfbottom{v4}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v4}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v4,v5}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0,whatever)}{vc1,vc3}
\fmffixed{(0,whatever)}{vb2,vb4}
\fmffixed{(0,whatever)}{vc2,vb2}
\fmffixed{(0,whatever)}{vc4,vb4}
\fmffixed{(0,whatever)}{vc1,vb1}
\fmffixed{(0,whatever)}{vc1,vb3}
\fmffixed{(whatever,0)}{vc1,vc2}
\fmffixed{(whatever,0)}{vb1,vb2}
\fmffixed{(whatever,0)}{vb3,vb4}
\fmffixed{(whatever,0)}{vc3,vc4}
%
\fmf{plain,tension=0.5,right=0.25}{v1,vc1}
\fmf{plain,tension=0.5,left=0.25}{v2,vc1}
\fmf{phantom,tension=0.5,right=0.25}{v2,vc2}
\fmf{plain,tension=0.5,left=0.25}{v3,vc2}
%
\fmf{plain,tension=0.5,left=0.25}{v4,vc3}
\fmf{phantom,tension=0.5,right=0.25}{v5,vc3}
\fmf{plain,tension=0.5,left=0.25}{v5,vc4}
\fmf{plain,tension=0.5,right=0.25}{v6,vc4}
%
\fmf{plain,tension=2,left=0}{vc1,vb1}
\fmf{plain,tension=2,left=0}{vb1,vb3}
\fmf{plain,tension=2,left=0}{vb3,vc3}
\fmf{plain,tension=2,left=0}{vc2,vb2}
\fmf{plain,tension=2,left=0}{vb2,vb4}
\fmf{plain,tension=2,left=0}{vb4,vc4}
\fmffreeze
\fmf{plain,tension=2,left=0}{vb1,vc2}
\fmf{plain,tension=2,left=0}{vb3,vb2}
\fmf{plain,tension=2,left=0}{vc3,vb4}
%
\fmf{plain,tension=0.5,right=0,width=1mm}{v4,v6}
}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\idzero}[1][black]{%
\fmftop{v1}
\fmfbottom{v2}
\fmfforce{(0.5w,h)}{v1}
\fmfforce{(0.5w,0)}{v2}
\fmffixed{(0.01w,0)}{v2l,v2}
\fmffixed{(0.01w,0)}{v2,v2r}
%
\fmf{plain,fore=#1}{v1,v2}
\fmf{plain,tension=0.5,right=0,width=1mm,fore=#1}{v2l,v2r}
%\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__v2)}
}

\newcommand{\idone}[1][black]{%
\fmftop{v1}
\fmfbottom{v3}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v3}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v3,v4}
%
\fmf{plain,fore=#1}{v1,v3}
\fmf{phantom}{v1,vc1}
\fmf{phantom}{vc1,v3}
\fmf{plain,fore=#1}{v2,v4}
\fmf{plain,tension=0.5,right=0,width=1mm,fore=#1}{v3,v4}
%\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__v3)}

\fmfiset{p2}{vpath(__v2,__v4)}
}

\newcommand{\idtwo}[1][black]{%
\fmftop{v1}
\fmfbottom{v4}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v4}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v4,v5}
\fmffixed{(0.25w,0)}{v5,v6}
%
\fmf{plain,fore=#1}{v1,v4}
\fmf{plain,fore=#1}{v2,v5}
\fmf{plain,fore=#1}{v3,v6}
\fmf{plain,tension=0.5,right=0,width=1mm,fore=#1}{v4,v6}
%\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__v4)}
\fmfiset{p2}{vpath(__v2,__v5)}
\fmfiset{p3}{vpath(__v3,__v6)}
}


\newcommand{\vacpol}[3][black]{%
\fmfcmd{
begingroup;
save t,v,tv,do,di,ppol,pstr,dia;
path ppol,pstr;
pair v[],tv[],do[],di[];
ppol=vpath(__#2,__#3);
t1=arctime (1/3*arclength ppol) of ppol;
t2=arctime (2/3*arclength ppol) of ppol;
v1=point t1 of ppol;
v2=point t2 of ppol;
pstr=v1--v2;
t3=arctime (0.5*arclength pstr) of pstr;
v3=point t3 of pstr;
dia=arclength pstr; 
fill(fullcircle scaled dia shifted v3) withcolor #1;
endgroup;
}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  chiral parts of the wrapping graphs with gluons
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\Woneplain}{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
%\fmffixed{(0,0.9w)}{v1,v5}
%
\fmf{plain,tension=0.5,right=0.25}{v2,vc1}
\fmf{plain,tension=0.5,left=0.25}{v3,vc1}
  \fmf{plain}{vc1,vc2}
\fmf{plain,tension=0.5,left=0.125}{vc3,vc2}
\fmf{plain,tension=0.5,left=0.25}{v6,vc2}
\fmf{plain,tension=0.5,right=0.25}{v7,vc2}
\fmf{plain}{v1,v5}
\fmf{plain}{v4,v8}
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v8}
%\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__v5)}
\fmfiset{p2}{vpath(__v2,__vc1)}
\fmfiset{p3}{vpath(__v3,__vc1)}
\fmfiset{p4}{vpath(__vc1,__vc2)}
\fmfiset{p5}{vpath(__v6,__vc2)}
\fmfiset{p6}{vpath(__v7,__vc2)}
\fmfiset{p7}{vpath(__v4,__v8)}
}

\newcommand{\WoneplainB}{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
%\fmffixed{(0,0.9w)}{v1,v5}
%
\fmf{plain,tension=0.5,right=0.25}{v2,vc1}
\fmf{plain,tension=0.5,left=0.25}{v3,vc1}
  \fmf{plain}{vc1,vc2}
\fmf{plain,tension=0.5,left=0.125}{vc3,vc2}
\fmf{plain,tension=0.5,left=0.25}{v6,vc2}
\fmf{plain,tension=0.5,right=0.25}{v7,vc2}
%\fmf{plain}{v1,v5}
%\fmf{plain}{v4,v8}
%\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v8}
%\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__v5)}
\fmfiset{p2}{vpath(__v2,__vc1)}
\fmfiset{p3}{vpath(__v3,__vc1)}
\fmfiset{p4}{vpath(__vc1,__vc2)}
\fmfiset{p5}{vpath(__v6,__vc2)}
\fmfiset{p6}{vpath(__v7,__vc2)}
\fmfiset{p7}{vpath(__v4,__v8)}
}

\newcommand{\Wtwoplain}{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
%\fmffixed{(0,0.9w)}{v1,v5}
%
\fmf{plain,tension=0.5,right=0.25}{v1,vc1}
\fmf{plain,tension=0.5,left=0.25}{v2,vc1}
\fmf{plain,tension=0.5,right=0.25}{v3,vc2}
\fmf{plain,tension=0.5,left=0.25}{v4,vc2}
  \fmf{plain}{vc1,vc3}
  \fmf{plain}{vc2,vc4}
\fmf{plain,tension=0.5,left=0.25}{v5,vc3}
\fmf{plain,tension=0.5,right=0.25}{v6,vc3}
\fmf{plain,tension=0.5,left=0.25}{v7,vc4}
\fmf{plain,tension=0.5,right=0.25}{v8,vc4}
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v8}
%\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__vc1)}
\fmfiset{p2}{vpath(__v2,__vc1)}
\fmfiset{p3}{vpath(__vc1,__vc3)}
\fmfiset{p4}{vpath(__v5,__vc3)}
\fmfiset{p5}{vpath(__v6,__vc3)}
\fmfiset{p6}{vpath(__v3,__vc2)}
\fmfiset{p7}{vpath(__v4,__vc2)}
\fmfiset{p8}{vpath(__vc2,__vc4)}
\fmfiset{p9}{vpath(__v7,__vc4)}
\fmfiset{p10}{vpath(__v8,__vc4)}
}

\newcommand{\Wthreeplain}{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
%\fmffixed{(0,0.9w)}{v1,v5}
\fmffixed{(0,whatever)}{vc1,vc2}
\fmffixed{(0,whatever)}{vc3,vc4}
%
\fmf{plain,tension=0.5,right=0.25}{v2,vc1}
\fmf{plain,tension=0.5,left=0.25}{v3,vc1}
\fmf{plain,right=0.25}{v1,vc3}
\fmf{plain,tension=0.5,left=0.25}{v5,vc4}
\fmf{plain,tension=0.5,right=0.25}{v6,vc4}
\fmf{plain,right=0.25}{v7,vc2}
\fmf{plain}{v8,v4}
  \fmf{plain,tension=1}{vc1,vc2}
  \fmf{plain,tension=0.5}{vc2,vc3}
  \fmf{plain,tension=1}{vc3,vc4}
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v8}
%\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__vc3)}
\fmfiset{p2}{vpath(__vc3,__vc4)}
\fmfiset{p3}{vpath(__v5,__vc4)}
\fmfiset{p4}{vpath(__v3,__vc1)}
\fmfiset{p5}{vpath(__vc1,__vc2)}
\fmfiset{p6}{vpath(__v7,__vc2)}
\fmfiset{p7}{vpath(__v4,__v8)}
}

\newcommand{\Wfourplain}{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
%\fmffixed{(0,0.9w)}{v1,v5}
\fmffixed{(0,whatever)}{vc1,vc2}
\fmffixed{(0,whatever)}{vc3,vc4}
\fmffixed{(0,whatever)}{vc5,vc6}
%
\fmf{plain,tension=0.25,right=0.25}{v1,vc1}
\fmf{plain,tension=0.25,left=0.25}{v2,vc1}
\fmf{plain,left=0.25}{v5,vc2}
\fmf{plain,tension=1,left=0.25}{v3,vc3}
\fmf{plain,tension=1,left=0.25}{v4,vc5}
\fmf{plain,left=0.25}{v6,vc4}
\fmf{plain,tension=0.25,left=0.25}{v7,vc6}
\fmf{plain,tension=0.25,right=0.25}{v8,vc6}
  \fmf{plain,tension=0.5}{vc1,vc2}
  \fmf{plain,tension=0.5}{vc2,vc3}
  \fmf{plain,tension=0.5}{vc3,vc4}
  \fmf{plain,tension=0.5}{vc4,vc5}
  \fmf{plain,tension=0.5}{vc5,vc6}
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v8}
%\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__vc1)}
\fmfiset{p2}{vpath(__vc1,__vc2)}
\fmfiset{p3}{vpath(__v5,__vc2)}
\fmfiset{p4}{vpath(__v4,__vc5)}
\fmfiset{p5}{vpath(__vc5,__vc6)}
\fmfiset{p6}{vpath(__v8,__vc6)}
}

\newcommand{\Wfiveplain}{%
\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
%\fmffixed{(0,0.9w)}{v1,v5}
\fmffixed{(0,whatever)}{vc1,vc3}
\fmffixed{(0,whatever)}{vc2,vc4}
\fmffixed{(0,whatever)}{vc5,vc6}
%
\fmf{plain,tension=1,right=0.25}{v1,vc1}
\fmf{plain,tension=1,left=0.25}{v2,vc1}
\fmf{plain,tension=1,right=0.25}{v3,vc2}
\fmf{plain,tension=1,left=0.25}{v4,vc2}
\fmf{plain,tension=1,left=0.125}{v5,vc3}
\fmf{plain,tension=0.25,left=0.25}{v6,vc6}
\fmf{plain,tension=0.25,right=0.25}{v7,vc6}
\fmf{plain,tension=1,right=0.125}{v8,vc4}
  \fmf{plain,tension=4}{vc1,vc3}
  \fmf{plain,tension=4}{vc2,vc4}
  \fmf{plain,tension=0.5}{vc3,vc5}
  \fmf{plain,tension=0.5}{vc4,vc5}
  \fmf{plain,tension=1}{vc5,vc6}
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v8}
%\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__vc1)}
\fmfiset{p2}{vpath(__vc1,__vc3)}
\fmfiset{p3}{vpath(__v5,__vc3)}
\fmfiset{p4}{vpath(__v4,__vc2)}
%\fmfiset{p5}{vpath(__vc2,__vc6)}
\fmfiset{p5}{vpath(__vc2,__vc4)}
\fmfiset{p6}{vpath(__v8,__vc4)}
}

\newcommand{\Wsixplain}{%

\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0.125w,h)}{v1}
\fmfforce{(0.125w,0)}{v5}
\fmffixed{(0.25w,0)}{v1,v2}
\fmffixed{(0.25w,0)}{v2,v3}
\fmffixed{(0.25w,0)}{v3,v4}
\fmffixed{(0.25w,0)}{v5,v6}
\fmffixed{(0.25w,0)}{v6,v7}
\fmffixed{(0.25w,0)}{v7,v8}
%\fmffixed{(0,0.9w)}{v1,v5}
\fmffixed{(0,whatever)}{vc1,vc3}
\fmffixed{(0,whatever)}{vc2,vc4}
\fmffixed{(0,whatever)}{vc5,vc6}
%
\fmf{plain,tension=1,left=0.25}{v5,vc1}
\fmf{plain,tension=1,right=0.25}{v6,vc1}
\fmf{plain,tension=1,left=0.25}{v7,vc2}
\fmf{plain,tension=1,right=0.25}{v8,vc2}
\fmf{plain,tension=1,right=0.125}{v1,vc3}
\fmf{plain,tension=0.25,right=0.25}{v2,vc6}
\fmf{plain,tension=0.25,left=0.25}{v3,vc6}
\fmf{plain,tension=1,left=0.125}{v4,vc4}
  \fmf{plain,tension=4}{vc1,vc3}
  \fmf{plain,tension=4}{vc2,vc4}
  \fmf{plain,tension=0.5}{vc3,vc5}
  \fmf{plain,tension=0.5}{vc4,vc5}
  \fmf{plain,tension=1}{vc5,vc6}
\fmf{plain,tension=0.5,left=0,width=1mm}{v5,v8}
%\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__vc3)}
\fmfiset{p2}{vpath(__vc3,__vc1)}
\fmfiset{p3}{vpath(__v5,__vc1)}
\fmfiset{p4}{vpath(__v4,__vc4)}
\fmfiset{p5}{vpath(__vc4,__vc2)}
\fmfiset{p6}{vpath(__v8,__vc2)}
}

\newcommand{\Wsevenplain}{

\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0w,h)}{v1}
\fmfforce{(0w,0)}{v5}
\fmffixed{(0.2w,0)}{v1,v2}
\fmffixed{(0.2w,0)}{v2,v3}
\fmffixed{(0.2w,0)}{v3,v4}
\fmffixed{(0.2w,0)}{v4,v9}
\fmffixed{(0.2w,0)}{v5,v6}
\fmffixed{(0.2w,0)}{v6,v7}
\fmffixed{(0.2w,0)}{v7,v8}
\fmffixed{(0.2w,0)}{v8,v10}
%\fmffixed{(0,0.9w)}{v1,v5}
\fmffixed{(0,whatever)}{vc1,vc2}
\fmffixed{(0,whatever)}{vc3,vc4}
\fmffixed{(whatever,0)}{vc1,vc6}
\fmffixed{(whatever,0)}{vc2,vc4}
%
\fmf{plain,tension=0.5,right=0.25}{v2,vc1}
\fmf{plain,tension=0.5,left=0.25}{v3,vc1}
\fmf{plain,right=0.25}{v1,vc3}
\fmf{plain,tension=0.5,left=0.25}{v5,vc4}
\fmf{plain,tension=0.5,right=0.25}{v6,vc4}
\fmf{plain,right=0.25}{v7,vc2}
\fmf{plain,tension=0.5,left=0.25}{v8,vc5}
\fmf{plain,tension=0.5,right=0.25}{v10,vc5}
\fmf{plain,tension=0.5,right=0.25}{v4,vc6}
\fmf{plain,tension=0.5,left=0.25}{v9,vc6}
\fmf{plain}{vc5,vc6}
\fmf{plain,tension=1}{vc1,vc2}
\fmf{plain,tension=0.5}{vc2,vc3}
\fmf{plain,tension=1}{vc3,vc4}
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v10}
%\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__vc3)}
\fmfiset{p2}{vpath(__vc3,__vc4)}
\fmfiset{p3}{vpath(__v5,__vc4)}
\fmfiset{p4}{vpath(__v3,__vc1)}
\fmfiset{p5}{vpath(__vc1,__vc2)}
\fmfiset{p6}{vpath(__v7,__vc2)}
\fmfiset{p7}{vpath(__v4,__v8)}
\fmfiset{p8}{vpath(__v4,__vc6)}
\fmfiset{p9}{vpath(__vc6,__vc5)}
\fmfiset{p10}{vpath(__v8,__vc5)}
}

\newcommand{\Weightplain}{

\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0w,h)}{v1}
\fmfforce{(0w,0)}{v5}
\fmffixed{(0.2w,0)}{v1,v2}
\fmffixed{(0.2w,0)}{v2,v3}
\fmffixed{(0.2w,0)}{v3,v4}
\fmffixed{(0.2w,0)}{v4,v9}
\fmffixed{(0.2w,0)}{v5,v6}
\fmffixed{(0.2w,0)}{v6,v7}
\fmffixed{(0.2w,0)}{v7,v8}
\fmffixed{(0.2w,0)}{v8,v10}
%\fmffixed{(0,0.9w)}{v1,v5}
\fmffixed{(0,whatever)}{vc1,vc2}
\fmffixed{(0,whatever)}{vc3,vc4}
\fmffixed{(whatever,0)}{vc1,vc6}
\fmffixed{(whatever,0)}{vc2,vc4}
%
\fmf{plain,tension=0.5,right=0.25}{v4,vc1}
\fmf{plain,tension=0.5,left=0.25}{v9,vc1}
\fmf{plain,right=0.25}{v3,vc3}
\fmf{plain,tension=0.5,left=0.25}{v7,vc4}
\fmf{plain,tension=0.5,right=0.25}{v8,vc4}
\fmf{plain,right=0.25}{v10,vc2}
\fmf{plain,tension=0.5,left=0.25}{v5,vc5}
\fmf{plain,tension=0.5,right=0.25}{v6,vc5}
\fmf{plain,tension=0.5,right=0.25}{v1,vc6}
\fmf{plain,tension=0.5,left=0.25}{v2,vc6}
\fmf{plain}{vc5,vc6}
\fmf{plain,tension=1}{vc1,vc2}
\fmf{plain,tension=0.5}{vc2,vc3}
\fmf{plain,tension=1}{vc3,vc4}
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v10}
%\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v3,__vc3)}
\fmfiset{p2}{vpath(__vc3,__vc4)}
\fmfiset{p3}{vpath(__v7,__vc4)}
\fmfiset{p4}{vpath(__v9,__vc1)}
\fmfiset{p5}{vpath(__vc1,__vc2)}
\fmfiset{p6}{vpath(__v10,__vc2)}
\fmfiset{p7}{vpath(__v1,__v8)}
\fmfiset{p8}{vpath(__v2,__vc6)}
\fmfiset{p9}{vpath(__vc6,__vc5)}
\fmfiset{p10}{vpath(__v6,__vc5)}
}

\newcommand{\Wnineplain}{

\fmftop{v1}
\fmfbottom{v5}
\fmfforce{(0w,h)}{v1}
\fmfforce{(0w,0)}{v5}
\fmffixed{(0.2w,0)}{v1,v2}
\fmffixed{(0.2w,0)}{v2,v3}
\fmffixed{(0.2w,0)}{v3,v4}
\fmffixed{(0.2w,0)}{v4,v9}
\fmffixed{(0.2w,0)}{v5,v6}
\fmffixed{(0.2w,0)}{v6,v7}
\fmffixed{(0.2w,0)}{v7,v8}
\fmffixed{(0.2w,0)}{v8,v10}
%\fmffixed{(0,0.9w)}{v1,v5}
\fmffixed{(0,whatever)}{vc1,vc2}
\fmffixed{(0,whatever)}{vc3,vc4}
%\fmffixed{(whatever,0)}{vc1,vc6}
%\fmffixed{(whatever,0)}{vc2,vc4}
%
\fmf{plain,tension=0.5,right=0.25}{v1,vc1}
\fmf{plain,tension=0.5,left=0.25}{v2,vc1}
%\fmf{plain,right=0.25}{v3,vc3}
\fmf{plain,tension=0.5,left=0.25}{v5,vc2}
\fmf{plain,tension=0.5,right=0.25}{v6,vc2}
%\fmf{plain,right=0.25}{v10,vc2}
\fmf{plain,tension=0.5,left=0.25}{v8,vc4}
\fmf{plain,tension=0.5,right=0.25}{v10,vc4}
\fmf{plain,tension=0.5,right=0.25}{v4,vc3}
\fmf{plain,tension=0.5,left=0.25}{v9,vc3}
\fmf{plain,tension=1}{vc1,vc2}
\fmf{plain,tension=1}{vc3,vc4}
\fmf{plain,tension=1}{v3,v7}
\fmf{plain,tension=0.5,right=0,width=1mm}{v5,v10}
%\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v2,__vc1)}
\fmfiset{p2}{vpath(__vc1,__vc2)}
\fmfiset{p3}{vpath(__v6,__vc2)}
\fmfiset{p4}{vpath(__v3,__v7)}
\fmfiset{p5}{vpath(__v4,__vc3)}
\fmfiset{p6}{vpath(__vc3,__vc4)}
\fmfiset{p7}{vpath(__vc4,__v8)}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\cvert}[7][]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-8\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(1,2)(1,2){%
\begin{fmfchar*}(12,12)
\fmfright{v3,v2}
%\fmfleft{v1}
\fmfpoly{phantom}{v1,v3,v2}
\fmf{#2}{v1,vc1}
\fmf{#3}{vc1,v2}
\fmf{#4}{vc1,v3}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__vc1)}
\fmfiset{p2}{vpath(__vc1,__v2)}
\fmfiset{p3}{vpath(__vc1,__v3)}
{#1}
\fmfis{phantom,ptext.clen=6,ptext.hout=3,ptext.oout=12,ptext.out=#5,ptext.sep=;}{p1}
\fmfis{phantom,ptext.clen=6,ptext.hin=3,ptext.oin=14,ptext.in=#6,ptext.sep=;}{p2}
\fmfis{phantom,ptext.side=right,ptext.clen=6,ptext.hin=-3,ptext.oin=14,ptext.in=#7,ptext.sep=;}{p3}
%\fmfiv{label=$\scriptstyle \alpha$,l.dist=2}{vloc(__v1)}
%\fmfiv{label=$\scriptstyle \beta$,l.dist=2}{vloc(__v2)}
%\fmfiv{label=$\scriptstyle \gamma$,l.dist=2}{vloc(__v3)}
\end{fmfchar*}}}%}
}

\newcommand{\qvert}[8]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-8\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(1,2)(1,2){%
\begin{fmfchar*}(12,12)
\fmfleft{v2,v1}
\fmfright{v3,v4}
\fmfforce{(0,0)}{v1}
\fmfforce{(0,h)}{v2}
\fmfforce{(w,h)}{v3}
\fmfforce{(w,0)}{v4}
\fmf{#1}{v1,vc1}
\fmf{#2}{v2,vc1}
\fmf{#3}{vc1,v3}
\fmf{#4}{vc1,v4}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__vc1)}
\fmfiset{p2}{vpath(__v2,__vc1)}
\fmfiset{p3}{vpath(__vc1,__v3)}
\fmfiset{p4}{vpath(__vc1,__v4)}
\fmfis{phantom,ptext.clen=6,ptext.hout=3,ptext.oout=12,ptext.out=#6,ptext.sep=;}{p2}
\fmfis{phantom,ptext.clen=6,ptext.hout=3,ptext.oout=12,ptext.out=#5,ptext.sep=;}{p1}
%\fmfis{phantom,ptext.side=right,ptext.clen=6,ptext.hout=-3,ptext.oout=12,ptext.out=#6,ptext.sep=;}{p2}
\fmfis{phantom,ptext.side=right,ptext.clen=6,ptext.hin=-3,ptext.oin=12,ptext.in=#8,ptext.sep=;}{p4}
%\fmfis{phantom,ptext.clen=6,ptext.hin=3,ptext.oin=12,ptext.in=#7,ptext.sep=;}{p3}
\fmfis{phantom,ptext.clen=6,ptext.hin=3,ptext.oin=12,ptext.in=#7,ptext.sep=;}{p3}
%\fmfiv{label=$\scriptstyle \alpha$,l.dist=2}{vloc(__v1)}
%\fmfiv{label=$\scriptstyle \beta$,l.dist=2}{vloc(__v2)}
%\fmfiv{label=$\scriptstyle \gamma$,l.dist=2}{vloc(__v3)}
\end{fmfchar*}}}%}
}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\cVat}[7][]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-13\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(2,1)(2,1){%
\begin{fmfchar*}(21,24)
\fmftop{v2}
\fmfbottom{v3}
%\fmfleft{v1}
\fmfforce{(w,h)}{v2}
\fmfforce{(w,0)}{v3}
%\fmfforce{(0w,0.5h)}{v1}
%\fmffixed{(0,0.75h)}{v2,vg3}
%\fmffixed{(0,0.25h)}{v2,vg2}
%\fmffixed{(0.5w,0h)}{v1,vg1}
\fmfpoly{phantom}{v1,v3,v2}
\fmf{phantom,tension=2}{v1,vg1}
\fmf{phantom}{v2,vg1}
\fmf{phantom}{v3,vg1}
\fmffreeze
%
\fmf{phantom,tension=1}{vg2,v2}
\fmf{phantom,tension=1}{vg3,v3}
\fmf{phantom}{vg1,v1}
\fmf{phantom,tension=0.5}{vg1,vg2}
\fmf{phantom,tension=0.5}{vg1,vg3}
\fmffreeze
\fmfposition
\fmf{phantom}{vg2,vg3}
\fmfipath{p[],pca}
\fmfipair{vm[],vo[],vi[]}
\fmfiset{p1}{vpath(__vg1,__v1)}
\fmfiset{p2}{vpath(__vg2,__v2)}
\fmfiset{p3}{vpath(__vg3,__v3)}
\fmfiset{p4}{vpath(__vg2,__vg3)}
\fmfiset{p6}{vpath(__vg3,__vg1)}
\fmfiset{p5}{vpath(__vg1,__vg2)}
%
{#1}
\fmfis{#2,ptext.clen=7,ptext.hin=3,ptext.hout=3,ptext.oin=6,ptext.oout=6,ptext.sep=;}{reverse(p1)}
\fmfis{#3,ptext.clen=7,ptext.hin=3,ptext.hout=3,ptext.oin=6,ptext.oout=6,ptext.sep=;}{p5}
\fmfis{#4,ptext.clen=7,ptext.hin=-9,ptext.hout=-9,ptext.oin=6,ptext.oout=6,ptext.sep=;}{p6}
\fmfis{#5,ptext.clen=7,ptext.hin=3,ptext.hout=3,ptext.oin=6,ptext.oout=6,ptext.sep=;}{p4}
\fmfis{#6,ptext.clen=7,ptext.hin=3,ptext.hout=3,ptext.oin=6,ptext.oout=6,ptext.sep=;}{p2}
\fmfis{#7,ptext.clen=7,ptext.hin=-9,ptext.hout=-9,ptext.oin=6,ptext.oout=6,ptext.sep=;}{p3}
\end{fmfchar*}}}%}
}

\newcommand{\cVatg}[6][]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-13\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(2,1)(2,1){%
\begin{fmfchar*}(21,24)
\fmftop{v3}
\fmfbottom{v2}
%\fmfleft{v1}
\fmfforce{(w,h)}{v3}
\fmfforce{(w,0)}{v2}
%\fmfforce{(0w,0.5h)}{v1}
%\fmffixed{(0,0.75h)}{v2,vg3}
%\fmffixed{(0,0.25h)}{v2,vg2}
%\fmffixed{(0.5w,0h)}{v1,vg1}
\fmfpoly{phantom}{v1,v2,v3}
\fmf{phantom,tension=2}{v1,vg1}
\fmf{phantom}{v2,vg1}
\fmf{phantom}{v3,vg1}
\fmffreeze
%
\fmf{phantom,tension=1}{vgo2,v2}
\fmf{phantom,tension=1}{vgo3,v3}
\fmf{phantom}{vg1,vgm1}
\fmf{phantom}{vgm1,v1}
\fmf{#3,tension=0.5}{vg1,vgi2}
\fmf{#4,tension=0.5}{vg1,vgi3}
\fmf{#3,tension=1}{vgi2,vgm2}
\fmf{#3,tension=1}{vgm2,vgo2}
\fmf{#4,tension=1}{vgi3,vgm3}
\fmf{#4,tension=1}{vgm3,vgo3}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfipair{vm[],vo[],vi[]}
\fmfiset{p1}{vpath(__vg1,__v1)}
\fmfiset{p2}{vpath(__vgo2,__v2)}
\fmfiset{p3}{vpath(__vgo3,__v3)}
\fmfiset{vm1}{vloc(__vgm1)}
\fmfiset{vi2}{vloc(__vgi2)}
\fmfiset{vm2}{vloc(__vgm2)}
\fmfiset{vo2}{vloc(__vgo2)}
\fmfiset{vi3}{vloc(__vgi3)}
\fmfiset{vm3}{vloc(__vgm3)}
\fmfiset{vo3}{vloc(__vgo3)}
%
{#1}
\fmfis{#2,ptext.clen=7,ptext.hout=3,ptext.oin=0,ptext.oout=6,ptext.sep=;}{reverse(p1)}
\fmfis{#5,ptext.clen=7,ptext.hin=3,ptext.oin=6,ptext.oout=0,ptext.sep=;}{p2}
\fmfis{#6,ptext.clen=7,ptext.hin=-8,ptext.oin=6,ptext.oout=0,ptext.sep=;}{p3}
\end{fmfchar*}}}%}
}


\newcommand{\cVatt}[9][]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-14\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(2,2)(2,2){%
\begin{fmfchar*}(21,24)
\fmftop{v3}
\fmfbottom{v2}
%\fmfleft{v1}
\fmfforce{(w,h)}{v2}
\fmfforce{(w,0)}{v3}
%\fmfforce{(0w,0.5h)}{v1}
%\fmffixed{(0,0.75h)}{v2,vg3}
%\fmffixed{(0,0.25h)}{v2,vg2}
%\fmffixed{(0.5w,0h)}{v1,vg1}
\fmfpoly{phantom}{v1,v3,v2}
\fmf{phantom,tension=2}{v1,vg1}
\fmf{phantom}{v2,vg1}
\fmf{phantom}{v3,vg1}
\fmffreeze
%
\fmf{phantom,tension=1}{vgo2,v2}
\fmf{phantom,tension=1}{vgo3,v3}
\fmf{phantom}{vg1,v1}
\fmf{#3,tension=0.5}{vg1,vgi2}
\fmf{#4,tension=0.5}{vg1,vgi3}
\fmf{#6,tension=0.5}{vgi2,vgo2}
\fmf{#7,tension=0.5}{vgi3,vgo3}
\fmffreeze
\fmfposition
\fmf{#5,label=$ $}{vgi2,vgi3}
\fmf{#5}{vgo2,vgo3}
\fmfipath{p[],pca}
\fmfipair{vm[],vo[],vi[]}
\fmfiset{p1}{vpath(__vg1,__v1)}
\fmfiset{p2}{vpath(__vgo2,__v2)}
\fmfiset{p3}{vpath(__vgo3,__v3)}
\fmfiset{p4}{vpath(__vgi2,__vgi3)}
\fmfiset{p5}{vpath(__vgi3,__vg1)}
\fmfiset{p6}{vpath(__vg1,__vgi2)}
\fmfiset{p7}{vpath(__vgo2,__vgo3)}
\fmfiset{p8}{vpath(__vgo3,__vgi3)}
\fmfiset{p9}{vpath(__vgi2,__vgo2)}
%
{#1}
\fmfis{#2,ptext.clen=7,ptext.hout=3,ptext.oin=0,ptext.oout=6,ptext.sep=;}{reverse(p1)}
\fmfis{#8,ptext.clen=7,ptext.hin=3,ptext.oin=6,ptext.oout=0,ptext.sep=;}{p2}
\fmfis{#9,ptext.clen=7,ptext.hin=-8,ptext.oin=6,ptext.oout=0,ptext.sep=;}{p3}
\end{fmfchar*}}}%}
}


\newcommand{\cVab}[7][plain]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-13\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(2,1)(6,1){%
\begin{fmfchar*}(21,24)
\fmftop{v3}
\fmfbottom{v2}
%\fmfleft{v1}
\fmfforce{(w,h)}{v3}
\fmfforce{(w,0)}{v2}
%\fmfforce{(0w,0.5h)}{v1}
\fmffixed{(0,0.75h)}{v2,vg3}
\fmffixed{(0,0.25h)}{v2,vg2}
\fmfpoly{phantom}{v1,v2,v3}
%
\fmf{#1}{vg2,vg3}
\fmf{#1}{vg2,v2}
\fmf{#1}{vg3,v3}
\fmf{photon}{vg #2,v1}
\fmf{photon,left=#3}{vg3,vg2}
\fmffreeze
\fmfposition
\fmfipath{p[],pca}
\fmfipair{vm[],vo[],vi[]}
\fmfiset{p1}{vpath(__vg #2,__v1)}
\fmfiset{p2}{vpath(__vg2,__v2)}
\fmfiset{p3}{vpath(__vg3,__v3)}
\fmfiset{p4}{vpath(__vg2,__vg3)}
\fmfiset{p5}{reverse(vpath(__vg3,__vg2))}
%
\fmfis{phantom,ptext.clen=6,ptext.hin=3,ptext.hout=3,ptext.oin=6,ptext.oout=6,ptext.out=#4,ptext.sep=;}{reverse(p1)}
\fmfis{phantom,ptext.clen=6,ptext.hin=3,ptext.hout=3,ptext.oin=6,ptext.oout=6,ptext.out=#5,ptext.sep=;}{p3}
\fmfis{phantom,ptext.clen=6,ptext.hin=-9,ptext.hout=-9,ptext.oin=6,ptext.oout=6,ptext.out=#6,ptext.sep=;}{p2}
\fmfis{phantom,ptext.clen=6,ptext.hin=-9,ptext.hout=-9,ptext.oin=8,ptext.oout=8,#7,ptext.sep=;}{p4}
\end{fmfchar*}}}%}
}

\newcommand{\cccV}[9][]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-13\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(2,1)(2,1){%
\begin{fmfchar*}(24,24)
\fmftop{v4}
\fmfleft{v1}
\fmfbottom{v2}
\fmfright{v2}
%\fmfleft{v1}
\fmfforce{(0.5w,h)}{v4}
\fmfforce{(0,0.5h)}{v1}
\fmfforce{(w,0.5h)}{v3}
\fmfforce{(0.5w,0)}{v2}
\fmffixed{(0.5w,0h)}{vs1,vs3}
\fmf{phantom}{vs1,v1}
\fmf{phantom}{vs2,v2}
\fmf{phantom}{vs3,v3}
\fmf{phantom}{vs4,v4}
\fmfpoly{phantom}{vs1,vs2,vs3,vs4}
\fmffreeze
%
\fmfposition
\fmfipath{p[],pca}
\fmfipair{vm[],vo[],vi[]}
\fmfiset{p1}{vpath(__vs1,__v1)}
\fmfiset{p2}{vpath(__vs2,__v2)}
\fmfiset{p3}{vpath(__vs3,__v3)}
\fmfiset{p4}{vpath(__vs4,__v4)}
\fmfiset{p5}{vpath(__vs1,__vs2)}
\fmfiset{p6}{vpath(__vs2,__vs3)}
\fmfiset{p7}{vpath(__vs3,__vs4)}
\fmfiset{p8}{vpath(__vs4,__vs1)}
%
{#1}
\fmfis{#2,ptext.clen=7,ptext.hin=3,ptext.hout=3,ptext.oin=5,ptext.oout=3,ptext.sep=;}{reverse(p1)}
\fmfis{#3,ptext.clen=7,ptext.hin=3,ptext.hout=3,ptext.oin=5,ptext.oout=3,ptext.sep=;}{p2}
\fmfis{#4,ptext.clen=7,ptext.hin=3,ptext.hout=3,ptext.oin=5,ptext.oout=3,ptext.sep=;}{p3}
\fmfis{#5,ptext.clen=7,ptext.hin=-8,ptext.hout=-8,ptext.oin=5,ptext.oout=3,ptext.sep=;}{p4}
\fmfis{#6,ptext.clen=7,ptext.hin=-8,ptext.hout=-8,ptext.oin=5,ptext.oout=5,ptext.sep=;}{p5}
\fmfis{#7,ptext.clen=7,ptext.hin=-8,ptext.hout=-8,ptext.oin=5,ptext.oout=5,ptext.sep=;}{p6}
\fmfis{#8,ptext.clen=7,ptext.hin=-8,ptext.hout=-8,ptext.oin=5,ptext.oout=5,ptext.sep=;}{p7}
\fmfis{#9,ptext.clen=7,ptext.hin=3,ptext.hout=3,ptext.oin=5,ptext.oout=5,ptext.sep=;}{reverse(p8)}
\end{fmfchar*}}}%}
}
\newcommand{\ccc}[9][]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-13\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(2,1)(2,1){%
\begin{fmfchar*}(24,24)
\fmftop{v4}
\fmfleft{v1}
\fmfbottom{v2}
\fmfright{v2}
%\fmfleft{v1}
\fmfforce{(0.5w,h)}{v4}
\fmfforce{(0,0.5h)}{v1}
\fmfforce{(w,0.5h)}{v3}
\fmfforce{(0.5w,0)}{v2}
\fmffixed{(0.5w,0h)}{vs1,vs3}
\fmf{phantom}{vs1,v1}
\fmf{phantom}{vs2,v2}
\fmf{phantom}{vs3,vs1}
\fmf{phantom}{vs2,vs1}
\fmffreeze
%
\fmfposition
\fmfipath{p[],pca}
\fmfipair{vm[],vo[],vi[]}
\fmfiset{p1}{vpath(__vs1,__v1)}
\fmfiset{p2}{vpath(__vs2,__v2)}
\fmfiset{p3}{vpath(__vs3,__v3)}
\fmfiset{p4}{vpath(__vs2,__vs1)}
\fmfiset{p5}{vpath(__vs3,__vs1)}
\svertex{vm1}{p1}
\dvertex{vu1}{vd1}{p1}
\svertex{vm2}{p2}
\dvertex{vu2}{vd2}{p2}
\svertex{vm3}{p3}
\dvertex{vu3}{vd3}{p3}
\svertex{vm4}{p4}
\dvertex{vd4}{vu4}{p4}
\svertex{vm5}{p5}
\dvertex{vd5}{vu5}{p5}
%
%
{#1}
\fmfis{#2,ptext.clen=7,ptext.hin=3,ptext.hout=3,ptext.oin=5,ptext.oout=3,ptext.sep=;}{reverse(p1)}
\fmfis{#3,ptext.clen=7,ptext.hin=3,ptext.hout=3,ptext.oin=5,ptext.oout=3,ptext.sep=;}{p2}
\fmfis{#4,ptext.clen=7,ptext.hin=3,ptext.hout=3,ptext.oin=5,ptext.oout=3,ptext.sep=;}{p3}
\fmfis{#5,ptext.clen=7,ptext.hin=-8,ptext.hout=-8,ptext.oin=5,ptext.oout=3,ptext.sep=;}{p4}
\fmfis{#6,ptext.clen=7,ptext.hin=-8,ptext.hout=-8,ptext.oin=5,ptext.oout=5,ptext.sep=;}{p5}
\fmfis{#7,ptext.clen=7,ptext.hin=-8,ptext.hout=-8,ptext.oin=5,ptext.oout=5,ptext.sep=;}{p6}
\fmfis{#8,ptext.clen=7,ptext.hin=-8,ptext.hout=-8,ptext.oin=5,ptext.oout=5,ptext.sep=;}{p7}
\fmfis{#9,ptext.clen=7,ptext.hin=3,ptext.hout=3,ptext.oin=5,ptext.oout=5,ptext.sep=;}{reverse(p8)}
\end{fmfchar*}}}%}
}

% \newcommand{\cVaeff}[7]{%
% \settoheight{\eqoff}{$\times$}%
% \setlength{\eqoff}{0.5\eqoff}%
% \addtolength{\eqoff}{-13\unitlength}%
% %\smash[b]{%
% \raisebox{\eqoff}{%
% \fmfframe(1,1)(2,1){%
% \begin{fmfchar*}(21,24)
% \fmftop{v3}
% \fmfbottom{v2}
% %\fmfleft{v1}
% \fmfforce{(w,h)}{v3}
% \fmfforce{(w,0)}{v2}
% %\fmfforce{(0w,0.5h)}{v1}
% %\fmffixed{(0,0.75h)}{v2,vg3}
% %\fmffixed{(0,0.25h)}{v2,vg2}
% %\fmffixed{(w,0h)}{v1,vg}
% \fmfpoly{phantom}{v1,v2,v3}
% %
% \fmf{#4}{vg,v3}
% \fmf{#3}{vg,v2}
% \fmf{#2}{vg,v1}
% \fmffreeze
% \fmfposition
% \fmfipath{p[],pca}
% \fmfipair{vm[],vo[],vi[]}
% \fmfiset{p1}{vpath(__vg,__v1)}
% \fmfiset{p2}{vpath(__vg,__v2)}
% \fmfiset{p3}{vpath(__vg,__v3)}
% %
% \fmfis{phantom,ptext.clen=7,ptext.hout=5,ptext.oin=0,ptext.oout=18,ptext.out=#5,ptext.sep=;}{reverse(p1)}
% \fmfis{phantom,ptext.clen=7,ptext.hin=3,ptext.oin=18,ptext.oout=0,ptext.in=#6,ptext.sep=;}{p2}
% \fmfis{phantom,ptext.clen=7,ptext.hin=-10,ptext.oin=18,ptext.oout=0,ptext.in=#7,ptext.sep=;}{p3}
% \fmfcmd{fill fullcircle scaled 8 shifted vloc(__vg) withcolor black ;}
% \fmfiv{plain,label=$\scriptstyle\textcolor{white}{#1}$,l.dist=0}{vloc(__vg)}
% \end{fmfchar*}}}%}
% }




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% D-algebra  
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\vacpolD}[9]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-10\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(1,0)(1,0){%
\begin{fmfchar*}(30,20)
\fmfleft{v1}
\fmfright{v2}
\fmffixed{(0.4w,0)}{vc1,vc2}
\fmf{#1}{v1,vc1}
\fmf{#3}{vc2,v2}
\fmf{#2,tension=0.5,left=1}{vc1,vc2}
\fmf{#2,tension=0.5,left=1}{vc2,vc1}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__vc1)}
\fmfiset{p2}{vpath(__vc1,__vc2)}
\fmfiset{p3}{reverse vpath(__vc2,__vc1)}
\fmfiset{p4}{vpath(__vc2,__v2)}
\fmfis{phantom,ptext.clen=6,ptext.hout=3,ptext.oout=12,ptext.out=#4,ptext.sep=;}{p1}
\fmfis{phantom,ptext.clen=6,ptext.hin=3,ptext.hout=3,ptext.oin=10,ptext.oout=10,ptext.in=#5,ptext.out=#6,ptext.sep=;}{p2}
\fmfis{phantom,ptext.side=right,ptext.clen=6,ptext.hin=-4,ptext.hout=-4,ptext.oin=10,ptext.oout=10,ptext.in=#7,ptext.out=#8,ptext.sep=;}{p3}
\fmfis{phantom,ptext.clen=6,ptext.hin=3,ptext.oin=12,ptext.in=#9,ptext.sep=;}{p4}
\end{fmfchar*}}}}%}

% one- and two-loop self energy diagrams

\newcommand{\seone}[9][]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-7.5\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(1,0)(1,0){%
\begin{fmfchar*}(20,15)
\fmfleft{v1}
\fmfright{v2}
\fmffixed{(0.66w,0)}{vc1,vc2}
\fmf{#4}{v1,vc1}
\fmf{#5}{vc2,v2}
\fmf{phantom,left=#2}{vc1,vc2}
\fmf{phantom,left=#3}{vc2,vc1}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfipair{vm[]}
\fmfiset{p1}{vpath(__vc1,__vc2)}
\fmfiset{p2}{vpath(__vc2,__vc1)}
\fmfi{#6}{subpath (0,length(p1)/2) of p1}
\fmfi{#7}{subpath (length(p1)/2,length(p1)) of p1}
\fmfi{#9}{subpath (0,length(p2)/2) of p2}
\fmfi{#8}{subpath (length(p2)/2,length(p2)) of p2}
\svertex{vm1}{p1}
\svertex{vm2}{p2}
{#1}
\end{fmfchar*}}}}%}

\newcommand{\seonetwo}[6]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-7.5\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(1,0)(1,0){%
\begin{fmfchar*}(20,15)
\fmfleft{v1}
\fmfright{v2}
\fmffixed{(0.66w,0)}{vc1,vc2}
\fmf{#3}{v1,vc1}
\fmf{#4}{vc2,v2}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfipair{vm[]}
\fmfi{#5}{vloc(__vc1){dir 90}..{dir 90}vloc(__vc2)}
\fmfi{#6}{vloc(__vc1){dir -90}..{dir -90}vloc(__vc2)}
\end{fmfchar*}}}}%}

% two-loop chiral self energy diagrams

\newcommand{\swfoneone}{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-7.5\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(1,0)(1,0){%
\begin{fmfchar*}(20,15)
\fmfleft{v1}
\fmfright{v2}
\fmffixed{(0.66w,0)}{vc1,vc2}
\fmf{plain}{v1,vc1}
\fmf{plain}{vc2,v2}
\fmf{plain,left=1}{vc1,vc2}
\fmf{plain,left=1}{vc2,vc1}
\fmffreeze
\fmfposition
\fmfipair{vm[]}
\svertex{vm1}{vpath(__vc1,__vc2)}
\svertex{vm2}{vpath(__vc2,__vc1)}
%\fmfi{photon}{vm1--vm2}
\end{fmfchar*}}}}%}


\newcommand{\swftwoone}{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-7.5\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(1,0)(1,0){%
\begin{fmfchar*}(20,15)
\fmfleft{v1}
\fmfright{v2}
\fmffixed{(0.66w,0)}{vc1,vc2}
\fmf{plain}{v1,vc1}
\fmf{plain}{vc2,v2}
\fmf{plain,left=1}{vc1,vc2}
\fmf{plain,left=1}{vc2,vc1}
\fmffreeze
\fmfposition
\fmfipair{vm[]}
\svertex{vm1}{vpath(__vc1,__vc2)}
\svertex{vm2}{vpath(__vc2,__vc1)}
\fmfi{photon}{vm1--vm2}
\end{fmfchar*}}}}%}

\newcommand{\swftwotwo}[3][plain]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-7.5\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(1,0)(1,0){%
\begin{fmfchar*}(20,15)
\fmfleft{v1}
\fmfright{v2}
\fmffixed{(0.33w,0)}{vc1,vc2}
\fmffixed{(0.33w,0)}{vc2,vc3}
\fmf{#1}{v1,vc1}
\fmf{#1}{vc1,vc2}
\fmf{#1}{vc2,vc3}
\fmf{#1}{vc3,v2}
\fmf{photon,tension=0.5,left=#3}{vc1,vc2}
\fmf{photon,tension=0.5,left=#2}{vc1,vc3}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\end{fmfchar*}}}}%}

\newcommand{\swftwotwor}[3][plain]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-7.5\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(1,0)(1,0){%
\begin{fmfchar*}(20,15)
\fmfleft{v1}
\fmfright{v2}
\fmffixed{(0.33w,0)}{vc1,vc2}
\fmffixed{(0.33w,0)}{vc2,vc3}
\fmf{#1}{v1,vc1}
\fmf{#1}{vc1,vc2}
\fmf{#1}{vc2,vc3}
\fmf{#1}{vc3,v2}
\fmf{photon,tension=0.5,left=#3}{vc2,vc3}
\fmf{photon,tension=0.5,left=#2}{vc1,vc3}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\end{fmfchar*}}}}%}

\newcommand{\swftwothree}[3][plain]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-7.5\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(1,0)(1,0){%
\begin{fmfchar*}(20,15)
\fmfleft{v1}
\fmfright{v2}
\fmffixed{(0.33w,0)}{vc1,vc2}
\fmffixed{(0.33w,0)}{vc2,vc3}
\fmf{#1}{v1,vc1}
\fmf{#1}{vc1,vc2}
\fmf{#1}{vc2,vc3}
\fmf{#1}{vc3,v2}
\fmf{photon,tension=0.5,left=#3}{vc2,vc3}
\fmf{photon,tension=0.5,left=#2}{vc1,vc3}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\end{fmfchar*}}}}%}

\newcommand{\swftwofour}[3][plain]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-7.5\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(1,0)(1,0){%
\begin{fmfchar*}(20,15)
\fmfleft{v1}
\fmfright{v2}
\fmffixed{(0.33w,0)}{vc1,vc2}
\fmffixed{(0.33w,0)}{vc2,vc3}
\fmf{#1}{v1,vc1}
\fmf{#1}{vc1,vc2}
\fmf{#1}{vc2,vc3}
\fmf{#1}{vc3,v2}
\fmf{photon,tension=0.5,left=#2}{vc1,vc2}
\fmf{photon,tension=0.5,left=#3}{vc2,vc3}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\end{fmfchar*}}}}%}

\newcommand{\swftwofive}[2][plain]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-7.5\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(1,0)(1,0){%
\begin{fmfchar*}(20,15)
\fmfleft{v1}
\fmfright{v2}
\fmffixed{(0.22w,0)}{vc1,vc2}
\fmffixed{(0.22w,0)}{vc2,vc3}
\fmffixed{(0.22w,0)}{vc3,vc4}
\fmf{#1}{v1,vc1}
\fmf{#1}{vc1,vc2}
\fmf{#1}{vc2,vc3}
\fmf{#1}{vc3,vc4}
\fmf{#1}{vc4,v2}
\fmf{photon,tension=0.5,left=#2}{vc1,vc3}
\fmf{photon,tension=0.5,left=-(#2)}{vc2,vc4}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__vc1)}
\fmfiset{p2}{vpath(__vc1,__vc2)}
\fmfiset{p3}{vpath(__vc2,__vc3)}
\fmfiset{p4}{vpath(__vc3,__vc4)}
\fmfiset{p5}{vpath(__vc4,__v2)}
\end{fmfchar*}}}}%}


\newcommand{\swftwosix}[3]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-7.5\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(1,0)(1,0){%
\begin{fmfchar*}(20,15)
\fmfleft{v1}
\fmfright{v2}
\fmffixed{(0.33w,0)}{vc1,vc2}
\fmffixed{(0.33w,0)}{vc2,vc3}
\fmffixed{(0,0.33w)}{vc2,vc}
\fmf{plain}{v1,vc1}
\fmf{plain}{vc1,vc2}
\fmf{plain}{vc2,vc3}
\fmf{plain}{vc3,v2}
\fmf{#1,tension=0.5,left=0.5}{vc1,vc}
\fmf{#2,tension=0.5,left=0.5}{vc,vc3}
\fmffreeze
\fmf{#3,tension=0.5}{vc2,vc}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__vc1)}
\fmfiset{p2}{vpath(__vc1,__vc2)}
\fmfiset{p3}{vpath(__vc2,__vc3)}
\fmfiset{p4}{vpath(__vc3,__v2)}
\fmfiset{p5}{vpath(__vc1,__vc)}
\fmfiset{p6}{vpath(__vc2,__vc)}
\fmfiset{p7}{vpath(__vc,__vc3)}
% \fmfis{phantom,ptext.clen=6,ptext.hout=-9,ptext.oout=6,ptext.out=$\scriptstyle\D^2$,ptext.sep=;}{p1}
% \fmfis{phantom,ptext.clen=6,ptext.hin=-9,ptext.hout=-9,ptext.oin=6,ptext.oout=6,ptext.in=$\scriptstyle\barD^2$,ptext.out=$\scriptstyle\D^2$,ptext.sep=;}{p2}
% \fmfis{phantom,ptext.clen=6,ptext.hin=-9,ptext.hout=-9,ptext.oin=6,ptext.oout=6,ptext.in=$\scriptstyle\barD^2$,ptext.out=$\scriptstyle\D^2$,ptext.sep=;}{p3}
% \fmfis{phantom,ptext.clen=6,ptext.hin=-9,ptext.oin=6,ptext.in=$\scriptstyle\barD^2$,ptext.sep=;}{p4}
% \fmfis{phantom,ptext.clen=6,ptext.hin=3,ptext.oin=4,ptext.in=$\scriptstyle\D;\scriptstyle\barD$,ptext.sep=;}{p5}
% \fmfis{phantom,ptext.clen=6,ptext.hin=3,ptext.oin=4,ptext.in=$\scriptstyle\D;\scriptstyle\barD$,ptext.sep=;}{p6}
% \fmfis{phantom,ptext.clen=6,ptext.hout=3,ptext.oout=4,ptext.out=$\scriptstyle\D;\scriptstyle\barD$,ptext.sep=;}{p7}
\end{fmfchar*}}}}%}


\newcommand{\swftwoseven}[1]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-7.5\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(1,0)(1,0){%
\begin{fmfchar*}(20,15)
\fmfleft{v1}
\fmfright{v2}
\fmffixed{(0.66w,0)}{vc1,vc2}
\fmf{plain}{v1,vc1}
\fmf{plain}{vc2,v2}
\fmf{plain,tension=0.5}{vc1,vc2}
\fmf{photon,tension=0.5,left=#1}{vc1,vc2}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__vc1)}
\fmfiset{p2}{vpath(__vc1,__vc2)}
\fmfiset{p3}{reverse vpath(__vc2,__vc1)}
\fmfiset{p4}{vpath(__vc2,__v2)}
\vacpol{vc1}{vc2}
% \fmfis{phantom,ptext.clen=6,ptext.hout=3,ptext.oout=12,ptext.out=#4,ptext.sep=;}{p1}
% \fmfis{phantom,ptext.clen=6,ptext.hin=3,ptext.hout=3,ptext.oin=10,ptext.oout=10,ptext.in=#5,ptext.out=#6,ptext.sep=;}{p2}
% \fmfis{phantom,ptext.side=right,ptext.clen=6,ptext.hin=-4,ptext.hout=-4,ptext.oin=10,ptext.oout=10,ptext.in=#7,ptext.out=#8,ptext.sep=;}{p3}
% \fmfis{phantom,ptext.clen=6,ptext.hin=3,ptext.oin=12,ptext.in=#9,ptext.sep=;}{p4}
\end{fmfchar*}}}}%}

\newcommand{\swftwoeight}[1]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-7.5\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(1,0)(1,0){%
\begin{fmfchar*}(20,15)
\fmfleft{v1}
\fmfright{v2}
\fmf{plain}{v1,vc1}
\fmf{plain}{vc1,v2}
\fmf{photon,tension=1,right=#1}{vc1,vc1}
%\fmf{photon,tension=0.5,left=#1}{vc1,vc2}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfiset{p1}{vpath(__v1,__vc1)}
\fmfiset{p2}{vpath(__vc1,__v2)}
\fmfiset{p3}{reverse vpath(__vc1,__vc1)}
\vacpol{vc1}{vc1}
% \fmfis{phantom,ptext.clen=6,ptext.hout=3,ptext.oout=12,ptext.out=#4,ptext.sep=;}{p1}
% \fmfis{phantom,ptext.clen=6,ptext.hin=3,ptext.hout=3,ptext.oin=10,ptext.oout=10,ptext.in=#5,ptext.out=#6,ptext.sep=;}{p2}
% \fmfis{phantom,ptext.side=right,ptext.clen=6,ptext.hin=-4,ptext.hout=-4,ptext.oin=10,ptext.oout=10,ptext.in=#7,ptext.out=#8,ptext.sep=;}{p3}
% \fmfis{phantom,ptext.clen=6,ptext.hin=3,ptext.oin=12,ptext.in=#9,ptext.sep=;}{p4}
\end{fmfchar*}}}}%}


% special four-loop self energy diagrams

\newcommand{\csefour}[9][]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-7.5\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(1,0)(1,0){%
\begin{fmfchar*}(20,15)
\fmfleft{v1}
\fmfright{v2}
\fmffixed{(0.66w,0)}{vc1,vc2}
\fmf{#2}{v1,vc1}
\fmf{#3}{vc2,v2}
\fmf{phantom,left=1}{vc1,vc2}
\fmf{phantom,left=1}{vc2,vc1}
\fmffreeze
\fmfposition
\fmfipath{p[]}
\fmfipair{vm[],vl[],vr[]}
\fmfiset{p1}{vpath(__vc1,__vc2)}
\fmfiset{p2}{vpath(__vc2,__vc1)}
\fmfi{#4}{subpath (0,length(p1)/3) of p1}
\fmfi{#5}{subpath (length(p1)/3,2length(p1)/3) of p1}
\fmfi{#6}{subpath (2length(p1)/3,length(p1)) of p1}
\fmfi{#7}{subpath (0,length(p2)/3) of p2}
\fmfi{#8}{subpath (length(p2)/3,2length(p2)/3) of p2}
\fmfi{#9}{subpath (2length(p2)/3,length(p2)) of p2}
\dvertex{vl1}{vr1}{p1}
\dvertex{vr2}{vl2}{p2}
\fmfiset{p3}{vl1--vl2}
\fmfiset{p4}{vr1--vr2}
\svertex{vm1}{p3}
\svertex{vm2}{p4}
{#1}
\end{fmfchar*}}}}%}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% two-loop integrals

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\Itwot}[6][(20,15)]{%
%\smash[b]{%
\fmfframe(1,-1)(1,-1){%
\begin{fmfchar*}#1
  \fmfleft{in}
  \fmfright{out}
\fmf{phantom}{in,v1}
\fmf{phantom}{out,v4}
\fmfforce{(0,0.5h)}{in}
\fmfforce{(w,0.5h)}{out}
\fmffixed{(0,0.75h)}{v3,v2}
\fmfpoly{phantom}{v1,v3,v2}
\fmfpoly{phantom}{v2,v3,v4}
\fmf{plain,label=$\brotate{30}{$\scriptscriptstyle #2$}$,l.side=left,l.dist=4}{v1,v2}
\fmf{plain,label=$\brotate{-30}{$\scriptscriptstyle #5$}$,l.side=right,l.dist=4}{v1,v3}
\fmf{plain,label=$\brotate{90}{$\scriptscriptstyle #6$}$,l.side=right,l.dist=4}{v2,v3}
\fmf{plain,label=$\brotate{30}{$\scriptscriptstyle #4$}$,l.side=right,l.dist=4}{v3,v4}
\fmf{plain,label=$\brotate{-30}{$\scriptscriptstyle #3$}$,l.side=left,l.dist=4}{v2,v4}
\end{fmfchar*}}%}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% four-loop integrals

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\intsixpo}[2][20]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-0\unitlength}%
\raisebox{\eqoff}{%
\fmfframe(0,0)(0,-10){%
\begin{fmfchar*}(#1,20)
  \fmfleft{in}
  \fmfright{out1}
\fmf{phantom}{in,v1}
\fmf{phantom}{out,v4}
\fmfforce{(0,0.5h)}{in}
\fmfforce{(h,0.5h)}{out}
\fmfpoly{phantom}{v1,v6,v5,v4,v3,v2}
\fmffixed{(0.05h,0)}{in,v1}
\fmffixed{(0.9h,0)}{v1,v4}
%\fmf{#2}{v1,v2}
%\fmf{#3}{v2,v3}
%\fmf{plain}{v3,v4}
%\fmf{plain}{v4,v5}
%\fmf{#5}{v5,v6}
%\fmf{#5}{v6,v1}
\fmf{phantom}{vc,v1}
\fmf{phantom}{vc,v2}
\fmf{phantom}{vc,v3}
\fmf{phantom}{vc,v4}
\fmf{phantom}{vc,v5}
\fmf{phantom}{vc,v6}
\fmffreeze
{#2}
\end{fmfchar*}}}
}

\newcommand{\intsixp}[2][20]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-10\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(0,0)(0,0){%
\begin{fmfchar*}(#1,20)
  \fmfleft{in}
  \fmfright{out1}
\fmf{phantom}{in,v1}
\fmf{phantom}{out,v4}
\fmfforce{(0,0.5h)}{in}
\fmfforce{(h,0.5h)}{out}
\fmfpoly{phantom}{v1,v6,v5,v4,v3,v2}
\fmffixed{(0.05h,0)}{in,v1}
\fmffixed{(0.9h,0)}{v1,v4}
%\fmf{#2}{v1,v2}
%\fmf{#3}{v2,v3}
%\fmf{plain}{v3,v4}
%\fmf{plain}{v4,v5}
%\fmf{#5}{v5,v6}
%\fmf{#5}{v6,v1}
\fmf{phantom}{vc,v1}
\fmf{phantom}{vc,v2}
\fmf{phantom}{vc,v3}
\fmf{phantom}{vc,v4}
\fmf{phantom}{vc,v5}
\fmf{phantom}{vc,v6}
\fmffreeze
{#2}
\end{fmfchar*}}}%}
}

\newcommand{\fourlint}[9][]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-10\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(0,0)(0,0){%
\begin{fmfchar*}(20,20)
  \fmfleft{in}
  \fmfright{out1}
\fmf{phantom}{in,v1}
\fmf{phantom}{out,v3}
\fmfforce{(0,0.5h)}{in}
\fmfforce{(w,0.5h)}{out}
\fmfpoly{phantom}{v1,v4,v3,v2}
\fmffixed{(0.9w,0)}{v1,v3}
\fmffixed{(0.05w,0)}{in,v1}
\fmf{#2}{v1,v2}
\fmf{#3}{v2,v3}
\fmf{#4}{v3,v4}
\fmf{#5}{v4,v1}
\fmf{phantom}{vc,v1}
\fmf{phantom}{vc,v3}
\fmffreeze
\fmf{#6}{vc,v1}
\fmf{#7}{vc,v2}
\fmf{#8}{vc,v3}
\fmf{#9}{vc,v4}
{#1}
\end{fmfchar*}}}%}
}

\newcommand{\fivelint}[6][]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-12\unitlength}%
%\smash[b]{%
\raisebox{\eqoff}{%
\fmfframe(-2,-2)(-2,-2){%
\begin{fmfchar*}(30,30)
  \fmfleft{in}
  \fmfright{out}
\fmf{phantom}{in,v1}
\fmf{phantom}{out,v3}
\fmfforce{(0,0.5h)}{in}
\fmfforce{(w,0.5h)}{out}
\fmfpoly{phantom}{v1,v5,v4,v3,v2}
\fmffixed{(.75w,0)}{v1,v4}
\fmf{#2}{v1,v2}
\fmf{#3}{v2,v3}
\fmf{#4}{v3,v4}
\fmf{#5}{v4,v5}
\fmf{#6}{v5,v1}
\fmf{phantom}{vc,v1}
\fmf{phantom}{vc,v2}
\fmf{phantom}{vc,v3}
\fmf{phantom}{vc,v4}
\fmf{phantom}{vc,v5}
\fmffreeze
{#1}
\end{fmfchar*}}}%}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% graphs for triangle rules  
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\IBPtriangle}[7][1]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-6.5\unitlength}%
\raisebox{\eqoff}{%
\fmfframe(1,1)(#1,1){%
\begin{fmfchar*}(20,10)
\fmfleft{vi}
\fmfright{vo1,vo2}
\fmfforce{(0,0.5h)}{vi}
\fmfforce{(w,h)}{vo1}
\fmfforce{(w,0)}{vo2}
\fmffixed{(0,h)}{v3,v2}
\fmffixed{(0.5w,0)}{v2,vo1}
\fmffixed{(0.5w,0)}{v3,vo2}
\fmfpoly{phantom}{v1,v3,v2}
\fmf{#2}{vi,v1}
\fmf{#6}{vo1,v2}
\fmf{#7}{v3,vo2}
\fmffreeze
\fmf{#3}{v1,v2}
\fmf{#4}{v1,v3}
\fmf{#5}{v2,v3}
\end{fmfchar*}}}
}

\newcommand{\IBPvertex}[5][1]{%
\settoheight{\eqoff}{$\times$}%
\setlength{\eqoff}{0.5\eqoff}%
\addtolength{\eqoff}{-6\unitlength}%
\raisebox{\eqoff}{%
\fmfframe(1,1)(#1,1){%
\begin{fmfchar*}(20,10)
\fmfleft{vi}
\fmfright{vo1,vo2}
\fmfforce{(0,0.5h)}{vi}
\fmfforce{(w,h)}{vo1}
\fmfforce{(w,0)}{vo2}
\fmffixed{(0,h)}{v3,v2}
\fmffixed{(0.5w,0)}{v2,vo1}
\fmffixed{(0.5w,0)}{v3,vo2}
\fmfpoly{phantom}{v1,v3,v2}
\fmf{#2}{vi,v1}
\fmffreeze
\fmf{#3}{v1,vc}
\fmf{#4}{vo1,vc}
\fmf{#5}{vc,vo2}
\end{fmfchar*}}}
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\nvml}[3][1]{%
%begingroup;
\fmfcmd{%
begingroup;
save a, vp, tvp, nvp, tv, nv, ip, ts, tt, is, it, n, m, scale, t, r, s, ttpr, tnpr, ep, mm;
path lcirc;
pair vp[][], tvp[][], tv[][], nvp[][], nv[][], ip[][], ts[], is[], tt[], it[], ep[], mid;
% number of path segments
n := #2;
% number of points along path segment
m:=3;
for i=1 upto n:
for j=1 upto m:
a[i][j] := arctime ((j-1)/(m-1)*arclength pm[i]) of pm[i];
vp[i][j] := point a[i][j] of pm[i];
tvp[i][j] := unitvector direction a[i][j] of pm[i];
nvp[i][j] := tvp[i][j] rotated -90;
endfor;
endfor;
% decision if path is closed or open
if(vp[1][1]=vp[n][m]):
% closed path
vp[n+1][1] := vp[1][1];
tvp[0][m] := tvp[n][m];
nvp[0][m] := nvp[n][m];
tvp[n+1][1] :=tvp[1][1];
nvp[n+1][1] :=nvp[1][1];
else:
% open path
vp[n+1][1] := vp[n][m];
tvp[0][m] := (0,0);
nvp[0][m] := (0,0);
tvp[n+1][1] :=tvp[n][m];
nvp[n+1][1] :=nvp[n][m];
fi;
% initialization of number of points in the momentum path
s := 1;
for i=1 upto n:
for j=1 upto m:
% tangent and normal vectors at corners (j=1,m)
if (j=1):
tv[i][1] := (tvp[i-1][m]+tvp[i][1]);
nv[i][1] := (nvp[i-1][m]+nvp[i][1]);
if (not(tv[i][1]=(0,0))):
tv[i][1] := unitvector tv[i][1];
fi;
if (not(nv[i][1]=(0,0))):
nv[i][1] := unitvector nv[i][1];
fi;
% inner products of tangent tangent and tangent normal vectors at corners j=1
ttpr := tvp[i][1] dotprod tvp[i-1][m];
tnpr := tvp[i][1] dotprod nvp[i-1][m];
elseif (j=m):
tv[i][m] := (tvp[i][m]+tvp[i+1][1]);
nv[i][m] := (nvp[i][m]+nvp[i+1][1]);
if (not(tv[i][m]=(0,0))):
tv[i][m] := unitvector tv[i][m];
fi;
if (not(nv[i][m]=(0,0))):
nv[i][m] := unitvector nv[i][m];
fi;
ttpr := tvp[i][m] dotprod tvp[i+1][1];
tnpr := -tvp[i][m] dotprod nvp[i+1][1];
else:
nv[i][j] :=nvp[i][j];
tv[i][j] :=tvp[i][j];
fi;
%if(tvp[i-1][m]=tvp[i][1]):
scale := 25;
%else:
%scale := 150/(sqrt(abs(tvp[i-1][m]-tvp[i][1])));
%fi;
if ((j=1) or (j=m)):
 if ((tnpr<=0) and not((tv[i][j]=(0,0)) or (nv[i][j]=(0,0)))):
  ip[i][j] := vp[i][j] shifted(0.15*scale*nvp[i][j]);
  ts[s] := tvp[i][j];
  is[s] := ip[i][j];
  s:=s+1;
 else:
  if ((j=1) and (ttpr>0)):
%   ip[i][j] := vp[i][j] shifted(0.15*scale*nv[i][j]);
%   ts[s] := tv[i][j];
%   is[s] := ip[i][j];
%   s:=s+1;
  fi;
 fi;
else:
 ip[i][j] := vp[i][j] shifted(0.15*scale*nv[i][j]);
 ts[s] := tv[i][j];
 is[s] := ip[i][j];
 s:=s+1;
fi;
endfor;
endfor;
if(vp[1][1]=vp[n][m]):
% closed path
ts[s] := ts[1];
is[s] := is[1];
else:
% open path
tv[n+1][1] := unitvector (tvp[n][m]+tvp[n+1][1]);
nv[n+1][1] := unitvector (nvp[n][m]+nvp[n+1][1]);
ip[n+1][1] := vp[n+1][1] shifted(0.15*scale*nv[n+1][1]);
ts[s] := tv[n+1][1];
is[s] := ip[n+1][1];
fi;
% tension of the momentum loop
t=#1;
lcirc:=is[1];
%draw(is[1]) withpen pencircle scaled 2;
for k=2 upto s:
lcirc := lcirc{ts[k-1]}..tension t..{ts[k]}is[k];
%lcirc := lcirc{ts[k-1]}...{ts[k]}is[k];
%draw(is[k]) withpen pencircle scaled 2;
endfor;
mm := arctime (0.5* arclength lcirc) of lcirc;
if(vp[1][1]=vp[n][m]):
% closed path
ep1 := point arctime (0* arclength lcirc) of lcirc of lcirc;
ep2 := point mm of lcirc;
mid := 1/2[ep1,ep2];
else:
% open path
ep1 := point mm of lcirc;
ep2 :=unitvector direction mm of lcirc rotated -90;
mid:= ep1 shifted(0.2*scale*ep2);
fi;
%lcirc:=is[1]..tension t..{ts[2]}is[2];
%%lcirc:=is[1]...{ts[2]}is[2];
%for k=3 upto (m-1)*n+1:
%if (floor((k-1)/(m-1))=(k-1)/(m-1)):
%lcirc := lcirc{ts[k-1]}..tension t..is[k];
%%lcirc := lcirc{ts[k-1]}...is[k];
%elseif (floor((k-2)/(m-1))=(k-2)/(m-1)):
%lcirc := lcirc..tension t..{ts[k]}is[k];
%%lcirc := lcirc...{ts[k]}is[k];
%else:
%lcirc := lcirc{ts[k-1]}..tension t..{ts[k]}is[k];
%lcirc := lcirc{ts[k-1]}...{ts[k]}is[k];
%fi;
%endfor;
%lcirc:=it[1]{tt[1]}..tension t..{tt[2]}it[2];
%lcirc:=it[1]{tt[1]}...{tt[2]}it[2];
%for k=2 upto (m-2)*n-1:
%lcirc := lcirc;
%lcirc := lcirc{tt[k]}..tension t..{tt[k+1]}it[k+1];
%lcirc := lcirc{tt[k]}...{tt[k+1]}it[k+1];
%endfor;
%lcirc:=lcirc..tension t..{tt[1]}it[1];
draw(lcirc) withpen pencircle scaled 0.25;
drawarrow(subpath(mm*0.8,mm*1.1) of lcirc) withpen pencircle scaled 0.25;
endgroup;
}
\fmfiv{label=#3,l.dist=0}{mid}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\DeclareMathOperator{\ad}{ad} 
\DeclareMathOperator{\sn}{span}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\str}{str}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\diag}{diag}
%\DeclareMathOperator{\de}{d}
\DeclareMathOperator{\res}{Res}
\DeclareMathOperator{\R}{R}
\DeclareMathOperator{\Sop}{S}
\DeclareMathOperator{\T}{T}
\DeclareMathOperator{\re}{Re}
\DeclareMathOperator{\im}{Im}
\DeclareMathOperator{\vol}{vol}
\DeclareMathOperator{\perm}{P}
\DeclareMathOperator{\dperm}{\mathbf{P}}
\DeclareMathOperator{\depsilon}{\bm{\epsilon}}
\DeclareMathOperator{\ds}{d}
\DeclareMathOperator{\Lam}{\Lambda}
\DeclareMathOperator{\Top}{T}
\DeclareMathOperator{\Kop}{K}
\DeclareMathOperator{\Rop}{R}
\DeclareMathOperator{\Fop}{F}
\DeclareMathOperator{\chiop}{\mathbf{\chi}}
\DeclareMathOperator{\D}{D}
\DeclareMathOperator{\barD}{\vphantom{\D}\smash[t]{\bar{\mathrm{D}}}}
\DeclareMathOperator{\Ld}{L}
\DeclareMathOperator{\spinp}{\Pi}
%
\newcommand{\cone}[4]{%
\chiop(\sideset{_{\smash{#2}}}{_{\smash{#3}}}{\operatorname{\text{$\overset{#4}{#1}$}}})
}
\newcommand{\ctwo}[8]{%
\chiop(\sideset{_{\smash{#3}}}{_{\smash{#4}}}{\operatorname{\text{$\overset{#5}{#1}$}}},
\sideset{_{\smash{#6}}}{_{\smash{#7}}}{\operatorname{\text{$\overset{#8}{#2}$}}})
}
\newcommand{\chiimp}[4][]{%
\smash[t]{#1{\chiop}}_{(#2)}^{(#3)}(#4)
}
%
\newlength{\eqoff}
\newlength{\eqofftwo}
\newlength{\eqoffthree}
\newlength{\eqofffour}
\newlength{\unit}
%\setlength{\unit}{0.0125\textwidth}
\setlength{\unit}{0.5\textwidth}
\psset{xunit=\unit,yunit=\unit,runit=\unit}
\newlength{\linew}
\setlength{\linew}{0.25pt}
\psset{linewidth=\linew}
%

fourloops5.bbl
\providecommand{\href}[2]{#2}\begingroup\raggedright\begin{thebibliography}{10}

\bibitem{Beisert:2010jr}
N.~Beisert {\em et.~al.}, {\it {Review of AdS/CFT Integrability: An Overview}},
   \href{http://xxx.lanl.gov/abs/1012.3982}{{\tt arXiv:1012.3982}}.

\bibitem{Leigh:1995ep}
R.~G. Leigh and M.~J. Strassler, {\it {Exactly marginal operators and duality
  in four-dimensional $\mathcal{N}=1$ supersymmetric gauge theory}},  {\em
  Nucl. Phys.} {\bf B447} (1995) 95--136,
  [\href{http://xxx.lanl.gov/abs/hep-th/9503121}{{\tt hep-th/9503121}}].

\bibitem{Zoubos:2010kh}
K.~Zoubos, {\it {Review of AdS/CFT Integrability, Chapter IV.2: Deformations,
  Orbifolds and Open Boundaries}},
  \href{http://xxx.lanl.gov/abs/1012.3998}{{\tt arXiv:1012.3998}}.

\bibitem{Khoze:2005nd}
V.~V. Khoze, {\it {Amplitudes in the beta-deformed conformal Yang-Mills}},
  {\em JHEP} {\bf 02} (2006) 040,
  [\href{http://xxx.lanl.gov/abs/hep-th/0512194}{{\tt hep-th/0512194}}].

\bibitem{Mauri:2005pa}
A.~Mauri, S.~Penati, A.~Santambrogio, and D.~Zanon, {\it {Exact results in
  planar $\mathcal{N}=1$ superconformal Yang-Mills theory}},  {\em JHEP} {\bf
  11} (2005) 024, [\href{http://xxx.lanl.gov/abs/hep-th/0507282}{{\tt
  hep-th/0507282}}].

\bibitem{Roiban:2003dw}
R.~Roiban, {\it {On spin chains and field theories}},  {\em JHEP} {\bf 09}
  (2004) 023, [\href{http://xxx.lanl.gov/abs/hep-th/0312218}{{\tt
  hep-th/0312218}}].

\bibitem{Berenstein:2004ys}
D.~Berenstein and S.~A. Cherkis, {\it {Deformations of $\mathcal{N}=4$ SYM and
  integrable spin chain models}},  {\em Nucl. Phys.} {\bf B702} (2004) 49--85,
  [\href{http://xxx.lanl.gov/abs/hep-th/0405215}{{\tt hep-th/0405215}}].

\bibitem{Beisert:2005if}
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\end{thebibliography}\endgroup
