We resolve the mixing of the scalar operators of naive dimension $4$ belonging to the representation ${20^{'}}$ of the $SU(4)$ R--symmetry in ${\cal N}=4$ SYM. We compute the order $g^2$ corrections to their anomalous dimensions and show the absence of instantonic contributions thereof. Ratios of the resulting expressions are irrational numbers, even in the large $N$ limit where, however, we observe the expected decoupling of double-trace operators from single-trace ones. We briefly comment on the generalizations of our results required in order to make contact with the double scaling limit of the theory conjectured to be holographically dual to type IIB superstring on a pp-wave.
0205321.tex
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\begin{document}  
 
 
\input FEYNMAN 
 
\thispagestyle{empty} 
\begin{flushright} 
ROM2F/2002/15  
\end{flushright} 
\vspace{1.5cm} 
\begin{center} 
{\LARGE {\bf On Operator Mixing in ${\cal N}=4$ SYM}} \\ 
\vspace{1cm}  
{Massimo Bianchi, Burkhard Eden,  
Giancarlo Rossi and Yassen  S. Stanev$^{\dagger}$} \\  
\vspace{0.6cm}  
{{\it Dipartimento di Fisica, \ Universit{\`a} di Roma \   
``Tor Vergata''}} \\  {{\it I.N.F.N.\ -- \ Sezione di Roma \  
``Tor Vergata''}} \\ {{\it Via della Ricerca  Scientifica, 1}}  
\\ {{\it 00133 \ Roma, \ ITALY}} \\  
\end{center} 
\vspace{1cm} 
 
\begin{abstract} 
We resolve the mixing of the scalar operators of naive dimension $4$  
belonging to the representation ${\bf 20^{'}}$ of the $SU(4)$ R--symmetry 
in ${\cal N}=4$ SYM.  
We compute the order $g^2$ corrections to their anomalous  
dimensions and show the absence of instantonic contributions thereof.  
Ratios of the resulting expressions are irrational numbers, even in the  
large $N$ limit where, however, we observe the expected decoupling  
of double-trace operators from single-trace ones. We briefly comment on the  
generalizations of our results required in order to make contact with  
the double scaling limit of the theory conjectured to  
be holographically dual to type IIB superstring on a pp-wave.  
 
 
\end{abstract} 
\vspace{4cm} 
\noindent 
\rule{6.5cm}{0.4pt}  
 
{\footnotesize ${}^{\dagger}$~On leave of absence from Institute for  
Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences,  
BG-1784, Sofia, Bulgaria} 
\newpage  
 
\setcounter{page}{1} 
 
\section{Introduction and summary} 
 
The holographic correspondence between superstring theory in anti de Sitter  
(AdS) spaces and superconformal theories (CFT)~\cite{magoo,dzfphl,dhf} has   
renewed the interest in ${\cal N}=4$ supersymmetric Yang-Mills theory  
(SYM) and triggered the discovery of new unexpected properties of its  
superconformal phase.  
 
Until very recently, most of the available results  
on the superstring side were confined to the  
low energy (weak curvature $L^{2} >> \alpha^{\prime}$)   
approximation where supergravity takes over. This limit   
corresponds to the strong `t Hooft coupling  ($g^{2}N >> 1$) regime 
of the dual gauge theory which is obviously inaccessible by perturbative  
means. 
Barring few important exceptions, perturbative and non-perturbative  
tests~\cite{mbtest} were thus restricted to protected  
quantities, \ie observables which are actually independent of the  
coupling constant. By now there is quite a long list of such  
protected quantities that  
includes dimensions of operators belonging to short  
multiplets~\cite{twopoint}, certain OPE coefficients of  
chiral primary operators (CPO's)~\cite{threepoint}, extremal~\cite{extr} and  
next-to-extremal~\cite{next} correlators.  
 
The first truly dynamical test of the correspondence  
emerged from the remarkable agreement between SYM instanton effects and  
D-instanton corrections to higher derivative terms in the type IIB  
superstring effective action~\cite{bgkretal}. Bonus  
symmetry~\cite{intri} of up to four-point functions of protected operators  
was another suggestive hint to the underlying type IIB string description  
of ${\cal N}=4$ SYM. Another class of observables that should clearly display   
stringy behaviour are Maldacena-Wilson loops~\cite{mwloop}. So far  
non-trivial string predictions~\cite{gross} have only received  
partial support from perturbation theory~\cite{esz,ps} and seem to  
require a deeper understanding of D-instanton effects in order to  
accommodate SYM instanton corrections~\cite{bgk}.  
Among the other achievements of ${\cal N}=4$ super-instanton calculus  
the `two-line proof'~\cite{mbtest} of 
the partial non-renormalization of the four-point  
function of operators in the supercurrent multiplet~\cite{partial} stands out  
for its simplicity. However, partial non-renormalization is essentially  
a consequence of $SU(2,2|4)$ superconformal symmetry that  
severely constrains the dynamics, though it does  
not completely trivialize it~\cite{partial,dolosb}.  
 
The main purpose of this paper is to resolve the mixing of the scalar 
primary operators of naive dimension $\Delta_{0} = 4$ in the (real) 
representation ${\bf 20^{'}}$ of the $SU(4)$  
R-symmetry group and compute their anomalous dimensions 
at order $g^{2}$. 
 
Relying on previous  
results on four-point functions of lowest CPO's ${\cal Q}$ with~\cite{kon} or  
without~\cite{fourpoint,bkrs1,bkrs2} insertion of the lowest Konishi scalar  
${\cal K}$, we  
disentangle the mixing among the scalar operators in the ${\bf 20^{'}}$  
representation. The vanishing of instanton contributions to the relevant  
four-point functions~\cite{kon} implies the absence of  
non-perturbative corrections to the mixing coefficients and anomalous 
dimensions that we compute.  
Our analysis is further simplified by the observation that  
operators which belong to the Konishi multiplet and whose leading terms are 
generalized 
Yukawa couplings, decouple at the order at which we  
work. The remaining operators show an intricate pattern of mixing at  
finite $N$ that simplifies significantly as $N\rightarrow \infty$.  
In this analysis we exploit the  
vanishing of the anomalous dimension of the operator ${\cal D}_{\bf 20^{'}}$  
that appears in  
OPE of two ${\cal Q}$'s. This property may  
be viewed as resulting from a generalized shortening condition of the  
`linear' type that survives  
interaction as a consequence of certain differential constraints satisfied by  
three point functions involving two protected  
operators~\cite{esok,hh}~\footnote{It should be kept in  
mind, however, that there are operators that satisfy the same  
shortening condition, \ie saturate the same unitarity bound at tree 
level, but violate it after inclusion of radiative  
corrections~\cite{bps1, howe}.}. 
Our results for the one-loop anomalous dimensions definitely exclude the  
possibility~\cite{aft} that ratios of anomalous  
dimensions be rational even in the large $N$ limit.  
This is not in conflict with  
any basic principle but rather suggests that the  
theory behaves in a highly non trivial fashion.  
Still, it is reassuring to find that multi-trace operators that are dual  
to multi-particle bound states  
have in the limit $N\rightarrow\infty$ with $g^2 N$ fixed 
anomalous dimensions that are given by the sums of the anomalous  
dimensions of their constituents.  
This suggests that the dual bound states are at threshold.  
Moreover, as observed in~\cite{kon}, the absence of non-perturbative  
instanton corrections 
for the anomalous dimensions and OPE coefficients of the 
operators which we study is in line with S-duality that maps operators 
dual to string excitations into operators dual to dyonic string excitations. 
 
The pattern of intricate mixings and irrational anomalous dimensions  
that we find, may not necessarily prove to be an  
unsurmountable obstacle towards the extrapolation of the string  
spectrum and interactions from low energy (strong coupling) to  
large curvature (weak coupling) at least in the Penrose limit of  
$AdS_{5}\times S^{5}$ which gives rise to a maximally supersymmetric  
pp-wave~\cite{ppwaves,bmn}. String loop corrections, which seem to be  
calculable in the pp-wave background~\cite{danetal}, may play a crucial role  
in quantitatively establishing this  
correspondence~\cite{plefkaetal,grossetal}. We will argue  
that our results can be generalized to yield further insight into the  
properties of the set of operators dual to the low-lying string excitations. 
 
The plan of the paper is as follows:  
After recalling some basic definitions and establishing our notation  
in Section 2, in Section 3 we briefly describe unitary irreducible 
representations (UIR's) of the superconformal group $SU(2,2|4)$  
and discuss the emergence of multiplet shortening.  
In Section 4 we identify the scalar composite operators of naive dimension  
$\Delta_{0} =4$ belonging to  
the representation ${\bf 20^{'}}$. In Section 5 we  
perform to order $g^2$ the orthogonalization  
of the two-point functions and compute the anomalous dimensions of  
these operators. In Sections 6 and 7 we  
follow a different route to the same results that requires the computation  
of the four-point function of two ${\cal Q}$'s and two ${\cal K}$'s  
at order $g^{4}$, thus extending similar results previously obtained  
at order $g^{2}$~\cite{kon}. In Section 8 we briefly comment on  
possible generalizations of our results to  
the double scaling limit of the theory which is conjectured to  
be holographically dual to type IIB superstring on a pp-wave~\cite{ppwaves}. 
In the Appendix we gather unwieldy formulae. 
 
\section{Notation and conventions} 
 
In this section, we summarize our notations and conventions and we 
recall some relevant results of~\cite{kon} concerning the structure  
and the renormalization properties of the unprotected Konishi supermultiplet. 
 
The field content of ${\cal N}=4$ SYM~\cite{n4sym} comprises a 
vector, $A_{\mu}$, four Weyl spinors, $\psi^{A}$ ($A=1,2,3,4$), and 
six real scalars, $\varphi^{i}$ ($i=1,2,\ldots,6$), all in the adjoint 
representation of the gauge group, that we take to be $SU(N)$ for  
definiteness. In the ${\cal N}=1$ approach 
that we shall follow the fundamental fields can be arranged into a 
vector superfield, $V$, and three chiral superfields, $\Phi^{I}$ ($I=1,2,3$). 
The six real scalars, $\varphi^{i}$, are combined into three complex 
fields, $\phi^{I}= (\varphi^{I} + i  
\varphi^{I+3})/\sqrt{2}$ and $\phi^{\dagger}_{I} = (\varphi^{I} - i  
\varphi^{I+3})/\sqrt{2}$ that are the lowest 
components of the chiral and antichiral superfields, $\Phi^{I}$ and 
$\Phi^{\dagger}_{I}$, respectively. Three of the Weyl fermions, $\psi^{I}$,  
are the spinors of the chiral multiplets. The fourth spinor,  
$\lambda = \psi^{4}$, together with the 
vector, $A_{\mu}$, form the vector multiplet. In this way only an 
$SU(3)\otimes U(1)$ subgroup of the full $SU(4)$ R-symmetry is manifest. 
 
The complete ${\cal N}=4$ SYM action in the ${\cal N}=1$ superfield 
formulation has a non-polynomial form, as we do not work in the  
Wess--Zumino gauge. A gauge fixing term must anyway be  added to the  
classical action. We shall use the Fermi-Feynman 
gauge, as it makes corrections to the propagators of the fundamental 
superfields vanish at order $g^{2}$~\cite{finite,zanon,kov}. Actually a  
stronger result has been proved in these papers, namely the vanishing of  
the anomalous dimensions of the fundamental fields up to O($g^4$). With  
the Fermi-Feynman gauge choice the terms relevant for the calculation of  
the Green functions we are interested in are 
\begin{eqnarray} 
S &=& \int d^{4}x\:d^{2}\theta d^{2}\bar\theta\, 
\left\{\rule{0pt}{18pt} 
V^{a} \Box  V_{a}  - \Phi^{a\dagger}_{I}\Phi^{I}_{a} - 2 ig 
f_{abc}{\Phi^{\dagger}}^{a}_{I} V^{b}\Phi^{Ic} + 2 g^{2} f_{abe} 
f_{ecd}{\Phi^{\dagger}}^{a}_{I} 
V^{b} V^{c} \Phi^{Id}  \right.  \nonumber \\
&& \left. - \frac{ig \sqrt{2}}{3!} f^{abc} \left[ 
\varepsilon_{IJK} 
\Phi_{a}^{I} \Phi_{b}^{J} \Phi_{c}^{K} \delta^{(2)}({\overline \theta}) 
- \varepsilon^{IJK} \Phi^{\dagger}_{aI}\Phi^{\dagger}_{bJ} 
\Phi^{\dagger}_{cK}\delta^{(2)}(\theta) \right] + 
\ldots \right\} \, , 
\label{actionsuper} 
\end{eqnarray} 
where $f_{abc}$ are the structure constants of the gauge group. As neither 
the cubic and quartic vector interactions nor the ghost terms will contribute 
to the calculations we will present in this paper, we have omitted them 
in eq.~(\ref{actionsuper}). 
 
Since all superfields are massless, their propagators have an equally 
simple form in momentum and in coordinate space and thus we choose to 
work in the latter which is more suitable for the study of 
conformal field theories. In Euclidean coordinate space one finds 
\be 
\langle \Phi^{\dagger}_{Ia} (x_i,\theta_i, \bar \theta_i) 
\Phi^J_b (x_j,\theta_j, \bar \theta_j) \rangle = 
 {{\delta_{I}}^{J} \delta_{ab} \over 4 \pi^2} 
{\rm e}^{\left( \xi_{ii} +\xi_{jj} -2 \xi_{ji} \right) \cdot \partial_j} 
{1\over x_{ij}^2} \, , 
\label{propfi} 
\ee 
\be 
\langle V_a (x_i,\theta_i, \bar \theta_i) 
V_b (x_j,\theta_j, \bar \theta_j) \rangle = 
- {\delta_{ab} \over 8 \pi^2} 
{\delta^{(2)}(\theta_{ij}) \delta^{(2)}(\bar \theta_{ij}) \over x_{ij}^2} \, , 
\label{propv} 
\ee 
where 
$x_{ij} = x_{i}-x_{j}$, $\theta_{ij}= \theta_{i}- \theta_{j}$, 
 $\xi^{\mu}_{ij}= \theta_i^{\alpha} \sigma^{\mu}_{\alpha {\dot 
\alpha}} \bar \theta_j^{\dot \alpha}$. 
 
The simplest protected (dimension two) CPO's  
\be 
{\cal{Q}}^{(ij)}_{\bf 20^{'}} = 
\tr \left(\varphi^i \varphi^j -{\delta^{ij} \over 6} 
\sum_k \varphi^k \varphi^k \right)\, , 
\label{defQ} 
\ee 
belong to the representation ${\bf 20^{'}}$ of $SU(4)$ and 
are the lowest component of the ${\cal N}=4$ supercurrent multiplet. 
 
In terms of $SU(3)\otimes U(1)$ the ${\cal Q}_{\bf 20^{'}}^{(ij)}$'s 
decompose in 
\be 
{\cal{C}}^{IJ}(x)=\tr(\phi^I(x)\phi^J(x)) \, , 
\qquad {\cal{C}}^{\dagger}_{IJ} (x) = 
\tr(\phi^{\dagger}_I (x) \phi^{\dagger}_J (x)) 
\ee 
and 
\be 
{\cal{V}}^I_J = \tr\left({\rm{e}}^{-2gc(x)} 
\phi^{\dagger}_J(x){\rm{e}}^{2gc(x)}\phi^I(x)\right)- 
{\delta^I_J\over 3}\tr\left({\rm{e}}^{-2gc(x)} \phi^{\dagger}_L(x) 
{\rm{e}}^{2gc(x)} \phi^L(x)\right) \, , 
\ee 
where the exponentials ($c(x)$ is the lowest component of the vector  
superfield) are included to ensure gauge invariance and 
regularization of the operators is understood, \eg by point-splitting  
(see below). Note that no  
normal-ordering is needed because the {\it vev}'s of all the above operators  
vanish, none of them being an $SU(4)$ singlet. 
 
The ${\cN} = 4$ Konishi multiplet is a long multiplet of  
$SU(2,2|4)$~\cite{af}. Its lowest component, $\cK$, is a scalar operator of 
(naive) conformal dimension $\Delta_0=2$, which is a singlet of the $SU(4)$  
R-symmetry group. The (naive) definition of $\cK$ is 
\be 
\cK(x)\Big{|}_{\rm naive}  = {1\over 2} 
\sum_{i=1}^{6} : \tr(\varphi^{i}(x)\varphi^{i}(x)):\, , 
\label{K1naivest} 
\ee 
where the trace is over colour indices and the symbol $::$ stands for  
normal ordering. As usual, normal ordering means subtracting the operator  
{\it vev} or, in other words, requiring $\langle \cK\rangle=0$. 
In terms of ${\cN} = 1$ superfields $\cK$ can be written in the form 
\be 
{\cK}(x) \Big{|}_{\rm formal}  = \sum_{I=1}^{3} : \tr({\rm{e}}^{-2gc(x)} 
\phi^{\dagger}_I(x) {\rm{e}}^{2gc(x)} \phi^I(x)):\, . 
\label{K1naive2} 
\ee 
Since the operator ${\cK}$ has an anomalous dimension, 
$\gamma^{\cal K}(g^2)$,  
it will suffer a non-trivial renormalization. We assume (as is always the 
case in perturbation theory) that $\gamma^{\cal K}(g^2)$ is small and  
represented by the series expansion 
\be  
\gamma^{\cal K}(g^2)= 
g^2\gamma^{\cal K}_1+g^4\gamma^{\cal K}_2+\ldots  \label{gammak} \, .  
\ee 
{}From the results of refs.~\cite{ans} and~\cite{bkrs1,bkrs2}, one gets  
for the first two coefficients of the expansion  
\ba 
&&\gamma^{\cal K}_1=\frac{3N}{4\pi^2}\label{gammak11}\\ 
&&\gamma^{\cal K}_2=-\frac{3N^{2}}{16\pi^4}\label{gammak12}\, . 
\ea 
It is convenient to regularize operators by point splitting.  
In particular for $\cK$ we write 
\be 
{\cK}(x)\Big{|}_{\rm reg}   =   
a^{\cal K}(g^2)\sum_{I=1}^{3}:\tr({\rm{e}}^{-2gc(x)} 
\phi^{\dagger}_I(x+{\epsilon \over 
2}){\rm{e}}^{2gc(x)}\phi^I(x-{\epsilon \over 2})):\, , 
\label{K1ansatz}  
\ee  
where $\epsilon$ is an infinitesimal, but otherwise arbitrary,  
four-vector. Note that, due to our choice of gauge-fixing, there is no need 
to ``point-split'' the vector field in the exponents,  
because the $c$-field has vanishing propagator. Finally, the renormalized  
operator has the form 
\be 
{\cK}(x)\Big{|}_{\rm ren}   =  \lim_{\epsilon \rightarrow 0} 
{a^{\cal K}(g^2) \over (\epsilon^2)^{{1 \over 
2}\gamma^{\cal K}(g^2)}}\sum_{I=1}^{3}:\tr \left( {\rm{e}}^{-2gc(x)} 
\phi^{\dagger}_I(x+{\epsilon \over 2}){\rm{e}}^{2gc(x)} 
\phi^I(x-{\epsilon \over 2}) \right):\, , 
\label{K1ren} \ee  
where $a^{\cal K}(g^2)$ is a normalization factor that we  
choose of the form $a^{\cal K}(g^2)=1+g^2 a^{\cal K}_1+g^4 a^{\cal K}_2  
+\ldots $. Unlike  
the operators corresponding to symmetry generators (like the R-symmetry  
currents or the stress-energy tensor), the Konishi scalar $\cK$ has  
no intrinsic normalization, so we shall use this freedom   
in the normalization factor $a^{\cal K}(g^2)$ of eq.~(\ref{K1ren}), to make  
the two-point function of $\cK$ depend on $g^2$ only through $\gamma^{\cal K}$. 
This is achieved at the order we work by setting $a^{\cal K}_1 =  
{3 N /8 \pi^2 }$  
(other coefficients would require higher order computations to be fixed).  
With this choice one gets 
\be \langle {\cK}(x_1){\cK}(x_2)\rangle = {3(N^2-1) \over 4 (4 \pi^2)^2} 
{1 \over({x_{12}}^2)^{2+\gamma^{\cal K}(g^2)}} \, .  
\label{2pK1} 
\ee 
 
\section{Comments on the UIR's of $SU(2,2|4)$} 
 
The unitary irreducible representations (UIR's) of $SU(2,2|4)$ have been  
classified 
in~\cite{bps2}~\footnote{In (perturbative) ${\cal N} =4$ SYM theory only  
UIR's of $PSU(2,2|4)$ are actually relevant. They are characterized  
by the vanishing of the $U(1)_C$ central charge that extends $PSU(2,2|4)$ 
to $SU(2,2|4)$.}.  
A general UIR is specified by a set of quantum 
numbers comprising the dilation weight $\Delta$, the Lorentz spins 
$(j_1,j_2)$ and the Dynkin labels $[k,l,m]$  
of the $SU(4)$ $R$-symmetry. There are three ``unitary 
series'', which are distinguished by different relations between the dilation 
weight and the other quantum numbers. 
 
It has been known for some time that generic UIR's can be obtained by 
tensoring the so-called ``singleton''  
representations~\cite{FlaFro}~\footnote{Alternatively the UIR's  
may be built by using the 
oscillator method (see~\cite{Gunaydin4d} and references therein) or they can 
be realized as ``analytic tensor fields'' on analytic superspace~\cite{hh}.}. 
In~\cite{bps3} this has been elaborated in full detail using the 
technique of harmonic superspace~\cite{EmeryMin}. Within this approach,  
in addition to the 
usual ${\cal N}=4$ super Minkowski space one introduces $4\times 4$ matrices, 
$u^A_r$, parameterizing the coset $SU(4)/U(1)^3$.  
We omit the details of the construction, but rather remark 
that these matrices should be contracted on all free $SU(4)$ indices in the 
constraints defining the ``Grassmann analytic''  
superfields~\cite{SohniusSohniusSiegelHST}  
\begin{equation} 
W^{[1\ldots k]} \,, \qquad 1\leq k\leq 3 
\end{equation} 
which read 
\begin{eqnarray} 
& D^A_\alpha W^{[1\ldots k]} = 0 \,, & \qquad 1\leq A \leq k \, , \label{Wcon}  
\\ 
& \bar D_A^{\dot\alpha}  W^{[1\ldots k]} = 0 \,, & \qquad k+1\leq A \leq 4 
\nonumber \, .  
\end{eqnarray} 
In eq. (\ref{Wcon}) $D^A_\alpha = u^A_r D^r_\alpha$ are projected ${\cal N}=4$ 
supercovariant derivatives. The constraints 
express the fact that the $W$ superfields depend on half of the spinor  
coordinates, 
\ie they are 1/2 BPS objects. 
 
Besides the $W$'s, the list of singletons additionally includes  
${\cal N} =4$ chiral superfields, 
which - unlike the former - may have either left or right 
handed spinor indices, but cannot carry a non-trivial 
$SU(4)$ representation. For the present purpose we only need to  
introduce the scalar chiral 
superfield, $\Psi$, which satisfies  
\begin{eqnarray} 
\bar D_A^{\dot\alpha} \Psi & = & 0 \, ,\label{Psicon} \\   
D^{\alpha (A} D^{B)}_\alpha \Psi & = & 0 \, , \nonumber 
\end{eqnarray} 
where the second (`linear') constraint is a sort of `field equation'.  
In the tensoring procedure 
it is assumed that the chiral superfield is `on shell'. 
 
The statement that any UIR can be obtained as a product of singletons is 
formal in that none of the multiplets $W^1, W^{[123]}$ and $\Psi$ can be  
expressed in terms of elementary fields. On the contrary,  
$W^{[12]}$ is the fundamental ${\cal N}=4$ SYM 
multiplet and its square is the supercurrent multiplet. 
 
An important observation is that the component field 
content of the Konishi multiplet in the non-interacting theory 
is correctly reproduced by 
the product of a $\Psi$ field satisfying~(\ref{Psicon}) with its complex 
conjugate:  
\begin{equation} 
{\cK}|_{g^0} = \Psi \bar\Psi 
\end{equation} 
Using~(\ref{Psicon}), one may verify that~\footnote{In the interacting  
theory the situation is more complicate, as the r.h.s. of the equations 
in (\ref{Kcon}) do not vanish~\cite{howe}. For instance, the first becomes 
$D^{\alpha (A} D^{B)}_\alpha {\cK} \propto 
g \tr([W^{AC},W^{BD}] \, \bar{W}_{CD})$. These considerations are at 
the basis of the derivation of the Konishi anomaly~\cite{konishi}.}
\begin{equation} 
D^{\alpha (A} D^{B)}_\alpha {\cK}|_{g^0} = 0 \, ,\qquad \qquad 
\bar D_{\dot\alpha (A} \bar D_{B)}^{\dot\alpha} {\cK}|_{g^0}  = 0 \, .  
\label{Kcon} 
\end{equation} 
By acting with the products of $D^{\alpha (A} D^{B)}_\alpha$ and  
$\bar D_{\dot\alpha (A} \bar D_{B)}^{\dot\alpha}$ on  
${\cK}|_{g^0}$ and performing some $D$-algebra, 
one realizes that both 
the singlet and the \textbf{15} components of the current 
\begin{equation} 
K^A_{\mu \, B} \, = \,  
\bar\sigma^{\dot\alpha\alpha}_\mu 
[ D^A_\alpha, \bar D_{\dot\alpha \, B} ] \, {\cK}|_{g^0, \, \theta, 
\bar\theta = 0}  
\end{equation} 
are conserved. There are similar constraints on some of the higher 
components. Also, each of the linear constraints~(\ref{Kcon}) separately 
implies the absence of some component fields in the supermultiplet. 
 
In the interacting theory the Konishi multiplet has  
an anomalous dimension, so we may formally 
write 
\begin{equation} 
{\cK} = (\Psi \bar\Psi)^{(1+\gamma)} \, . 
\end{equation} 
For $\gamma \neq 0$ it does not satisfy any differential constraint,  
consistently with it being a long multiplet. 
 
Next, we focus on operators of naive dimension 4 in the 
${\bf 20^{'}}$. Operators in this representation can be either  
single or double trace composites of the  
fields in the fundamental ${\cal N}=4$ SYM multiplet. 
Independently of their actual expression, the formalism of~\cite{bps3} 
allows to determine the 
component field content multiplets of this kind by representing it in the form 
\begin{equation} 
{\cal O}_{\bf {20^{'}}} =  (\Psi \bar\Psi)^{(1+\gamma)} (W^{[12]})^2 \, . 
\label{masterequation} 
\end{equation} 
Eq.~(\ref{masterequation}) shows that  
it has the same field content as a product of the Konishi 
and supercurrent multiplets. 
 
If and only if $\gamma=0$, the operator satisfies differential constraints 
which are in the intersection of the conditions~(\ref{Wcon})  
and~(\ref{Kcon}), \ie 
\begin{eqnarray} 
D^{\alpha A} D^B_\alpha {\cal O}_{\bf {20^{'}}} & = & 0 \,, \qquad 1 \leq A,B 
\leq 2 \, , \label{Ocon} \\ 
\bar D_{\dot\alpha A} \bar D_B^{\dot\alpha} {\cal O}_{\bf {20^{'}}} &=& 0 \, , 
\qquad 3 \leq A,B \leq 4 \,. \nonumber 
\end{eqnarray} 
Note that the derivatives $D^1,D^2,\bar D_3,\bar D_4$ mutually anticommute, 
so that we can never derive a condition containing a space time derivative from 
them. Hence, even in the non-interacting case, such multiplets do not 
contain conserved tensor currents. The constraints~(\ref{Ocon}) merely 
express the absence of some component fields. 
 
If the conditions~(\ref{Ocon}) happen to be enforced by some mechanism, 
\eg because the field occurs in the OPE of two supercurrent  
multiplets~\cite{esok}, then the operator will have protected dimension. 
This is exactly what happens for the double-trace operator  
${\cal D}_{\bf {20^{'}}}$. Its explicit expression, obtained 
in~\cite{kon}, is given for convenience in eq.~(\ref{defDQ}). 
It may be argued that the reason why this operator is protected 
is to be ascribed to the fact  
that it obeys constraints not involving SYM covariant  
superderivatives~\cite{howe}. 
 
\section{Scalar operators in the ${\bf 20^{'}}$ representation} 
 
For a sufficiently large number of colours, $N \geq 4$, there are 6  
distinct scalar primary operators of naive scale dimension $\Delta_0=4$  
in the real  
representation ${\bf 20^{'}}$ of $SU(4)$. Two of them  
${\cal K}_{{\bf 20}^{'}}^{\pm}$ belong to the Konishi multiplet. At leading  
order  
in $g$ they correspond to generalized Yukawa couplings 
\be 
{\cal K}^{ij}_{{\bf 20'}} = t^{(i}_{AB} 
 Tr( \varphi^{j)} [\lambda^{A},\lambda^{B}]) + \ldots \, , 
\ee 
which are  
bilinear in the fermions. Hence they do not contribute  
at tree level and, in general, to the leading logarithms,  
$g^{2n}(\log x_{12}^2)^{n}$,  
 in the correlation functions that we shall analyze. 
The other four operators are quartic in the fundamental scalars (at  
leading order in $g$) and naively can be written in the  
form~\footnote{We shall omit the  
${\bf {20}^{'}}$ label from now on and choose to deal with  
one of the 20 components of the multiplet that we take to be the one  
belonging to the representation ${\bf 6}_{+2}$ of $SU(3)\otimes U(1)$  
in the decomposition ${\bf{20}^{'}} \rightarrow {\bf 8}_{0}\oplus  
{\bf 6}_{+2}\oplus{\bf 6}^{*}_{-2}$, namely ${\cal O}^{11}$.}  
\be 
{\cal O}_{\ell}^{11} (x) \vert_{\rm naive} = \sum_{a,b,c,d} 
: \phi^{1}_a(x) \phi^{1}_b(x)   
\sum_{I=1}^{3} \left( \phi^{I}_c(x) \phi^{\dagger}_{I,d}(x) +  
\phi^{I}_d(x) \phi^{\dagger}_{I,c}(x) 
\right): X_{\ell}^{abcd} \quad  , 
\label{O_naive} 
\ee 
where the index $\ell=1,2,3,4$ labels the four different operators  
characterized  
by the four colour tensors $X_{\ell}^{abcd}$, given below  
(eqs.~(\ref{X_D}) to~(\ref{X_L})).  
 
To properly define renormalized operators one has to make three  
modifications to the naive formula~(\ref{O_naive}). First, in order  
to ensure gauge invariance the fundamental scalars have to be replaced by   
\ba 
&& \phi^{\dagger}_{I}(x)  \rightarrow {\rm e}^{-gc(x)}  
\phi^{\dagger}_{I}(x) {\rm e}^{gc(x)} \, , \nonumber \\ 
&& \phi^{I}(x) \rightarrow {\rm e}^{gc(x)} \phi^{I}(x) {\rm  
e}^{-gc(x)} \, ,  
\label{O_gauge} 
\ea 
respectively. Second, one has to regularize the operator.  
We choose to do this by point splitting, separating the arguments  
of the four scalars in~(\ref{O_naive}) by a small distance. There are  
several, essentially equivalent, ways to do this,  
the most compact one is to position the four scalars at the vertices of a  
tetrahedron so that all the separations have equal length which we shall  
denote by $\epsilon$. As remarked after eq.(\ref{K1ansatz}), there is  
no need to point split the different gauge exponents in~(\ref{O_gauge}).  
Still one has to separate their arguments from the arguments of the  
fundamental scalars. We decided to put them all at the center of the 
tetrahedron. Finally, as in the case of ${\cal K}$,  
operators that acquire anomalous  
dimension have to be renormalized. Within the regularization  
prescription we chose this amounts to writing  
\be 
{\cal O}_{\ell}^{11} (x) \vert_{\rm ren} =  
\lim_{\epsilon \rightarrow 0} 
{a^{\ell}(g^2) \over {(\epsilon^2)^{{1 \over 2} \gamma^{O_{\ell}}(g^2)}}}  
\left( {\cal O}_{\ell}^{11} (x) \vert_{\rm reg} + \dots \right)\, , 
\label{O_renorm} 
\ee 
where dots in the r.h.s. denote possible subleading (in $g$) mixings 
which will not contribute at the order we work. Here 
$\gamma^{O_{\ell}}(g^2)$ is the anomalous dimension of the  
operator $O_{\ell}$ for which as before we assume the power series expansion  
\be 
\gamma^{O_{\ell}}(g^2) = g^2 \gamma^{O_{\ell}}_1 +g^4 \gamma^{O_{\ell}}_2  
+ \dots \, , 
\label{gammaexp} 
\ee 
while  $a^{\ell}(g^2)$ is a finite renormalization of the form 
\be 
a^{\ell}(g^2) =1+ g^2 a^{\ell}_{1} +g^4 a^{\ell}_{2} + \dots \, , 
\label{aexp} 
\ee 
which depends on the regularization prescription.  
 
Let us now list the four (for $N \geq 4$) colour tensors   
$X_{\ell}^{abcd}$.  
The protected operator, ${\cal D}$, is double trace  
and corresponds to the tensor   
\be 
X_{{\cal D}}^{abcd} = {1 \over 2} \delta^{ac}\delta^{bd}- 
{1 \over 6} \delta^{ab}\delta^{cd} \, . 
\label{X_D} 
\ee 
Note that ${\cal D}$ can be expressed in terms of only the protected 
$\Delta =2$  CPO's  
${\cal{Q}}^{(ij)}$ of eq.~(\ref{defQ}) as 
\be 
{\cal D}^{(ij)} = \sum_k {\cal{Q}}^{(ik)}{\cal{Q}}^{(jk)} -  
{\delta^{ij} \over 6 }\sum_{k,\ell}{\cal{Q}}^{(k\ell)}{\cal{Q}}^{(k\ell)} \ . 
\label{defDQ} 
\ee  
We have to choose an appropriate basis for the remaining three operators,  
which we denote  
by ${\cal M}$, ${\cal P}$ and ${\cal L}$. This choice is at this point  
purely conventional, since, as we shall demonstrate, the correct  
operators, \ie~those that have well defined (anomalous) dimensions,  
will turn out to be complicated linear combinations of the former.  
We denote by ${\cal M}$ the double trace operator corresponding  
to the colour tensor 
\be 
X_{{\cal M}}^{abcd} = {1 \over 2} \delta^{ab}\delta^{cd} \, , 
\label{X_M} 
\ee 
Note that ${\cal M}$ is the product of the lowest   
chiral primary operator  
${\cal{Q}}^{(ij)}$  of eq.~(\ref{defQ}) 
and the lowest operator in the Konishi supermultiplet, $\cK$,  
given in eq.~(\ref{K1naivest}), \ie 
\be 
{\cal M}^{(ij)} = {\cal{Q}}^{(ij)} \cK \ . 
\label{defMQK} 
\ee  
We denote by  ${\cal P}$ the single trace operator corresponding to 
\be 
X_{{\cal P}}^{abcd} = -{1 \over 2} \sum_e f_{ace}f_{bde} \, . 
\label{X_P} 
\ee 
Note that for  
$SU(2)$ ${\cal P}$ is a linear combination of ${\cal D}$ and ${\cal M}$. 
The fourth operator, ${\cal L}$, is constructed by saturating colour  
indices with the (traceless) quartic Casimir operator of $SU(N)$  
\ba 
X_{{\cal L}}^{abcd} &=&   {\rm Tr}(T^a T^b T^c T^d) + {\rm permutations  
\ of} \ b,c,d \nonumber  \\ 
 &-& {1 \over 2}{2N^2-3 \over N(N^2+1)} \left( \delta^{ab}  
\delta^{cd} + \delta^{ac} \delta^{bd}+ \delta^{ad} \delta^{bc} \right)\, ,  
\label{X_L} 
\ea 
where $T^a$ are the generators of $SU(N)$ in the fundamental  
representation. $X_{{\cal L}}$ vanishes both for $SU(2)$ and $SU(3)$. 
 
{}From this general discussion it is clear that one  
has to treat separately the two low rank cases of $SU(2)$ and $SU(3)$,  
since the number of independent operator varies with $N$. 
 
\section{Orthogonalization of the scalar operators} 
 
We shall consider only tree-level and order $g^2$  
constraints, since higher loop quantities will depend also on  
terms omitted in equation~(\ref{O_renorm}). The correctly renormalized 
operators (\ie~those having well defined  
anomalous dimensions) must satisfy the following three orthogonality  
properties. 
 
1. They have to be orthogonal to (\ie~they must have vanishing two-point  
functions with) the two operators belonging to the Konishi multiplet,  
${\cal K}_{{\bf{20}^{'}}}^{\pm}$. This is automatic both at tree  
level and for the leading logarithms at order $g^2$, because  
${\cal K}_{{\bf{20}^{'}}}^{\pm}$ are bilinear in the fermions at leading order. 
 
2. They have to be orthogonal to the protected operator ${\cal D}$.  
An explicit calculation shows that both at tree level and at order $g^2$  
the operator ${\cal L}$ already enjoys this property, while for the other  
two one needs to introduce the definitions  
\ba 
\widehat {\cal M}  &=& {\cal M} - {6 \over 3 N^2-2 } {\cal D} \\ 
\widehat {\cal P}  &=& {\cal P} + {2 \over 3}{N \over N^2-2 } {\cal  
M}- {N \over N^2-2 } {\cal D}\, . 
\label{MPhat} 
\ea 
Indeed any linear combination of $\widehat {\cal M}$ and $\widehat  
{\cal P}$ is orthogonal (in the sense explained above) to ${\cal D}$.  
Our choice is such that they are mutually orthogonal  
at tree level for any $N$ and the operator $\widehat {\cal P}$ vanishes  
for $SU(2)$. Thus for $SU(2)$ the only relevant operator is  
$\widehat {\cal M}$. {}From the presence of logarithmic terms at short  
distances in the expansion of the four-point function  
$\langle {\cal C}^{11}(x_1) {\cal C}_{11}^{\dagger}(x_2) {\cal K}_1(x_3)  
{\cal K}_1(x_4) \rangle$  at order $g^2$ it follows that $\widehat {\cal M}$  
has a non-vanishing one-loop anomalous dimension.  
The precise value was computed in~\cite{kon} with the result 
\be 
\gamma_1^{\widehat {\cal M}}=5\times{N g^2\over 4\pi^2} \ \, , \qquad N=2 \, .  
\label{gmh1} 
\ee 
 
3. Finally they have to be mutually orthogonal both at tree level and  
at one loop. In other words we have to define new operators 
\ba 
{\cal O}_1 &=& \alpha_{1 {\cal M}} \widehat {\cal M}  
+\alpha_{1 {\cal P}} \widehat {\cal P} 
+\alpha_{1 {\cal L}}  {\cal L} \nonumber \\ 
{\cal O}_2 &=& \alpha_{2 {\cal M}} \widehat {\cal M}  
+\alpha_{2 {\cal P}} \widehat {\cal P} 
+\alpha_{2 {\cal L}}  {\cal L}  \label{theops} \\ 
{\cal O}_3 &=& \alpha_{3 {\cal M}} \widehat {\cal M}  
+\alpha_{3 {\cal P}} \widehat {\cal P} 
+\alpha_{3 {\cal L}}  {\cal L} \nonumber  
\ea 
and require   
\be 
\langle {\cal O}_i(x) {\cal O}_j(y) \rangle =  0 \quad {\rm if} \ i \neq j 
\label{ort} 
\ee 
both at tree level and at order $g^2$. 
 
It follows from~(\ref{O_renorm}) that  
the order $g^2$ correction to the anomalous dimension of the operator  
${\cal O}_i$ is equal to the ratio of the coefficient of the 
$\ln (x-y)^2$ term 
at order $g^2$ and the tree level normalization of the two-point  
function of ${\cal O}_i$, \ie 
\be 
\gamma_1^{{\cal O}_i} =   
-{\langle {\cal O}_i(x) {\cal O}_i(y) \rangle \vert_{g^2, \ln (x-y)^2} \over  
\langle {\cal O}_i(x) {\cal O}_i(y) \rangle \vert_{0} } \ . 
\label{anomR} 
\ee 
Hence it suffices to solve the orthogonality relations~(\ref{ort}) 
to get the explicit expressions of the ${\cal O}_i$'s anomalous dimensions. 
 
Before proceeding to this rather long calculation  
let us make two comments. First, we notice that the order $g^2$ contributions 
to these  
two-point functions come from two different types of diagrams,  
corresponding to either chiral or vector internal lines. 
Both types of diagrams can be expressed in terms of the massless box integral  
(see the Appendix). At any order in $g^2$  
each orthogonality relation~(\ref{ort}) leads to one equation 
for the coefficients $\alpha$ in~(\ref{theops}).  
These equations ensure the cancellation of all $\ln(\epsilon)$-singularities.  
At this point, 
by a suitable choice of the coefficients $a_{1}^{\ell}$ in~(\ref{aexp}) one can 
also cancel the finite corrections at the same order.  
Second, since the overall normalization of the operators  
${\cal O}_i$ is arbitrary, 3 out of the 9 parameters in~(\ref{theops}) can be 
chosen without loss of generality to be equal to 1. Thus the orthogonality  
constraint leads to 6 quadratic equations for 6 variables. A careful analysis 
shows that up to permutation symmetry there is only one acceptable  
(\ie real) solution.  
 
The two-point function of the protected operator ${\cal D}$ is given by its  
tree-level value 
\be 
\langle {\cal D} {\cal D} \rangle \vert_0 \, = \,  
\langle {\cal D}(x_1) {\cal D}(x_2)\rangle \, = \,   
{10 \over 9} (N^2-1) (3N^2-2) {\cal I}_0(x_{12}) \label{dd}  \, , 
\ee 
where 
\be 
{\cal I}_0(x_{12}) = 
{1 \over (4 \pi^2)^4 (x_{12}^2)^4} \, . 
\ee 
For the other operators at tree level we find 
\ba 
\langle \widehat {\cal M} \widehat {\cal M} \rangle \vert_0 &=&  
{18 (N^2-1)(N^2-2)(N^2+1)\over (3N^2-2)}  \; {\cal I}_0(x_{12}) \, ,\\ 
\langle \widehat {\cal P} \widehat {\cal P} \rangle \vert_0 &=& 
{3 N^2(N^2-1)(N^2-4) \over (N^2-2)}  \; {\cal I}_0(x_{12}) \, ,\\ 
\langle  {\cal L}  {\cal L} \rangle \vert_0 &=& 
{15 (N^2-1) (N^2-4) (N^2-9) \over (N^2+1)} \; {\cal I}_0(x_{12}) \, , 
\label{2pt_tree} 
\ea 
while the off-diagonal functions vanish by construction.  
 
At order $g^2$ we obtain  
\ba 
\langle \widehat {\cal M} \widehat {\cal M} \rangle \vert_{g^2} &=&  
18 N (N^2-1)(N^2+1) \; {\cal I}(x_{12}) \, , \\ 
\langle \widehat {\cal P} \widehat {\cal M} \rangle \vert_{g^2} &=&  
-{18 N^2 (N^2-1)(N^2-4) \over (N^2-2)} \; {\cal I}(x_{12}) \, , \\ 
\langle  {\cal L} \widehat {\cal M} \rangle \vert_{g^2} &=&  0 \, ,\\ 
\langle \widehat {\cal P} \widehat {\cal P} \rangle \vert_{g^2} &=& 
{2 N^3 (N^2-1)(N^2-4)(5 N^2-16) \over (N^2-2)^2} \; {\cal I}(x_{12}) \, , \\ 
\langle  {\cal L} \widehat {\cal P} \rangle \vert_{g^2} &=& 
-{5 N (N^2-1) (N^2-4) (N^2-9) \over (N^2+1)} \; {\cal I}(x_{12}) \, , \\ 
\langle  {\cal L}  {\cal L} \rangle \vert_{g^2} &=& 
{25 N (N^2-1) (N^2-4) (N^2-9) \over (N^2+1)} \; {\cal I}(x_{12}) \, , 
\label{2pt_g^2} 
\ea 
where ${\cal I}(x_{12})$ is proportional to the short distance limit 
of the box integral. 
Using eq.~(\ref{limitB}) in the Appendix, one finds 
\be 
{\cal I}(x_{12}) \, = \, -\frac{1}{(4\pi)^5(x_{12}^2)^{4}} \,  
\left({\rm\ln}(x_{12}^2) + 1 
\right) \, .
\ee 
 
Let us start by analyzing the generic case $N \geq 4$. 
The cases $N=3$ and $N=2$  are much simpler and we shall 
discuss them in Sect.~7. Since, as we  
said, all 
coefficients in~(\ref{theops}) turn out to be non-vanishing,  
without loss of generality we shall set $\alpha_{i {\cal P}}=1$.  
With this choice the orthogonality relations lead to the equation 
\be 
\alpha_{i {\cal M}} = {\zeta_i \over (N^2+1)(N^2-2)}\, , 
\ee 
where $\zeta_i$ are the three roots of the cubic equation  
\ba 
&& 8N \zeta^3+(-N^6+2N^4+68N^2-40)\zeta^2+(-3N^9-16N^7+132N^5-80N^3)\zeta  
\nonumber \\ 
&& -84N^{10}+64N^4+244N^8-224N^6+9N^{12} = 0 \, . 
\label{cubic} 
\ea 
For the coefficients $\alpha_{i {\cal L}}$ we get  
\ba 
\alpha_{i {\cal L}} &=&  
-{24 \over 5} \alpha_{i {\cal M}}^2 {(N^2+1)^3 \over  
(3N^2-2)(N^2-4)(N^2-9)} \nonumber \\ 
&+& {3 \over 5} \alpha_{i {\cal  
M}}{(N^2-2N+2)(N^2+2N+2)(N^4-19N^2+10)(N^2+1) \over  
N(N^2-2)(3N^2-2)(N^2-4)(N^2-9)} \nonumber \\ 
&+& {4 \over 5} {(N^2+1)^2 N^2 \over (N^2-2)^2(N^2-9)}\, . 
\ea 
Inserting all these formulae into equations~(\ref{anomR}), we obtain  
for the order $g^2$ corrections to the anomalous dimensions of the operators  
${\cal O}_i$ the following expressions   
\be 
\gamma_1^{{\cal O}_i} = {g^2 N \over 4 \pi^2}  
{ 
8 \zeta_i^2 N-(N^6-2N^4-68N^2+40)\zeta_i+6N^9-64N^7+184N^5-96N^3 \over 
N^3(N^2-4)(3N^2-2)(N^2-2)} \, , 
\label{anomf} 
\ee 
where again the $\zeta_i$'s are the  
three roots of the cubic equation~(\ref{cubic}). 
 
In Fig.~1 we plot the ratios $\gamma_1^{{\cal O}_i}/\gamma_1^{\cal K}$, 
$i=1,2,3$, as functions of $N$, where $\gamma_1^{\cal K}$ is the order $g^2$ 
coefficient of the Konishi multiplet anomalous dimension, displayed in  
eq. (\ref{gammak11}). 
 
 
\begin{figure}[!htbp] 
 %\begin{minipage}[t]{\linewidth} 
    \centering 
    \includegraphics[width=0.85\linewidth]{plot_apoint08.eps} 
    \caption{The ratios $\gamma_1^{{\cal O}_i}/\gamma_1^{\cal K}$, 
$i=1,2,3$, as functions of $N$. $\gamma_1^{\cal K}$ is the order $g^2$ 
coefficient of the Konishi multiplet anomalous dimension, displayed in  
eq. (\ref{gammak11}).} 
    \label{diagrA} 
 %\end{minipage} \\ [2.5cm] 
\end{figure} 
 
\subsection{The $N\rightarrow \infty$ limit} 
\label{largen} 
 
We would like to end this section by briefly discussing the  
peculiar properties of the above orthogonalization procedure  
in the $N \rightarrow \infty$ limit. To this purpose we notice that 
 
1) multiple trace operators with different number of traces are   
mutually orthogonal in this limit.  
 
2) in the space of multiple $n$-trace operators one can choose  
a basis of ``product'' operators of 
the form  
\be 
 \prod_{j=1\ldots n} \tilde {\cal O}_j \ , 
\label{prodop}  
\ee 
where the $\tilde {\cal O}_j$'s are mutually orthogonal  
single trace operators of well defined anomalous dimensions, 
$\gamma(\tilde {\cal O}_j)$.  
 
Note that both   
double trace operators  defined above, \ie  
${\cal D}$ of eq.~(\ref{defDQ}) and ${\cal M}$ of 
eq.~(\ref{defMQK}), have the form~(\ref{prodop})  
(the $SU(4)$ trace subtraction  
in ${\cal D}$ does not affect the following argument).  
Since for large $N$ the two-point function of a multiple trace  
``product'' operator 
is dominated by the most disconnected part (i.e. by the  products of the 
two point functions of the constituent operators),  
different ``product'' operators will   
be orthogonal for large $N$ due to the  
orthogonality of the $\tilde {\cal O}_j$'s.  Moreover, it follows that  
in the $N \rightarrow \infty$ limit the anomalous dimension of the 
``product" operator 
is the sum of the  anomalous dimensions of its constituent operators, 
namely 
\be 
\lim_{N \rightarrow \infty}\gamma(\prod_{j=1\ldots n} \tilde {\cal O}_j) =  
\sum_{j=1\ldots n}  \lim_{N \rightarrow \infty} \gamma(\tilde {\cal O}_j) \ . 
\label{sumgamma} 
\ee  
 
Let us now return to the case at hand. 
{}From Fig.~1 one can see that for large $N$ the anomalous dimension 
of one of the three operators tends to the anomalous dimension of the  
Konishi supermultiplet, $\gamma^{\cal K}_1$. 
An explicit  calculation confirms that this operator is indeed ${\cal M}$  
(see eq.~(\ref{defMQK})). 
The anomalous dimensions of the other two operators,  
which in line with the above discussion for large $N$ will  
be dominantly single trace ones, tend to  
\be 
\lim_{N \rightarrow \infty} \gamma_{\pm} = {1 \over 6}(5 \pm \sqrt{5})  
\gamma^{\cal K}_1 \ . 
\label{dominant} 
\ee 
The corresponding colour tensors $X^{abcd}$ entering eq.~(\ref{O_naive}) are 
in the $N \rightarrow \infty$ limit 
\be 
X_{\pm}^{abcd} = \sqrt{\sqrt{5} \pm 1} \ {\rm Tr}(T^a T^b T^c T^d)  
\mp \sqrt{\sqrt{5} \mp 1} \ {\rm Tr}(T^a T^c T^b T^d) \ . 
\label{XNinf} 
\ee 
  
\section{The four point function $\langle {\cal C}^{11}  {\cal  
C}_{11}^{\dagger}  {\cal K}_1  {\cal K}_1 \rangle$  at order $g^4$} 
 
Given the complexity of the results we got in the previous section, 
it is important to rederive them from an alternative point of view,  
\ie~by performing an OPE analysis of appropriate four-point  
functions. 
 
Since the only operator of dimension 4 in the ${\bf 20^{'}}$ that appears  
in the OPE of two CPO's ${\cal Q}$ is the protected operator 
${\cal D}_{{\bf 20^{'}}}$ (see eqs.~(\ref{O_naive}) and~(\ref{X_D})) 
\cite{aft}, little mileage can be gained by studying 
only the correlation function of four ${\cal Q}$'s. 
The next simplest four-point functions involve the lowest 
component of the Konishi multiplet. 
Correlation functions involving two Konishi operators like 
$\langle {\cal Q}{\cal Q}{\cal K}{\cal K} \rangle $ meet all the necessary  
requirements. 
 This kind of correlators have been  
calculated at lowest order in $g^2$ in~\cite{kon}, where it was also shown 
that they receive no instanton corrections. 
 
Since there is only one $SU(4)$ tensor structure, without loss of  
generality we can use the following representative for this correlator 
\be 
G(x_{1}, x_{2}, x_{3}, x_{4}) =  
\langle {\cal C}^{11}(x_1) {\cal C}_{11}^{\dagger}(x_2)  {\cal  
K}_1(x_3)  {\cal K}_1(x_4) \rangle \ ,  
\label{G0} 
\ee 
which with the help of the identity  
(no summation on $J$)  
\be 
:tr( \phi^{\dagger}_J(x) \phi^{J}(x)): =  
{\cal V}^J_J(x) + {1 \over 3} {\cal K}_1(x) . 
\ee  
can be written also as  
\ba 
&& G(x_{1}, x_{2}, x_{3}, x_{4}) = 
\label{cckk1} \\ 
&& 9 \langle {\cal C}^{11}(x_1) {\cal  
C}_{11}^{\dagger}(x_2)  
:tr( \phi^{\dagger}_2(x_3) \phi^{2}(x_3)):    
:tr( \phi^{\dagger}_3(x_4) \phi^{3}(x_4)): \rangle -  
\nonumber \\  
&& 3 \langle {\cal C}^{11}(x_1) {\cal  
C}_{11}^{\dagger}(x_2)  {\cal K}_1(x_3)   
{\cal V}^3_3(x_4) \rangle 
-3\langle {\cal C}^{11}(x_1) {\cal C}_{11}^{\dagger}(x_2)  {\cal  
V}^2_2(x_3)  {\cal K}_1(x_4) \rangle -  \nonumber \\ 
&&  9\langle {\cal C}^{11}(x_1) {\cal  
C}_{11}^{\dagger}(x_2)  {\cal V}^2_2(x_3)   
{\cal V}^3_3(x_4) \rangle+   
\langle {\cal  
C}^{11}(x_1) {\cal C}_{11}^{\dagger}(x_2)\rangle 
 \langle {\cal K}_1(x_3)  {\cal K}_1(x_4) \rangle 
 \nonumber \ . 
\ea 
For the purposes of our analysis only the first (connected) term  
in the r.h.s. of eq.~(\ref{cckk1}) has really to be computed at  
order $g^4$. In fact in the OPE limit  
$x_{13} \rightarrow 0$, the leading $\ln^2(x_{13}^2)$ contributions 
in the ${\bf 20^{'}}$ channel that comes from the other correlators in  
eq.~(\ref{cckk1}) 
can be obtained simply by using tree level and order $g^2$ data.  
 
In order to simplify the calculation it is convenient to make use of  
conformal invariance to map one of the coordinate points to infinity,  
while at the same time appropriately rescaling the corresponding 
field operator. Introducing the abbreviated notation 
\be 
{\cal C}_{11}^{\dagger}(\infty) = \lim_{x_2 \rightarrow \infty} 
x_2^4 \ {\cal C}_{11}^{\dagger}(x_2)\, , 
\label{limitinf} 
\ee 
we find after a lengthy calculation (which is sketched in the Appendix) 
\ba 
&& \langle {\cal C}^{11}(x_1) {\cal C}_{11}^{\dagger}(\infty)  
:tr( \phi^{\dagger}_2(x_3) \phi^{2}(x_3)):    
:tr( \phi^{\dagger}_3(x_4) \phi^{3}(x_4)): \rangle\vert_{g^4} =  
\nonumber \\ 
&& \nonumber \\ 
&& \qquad {N^2(N^2-1) \over \ 16 (4 \pi^2)^6} \left [  
{1 \over 2 x_{34}^4 } B \left( {x_{13}^2 \over x_{34}^2},  {x_{14}^2  
\over x_{34}^2} \right) 
\left ( 
{x_{13}^2 \over x_{14}^2}+{x_{14}^2 \over x_{13}^2}- 
{x_{34}^2 \over x_{13}^2}-{x_{34}^2 \over x_{14}^2}-2  \right)  +  
\right. \nonumber \\ 
&& \qquad 
{1 \over 2 x_{13}^2 x_{34}^2} \ln\left( {x_{13}^2 \over  
x_{14}^2}\right) 
  \ln\left( {x_{14}^2 \over x_{34}^2} \right) + 
{1 \over 2 x_{14}^2 x_{34}^2} \ln\left( {x_{14}^2 \over  
x_{13}^2}\right)  
  \ln\left( {x_{13}^2 \over x_{34}^2} \right) + 
\nonumber \\ 
&& \qquad 
{1 \over  x_{13}^2 x_{34}^2}\left(\ln( x_{34}^2)+1\right)  \left(  
\ln\left(  {x_{13}^2 x_{34}^2 \over x_{14}^2}\right)+1 \right) + 
\nonumber \\ 
&& \qquad 
{1 \over  x_{14}^2 x_{34}^2}\left(\ln( x_{34}^2)+1\right)  \left(  
\ln\left(  {x_{14}^2 x_{34}^2 \over x_{13}^2}\right)+1 \right) + 
\nonumber \\ 
&& \qquad \left. 
{4 \over  x_{34}^4 } B \left( {x_{13}^2 \over x_{34}^2},  {x_{14}^2  
\over x_{34}^2} \right) 
\left(\ln( x_{34}^2)+1\right) 
+ {1 \over  x_{13}^2 x_{14}^2} \left(\ln( x_{13}^2)+1\right) 
\left(\ln( x_{14}^2)+1\right) \right] 
\label{W} \, , 
\ea 
where $B$ is the box integral given in eq.~(\ref{Brsf}).  
 
{}From eq.(\ref{W}) we can now determine the coefficient 
of the leading logarithmic terms in $G$ associated to the scalar  
operators of (naive) dimension $\Delta_0=4$. This requires 
to identify also all the other operators with (naive) dimension  
smaller or equal to 4 and to subtract their contributions.  
In particular for the coefficient of $\ln^2(x_{13}^2)$  
in~(\ref{W}), we find the factor $-N^2(N^2-1)/(32 (4 \pi^2)^6)$.
Looking at the other correlators in eq.~(\ref{cckk1}),  
we see that in the functions $\langle {\cal C}^{11} {\cal C}_{11}^{\dagger}   
{\cal K}_1 {\cal V}^3_3 \rangle$ and $\langle {\cal C}^{11} {\cal  
C}_{11}^{\dagger}  {\cal V}^2_2 {\cal K}_1 \rangle$ only the protected  
operators of dimension $\Delta_0=2$ and $\Delta_0=4$ can appear 
in the relevant intermediate channel, while in  
$\langle {\cal C}^{11} {\cal C}_{11}^{\dagger}   
{\cal V}^2_2 {\cal V}^3_3 \rangle$  
the component ${\cal K}_{84}$ of the Konishi multiplet alone contributes.  
Finally the disconnected parts in eq.~(\ref{cckk1})  
give no contribution to the $\ln^2(x_{13}^2)$ term. 
The explicit calculation of the residual coefficients after  
subtracting the known contribution of all these operators confirms  
that for $N=2$ at level $\Delta_0=4$ in the  
representation ${\bf 20^{'}}$ of $SU(4)$ there is only one unprotected 
scalar operator, namely $\widehat{\cal M}$, having anomalous  
dimension given by eq.~(\ref{gmh1}).   
This fact will allow us to compute in the next section the anomalous  
dimensions of all the scalar operators appearing at the same level in the  
representation ${\bf 20^{'}}$ of $SU(4)$ for $N>2$.  
 
 
\section{A second derivation of the order $g^2$ anomalous dimension of  
the ${\bf 20^{'}}$ operators} 
 
In this section we present an alternative computation of the  
anomalous dimensions of the scalar operators with naive dimension   
$\Delta_0=4$  
belonging to the representation ${\bf 20^{'}}$ of $SU(4)$,  
that does not require their explicit construction  
in terms of fundamental fields. 
 
The basic assumption is that, according to conformal invariance,  
logarithmic terms exponentiate to a power law. This has been confirmed  
in all order $g^2$- and $g^4$- calculations performed so far. 
 
Another important ingredient is the observation that four-point  
functions of single trace operators quadratic in the fundamental fields   
(like the protected operators ${\cal C}^{11}$,  ${\cal C}_{11}^{\dagger}$ as  
well as the lowest component of the Konishi multiplet  ${\cal K}$) have a  
particularly simple  
polynomial behaviour in $N$. Indeed, expanding the interaction  
Lagrangian in  
powers of $g$ one observes that the coefficient of $g^{2n}$ contains  
the product of  
exactly $2n$ $SU(N)$ structure constants $f^{abc}$, hence the connected 
part of the perturbative amplitude can be represented as  
a single trace of $2n$ $SU(N)$ matrices in the adjoint representation. 
It is a rather long, but straightforward, computation to evaluate all  
the traces we shall need (i.e. those that can appear in perturbation theory  
up to order $g^{10}$).  
Below we list the types of $N$ behaviour that one encounters. The 
coefficients with which these power behaviours appear depend on the  
particular trace one is considering and will be of no interest here. One gets 
\begin{itemize} 
\item order $g^2$: only $N(N^2-1)$; 
 
\item order $g^4$: only $N^2(N^2-1)$; 
 
\item order $g^6$: only $N^3(N^2-1)$; 
 
\item order $g^8$: linear combination of $N^4(N^2-1)$ and $N^2(N^2-1)$; 
 
\item order $g^{10}$: linear combination of $N^5(N^2-1)$ and $N^3(N^2-1)$ 
\end{itemize} 
 
In the computation of the four-point function $G$  
there are both connected and disconnected contributions 
(see eq.~(\ref{cckk1})). The disconnected piece 
is effectively double trace, so it is multiplied by one more factor of  
$N^2-1$ with respect to the connected term.  
 
We consider the double OPE $x_{13} \rightarrow 0$, $x_{24} \rightarrow  
0$, \ie the 
exchange of operators between the products 
${\cal C}^{11}(x_1) {\cal K}_1(x_3)$ and ${\cal C}_{11}^{\dagger}(x_2)   
{\cal K}_1(x_4)$, and concentrate on the leading $\ln(x^2_{34})$ 
behaviour only.  
To be more precise we shall analyze the coefficient of the terms  
$g^{2n}\ln^n (x^2_{34})$ in the correlator~(\ref{G0}). 
A convenient normalization is to extract a common factor equal to 
\be 
{(-\gamma_1^{\cal K})^n \over n!}  \ { 
\langle {\cal C}^{11}  {\cal C}_{11}^{\dagger}\rangle_0 \  
\langle  {\cal K}_1  {\cal K}_1  \rangle_0 \over (N^2-1) }\, ,   
\label{commonf} 
\ee 
where  
\ba 
\langle {\cal C}^{11}  {\cal C}_{11}^{\dagger}\rangle_0 &=& {(N^2-1) \over  
2 (4 \pi^2)^2}   \ , \\  
\langle  {\cal K}_1  {\cal K}_1  \rangle_0 &=& {3 (N^2-1) 
\over 4 (4 \pi^2)^2}   
\ea 
are the tree level normalizations of the two-point functions of the operators 
${\cal C}^{11}$ and ${\cal K}_1$, respectively, while  
$\gamma_1^{\cal K}= 3N/(4 \pi^2)$ is the order $g^2$ correction to the  
anomalous dimension of the Konishi multiplet (eq.~(\ref{gammak11})). 
 
With these choices the contribution from the disconnected diagrams  
to the $g^{2n}\ln^n (x^2_{34})$ term is $N^2-1$ for any value of $n$. 
It is convenient to ``measure'' anomalous dimensions in units of the  
anomalous dimension of the Konishi multiplet, by defining 
\be 
\eta_i = {\gamma_1^{{\cal O}_i} \over \gamma_1^{\cal K}} 
\label{relative} 
\ee 
and to introduce the ratio  
\be 
F_{i} = {N^2-1 \over  
\langle {\cal C}^{11}  {\cal C}_{11}^{\dagger}\rangle_0 \  
\langle  {\cal K}_1  {\cal K}_1  \rangle_0  }{ \langle {\cal C}^{11}   
{\cal K}_1 {\cal O}_i \rangle_0 \  
\langle  {\cal C}_{11}^{\dagger}  {\cal K}_1   {\cal O}_i^{\dagger}  
\rangle_0  
\over  
\langle {\cal O}_i  {\cal O}_i^{\dagger}  \rangle_0 } 
\label{F_i} 
\ee   
to normalize the contributions of the different operators to the  
four-point function~(\ref{cckk1}). In eq.~(\ref{F_i}) the notation 
$\langle X \rangle_0$ means that the corresponding expectation value 
has been evaluated at tree level. We thus obtain the following conditions. 
 
$\bullet$ At tree level 
\be 
F_1 + F_2 + F_3 \equiv P_0 =  {3(N^2-2) (N^2+1) \over (3 N^2 -2)}\, . 
\label{eqtree} 
\ee 
The complicated form of the r.h.s. is due to the subtraction of the  
contribution of the protected operator ${\cal D}$. 
 
$\bullet$ At order $g^2$ and $g^4$ the OPE analysis of~(\ref{W}) lead  
to the equations 
\ba 
g^2 &:& F_1 \eta_1 + F_2 \eta_2 + F_3 \eta_3 \equiv P_1 =  N^2+1 \, ,  
\label{eqg24}\\ 
g^4 &:& F_1 \eta_1^2 + F_2 \eta_2^2 + F_3 \eta_3^2 \equiv P_2 =  N^2+{13  
\over 3}\, . \nonumber 
\ea 
 
$\bullet$ At order $g^6$ and $g^8$ and $g^{10}$ one gets 
\ba 
g^6 &:& F_1 \eta_1^3 + F_2 \eta_2^3 + F_3 \eta_3^3 \equiv P_3 = N^2+b_3  
\nonumber \, , \\ 
g^8 &:& F_1 \eta_1^4 + F_2 \eta_2^4 + F_3 \eta_3^4 \equiv P_4 = N^2+b_4  
+{c_4 \over N^2} \label{eqg6810} \, , \\ 
g^{10} &:& F_1 \eta_1^5 + F_2 \eta_2^5 + F_3 \eta_3^5 \equiv P_5 = N^2 +b_5  
+{c_5 \over N^2}\, , \nonumber 
\ea 
where $b_i$ and $c_i$ are for the moment unknown coefficients. 
 
It is convenient to eliminate the $F_i$ from the above equations in favour of 
the $P_L$, obtaining the system  
\be 
P_{L+3} - (\eta_1+\eta_2+\eta_3) P_{L+2} +  
(\eta_1 \eta_2 +\eta_1 \eta_3 +\eta_2 \eta_3)  P_{L+1}-(\eta_1 \eta_2  
\eta_3) P_{L}  =  0  
\label{newsys} 
\ee 
for $L=0,1,2$. Note that this equation actually holds in general  
(i.e. for any value of $L$). Hence 
the knowledge of $P_0 \dots P_5$ completely determines all $P_L$.  
 
In order to compute the unknown coefficients $b_i$ and $c_i$ recall 
that for $N=2$ 
there is only one possible operator, $\widehat{\cal M}$. {}From  
eq.~(\ref{eqg24}) its relative anomalous dimension is determined   
to be $\eta_{\widehat{\cal M}}=5/3$ for $N=2$ in agreement with  
eq.~(\ref{gmh1}).  
Then from the system~(\ref{eqg6810}) for $N=2$ we find $b_3=89/9$, 
so that $P_3$ is completely determined, and we can express  
$b_4$ and $b_5$ in terms of $c_4$ and $c_5$ as follows:  
$b_4 = 517/27-c_4/4$, $b_5 = 2801/81-c_5/4$. 
 
The knowledge of $P_3$ allows us to solve the case $N=3$,  
where only two operators are present 
with anomalous dimensions 
$(19\pm \sqrt{61})/18 $. Then we can also fix the coefficients 
entering $P_4$ and $P_5$ to be $c_4=40/9$ and $c_5=1720/81$. 
 
Substituting back these numbers into the system~(\ref{newsys}) gives for 
generic $N \geq 4$ the equations  
\ba 
&& \eta_1+\eta_2+\eta_3 = {8 \over 3} \, , \nonumber\\ 
&& \eta_1 \eta_2 +\eta_1 \eta_3 +\eta_2 \eta_3 = {10(2N^2-1) \over 9  
N^2} \, , \label{gammasol} \\ 
&& \eta_1 \eta_2 \eta_3 = {5(3N^2-2) \over 27 N^2}\, , \nonumber 
\ea 
whose solutions precisely yield the values of the anomalous  
dimension~(\ref{anomf}) previously found in Sect.~5. 
 
\section{Relation with the Penrose double scaling limit} 
 
In this admittedly more speculative section we would like to  
discuss how one might generalize our calculations in order to make contact  
with the double scaling limit that corresponds  
to type IIB superstring around a pp-wave supported by a   
RR 5-form flux~\cite{ppwaves}.  
{}From the supergravity perspective this corresponds to performing a 
Penrose limit around a null geodesic at the center of $AdS_{5}\times S^{5}$.  
The resulting maximally supersymmetric geometry~\footnote{In our  
conventions, the indices of $\vec X$ run over $1,4,7,8$ and those of  
$\vec{Y}$ over $2,3,5,6$.} 
\be 
ds^{2} = - 4 dx^{+} dx^{-} - \mu^{2} (|\vec X|^{2} + |\vec{Y}|^{2})  
(dx^{+})^{2} + (|d\vec X|^{2} + |d\vec{Y}|^{2})  
\ee 
\be 
F_{+2356} = F_{+1478} = \mu \quad , \quad e^{\Phi} = g_{s} \, , 
\ee 
with all other fields set to zero,  
admits an exactly solvable worldsheet description in the  
Green-Schwarz formalism in the light-cone gauge~\cite{bmn,gs},  
see however~\cite{berko}. In the Penrose limit the $SU(2,2|4)$ super-isometry  
of $AdS_{5}\times S^{5}$ undergoes an Inon\"u-Wigner contraction. 
In particular $SO(4,2)\rightarrow SO(4)_{X}\times U(1)_{\Delta}$ and  
$SO(6)\rightarrow SO(4)_{Y}\times U(1)_{J}$ but at the same time a Heisenberg  
group, $H(8)$, emerges so that the total number of generators remains  
equal to 30 as for $AdS_{5}\times S^{5}$. A similar rearrangement takes place 
for the 32 supersymmetry charges.  
In addition to the standard identifications 
\be 
g_{s} = {g^{2} \over 4\pi} \quad , \qquad {L^{2}\over \alpha^{\prime}} =  
\sqrt{g^{2} N} \quad , 
\ee 
at large $N$ and large $J$, with $J\approx \sqrt{N}$, the relevant 
coupling turns out to be~\cite{bmn} 
\be 
\lambda^{\prime} = {g^{2} N \over 4\pi J^{2}} \, , 
\ee 
where $J$ is the $U(1)_{J}$  
charge that appears in the above decomposition of  
$SO(6)$~\footnote{This $U(1)_J$ does not coincide with the $U(1)_{R}$ in  
the ${\cal N}=1$ decomposition of ${\cal N}=4$ SYM used so far.}. 
The relevant null geodesic is identified by the light-cone coordinates  
$x^{\pm} = \mu^{\mp 1} (\tau\pm\psi)/2L^{2}$, where $\tau$ is the global  
time in $AdS_{5}$~\cite{dhf}, conjugate to $\Delta$, and $\psi$ is an angular  
coordinate in $S^{5}$, conjugate to $J$. Thus in the Penrose limit 
the light cone momentum $P^{+}$ is essentially proportional to $J$,  
\ie 
\be 
p^{+} = {\Delta + J \over 2\mu L^{2}} \approx {J \over \mu L^{2}} \, .  
\ee 
Operators with $\Delta = J$ and $\Delta = J+1$ are known to be  
protected, as a consequence of $SU(2,2|4)$ shortening conditions  
of BPS type that survive the relevant Inon\"u-Wigner contraction. 
The simplest nearly protected operators, that are expected to  
correspond to the lowest type IIB superstring excitations $Y^{a}_{n}  
Y^{b}_{-n}|p^{+}\rangle$,  are of the form  
\be 
{\cal A}^{ab}_{n} = \sum_{\ell=0}^{J} q_{n}^{\ell}  
tr(Z^{J-\ell} Y^{a} Z^{\ell} Y^{b}) 
\label{bmnop} 
\ee 
where $q_{n} = \exp[2\pi i n/(J+1)]$ and, in our previous notation,   
$Z=\phi^{1}$ and $Y^{a} = \varphi^{a}$ for $a=2,3,5,6$~\footnote{Notice that, 
at variance with what is done in 
refs.~\cite{bmn,danetal,plefkaetal,grossetal}, 
we defined the Fourier coefficients in~(\ref{bmnop}) to be  
$q_{n} = \exp[2\pi i n/(J+1)]$, and not $q_{n} = \exp[2\pi i n/J]$,  
in order to have a formula yielding $J+1$, and not $J$, linearly  
independent operators. This modification is at the origin of a number of  
further useful implications.}.  
 
The knowledge of the free spectrum of the light cone Hamiltonian $P^{-}$ 
gives a prediction for  
the `planar' contributions to the anomalous dimensions of the  
operators ${\cal A}^{ab}_{n}$, \ie 
\be 
 p^{-} = {\mu \over 2} (\Delta - J) =  
\mu \sqrt{1 + {4\pi g_{s} N n^{2} \over J^{2}}} + {\rm (non-planar)} \, . 
\ee 
As noticed by~\cite{bmn}, the effective string loop counting parameter is  
$J^{2}/N$. In~\cite{danetal} the first `non-planar'  
corrections in SYM theory have been explicitly computed and matched  
with string one-loop corrections. 
 
The obvious difference between ${\cal A}^{ab}_{n}$ and the operators we have  
considered in the previous sections is the presence of 
a large number ($J\approx \sqrt{N}$) of $Z$'s that  
account for their large $U(1)_{J}$ charge.  
Aside from combinatorial factors that can be elegantly deduced  
by resorting to a gaussian matrix model~\cite{plefkaetal}, we believe that 
most of the $Z$ fields are `spectator' to the perturbative 
order one can reliably work.  
A subtler difference pertains to the $SO(4)$ transformation properties.  
In the decomposition ${\bf 20}^{'} = (1,1)_{0} + (3,3)_{0} + (2,2)_{+1}  
+(2,2)_{-1} +(1,1)_{+2} +(1,1)_{-2}$, the operators ${\cal O}_\ell^{11}$,  
we have mostly concentrated our  
attention on, transform in the $(1,1)_{+2}$. On the other hand 
${\cal A}^{ab}_{n}$ belong to  
$(3,3)_{J} + (3,1)_{J} +(1,3)_{J} +(1,1)_{J}$. The $SU(4)$ R-symmetry  
of the theory should help disposing of this problem.  
 
For generic $J$, ${\cal A}^{ab}_{n}$  
can belong only to  the representations which appear in the decomposition 
\ba  
{}[0,J,0]\otimes [0,1,0]\otimes [0,1,0] = [0,J+2,0] +  
{}[2,J-2,2] + 2 [1,J,1]  + 
\nonumber \\ 
{}[2,J-1,0] + [0,J-1,2] + 2 [1,J-2,1] + 3 [0,J,0] + [0,J-2,0]  \, . 
\ea 
Note that only the last representation in the r.h.s. does not saturate any  
unitarity bound and necessarily belongs to a long multiplet.  
The first five, if super-primary, are protected  
(1/2, 1/4, 1/4, 1/8 and 1/8 BPS, respectively). Only super-descendants in  
these representations can acquire anomalous dimensions. 
The sixth and seventh representations saturate the bound  
$\Delta \ge 2 + k + l +m$ that leads to 
a linear type shortening condition of the type~(\ref{Ocon}). 
 
Although a detailed resolution of the mixing of these operators goes  
beyond the scope of the present investigation, we would like to  
argue that our computations are the building blocks for the  
study of their mixing properties. The ${\cal N}=1$ formalism makes more  
transparent many laborious cancellations found to take place  
in~\cite{danetal,plefkaetal,grossetal}. For instance, the operator  
${\cal A}^{ab}_{0}$, being totally symmetric in the $SU(4)$ indices,  
is a CPO belonging to the representation $[0,J+2,0]$, hence it is protected  
(1/2 BPS) with $\Delta=J+2$. Moreover the symmetric, ${\cal A}^{(ab)}_{n}$,  
and the antisymmetric, ${\cal A}^{[ab]}_{n}$, parts of the operators defined  
in eq.~(\ref{bmnop}) are mutually orthogonal at tree-level for all values of  
$n$, $N$ and $J$. 

Actually one can go one step further and include also multiple 
trace operators. A preliminary analysis of the symmetric case for $J=4$, 
including all possible mixings, 
shows the existence, for any $N$, of eight protected and three 
unprotected operators. The latter have exactly the anomalous dimensions 
reported in eq.~(\ref{anomf}) and turn out to be  
super-descendants of the operators, ${\cal O}_i$, identified in Sect.~5. 
Even though multiple trace operators decouple from  
single trace ones at large $N$, mixing effects may compete with `non-planar'  
$(J^{2}/N)^{2}$ corrections that are dual to string loop corrections  
to masses. Only after disentangling operator mixing, can the comparison 
between gauge theory and string theory results be sensible. 
We will come back to this and related issues in a forthcoming publication.

Despite the success of the proposal of ref.~\cite{bmn}, the  
way holography is realized in the pp-wave background  
is still a matter of debate~\cite{reyetal}. This prevents a naive  
application of the procedure that for (asymptotically) AdS spaces 
has lead to `holographic renormalization'~\cite{holoren}. However conformal  
flatness of the background, that is made manifest by the coordinate  
transformation 
\cite{bernas} 
\ba 
&& u = \tan(x^{+}) \, , \quad   
 v = x^{-} - {1\over 2}  (|\vec{X}|^{2} +  
|\vec{Y}|^{2}) \tan(x^{+}) \, ,\quad \\  
&& {\vec{X} \ }^\prime = {\vec{X}\over \cos(x^{+})}  
\, ,  
\quad 
{\vec{Y} \ }^\prime = {\vec{Y}\over \cos(x^{+})} \, , \nonumber 
\ea 
guarantees the absence of higher derivative   
corrections to amplitudes with fewer than four insertions, 
much in the same way as in $AdS_{5}\times S^{5}$~\cite{tbmbg},  
and makes one hope  
that a viable string description along the lines of~\cite{berko} could  
be not far from reach. It would then be interesting to study  
non-perturbative effects induced by instanton-like D-brane  
solutions~\cite{ksmtr} in the double scaling limit.  
In particular, it should be possible to extend  
our proof of the absence of instanton contributions  
to the anomalous dimensions of operators dual to string excitations.  
We plan to come back to these and related issues in the near future. 
 
\section{Conclusions} 
 
The centerpiece of this article is the resolution of the mixing of scalar 
operators of naive dimension 4 in the ${\bf 20^{'}}$ of $SU(4)$ at order $g^2$ 
in perturbation theory. The problem is investigated in two different ways. 
First, by a direct orthogonalization of all possible single and  
double trace structures. Second, via a OPE analysis of a four point  
correlator 
computed up to order $g^4$. The agreement of the two approaches was for us 
an important consistency check, because the expressions found for the mixing 
matrix and anomalous dimensions are not rational in $N$. This certainly comes 
as an unexpected feature, but it is not in conflict with any fundamental  
property of the field theory. 
 
Of the three unprotected operators not belonging to the Konishi multiplet 
that we have identified, 
there is one whose anomalous dimension tends to $\gamma_{1}^{\cal K}$ 
in the large $N$ limit. In this 
limit the operator under consideration can be identified with the product  
of the lowest Konishi operator and the lowest component of the  
supercurrent multiplet. We find it noteworthy that its anomalous dimension is 
the sum of the anomalous dimensions of its two factors, thus perhaps  
hinting at a deeper physical significance of singleton multiplication  
as a way of generating generic UIR's of $SU(2,2|4)$. 
 
 
\section*{Acknowledgements} 
We are grateful to Matthias Blau, Hugh Osborn, Augusto Sagnotti, Kostas 
Skenderis and Marika Taylor for 
useful conversations. M.B. thanks Soo-Jong Rey and Dan Freedman for  
clarifying e-mail exchanges. B.E. acknowledges many useful discussions with 
Emery Sokatchev and Christian Schubert. This work was supported in part by 
I.N.F.N., by the EC contract HPRN-CT-2000-00122, by the EC contract 
HPRN-CT-2000-00148, by the INTAS contract 99-0-590 and by the MURST-COFIN 
contract 2001-025492. 
 
 
\section*{Appendix} 
  
\subsection*{The explicit calculation of the correlation function~(\ref{W})} 
 
In this appendix we sketch the calculation of the connected part of the 
amplitude 
\begin{equation} 
A(x_1, x_2, x_3, x_4) = \langle {\cal C}^{11}(x_1) {\cal  
C}_{11}^{\dagger}(x_2)  
:tr( \phi^{\dagger}_2(x_3) \phi^{2}(x_3)):    
:tr( \phi^{\dagger}_3(x_4) \phi^{3}(x_4)): \rangle 
\label{ftc} 
\end{equation} 
at order $g^4$. 
There are no connected diagrams with two vector lines, since  
$A$ vanishes at tree level. 
A direct calculation shows also that the   
diagrams with one chiral  and one vector line  sum up 
to zero.  Hence we shall consider only diagrams with  all lines 
corresponding to chiral propagators. We project on the lowest components 
of the supermultiplets, so graphs with a cubic chiral vertex with all three 
lines attached to external points do not contribute. Moreover, 
each internal chiral line gives rise to a delta function.  
The relevant order $g^4$ diagrams are shown in Fig.~2. 
\vskip 0.5 in 
 \begin{center} 
 \begin{picture}(38000,8000)(0,-3300) 
  
 \drawline\fermion[\E\REG](0,0)[6000] \global\advance\pfrontx by 
 400 \global\advance\pfronty by 200 \put(\pfrontx,\pfronty){2} 
 \global\advance\pmidx by -700 \global\Yone=-1500 
 \put(\pmidx,\Yone){$\textbf{(a)}$} 
 \drawline\fermion[\N\REG](\pbackx,\pbacky)[6000] 
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% \vskip -0.8 cm  
The topologies~\textbf{(a)}, \textbf{(b)} and~\textbf{(c)} also  
occur with points 3 and 4 flipped. All 
diagrams have the same combinatorial weight and group factor. In 
order to evaluate the integral coming from diagram \textbf{(a)},  
we send the argument of one of the dimension two operators, say ${\cal 
C}^\dag_{11}$, to infinity as in eq.~(\ref{limitinf}).  
Diagram~\textbf{(d)} 
reduces to the product of two box integrals, so we shall 
concentrate on the graphs~\textbf{(a)} and the sum of~\textbf{(b)}  
and~\textbf{(c)}. 
 
\subsection*{Point splitting regularization} 
We shall first sketch the calculation of the most complicated  
diagram~\textbf{(a)} regularized by point-splitting as 
described in Sect.~2.  
The basic object is the massless scalar box integral defined by 
\be 
\int {d^4 x_5 \over x_{15}^2 x_{25}^2 x_{35}^2 x_{45}^2} =  
{\pi^2 \over x_{13}^2 x_{24}^2 }  B(r,s) \ . 
\label{massbox} 
\ee 
A more explicit expression of $B$ is given by  
\ba B(r,s) & = & {1 \over \sqrt{p}} \left \{\ln (r)\ln 
(s)  - \left [\ln \left({r+s-1 -\sqrt {p} \over 2}\right) 
\right]^{2} \right. \nonumber \\ 
&& \left. -2 {\rm Li}_2 \left({2 \over 1+r-s+\sqrt {p}}\right ) - 
2 {\rm Li}_2 \left({2 \over 1-r+s+\sqrt {p}}\right )\right \}\, ,   
\label{Brsf} 
\ea 
where  
\be 
p = 1 + r^{2} + s^{2} - 2r - 2s - 2rs \,  
\label{pdef} 
\ee 
is a function of the two conformally invariant ratios 
\be 
r = { x_{12}^2 x_{34}^2 \over x_{13}^2 x_{24}^2 } \quad  , \qquad 
s ={x_{14}^2 x_{23}^2\over x_{13}^2 x_{24}^2} \,  
\label{rs} 
\ee 
and the dilogarithm function is defined as 
${\rm Li}_2 (z)=\sum_{n=1}^{\infty}{z^n\over n^2}$. 
 
The small $r$ limit of $B(r,s)$ is  
\be 
\lim_{r\rightarrow 0 } B(r,s) = -{\rm{\ln}}(r) + 2 +O(r) \ . 
\label{limitB} 
\ee 
The partial derivatives of  $B(r,s)$ have the following form 
\ba 
\partial_r  B(r,s) &=& -{r-s-1 \over p}  B(r,s) - {r+s-1 \over p r }  
{\rm \ln}(s) +  
{2 \over p}{\rm \ln}(r) \ , \\ 
\partial_s  B(r,s) &=& -{s-r-1 \over p}  B(r,s) - {r+s-1 \over p s}  
{\rm \ln}(r) +  
{2 \over p}{\rm \ln}(s) \ . \nonumber 
\label{partB} 
\ea 
 
If we introduce the notation  
\ba 
(x_{4^{\pm}5})^2 &=& (x_4 \pm {\epsilon_4 \over 2} - x_5)^2 \ , \\ 
(x_{3^{\pm}6})^2 &=& (x_3 \pm {\epsilon_3 \over 2} - x_6)^2 \ , \nonumber 
\ea 
the integral corresponding to diagram \textbf{(a)} has the form 
\be 
I_a = \int{ d^4 x_6 d^4 x_5 \over (x_{3^{+}6})^2 (x_{3^{-}6})^2  
 x_{56}^2  x_{15}^2 (x_{4^{+}5})^2 (x_{4^{-}5})^2} \ . 
\label{I_a} 
\ee 
In order to evaluate it we shall add and subtract from $I_a$  the auxiliary  
factorized integral 
\be 
I_1 = x_{34}^2 \int{ d^4 x_6 \over x_{46}^2 (x_{3^{+}6})^2 (x_{3^{-}6})^2 } 
\int{ d^4 x_5  \over  x_{15}^2 
(x_{4^{+}5})^2 (x_{4^{-}5})^2 x_{35}^2} \ , \  
\ee 
which in the limit of small $\epsilon_3$ and $\epsilon_4$  
can be computed by the use of 
eqs.~(\ref{massbox})-(\ref{limitB}) to give 
\be 
\lim_{\epsilon_{3,4} \rightarrow 0} I_1 = 
{\pi^4 \over x_{14}^2 x_{34}^2} \left[ -{\rm \ln}\left({\epsilon_3^2 \over  
x_{34}^2}\right) 
 + 2 \right] 
\left[ -{\rm \ln}\left({\epsilon_4^2 x_{13}^2 \over x_{14}^2x_{34}^2}\right) 
 + 2 \right] \ . 
\label{I_a1} 
\ee 
Note that in this limit $I_1$ has   
logarithmic divergences,  
which are cancelled by the factors $(\epsilon_{3,4}^2)^{-{1 \over 
2}\gamma^{\cal K}(g^2)}$ 
in the definition of the renormalized Konishi operator,  eq.~(\ref{K1ren}).  
We can evaluate the  
integral over  $x_6$ in the difference $I_a - I_1$  and  
take the limit $\epsilon_4 \rightarrow 0 $, using again  
eqs.~(\ref{massbox})-(\ref{limitB}), obtaining  
\ba   
&&\lim_{\epsilon_4 \rightarrow 0 } (I_a - I_1)  =  
-{\pi^2 \over x_{14}^2} {\cal J} \ , \\ 
&& {\cal J} = \int { d^4 x_6 \over x_{46}^2 (x_{3^{+}6})^2 (x_{3^{-}6})^2 } 
{\rm \ln}\left({x_{16}^2 x_{34}^2\over x_{46}^2 x_{13}^2} \right) \ . 
\label{resI} 
\ea    
In order to calculate ${\cal J}$ let us first prove that it is finite in  
the limit 
$\epsilon_3 \rightarrow 0$. 
To this end we compute the derivative of ${\cal J}$ with respect 
to $x_1$.  
Since under derivation the logarithm disappears,  
we can express the derivative of ${\cal J}$ in terms of the standard  
box integral and take the limit $\epsilon_3 \rightarrow 0$ with the  
finite result  
\be 
\lim_{\epsilon_3 \rightarrow 0} \partial_{x_1^{\mu}}  {\cal J} =  
2 x_{13 \mu} {\pi^2 \over  x_{13}^2  x_{34}^2} {\rm \ln}\left({x_{13}^2   
\over x_{14}^2 } \right) 
- \partial_{x_3^{\mu}} \left[ {\pi^2 \over  x_{34}^2} B\left({x_{13}^2   
\over x_{34}^2 },   
{x_{14}^2  \over x_{34}^2 }\right) \right] \ . 
\label{partJ} 
\ee 
Hence, if present, the divergent (for $\epsilon_3 \rightarrow 0$)  
part of ${\cal J}$ will be independent of $x_1$.  
But in the limit $x_1 \rightarrow x_4$, ${\cal J}$ is equal to zero, so it has  
a finite $\epsilon_3 \rightarrow 0$ limit. 
Scale invariance then allows us to  write it as 
\be 
{\cal J} = {\pi^2 \over  x_{34}^2} 
 f\left({x_{13}^2  \over x_{34}^2 },   
{x_{14}^2  \over x_{34}^2 }\right)  \, .  
\label{Jrep} 
\ee 
Formula~(\ref{partJ}) gives rise to two equations for the two partial  
derivatives of  the function $f(a,b)$ defined above.  
These,  together with the initial condition $f(1,0)=0$, 
determine completely the function $f$, yielding 
\be 
f(a,b) = - {1 \over 2 } \left[(a-b-1) B(a,b) + {\rm \ln}(a)  
{\rm \ln}\left({b \over a}\right) 
\right] \, . 
\ee 
Inserting this solution in eqs.~(\ref{Jrep}), (\ref{resI}) and taking  
into account eq.~(\ref{I_a1}), we obtain the complete expression  
of  $I_a$ (eq.~(\ref{I_a})).   
 
The computation of the other three diagrams~\textbf{(b)}, \textbf{(c)}  
and~\textbf{(d)} is simpler and  
makes use of eqs.~(\ref{massbox})-(\ref{limitB}) only.  
The final result of the calculation is given in eq.~(\ref{W}). 
 
\subsection*{Dimensional regularization} 
 
The calculation is done in the Euclidean regime and with the 
space-time dimension changed to   $d = 4 + 2 \epsilon$. In order to preserve 
the validity of equation $\nabla^{2} \, G_{0} = - \delta^{(d)}$,  
one has to appropriately modify the scalar 
propagator: 
\begin{equation} 
G_{0} = \frac{1}{4 \pi^2 x^2} \, \rightarrow \, 
\frac{\Gamma(\frac{d}{2})} {(d - 2) 2 \pi^\frac{d}{2}} \, 
\biggl(\frac{1}{x^2}\biggr)^{(\frac{d}{2} - 1)} = \frac{1}{4 
\pi^2} \, \frac{\Gamma(1 + \epsilon)}{\pi^\epsilon} \, 
\biggl(\frac{1}{x^2}\biggr)^{(1 + \epsilon)} 
\end{equation} 
The regulator is introduced in the spirit of dimensional  
reduction~\cite{SiegelGuess}, 
\ie the $x$ dependence of the superpropagators is 
changed as indicated above, but their $\theta$-dependence is not.  
One can perform the graph calculation formally since the 
regulator preserves the $\theta$ algebra by 
definition and is compatible with partial integration. 
 
Renormalization introduces a renormalization factor $Z(g^2)$ 
for the Konishi operator in the operator sums $:tr( \phi^{\dagger}_2(x) 
\phi^{2}(x)): \ = {\cal V}^2_2 + Z(g^2) {\cal K}_1/3$ and  
$:tr( \phi^{\dagger}_3(x) \phi^{3}(x)): = \ {\cal V}^3_3 + Z(g^2) {\cal 
K}_1/3$, with $Z|_{g^0} = 1$. The order $g^2$ part of the $Z$  
factor can be found 
by reexpressing~(\ref{2pK1}) in dimensional regularization. 
The order $g^4$ piece of $Z$ does not play any role in the current 
calculation because there are no connected tree diagrams between ${\cal 
C}^{11},{\cal C}^\dag_{11},{\cK}$ and $:tr( \phi^{\dagger}_I(x) 
\phi^{I}(x)):$ for $I=2,3$. However, in order to obtain a finite expression 
for the correlator~(\ref{ftc}) we have to take into account the 
subtractions at order $g^4$, coming from its expression  
at tree level and order $g^2$. 
 
\subsubsection*{Diagram~\textbf{(a)}} 
The most complicated calculation concerns the contribution of this  
diagram. We need to compute the $x$-space integral 
\begin{equation} 
J_a(1,3,4) = \int \frac{d^4x_5}{x^2_{15} \, x^4_{45}} \int 
\frac{d^4x_6}{x^2_{56} x^4_{36}}\, . 
\end{equation} 
We shall first evaluate the second subintegral, e.g. by the standard 
Feynman parameter trick. We find 
\begin{equation} 
\int \frac{d^{(4 + 2 \epsilon)}x_6}{(x_{56}^2 \, x_{36}^4)^{(1 + 
\epsilon)}} = \, \frac{\rho}{\epsilon} \, 
\biggl(\frac{1}{x^2_{35}}\biggr)^{(1+2\epsilon)} \label{firstStep} \; , 
\end{equation} 
where 
\begin{equation} 
\rho = \frac{-\pi^{(2+\epsilon)}}{(1+2\epsilon)\Gamma(1+\epsilon)} \; . 
\label{alp} 
\end{equation} 
Note that the power of the propagator in~(\ref{firstStep}) is not the standard 
$1+\epsilon$. We split $(x_{35}^2)^{-(1+2\epsilon)} = (x^2_{35})^{-(1+ 
\epsilon)} (x^2_{35})^{(-\epsilon)}$ and expand the last factor, thus 
obtaining three contributions to $J_a$
 
\begin{eqnarray} 
J_a(1,3,4) & = & \frac{\rho}{\epsilon} \int \frac{d^4x_5}{(x^2_{15} \, 
x^4_{45} \, x^2_{35})^{(1+\epsilon)}} - \rho \int \frac{d^4x_5 \, 
\ln(x^2_{35})}{(x^2_{15} \, x^4_{45} \, x^2_{35})^{(1+\epsilon)}}  + \\ 
& & + \frac{\rho \epsilon}{2} \int \frac{d^4x_5 \, \ln^2(x^2_{35})}{(x^2_{15} 
\, x^4_{45} \, x^2_{35})^{(1+\epsilon)}} + \ldots \nonumber \\  
& \equiv & J_{a1} \, + \, J_{a2} \, + \, J_{a3} \, + \ldots \nonumber \, . 
\end{eqnarray} 
In all three pieces the remaining integral diverges and yields another 
simple pole in $\epsilon$. Since we need to know $J_a$ up to 
$O(\epsilon)$, we have to compute the first of these integrals 
at $O(1/\epsilon),O(1)$ and $O(\epsilon)$, the second at $O(1/\epsilon),O(1)$, 
but only the pole part of the third is needed and all higher terms can be 
ignored. 
 
One can compute the integrals $J_{ai}$ by the method of Gegenbauer 
polynomials (see~\cite{tellyouwhatitiswhenifindthepaper} and references 
therein). This approach is particularly suited to 
three point integrals in which only one propagator has an unusual 
exponent. The other two propagators are expanded in orthogonal 
polynomials and the angular and radial integrations are carried out. This 
produces two infinite series, which, however, can be written as an 
expansion in orthogonal polynomials of a single propagator at the 
expense of introducing a parametric integral~\cite{diffren}. 
 
We first consider the contribution $J_{a1}$. Let $x 
= x_{14}, \, z = x_{34}$, then for $|x|<|z|$: 
\begin{eqnarray} 
J_{a1} & \equiv & \frac{\rho}{\epsilon} \, I_{a1} \, = \,  
\frac{\rho}{\epsilon} \,  
\int \frac{d^{(4 + 2 \epsilon)}y}{(y^4 \, (x-y)^2 \, 
(z-y)^2)^{(1 + \epsilon)}} \label{J1} \\ & = & \frac{\rho^2}{2 \, \epsilon} 
\int_0^1 dt  [ \, (\frac{1}{z^2})^{(1 + 2\epsilon)} 
t^{(1 + 3 \epsilon)} - \, (\frac{1}{x^2})^{(1 + 2\epsilon)} t^{-(1 
+ \epsilon)} \, ] \, \frac{1}{((z - x t)^2)^{(1 + \epsilon)}} 
\nonumber 
\end{eqnarray} 
The last term in the denominator can be factorized as 
\begin{equation} 
((z - x t)^2)^{(1 + \epsilon)} = (x^2)^{(1 + \epsilon)} ((t - 
t_-)(t - t_+))^{(1 + \epsilon)} \, , 
\end{equation} 
where $t_\pm = (x\cdot z \pm \sqrt{(x\cdot z)^2 -  
x^2 z^2})/x^2$~\cite{diffren}. The first parameter integral in~(\ref{J1})  
is regular, while the second may be performed after subtracting the 
singularity. The result reads 
\begin{equation} 
J_{a1} = \frac{\rho^2}{\epsilon^2} \frac{(x^2_{13})^\epsilon}{(x_{14}^2 
x_{34}^2)^{(1+2\epsilon)}} + O(1) \label{JJ1} \, . 
\end{equation} 
The finite part of the integral contains the box integral $B$ and some 
$\ln^2$ terms. 
 
Remarkably, the knowledge of $J_{a1}$ is only necessary in as 
much as its infinities are concerned. The renormalization of 
the Konishi operator produces a term, $Z|_{g^2} \, I_{a1}$, 
equal to the product of $Z|_{g^2}$ and the order $g^{2}$ contribution  
to $A$, which exactly cancels the contribution $J_{a1}$~\footnote{The  
finite part of the $Z$ factor leads to a term of the type $b \, I_{a1}$,  
which also produces a first order 
pole in $\epsilon$. By a judicious choice of $b$  
this term is cancelled by $J_{a1}$.}. 
 
Let us now focus on $J_{a2}$. We shall write the integral as  
\begin{equation} 
J_{a2} \, = \, - \rho \,  
\int \frac{d^{(4+2\epsilon)}x_5 \, \ln(x^2_{35})}{(x^2_{15} \, 
x^2_{35} \, x^4_{45})^{(1+\epsilon)}} = \frac{\rho}{1+\epsilon} \, 
\partial_\beta \, \int 
\frac{d^{(4+2\epsilon)}x_5}{(x^2_{15} \, (x^2_{35})^\beta \, 
x^4_{45})^{(1+\epsilon)}}  
\label{J2} \vert_{\beta=1}\, . 
\end{equation} 
Define 
\begin{equation} 
K_{a2} \, = \, \int 
\frac{d^{(4+2\epsilon)}x_5}{(x^2_{15})^{(1+\epsilon)} 
(x^2_{45})^{(1+2\epsilon)} (x^2_{35})^{\beta(1+\epsilon)}} \, . 
\end{equation} 
We can isolate the divergence in $J_{a2}$ by writing 
\begin{equation} 
\biggl(\frac{x_{14} \partial_4}{1+2\epsilon} - 1\biggr) \, K_{a2} \, = \, 
(x_{14} \partial_4 - 1) \int \frac{d^4x_5}{x^2_{15} \, x^2_{45} \, 
(x^2_{35})^\beta} + O(\epsilon) \, , \label{eeqq1} \quad \; 
\end{equation} 
since the integral is finite if $\beta$ is close to one. Swapping integration 
and differentiation in the l.h.s. we derive (the integrals are finite as long 
as the regulator is not removed) 
\begin{eqnarray} 
&& \biggl(\frac{x_{14} \partial_4}{1+2\epsilon} - 1\biggr) \, K_{a2} 
\label{eeqq2} \\ 
&& \; = x_{14}^2 \int \frac{d^{(4+2\epsilon)}x_5}{(x^2_{15} \, 
x^4_{45})^{(1+\epsilon)} (x^2_{35})^{\beta(1+\epsilon)}} - \int 
\frac{d^{(4+2\epsilon)}x_5}{(x^2_{15})^\epsilon 
(x^4_{45})^{(1+\epsilon)} (x^2_{35})^{\beta(1+\epsilon)}} \, . \nonumber 
\end{eqnarray} 
Let us call the second term in the last line $L_{a2}$. Expanding 
$(x^2_{15})^{-\epsilon}$ up to first order in $\epsilon$ and 
recombining the pole parts of the resulting two integrals one finds 
\begin{equation} 
L_{a2} \, = \, 
-\frac{1}{(x^2_{14})^\epsilon} \int 
\frac{d^{(4+2\epsilon)}x_5}{(x^4_{45})^{(1+\epsilon)} 
(x^2_{35})^{\beta(1+\epsilon)}} + O(\epsilon) \, . 
\end{equation} 
We substitute this expression into~(\ref{eeqq2}) and equate 
with~(\ref{eeqq1}). After dividing by $x^2_{14}$, we obtain 
\begin{eqnarray} 
 \int 
\frac{d^{(4+2\epsilon)}x_5}{(x^2_{15} \, (x^2_{35})^\beta \, 
x^4_{45})^{(1+\epsilon)}} &=& \frac{1}{(x^2_{14})^{(1+\epsilon)}} 
\int \frac{d^{(4+2\epsilon)}x_5}{( (x^2_{35})^\beta \, x^4_{45} 
)^{(1+\epsilon)}} \\ 
&+& \frac{x_{14} \partial_4 - 1}{x^2_{14}} \int 
\frac{d^4x_5}{x^2_{15} \, (x^2_{35})^\beta \, x^2_{45}} \, + O(\epsilon) \, . 
\end{eqnarray} 
The first term in the r.h.s. contains a divergence, but as a two-point 
structure it is readily calculable. We compute the second term in the r.h.s. 
again by means of Gegenbauer polynomials. As above, we introduce a one 
parameter integral to rewrite the result of the angular 
integration in closed form. One finds for $\beta = 1 + \Delta$ 
\begin{equation} 
\int \frac{d^4x_5}{x^2_{15} \, x^2_{45} \, 
(x^2_{35})^{(1+\Delta)}} = \biggl(1- 
\frac{\Delta}{2}\bigl(\ln(x^2_{13}) + \ln(x^2_{34})\bigr)\biggr) 
\int \frac{d^4x_5}{x^2_{15} \, x^2_{35} \, x^2_{45}} + O(\Delta^2)  
\, , 
\end{equation} 
which is sufficient to calculate the first order parametric 
derivative in~(\ref{J2}). Finally, we employ the identity 
\begin{equation} 
\frac{x_{14} \partial_4 - 1}{x^2_{14}} \int \frac{d^4x_5}{x^2_{15} 
\, x^2_{35} \, x^2_{45}} = \frac{\pi^2}{x^2_{14} \, x^2_{34}} \, 
\ln\biggl(\frac{x^2_{14}}{x^2_{13}}\biggr) \, . 
\end{equation} 
Collecting terms we find  
\begin{eqnarray} 
J_{a2} & = & - \frac{\rho^2}{\epsilon} \,  
\frac{\ln(x^2_{34})}{(x^2_{14})^{(1+\epsilon)} 
(x^2_{34})^{(1+2\epsilon)}} + \nonumber \\  
\bigskip 
& & \frac{\pi^4}{2} 
\frac{1}{x^2_{14} \, x^2_{34}} \biggl[ 
\biggl(1-\frac{x^2_{14}}{x^2_{13}} 
-\frac{x^2_{34}}{x^2_{13}}\biggr) \, 
B\biggl(\frac{x^2_{14}}{x^2_{13}},\frac{x^2_{34}}{x^2_{13}}\biggr) + 
\\  
\bigskip 
& & 
 \bigl(\ln(x^2_{13}) + \ln(x^2_{34})\bigr) 
\ln\biggl(\frac{x^2_{14}}{x^2_{13}}\biggr) \biggr] + O(\epsilon) \, . \nonumber 
\end{eqnarray} 
As for $J_{a3}$, since we learned above how to extract a simple pole, without  
any further calculation we get  
\begin{equation} 
J_{a3} = \frac{\pi^4}{2} \, 
\frac{\ln(x^2_{34})^2}{x^2_{14} \, x^2_{34}} + O(\epsilon) \, . 
\end{equation} 
 
\subsubsection*{Diagrams~\textbf{(b)} and~\textbf{(c)}} 
It is convenient to compute directly the sum of these graphs, 
since compensations between the two terms lead to a 
simpler result.  
The $x$-space integrals are of the form 
\begin{equation} 
J_b + J_c \, = \, 
\int \frac{d^4x_5}{x^2_{15} \, x^2_{35}} \int 
\frac{d^4x_6}{x^2_{56} x^2_{36} x^4_{46}} + (3 \leftrightarrow 4) 
\, . \label{bc} 
\end{equation} 
The second subintegral is divergent. It is given by  
$I_{a1} = \epsilon J_{a1}/\rho $ 
which was calculated already in eqs.~(\ref{J1})-(\ref{JJ1}). 
Its coordinate behaviour at 
orders $O(1/\epsilon)$ and $O(1)$ is essentially $1/(x^2_{34} x^2_{45})$, 
which does not lead to overlapping divergences in the second  
integration. We can then expand $I_{a1}$ 
obtaining (one can neglect terms of order $O(\epsilon)$ and higher) 
\begin{eqnarray} 
J_b \, + \, J_c &=&\frac{\rho}{\epsilon} \frac{1}{(x^2_{34})^{(1+2\epsilon)}} 
\int \frac{d^{(4+2\epsilon)}x_5}{(x^2_{15} \, x^2_{35} \, 
x^2_{45})^{(1+\epsilon)}} 
\biggl[\biggl(\frac{x^2_{45}}{x^2_{35}}\biggr)^\epsilon + 
\biggl(\frac{x^2_{35}}{x^2_{45}}\biggr)^\epsilon\biggr] 
\\ 
& = & \frac{\rho}{\epsilon} 
\frac{2}{(x^2_{34})^{(1+\epsilon)}} \int 
\frac{d^{(4+2\epsilon)}x_5}{(x^2_{15} \, x^2_{35} \, 
x^2_{45})^{(1+\epsilon)}} + 2 \pi^2 \, 
\frac{\ln(x^2_{34})}{x^2_{34}} \int \frac{d^4x_5}{x^2_{15} \, 
x^2_{35} \, x^2_{45}} \nonumber 
\end{eqnarray} 
Note that the first term contains trilogarithms but is again identically 
cancelled by $Z|_{g^2}$ times an order $g^2$ diagram. The second integral is 
once again the box $B$. The end result of the calculation is in exact agreement 
with~(\ref{W}). 
 
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\end{document}


\bibitem{fut} 
M. Bianchi, B. Eden, G.C. Rossi and Ya.S. Stanev, in preparation.


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%  \begin{picture}(10000,18000)
%  \drawline\gluon[\S\REG](0,16000)[8]
%  \drawline\fermion[\SW\REG](\gluonbackx,\gluonbacky)[2000]
%  \drawline\fermion[\SE\REG](\gluonbackx,\gluonbacky)[2000]
%  \end{picture}
%  \end{document}
%
%**************************************************************************
%
\message{FEYNMAN:  For generating Feynman Diagrams in LaTex}
\message{Mark 1.0 Last Altered by MJSL 2/89}
\textheight 650pt \textwidth 400pt  % Page size set.
\setlength{\unitlength}{0.01pt}
\gdef\Feynmanlength{\setlength{\unitlength}{0.01pt}}  % Say \Feynmanlength
\gdef\unlock{\catcode`\@=11}
%  Allows use of "@" in macro names, like PLAIN.TEX does.
\gdef\lock{\catcode`\@=12}%  Change @'s back to their normal category code.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% ESTABLISH SOME COUNTERS TO INDICATE WHICH TYPE OF PARTICLE LINE IS
% BEING DRAWN.
%
% EACH TYPE OF  PARTICLE LINE HAS A CODE OF 3 1-DIGIT NUMBERS.
% THE FIRST DIGIT INDICATES THE BASIC PARTICLE TYPE (GLUON, PHOTON ETC.)
% THE SECOND INDICATES THE DIRECTION (HORIZONTAL, ANGLED, VERTICLE)
% THE THIRD INDICATES SUB-CASES (DIFFERENT SIZES ETC.)
%
%                             THE CODE
%                             ========
%
%  FIRST DIGIT:  \LINETYPE or \LTYPE
%
%    1   FERMION              (SOLID LINE)
%    2   SCALAR               (DASHED LINE)
%    3   PHOTON/VECTOR BOSON  (UNDULATING LINE) 
%    4   GLUON                (CURLY, LOOPING LINE)
%    5   ESPECIAL             (USER-DEFINED)
%
%  SECOND DIGIT:  \LINEDIRECTION or \LDIR
%
%    Eight directions are possible...the eight compass points.
%    North is at the top of the page
%
%    0   \N   North        (Vertical;  bottom to top)
%    1   \NE  North-East   (45 degrees; to the upper right)
%    2   \E   East         (Horizontal;  left to right)
%    3   \SE  South-East   (45 degrees; to the lower right)
%    4   \N   South        (Vertical;  top to bottom)
%    5   \SW  South-West   (45 degrees; to the lower left)
%    6   \W   West         (Horizontal;  right to left)
%    7   \NE  North-West   (45 degrees; to the upper left)
%
%  THIRD DIGIT:  \LINECONFIGURATION or \LCONFIG
%
%    0   STANDARD CONFIGURATION
%    1   FLIPPED (INVERTED; UPSIDE-DOWN) CONFIGURATION (PHOTONS & GLUONS)
%    2   CURLY
%    3   FLIPPEDCURLY
%    4   FLAT (PHOTONS AND GLUONS - CERTAIN DIRECTIONS ONLY)
%    5   FLIPPEDFLAT
%    6   CENTRAL (GLUONS ONLY);  LONGPHOTON (PHOTONS ONLY)
%    7   FLIPPEDCENTRAL; FLIPPEDLONG
%    8   SQUASHED (GLUONS ONLY - ONLY `EASTBOUND')
%    9   FLIPPEDSQUASHED
%
%.............................................................
%
\global\newcount\LINETYPE                     
\global\newcount\LINEDIRECTION
\global\newcount\LINECONFIGURATION
\newcommand{\LTYPE}{\LINETYPE}
\newcommand{\LDIR}{\LINEDIRECTION}
\newcommand{\LCONFIG}{\LINECONFIGURATION}
%DEFAULTS:  Horizontal fermion.   
\global\LINETYPE=1  \global\LINEDIRECTION=0  \global\LINECONFIGURATION=0
%  The parametric code names.  Don't change these.
\global\newcount\fermion    \fermion=1
\global\newcount\scalar     \scalar=2
\global\newcount\photon     \photon=3
\global\newcount\gluon      \gluon=4
\global\newcount\especial   \especial=5
\gdef\N{0}  \gdef\NE{1}  \gdef\E{2}   \gdef\SE{3}
\gdef\S{4}  \gdef\SW{5}  \gdef\W{6}   \gdef\NW{7}
\global\newcount\REG            \global\REG=0
\global\newcount\FLIPPED        \global\FLIPPED=1
\global\newcount\CURLY          \global\CURLY=2
\global\newcount\FLIPPEDCURLY   \global\FLIPPEDCURLY=3
\global\newcount\FLAT           \global\FLAT=4
\global\newcount\FLIPPEDFLAT    \global\FLIPPEDFLAT=5
\global\newcount\CENTRAL        \global\CENTRAL=6
\global\newcount\FLIPPEDCENTRAL \global\FLIPPEDCENTRAL=7
\gdef\LONGPHOTON{6}             \gdef\FLIPPEDLONG{7}
\global\newcount\SQUASHEDGLUON  \global\SQUASHEDGLUON=8
\gdef\SQUASHED{\SQUASHEDGLUON}
%\global\newcount\FLIPPEDSQUASHEDGLUON  \FLIPPEDSQUASHEDGLUON=9
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SOME COUNTERS AND DEFINITIONS FOR POSITIONS AND LENGTHS OF LINES & FEATURES %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcount\adjx \adjx=0
\newcount\adjy \adjy=0
\global\newdimen\BIGPHOTONS     \BIGPHOTONS=0pt  %  DEFAULT:  10 & 11-PT PHOTONS
\gdef\bigphotons{\global\BIGPHOTONS=12pt}%FOR 12-PT DOCS DRAWING E-W PHOTONS.
\global\newdimen\THICKPHOTONS     \THICKPHOTONS=0pt  %  FOR E-W PHOTONS 
\global\newdimen\THICKPHOTONSWITCH    \THICKPHOTONSWITCH=0pt
\gdef\THICKPHOTONTEST{
\THICKPHOTONSWITCH=0pt
\ifdim\THICKPHOTONS=0pt \relax
  \else \ifnum\LTYPE=3
           \ifnum\LDIR=2 \THICKPHOTONSWITCH=1pt \fi % THICK \E PHOTON
           \ifnum\LDIR=6 \THICKPHOTONSWITCH=1pt \fi % THICK \W PHOTON
        \fi
\fi
}  % end of THICKPHOTONTEST
\gdef\THICKLINES{\thicklines  \THICKPHOTONS=1pt}
\gdef\THINLINES{\thinlines  \THICKPHOTONS=0pt}
\global\newcount\phantomswitch   \global\phantomswitch=0
\global\newcount\stemlength   \global\stemlength=275   % Default STEM length.
\global\newcount\absstemlength        % A copy of STEM length.
\global\newcount\stemlengthx          % FOR STEMS on particle lines
\global\newcount\stemlengthy          % FOR STEMS on particle lines
\newdimen\FRONTSTEM  \FRONTSTEM=0pt   % FOR STEMS
\newdimen\BACKSTEM   \BACKSTEM=0pt    % FOR STEMS
\newdimen\EITHERSTEM \EITHERSTEM=0pt  % FOR STEMS
\gdef\frontstemmed{\FRONTSTEM=1pt}            % FOR STEMS
\gdef\backstemmed{\BACKSTEM=1pt}              % FOR STEMS
\gdef\stemmed{\FRONTSTEM=1pt  \BACKSTEM=1pt}    % FOR STEMS
\global\newcount\arrowlength                % FOR ARROWS
\global\newdimen\ATTIP   \global\ATTIP=0pt  % FOR ARROWS
\global\newdimen\ATBASE  \global\ATBASE=1pt % FOR ARROWS
\global\newcount\unitboxnumber  % SHOWS THE NUMBER OF `UNIT BOXES' IN LINE
\global\newcount\unitboxnumberpo  % One more than \unitboxnumber (in GLUONSETUP)
\global\newcount\particlelengthx  % THE X-LENGTH OF THE PARTICLE LINE
\gdef\plengthx{\particlelengthx}
\global\newcount\particlelengthy  % THE Y-LENGTH OF THE PARTICLE LINE
\gdef\plengthy{\particlelengthy}  
\global\newcount\boxlengthx  % THE X-LENGTH OF THE BOX:  abs(plengthx) usually
\global\newcount\boxlengthy  % THE y-LENGTH OF THE box:  abs(plengthy) usually
\global\newcount\particleadjustx  % Replaces \gluonadjustx, \scalaradjustx etc.
\global\newcount\particleadjusty  % Replaces \gluonadjusty, \scalaradjusty etc.
\global\newcount\particlelength   % The LENGTH of a particle line BOX (x)
\global\newcount\particlefrontx
\gdef\pfrontx{\particlefrontx}
\global\newcount\PFRONTx
\global\newcount\particlefronty
\gdef\pfronty{\particlefronty}
\global\newcount\PFRONTy
\global\newcount\particlebackx
\gdef\pbackx{\particlebackx}
\global\newcount\particlebacky
\gdef\pbacky{\particlebacky}
\global\newcount\particlemidx
\gdef\pmidx{\particlemidx}
\global\newcount\particlemidy
\gdef\pmidy{\particlemidy}
% SOME SPECIAL DEFS FOR \SCALARs:
\global\newcount\seglength  \global\newcount\gaplength
\global\gaplength=850  %default
\global\seglength=1416  % Length of each seg not including `ends' for attachment
% Now some storage locations for the user:
\global\newcount\Xone    \global\newcount\Yone    % user co-ords (\Xone,\Yone)
\global\newcount\Xtwo    \global\newcount\Ytwo    % user co-ords (\Xtwo,\Ytwo)
\global\newcount\Xthree  \global\newcount\Ythree  % user's (\Xthree,\Ythree)
\global\newcount\Xfour   \global\newcount\Yfour   % user co-ords (\Xfour,\Yfour)
\global\newcount\Xfive   \global\newcount\Yfive   % user co-ords (\Xfive,\Yfive)
\global\newcount\Xsix    \global\newcount\Ysix    % user co-ords (\Xsix,\Ysix)
\global\newcount\Xseven  \global\newcount\Yseven  % user's (\Xseven,\Yseven)
\global\newcount\Xeight  \global\newcount\Yeight  % user's (\Xeight,\Yeight)
%
%  SOME COUNTERS IDENTIFYING VARIOUS LINE PORTIONS AND DIMENSIONS:
%
\newsavebox{\lastline}  %  Default name for an unnamed particle line.
\global\newcount\numlineparts   % Num of pieces each unitbox of the line needs
\global\newcount\upperlineadjx  \upperlineadjx=0  %Default
\global\newcount\upperlineadjy  \upperlineadjy=0  %Default
\global\newcount\lowerlineadjx  \lowerlineadjx=0  %Default
\global\newcount\lowerlineadjy  \lowerlineadjy=0  %Default
\global\newcount\thirdlineadjx  \thirdlineadjx=0  %Default
\global\newcount\thirdlineadjy  \thirdlineadjy=0  %Default
\global\newcount\fourthlineadjx \fourthlineadjx=0  %Default
\global\newcount\fourthlineadjy \fourthlineadjy=0  %Default
\global\newcount\unitboxwidth   \unitboxwidth=1000%Default
\global\newcount\unitboxheight  \unitboxheight=0  %Default
\global\newcount\numupperunits  \numupperunits=8  %Default
\global\newcount\numlowerunits  \numlowerunits=8  %Default
\global\newcount\numthirdunits  \numthirdunits=8  %Default
\global\newcount\numfourthunits \numfourthunits=8  %Default
%  Some counters.  =0 until a line-type is drawn. Then=1.
\global\newcount\fermioncount   \global\fermioncount=0    
\global\newcount\scalarcount    \global\scalarcount=0    
\global\newcount\photoncount    \global\photoncount=0    
\global\newcount\gluoncount     \global\gluoncount=0    
\global\newcount\especialcount  \global\especialcount=0    
\global\newcount\vertexcount    \global\vertexcount=-1
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%     AUXILIARY ROUTINES FOR SETTING PARTICLE DIRECTIONS     %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\global\newcount\XDIR
\global\newcount\YDIR
\gdef\SETDIR{  % SETS THE DIRECTIONS
\ifcase\LDIR 
     \global\XDIR=0  \global\YDIR=1   %\N  case.
\or  \global\XDIR=1  \global\YDIR=1   %\NE case.
\or  \global\XDIR=1  \global\YDIR=0   %\E  case.
\or  \global\XDIR=1  \global\YDIR=-1  %\SE case.
\or  \global\XDIR=0  \global\YDIR=-1  %\S  case.
\or  \global\XDIR=-1 \global\YDIR=-1  %\SW case.
\or  \global\XDIR=-1 \global\YDIR=0   %\W  case.
\or  \global\XDIR=-1 \global\YDIR=1   %\NW case.
\else\DIRECTERROR 
\fi}  % END OF \SETDIR
%
\gdef\moduloeight#1{
\ifnum#1>7 \global\advance #1 by -8 
\relax
\moduloeight#1 
\relax
\else \relax  
\fi}
\gdef\multroothalf#1{\global\multiply #1 by 7071 \global\divide #1 by 10000}
\gdef\negate#1{\global\multiply #1 by -1}
\gdef\double#1{\global\multiply #1 by 2}
\gdef\slanttest(#1,#2){ 
\ifodd\LDIR
\multiply #1 by 7071  \divide #1 by 10000
\multiply #2 by 7071  \divide #2 by 10000
\fi
}
\gdef\gslanttest(#1,#2){
\ifodd\LDIR
\multroothalf#1
\multroothalf#2
\fi
}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% AUXILIARY ROUTINES FOR SETTING PARTICLE LENGTHS & POSTIONS %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\gdef\setplength{ % calcs length of particle line
\global\particlelengthx=\unitboxwidth
\global\particlelengthy=\unitboxheight
\global\multiply \particlelengthx by \unitboxnumber
\global\multiply \particlelengthy by \unitboxnumber
\global\advance \particlelengthx by \particleadjustx
\global\advance \particlelengthy by \particleadjusty
}
\gdef\boxlengthdefault{  % DEFAULT FOR BOX SIZES IN \drawas
\global\boxlengthx=\plengthx
\global\boxlengthy=\plengthy
\ifnum\plengthx<0 \global\multiply\boxlengthx by -1 \fi
\ifnum\plengthy<0 \global\multiply\boxlengthy by -1 \fi
}
\gdef\rearcoords{  %  CALCULATES THE CO-ORDINATES OF THE BACK OF PARTICLE LINE
\global\particlebacky=\particlefronty 
\global\particlebackx=\particlefrontx 
\global\advance \particlebackx by \particlelengthx
\global\advance \particlebacky by \particlelengthy
}
\gdef\midcoords{  %  CALCULATES THE CO-ORDINATES OF THE MID OF PARTICLE LINE
\global\particlemidy=\particlefronty
\global\particlemidx=\particlefrontx
\global\stemlengthx=\particlelengthx  % Convenient variables not being used
\global\stemlengthy=\particlelengthy  
\global\divide\stemlengthx by 2
\global\divide\stemlengthy by 2
\global\advance \particlemidx by \stemlengthx
\global\advance \particlemidy by \stemlengthy
}
\gdef\setparticle{\setplength\rearcoords\midcoords\boxlengthdefault}  %sets line
%
\gdef\setcoords(#1,#2,#3)(#4,#5,#6)[#7,#8]{  
% Sets co-ords of first 3 line-parts of a line and the unitbox height and width
% Used by photons and gluons.
\global\upperlineadjx=#1
\global\lowerlineadjx=#2
\global\thirdlineadjx=#3
\global\upperlineadjy=#4
\global\lowerlineadjy=#5
\global\thirdlineadjy=#6
\global\unitboxwidth=#7
\global\unitboxheight=#8
}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                     ROUTINES FOR DRAWING LINES                              %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% **************   ROUTINE FOR DRAWING STORED LINES AND PICTURES   *************
%
\gdef\drawoldpic#1(#2,#3){  % DRAWS PRE-SAVED PICTURE
\global\particlefrontx=#2
\global\particlefronty=#3
\rearcoords  
\midcoords
\put(#2,#3){\usebox{#1}}
}
%
\gdef\drawsavedline`#1' as #2[#3#4](#5,#6)[#7]{
\global\LINETYPE=#2
\global\LINEDIRECTION=#3
\global\LINECONFIGURATION=#4
\global\particlefrontx=#5
\global\particlefronty=#6
\global\unitboxnumber=#7  
% Formerly called \numhalfwiggles,\numdashes, \numloops, \fermionlength
% #1 is saved linename;   #2 is \LINETYPE;    #3 is \LINEDIRECTION
% #4 is \LINECONFIGURATION (#5,#6)=(x,y) co-ords;  #7 is linelength (eg#wiggles)
\selectcase
\rearcoords% moved from before selectcase.
\midcoords
\ifnum\phantomswitch=0 \drawas{#1}\fi
% if \phantomswitch=1 then just set the line up and don't draw it.
}

\gdef\startphantom{\phantomswitch=1} % BEGIN PHANTOM MODE.
\gdef\stopphantom{\phantomswitch=0}  % END PHANTOM MODE.
% USE AS: 
% \startphantom...\drawline\gluon[...]...\drawvertex\photon...\stopphantom

\gdef\drawas#1{
\global\savebox{#1}(\boxlengthx,\boxlengthy){
\setlength{\unitlength}{0.01pt}
\begin{picture}(\boxlengthx,\boxlengthy)
\multiput(\upperlineadjx,\upperlineadjy)(\unitboxwidth,\unitboxheight)
{\numupperunits}{\upperunitbox}
\ifnum\numlineparts > 1  %  If the line needs 2 parts per unit or more
\multiput(\lowerlineadjx,\lowerlineadjy)(\unitboxwidth,\unitboxheight)
{\numlowerunits}{\lowerunitbox}  
\fi
\ifnum\numlineparts > 2  %  If the line needs 3 parts per unit or more
\multiput(\thirdlineadjx,\thirdlineadjy)(\unitboxwidth,\unitboxheight)
{\numthirdunits}{\thirdunitbox}  
\fi
\ifnum\numlineparts > 3  %  If the line needs 4 parts per unit or more
\multiput(\fourthlineadjx,\fourthlineadjy)(\unitboxwidth,\unitboxheight)
{\numfourthunits}{\lowerunitbox}  
\fi
\end{picture} }
% CHECK STEMS
\global\PFRONTx=\pfrontx  \global\PFRONTy=\pfronty   %save this value
\SETFRONTSTEM
% Now take into account the possibility of THICK E-W photons (drawn twice)
\THICKPHOTONTEST
\ifdim\THICKPHOTONSWITCH=1pt\global\advance\PFRONTy by 20  \fi
\put(\PFRONTx,\PFRONTy) {\usebox{#1}}   %\pfrontX,Y=\particlefrontx,y
%\put(\particlefrontx,\particlefronty) {\usebox{#1}}
\ifdim\THICKPHOTONSWITCH=1pt
\global\advance\PFRONTy by -40
\put(\PFRONTx,\PFRONTy) {\usebox{#1}}   % The second \E or \W photon ->thicker
\global\advance \PFRONTy by 20  %re-adjust:  advanced by -20 in total above.
\fi  %End of \ifdim\THICKPHOTONSWITCH=1
\SETBACKSTEM
\seglength=1416   \gaplength=850   % Re-set \SCALR defaults.
}
%
% *********   ROUTINES FOR STORING LINES  *******
%

\gdef\drawandsaveline`#1' as #2[#3#4](#5,#6)[#7]{
% #1 is saved linename;   #2 is \LINETYPE;    #3 is \LINEDIRECTION
% #4 is \LINECONFIGURATION (#5,#6)=(x,y) co-ords;  #7 is linelength (eg#wiggles)
\global\newsavebox{#1}
\drawsavedline`#1' as #2[#3#4](#5,#6)[#7]
}

\gdef\drawline#1[#2#3](#4,#5)[#6]{   % Draw line but don't name it.
\drawsavedline`\lastline' as #1[#2#3](#4,#5)[#6]}

\gdef\saveas#1{  %  For saving a line after the fact.
\global\newsavebox#1
\drawas#1}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                      %
%                           C A S E S                                  %
%                           ---------                                  % 
%                                                                      %       
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% ERROR MESSAGES FOR INCORRECT CASE SPECIFICATION:
\gdef\TYPEERROR{\message{*** ERROR IN PARTICLE TYPE SELECTION ***}
\message{+++ Try with line type \fermion,\scalar,\photon,\gluon 
(see manual) +++}\SETERR}
\gdef\DIRECTERROR{\SETERR\message{*** ERROR IN PARTICLE DIRECTION SELECTION ***}
\message{+++ Try again with direction N, NE, E, SE  etc. or see manual +++}}
\gdef\UNIMPERROR{\message{*** ERROR IN PARTICLE OPTIONS SELECTION ***}
\message{
+++ The requested options combination has not yet been implemented +++}\SETERR}
\gdef\SETERR{\gdef\upperunitbox{{\tiny Error}}  % PRINTS `error' in diagram.
\gdef\lowerunitbox{\relax}
\gdef\thirdunitbox{\relax}
}
\gdef\neglengthcheck{\ifnum\unitboxnumber < 1 
\message{   *** ERROR:  PARTICLE OF NEGATIVE OR ZERO LENGTH REQUESTED. ***   }
\message{   ***         TAKING ABSOLUTE VALUE. ***   }\negate\unitboxnumber \fi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\gdef\selectcase{  
\neglengthcheck   %  check for particles of negative length.
% select PARTICLE alignment:
\SETDIR  
%  Select particle type
\ifcase\LINETYPE
\TYPEERROR  % \LINETYPE=0 case.
\or \selectfermion  % \LINETYPE=1 case.
\or \selectscalar   % \LINETYPE=2 case.
\or \selectphoton   % \LINETYPE=3 case.
\or \selectgluon    % \LINETYPE=4 case.
\or \selectespecial % \LINETYPE=5 case.
\else \TYPEERROR \fi  }
%%%%%%%%%%% (1) FERMIONS %%%%%%%%%%% 
\gdef\selectfermion{
% Input fermion-setup stuff ONLY IF HAVE NOT DONE SO YET.
% This avoids having to process a fermion if none are drawn.
\ifnum\fermioncount=0 \input FERMIONSETUP \fi   
%                  CONTAINS fermion DEFINITIONS.
\global\advance\fermioncount by 1  % Counts number of fermions drawn. 
\ALLfermion   
}
%%%%%%%%%%% (2) SCALARS %%%%%%%%%%% 
\gdef\selectscalar{
% Input scalar-setup stuff ONLY IF HAVE NOT DONE SO YET.
% This avoids having to process a scalar if none are drawn.
\ifnum\scalarcount=0 \input SCALARSETUP \fi   
%                 CONTAINS scalar DEFINITIONS.
\global\advance\scalarcount by 1  % Counts number of scalars drawn. 
\ALLscalar
}
%%%%%%%%%%% (3) PHOTONS %%%%%%%%%%% 
\gdef\selectphoton{   % RECURSIVELY RE-DEFINED IN PHOTONSETUP(23+).TEX.
% Input photon-setup stuff ONLY IF HAVE NOT DONE SO YET.
% This avoids having to process a photon if none are drawn.
\ifnum\photoncount=0 \input PHOTONSETUP  \fi   
\selectphoton
%CONTAINS PHOTON DEFINITIONS. 
}
%%%%%%%%%%% (4) GLUONS %%%%%%%%%%% 
\gdef\selectgluon{   % RECURSIVELY RE-DEFINED IN GLUONSETUP(25+).TEX.
% Input gluon-setup stuff ONLY IF HAVE NOT DONE SO YET.
% This avoids having to process a gluon if none are drawn.
\ifnum\gluoncount=0 \input GLUONSETUP  \fi
\selectgluon
%                  CONTAINS gluon DEFINITIONS.
}
%%%%%%%%%%% (5) SPECIAL - USER DEFINED %%%%%%%%%%% 
\gdef\selectespecial{\UNIMPERROR}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                   ROUTINES FOR DRAWING VERTICES                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Input vertex-setup stuff ONLY IF HAVE NOT DONE SO YET.                      %
% This avoids having to process a vertex if none are drawn. 
\gdef\checkvertex{ %immediately re-defines \drawvertex,\vertexcap,\linkvertex...
\ifnum\vertexcount=-1   \input VERTEX  \fi}
% RECURSIVE DEFINITIONS:
\gdef\drawvertex#1[#2#3](#4,#5)[#6]{\checkvertex\drawvertex#1[#2#3](#4,#5)[#6]}
\gdef\vertexcap#1{\checkvertex\vertexcap#1}
\gdef\vertexcaps{\checkvertex\vertexcaps}
\gdef\vertexlink#1{\checkvertex\vertexlink#1}
\gdef\vertexlinks{\checkvertex\vertexlinks}
\gdef\stemvertex#1{\checkvertex\stemvertex#1}
\gdef\stemvertices{\checkvertex\stemvertices}
\gdef\flipvertex{\checkvertex\flipvertex}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                   ROUTINES FOR DRAWING ARROWS                               %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% SYNTAX:  \drawarrow[\NW\ATBASE](\pmidx,\pmidy)  etc.
\global\arrowlength=349  % Length of arrow
\gdef\drawarrow[#1#2](#3,#4){
\global\LDIR=#1
\SETDIR
\global\boxlengthx=#3  % Just a convenient variable name.  No significance.
\global\boxlengthy=#4  % The Arrow co-ordinates.
\ifdim#2=1pt  % CASE \ATBASE WHERE THE CO-ORDS ARE AT THE ARROWS BASE.
   %   #2 IS either \ATTIP or \ATBASE...Depending where it is to be positioned.
\adjx=\arrowlength      \adjy=\arrowlength
\multiply\adjx by \XDIR \multiply\adjy by \YDIR  % Set in \SETDIR
\slanttest(\adjx,\adjy)
\global\advance\boxlengthx by \adjx    \global\advance\boxlengthy by \adjy
\fi
\ifnum\phantomswitch=0\put(\boxlengthx,\boxlengthy){\vector(\XDIR,\YDIR){0}}\fi
}  % END OF \drawarrow.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                     ROUTINES FOR DRAWING STEMS                              %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\gdef\SETFRONTSTEM{
\EITHERSTEM=\FRONTSTEM   \advance\EITHERSTEM by \BACKSTEM
\ifdim\EITHERSTEM>0pt
\global\stemlengthx=\stemlength   \global\stemlengthy=\stemlength   
\global\absstemlength=\stemlength   
\SETDIR
\gslanttest(\stemlengthx,\stemlengthy)
\gslanttest(\absstemlength,\REG)  % the \REG is to use up the parameter space.
\ifnum\XDIR=0 \stemlengthx=0 \fi
\ifnum\YDIR=0 \stemlengthy=0 \fi
\global\multiply\stemlengthx by \XDIR
\global\multiply\stemlengthy by \YDIR
\ifdim\FRONTSTEM=1pt 
\ifnum\phantomswitch=0
          \put(\pfrontx,\pfronty){\line(\XDIR,\YDIR){\absstemlength}}\fi
\global\advance\plengthx by \stemlengthx
\global\advance\plengthy by \stemlengthy
\global\advance\PFRONTx by \stemlengthx   
\global\advance\PFRONTy by \stemlengthy
\global\advance\pmidx by \stemlengthx
\global\advance\pmidy by \stemlengthy
\global\advance\pbackx by \stemlengthx
\global\advance\pbacky by \stemlengthy
% FOR STEMMED PHOTONS AND GLUONS, \photonfront,back(x,y) are for the
% photon proper (no stem) while \pbackx,y include the stems.
\ifnum\LTYPE=3
\global\photonfrontx=\PFRONTx  \global\photonfronty=\PFRONTy
\global\photonbackx=\pbackx    \global\photonbacky=\pbacky
\fi  % END LTYPE
\ifnum\LTYPE=4
\global\gluonfrontx=\PFRONTx  \global\gluonfronty=\PFRONTy
\global\gluonbackx=\pbackx    \global\gluonbacky=\pbacky
\fi  % END LTYPE
\fi  % END FRONTSTEM
\fi  % END EITHERSTEM
}    % end of \SETFRONTSTEM
%
\gdef\SETBACKSTEM{
\ifdim\BACKSTEM=1pt 
\ifnum\phantomswitch=0
       \put(\pbackx,\pbacky){\line(\XDIR,\YDIR){\absstemlength}}\fi
\global\advance\plengthx by \stemlengthx
\global\advance\plengthy by \stemlengthy
\global\advance\pbackx by \stemlengthx
\global\advance\pbacky by \stemlengthy
\fi  % END BACKSTEM
\global\stemlength=275  \FRONTSTEM=0pt  \BACKSTEM=0pt % Reset default switches.
}    % END OF \SETBACKSTEM 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                              LOOPS                                        %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\gdef\drawloop#1[#2#3](#4,#5){  %RECURSIVE.  
\input LOOPS  % contains loops definitions
\drawloop#1[#2#3](#4,#5)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Feynmanlength  % Set length scale to centipoints.

hep-th0205321.extracted_bib
\begin{thebibliography}{123} 
         
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H. Liu and A.A. Tseytlin, 
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J. Erdmenger and M. Perez-Victoria, 
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M. Bianchi, M.B. Green, S. Kovacs and G.C. Rossi,  
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K. Intriligator,  
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M. Bianchi, M.B. Green and  S. Kovacs, 
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B. Eden, A.C. Petkou, C. Schubert and E. Sokatchev, 
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M. Bianchi, S. Kovacs, G.C. Rossi and Ya.S. Stanev, 
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B. Eden, P.S. Howe, C. Schubert, E. Sokatchev and P.C. West, 
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B. Eden, P.S. Howe, C. Schubert, E. Sokatchev and P.C. West, 
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B. Eden, P.S. Howe, A. Pickering, E. Sokatchev and P.C. West, 
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{\it Nucl. Phys.} {\bf B581} (2000) 523, {\tt hep-th/0001138}; 
B. Eden, C. Schubert and E. Sokatchev, 
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{\it Phys. Lett.} {\bf B482} (2000) 309, {\tt hep-th/0003096}. 
 
\bibitem{bkrs1} 
M. Bianchi, S. Kovacs, G.C. Rossi and Ya.S. Stanev,  
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{\it JHEP} {\bf 9908} (1999) 020, {\tt hep-th/9906188}. 
 
\bibitem{bkrs2} 
M. Bianchi, S. Kovacs, G.C. Rossi and Ya.S. Stanev, 
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{\it Nucl. Phys.} {\bf B584} (2000) 216, {\tt  hep-th/0003203}. 
 
\bibitem{esok} 
G. Arutyunov, B. Eden, A.C. Petkou and E. Sokatchev, 
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{\it Nucl. Phys.} {\bf B620} (2002) 380, {\tt hep-th/0103230}; 
G. Arutyunov, B. Eden and E. Sokatchev, 
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{\it Nucl. Phys.} {\bf B619} (2001) 359, {\tt hep-th/0105254}; 
B. Eden and E. Sokatchev, 
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{\it Nucl. Phys.} {\bf B618} (2001) 259, {\tt hep-th/0106249}; 
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{\it Nucl. Phys.} {\bf B626} (2002) 265, {\tt hep-th/0107212}. 
 
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P.J. Heslop and P.S. Howe,  
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\bibitem{bps1} 
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P.J. Heslop and P.S. Howe, 
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{\it Phys. Lett.} {\bf B516} (2001) 367, {\tt hep-th/0106238} 
 
\bibitem{aft}  
G. Arutyunov, S. Frolov and A.C. Petkou,  
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\bibitem{ppwaves} 
M. Blau, J. Figueroa-O'Farrill, C. Hull and G. Papadopoulos, 
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D. Berenstein, J.M. Maldacena and H. Nastase, 
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N.R. Constable, D.Z. Freedman, M. Headrick, S. Minwalla,  
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\bibitem{berko} 
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\end{thebibliography}