\boldmath High-Energy Limit of Quantum Electrodynamics beyond Sudakov Approximation \unboldmath
Alexander A. Penin
We study the high-energy behavior of the scattering amplitudes in quantum electrodynamics beyond the leading order of the small electron mass expansion in the leading logarithmic approximation. In contrast to the Sudakov logarithms, the mass-suppressed double-logarithmic radiative corrections are induced by a soft electron pair exchange and result in enhancement of the power-suppressed contribution. Possible applications of our result to the analysis of the high-energy processes in quantum chromodynamics is also discussed.
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\begin{document}

\title{
\boldmath High-Energy Limit of Quantum Electrodynamics beyond Sudakov
Approximation
\unboldmath}
\author{Alexander A. Penin}
\affiliation{Department of Physics, University of Alberta, Edmonton, Alberta T6G
2J1, Canada}
\affiliation{Institut f\"ur Theoretische Teilchenphysik,
Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany}
%\preprint{ALBERTA-THY-18-14}
\begin{abstract}
We study the high-energy behavior of the scattering amplitudes in  quantum
electrodynamics beyond the leading order of the small electron mass expansion in
the leading logarithmic approximation. In contrast to the  Sudakov logarithms,
the mass-suppressed double-logarithmic radiative corrections  are induced by a
soft electron pair exchange and result in enhancement of the power-suppressed
contribution. Possible applications of our result to the analysis of the
high-energy processes in quantum chromodynamics is also discussed.

\end{abstract}
\pacs{11.15.Bt, 12.20.Ds, 12.38.Bx, 12.38.Cy}

\maketitle
In a renowned paper \cite{Sudakov:1954sw} V.V.~Sudakov derived the leading
asymptotic behavior of an electron scattering amplitude in quantum
electrodynamics (QED) at high energy. It is determined by the ``Sudakov''
radiative corrections, which include the second power of the  large logarithm of
the electron mass $m_e$ divided by a characteristic momentum transfer of the
process per each power of the fine structure constant $\alpha$. Sudakov double
logarithms exponentiate and result in a strong universal suppression of any
electron scattering amplitude with a fixed number of emitted photons in the
limit when all the kinematic invariants of the process are large. This result
plays a fundamental role in particle physics. Within different approaches it has
been  extended to the nonabelian gauge theories and to the subleading logarithms
\cite{Frenkel:1976bj,Mueller:1979ih,Collins:1980ih,Sen:1981sd,Sterman:1986aj,
Catani:1998bh}, which is crucial for a wide class of applications  from deep
inelastic scattering to Drell-Yan processes and the Higgs boson production. At
the same time no significant progress has been achieved in the study of the
logarithmically enhanced corrections to the subleading contributions suppressed
by a power of electron mass at high energies. However, the power-suppressed
contributions are of great interest. They can become asymptotically dominant at
very high energies due to Sudakov suppression of the leading  terms. At the
intermediate energies the power corrections in many cases are phenomenologically
important
\cite{Bernreuther:2004ih,Bernreuther:2005gw,Bonciani:2007eh,Bonciani:2008ep}.
Moreover, in contrast to the Euclidean operator product expansion
\cite{Wilson:1969zs} or nonrelativistic threshold dynamics \cite{Caswell:1985ui}
very little is known about the general all-order structure of the large
logarithms beyond the leading-power approximation in the high-energy limit,
which is a real challenge for the effective field theory approach. This problem
is now actively discussed in various contexts (see {\it e.g.}
\cite{Feucht:2004rp,Laenen:2010uz,Becher:2013iya,Banfi:2013eda,deFlorian:2014vta,
Anastasiou:2014lda}). In this Letter we make the first step toward the solution
of the problem and generalize the result of Ref.~\cite{Sudakov:1954sw} to the
leading power-suppressed contribution. We present a detailed analysis of the
electron scattering in the external field and later discuss the extension of the
result to more complex processes.

The  amplitude  ${\cal F}$  of the electron scattering in an external
field can be parameterized in the standard way  by the  Dirac and Pauli
form factors
%\begin{equation}
%{\cal F}=F_1\bar{\psi}(p_1)\gamma_\mu\psi(p_2) +\ldots
%\label{eq::Dirac}
%\end{equation}
\begin{equation}
{\cal F}=\bar{\psi}(p_1)\left(\gamma_\mu F_1+{i\sigma_{\mu\nu}q^\nu \over 2m_e}
F_2\right)\psi(p_2).
\label{eq::Dirac}
\end{equation}
The Pauli form factor $F_2$ does not contribute in the approximation discussed
in this Letter and we mainly focus on the high-energy behavior of the Dirac form
factor $F_1$. We consider the limit of the on-shell electron $p_1^2=p_2^2=m_e^2$
and the large Euclidean momentum transfer $Q^2=-(p_2-p_1)^2$ when the ratio
$\rho\equiv {m_e^2/Q^2}$ is positive and small. The Dirac form factor can then
be expanded in an asymptotic series in $\rho$
\begin{equation}
F_1=S_\lambda\sum_{n=0}^\infty \rho^n F^{(n)}_1,
\label{eq::F1}
\end{equation}
where $F^{(n)}_1$ are given by the power series in $\alpha$ with the
coefficients depending on $\rho$ only logarithmically. The factor
$S_\lambda=\exp{\left[-{\alpha\over 2\pi}B(\rho)\,
\ln\left({\lambda^2/m_e^2}\right)\right]}$ with $B(\rho)=\ln\rho+{\cal O}(1)$
accounts for the universal singular dependence of the amplitude on the auxiliary
photon mass $\lambda$ introduced to regulate the infrared divergences
\cite{Yennie:1961ad}. In the double-logarithmic approximation the leading term
is given by the Sudakov exponent $F^{(0)}_1=e^{-x}$, with $x={\alpha\over
4\pi}\ln^2\rho$ \cite{Jackiw:1968zz}. Let us outline our approach for the
analysis of the power-suppressed double-logarithmic contributions. We use the
expansion by regions method \cite{Beneke:1997zp,Smirnov:1997gx,Smirnov:2002pj}
to get a systematic expansion of the Feynman integrals in $\rho$. In this method
the coefficients $F^{(n)}_1$  are given by the sum over contributions of
different virtual momentum regions. Each contribution is represented by a
Feynman integral which in general is divergent. These spurious divergences
result from the process of  scale separation and have to be dimensionally
regulated. The singular terms cancel out in the sum of all regions but can be
used to determine the logarithmic contributions to $F^{(n)}_1$.  The
double-logarithmic contributions are determined by the leading singular behavior
of the integrals and can be found by the method developed in Refs.
\cite{Sudakov:1954sw,Jackiw:1968zz,Gorshkov:1966ht}. Though the method is blind
to the power corrections, it can be applied in this case since the expansion by
regions provides the integrals which are {\it homogeneous} in the expansion
parameter. Sudakov logarithms are produced by the soft virtual photons, which
are collinear to either $p_1$ or $p_2$.  We have found that such a configuration
of virtual momenta does not produce  double logarithms in the first order in
$\rho$. This observation agrees with the analysis \cite{Korchemsky:1987wg} of 
the cusp anomalous dimension, which determines the double-logarithmic
corrections to the light-like Wilson line with a cusp. For the large cusp angle
corresponding to the limit $\rho\to 0$  from the result of
Ref.~\cite{Korchemsky:1987wg} one gets
\begin{equation}
\Gamma_{cusp}=-{\alpha\over\pi}\ln\rho\left(1+{\cal O}(\rho^2)\right),
\end{equation}
with vanishing first-order term in $\rho$. Nevertheless, the ${\cal O}(\rho)$
double-logarithmic contribution does exist but originates from completely
different virtual momentum configuration described  below. Let us consider an
electron propagator $S={\slashed{p}_i-\slashed{l}+m_e\over (p_i-l)^2-m_e^2}$,
where $l$ is the momentum of a virtual photon with the propagator
$D={-g_{\mu\nu}\over l^2-\lambda^2}$. In the soft-photon limit $l\to 0$ the
electron propagator becomes eikonal $S\approx-{\slashed{p}_i+m_e\over 2p_il}$
and develops  a collinear singularity when $l$ is parallel to $p_i$.
Alternatively, we may consider the soft-electron limit $l'\to 0$, where
$l'=p_i-l$. Then the electron propagator becomes scalar $S\approx{m_e\over
l'^2-m_e^2}$ while the photon propagator becomes eikonal
$D\approx{g_{\mu\nu}\over 2p_il'-m_e^2+\lambda^2}$. Thus the roles of the
electron and photon propagators are exchanged. The existence of non-Sudakov
double-logarithmic contributions due to soft electron exchange has actually been
known for a long time \cite{Gorshkov:1966ht,Fadin:1997sn}. However in our case
this virtual momentum configuration does not produce a  double-logarithmic
contribution in one loop because the  momentum shift distorts  the eikonal
structure of the second electron propagator and removes the soft singularity at
small $l'$ necessary to get  the second power of the large logarithm.
\begin{figure}[t]
\includegraphics[width=1.5cm]{sud1.eps}
\hspace*{10mm}\includegraphics[width=1.5cm]{sud2.eps}
\hspace*{10mm}\includegraphics[width=1.5cm]{sud3.eps}
\caption{\label{fig::diagram2loop}  Different representations of the
two-loop Feynman diagram giving the leading power-suppressed double-logarithmic
contribution. In figure (c) the double line arrow represents the soft electron
pair propagator and the empty blobs represent  the nonlocal interaction of the
soft electron pair to the eikonal electrons  and positrons.
}
\end{figure}
This may be avoided only in the two-loop diagram  of  nonplanar topology,
Fig.~\ref{fig::diagram2loop}(a). After shifting the photon virtual momenta by
$p_1$ and $p_2$  the diagram can be twisted into the shape of
Fig.~\ref{fig::diagram2loop}(b,c) with soft  electron pair exchange between the
eikonal lines.  The corresponding contribution has an explicit suppression
factor $m_e^2$ from two soft electron propagators. Hence the integration over
the virtual momenta can be performed in the leading order of the small electron
mass expansion. Note that in the case under consideration the electron mass
regulates both soft and collinear  divergences and we can put $\lambda=0$.  The
calculation is conveniently performed by using the light-cone coordinates where
$p_1\approx {p_1}_-$ and $p_{2}\approx{p_2}_+$.  In this representation only the
interaction of the transverse photons to soft electrons is not mass-suppressed
and  we can use ${g^\perp_{kl}\over 2p_il}$ for the eikonal photon propagators.
To get the double-logarithmic part of the correction we use  Sudakov
parametrization  of a virtual photon momentum $l=up_1+vp_2+l_\perp$. After
integrating over the transverse components $l_\perp$ we get the following
representation of the two-loop power-suppressed form factor
\begin{equation}
\left.F_1^{(1)}\right|_{2-loop}=-4x^2\!\!\int\!\! K(\eta_1,\eta_2,\xi_1,\xi_2)
 {\rm d}\eta_1{\rm d}\eta_2{\rm d}\xi_1{\rm d}\xi_2,
\label{eq::F12loop}
\end{equation}
where  $\eta=\ln v/\ln\rho $, $\xi=\ln u/\ln\rho$,  the integration goes over
the four-dimensional cube $0<\eta_i,~\xi_i<1$, and the kernel
\begin{eqnarray}
 K(\eta_1,\eta_2,\xi_1,\xi_2)&=&
 \theta(1-\eta_1-\xi_1)\theta(1-\eta_2-\xi_2)
 \nonumber\\
 && \times\theta(\eta_2-\eta_1) \theta(\xi_1-\xi_2)
 \label{eq::kernel2loop}
\end{eqnarray}
selects the kinematically allowed region of double-logarithmic integration. This
gives $F^{(1)}_1=-{x^2\over 3}+{\cal O}(x^3)$, in agreement with
\cite{Bernreuther:2004ih,Mastrolia:2003yz}.
\begin{figure}[t]
\includegraphics[width=1.5cm]{sud4.eps}
\hspace*{20mm}\includegraphics[width=1.5cm]{sud5.eps}\\
\includegraphics[width=1.5cm]{sud6.eps}\hspace*{20mm}\includegraphics[width=1.5cm]{sud7.eps}
\caption{\label{fig::diagram3loop} Feynman diagrams contributing to the
double-logarithmic correction factors $\phi^{a,b,c,d}$, Eq.~(\ref{eq::phi}).
}
\end{figure}
The higher-order double-logarithmic corrections are generated in a usual way
through the exchange of soft photons with the propagator ${-g_{+-}\over
l^2-\lambda^2}$.  The relevant topologies of the three-loop diagrams   are given
in Fig.~\ref{fig::diagram3loop}. Note that the soft photon exchange between the
soft and internal eikonal electron lines does not produce the
double-logarithmic correction. By using the factorization properties of the soft
photon contribution \cite{Sudakov:1954sw} we find the following representation
of the all-order double-logarithmic result
\begin{eqnarray}
 F^{(1)}_1&=&-4x^2\!\!\int\!\! {\phi}^b(\eta_1,\xi_2)
 {\phi}^c(\eta_1,\xi_1){\phi}^c(\xi_2,\eta_2)
{\phi}^d(\eta_2){\phi}^d(\xi_1) \nonumber  \\
 &&
 \times\Big[{\phi}^a(\eta_2,\xi_1)K_1(\eta_1,\eta_2,\xi_1,\xi_2)
 \nonumber  \\
 &&
 +K_2(\eta_1,\eta_2,\xi_1,\xi_2)\Big]
 {\rm d}\eta_1{\rm d}\eta_2{\rm d}\xi_1{\rm d}\xi_2,
\label{eq::F1integral}
 \end{eqnarray}
where the Sudakov correction factors corresponding to
Fig.~\ref{fig::diagram3loop}(a-d) are
\begin{eqnarray}
{\phi}^a(\eta,\xi)&=&\exp{\big[-x\left(1-\eta-\xi\right)^2\big]},
\nonumber\\
{\phi}^b(\eta,\xi)&=&\exp{\big[-2x\eta\xi\big]},
\nonumber\\
{\phi}^c(\eta,\xi)&=&\exp{\big[-x\eta\left(\eta+2\xi-2\right)\big]},
\nonumber\\
{\phi}^d(\eta)&=&\exp{\big[-x\left(1-\eta\right)^2\big]},
\label{eq::phi}
\end{eqnarray}
respectively, and the new kernels read
\begin{eqnarray}
K_1(\eta_1,\eta_2,\xi_1,\xi_2)&=&
 \theta(1-\eta_2-\xi_1)\theta(1-\eta_2-\xi_2)
 \nonumber\\
 && \times\theta(\eta_2-\eta_1) \theta(\xi_1-\xi_2),
 \nonumber \\
 K_2(\eta_1,\eta_2,\xi_1,\xi_2)&=&
 \theta(1-\eta_1-\xi_1)\theta(1-\eta_2-\xi_2)
 \nonumber\\
 && \times\theta(\eta_2-\eta_1) \theta(\xi_1+\eta_2-1),
\label{eq::kernel}
 \end{eqnarray}
with $K=K_1+K_2$. We are not able to find the result for the  four-fold
integral~(\ref{eq::F1integral}) in a closed analytic form. However,  the
coefficients of the series
\begin{table}[t]
  \begin{ruledtabular}
    \begin{tabular}{c|c|c|c|c|c|c|c}
      $n$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$
%      &$8$
      \\
      \hline
      $(-1)^nn!c_n$ &  ${29\over 30}$ & ${257\over  210}$ & ${1231\over 630}$
      & ${396581\over 103950}$ & ${5531381\over 630630}$ &
      ${72078311\over 3153150}$ & ${4510839803\over 68918850}$
%      &${100062367\over 50791104000}$
      \\
      %\hline
    \end{tabular}
    \end{ruledtabular}
    \caption{\label{tab::cn}
      The   normalized coefficients of the series~(\ref{eq::F1series}) up to
      $n=7$.}
\end{table}
\begin{equation}
F^{(1)}_1=-{x^2\over 3}\left(1+\sum_{n=1}^\infty c_nx^n\right)
\label{eq::F1series}
\end{equation}
can in principle be analytically computed for any given $n$ corresponding to the
$(n+2)$-loop double-logarithmic contribution.  The first seven coefficients of
the series are listed in Table~\ref{tab::cn}. The series~(\ref{eq::F1series}) is
useful only for sufficiently small $x$. For large $x$ the
integral~(\ref{eq::F1integral}) can be computed numerically. The result of the
numerical evaluation is presented in Fig.~\ref{fig::fx} for the function
$f(x)=-3F^{(1)}_1$. The function rapidly grows at $x\sim 1$ and then
monotonically approaches the limit $f(\infty)=1.33496\ldots$ corresponding to
\begin{equation}
F^{(1)}_1(x=\infty)=-0.444988\ldots\, .
\label{eq::F1asymptotic}
\end{equation}
Thus the power-suppressed amplitude  is {\em enhanced} by the double-logarithmic
corrections at high energy. A similar effect has been observed before {\it e.g.}
for the  electron-muon backward scattering \cite{Gorshkov:1966ht}. In the case
under consideration the enhancement may be related to a specific structure of
the process with the soft electron pair exchange. Through the pair emission an
eikonal electron is converted into an eikonal positron with approximately the
same momentum but opposite electric charge, Fig.~\ref{fig::diagram2loop}(c). As
a result the double-logarithmic contribution of the topology of
Fig.~\ref{fig::diagram3loop}(c)  has an opposite sign with respect to the one of
Fig.~\ref{fig::diagram2loop}(a,b,d) and cancels the standard Sudakov
suppression. The double-logarithmic approximation of $F_1^{(1)}$ given above is 
valid for $\alpha\ll x\ll1/\alpha$, which covers the energy interval
$1\ll|\ln\rho|\ll 1/\alpha$ sufficient for any practical applications.  For
higher energies corresponding to $x\sim 1/\alpha$ the subleading terms
proportional to powers of $\alpha\ln\rho\sim 1$ have to be resumed to all
orders.
\begin{figure}[t]
\vspace*{-15pt}
\begin{center}
\includegraphics[width=8cm]{sud8.eps}\\[-15pt]
\hspace*{15pt}$x$\\
\vspace*{-92pt}
\hspace*{-220pt}$f(x)$
\vspace*{70pt}
\end{center}
\caption{\label{fig::fx}  The result of the numerical evaluation of the function
$f(x)=-3F^{(1)}_1$.
}\end{figure}

Unlike the  Sudakov double logarithms,  the  leading power-suppressed
double-logarithmic corrections depend not only on the charges of the initial and
final states but also on the details of the scattering process. For example, the
non-singlet ({\it i.e.} without a closed electron loop) ${\cal O}(\rho)$
double-logarithmic corrections to the scalar form factor vanish to all orders in
$\alpha$ due to a specific Lorentz and Dirac structure of the soft electron pair
interaction with the eikonal electrons and positrons. A less trivial example is
the Pauli form factor. The expansion of  $F_2$ in $\rho$ ({\it cf.}
Eq.~(\ref{eq::F1}))  starts with the first order term $F^{(1)}_2$. In the
double-logarithmic approximation $F^{(1)}_2=0$  and for the leading mass
correction from the soft electron pair exchange  we obtain
$F^{(2)}_2=4F^{(1)}_1$, in agreement with
\cite{Bernreuther:2004ih,Mastrolia:2003yz}. Thus the ${\cal O}(\rho)$
corrections are universally related to the soft electron pair exchange and can
be be obtained as a straightforward generalization of our analysis for more
complicated processes such as Bhabha scattering, where only the leading result
of the small electron mass expansion is available  in two loops
\cite{Penin:2005eh,Penin:2005kf}. Moreover, up to two loops the structure of the
${\cal O}(\rho)$ double-logarithmic correction in quantum chromodynamics (QCD)
is similar to the one in QED. In particular, the double-logarithmic
power-suppressed term in two-loop corrections to the heavy-quark vector form
factor  differs from the QED result only by the $C_F^2-C_AC_F/2$ color factor of
the diagram in Fig.~\ref{fig::diagram2loop}. Thus our method can be applied to
the calculation of the dominant two-loop power-suppressed corrections to the
high-energy processes involving heavy quarks. For the energies ranging from
approximately 10 to 100 times the  heavy-quark mass we have $|\ln\rho|\gg 1$ and
$\rho\ln^4\rho\sim 1$, {\it i.e.} the double-logarithmic terms saturate the
power-suppressed contribution and are comparable in magnitude  to the
nonlogarithmic leading-power corrections in the strong coupling constant,  which
are phenomenologically significant. Beyond the two-loop approximation our result
is not directly applicable to the QCD amplitudes since the eikonal gluons in
Fig.~\ref{fig::diagram2loop}(b) can radiate  soft gluons producing additional
double-logarithmic corrections.  As a consequence,  the leading power-suppressed
double-logarithmic corrections to the heavy-quark vector form factor get a
nonabelian contribution in every order of perturbation theory in contrast to the
purely abelian Sudakov double logarithms. 

To summarize, we have generalized the result of Sudakov~\cite{Sudakov:1954sw} to
the leading power-suppressed contribution.  This is an important step towards a
systematic renormalization group analysis of the  high energy behavior of the
gauge theory amplitudes beyond the leading power approximation.  The  leading
power-suppressed double-logarithmic corrections reveal a few characteristic
features which distinguish them from the Sudakov double logarithms. In
particular, they  are induced by a soft electron pair exchange and result in an
enhancement of the power-suppressed contribution. In QCD our method can be used
for the analysis of the high-energy processes involving heavy quarks up to two
loops. Extending the analysis to the higher orders of perturbative  QCD and to
subleading logarithms is  a very interesting problem which is beyond the scope
of this Letter.

\begin{acknowledgements}
I would like to thank R. Bonciani,  K. Melnikov,  V. Smirnov, and  N. Zerf
for  useful discussions and collaboration. This research  was supported in part
by NSERC and  Perimeter Institute for Theoretical Physics.
Research at Perimeter Institute is supported by the Government of Canada
through Industry Canada and by the Province of Ontario through the Ministry
of Research and Innovation.
\end{acknowledgements}



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

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  %``Two loop Sudakov form-factor in a theory with mass gap,''
  Phys.\ Rev.\ Lett.\  {\bf 93}, 101802 (2004).
  %[hep-ph/0404082].
  %%CITATION = HEP-PH/0404082;%%

 %\cite{Laenen:2010uz}
\bibitem{Laenen:2010uz}
  E.~Laenen, L.~Magnea, G.~Stavenga and C.~D.~White,
  %``Next-to-eikonal corrections to soft gluon radiation: a diagrammatic
  %approach,''
  JHEP {\bf 1101}, 141 (2011).
  %[arXiv:1010.1860 [hep-ph]].
  %%CITATION = ARXIV:1010.1860;%%

%\cite{Banfi:2013eda}
\bibitem{Banfi:2013eda}
  A.~Banfi, P.~F.~Monni and G.~Zanderighi,
  %``Quark masses in Higgs production with a jet veto,''
  JHEP {\bf 1401}, 097 (2014).
  %[arXiv:1308.4634 [hep-ph]].
  %%CITATION = ARXIV:1308.4634;%%

%\cite{Becher:2013iya}
\bibitem{Becher:2013iya}
  T.~Becher and G.~Bell,
  %``Enhanced non-perturbative effects through the collinear anomaly,''
  Phys.\ Rev.\ Lett.\  {\bf 112}, 182002 (2014).
  %[arXiv:1312.5327 [hep-ph]].
%%CITATION = ARXIV:1312.5327;%%

%\cite{deFlorian:2014vta}
\bibitem{deFlorian:2014vta}
  D.~de Florian, J.~Mazzitelli, S.~Moch and A.~Vogt,
  %``Approximate N$^{3}$LO Higgs-boson production cross section using physical-kernel constraints,''
  JHEP {\bf 1410}, 176 (2014).
  %%CITATION = ARXIV:1408.6277;%%
  %5 citations counted in INSPIRE as of 27 Nov 2014lda

%\cite{Anastasiou:2014lda}
\bibitem{Anastasiou:2014lda}
  C.~Anastasiou, C.~Duhr, F.~Dulat, E.~Furlan, T.~Gehrmann, F.~Herzog and B.~Mistlberger,
  %``Higgs boson gluon-fusion production beyond threshold in N3LO QCD,''
  arXiv:1411.3584 [hep-ph].
  %%CITATION = ARXIV:1411.3584;%%
  %1 citations counted in INSPIRE as of 27 Nov 2014


  %\cite{Yennie:1961ad}
\bibitem{Yennie:1961ad}
  D.~R.~Yennie, S.~C.~Frautschi and H.~Suura,
  %``The infrared divergence phenomena and high-energy processes,''
  Annals Phys.\  {\bf 13}, 379 (1961).
  %%CITATION = APNYA,13,379;%%


%\cite{Jackiw:1968zz}
\bibitem{Jackiw:1968zz}
  R.~Jackiw,
  %``Dynamics at high momentum and the vertex function of spinor
  %electrodynamics,''
  Annals Phys.\  {\bf 48}, 292 (1968).
  %%CITATION = APNYA,48,292;%%


%\cite{Beneke:1997zp}
\bibitem{Beneke:1997zp}
  M.~Beneke and V.~A.~Smirnov,
  %``Asymptotic expansion of Feynman integrals near threshold,''
  Nucl.\ Phys.\ B {\bf 522}, 321 (1998).
  %[hep-ph/9711391].
  %%CITATION = HEP-PH/9711391;%%
  %354 citations counted in INSPIRE as of 11 Nov 2014

  %\cite{Smirnov:1997gx}
\bibitem{Smirnov:1997gx}
  V.~A.~Smirnov,
  %``Asymptotic expansions of two loop Feynman diagrams in the Sudakov limit,''
  Phys.\ Lett.\ B {\bf 404}, 101 (1997).
  %[hep-ph/9703357].
  %%CITATION = HEP-PH/9703357;%%

%\cite{Smirnov:2002pj}
\bibitem{Smirnov:2002pj}
  V.~A.~Smirnov,
  {\it Applied asymptotic expansions in momenta and masses},
  Springer Tracts Mod.\ Phys.\  {\bf 177 } (2002)  1.

  %\cite{Gorshkov:1966ht}
\bibitem{Gorshkov:1966ht}
  V.~G.~Gorshkov, V.~N.~Gribov, L.~N.~Lipatov and G.~V.~Frolov,
  %``Doubly logarithmic asymptotic behavior in quantum electrodynamics,''
  Sov.\ J.\ Nucl.\ Phys.\  {\bf 6}, 95 (1968)
  [Yad.\ Fiz.\  {\bf 6}, 129 (1967)].
  %%CITATION = SJNCA,6,95;%%

%\cite{Korchemsky:1987wg}
\bibitem{Korchemsky:1987wg}
  G.~P.~Korchemsky and A.~V.~Radyushkin,
  %``Renormalization of the Wilson Loops Beyond the Leading Order,''
  Nucl.\ Phys.\ B {\bf 283}, 342 (1987).
  %%CITATION = NUPHA,B283,342;%%

%\cite{Fadin:1997sn}
\bibitem{Fadin:1997sn}
  V.~S.~Fadin, V.~A.~Khoze and A.~D.~Martin,
  %``Higgs studies in polarized gamma gamma collisions,''
  Phys.\ Rev.\ D {\bf 56}, 484 (1997).
  %[hep-ph/9703402].
  %%CITATION = HEP-PH/9703402;%%

  %\cite{Mastrolia:2003yz}
\bibitem{Mastrolia:2003yz}
  P.~Mastrolia and E.~Remiddi,
  %``Two loop form-factors in QED,''
  Nucl.\ Phys.\ B {\bf 664}, 341 (2003).
  %[hep-ph/0302162].
  %%CITATION = HEP-PH/0302162;%%


  %\cite{Penin:2005kf}
\bibitem{Penin:2005kf}
  A.~A.~Penin,
  %``Two-loop corrections to Bhabha scattering,''
  Phys.\ Rev.\ Lett.\  {\bf 95}, 010408 (2005).
  %[hep-ph/0501120].
  %%CITATION = HEP-PH/0501120;%%



  %\cite{Penin:2005eh}
\bibitem{Penin:2005eh}
  A.~A.~Penin,
  %``Two-loop photonic corrections to massive Bhabha scattering,''
  Nucl.\ Phys.\ B {\bf 734}, 185 (2006).
  %[hep-ph/0508127].
  %%CITATION = HEP-PH/0508127;%%




\end{thebibliography}

\end{document}


1412.0671.extracted_bib
{Sudakov:1954sw V.~V.~Sudakov, %``Vertex parts at very high-energies in quantum electrodynamics,'' Sov.\ Phys.\ JETP {3, 65 (1956) [Zh.\ Eksp.\ Teor.\ Fiz.\ {30, 87 (1956)]. %%CITATION = SPHJA,3,65;%% %\cite{Frenkel:1976bj
1412.0671.extracted_bib
{Frenkel:1976bj J.~Frenkel and J.~C.~Taylor, %``Exponentiation of Leading Infrared Divergences in Massless Yang-Mills %Theories,'' Nucl.\ Phys.\ B {116, 185 (1976). %%CITATION = NUPHA,B116,185;%% %\cite{Mueller:1979ih
1412.0671.extracted_bib
{Mueller:1979ih A.~H.~Mueller, %``On the Asymptotic Behavior of the Sudakov Form-factor,'' Phys.\ Rev.\ D {20, 2037 (1979). %%CITATION = PHRVA,D20,2037;%% %\cite{Collins:1980ih
1412.0671.extracted_bib
{Collins:1980ih J.~C.~Collins, %``Algorithm to Compute Corrections to the Sudakov Form-factor,'' Phys.\ Rev.\ D {22, 1478 (1980). %%CITATION = PHRVA,D22,1478;%% %\cite{Sen:1981sd
1412.0671.extracted_bib
{Sen:1981sd A.~Sen, %``Asymptotic Behavior of the Sudakov Form-Factor in QCD,'' Phys.\ Rev.\ D {24, 3281 (1981). %%CITATION = PHRVA,D24,3281;%% %\cite{Sterman:1986aj
1412.0671.extracted_bib
{Sterman:1986aj G.~F.~Sterman, %``Summation of Large Corrections to Short Distance Hadronic Cross-Sections,'' Nucl.\ Phys.\ B {281, 310 (1987). %%CITATION = NUPHA,B281,310;%% %\cite{Catani:1998bh
1412.0671.extracted_bib
{Catani:1998bh S.~Catani, %``The Singular behavior of QCD amplitudes at two loop order,'' Phys.\ Lett.\ B {427, 161 (1998). %[hep-ph/9802439]. %%CITATION = HEP-PH/9802439;%% %\cite{Bernreuther:2004ih
1412.0671.extracted_bib
{Bernreuther:2004ih W.~Bernreuther, R.~Bonciani, T.~Gehrmann, R.~Heinesch, T.~Leineweber, P.~Mastrolia and E.~Remiddi, %``Two-loop QCD corrections to the heavy quark form-factors: The Vector %contributions,'' Nucl.\ Phys.\ B {706, 245 (2005). %[hep-ph/0406046]. %%CITATION = HEP-PH/0406046;%% %\cite{Bernreuther:2005gw
1412.0671.extracted_bib
{Bernreuther:2005gw W.~Bernreuther, R.~Bonciani, T.~Gehrmann, R.~Heinesch, P.~Mastrolia and E.~Remiddi, %``Decays of scalar and pseudoscalar Higgs bosons into fermions: Two-loop QCD %corrections to the Higgs-quark-antiquark amplitude,'' Phys.\ Rev.\ D {72, 096002 (2005). %[hep-ph/0508254]. %%CITATION = HEP-PH/0508254;%% %20 citations counted in INSPIRE as of 01 Nov 2014 %\cite{Bonciani:2007eh
1412.0671.extracted_bib
{Bonciani:2007eh R.~Bonciani, A.~Ferroglia and A.~A.~Penin, %``Heavy-flavor contribution to Bhabha scattering,'' Phys.\ Rev.\ Lett.\ {100, 131601 (2008). %[arXiv:0710.4775 [hep-ph]]. %%CITATION = ARXIV:0710.4775;%% %\cite{Bonciani:2008ep
1412.0671.extracted_bib
{Bonciani:2008ep R.~Bonciani, A.~Ferroglia and A.~A.~Penin, %``Calculation of the Two-Loop Heavy-Flavor Contribution to Bhabha Scattering,'' JHEP {0802, 080 (2008). %[arXiv:0802.2215 [hep-ph]]. %%CITATION = ARXIV:0802.2215;%% %\cite{Wilson:1969zs
1412.0671.extracted_bib
{Wilson:1969zs K.~G.~Wilson, %``Nonlagrangian models of current algebra,'' Phys.\ Rev.\ {179, 1499 (1969). %%CITATION = PHRVA,179,1499;%% %\cite{Caswell:1985ui
1412.0671.extracted_bib
{Caswell:1985ui W.~E.~Caswell and G.~P.~Lepage, %``Effective Lagrangians for Bound State Problems in QED, QCD, and Other Field %Theories,'' Phys.\ Lett.\ B {167, 437 (1986). %%CITATION = PHLTA,B167,437;%% %\cite{Feucht:2004rp
1412.0671.extracted_bib
{Feucht:2004rp B.~Feucht, J.~H.~Kuhn, A.~A.~Penin and V.~A.~Smirnov, %``Two loop Sudakov form-factor in a theory with mass gap,'' Phys.\ Rev.\ Lett.\ {93, 101802 (2004). %[hep-ph/0404082]. %%CITATION = HEP-PH/0404082;%% %\cite{Laenen:2010uz
1412.0671.extracted_bib
{Laenen:2010uz E.~Laenen, L.~Magnea, G.~Stavenga and C.~D.~White, %``Next-to-eikonal corrections to soft gluon radiation: a diagrammatic %approach,'' JHEP {1101, 141 (2011). %[arXiv:1010.1860 [hep-ph]]. %%CITATION = ARXIV:1010.1860;%% %\cite{Banfi:2013eda
1412.0671.extracted_bib
{Banfi:2013eda A.~Banfi, P.~F.~Monni and G.~Zanderighi, %``Quark masses in Higgs production with a jet veto,'' JHEP {1401, 097 (2014). %[arXiv:1308.4634 [hep-ph]]. %%CITATION = ARXIV:1308.4634;%% %\cite{Becher:2013iya
1412.0671.extracted_bib
{Becher:2013iya T.~Becher and G.~Bell, %``Enhanced non-perturbative effects through the collinear anomaly,'' Phys.\ Rev.\ Lett.\ {112, 182002 (2014). %[arXiv:1312.5327 [hep-ph]]. %%CITATION = ARXIV:1312.5327;%% %\cite{deFlorian:2014vta
1412.0671.extracted_bib
{deFlorian:2014vta D.~de Florian, J.~Mazzitelli, S.~Moch and A.~Vogt, %``Approximate N$^{3$LO Higgs-boson production cross section using physical-kernel constraints,'' JHEP {1410, 176 (2014). %%CITATION = ARXIV:1408.6277;%% %5 citations counted in INSPIRE as of 27 Nov 2014lda %\cite{Anastasiou:2014lda
1412.0671.extracted_bib
{Anastasiou:2014lda C.~Anastasiou, C.~Duhr, F.~Dulat, E.~Furlan, T.~Gehrmann, F.~Herzog and B.~Mistlberger, %``Higgs boson gluon-fusion production beyond threshold in N3LO QCD,'' arXiv:1411.3584 [hep-ph]. %%CITATION = ARXIV:1411.3584;%% %1 citations counted in INSPIRE as of 27 Nov 2014 %\cite{Yennie:1961ad
1412.0671.extracted_bib
{Yennie:1961ad D.~R.~Yennie, S.~C.~Frautschi and H.~Suura, %``The infrared divergence phenomena and high-energy processes,'' Annals Phys.\ {13, 379 (1961). %%CITATION = APNYA,13,379;%% %\cite{Jackiw:1968zz
1412.0671.extracted_bib
{Jackiw:1968zz R.~Jackiw, %``Dynamics at high momentum and the vertex function of spinor %electrodynamics,'' Annals Phys.\ {48, 292 (1968). %%CITATION = APNYA,48,292;%% %\cite{Beneke:1997zp
1412.0671.extracted_bib
{Beneke:1997zp M.~Beneke and V.~A.~Smirnov, %``Asymptotic expansion of Feynman integrals near threshold,'' Nucl.\ Phys.\ B {522, 321 (1998). %[hep-ph/9711391]. %%CITATION = HEP-PH/9711391;%% %354 citations counted in INSPIRE as of 11 Nov 2014 %\cite{Smirnov:1997gx
1412.0671.extracted_bib
{Smirnov:1997gx V.~A.~Smirnov, %``Asymptotic expansions of two loop Feynman diagrams in the Sudakov limit,'' Phys.\ Lett.\ B {404, 101 (1997). %[hep-ph/9703357]. %%CITATION = HEP-PH/9703357;%% %\cite{Smirnov:2002pj
1412.0671.extracted_bib
{Smirnov:2002pj V.~A.~Smirnov, {Applied asymptotic expansions in momenta and masses, Springer Tracts Mod.\ Phys.\ {177 (2002) 1. %\cite{Gorshkov:1966ht
1412.0671.extracted_bib
{Gorshkov:1966ht V.~G.~Gorshkov, V.~N.~Gribov, L.~N.~Lipatov and G.~V.~Frolov, %``Doubly logarithmic asymptotic behavior in quantum electrodynamics,'' Sov.\ J.\ Nucl.\ Phys.\ {6, 95 (1968) [Yad.\ Fiz.\ {6, 129 (1967)]. %%CITATION = SJNCA,6,95;%% %\cite{Korchemsky:1987wg
1412.0671.extracted_bib
{Korchemsky:1987wg G.~P.~Korchemsky and A.~V.~Radyushkin, %``Renormalization of the Wilson Loops Beyond the Leading Order,'' Nucl.\ Phys.\ B {283, 342 (1987). %%CITATION = NUPHA,B283,342;%% %\cite{Fadin:1997sn
1412.0671.extracted_bib
{Fadin:1997sn V.~S.~Fadin, V.~A.~Khoze and A.~D.~Martin, %``Higgs studies in polarized gamma gamma collisions,'' Phys.\ Rev.\ D {56, 484 (1997). %[hep-ph/9703402]. %%CITATION = HEP-PH/9703402;%% %\cite{Mastrolia:2003yz
1412.0671.extracted_bib
{Mastrolia:2003yz P.~Mastrolia and E.~Remiddi, %``Two loop form-factors in QED,'' Nucl.\ Phys.\ B {664, 341 (2003). %[hep-ph/0302162]. %%CITATION = HEP-PH/0302162;%% %\cite{Penin:2005kf
1412.0671.extracted_bib
{Penin:2005kf A.~A.~Penin, %``Two-loop corrections to Bhabha scattering,'' Phys.\ Rev.\ Lett.\ {95, 010408 (2005). %[hep-ph/0501120]. %%CITATION = HEP-PH/0501120;%% %\cite{Penin:2005eh
1412.0671.extracted_bib
{Penin:2005eh A.~A.~Penin, %``Two-loop photonic corrections to massive Bhabha scattering,'' Nucl.\ Phys.\ B {734, 185 (2006). %[hep-ph/0508127]. %%CITATION = HEP-PH/0508127;%%