%Title: Recent Results on Sudakov Logarithms in Electroweak Processes
%Author: Johann H. Kuehn
%Published: <EM> Proceedings of ICHEP 2002, Elsevier Science B.V., Bentvelsen et al. Eds.  </EM> <B>  </B> (2002) 220-223. 
%%%%%%%%%% EW-4-4.tex %%%%%%%%%%
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% Recent Results on Sudakov Logarithms in Electroweak Processes
% by Johann H. Kuehn
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% proceedings
% for parallel session talk EW-4-4 at ICHEP Amsterdam, July 26, 2002
% "Sudakov Logarithms in Electroweak Processes at High Energy"
%
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\title{Recent Results on Sudakov Logarithms in Electroweak Processes}

\author{%
J.H.~K\"uhn%
\address{Institut f\"ur Theoretische Teilchenphysik,
Universit\"at Karlsruhe, 76128 Karlsruhe, Germany}
}

\begin{document}

\begin{abstract}
Recent results for the asymptotic behavior of 
fermion scattering amplitudes in the Sudakov
limit are discussed including next-to-leading logarithmic (NLL)
and NNLL corrections.
These are used for  the analysis of the
dominant electroweak corrections to the fermion-antifermion pair
production in $e^+e^-$ annihilation at high energy.
The importance of angular dependent as well as subleading terms is emphasized.
%\vspace{1pc}
\vspace{1.5ex}
\end{abstract}

% typeset front matter (including abstract)
\maketitle


\section{Introduction}
Four fermion processes are generally considered as
benchmark processes at high energy colliders, with electron
positron annihilation into muon or quark pairs at LEP and the 
Drell Yan
process at hadron colliders as characteristic examples. Within the
presently accessible energy region, typically up to 200~GeV, radiative
corrections are dominated by the shift in the $W$ and $Z$ masses as
parametrized by the $\rho$ parameter and by the running of the 
coupling
constant. Vertex corrections and box diagrams involving gauge 
bosons are
generally of minor importance. In the TeV region, accessible at future
colliders like the LHC or TESLA, this picture changes drastically. 
A new
class of effects starts to become relevant and rapidly dominant which
are generally denoted as double logarithmic corrections and which were
first observed by Sudakov
in the context of quantum electrodynamics for
reactions with a tight cut on the radiated energy of the photons.
For electroweak interactions large negative corrections arise from the
exchange of gauge bosons which remain uncompensated if one restricts 
the
analysis to exclusive final states, consisting e.g.\ of a fermion
antifermion pair only.
The discussion of double logarithmic corrections is fairly
straightforward for a theory with massive gauge bosons only.
An important complication arises from the presence of massless photons
in the final state. Events with soft and hard photon radiation are
normally included in the sample -- whence a
``semi-inclusive'' definition of the cross section is closest to the
the experimental analysis.

One loop corrections to the four fermion process are available since
long (see e.g.\ \cite{Hol}). In this case the separation of photonic
and weak corrections is still possible and, employing  axial 
gauge, the
leading logarithms can be trivially attributed to fermion self energy
diagrams with virtual $W$ or $Z$ boson emission \cite{KuhPen}.

Results for the leading logarithmic corrections in higher orders have
been presented in \cite{KuhPen,Fad}.
In \cite{KPS}, which includes NLL corrections and angular dependent terms,
it was demonstrated that the approach developed in
\cite{KuhPen}, where axial gauge was adopted and
both real and virtual photon emmission neglected, is
numerically nearly identical to the results of \cite{Fad} where 
virtual
photon corrections and real photon emmission are included -- a
consequence of the smallness of the weak mixing angle, $\sin^2\theta_W
=0.23$. The analysis of subleading logarithms for the form
factor, for the four fermion scattering in a spontaneously broken
$SU(2)$ gauge theory and, last not least, for the Standard Model with
$W$, $Z$ and the massless photon was performed in \cite{KPS}.

Before presenting a brief review (an extended review of this field can also be
found in \cite{Melles1})
of the techniques and some recent results
let us emphasize the importance of subleading corrections.
In the energy region relevant at LHC or TESLA, i.e.\ around 1 to 2~TeV,
this is already evident from the one-loop vertex correction
\be
|F_B + \delta F|^2 \approx F_B \left(1+
2\frac{\alpha_W}{\pi} C_F \rho(s/M^2)\right),
\ee
\[
\rho(u) =
-\frac14\ln^2 u  + \frac 34\ln u + \frac{\pi^2}{12} - \frac 78 .
\]
For the timelike region and for characteristic values of the coupling, 
energy and mass of
$\alpha_W/\pi = 10^{-2}$, $C_F = 3/4$ and $s/M^2=10^2$ one finds
\bea
2\frac{\alpha_W}{\pi} C_F \rho &\!\!\!=\!\!\!&1.5 \times 10^{-2} 
( - 5.30 + 3.45 - 0.05)
%\nn \\
%&\!\!\!=\!\!\!& -0.029.
\eea
Large compensations between leading and subleading terms are observed
and, in fact, this pattern will reappear for the Standard Model as
discussed below.

The calculation of the leading logarithmic terms in~\cite{Fad}
is based on evolution equations that govern
the dynamics of the amplitudes in the Sudakov limit.
In \cite{KPS} and \cite{NNLL} this approach was applied to the
next-to-leading logarithmic and the NNLL analysis respectively
of the Abelian form factor
and the four fermion amplitude in the $SU(N)$ gauge theory.
Functions that enter the evolution equations
in the  next-to-leading logarithmic approximation
were evaluated by using, as an
input, asymptotic expansions of one-loop diagrams.
The solution of these equations lead to a summation of the leading
and subleading Sudakov logarithms.
The  expansion of one-loop diagrams through the so-called
generalized strategy of regions \cite{BS,SR}
identifies in a systematic way the nature
of various contributions and the origin of logarithms.
\section{The  Abelian form factor in the Sudakov limit}
Let us first analyze
the (vector) form factor  which determines the  amplitude of the
fermion scattering in the external Abelian field. In Born
approximation
\be
\F_B=\bar\psi(p_2)\gm_\mu\psi(p_1) .
\ee
We consider the limit
$s=(p_1-p_2)^2 \to-\infty$ with
on-shell massless fermions, $p_1^2=p_2^2=0$, and
gauge bosons with a small non-zero mass $M^2\ll -s$.
For convenience
$p_{1,2} = (Q/2,0,0,\mp Q/2)$ so that  $2 p_1 p_2 = Q^2=-s$.
The asymptotic behavior of the form factor
is obtained from
\bea
\label{evolsolf}%
\F=F_0(\al(M^2))\exp \Bigl\{\int_{M^2}^{Q^2}{\dd x\over x}
\nn \\[-2ex]
\eea
\vspace{-6ex}
\[\times
\Bigl[\int_{M^2}^{x}{\dd x'\over x'}\gm(\al(x'))+\zeta(\al(x))
+\xi(\al(M^2))\Bigr]\Bigr\} \F_B .
\]
The  NLL approximation includes all the terms of the
form $\al^n\log^{2n-m}(Q^2/M^2)$ with $m=0,~1$.
This requires the one-loop values of $\gm(\al)$ and $\zeta(\al)+\xi(\al)$
and using the one-loop running of $\al$ in $\gm(\al)$.
The  NNLL approximation includes all the terms of the
form $\al^n\log^{2n-m}(Q^2/M^2)$ with $m=0,~1,~2$.
In this case
$\gm(\al)$ is required up to ${\cal O}(\al^2)$,
$\zeta(\al)$, $\xi(\al)$ and $F_0(\al)$
up to ${\cal O}(\al)$ together with the two-loop running of $\al$
in $\gm(\al)$ and one-loop running of $\al$
in $\zeta(\al)$.

The expansion by regions consists of the following steps:
(i) consider various regions of a loop momentum $k$ and expand, in
every region, the integrand in Taylor series with respect to the
parameters that are there considered small,
(ii) integrate the  expanded integrand
over the whole integration domain of the loop momenta,
(iii) put to zero any scaleless integral.
In step (ii) dimensional regularization with $d=4-2\ep$ space-time
dimensions  is used to handle the divergences.
The following regions are of relevance in this context:
\bea
%\label{h}
\mbox{{\em hard} (h):} &\hspace{-1em}& k\sim Q,
\nn \\
%\label{1c}
\mbox{{\em 1-collinear} (1c):} &\hspace{-1em}& k_+\sim Q,\,\,k_-\sim M^2/Q,
\,\, \uk \sim M,
\nn \\
%\label{2c}
\mbox{{\em 2-collinear} (2c):} &\hspace{-1em}& k_-\sim Q,\,\,k_+\sim M^2/Q,
\,\,\uk \sim M,
\nn \\
\label{soft}
\mbox{{\em soft} (s):} &\hspace{-1em}& k\sim M.
%\label{us}
\eea
with $k_{\pm} =k_0\pm k_3, \, \uk=(k_1,k_2)$.

% no 2
From the one-loop result
and the hard part in two-loop approximation one finds
\bea
\gm^{(1)}&\!\!\!=\!\!\!&-2C_F \,,\quad
\zeta^{(1)} = 3C_F\,,\quad
\xi^{(1)} = 0 \,,
\nn \\
F_0^{(1)}&\!\!\!=\!\!\!& -C_F
\left({7\over 2}+{2\pi^2\over 3}\right) \,,
%\label{1loop}
\\
\gm^{(2)}&\!\!\!=\!\!\!&-2C_F\left[\left({67\over 9}-{\pi^2\over 3}\right)C_A
-{20\over 9}T_Fn_f\right] \,,
\nn
%\label{2loop}
\eea
with $\gamma = \left(\frac{\alpha}{4\pi}\right) \gamma^{(1)} +
\left(\frac{\alpha}{4\pi}\right)^2 \gamma^{(2)} + \ldots$
and similarly for $\zeta$, $\xi$ and $F_0$.

% no 3
Let us also consider the two-loop corrections.
The LL, NLL and NNLL approximations are given respectively by
\bea
{\cal F}^{(2)}_{LL}&\!\!\!=\!\!\!&
{1\over 8}{(\gm^{(1)})^2}\ln^4\left({Q^2\over M^2}\right){\cal F}_B\,,
\label{llf}\\
{\cal F}^{(2)}_{NLL}&\!\!\!=\!\!\!&
{1\over 2}\left(\zeta^{(1)}-{1\over 3}\bt_0\right)
\gm^{(1)}
\ln^3\left({Q^2\over M^2}\right){\cal F}_B\,,
\label{nllf}
\nn \\ \\
{\cal F}^{(2)}_{NNLL}&\!\!\!=\!\!\!&{1\over 2}\biggl(\gm^{(2)}+
\left(\zeta^{(1)}-\bt_0\right)\zeta^{(1)}
\nn \\ && \qquad
+F_0^{(1)}\gm^{(1)}\biggr)
\ln^2\left({Q^2\over M^2}\right){\cal F}_B\,.
\label{nnllf}
\eea

% no 4
The relatively
small coefficient  of the  LL terms and the
large coefficient of the  NNLL terms in the
form factor are clearly indicative
of the  importance of the NNLL corrections and, as we will see,
reflect the general structure of the logarithmically enhanced
electroweak corrections.


% no 5
\section{The four fermion amplitude}
Let us now investigate the four fermion scattering
at fixed angles  in the limit when
all the  invariant energy and momentum transfers
of the process are far larger than the
gauge boson mass,  $|s|\sim |t| \sim |u| \gg M^2$.
The analysis of the four
fermion amplitude is  complicated  by the extra kinematical variable and
the presence of different
``color'' and Lorentz structures.
%
The complete  basis  consists
of four independent  chiral amplitudes, each of them
of two possible color structures.

The collinear  divergences in the hard part of the
virtual corrections and the corresponding ``collinear''
logarithms are  known to
factorize. They are
responsible, in particular, for the double logarithmic
contribution, depend only on the
properties of the external on-shell particles
but not on  a specific process
%
and are the same as for the form factor ${\cal F}$ discussed in the
previous section.   Let us denote by
$\tilde {\cal A}$   the amplitude with
the collinear logarithms  factored out.
%
It contains
the logarithms of ``soft''  nature  corresponding to
the soft divergences of the hard region contribution 
and the renormalization group   logarithms.
It can be  represented as a vector
in the color/chiral basis
and satisfies the evolution equation 
\be
{\partial \over \partial \ln{Q^2}}\tilde {\cal A}=
{\bf \chi}(\al(Q^2))\tilde {\cal A} \, ,
\label{evoleqa}
\ee
where  $\chi(\al)$ is the matrix of the soft
anomalous dimensions.  It can be extracted from 
the coefficient in front of the single pole of the hard
contribution to  $\tilde {\cal A}$.
The matrix elements of $\chi(\al)$ depend on the scattering angle.
These angular dependent terms as well as those in the starting values of
$\tilde {\cal A}$ discussed below
are important for the calculation of NLL (see also \cite{Melles2})
and NNLL terms in cross sections
and asymmetries.
The solution of eq.~(\ref{evoleqa}) reads
\be
\tilde {\cal A}=
\sum_i\tilde {\cal A}_{0i}(\al(M^2))\exp\biggl[\int_{M^2}^{Q^2}
{\dd x\over x}\chi_i(\al(x))\biggr] ,
\label{evolsola}
\ee
where $\chi_i(\al)$ are eigenvalues of $\chi(\al)$ and
$\tilde {\cal A}_{0i}(\al)$ 
are $Q$-independent  eigenvectors of $\chi(\al)$
which determine
the initial conditions for the evolution equation at $Q=M$.

To illustrate the significance
of the subleading  contributions
let us again  discuss the standard model inspired
example  considered in the  previous section.
%
For the  annihilation process one has to
make the analytical continuation to the Minkowskian region
of negative $Q^2=-s$.
%
After integrating the  differential cross section
over all angles
one obtains the
result with logarithmic accuracy,
e.g.\ for the case of $SU(2)_L$ group
with $n_f = 6$ and one scalar boson
\bea
\sigma^{(1)}&\!\!\!=\!\!\!&\biggl[-3\ln^2\left({s\over M^2}\right)
+{80\over 3}\ln\left({s\over M^2}\right)
\nn \\ &\!\!\!\!\!\!&
-\left({25\over 9}+3\pi^2\right)\biggr]\sigma_B\,,
\nn \\
\sigma^{(2)}&\!\!\!=\!\!\!&\biggl[{9\over 2}\ln^4\left({s\over M^2}\right)
-{449\over 6}\ln^3\left({s\over M^2}\right)
\nn \\ &\!\!\!\!\!\!&
+\left({4855\over 9}+{37\pi^2\over 3}\right)\ln^2\left({s\over M^2}\right)
\biggr]\sigma_B\,,
\label{sigsu2s}
\eea
if initial and final state fermions have the same
isospin.
%
Again we observe a relatively
small coefficient  of the  LL  and a
large coefficient of the  NNLL terms.


\section{Electroweak processes}
To analyze the  electroweak correction to the process
$f'\bar f'\rightarrow f\bar f$
the approximation with
the $W$ and $Z$ bosons of the same mass  $M$, the Higgs boson
of the mass $M_H\sim M$
and massless quarks and leptons
is employed.
A fictitious photon mass $\lm$ has to be  introduced
to regularize the infrared divergences.
%
The NNLL approximation
is not sensitive to the fine details
of the mass generation because
we need only the hard
part of the   potentially dangerous  self-energy
insertion to the gauge boson propagator.

The photon is, however, massless and
the corresponding  infrared divergent contributions  should be
accompanied by  the real soft photon radiation
integrated to some resolution energy  $\omega_{res}$
to get an infrared safe cross
section independent on an auxiliary photon mass.
%
To study the virtual corrections in the limit of the vanishing
photon mass the
general approach of the infrared evolution equations
developed in~\cite{Fad} (see also  references therein)
is particularly useful.

% additional
Explicit expressions for the NNLL results are given in \cite{NNLL}.
Numerically, for the reaction $e^+e^-\rightarrow \mu^+\mu^-$
as a typical example, one obtains a correction factor
\bea
  R^{\mu^+\mu^-} =
    1 - 1.39\,L(s) + 10.12\,\ell(s) - 31.33\,a
\nn \\
    + 1.42\,L^2(s) - 18.43\,L(s)\ell(s)
    + 99.89\,\ell^2(s)
\eea
with $a = g^2/16\pi^2 = 2.69 {\cdot} 10^{-3}$,
$\ell(s) = a\,\ln s/M^2$, $L(s) = a\,\ln^2 s/M^2$.
%
To get the infrared safe result for the semi-inclusive cross sections one has to
include the standard QED corrections due to soft photon emission and the pure
QED virtual corrections.

In view of the large subleading terms
the evaluation of two-loop terms linear in the logarithm
or even of the constant terms seems desirable.
As a first step the fermionic corrections to the form factor,
i.e.\ terms resulting from the inclusion of massless fermions in loop insertions
in the boson propagator,
have been calculated for the Abelian form factor~\cite{FKMPS}:
\bea
&& \hspace{-2em}  F_{2,n_f} =
%  n_f \left(\frac{g^2}{16\pi^2}\right)
  n_f \frac{g^2}{16\pi^2}
  \biggl\{
    -\frac{4}{9}\ln^3\biggl(\frac{-s}{M^2}\biggr)
%\nn \\ && \qquad
    +\frac{38}{9}\ln^2\biggl(\frac{-s}{M^2}\biggr)
\nn \\ && \qquad
    -\frac{34}{9}\ln\biggl(\frac{-s}{M^2}\biggr)
%\nn \\ && \qquad
    +\frac{16}{27}\pi^2 + \frac{115}{9}
    \biggr\}
\eea
The oscillating pattern of the signs in subleading terms continues,
and the linear logarithm as well as the constant terms are sizeable.
These results indicate the need for the evaluation of two-loop corrections
for realistic reactions,
including the linear logarithm and the constant term.

In addition to four fermion processes, which are clearly the benchmark reactions
at high energies, also other reactions have been investigated in the high energy
limit.
In the case of gauge boson pair production LL and NLL terms have been evaluated
by several authors~\cite{MellesDP}.
Exclusive production of transversly and longitudinally polarized $W$
is suppressed by $\exp(-C_A\,\ln^2 s/M^2)$ and $\exp(-C_F\,\ln^2 s/M^2)$
respectively, as expected from the equivalence theorem.
The implications of these results for gauge boson pair production at hadron
colliders were studied in~\cite{ADP}.
As noted in~\cite{CCC},
enhanced electroweak corrections are also present in inclusive boson fusion
processes, a consequence of the lack of compensation between virtual and real
emission due to the non-Abelian charges of the incoming particles.

\noindent
\emph{Acknowledgement:}
Pleasant collaboration with 
B.~Feucht, S.~Moch, A.A~Penin and V.A.~Smirnov is gratefully acknowledged.


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\bibitem{KuhPen} J.H. K\"uhn and A.A. Penin, Preprint TTP/99--28,
                 hep-ph/9906545.
                 %Preprint TTP/99--28

\bibitem{Fad} V.S.~Fadin, L.N.~Lipatov, A.D.~Martin and M.~Melles,
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              %Preprint DESY/00-008, TTP/99--54

\bibitem{Melles1} M.~Melles, hep-ph/0104232.

\bibitem{NNLL}
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\bibitem{SR}
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\bibitem{Melles2}
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\bibitem{FKMPS}
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\bibitem{MellesDP}
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\bibitem{ADP} E.~Accomando, A.~Denner and S.~Pozzorini,
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\bibitem{CCC} M.~Ciafaloni, P.~Ciafaloni and D.~Comelli,
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\end{thebibliography}


\end{document}

