%% slacpub7176: page file slacpub7176003.tcx.
%% section 3 Resummation of soft emission and Wilson lines [slacpub7176003 in slacpub7176003: slacpub7176004]
%%%% latex2techexplorer block:
%% latex2techexplorer page setup:
\iftechexplorer
\setcounter{section}{2}
\fi
\iftechexplorer
\setcounter{secnumdepth}{2}
\setcounter{tocdepth}{2}
\def\thepart#1{}%
\def\thechapter#1{}%
\newcommand{\partLink}[3]{\docLink{#1.tcx}[part::#2]{#3}\\}
\newcommand{\chapterLink}[3]{\docLink{#1.tcx}[chapter::#2]{#3}\\}
\newcommand{\sectionLink}[3]{\docLink{#1.tcx}[section::#2]{#3}\\}
\newcommand{\subsectionLink}[3]{\docLink{#1.tcx}[subsection::#2]{#3}\\}
\newcommand{\subsubsectionLink}[3]{\docLink{#1.tcx}[subsubsection::#2]{#3}\\}
\newcommand{\paragraphLink}[3]{\docLink{#1.tcx}[paragraph::#2]{#3}\\}
\newcommand{\subparagraphLink}[3]{\docLink{#1.tcx}[subparagraph::#2]{#3}\\}
\newcommand{\partInput}{\partLink}
\newcommand{\chapterInput}{\chapterLink}
\newcommand{\sectionInput}{\sectionLink}
\else
\newcommand{\partInput}[3]{\input{#2.tcx}}
\newcommand{\chapterInput}[3]{\input{#2.tcx}}
\newcommand{\sectionInput}[3]{\input{#2.tcx}}
\fi
\newcommand{\subsectionInput}[3]{\input{#2.tcx}}
\newcommand{\subsubsectionInput}[3]{\input{#2.tcx}}
\newcommand{\paragraphInput}[3]{\input{#2.tcx}}
\newcommand{\subparagraphInput}[3]{\input{#2.tcx}}
\aboveTopic{slacpub7176.tcx}%
\previousTopic{slacpub7176002.tcx}%
\nextTopic{slacpub7176004.tcx}%
\bibfile{slacpub7176005u2.tcx}%
\newmenu{slacpub7176::context::slacpub7176003}{
\docLink{slacpub7176.tcx}[::Top]{Top}%
\sectionLink{slacpub7176002}{slacpub7176002}{Previous: 2. IR sensitivity of LLA and soft gluon emission at large angles}%
\sectionLink{slacpub7176004}{slacpub7176004}{Next: 4. Top quark production at the TEVATRON}%
}
%%%% end of latex2techexplorer block.
%%%% code added by add_nav perl script
\docLink{slacpub7176.tcx}[::Top]{Top of Paper}%

\docLink{pseudo:previousTopic}{Previous Section}%
\bigskip%
%%%% end of code added by add_nav
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% author definitions added by nc_fix
\def\Journal#1#2#3#4{{#1} {\bf #2}, #3 (#4)}
\def\NCA{\em Nuovo Cimento}
\def\NIM{\em Nucl. Instrum. Methods}
\def\NIMA{{\em Nucl. Instrum. Methods} A}
\def\NPB{{\em Nucl. Phys.} B}
\def\PLB{{\em Phys. Lett.} B}
\def\PRL{\em Phys. Rev. Lett.}
\def\PRD{{\em Phys. Rev.} D}
\def\ZPC{{\em Z. Phys.} C}
\def\st{\scriptstyle}
\def\sst{\scriptscriptstyle}
\def\mco{\multicolumn}
\def\ra{\rightarrow}
\def\vp{{\bf p}}
\def\al{\alpha}
\def\ab{\bar{\alpha}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\CPbar{\hbox{{\rm CP}\hskip1.80em{/}}}%temp replacement due to no font
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%% end of definitions added by nc_fix
\section{\usemenu{slacpub7176::context::slacpub7176003}{Resummation of soft emission and Wilson lines}}\label{section::slacpub7176003}
Exponentiation of large logarithms occurs for both,
collinear emission and soft gluon emission, including emission
at large angles. Thus, after a complete treatment of
resummation of soft gluon emission,
the apparent $1/Q$ sensitivity in LLA should be compensated by
other conributions to the exponent. The bestknown generalization
of the LLA formula Eq.~\docLink{slacpub7176001.tcx}[LLA]{2} was given by Sterman \cite{8}
\bea\label{form11}
\ln \omega_{DY}(N,\alpha_s) &=& \int_0^1 d z\,\frac{z^{N1}1}
{1z} \Bigg\{2\int_{Q^2 (1z)}^{Q^2 (1z)^2} \frac{d k_\perp^2}{k_\perp^2}\,
\Gamma_{\rm cusp}(\alpha_s(k_\perp)) \nonumber\\
&& + B(\alpha_s(\sqrt{1z} Q))
+\,C(\alpha_s((1z) Q)\Bigg\} + {\cal O}(1)
\eea
This expression involves three ``anomalous dimensions''
$\Gamma_{\rm cusp},B$ and $C$.
The LLA corresponds to taking into account the leading term $O(\alpha_s)$
in the expansion of $\Gamma_{\rm cusp}$ and neglecting $B,C$. The
nexttoleading logarithmic accuracy requires two terms in
$\Gamma_{\rm cusp}$ and the leading $O(\alpha_s)$ terms in $B,C$.
In general, higher order corrections to $\Gamma_{\rm cusp},B,C$ generate
contributions with less and less powers of $\ln N$ for a given
power of $\alpha_s$.
The three terms in Eq.~\docLink{slacpub7176003.tcx}[form11]{7} have the following origin:
The function ``$B$'' comes from subtracting the DIS cross section
(squared) and is irrelevant for our discussion.
The term with a double integral resums soft
and collinear gluon radiation to all orders. This contribution
is universal for all hard processes.
Finally, ``$C$'' corrects for soft gluon radiation at large angles and
is processdependent.
This term starts with $O(\alpha_s^2)$
in agreement with the common wisdom that the radiation at large angles
is suppressed by two powers of $\ln N$, but taking it into account
can be crucial to recover the correct IR behavior.
Thus, the $1/Q$ IR sensitivity of the generalized LLA
expression given by the first term in Eq.~\docLink{slacpub7176003.tcx}[form11]{7} should be cancelled
by the ``$C$''term. To test how this happens, one needs some
approximation to calculate the anomalous dimensions to all orders
in perturbation theory. A convenient formal parameter is $N_f$, the number
of light fermion flavors. The leading contribution in the
large$N_f$ limit corresponds to a chain of fermion loops inserted into
the single gluon line. Taking into account the chain of loops has two
effects: First, it generates the correct argument of the running coupling
in Eq.~\docLink{slacpub7176003.tcx}[form11]{7} which is the gluon transverse momentum. [As usual,
we tacitly assume that $N_f$ can be used to reconstruct the
full oneloop running, determined by $\beta_0$.] Second,
counter\terms for individual fermion loops produce nontrivial
anomalous dimensions $\Gamma_{\rm cusp},B,C$ to all orders in $\alpha_s$.
The corresponding calculation is technical and can be found in Ref.~1.
The result is that all $1/Q$ IR contributions generated by the
generalized LLA term are cancelled by the IR contributions related to
factorial divergence of the perturbation expansion of ``$C$''.
This tells us that although the resummation formula Eq.~\docLink{slacpub7176003.tcx}[form11]{7} is
valid, it involves significant cancellations between different
contributions and the true IR behaviour is restored only after summation of
an {\em infinite} number of terms in the expansion of ``$C$''.
This contradicts the logic of resummation to resum an
infinite number of large logarithms by calculating
only a finite number of terms in the anomalous dimensions.
The simplest remedy would be to use the exact phase space factor
for the oneloop gluon emission and replace the first term in
Eq.~\docLink{slacpub7176003.tcx}[form11]{7} by
\be\label{form11mod}
\ln \omega_{DY}(N,\alpha_s) = 2\!\int_0^1 \!\!d z\,[z^{N1}1]
\int_{Q^2 (1z)}^{Q^2 (1z)^2}\! \frac{d k_\perp^2}{k_\perp^2}\,
\frac{\Gamma_{\rm cusp}(\alpha_s(k_\perp))}{\sqrt{(1z)^2+4k_\perp^2/Q^2}}
+\ldots
\ee
With this substitution the $1/Q$ IR sensitivity disappears and the
functions $B,C$ are modified
starting $O(\alpha_s^2)$ so that the undesired
behavior of $C$ in large orders is removed.
A different approach to soft gluon resummation emphasizes the
renormalization of Wilson lines \cite{9}. Its theoretical advantage
is the operator language that avoids the separation of
smallangle and largeangle soft emission. $1/Q$ IR contributions
never appear. The starting point is the wellknown fact that
soft gluon emission from a fast quark can be described by a
Wilson line operator along the classical trajectory of the quark.
The product of Wilson lines for the annihilating
quark and antiquark is denoted by $U_{\rm DY}(x)$,
where $x$ is the annihilation spacetime
point. Up to corrections which vanish
as $z\to 1$, the partonic DrellYan cross section is given by \cite{9}
\be\label{softfactorization}
\omega_{\rm DY}(z,\alpha_s) = H_{\rm DY}(\alpha_s)\,
W_{\rm DY}(z,\alpha_s)\,.
\ee
$H_{\rm DY}=1+{\cal O}(\alpha_s)$ is a shortdistance dominated function,
independent of $z$.
$W_{\rm DY}$ is the square of the
matrix element $\langle nT U_{\rm DY}(0)0\rangle$,
summed over all final states:
\be
W_{\rm DY}(z,\alpha_s) = \frac{Q}{2}\int_{\infty}^
\infty\frac{d y_0}{2\pi}\,e^{i y_0 Q (1z)/2}\,
\langle 0\bar{T}\,U^\dagger_{\rm DY}(y)
\,T\,U_{\rm DY}(0)0\rangle
\ee
The Fourier transform is taken with respect to the energy of soft
partons and $y=(y_0,\vec{0})$.
The crucial observation is that the Wilson line depends only on
the ratio $(\mu N)/(Q N_0)$ (taking moments of $W_{\rm DY}(z,\alpha_s)$),
where $\mu$ is a
cutoff separating soft and hard emission (the renormalization scale
for the Wilson line) and $N_0$ is a suitable constant.
Hence the $N$dependence of the DrellYan cross section in the soft
limit can be obtained from the $\mu$dependence of Wilson lines,
which is given by the renormalization group equation (here
$\alpha_s=\alpha_s(\mu)$)
\be\label{evolutionequation}
{}\!\!\left(\!\mu^2\!\frac{\partial}{\partial\mu^2} + \beta(\alpha_s)
\frac{\partial}{\partial \alpha_s}\!\!\right)\,\!\ln W_{\rm DY}\!
\!\left(\frac{\mu^2 N^2}{Q^2 N_0^2},\alpha_s\!\right) =
\Gamma_{\rm cusp}(\alpha_s)\,\ln\frac{\mu^2 N^2}{Q^2 N_0^2}
+ \Gamma_{\rm DY}(\alpha_s)\,.
\ee
It involves two anomalous dimensions $\Gamma_{\rm cusp}(\alpha_s)$
and $\Gamma_{\rm DY}(\alpha_s)$ related to the cusp
and to presence of lightlike segments on the Wilson line, respectively.
The general solution of Eq.~\docLink{slacpub7176003.tcx}[evolutionequation]{11} is given by
\bea\label{sol1}
\lefteqn{\ln W_{\rm DY}\!
\left(\frac{ N^2}{ N_0^2},\alpha_s(Q)\right)=
\ln W_{\rm DY}(1,\alpha_s(Q N_0/N))+}
\nonumber\\&&
+\!\int_{Q^2 N_0^2/N^2}^{Q^2}
\!\frac{d k_\perp^2}{
k_\perp^2}\left[\Gamma_{\rm cusp}(\alpha_s(k_\perp)\,\ln\frac{k_\perp^2 N^2}{
Q^2 N_0^2} + \Gamma_{\rm DY}(\alpha_s(k_\perp))\right],
\eea
where we set $\mu=Q$.
The inhomogeneous second term in Eq.~\docLink{slacpub7176003.tcx}[sol1]{12} can be rewritten (identically)
in a more familiar form, which resembles the first term in Eq.~\docLink{slacpub7176003.tcx}[form11]{7}:
\be
2\int_0^{1N_0/N}\!\!\frac{d z}{1z}\!\Bigg[\int_{(1z)^2 Q^2}^{Q^2}
\frac{d k_\perp^2}{k_\perp^2}\,\Gamma_{\rm cusp}(\alpha_s(k_\perp))
+\,\Gamma_{\rm DY}(\alpha_s((1z) Q)\Bigg]
\ee
Note presence of the
{\em initial condition} $W_{\rm DY}(1,\alpha_s(Q N_0/N))$.
Its expansion in $\alpha_s$
produces subdominant logarithms $\alpha_s^k\ln^{k1} N$ which can be absorbed
into a redefinition of $\Gamma_{\rm DY}$:
\be\label{redef}
\Gamma_{\rm DY}(\alpha_s) \longrightarrow \tilde{C}(\alpha_s)=
\Gamma_{\rm DY}(\alpha_s)
 \beta(\alpha_s)\frac{d}{d\alpha_s}\,\ln W_{\rm DY}(1,\alpha_s).
\ee
$\tilde{C}(\alpha_s)$ starts at order $\alpha_s^2$.
It does not affect resummation of large
logarithms in $N$ to nexttoleading accuracy $\alpha_s^k\ln^k N$.
It remains to subtract the DIS cross section, which can also be
implemented in the language of Wilson lines, see Ref.~9 for details.
Finally, we get
\bea\label{form2}
\ln \omega_{\rm DY}(N,\alpha_s) &=& \int_0^{1N_0/N}\!\!
d z\,\frac{1}{1z} \Bigg\{2\int_{Q^2 (1z)}^{Q^2 (1z)^2}
\frac{d k_\perp^2}{k_\perp^2}\,\Gamma_{\rm cusp}(\alpha_s(k_\perp))
\nonumber\\&&{} +
\tilde{B}(\alpha_s(\sqrt{1z} Q))
+\,\tilde{C}(\alpha_s((1z) Q))\Bigg\} + {\cal O}(1)\,.
\eea
($N_0=\exp(\gamma_E)$ in the $\overline{\rm MS}$ scheme.)
This form of the resummed cross section is as
legitimate in the framework of the perturbation theory as
the more conventional expression in Eq.~\docLink{slacpub7176003.tcx}[form11]{7}. They have different
properties, however, as far as sensitivity to the IR behavior
of the coupling is concerned, which becomes
important when the anomalous dimensions $\Gamma_{\rm cusp},\ldots$ are
truncated to finite order.
Since the region of very large $z\to 1$
is removed in Eq.~\docLink{slacpub7176003.tcx}[form2]{15}, this expression shows no IR sensitivity
at all unless $N>Q/\Lambda_{\rm QCD}$.
Loosely speaking, this is so because the Wilson line approach
treats small and largeangle gluon emission in a coherent way.
Because this technique can also treat subleading logarithms in a
systematic way, it is preferred over, for example, the modification
of the phase space as in Eq.~\docLink{slacpub7176003.tcx}[form11mod]{8}.
It is natural to expect (and explicit calculation \cite{1} in
the large$N_f$ limit confirms this) that the anomalous dimensions
$\Gamma_{\rm cusp}(\alpha_s)$
and $\Gamma_{\rm DY}(\alpha_s)$ in the $\overline{MS}$ scheme
are analytic functions of the coupling
at $\alpha_s=0$. Then, all power
corrections to the resummed cross section
(to all orders in $N\Lambda_{\rm QCD}/Q$)
originate exclusively from the initial condition
for the evolution equation for the Wilson line, and are not
created (or modified) by the evolution, i.e. by soft gluon
resummation.
Thus, if the resummation of soft gluon emission is done coherently
for all angles, the only effect of soft gluons on power
corrections is a {\em change of scale}, the replacement
$Q\to Q/N$ as the parameter of the power expansion.
This suggests that, in general, there is no reason to suspect
new nonperturbative contributions in resummed cross sections
as compared to finiteorder calculations. The conclusion that
power corrections to DrellYan production are suppressed by $1/Q^2$
is then consistent with the analysis of power corrections
at tree level by Qiu and Sterman \cite{10}.
The redefinition in Eq.~\docLink{slacpub7176003.tcx}[redef]{14} transforms the
IR sensitivity (and potential power corrections) of the
initial condition for the evolution equation for
the Wilson line into IR sensitivity of the function
$\tilde{C}$ in Eq.~\docLink{slacpub7176003.tcx}[form2]{15}. As mentioned above, this function
becomes important precisely when one starts to be sensitive to
gluon radiation at large angles, and the conclusions on power corrections
depend sensitively on this region. Because of this,
we are sceptical that universality of nonperturbative corrections to
resummed cross sections could be deduced from the universality of
softcollinear gluon emission as embodied by the eikonal (cusp)
anomalous dimension, an idea originally put forward in Refs.~3,11.
In the Wilson line technique the solution
of the evolution equation never involves the QCD coupling integrated
over the IR Landau pole as long as $N