%Title: Electroweak corrections to large transverse momentum production of Z bosons and photons at the LHC 
%Author: A.Kulesza, S.Pozzorini and M. Schulze
%Published:     Contribution to the <EM> Physics at TeV Colliders </EM> Workshop Proceedings, Les Houches, May 2005. 
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%\bibliographystyle{lesHouches}


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\newcommand{\ord}{{\cal O}}
\def\nl{\nonumber\\}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\beqar{\begin{eqnarray}}
\def\eeqar{\end{eqnarray}}
\newcommand{\shat}{{\hat s}}
\newcommand{\that}{{\hat t}}
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\def\de{\delta}
\def\veps{\varepsilon}
\def\la{\lambda}
\def\si{\sigma}

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\newcommand{\thw}{{\theta_\mathrm{w}}}
\newcommand{\sw}{{s_{\mathrm{\scriptscriptstyle W}}}}
\newcommand{\cw}{{c_{\mathrm{\scriptscriptstyle W}}}}

\newcommand{\cew}{C^{\mathrm{ew}}}

\begin{document}
\bibliographystyle{unsrt}
\title{Electroweak corrections to large transverse momentum
  production of \mbox{Z bosons} and photons at the
  LHC\footnote{Contribution to the {\em Physics at TeV Colliders}
Workshop
Proceedings, Les Houches, May 2005.}}

\author{A.Kulesza$^1$, S.Pozzorini$^2$, M. Schulze$^1$}
\institute{$^1$Institut f\"ur Theoretische Teilchenphysik,
  Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany
\\$^2$Max-Planck-Institut f\"ur Physik (Werner-Heisenberg-Institut),
F\"ohringer Ring 6, D-80805 M\"unchen, Germany}

\maketitle

\begin{abstract}
We investigate the electroweak corrections to the 
$Z$-boson and direct photon production
at large transverse momentum at the LHC. Analytical results
for the high energy next-to-next-to-leading
logarithmic approximation of the one-loop weak corrections and  
the two-loop electroweak corrections in the next-to-leading
logarithmic approximation are presented. 
Numerical results showing the relevance of these corrections are discussed. 
\end{abstract}

\section{INTRODUCTION}

The Large Hadron Collider (LHC) with its 
high center-of-mass energy and design luminosity
will offer a unique possibility to explore production of gauge bosons with very large
transverse momentum ($\pT$). Being embedded in the
environment of hadronic collisions, the reaction necessarily
involves hadronic physics, like parton distributions, and
depends on the strong coupling constant. In turn, the cross section
for the $Z$-boson or direct photon production at large $\pT$ is an important means
to constrain information on the parton distribution functions (PDFs).
For this class of production processes at the LHC, 
electroweak (EW) corrections from virtual boson exchange become 
important. This is due to the presence of the logarithmic 
terms of the form $\alpha^k\log^{2k-i}(\hat{s}/M_W^2)$ (with $i=0$ for
the leading logarithmic (LL) terms, $i=1$ for the next-to-leading
logarithmic (NLL) terms, etc.) at the $k$-loop level.
These corrections, also known as electroweak Sudakov logarithms, may
well amount to several tens of per cent. A recent survey of the literature on 
logarithmic EW corrections
can be found in Ref.~\cite{Hollik:2004dz}.
Specifically, full one-loop weak corrections to the hadronic $Z$-boson and
prompt photon production at large $\pT$ have been studied 
in Refs.~\cite{Kuhn:2005az,Kuhn:2005gv}. 
Numerical results for the one-loop corrections to these two processes
can be also found in Ref.~\cite{Maina:2004rb}. In the following we
briefly discuss the calculation of Refs.~\cite{Kuhn:2005az,Kuhn:2005gv}
and present compact analytic expressions
for the $Z$-boson and direct photon production in the high energy
approximation, derived from the exact one-loop results
in Refs.~\cite{Kuhn:2005az,Kuhn:2005gv}. Moreover, we also present the NLL contributions
to the two-loop EW corrections,
calculated in Refs.~\cite{Kuhn:2005gv,Kuhn:2004em}. Corresponding numerical
results are discussed.


Of course, to achieve reliable predictions at high $\pT$, the QCD corrections in
next-to-leading order (NLO) need to be taken into
account as they can amount to several tens of per cent correction for both processes.
However, in the following we are interested
only in the EW corrections to the leading order (LO) in the
strong coupling constant $\alpha_s$  cross section for the  process $pp \to V j$ \mbox{($V=\gamma,Z;\
j=\mathrm{jet}$)}.



\section{CALCULATION}
In the calculation of the one-loop
corrections to the hadronic production of $Z$ bosons~\cite{Kuhn:2005az} and
the hadronic production of photons~\cite{Kuhn:2005gv} at large $\pT$
we consider only weak contributions, i.e. we do not include electromagnetic
corrections.
%The calculation of the one-loop
%corrections to the hadronic production of $Z$ bosons~\cite{Kuhn:2005az} and the hadronic production 
%of photons~\cite{Kuhn:2005gv} at large $\pT$ involves only weak
%contributions, i.e. no electromagnetic corrections are considered.
The quarks are assumed massless and diagrams involving couplings of
quarks to Higgs bosons or would-be-Goldstone bosons are neglected. Moreover,
we omit quark-mixing effects in the calculations. 
The calculation for
the $\bar q q \to V g$ ($V=\gamma,Z$) subprocess is performed at the  
level of matrix elements and allows for full
control over polarization effects. Results for the other contributing subprocesses,
$q\bar q\to V g$,
$g q \to V  q$,
$q g \to V  q$,
$\bar q g  \to V  \bar q$
and
$g \bar q   \to V  \bar q$
are easily obtained from CP symmetry and crossing transformations.
The one-loop amplitude is split
into two parts: Abelian and non-Abelian, 
as the structure of the gauge group generators in front of each term
contributing to the amplitude can
 be classified as either
Abelian (characteristic for Abelian theories) or non-Abelian
(originating from non-commutativity of weak interactions). 
Tensor loop integrals, appearing in
the expressions for one-loop corrections are reduced to scalar integrals
by means of Passarino-Veltman technique. The
renormalization of $Z$- and photon-production amplitudes 
is perfomed in the $\overline{\rm MS}$ and in the on-shell (OS)
scheme, respectively. Further
details of the calculation can be found in Refs.~\cite{Kuhn:2005az,Kuhn:2005gv}.


\subsection{Analytical results}

The $\pT$ distribution for the partonic reaction $\bar q q \to V g$
($V=\gamma,\;Z$) reads 
\beqar\label{partoniccs1}%\refeq{partoniccs1}
\frac{\rd \hat{\si}^{\bar q q \to V g}}{\rd \pT}
&=&
\frac{\pT}{8\pi N_c^2\shat|\that-\uhat|}
\left[
\overline{\sum}|\M^{\bar q q \to V g}|^2+(\that\leftrightarrow \uhat)
\right]
,
\eeqar
where $N_c=3$,
$\overline{\sum}=
\frac{1}{4}
\sum_{\mathrm{pol}} 
\sum_{\mathrm{col}}
$, 
$\shat=(p_{\bar q}+ p_q)^2$,
$\that=(p_{\bar q}- p_V)^2$,
$\uhat=(p_q-p_V)^2$ 
and $p_V^2=M_V^2$.
To  $\ord(\alpha^2\alpha_\rS)$ for the unpolarized squared matrix element 
we have
\beqar\label{generalresult}%\refeq{generalresult}
\lefteqn{
\overline{\sum}
|\M_1^{\bar q q \to V g}|^2 =
8 \pi^2 \alpha \alpha_\rS (N_\mathrm{c}^2-1)\,    
\sum_{\lambda=\mathrm{R,L}}  
\Bigg\{  
\left(I^V_{q_\la}\right)^2  \bigg[ 
H_0^V \left( 1+2\delta C^{V,\mathrm{A}}_{q_{\lambda}} \right)} 
\\&&
+ \frac{\alpha}{2\pi}\sum_{V'=\mathrm{Z,W^{\pm}}} \left(I^{V'} I^{\bar{V'}}\right)_{q_{\lambda}}
\,H_1^{V,\mathrm{A}}(M_{V'}^2)
\bigg]
+  \frac{U_{V {W^3}}}{\sw} T_{q_\la}^3 I^V_{q_\la} 
\bigg[
2 H_0^V\delta C^{V,\mathrm{N}}_{q_{\lambda}}
+ \frac{\alpha}{2\pi}
\frac{1}{\sw^2}H_1^{V,\mathrm{N}}(M_W^2)
\bigg]\Bigg\} , \nonumber
\eeqar
where $I^V_{q_\la}$ represents the 
coupling of an electroweak gauge boson $V$ 
to right-handed ($\la=\rR$) or left-handed ($\la=\rL$) quarks.
In terms of the electric charge $Q_q$, the weak isospin $T^3_{q_\la}$
and the weak hypercharge $Y_{q_\la}$ we have $I^{\gamma}_{q_\la}=-Q_q$,
$I^Z_{q_\la}=\cw/\sw T^3_{q_\la}-\sw{Y_{q_\la}}/(2\cw)$,
with the shorthands $\cw=\cos{\thw}$ and  $\sw=\sin{\thw}$
for the  weak mixing angle $\thw$. Moreover the relevant elements of
the Weinberg rotation matrix $U$ are given by $U_{\gamma {W^3}} = -\sw$ and 
$U_{Z {W^3}} = \cw$.
The term 
$H_0^V=({\that^2+\uhat^2+2\shat M_V^2})/({\that\uhat})$ in \refeq{generalresult}
represents the Born contribution.
$\delta C^{V,\mathrm{A}/\mathrm{N}}_{q_{\lambda}}$
summarize the counterterms associated with the renormalization
of the couplings and the gauge-boson wave function.
The contributions resulting from the bare loop diagrams 
and the fermionic wave-function renormalization correspond to the
functions $H_1^{V,\mathrm{A/N}}$.
The complete analytic expressions for these functions 
and for the counterterms, as well as details concerning the choice of
the renormalization scheme, can be found in Ref.~\cite{Kuhn:2005az}
for the $Z$-boson production and 
in Ref.~\cite{Kuhn:2005gv}
for photon production.
In the following we concentrate on the high-energy asymptotic
behaviour of the corrections, more precisely the
next-to-next-to-leading logarithmic (NNLL) approximation
of the full result.
Formally we consider the limit where $M_W^2/\shat\to 0$ and 
$\that/\shat$, $\uhat/\shat$ are constant.
In this limit the one-loop weak corrections are strongly enhanced by 
logarithms of the form $\log(\shat/M_W^2)$. 
The functions $H_1^{V,\mathrm{A/N}}$ assume the general form
\beq\label{nnllstracture}%\refeq{nnllstracture}
H_1^{V,\mathrm{A/N}}(M_{V'}^2) \stackrel{\mathrm{NNLL}}{=}
\mathrm{Re }\,\left[
g_0^{V,\mathrm{A/N}}(M_{V'}^2)\,
\frac{ \that^2+\uhat^2}{\that\uhat}
+g_1^{V,\mathrm{A/N}}(M_{V'}^2)\,
\frac{ \that^2-\uhat^2}{\that\uhat}
+g_2^{V,\mathrm{A/N}}(M_{V'}^2)
\right].
\eeq
We find  $g_i^{\gamma,\mathrm{A}}=
g_i^{Z,\mathrm{A}}$ for $i=0,1,2$ and  $g_j^{\gamma,\mathrm{N}}=
g_j^{Z,\mathrm{N}}$ for $j=1,2$. For $V=\gamma,\;Z$ one has: 
\newcommand{\Deltamsbar}{\bar \Delta_{\mathrm{UV}}}
\beqar\label{heres1}%\refeq{heres1}
 g_0^{V,\mathrm{A}}(M_{V'}^2)&=& 
-\log^2\left(\frac{-\shat}{M_{V'}^2}\right)
+3\log\left(\frac{-\shat}{M_{V'}^2}\right)
+\frac{3}{2}\Biggl[
\log^2\left(\frac{\that}{\shat}\right)
+\log^2\left(\frac{\uhat}{\shat}\right)
\nl&&{}
+\log\left(\frac{\that}{\shat}\right)
+\log\left(\frac{\uhat}{\shat}\right)
\Biggr]
+\frac{7\pi^2}{3}-\frac{5}{2}
,\\ 
 g_1^{V,\mathrm{N}}(M_{V'}^2)&=&
- g_1^{V,\mathrm{A}}(M_{V'}^2)+
\frac{3}{2}\Biggl[
\log\left(\frac{\uhat}{\shat}\right)
-
\log\left(\frac{\that}{\shat}\right)
\Biggr]
=
\frac{1}{2}\Biggl[
\log^2\left(\frac{\uhat}{\shat}\right)
-
\log^2\left(\frac{\that}{\shat}\right)
\Biggr]
,\nl 
 g_2^{\mathrm{V,N}}(M_{V'}^2)&=&
 -g_2^{\mathrm{V,A}}(M_{V'}^2)=
-2\Biggl[
\log^2\left(\frac{\that}{\shat}\right)
+\log^2\left(\frac{\uhat}{\shat}\right)
+\log\left(\frac{\that}{\shat}\right)
+\log\left(\frac{\uhat}{\shat}\right)
\Biggr]
-4{\pi^2}
. \nonumber
\eeqar
The non-Abelian function $g_0^{V,\mathrm N}$ is given by
($\delta_{VV'}=1$ for $V=V'$, otherwise $\delta_{VV'}=0$):
\beqar
g_0^{V,\mathrm{N}}(M_W^2)&=& 
2 \left[ \Deltamsbar - \delta_{V \gamma}\log \left({M_W^2 \over M_Z^2}\right)\right]
+\log^2\left(\frac{-\shat}{M_W^2}\right)
-\log^2\left(\frac{-\that}{M_W^2}\right)
-\log^2\left(\frac{-\uhat}{M_W^2}\right)
\nl&+& \!\!\!\!
\log^2\left(\frac{\that}{\uhat}\right)
-\frac{3}{2}\Biggl[
\log^2\left(\frac{\that}{\shat}\right)
+\log^2\left(\frac{\uhat}{\shat}\right)
\Biggr] - 2\pi^2 + 2 \delta_{VZ} \left( -\frac{\pi^2}{9}
-\frac{\pi}{\sqrt{3}} +2  \right).
\eeqar
The ultraviolet singularity, 
$\Deltamsbar = 2/(4-D) - \gamma_{\mathrm{E}} +\log(4\pi \mu^2/{M_Z^2})$ 
is cancelled by the counterterms \cite{Kuhn:2005az, Kuhn:2005gv}. 


\newcommand{\logar}[2]{\mathrm{L}^{#1}_{#2}}
\newcommand{\newA}[2]{A^{#1,(#2)}}
The size of the logarithmically enhanced contributions grows with energy
and for transverse momenta of hundreds of GeV also the 
two-loop logarithms become important. At the NLL accuracy,
our results for the two-loop corrections include contributions with terms of the form  
$\alpha^2\logar{4}{\shat}$ and $\alpha^2\logar{3}{\shat}$
where $\logar{k}{\rhat}=\log^k(|\rhat|/ M_W^2)$. The expressions
presented here 
have been obtained using results of Refs.~\cite{Denner:2001jv,Denner:2003wi,Melles:2001gw}.
Since at two-loop level the purely weak corrections cannot be isolated 
from the complete electroweak corrections in a gauge-invariant way, we
have to consider the combination of weak and electromagnetic virtual 
corrections. The latter are regularized by means of a fictitious photon mass 
$\lambda=M_W$. 
For the unpolarized squared matrix element we have
$\overline{\sum}|\M_2^{\bar q q \to V g}|^2
-
\overline{\sum}|\M_1^{\bar q q\to V g}|^2 \stackrel{\mathrm{NLL}}{=}
2 {\alpha^3\alpha_\rS}(N_\rc^2-1)H_0^V
\newA{V}{2}
$ 
where
\beqar\label{twolooplogs}%\refeq{twolooplogs}
\newA{V}{2}&=&
\sum_{\la=\rL,\rR}\Biggl\{
\frac{1}{2}
\left(
I^V_{q_\la}\cew_{q_\la}
+ S^V_{q_\la}
\right)
\Biggl[
I^V_{q_\la}\cew_{q_\la}\left(\logar{4}{\shat}-6\logar{3}{\shat}\right)
\nl&&{}
+
S^V_{q_\la}
\left(\logar{4}{\that}+\logar{4}{\uhat}-\logar{4}{\shat}\right) 
\Biggr]
-(\delta_{VZ}-\delta_{V\gamma})\frac{T^3_{q_\la}Y_{q_\la}}{8\sw^4}
\left(\logar{4}{\that}+\logar{4}{\uhat}-\logar{4}{\shat}\right) 
\nl&&{}
+\frac{1}{6}I^V_{q_\la}
\Biggl[I^V_{q_\la}\left(
\frac{b_1}{\cw^2}\left(\frac{Y_{q_\la}}{2}\right)^2
+\frac{b_2}{\sw^2} C_{q_\la}
\right)
+S^V_{q_\la}b_2
\Biggr]\logar{3}{\shat}
\Biggr\}.
\eeqar
Here $\cew_{q_\la}=Y_{q_\la}^2/(4\cw^2)+C_{q_\la}/\sw^2$
is the electroweak Casimir operator 
for quarks, with $C_{q_\rL}=3/4$  and $C_{q_\rR}=0$. $S^V_{q_\la}$
is defined as $S^V_{q_\la} = C^{\rm ew}_{V V'} I_{q_\la}^{V'}/2$ where
$C^{\rm ew}_{V V'}$ is the electroweak Casimir operator in the adjoint
representation~\cite{Denner:2001jv,Pozzorini:2001rs} and $S^{\gamma}_{q_\la}= - T^3_
{q_\la}/\sw^2,\;S^{Z}_{q_\la}= \cw T^3_ {q_\la}/\sw^3$. 
The  one-loop $\beta$-function coefficients read $b_1=-41/(6\cw^2)$
and $b_2=19/(6\sw^2)$. We stress that although the above one- and two-loop results for the
photon and $Z$-boson production can be put in the same form, their
derivation requires separate calculation for each of the processes.



\subsection{Numerical results}
The hadronic cross sections are obtained using LO MRST2001 PDFs~\cite{Martin:2002dr}.
We choose $p_T^2$ as the factorization scale
and, similarly as the scale at which the running strong coupling constant
is evaluated. We also adopt the value 
$\alpha_S(M_Z^2)=0.13$ and use the one-loop running expression for 
$\alpha_S(\mu^2)$, in accordance with the LO 
PDF extraction method
of the MRST collaboration. 
%
We use the
following values of parameters~\cite{Eidelman:2004wy}:
$M_Z=91.19$ GeV, $M_W=80.39$ GeV and
$\alpha=1/137.0$, $\sw^2=1-\cw^2=1-M_W^2/M_Z^2$ for the $\gamma$ production
(OS scheme with $\alpha$ in the Thompson limit) or $\alpha=1/128.1$,
$\sw^2=0.2314$ for the $Z$-boson production ($\overline{{\rm MS}}$ scheme
with $M_Z^2$ as the renormalization scale).


First, we study the behaviour of the one-loop EW corrections to the transverse
momentum distributions of photons and $Z$ bosons produced at the LHC, see
Fig.~\ref{fig1}. 
The contribution
provided by the NLO correction is negative and increases in size with
$\pT$, reaching -37\%  (-17\%) for the $Z$-boson (photon) production at
$\pT=2 $ TeV. We also conclude that the NLL approximation works good,
at a per cent (or better) level of accuracy both for the photon and $Z$-boson production. 
In comparison, the quality of the NNLL approximation is excellent, at
the level of accuracy of $10^{-3}$ or better in the entire $\pT$ range
for both processes.
%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{Zetnnll}\includegraphics[width=0.505\textwidth]{gammannll}
 \caption{Upper plots: relative NLO (solid), NNL (dotted) and NNLL (dot-dashed) 
weak corrections wrt. the LO $\pT$ distribution for the process (a)
$pp\rightarrow Z j$ and (b) $pp\rightarrow \gamma j$ at
$\sqrt{s}=14$ TeV. Lower plots: NLL (dotted) and NNLL (dot-dashed) approximation
compared to the full NLO result for (c)
$pp\rightarrow Z j$ and (d) $pp\rightarrow \gamma j$ at
$\sqrt{s}=14$ TeV. } 
\label{fig1}
\end{center}
\end{figure}
%%%%%%%%%%%%%%


To demonstrate the relevance of the EW effects for the transverse
momentum distributions of the gauge bosons produced at the LHC, in
Fig.~\ref{fig2} we present 
the relative NLO and next-to-next-to-leading order (NNLO) 
corrections for the cross section,
integrated over $\pT$ starting from $p_T = p_T^{\rm cut}$, as a function of
$p_T^{\rm cut}$.
%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{Zet2loop}\includegraphics[width=0.5\textwidth]{gamma2loop}
 \caption{Relative NLO (solid) and NNLO (dotted) corrections 
wrt. the LO prediction  and statistical
  error (shaded area) for the unpolarized integrated cross section for
  (a) $pp\rightarrow Z j$ and (b) $pp\rightarrow \gamma j$ at $\sqrt{s}=14$ TeV as a
  function of $p_T^{\rm cut}$.}
\label{fig2}
\end{center}
\end{figure}
%%%%%%%%%%%%%%
This is compared with the statistical error, defined as
$\Delta \sigma_{\rm stat} / \sigma = 1 /\sqrt N$ with $N= {\cal L} \times
\sigma_{\rm LO}$. 
We assume a total integrated
luminosity ${\cal L} =300 \;{\rm fb}^{-1}$ for the LHC. 
It is clear from Fig.~\ref{fig2}, that the size of the one-loop
(two-loop logarithmic) corrections 
is much bigger than (comparable to) the statistical error for both the
$Z$-boson and the $\gamma$ production.


In Fig.~\ref{fig3}, we plot the ratio of the $\pT$ distribution for
the $\gamma$ production to the $\pT$ distribution for the $Z$-boson production.
Such ratio is expected to be less sensitive to theoretical errors
than the distributions themselves, since many
uncertainties such as the scale at which
$\alpha_\rS$ is calculated or the choice of PDFs 
cancel to a large extend in the ratio. Moreover, due to a similar
cancellation mechanism, the ratio should remain
stable against QCD corrections.
From Fig.~\ref{fig3} we observe that the EW 
corrections modify the production ratio considerably. The effect is
the strongest at high $\pT$. In this region, the LO photon cross
section is smaller than the cross section for $Z$-boson
production by about 25\%. The relatively large NLO corrections for
$Z$-boson production, as compared to $\gamma$ production, cause the full NLO production
rates to become equal at the highest $\pT$ considered here, i.e.
$p_T\sim 2$ TeV. The two-loop corrections modify the ratio and lead
to a few per cent decrease at
high $\pT$.

%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\includegraphics[width=0.6\textwidth]{ratio}
\caption{Ratio of the transverse momentum distributions
for the processes $pp\to \gamma j$ and  $pp\to Z j$ 
at $\sqrt{s}=14$ TeV for the LO (solid), NLO (dashed) and NNLO
(dotted) predictions.}
\label{fig3}
\end{center}
\end{figure}
%%%%%%%%%%%%%%

\section*{CONCLUSIONS}

At the LHC, the transverse momentum of $Z$ bosons or photons produced in the process 
$pp \to Vj$ ($V=\gamma,Z$) will reach the TeV scale. 
In this $\pT$ regime one-loop weak corrections are negative and large,
of the order of tens of per cent. In the high-energy limit these corrections are 
approximated with an excellent accuracy by the compact analytical expressions presented here. 
We also present expressions for the dominant logarithmic part of the two-loop EW
corrections and point out
that these  corrections are important for the 
correct interpretation of the measurements.
Moreover, we find that the EW corrections modify significantly the
ratio of the $Z$-boson 
and $\gamma$ transverse momentum distributions at high $\pT$.

\section*{ACKNOWLEDGEMENTS}
We are grateful to J.~H.~K\"uhn for the collaboration on the project. 
This work was supported in part by 
BMBF Grant No.~05HT4VKA/3
and by the Deutsche Forschungsgemeinschaft 
(SFB/TR-9 
``Computational Particle Physics''). 
M.~S. acknowledges financial 
support from the Graduiertenkolleg "Hochenergiephysik und Teilchenastrophysik".
A.K. would like to thank the organizers for the kind
hospitality and stimulating atmosphere during the workshop.



\bibliography{lhproc}
\end{document}
