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\usepackage{colordvi}
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\title{NLO Electroweak Corrections to Higgs Boson Production at Hadron Colliders}

\ShortTitle{
NLO Electroweak Corrections to Higgs Boson Production at Hadron Colliders}

\author{\speaker{Sandro Uccirati}
\thanks{In collaboration with S.Actis, G.Passarino, C.Sturm}\\
        Institut f\"ur Theoretische Teilchenphysik,
           Universit\"at Karlsruhe,\\ 
           76128 Karlsruhe, Germany\\
        E-mail: \email{uccirati@particle.uni-karlsruhe.de}}

%\author{Another Author\\
%        Affiliation\\
%        E-mail: \email{...}}

\abstract{
We present the complete two-loop electroweak corrections to $gg\to H$, the 
main partonic channel for the production of the Higgs boson at hadron colliders.
The evaluation is based on a numerical approach and the result covers all 
kinematical regions, including $WW$ and $ZZ$ thresholds, by introducing 
complex masses for the $W$ and the $Z$ bosons.
Furthermore the relation between physical observables measured at LHC and 
Tevatron and Higgs pseudo-observables (production cross section 
and partial decay widths) is revised by extensively using the notion of the 
Higgs complex pole.
Numerical effects for the Higgs production cross section are presented.
}

\FullConference{
RADCOR 2009 - 9th International Symposium on Radiative Corrections 
(Applications of Quantum Field Theory to Phenomenology) ,\\
		 October 25 - 30 2009\\
		 Ascona, Switzerland}

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\begin{document}

\vspace{-0.1cm}
\section{Introduction}
%--
\vspace{-0.2cm}
Gluon fusion is the main production channel of the SM Higgs-boson at hadron 
colliders, and both virtual and real corrections have been thoroughly 
investigated since the beginning of the 90's. 

The QCD next-to-leading order (NLO) radiative corrections to the total 
Higgs-production cross section have been first computed below the 
$t\overline{t}$ threshold in Ref.~\cite{Djouadi:1991tka}, and  using an 
effective-Lagrangian approach, where the top quark is integrated out, in 
Ref.~\cite{Dawson:1990zj}. 
Formally, the second method defines the heavy-top limit, and it has represented 
the starting point for subsequent higher order improvements. 
The result for the entire Higgs-mass range at NLO was derived in 
Ref.~\cite{Graudenz:1992pv}. 
Since QCD NLO corrections increase the cross section by more than $70\,\%$, 
there was a flurry of activity on higher order QCD effects. 
The next-to-next-to-leading order (NNLO) corrections to the total cross 
section have been calculated just in the heavy-top limit:
partial results have been obtained in Ref.~\cite{Harlander:2000mg} and the 
complete NNLO cross section has been derived in Ref.~\cite{Harlander:2002wh}.
An all-order resummed calculation of multiple soft-gluon emission at 
next-to-next-to-leading logarithmic (NNLL) accuracy has been performed 
in Ref.~\cite{Catani:2003zt} and including higher logarithmic orders in 
Ref.~\cite{Moch:2005ky}. 
Improvements with respect to the heavy-top limit have been recently obtained by 
the authors of Ref.~\cite{Marzani:2008az}. 

The electroweak NLO corrections to the total cross section were evaluated in 
Ref.~\cite{Djouadi:1994ge} in the heavy-top limit, and turned out 
to be less than $1\%$. The contribution due to the light fermions has been 
calculated analytically in Ref.~\cite{Aglietti:2004nj}, and found to be more 
sizable; the remaining component of the amplitude involving the top quark has been 
computed by means of a Taylor expansion in the Higgs external momentum in 
Ref.~\cite{Degrassi:2004mx} and is, for obvious reasons, valid in the kinematical
region below the $WW$ threshold. 
The complete electroweak corrections were computed in Ref.~\cite{Actis:2008ts} 
and are the topics of this article.
Under the hypothesis of factorization with respect to the dominant QCD soft and 
collinear radiation, the impact of the EW corrections to Higgs-production in 
proton-proton collisions has been estimated in Ref.~\cite{Aglietti:2006yd}.
The factorization hypothesis can be verified just by the direct computation 
of the EW$\times$QCD corrections. 
The contribution to these corrections coming from top-loops was evaluated in 
Ref.~\cite{Steinhauser:1998cm} in the heavy-top limit, while the authors of 
Ref.~\cite{Anastasiou:2008tj} used an effective theory, valid in the small 
Higgs mass range, to compute the corrections arising from light quarks, 
finding agreement with the factorization hypothesis.

\vspace{-0.1cm}
\section{Computation of the EW corrections}
%--
\vspace{-0.2cm}
The main problem that one encounters in the computation of electroweak 
processes at two-loop level is connected to the presence of many different 
mass scales. 
The common approach, which aims to write two-loop amplitudes in terms of 
analytic functions, relies on expansions in some of the scales 
of the problem restricting however the validity of the result to some 
regions of the parametric space. 
In order to have a result valid everywhere, we have abandoned the analytic 
way for a numerical evaluation of Feynman diagrams.
In the numerical approach the (many) heavy mass scales can be easily treated, 
but a special care has to be devoted to light particles. 
Their small masses are natural regulators for the collinear singularities, but 
generate instabilities in the numerical integration. 
Therefore we have developed some techniques to extract explicitly the divergent 
behavior in these small masses, which are of logarithmic type.
In the production process $gg\to H$ the total amplitude is free from 
collinear divergences and we could verify analytically that the logarithms 
in the small masses cancel when the contributions of all diagrams are summed 
up.

The strategy to obtain stable integral representations starts with Feynman 
parametrization. 
After the extraction of possible collinear logarithms the integrand 
always takes the form:
%--
\vspace{-.1cm}
\bq
z_1^{n_1}\,\cdots z_k^{n_k}V^\mu(z_1,\dots,z_k)\,\ln^m V(z_1,\dots,z_k),
\qquad\qquad
\mu= -1,-2,
\vspace{-.1cm}
\eq
%--
where $z_1,\dots,z_k$ are the Feynman variables, integrated in some 
finite domain ($\subseteq [0,1]^k$) and $n_1,\dots,n_k$ are natural numbers.
The function $V$ is quadratic with respect to a subset $\{z_i\}$ of 
$\{z_1,\dots,z_k \}$ and each $z_i^2$ is proportional to one external squared 
momentum.
This is a general feature of two-loop diagrams.
The idea is to manipulate the integral and put it in the following form:
%--
\vspace{-.2cm}
\bq
\!\!\!\int dz\,\frac{Q(z)}{U(z)}\,f\left( \frac{U(z)}{P(z)} \right)
\vspace{-.2cm}
\eq
%--
where the function $f$ must satisfy $f(0)= 0$ (possible candidates are the 
Spence functions $\ln^n(1\!+\!x)$, ${\rm Li}_n(x)$, ${\rm S}_{n,p}(x)$) and 
$Q$, $U$ and $P$ are polynomials.
In this way the possible zeros of the denominator $U$ are compensated 
by the ``regulating function'' $f$ and the integral has no unstable points 
anymore.

In the case where the quadratic is not complete (i.e. some external squared 
momenta vanish) the solution is simple. 
For the process $gg\to H$ the two external squared momenta corresponding 
to the two gluons are always zero and therefore the quadratic $V$ can be written 
as $V=a\,x\,y+b\,x+c\,y+d$, where $x,y$ are two integration variables and the 
parameters $a,b,c,d$ depend on the other variables and on the masses and 
momenta.
For $\mu= -1, -2$ and $m=0$ we can use the following relations (similar 
relations hold for $m>0$):
%--
\vspace{-.1cm}
\bqa
\frac{1}{A\,x+B} &=& 
\partial_x
\frac{1}{A}\,\ln\Big( 1 + \frac{A}{B}\,x \Big)
\qquad
A= a\,y+b,
\quad
B= c\,y+d
\nl
\frac{1}{(a\,x\,y+b\,x+c\,y+d)^2} &=& 
- \partial_x\,\partial_y\,
\frac{1}{a\,d-b\,c}\,
\ln\bigg\{ 1 + \frac{(a\,d-b\,c)\,x}{b\,(axy+bx+cy+d)} \bigg\}.
\vspace{-.1cm}
\eqa
%--
The two right-hand sides are exactly in the desired form: the zeros 
of the denominator correspond to the zeros of the logarithms.
After an integration by parts to get rid of the derivatives, the result 
is ready to be integrated numerically.

\vspace{-0.1cm}
\section{Behavior around $WW$ and $ZZ$ thresholds}
\label{NormTH}
%---
\vspace{-0.2cm}
As explained in details in Refs.~\cite{Actis:2008uh,Actis:2008ts}
the two-loop amplitude for the partonic process $gg\to H$ presents a singular 
behavior for $M_\ssH=2\,M_\ssW$ and $M_\ssH=2\,M_\ssZ$.
This singularity is of squared root type and is represented by terms 
proportional to the inverse power of the threshold factor $\beta_i$,
%%%
\vspace{-0.1cm}
\bq\label{eq:beta}
\beta_i = \sqrt{1 - 4\,M_i^2 \slash \mhs}, \qquad \text{with} 
\qquad M_i=\mw,\mz.
\vspace{-0.1cm}
\eq
%%%
The origin of this singularity is the derivative of the Higgs one-loop 
self-energy, associated with the Higgs wave function renormalization, as depicted 
in \fig{hasCdI}
%--
\begin{figure}[h!]
\vspace{-1.cm}
\bqas
\mbox{(1-loop diagrams) $\otimes$ (H wave-function FR)}
\quad &\Longrightarrow& \quad
\raisebox{0.1cm}{
\begin{picture}(80,40)(0,0)
 \SetWidth{1.2}
 \DashLine(-2,0)(25,0){3}           \Text(12,5)[cb]{$H$}
 \DashLine(55,0)(80,0){3}
  \CArc(40,0)(15,0,360)              \Text(40,17)[cb]{$M_W,M_Z$}
 \CCirc(40,-15){2.5}{1}{1}
\end{picture}
}
\,\,\times\,\,
\raisebox{0.1cm}{
\begin{picture}(80,40)(0,0)
 \SetWidth{1.1}
 \DashLine(0,0)(25,0){3}            \Text(10,5)[cb]{$H$}
 \Gluon(59,-20)(80,-20){2}{4}
 \Gluon(59,20)(80,20){2}{4}
 \Line(25,0)(59,20)
 \Line(59,20)(59,-20)
 \Line(59,-20)(25,0)
\end{picture}
}
\eqas
\vspace{-0.7cm}
\caption[]{Singular $\beta^{-1}$ behavior at the normal $m$ threshold 
coming from WFR.}
\label{hasCdI} 
\vspace{-0.4cm}
\end{figure}
%--

A pragmatic gauge-invariant solution to the problem of threshold 
singularities due to unstable particles has been introduced and formalized in 
Ref.~\cite{Passarino:2007fp} and is termed in the following as minimal 
complex-mass (MCM) scheme.
In this scheme the NLO amplitude is decomposed according to
%%%
\vspace{-0.2cm}
\bq\label{eq:MCMsplit}
{\cal A}^{\rm NLO}= 
\sum_{i=W,Z} \frac{A_{{\rm SR},i}}{\beta_i} + A_{{\rm REM}},
\vspace{-0.3cm}
\eq
%%%
where square-root-singular terms ($A_{{\rm SR},i}$ ) have been isolated from the 
component which is finite for $\beta_i\to 0$ ($A_{{\rm REM}}$). 
After proving that all coefficients in \eqn{eq:MCMsplit} satisfy separately the 
Ward identities, we minimally modify the amplitude introducing the complex-mass 
scheme of Ref.~\cite{Denner:2005fg} for the divergent terms:
the real masses of the $W$ and the $Z$ bosons are replaced by the corresponding 
complex poles in the threshold factors $\beta_i$, $i=W,Z$ and in the coefficients 
$A_{{\rm SR},i}$ and the real parts of the $W$ and the $Z$ self-energies stemming 
from mass renormalization at one loop are traded for the complete self-energies, 
including imaginary parts. 

The MCM scheme allows for a straightforward removal of unphysical infinities, 
but it does not deal with the artificial cusps associated with the crossing of 
normal thresholds, as shown in \fig{delta}, which represent a concrete problem in 
assessing the impact of two-loop electroweak corrections.
We have therefore undertaken the task of introducing the complete complex-mass (CM)
scheme of Ref.~\cite{Denner:2005fg}, where the procedure described for the 
divergent terms of \eqn{eq:MCMsplit} has been extended to the remainder 
$A_{{\rm REM}}$.
%In particular, all two-loop diagrams have been computed with complex masses for 
%the internal vector bosons relying on the techniques developed in 
%Ref.~\cite{Passarino:2001wv,newLong}.
%The scheme can be easily extended to the fermionic sector replacing
%also the top-quark real mass by its complex pole. Since the 
%behavior associated with the $t\overline{t}$ threshold is $\beta_t\,$-protected, 
%we do not purse this issue here.
In \fig{delta} are shown the two-loop electroweak percentage corrections for 
the partonic cross section $gg \to H$. 
The big contribution arising at large $M_H$ is basically due to the scalar sector 
of the SM, which is approaching the strong regime. 
This is a two-loop feature, being the one loop notoriously protected 
by Veltman's theorem.
%--
\begin{figure}[h!]
\vspace{-0.8cm}
$$
\!\!\!\!\!\!\!\!\!\!
\epsfig{file=deltaGluGlu.eps, scale=0.85}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\epsfig{file=deltaGluGlu1TeV.eps, scale=0.85}
$$
\vspace{-1.6cm}
\caption[]{\label{delta}
Two-loop electroweak percentage corrections for the partonic cross section 
$gg \to H$. The solid line denotes the total electroweak corrections in the MC 
scheme, including also top quarks, while the dotted line denotes the contribution 
coming from light fermion loops in the MCM scheme.}
\vspace{-0.5cm}
\end{figure}

\vspace{-0.1cm}
\section{Unstable particles and gauge invariance}
%--
\vspace{-0.2cm}
As pointed out in Ref.~\cite{Passarino:2007fp}, the 
doubly-contracted Ward identity for the amplitude of the decay 
$H\to\gamma\gamma$ is not satisfied at two-loop level above the $WW$ 
threshold: the on-shell mass renormalization of the Higgs boson introduces 
the real part of the one-loop self-energy which does not cancel with the 
corresponding term coming from the pure loop part, containing the full 
self-energy. 
Gauge invariance is then violated unless one restores it by hand. 
The origin of the problem is connected to the common definition of production 
cross section and decay width, which treats the Higgs boson as an asymptotic 
state.
But the Higgs boson is an unstable particle and should be removed from 
in/out bases in the Hilbert space and therefore, concepts as {\em production} 
of the Higgs boson or its {\em partial decay widths} do not have a precise 
meaning and should be replaced by conventionalized definitions which respect 
first principles of quantum field theory. 

The quest for a proper treatment of unstable particles dates back to
the sixties and to the work of Veltman~\cite{Veltman:1963th} (for earlier
attempts see Ref.~\cite{peierls}); more recently the question has been readdressed 
by Sirlin and collaborators~\cite{Grassi:2000dz}.
The staring point is the concept of complex pole.
The Higgs boson complex pole ($\cph$) is the solution of the equation
%--
\vspace{-0.1cm}
\bq
\cph - \mhss + \Sigma_{\ssHH}(\cph) = 0,
\label{eqCP}
\vspace{-0.1cm}
\eq
%-- 
where $\mhss$ is the lagrangian Higgs boson mass, real by construction, and 
$\Sigma_{\ssHH}(\cph)$ is the Higgs self-energy evaluated at $s=\cph$.
This definition is gauge invariant, while the usual on-shell real Higgs mass is 
not. 
The Dyson re-summed Higgs propagator is given by
%--
\vspace{-0.1cm}
\bq
\Delta_{\ssH }(s)= (s - \cph)^{-1}\,\Bigl[ 1 + \Pi_{\ssHH}(s)\Bigr]^{-1},
\qquad
Z_{\ssH} = 1 + \Pi_{\ssHH},
\qquad
\Pi_{\ssHH}(s) = \frac{\Sigma_{\ssHH}(s) - \Sigma_{\ssHH}(\cph)}{s-\cph}.
\label{tres}
\vspace{-0.1cm}
\eq
%--
At the parton level the $S\,$-matrix for a process $i \to f$ can be written 
as
%--
\vspace{-0.1cm}
\bq
S_{fi} = V_i(s)\,\Delta_{\ssH}(s)\,V_f(s) + B_{if}(s),
\label{Smat}
\vspace{-0.1cm}
\eq
%--
where $V_i$ is the production vertex $i \to H$ (e.g. $gg \to H$), $V_f$ 
is the decay vertex $H \to f$ (e.g. $H \to \gamma \gamma$) and $B_{if}$ is 
the non-resonant background (e.g. $gg \to \gamma \gamma$ boxes).
Using \eqn{tres} we can write
%--
\vspace{-0.1cm}
\bq
S_{fi} =
  \Big[ Z^{-{1\over2}}_{\ssH}(s)\,V_i(s)\Big]\,\frac{1}{s - \cph}\,
  \Big[ Z^{-{1\over2}}_{\ssH}(s)\,V_f(s)\Big] 
+ B_{if}(s) =
\frac{S(i\!\to\!H_c )\,S(H_c\!\to\!f)}{s - \cph} 
+ 
\ba{c}
\hbox{\small non resonant}\\[-0.2cm]
\hbox{\small terms}
\ea
\vspace{-0.1cm}
\eq
%--
where we have extracted the relevant pseudo-observable,
%--
\vspace{-0.1cm}
\bq
S(H_c\to f) = Z^{-1/2}_{\ssH}(\cph)\,V_f(\cph),
\label{PO}
\vspace{-0.2cm}
\eq
%--
which is gauge independent by construction.
A partial decay width can then be defined as
%--
\vspace{-0.2cm}
\bq
\srph\,\Gamma\lpar H_c \to f\rpar = \frac{(2\,\pi)^4}{2}\,\int\,
d\Phi_f\lpar P_{\ssH}\,,\,\{p_f\}\rpar\,
\sum_{\rm spins}\,\bmid S\lpar H_c \to f\rpar \bmid^2,
\label{GPO}
\vspace{-0.3cm}
\eq
%--
where the integration is over the phase space spanned by $| f >$, with the
constraint $P_{\ssH} = \sum\,p_f$. 
%One should not confuse phase space and the real value of $s= -P^2_{\ssH}$, 
%where the realistic observable is measured, with the complex value for $s$, 
%where gauge invariant loop corrections must be computed.
%The choice of $P^2_{\ssH}$ (phase space) where to define the 
%pseudo-observable is conventional, e.g. one can use the real part 
%of $\cph$.
Similarly one can define a production cross section using the 
pseudo-observable $S(i\!\to\!H_c)$.

In order to compute these pseudo-observables we rely on the guiding principle that 
Green functions involving unstable particles should smoothly approach the value 
for stable ones (the usual Feynman $-\,i\,0$ prescription) when the couplings 
of the theory (and therefore the imaginary parts of masses and momenta) tend to 
zero. 
To illustrate the procedure we consider the following one-loop scalar two-point 
function with complex masses and momenta:
%--
\vspace{-0.2cm}
\bq
\raisebox{0.11cm}{\scalebox{0.7}{
\begin{picture}(71,15)(0,0)
\DashLine(0,0)(20,0){3}              \Text(7,2.5)[cb]{$H$}
\CArc(35,0)(15,0,360)                \Text(36,6)[cb]{$m$}
\DashLine(50,0)(66,0){3}                
\end{picture}}}
\!=\;
\Delta - \!\!\int_0^1\!\!\!\!\! dx \ln\chi,
\quad
\chi\!= - s_\ssH\,x\,(1\!-\!x) + m^2,
\quad
s_\ssH= M_\ssH^2 - i\,\Gamma_{\!\!\!\ssH}\,M_\ssH,
\quad\!
m^2=\mu^2 \!\!-\! i\,\gamma\,\mu.
\;
\vspace{-0.1cm}
\eq
%--
Since both $\cph$ and $m^2$ are complex it can happen that $\chi$ is in the 
second quadrant, i.e. ${\rm Re}\chi<0$ and ${\rm Im}\chi>0$. 
In this case (and just in this case) our guiding principle is violated:
%--
\vspace{-0.2cm}
\bq
\lim_{\gamma,\Gamma_{\!\!\ssH}\to 0}{\rm Im}[\ln\chi] \quad=\quad \pi
\qquad\neq\qquad 
\ba{l}
\mbox{Feynman prescription for real masses}\\[-0.15cm]
(\mbox{i.e. } \gamma,\Gamma_{\!\!\ssH}=0, \; \mu^2\to \mu^2-i0)
\ea
 \quad=\quad - \pi
\vspace{-0.2cm}
\eq
%--
It is important to notice that this can happen only with complex momenta 
(i.e. complex $s_\ssH$): if just masses are complex, the imaginary 
part of the argument of the logarithm is always negative.
In order to restore the proper real limit, we define an analytical continuation of 
the logarithm on the second Riemann sheet, by adding $-2\,i\,\pi$ to it when 
$\chi$ is in the second quadrant.
In general let us consider $\ln(z)$, where $z$ is a complex quantity 
which in the real mass limit gets an imaginary part of $\pm\,i\,0$ from 
the Feynman prescription.
The proper analytical continuation of $\ln(z)$ is then given by:
%--
\vspace{-0.2cm}
\bq
\ln(z) \quad\to\quad
\ln^\pm(z)= 
\ln(z) 
\pm 2\,i\,\pi
\theta(- {\rm Re}z\,)\,\theta(\mp\,{\rm Im}\,z)
\vspace{-0.2cm}
\eq
%--
This is equivalent to moving the cut of the logarithm from the negative real 
axis to the positive (for $\ln^-$) or negative (for $\ln^+$) imaginary axis. 
If this is done at the integrand level (as in the present example) one has 
to take care that the integration contour does not cross the new cut.
In that case a contour deformation in the complex $x$ plane has to be performed 
in order to get the correct result.

Taking into consideration all these technical aspects we were able to estimate 
the gauge invariant production cross section to $gg \to H$ at one-loop level 
and compare it with the usual one in the complex mass scheme.
The effect, given in \fig{CP}, is negligible for a Higgs mass up to the 
$t\bar{t}$-threshold, but becomes large for an heavy Higgs (of the order of 
QCD corrections), making questionable the use of a perturbative description 
of the Higgs-resonant part for $pp\to H$.
%--
\vspace{-0.4cm}
\begin{figure}[h!]
$$
\epsfig{file=deltaGluGluCP.eps, scale=0.85}
$$
\vspace{-1.5cm}
\caption[]{\label{CP}
Comparison of the common production cross section $\sigma(gg\to H)$ with 
the gauge invariant pseudo-observable $\sigma_{CP}(gg\to H)$ based on the 
Higgs complex pole $\cph$: $\delta_{CP}= \sigma_{CP}/\sigma-1$.
}
\end{figure}

\vspace{-0.4cm}
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\end{thebibliography}

\end{document}


