Transverse-momentum resummation: Higgs boson production at the Tevatron and the LHC

Abstract

We consider the transverse-momentum (q_T) distribution of Standard Model
Higgs bosons produced by gluon fusion in hadron collisions. At small q_T
(q_T<<m_H, m_H being the mass of the Higgs boson), we resum the
logarithmically-enhanced contributions due to multiple soft-gluon emission to
all order in QCD perturbation theory. At intermediate and large values of q_T
(q_T <~m_H), we consistently combine resummation with the known fixed-order
results. We use the most advanced perturbative information that is available at
present: next-to-next-to-leading logarithmic resummation combined with the
next-to-leading fixed-order calculation. We extend previous results including
exactly all the perturbative terms up to order alphas^4 in our computation and,
after integration over q_T, we recover the known next-to-next-to-leading order
result for the total cross section. We present numerical results at the
Tevatron and the LHC, together with an estimate of the corresponding
uncertainties. Our calculation is implemented in an updated version of the
numerical code HqT.

Full text

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arXiv:1109.2109v1 [hep-ph] 9 Sep 2011
ZU-TH 17/11
Transverse-momentum resummation:
Higgs boson production at the Tevatron and the LHC
Daniel de Florian(a), Giancarlo Ferrera(b,c),
Massimiliano Grazzini(d)∗and Damiano Tommasini(b,d)
(a)Departamento de F´ ısica, FCEYN, Universidad de Buenos Aires,
(1428) Pabell´ on 1 Ciudad Universitaria, Capital Federal, Argentina
(b)Dipartimento di Fisica e Astronomia, Universit` a di Firenze and
INFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Florence, Italy
(c)Dipartimento di Fisica, Universit` a di Milano and
INFN, Sezione di Milano, I-20133 Milan, Italy
(d)Institut f¨ ur Theoretische Physik, Universit¨ at Z¨ urich, CH-8057 Z¨ urich, Switzerland
Abstract
We consider the transverse-momentum (qT) distribution of Standard Model Higgs
bosons produced by gluon fusion in hadron collisions.
mH, mH being the mass of the Higgs boson), we resum the logarithmically-
enhanced contributions due to multiple soft-gluon emission to all order in QCD
perturbation theory. At intermediate and large values of qT(qT∼<mH), we consistently
combine resummation with the known fixed-order results.
advanced perturbative information that is available at present: next-to-next-to-leading
logarithmic resummation combined with the next-to-leading fixed-order calculation.
We extend previous results including exactly all the perturbative terms up to order α4
in our computation and, after integration over qT, we recover the known next-to-next-
to-leading order result for the total cross section. We present numerical results at the
Tevatron and the LHC, together with an estimate of the corresponding uncertainties.
Our calculation is implemented in an updated version of the numerical code HqT.
At small qT (qT
≪
We use the most
S
September 2011
∗On leave of absence from INFN, Sezione di Firenze, Sesto Fiorentino, Florence, Italy.

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1Introduction
One of the major tasks of the physics program at high-energy hadron colliders, such as the Fermilab
Tevatron and the CERN LHC, is the search for the Higgs boson and the study of its properties.
Gluon–gluon fusion, through a heavy-quark (mainly top-quark) loop, is the main production
mechanism of the Standard Model (SM) Higgs boson over the entire range of Higgs boson masses
(100 GeV∼<mH∼<1 TeV) to be investigated at the LHC. At the Tevatron the gluon fusion process,
followed by the decay H → WW → l+l−ν¯ ν, gives the dominant contribution to the Higgs signal
in the range of mass 140 GeV∼<mH∼<180 GeV. In this mass region, first constraints beyond the
LEP lower bound of 114.4 GeV [1] were established: the SM Higgs boson was excluded at 95%
confidence level by CDF and D0 collaborations in the mass range 156 GeV < mH< 177 GeV [2].
The first results of the ATLAS and CMS collaborations presented at EPS 2011 conference [3], and
updated for Lepton Photon 2011 [4], dramatically extend the excluded region over most of the
mass range between 145 and 466 GeV.
The above exclusion relies on accurate theoretical predictions [5, 6] for the inclusive gg → H
cross section, which is now known up to next-to-next-to-leading order (NNLO) [7], with the
inclusion of soft-gluon contributions up to next-to-next-to-leading logarithmic accuracy (NNLL)
[8], and two-loop electroweak effects [9]†.
In this paper we consider the transverse momentum (qT) spectrum of the SM Higgs boson
H produced by the gluon fusion mechanism.
experimental search. A good knowledge of the qT spectrum can help to set up strategies to
improve the statistical significance. When studying the qT distribution of the Higgs boson in
QCD perturbation theory it is convenient to define two different regions of qT. In the large-qT
region (qT∼ mH), where the transverse momentum is of the order of the Higgs boson mass mH,
perturbative QCD calculations based on the truncation of the perturbative series at a fixed order
in αSare theoretically justified. In this region, the QCD radiative corrections are known up to
the next-to-leading order (NLO) [11, 12, 13] and QCD corrections beyond the NLO are evaluated
in Ref. [14], by implementing threshold resummation at the next-to-leading logarithmic (NLL)
level.
This observable is of direct importance in the
In the small-qT region (qT≪ mH), where the bulk of the events is produced, the convergence
of the fixed-order expansion is spoiled by the presence of large logarithmic terms, αn
To obtain reliable predictions, these logarithmically-enhanced terms have to be systematically
resummed to all perturbative orders [15, 16, 17, 18, 19]. It is then important to consistently
match the resummed and fixed-order calculations at intermediate values of qT, in order to obtain
accurate QCD predictions for the entire range of transverse momenta.
Slnm(m2
H/q2
T).
The resummation of the logarithmically enhanced terms is effectively (approximately) per-
formed by standard Monte Carlo event generators. In particular, MC@NLO [20] and POWEG
[21] combine soft-gluon resummation through the parton shower with the leading order (LO) result
valid at large qT, thus achieving a result with formal NLO accuracy.
The numerical program HqT [18] implements soft-gluon resummation up to NNLL accuracy
[22] combined with fixed-order perturbation theory up to NLO in the large-qT region [13]. The
†Updated predictions for the inclusive Higgs production cross sections at the LHC are presented in Ref. [10].
1

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program is used by the Tevatron and LHC experimental collaborations to reweight the qTspectrum
of the Monte Carlo event generators used in the analysis and is thus of direct relevance in the
Higgs boson search.
The program HqT is based on the transverse-momentum resummation formalism described in
Refs. [17, 18, 19], which is valid for a generic process in which a high-mass system of non strongly-
interacting particles is produced in hadron–hadron collisions. The method has so far been applied
to the production of the SM Higgs boson [18, 23, 24], single vector bosons [25, 26], WW [27] and
ZZ [28] pairs, slepton pairs [29], and Drell-Yan lepton pairs in polarized collisions [30].
In this paper we update and extend the phenomenological analysis presented in Ref. [18]. In
particular, we implement the exact value of the NNLO hard-collinear coefficients HH(2)
in Ref. [31, 32], and the recently derived value of the NNLL coefficient A(3)[33].
N
computed
We use the most advanced perturbative information that is available at present: NNLL re-
summation at small qT and the fixed-order NLO calculation at large qT. We present numerical
results for Higgs production at the Tevatron Run II and at the LHC and we perform a detailed
study of the perturbative uncertainties. We also consider the normalized qTspectrum and discuss
its theoretical uncertainties. Our calculation for the qT spectrum is implemented in the updated
version of the numerical code HqT, which can be downloaded from [34]. Other phenomenological
studies of the Higgs boson qTdistribution, which combine resummed and fixed-order perturbative
results at various levels of theoretical accuracy, can be found in Refs. [35]–[39].
The paper is organized as follows. In Sect. 2 we briefly review the resummation formalism of
Refs. [17, 18, 19] and its application to Higgs boson production. In Sect. 3 we present numerical
results for Higgs boson production at the Tevatron and the LHC. In Sect. 4 we summarize our
results.
2 Transverse-momentum resummation
In this section we briefly recall the main points of the transverse-momentum resummation approach
proposed in Refs. [17, 18, 19]. We consider the specific case of a Higgs boson H produced by
gluon fusion. As recently pointed out in Ref. [19], the gluon fusion qT-resummation formula has
a different structure than the resummation formula for q¯ q annihilation. The difference originates
from the collinear correlations that are a specific feature of the perturbative evolution of colliding
hadron into gluon partonic initial states. These gluon collinear correlations produce, in the small-
qT region, coherent spin correlations between the helicity states of the initial-state gluons and
definite azimuthal-angle correlations between the final-states particles of the observed high-mass
system. Both these kinds of correlations have no analogue for q¯ q annihilation processes in the
small-qT region. In the case of Higgs boson production, being H a spin-0 scalar particle, the
azimuthal correlations vanishes and only gluon spin correlations are present [19].
We consider the inclusive hard-scattering process
h1(p1) + h2(p2) → H(mH,qT) + X, (1)
where h1and h2are the colliding hadrons with momenta p1and p2, mH and qT are the Higgs
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boson mass and transverse momentum respectively, and X is an arbitrary and undetected final
state.
According to the QCD factorization theorem the corresponding transverse-momentum differ-
ential cross section dσH/dq2
?1
00
Tcan be written as
?1
dσH
dq2
T
(qT,mH,s) =
?
a,b
dx1
dx2fa/h1(x1,µ2
F)fb/h2(x2,µ2
F)dˆ σH,ab
dq2
T
(qT,mH, ˆ s;αS(µ2
R),µ2
R,µ2
F) ,
(2)
where fa/h(x,µ2
scale µF, dˆ σH,ab/dq2
of the hadronic (partonic) centre–of–mass energy, and µRis the renormalization scale‡.
F) (a = q, ¯ q,g) are the parton densities of the colliding hadron h at the factorization
Tare the perturbative QCD partonic cross sections, s (ˆ s = x1x2s) is the square
In the region where qT∼ mH, the QCD perturbative series is controlled by a small expansion
parameter, αS(mH), and fixed-order calculations are theoretically justified. In this region, the
QCD radiative corrections are known up to NLO [11, 12, 13]. In the small-qT region (qT≪ mH),
the convergence of the fixed-order perturbative expansion is spoiled by the presence of powers of
large logarithmic terms, αn
these terms have to be resummed to all orders.
Slnm(m2
H/q2
T) (with 1 ≤ m ≤ 2n − 1). To obtain reliable predictions
We perform the resummation at the level of the partonic cross section, which is decomposed as
dˆ σH,ab
dq2
T
=dˆ σ(res.)
dq2
H,ab
T
+dˆ σ(fin.)
dq2
H,ab
T
. (3)
The first term on the right-hand side contains all the logarithmically-enhanced contributions, at
small qT, and has to be evaluated to all orders in αS. The second term is free of such contributions
and can thus be computed at fixed order in perturbation theory. To correctly take into account
the kinematic constraints of transverse-momentum conservation, the resummation procedure has
to be carried out in the impact parameter space b. Using the Bessel transformation between the
conjugate variables qT and b, the resummed component dˆ σ(res.)
H,accan be expressed as
dˆ σ(res.)
H,ac
dq2
T
(qT,mH, ˆ s;αS(µ2
R),µ2
R,µ2
F) =
?∞
0
dbb
2J0(bqT) WH
ac(b,mH, ˆ s;αS(µ2
R),µ2
R,µ2
F) , (4)
where J0(x) is the 0th-order Bessel function. The resummation structure of WH
in exponential form considering the Mellin N-moments WH
z = m2
accan be organized
Nof WHwith respect to the variable
H/ˆ s at fixed mH§,
WH
N(b,mH;αS(µ2
R),µ2
R,µ2
F) = HH
× exp{GN(αS(µ2
N
?mH,αS(µ2
R);m2
R),L;m2
H/µ2
H/µ2
R,m2
R,m2
H/µ2
H/Q2)} ,
F,m2
H/Q2?
(5)
were we have defined the logarithmic expansion parameter L ≡ ln(Q2b2/b2
(γE= 0.5772... is the Euler number).
0), and b0 = 2e−γE
‡Throughout the paper we use parton densities f(x,µ2
scheme.
§For the sake of simplicity we are presenting the resummation formulae only for the specific case of the diagonal
terms in the flavour space. In general, the exponential is replaced by an exponential matrix with respect to the
partonic indeces (a detailed discussion of the general case can be found in Ref. [18]).
F) and running coupling αS(µ2
R) as defined in the MS
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The scale Q ∼ mH, appearing in the right-hand side of Eq. (5), named resummation scale [18],
parameterizes the arbitrariness in the resummation procedure. As a matter of fact the argument
of the resummed logarithms can always be rescaled as ln(m2
as Q ∼ mH and independent of b). Although WH
depend on Q when evaluated to all perturbative orders, its explicit dependence on Q appears when
WH
(see Eq. (6) below). As in the case of µRand µF, variations of Q around mHcan thus be used to
estimate the uncertainty from yet uncalculated logarithmic corrections at higher orders.
Hb2) = ln(Q2b2)+ln(m2
N(i.e., the product HH
H/Q2) (as long
N× exp{GN}) does not
Nis computed by truncation of the resummed expression at some level of logarithmic accuracy
The form factor exp{GN} is universal (process independent)¶and contains all the terms αn
with 1 ≤ m ≤ 2n, that order-by-order in αSare logarithmically divergent as b → ∞ (or, equiva-
lently, qT→ 0). Furthermore, due to the exponentiation property, all the logarithmic contributions
to GNwith n + 2 ≤ m ≤ 2n are vanishing. The exponent GNcan be systematically expanded as
GN(αS,L;m2
+αS
πg(3)
SLm
H/µ2
R,m2
H/Q2) = L g(1)(αSL) + g(2)
N(αSL;m2
H/µ2
R,m2
H/Q2)
N(αSL;m2
H/µ2
R,m2
H/Q2) + O(αn
SLn−2) (6)
where the term Lg(1)resums the leading logarithmic (LL) contributions αn
includes the NLL contributions αn
forth. The explicit form of the functions g(1), g(2)
SLn+1, the function g(2)
SLn−1[22, 33] and so
N
SLn[16], g(3)
Ncontrols the NNLL terms αn
Nand g(3)
Ncan be found in Ref. [18].
The process dependent function HH
all the perturbative terms that behave as constants as b → ∞. It can thus be expanded in powers
of αS= αS(µ2
Ndoes not depend on the impact parameter b and it includes
R):
HH
N(mH,αS;m2
H/µ2
R,m2
H/µ2
F,m2
H/Q2) = σ(0)
H(αS,mH)
?αS
π
?
1 +αS
πHH,(1)
N
(m2
H/µ2
F,m2
H/Q2)
+
?2
HH,(2)
N
(m2
H/µ2
R,m2
H/µ2
F,m2
H/Q2) + O(α3
S)
?
, (7)
where σ(0)
the second order HH,(2)
the large-Mtapproximation, are known.
H(αS,mH) is the partonic cross section at the Born level. The first order HH,(1)
N
[31, 32] coefficients in Eq. (7), for the case of Higgs boson production in
N
[40] and
To reduce the impact of unjustified higher-order contributions in the large-qTregion, the loga-
rithmic variable L in Eq. (5), which diverges for b → 0, is actually replaced by?L ≡ ln(Q2b2/b2
lead to a different behaviour of the form factor at small values of b. An additional and relevant
consequence of this replacement is that, after inclusion of the finite component (see Eq. (8)), we
exactly recover the fixed-order perturbative value of the total cross section upon integration of the
qT distribution over qT (i.e., the contribution of the resummed terms vanishes upon integration
over qT).
0+ 1)
[18, 23]. The variables L and?L are equivalent when Qb ≫ 1 (i.e. at small values qT), but they
The finite component of the transverse-momentum cross section dσ(fin.)
contain large logarithmic terms in the small-qT region, it can thus be evaluated by truncation of
H
(see Eq. (3)) does not
¶It only depends on the partonic channel that produces the Born cross section. It is thus usually called quark
or gluon Sudakov form factor.
4