The NNLO gluon fusion Higgs production cross-section with many heavy quarks

Abstract

We consider extensions of the Standard Model with a number of additional heavy quarks which couple to the Higgs boson via
top-like Yukawa interactions. We construct an effective theory valid for a Higgs boson mass which is lighter than twice the
lightest heavy quark mass and compute the corresponding Wilson coefficient through NNLO. We present numerical results for
the gluon fusion cross-section at the Tevatron for an extension of the Standard Model with a fourth generation of heavy quarks.
The gluon fusion cross-section at NLO and in the approximation of infinitely heavy quarks is enhanced by a factor of 9 with
respect to the Standard Model value. Tevatron experimental data can place stringent exclusion limits for the Higgs mass in
this model. We find that the relative magnitude of the higher order QCD corrections with respect to the LO is similar to the
Standard Model. Our result serves to carry out an exclusion analysis for a four-generation Standard Model with the same accuracy
as in the Standard Model analysis.

KeywordsHiggs Physics-NLO Computations

Full text

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arXiv:1003.4677v1 [hep-ph] 24 Mar 2010
Preprint typeset in JHEP style - PAPER VERSION
The NNLO gluon fusion Higgs production
cross-section with many heavy quarks.
Charalampos Anastasiou
Institute for Theoretical Physics, ETH Zurich,
8093 Zurich, Switzerland
E-mail: babis@phys.ethz.ch
Radja Boughezal
Institute for Theoretical Physics, University of Zurich
Winterthurerstr. 190, 8057 Zurich, Switzerland
E-mail: radja@physik.uzh.ch
Elisabetta Furlan
Institute for Theoretical Physics, ETH Zurich,
8093 Zurich, Switzerland
E-mail: efurlan@phys.ethz.ch
Abstract: We consider extensions of the Standard Model with a number of additional
heavy quarks which couple to the Higgs boson via top-like Yukawa interactions. We con-
struct an effective theory valid for a Higgs boson mass which is lighter than twice the lightest
heavy quark mass and compute the corresponding Wilson coefficient through NNLO. We
present numerical results for the gluon fusion cross-section at the Tevatron for an extension
of the Standard Model with a fourth generation of heavy quarks. The gluon fusion cross-
section is enhanced by a factor of roughly 9 with respect to the Standard Model value.
Tevatron experimental data can place stringent exclusion limits for the Higgs mass in this
model.

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1. Introduction
The discovery of the Higgs boson will introduce a new era in particle physics. The long
standing theoretical problem of understanding the mechanism for electroweak symmetry
breaking will be tackled for the first time using direct experimental findings. The measure-
ment of the Higgs boson mass and the production cross-sections of its various signatures will
be important constraints in formulating a theory of particle interactions at high energies.
The interaction of the Higgs boson and gluons is particularly important at hadron
collider experiments. In the Standard Model (SM), the gluon fusion cross-section is the
largest among all production cross-sections. The LHC will be able to discover the SM
Higgs boson in this production channel for the full range of its allowed mass values. The
branching ratios for the decays of the SM Higgs boson are dominated by other Higgs boson
interactions which involve the bottom quark and electroweak gauge bosons. However, the
Higgs-gluon interaction is still not negligible; a fraction of up to about 7% of Higgs bosons
may decay to gluons, depending on the Higgs boson mass.
Given that gluons are massless, a Higgs-gluon interaction arises as a loop effect via
other massive coloured particles which couple to the Higgs boson. Physics beyond the
Standard Model can alter significantly the strength of this interaction in various ways.
One possibility is that new coloured particles are not much heavier than the top quark,
and their contribution is therefore not suppressed. A second possibility is that new coloured
particles may be heavier than the top-quark but they have an enhanced Yukawa coupling
to the Higgs. A third possibility is that such particles are quite heavier than the top-quark,
but their multiplicity is large, thus building up a significant cumulative contribution.
A Higgs boson is often assumed to be lighter than about twice the mass of the top-quark
and twice the mass of new undiscovered particles which are hypothesized in extensions of
the Standard Model. Light new particles are hard to accommodate given the vigorous
experimental searches for new physics at LEP and the Tevatron. On the other hand, they
cannot be very heavy or, alternatively, they must have a rather strong Higgs coupling if
they contribute in reducing the fine tuning of the Higgs mass. Therefore, it is important
to calculate their contribution in the gluon fusion process as well as in the decay of a Higgs
boson to gluons.
Assuming a Higgs boson which is lighter than production thresholds of new heavy
particles, we can factorize the effect of new physics and QCD in the process gg → H,
by means of an effective field theory where the top-quark and all other possible heavy
coloured states which couple to the Higgs boson are integrated out. The effect of these
heavy particles is included in the Wilson coefficients of an effective theory with operators
of the Higgs boson and light quarks and gluons.
The possibilities for viable extensions of the Standard Model which alter the Higgs-
gluon interaction are many, and an equal number of matching calculations is required for
their study. This is a rather easy task at leading order in the strong coupling. However,
experience from the Standard Model shows that a precise estimate of the gluon-fusion
cross-section and the Higgs decay width to gluons requires calculations through next-to-
next-to-leading-order (NNLO) in the strong coupling.
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In this paper, we consider extensions of the Standard Model with additional heavy
quarks. We assume that these quarks have a Higgs Yukawa interaction of the same type
as Standard Model quarks. The existence of such quarks has dramatic implications for the
Higgs production cross-section in gluon fusion. At leading order, and in the limit where
the heavy quarks are much heavier than half the mass of the Higgs boson, the cross-section
scales as n2
h, where nh is the number of heavy quarks. Current measurements at the
Tevatron [1] and early data from the LHC can therefore constrain severely such models.
We first construct an effective Lagrangian integrating out the top-quark and the additional
heavy quarks. We compute the Wilson coefficient of the Higgs-gluon effective interaction
through NNLO in the strong coupling expansion. Finally, we present numerical results
for the gluon fusion cross-section at the Tevatron in a specific model with a fourth quark
generation.
2. The effective Lagrangian
We consider an arbitrary extension of the SM through new heavy quarks transforming
under the fundamental representation of the QCD gauge group SU(3). The number of
heavy quarks, including the top, is nh. We will denote their mass by mq, with q = 1...nh.
We assume that the new quarks, as the SM top, couple to the Higgs boson H through their
mass. Therefore, the Lagrangian we begin with is
L = Lnl
QCD+
nh
?
q=1
¯ψq(iD / − mq)ψq+ LY
,
LY = −H
v
nh
?
q=1
mq¯ψqψq.(2.1)
Here Dµ is the covariant derivative in the fundamental representation and Lnl
QCD Lagrangian with only the nlflavours of light quarks. We take these quarks to be
massless.
We focus on the changes that the heavy quarks induce on the Higgs production through
gluon fusion. When the quarks that couple to the Higgs boson are heavier than half the
Higgs boson mass, we can integrate them out. In this limit, we can replace the original
Lagrangian (2.1) with an effective Lagrangian
QCDis the
Leff= Leff,nl
QCD− C1H
vO1.(2.2)
C1is the Wilson coefficient [2] relative to the only dimension-four local operator O1that
arises when we integrate out the heavy quarks and all the quarks remaining are massless [3],
O1=1
4G′a
µνG′aµν.(2.3)
In this expression, G′a
Leff,nl
QCDdescribes the interactions among light quarks. It has the same form as Lnl
with different parameters and field normalizations because of the contributions from heavy
quarks loops. We relate the parameters in the effective theory to the parameters in the
full theory through multiplicative decoupling constants ζi. We will denote quantities in
µνis the field strength tensor in the effective theory. In Eq. (2.2),
QCD, but
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the effective theory with a prime. The derivation of the decoupling constants is reviewed
in [4]. In Section 4, we describe the main steps of their calculation and give the relevant
results.
3. Method
We compute the Wilson coefficient C1up to three loops. Diagrams containing both the
heavy mass scales appear for the first time at the three-loop order. We start from the bare
amplitude M0
gg→Hfor the process gg → H in the full theory,
M0
gg→H≡ M0,a1a2
µ1µ2(p1,p2)ǫµ1
a1ǫµ2
a2.(3.1)
Here, p1and p2are the momenta of the two gluons with polarizations ǫµ1
amplitude is related to the bare Wilson coefficient C0
a1and ǫµ2
a2. This
1by [4]
ζ0
3C0
v
1
=δa1a2(gµ1µ2(p1· p2) − pµ2
(N2− 1)(d − 2)(p1· p2)2
1pµ1
2)
M0,a1a2
µ1µ2(p1,p2)??p1=p2=0.(3.2)
N is the number of colours and d = 4 −2ǫ is the dimension of space-time. Bare quantities
are denoted by the superscript “0”. The factor ζ0
which the bare gluon field G′0,a
µ
is rescaled in the effective theory,
3is the bare decoupling coefficient by
G′0,a
µ
=
?
ζ0
3G0,a
µ
.(3.3)
We generate the Feynman diagrams F for the amplitude through three loops using
QGRAF [5]. We then perform an expansion of all diagrams in the external momenta
p1,p2, by applying the following differential operator [6] to their integrand:
DF =
∞
?
n=0
(p1· p2)n[DnF]p1=p2=0,(3.4)
with
D0= 1,
D1=1
d?12,
D2= −
1
2(d − 1)d(d + 2)
??11?22− d ?2
12
?,(3.5)
and ?ij≡ gµν
Differential operators of higher orders are not needed for the expansion in the external
momenta at leading order.
∂2
i∂pν
∂pµ
j.
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After Taylor expansion, all the Feynman diagrams are expressed in terms of one-, two-
and three-loop vacuum bubbles by using linear transformations on the loop-momenta ki:
I1[ν1] ≡
?ddk1
?ddk1ddk2
?ddk1ddk2ddk3
?ddk1ddk2ddk3
?ddk1ddk2ddk3
iπd/2
1
Pν1
1
,(3.6)
I2[ν1,ν2,ν5] ≡
(iπd/2)2
1
2Pν5
Pν1
1Pν2
5
,(3.7)
I3a[ν1,ν2,ν3,ν5,ν6,ν7] ≡
(iπd/2)3
1
Pν1
1Pν2
2Pν3
3Pν5
1
3Pν4
1
˜P˜ ν3
3Pν4
5Pν6
6Pν7
7
,(3.8)
I3b[ν1,ν2,ν3,ν4,ν5,ν6] ≡
(iπd/2)3
Pν1
1Pν2
2Pν3
4Pν5
5Pν6
6
,(3.9)
I3c[ν1, ˜ ν2, ˜ ν3,ν4,ν5,ν6] ≡
(iπd/2)3
Pν1
1
˜P˜ ν2
2
4Pν5
5Pν6
6
,(3.10)
with
P1= k2
P2= k2
P3= k2
P4= (k1− k2+ k3)2− m2
P5= (k1− k2)2,
P6= (k2− k3)2,
P7= (k3− k1)2,
1− m2
2− m2
3− m2
q,
q,
q,
˜P2= k2
˜P3= k2
2− m2
3− m2
q′ ,
q′ ,
q,
(3.11)
and νi, ˜ νipositive or negative integers. The third three-loop vacuum bubble I3ccontains
two heavy quarks of different mass, mqand mq′.
We perform a reduction of the above integral topologies to master integrals using the
algorithm of Laporta [7] and the program AIR [8]. We find five master integrals,
I1= I1[1]
= −?m2
I2= I3a[1,0,1,1,1,0]
?2−3ǫ Γ2(1 − ǫ)Γ(ǫ)Γ2(−1 + 2ǫ)Γ(−2 + 3ǫ)
I3= I3b[1,1,1,1,0,0] ,
q
?1−ǫΓ(−1 + ǫ) ,(3.12)
=?m2
q
Γ(2 − ǫ)Γ(−2 + 4ǫ)
,(3.13)
(3.14)
I4= I3c[1,1,1,1,0,0] , (3.15)
I5= I3c[2,1,1,1,0,0] .(3.16)
Single-scale master integrals appear in the calculation of the SM Wilson coefficient [9,10]
and can be computed with MATAD [11]. For the remaining two-scale master integrals we
used result from [12]. We checked all the master integrals independently through sector
decomposition with the program of Ref. [13].
– 4 –