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.
Institute for Theoretical Physics, ETH Zurich,
8093 Zurich, Switzerland
Institute for Theoretical Physics, University of Zurich
Winterthurerstr. 190, 8057 Zurich, Switzerland
Institute for Theoretical Physics, ETH Zurich,
8093 Zurich, Switzerland
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
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.
– 1 –
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  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
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
¯ψq(iD / − mq)ψq+ LY
LY = −H
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
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
C1is the Wilson coefficient  relative to the only dimension-four local operator O1that
arises when we integrate out the heavy quarks and all the quarks remaining are massless ,
In this expression, G′a
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),
– 2 –
the effective theory with a prime. The derivation of the decoupling constants is reviewed
in . In Section 4, we describe the main steps of their calculation and give the relevant
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
gg→Hfor the process gg → H in the full theory,
Here, p1and p2are the momenta of the two gluons with polarizations ǫµ1
amplitude is related to the bare Wilson coefficient C0
=δa1a2(gµ1µ2(p1· p2) − pµ2
(N2− 1)(d − 2)(p1· p2)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
is rescaled in the effective theory,
3is the bare decoupling coefficient by
We generate the Feynman diagrams F for the amplitude through three loops using
QGRAF . We then perform an expansion of all diagrams in the external momenta
p1,p2, by applying the following differential operator  to their integrand:
2(d − 1)d(d + 2)
??11?22− d ?2
and ?ij≡ gµν
Differential operators of higher orders are not needed for the expansion in the external
momenta at leading order.
– 3 –
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:
I3c[ν1, ˜ ν2, ˜ ν3,ν4,ν5,ν6] ≡
P4= (k1− k2+ k3)2− m2
P5= (k1− k2)2,
P6= (k2− k3)2,
P7= (k3− k1)2,
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  and the program AIR . We find five master integrals,
?2−3ǫ Γ2(1 − ǫ)Γ(ǫ)Γ2(−1 + 2ǫ)Γ(−2 + 3ǫ)
I3= I3b[1,1,1,1,0,0] ,
?1−ǫΓ(−1 + ǫ) ,(3.12)
Γ(2 − ǫ)Γ(−2 + 4ǫ)
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 . For the remaining two-scale master integrals we
used result from . We checked all the master integrals independently through sector
decomposition with the program of Ref. .
– 4 –
After these steps, the RHS of Eq. (3.2) becomes
−1 + ǫ?1 + 2log(m0
q) + log(m0
128+5975 3456+ O(ǫ)
q) + log(m0
q′)?+1051 + 27π2
q) + log(m0
q) + log2(m0
The first sum in this expression runs over all single-scale diagrams and corresponds to nh
copies of the SM Wilson coefficient. The second sum accounts for the 3-loop diagrams in
which either of the two massive quarks couples to the Higgs boson. We already symmetrized
it over q,q′. In Eq. (3.17) we introduced the factor
4. Decoupling and renormalization
The RHS of Eq. (3.17) contains the bare masses of the heavy quarks m0
coupling constant α0
the full theory is related to the bare strong coupling in the effective theory α
decoupling constants ζ0
qand the bare
sin the full theory; C0
q). The bare strong coupling in
s by the
Using these relations, we obtain the bare Wilson coefficient as a function of the bare
parameters in the effective theory and of the bare mass of the heavy quarks in the full
1(αThe bare parameters are related to the renormalized ones
– 5 –
through multiplicative renormalization constants Zias
q= Zmqmq(µ) .(4.4)
All the parameters in Eq. (4.3) are in the effective theory and all the parameters in Eq. (4.4)
are in the full theory. Finally, we renormalize the Wilson coefficient itself through a renor-
malization factor Z11[3,14,15],
4.1 Details of the calculation
A convenient way to compute the gluon field decoupling ζ0
3is through the relation
3= 1 + Π0
G(p = 0) ,(4.6)
quantity is computed at zero external momentum. Since we work in dimensional regu-
larization, only diagrams containing at least one massive quark contribute. We find only
one diagram per heavy flavour at one loop and seven at two loops. We employ the same
calculation techniques as in Section 3. Our result reads
Gis the transverse component of the gluon self-energy in the full theory. This
3= 1 +
+ ǫπ2+ 24log2(m0
32ǫ2−1 + 24log(m0
The bare decoupling parameter of the strong coupling constant, ζ0
g, can be computed
ghost field and of the gluon field respectively.
The bare decoupling constant of the ghost field is calculated from the ghost self-energy in
a similar way as ζ0
3. At one loop, there is no diagram contributing to the ghost decoupling.
At two loops, there is only one diagram per heavy flavour. We find
3are the bare decoupling constants of the gluon-ghost vertex, of the
3= 1 +
−89 + 6π2
– 6 –
The decoupling of the gluon-ghost vertex˜ζ0
1is given by
1= 1 + Γ0
¯ ηGη(0,0) ,(4.10)
and p′are the incoming four-momenta of ¯ η and G respectively. Again, this term receives
no contribution at one loop. At two loops, there are 5 non-massless diagrams for each
heavy flavour. Two of them vanish because of colour, and the other three add up to zero.
¯ ηGη(p,p′) is extracted from the 1PI amputated gluon-ghost Green function and p
1= 1 + O(α3
s) . (4.11)
Inserting Eqs. (4.11, 4.9, 4.7) into the relation (4.8) we find
g= 1 +
Here we introduced the notation
We now renormalize the mass of the heavy quarks in the full theory according to
Eq. (4.4). The mass renormalization constants in the full theory Zmqare obtained from
the one- and two-loop corrections to the quark propagator . We review here the main
steps of this calculation. Let us denote the sum of all the one-particle irreducible (1PI)
insertions into the quark propagator as −iΣ0(p),
−iΣ0(p) = −iΣ1L
0(p) − iΣ2L
0(p) + ... ,(4.14)
The full quark propagator then reads
0(p) in the sum of all the one-loop 1PI diagrams in the bare theory and so on.
/ p − m0q− Σ0(p).(4.15)
We can split Σ0(p) as
Σ0(p) = Σ10(p2) + (/ p − m0
conversely, the quantities
m0Σ10, Σ20are extracted from the bare self-energy Σ0as
4p2Tr(/ pΣ0) .(4.17)
Combining Eqs. (4.15) and (4.16) we obtain
– 7 –
The RHS of Eq. (4.18) is expressed in terms of renormalized quantities. On the other
hand, the one-loop result for Σ1L
10originally depends on the bare gauge parameter and on
the bare strong coupling and is divided itself by the bare mass. Therefore, one also needs
the one-loop renormalization of these three quantities in order to compute Zmqthrough
We compute the self-energy for arbitrary external momentum p, and set the mass of the
heavy quarks to zero. In the MS scheme we recover the result 1
Zmq= 1 −αs(µ)
?45 − 2nf
This relation holds both in the full theory, where the number of active flavours nf is
nf= nl+ nh, and in the effective theory. In this case, nf= nland we need to replace the
renormalized strong coupling in the full theory with the one of the effective theory.
So far, the coefficient for the mass decoupling and renormalization still depends on the
renormalized strong coupling in the full theory. We decouple it using the relation 
αs(µ) = ζ2
The strong coupling renormalization constants are related to the coefficients of the β func-
α= 1 −α′s(µ)
1denote the first two coefficients of the β function in the light-flavours theory,
Combining Eqs. (4.4 -4.21) we get
?2?45 − 2(nh+ nl)
+48Lm+ 10(nh+ nl) − 303
The next step is the renormalization of the bare strong coupling in the effective theory,
s. The relevant renormalization constant is given in Eq. (4.21).
We finally renormalize the bare Wilson coefficient C0
1(α′s,mq) using [3,14,15]
1 + α′s(µ)
1Note that in our conventions d = 4− 2ǫ, while in Ref.  d = 4+2ǫ. This explains the sign difference
in the 1/ǫ terms.
– 8 –
Our final result for the renormalized Wilson coefficient reads
This is the main result of our paper. Note that the second term in the square brackets
comes from diagrams containing two massive quarks loops. For nh= 1 we recover the SM
Wilson coefficient [9,10,17,18] through order O(α′s
At leading order in the heavy-quark expansion, and assuming a massless bottom quark,
the gluon fusion cross-section and the decay width of the Higgs boson to gluons are pro-
portional to the square of the Wilson coefficient. In this limit, their ratio with the corre-
sponding quantities in the Standard Model are:
σ(gg → H)(nh)
σ(gg → H)(SM)=Γ(H → gg)(nh)
Γ(H → gg)(SM)=
288nh(nh− 1) +
where mtthe mass of the top-quark. The O(α
very small. However, in a realistic phenomenological study [19,20] the exact quark mass
dependece of the cross-section as well as effects due to electroweak corrections need to be
accounted for through NLO.
s) term in the above expression is generally
5. Numerical Results for gluon fusion cross section at the Tevatron
In this Section, we present our numerical results for the cross-section at the Tevatron, in
a Standard Model with four generations. The Wilson coefficient for the fourth generation
model is obtained by considering three heavy quarks in Eq. (4.26). We set the top-quark
mt= 170.9GeV .
For the fourth generation we consider two scenarios, corresponding to fourth generation
down-quark masses of:
mB= 300GeV,mB= 400GeV
and an up-quark mass given by
mT− mB= 50GeV + 10log
This choice is permitted by constraints from electroweak precision tests, as described in
The calculation of the total cross-section differs from the Standard Model only in the
expression of the Wilson coefficient for the low energy effective Lagrangian. We combine
– 9 –
Eq. (4.26) with the known results for the Standard Model total cross-section at NNLO in
the large top-mass limit of Refs [22–24].
Adopting the same approach as in Ref. , we first compute the ratio of the NNLO
and LO cross-section in the effective theory. We estimate the contribution from Feynman
diagrams with only top-quark and fourth generation quarks to the total cross-section, by
multiplying this ratio with the exact leading order contributions of heavy quark diagrams
in the full theory.
These contributions are enhanced by roughly a factor of 9 with respect to the corresponding
Standard Model results, since
≃ 9 σLO;(t)
within a few percent.
Contributions from diagrams with bottom quark loops are small and we compute them
exactly through the NLO order approximation [25,26]. In comparison to their Standard
Model counterparts, the most important interference terms of diagrams with bottom quarks
only and diagrams with any of the heavier quarks are only enhanced by roughly a factor of
three. Therefore, these contributions are suppressed by roughly a factor of ∼ 3/9 in this
Finally, we include two-loop electroweak corrections from light-quark loops from the
first two generations  in the complex mass scheme, and the corresponding three-loop
mixed QCD and electroweak corrections as in Ref. . These are enhanced by a factor of
roughly 3 in comparisoin to the Standard Model, and are therefore suppressed by a factor
of 1/3 in this model. We ignore electroweak corrections with quarks from the third and
fourth generation, which are already found to be very small in the Standard Model  for
the Higgs mass range accessible at the Tevatron, when a complex mass scheme is employed.
In Table 1, we present the cross-section at a renormalization and factorization scale
µ = µf = µr = mH/2 and estimate the scale variation error by varying the common
scale µ in the interval
the parton distribution functions (including the parametric uncertainty of the value of αs)
according to Ref. .
The scale variation uncertainty and the uncertainty from the parton distributions are
the dominant uncertainties. Essentially, they are the same as in the Standard Model cross-
section with only three generations. We note that Ref.  preceded the release of Ref. 
where it became possible to include the uncertainty of the αsvalue in the fitted parton
densities and the corresponding parton density uncertainty was estimated to be smaller.
?. We use the MSTW2008 NNLO parton distribution
functions , and compute the uncertainty (with 90%CL) to the cross-section due to
In this paper, we constructed an effective field theory for extensions of the Standard Model
with many heavy quarks coupling to the Higgs boson via top-like Yukawa interactions. We
– 10 –
have computed the required Wilson coefficient of the −H
in the strong coupling expansion. We found the result
4vGµνGµνoperator through NNLO
This result can be utilized by the ongoing experimental studies at the Tevatron and the
LHC to constrain such models. The cross-section and the decay width are enhanced by
roughly the square of the number of heavy quarks with respect to the Standard Model.
The Tevatron has put stringent limits on the Standard Model Higgs boson gluon fusion
cross-section . Equivalent studies can be performed in models with additional quarks.
We have presented numerical results for the gluon fusion cross-section at the Tevatron in
a non-minimal Standard Model with four generations. The theoretical uncertainties of the
cross-section are practically independent of the number of heavy quarks and very similar
to the Standard Model.
We thank Achilleas Lazopoulos for many useful discussions and his help. We thank Giuliano
Panico for comments on the manuscript. This research is supported by the Swiss National
Science Foundation under contracts 200020-116756/2 and 200020-126632
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σ(pdf + αs)%
Table 1: The NNLO cross-section for Higgs production via gluon fusion at the TEVATRON. σ(1)
corresponds to mB= 300GeV and σ(2)to mB= 400GeV. The mass of the fourth generation up
quark is given by Eq. (5.1)
– 13 –