Page 1

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.

Page 2

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.

– 1 –

Page 3

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

– 2 –

Page 4

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.

– 3 –

Page 5

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 –

Page 6

After these steps, the RHS of Eq. (3.2) becomes

ζ0

3C0

v

1

=

nh

?

q=1

?1

3

?α0

sSǫ

π

??

−1 + ǫ?1 + 2log(m0

?

?

1

32ǫ2+1

ǫ

q)?

q) +π2

− 2ǫ2

log2(m0

q) + log(m0

24

?

+ O(ǫ3)

?

+

?α0

?α0

sSǫ

π

?2?

?3?

−1

4+ ǫlog(m0

q) +31

36

?

+ O(ǫ2)

?

+

sSǫ

π

−

?

3log(m0

16

q)

−223

576

?

+ nl

?

−

5

144

1

ǫ+103

864+5log(m0

q)

24

?

−9

16log2(m0

q) +223log(m0

q)

96

−π2

128+5975 3456+ O(ǫ)

??

−

?α0

+89

96

sSǫ

π

?3?

q>q′

?

1

16ǫ2−

1

16ǫ

?

3?log(m0

?log2(m0

q) + log(m0

q′)?+89

q′)?+1051 + 27π2

18

?

+1

2log(m0

q)log(m0

q′)

?log(m0

q) + log(m0

q′)?+5

16

q) + log2(m0

1728

+ O(ǫ)

?

.

(3.17)

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

Sǫ= e−ǫγE(4π)ǫ.(3.18)

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

g[4,10],

α

qand the bare

sin the full theory; C0

1= C0

1(α0

s,m0

q). The bare strong coupling in

′0

s by the

′0

s= (ζ0

g)2α0

s

.(4.1)

Similiarly,

α′

s= (ζg)2αs

.(4.2)

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

theory, C0

1(αThe bare parameters are related to the renormalized ones

1= C0

′0

s,m0

q).

– 5 –

Page 7

through multiplicative renormalization constants Zias

α

α0

′0

s= µ2ǫZ′

s= µ2ǫZααs(µ)

αα′

s(µ),(4.3)

,m0

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],

Z11C0

.

C1=

1

1.(4.5)

4.1 Details of the calculation

A convenient way to compute the gluon field decoupling ζ0

3is through the relation

ζ0

3= 1 + Π0

G(p = 0) ,(4.6)

where Π0

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

ζ0

3= 1 +

nh

?

?α0

q=1

??α0

?2?

sSǫ

π

??

1

6ǫ−log(m0

q)

3

+ ǫπ2+ 24log2(m0

q)

72

?

+

sSǫ

π

3

32ǫ2−1 + 24log(m0

q)

64ǫ

+3

4log2(m0

q) +1

16log(m0

q) +

91

1152+π2

64

??

(4.7)

.

The bare decoupling parameter of the strong coupling constant, ζ0

g, can be computed

as

ζ0

g=

˜ζ0

1

?ζ0

˜ζ0

33

, (4.8)

where˜ζ0

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

1,˜ζ0

3and ζ0

3are the bare decoupling constants of the gluon-ghost vertex, of the

˜ζ0

3= 1 +

?α0

sSǫ

π

?2 nh

?

q=1

?

−

3

64ǫ2+1

ǫ

?

5

128+316log(m0

q)

?

−89 + 6π2

768

−5

32log(m0

q) −3

8log2(m0

q)

?

.(4.9)

– 6 –

Page 8

The decoupling of the gluon-ghost vertex˜ζ0

1is given by

˜ζ0

1= 1 + Γ0

¯ ηGη(0,0) ,(4.10)

where Γ0

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.

Therefore

˜ζ0

¯ 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

ζ0

g= 1 +

?α0

?α0

sSǫ

π

??

?2?

−nh

12ǫ+L0

q

6

− ǫ

?

L0

2,q

6

+ nh

π2

144

??

+

sSǫ

π

n2

96ǫ2−nh

h

24ǫ

?

L0

q+3

4

?

+L0

q

8

+(L0

q)2

24

+nh

24

?

L0

2,q+11

6

?

+ n2

h

π2

576

?

.

(4.12)

Here we introduced the notation

L0

q=

nh

?

q=1

log(m0

q),L0

2,q=

nh

?

q=1

log2(m0

q) .(4.13)

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 [16]. 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)

where −iΣ1L

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.

i

/ p − m0q− Σ0(p).(4.15)

We can split Σ0(p) as

Σ0(p) = Σ10(p2) + (/ p − m0

q)Σ20(p2) ;(4.16)

conversely, the quantities

1

m0Σ10, Σ20are extracted from the bare self-energy Σ0as

1

m0Σ10=1

4Tr

?

1

m0Σ0+/ p

p2Σ0

?

,Σ20=

1

4p2Tr(/ pΣ0) .(4.17)

Combining Eqs. (4.15) and (4.16) we obtain

m0

q= mq(µ)

?

1 −

1

mqΣ1L

1 −

?1

mqΣ2L

1 +

1

mqΣ1L

1Σ1L

2

??

.(4.18)

– 7 –

Page 9

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

two loops.

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 [16]1

Zmq= 1 −αs(µ)

π

1

ǫ+

?αs(µ)

π

?2?1

ǫ2

?45 − 2nf

24

?

+1

ǫ

?

−101

48+572nf

??

.(4.19)

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 [4]

α′

s(µ) =Zα(ζ0

g)2

Z′α

αs(µ) = ζ2

gαs(µ) .(4.20)

The strong coupling renormalization constants are related to the coefficients of the β func-

tion as

α= 1 −α′s(µ)

πǫ

Z′

β′

0

+

?α′s(µ)

π

?2?

β

ǫ2−β′

′2

0

1

2ǫ

?

.(4.21)

Here β′

0and β′

1denote the first two coefficients of the β function in the light-flavours theory,

β′

0=1

4

?

11 −2

3nl

?

,β′

1=

1

16

?

102 −38

3nl

?

.(4.22)

Combining Eqs. (4.4 -4.21) we get

m0

q= mq(µ)

?

1 −α′s(µ)

π

1

ǫ+

?α′s(µ)

π

?2?45 − 2(nh+ nl)

24ǫ2

+48Lm+ 10(nh+ nl) − 303

144ǫ

??

,

(4.23)

with

Lm=

nh

?

q=1

log

?mq(µ)

µ

?

.(4.24)

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

α

′0

1(α′s,mq) using [3,14,15]

C1=

1

1 + α′s(µ)

?

∂

∂α′s(µ)logZ′α

C0

1

=

1 +α′s(µ)

π

β′

ǫ

0

+

?α′s(µ)

π

?2β′

1

ǫ

?

C0

1.(4.25)

1Note that in our conventions d = 4− 2ǫ, while in Ref. [16] d = 4+2ǫ. This explains the sign difference

in the 1/ǫ terms.

– 8 –

Page 10

Our final result for the renormalized Wilson coefficient reads

C1= −1

3

α′s(µ)

π

?

nh+11

4

α′s(µ)

π

nh−

?α′s(µ)

π

?2?

−1877

192nh+77

576n2

h+19Lm

8

+ nl

?67

96nh+2Lm

3

???

. (4.26)

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:

3).

σ(gg → H)(nh)

σ(gg → H)(SM)=Γ(H → gg)(nh)

?2

Γ(H → gg)(SM)=

?

n2

h−

?α′s(µ)

π

nh

77

288nh(nh− 1) +

?4

3nl+19

4

??

q

log

?mq(µ)

mt(µ)

??

+ O(α

′3

s).

(4.27)

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.

′2

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

mass to

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

?

mH

115GeV

?

GeV.(5.1)

This choice is permitted by constraints from electroweak precision tests, as described in

Ref. [21].

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 –

Page 11

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. [19], 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.

σNNLO;(t,B,T)

heavy

≃

?

σNNLO;(t,B,T)

σLO;(t,B,T)

?

effective

σLO;(t,B,T)

exact

(5.2)

These contributions are enhanced by roughly a factor of 9 with respect to the corresponding

Standard Model results, since

σLO;(t,B,T)

exact

≃ 9 σLO;(t)

exact

(5.3)

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

model.

Finally, we include two-loop electroweak corrections from light-quark loops from the

first two generations [27] in the complex mass scheme, and the corresponding three-loop

mixed QCD and electroweak corrections as in Ref. [19]. 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 [28] 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

?mH

the parton distribution functions (including the parametric uncertainty of the value of αs)

according to Ref. [30].

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. [19] preceded the release of Ref. [30]

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.

4,mH

?. We use the MSTW2008 NNLO parton distribution

functions [29], and compute the uncertainty (with 90%CL) to the cross-section due to

6. Conclusions

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 –

Page 12

have computed the required Wilson coefficient of the −H

in the strong coupling expansion. We found the result

4vGµνGµνoperator through NNLO

C1= −1

3

α′s(µ)

π

?

nh+11

4

α′s(µ)

π

nh−

?α′s(µ)

π

?2?

−1877

192nh+77

576n2

h+19Lm

8

+ nl

?67

96nh+2Lm

3

???

.(6.1)

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 [1]. 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.

Acknowledgments

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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– 12 –

Page 14

mH(GeV)

110

115

120

125

130

135

140

145

150

155

160

165

170

175

180

185

190

195

200

205

210

215

220

225

230

235

240

245

250

255

260

265

270

275

280

285

290

295

300

σ(1)(fb)

12384

10798

9449.9

8298.8

7314.0

6465.1

5731.4

5094.6

4540.5

4055.6

3630.2

3253.7

2924.1

2633.9

2376.7

2147.2

1943.9

1763.2

1601.8

1457.5

1328.1

1212.0

1107.7

1013.6

928.61

852.00

782.52

719.64

662.60

610.74

563.53

520.60

481.49

445.86

413.24

383.56

356.39

331.53

308.70

σ(2)(fb)

12308

10725

9384.3

8240.0

7258.7

6414.2

5684.1

5050.4

4498.5

4017.6

3595.1

3220.7

2893.2

2604.4

2348.9

2121.5

1919.7

1740.2

1580.0

1436.7

1308.4

1193.2

1089.6

996.33

912.21

836.33

767.44

705.19

648.81

597.51

550.90

508.52

469.93

434.72

402.68

373.28

346.53

322.04

299.71

δσ

σ(pdf + αs)%

+12%, −11%

+12%, −11%

+12%, −11%

+12%, −12%

+12%, −12%

+12%, −12%

+13%, −12%

+13%, −12%

+13%, −12%

+13%, −12%

+13%, −13%

+14%, −13%

+14%, −13%

+14%, −13%

+14%, −13%

+15%, −13%

+15%, −14%

+15%, −14%

+15%, −14%

+16%, −14%

+16%, −14%

+16%, −14%

+16%, −15%

+17%, −15%

+17%, −15%

+17%, −15%

+17%, −15%

+18%, −15%

+18%, −16%

+18%, −16%

+19%, −16%

+19%, −16%

+19%, −16%

+20%, −16%

+20%, −17%

+20%, −17%

+21%, −17%

+21%, −17%

+21%, −17%

δσ

σ(scale)%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

+12%, −8%

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 –