# Next-to-Leading Order Cross Sections for New Heavy Fermion Production at Hadron Colliders

**Abstract**

We evaluate the cross sections for new heavy fermion production at three
Large Hadron Collider energies accurate to next-to-leading order in
perturbative quantum chromodynamics. We treat the cases of pair production of
heavy quarks via strong interactions, single heavy quark production via
electroweak interactions, and the production of heavy leptons. Theoretical
uncertainties associated with the choice of the renormalization scale and the
parton distribution functions are specified. We derive a simple and useful
parameterization of our results which should facilitate phenomenological
studies of new physics models that predict new heavy quarks and/or leptons.

arXiv:0909.3555v2 [hep-ph] 11 May 2010

ANL-HEP-PR-09-93, EFI-09-24

Next-to-Leading Order Cross Sections for New Heavy Fermion

Production at Hadron Colliders

Edmond L. Berger∗

High Energy Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA

Qing-Hong Cao†

High Energy Physics Division, Argonne National

Laboratory, Argonne, Illinois 60439, USA and

Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA

Abstract

We evaluate the cross sections for new heavy fermion production at three Large Hadron Collider

energies accurate to next-to-leading order in perturbative quantum chromodynamics. We treat the

cases of pair production of heavy quarks via strong interactions, single heavy quark production via

electroweak interactions, and the production of heavy leptons. Theoretical uncertainties associated

with the choice of the renormalization scale and the parton distribution functions are speciﬁed. We

derive a simple and useful parametrization of our results which should facilitate phenomenological

studies of new physics models that predict new heavy quarks and/or leptons.

PACS numbers: 12.38.Bx, 12.38.Qk, 13.85.Ni, 14.65.Jk

∗Electronic address: berger@anl.gov

†Electronic address: caoq@hep.anl.gov

1

I. INTRODUCTION

Although the standard model (SM) of particle physics consists of three fermion genera-

tions, the number of these generations is not ﬁxed by theory. Asymptotic freedom in quan-

tum chromodynamics (QCD) limits the number of generations to fewer than nine. Neutrino

counting, based on data at the intermediate vector boson Z, shows that the number of gen-

erations having light neutrinos (mν≪mZ/2 ) is equal to 3. However, neutrino oscillations

suggest a new mass scale that is beyond that of the SM, and the possibility of additional

heavier neutrinos is open. In addition, it has recently been emphasized that the electroweak

oblique parameters do not exclude a fourth generation of chiral fermions [1–8]. In the era of

the Large Hadron Collider (LHC), the search for heavy quarks and leptons beyond those of

the SM should be kept in mind.

Observation of new heavy fermions requires knowledge of their expected production cross

sections and decay properties. In this work, we present inclusive cross sections of fourth

generation fermion production at three LHC energies, either in pairs or singly, calculated to

next-to-leading (NLO) order accuracy in QCD. Leading order Feynman diagrams for new

heavy fermion production are shown for illustrative purposes in Fig. 1. We do not show the

full set of NLO diagrams used in our calculations. Theoretical uncertainties associated with

higher-order perturbative contributions and with the choice of parton distribution functions

(PDFs) are speciﬁed. We show that the cross sections can be ﬁtted with a rather simple

analytic formula, and we present the functional form and values of the parameters for the

q

¯q

¯

Q

Q

q

b

q′

T

q

¯

q′T

¯

b(¯

B)

q

¯

q′E(e)

¯ν(¯

N)q

¯qN

¯

Nq

¯qE−

E+

ZZ/γ

(a) (b) (c)

(d) (e) (f)

W

gW

W

FIG. 1: Representative leading order Feynman diagrams for new heavy fermion production. Up-

percase (lowercase) letters denote new heavy (SM) fermions. The usual gauge bosons are denoted

by g, W, Z, γ.

2

heavy fermion production cross sections, and their dependence on the mass of the heavy

fermion.

II. HEAVY QUARK PAIR PRODUCTION IN QCD

Heavy quarks that carry the color charge exist in many models of new physics (NP).

Examples include models with a sequential fourth generation [2, 9], or with a vectorlike t

prime (T) [10, 11], or with a bprime (B) [12], or with a heavy top quark partner [13–16].

According to their physics motivation, the NP models can be further categorized into two

classes. In the ﬁrst group, heavy quarks are introduced to explain discrepancies between

data and SM predictions. In the second, heavy quarks are essential to solve theoretical

problems. For example, in Little Higgs models [13], the heavy top partner cancels the

quadratic divergence in the quantum corrections to the Higgs boson mass associated with

the SM top quark loop. The phenomenology of heavy quark identiﬁcation at the LHC has

been studied extensively [17–25].

Similar to the top quark in the SM, heavy quark Qcan be pair produced via the strong

interaction processes,

q¯q→g→QQand gg →QQ,

where gdenotes a gluon. In this work we assume the coupling g-Q-¯

Qis the same as the

gluon-quark interaction in the SM. The cross section for a diﬀerent coupling strength can

be obtained easily from our results by rescaling. In Fig. 2(a) we plot our calculated NLO

inclusive cross sections for QQ production at the LHC for three choices of the proton-proton

center mass (c.m.) energy. 1The square symbol denotes the results of our exact NLO

calculation whereas the curve is drawn from the simple ﬁtting functions discussed below. As

shown by the ratios presented in Fig. 2(b), an increase in the c.m. energy from 7 to 14 TeV

can enhance the total cross section markedly, depending on the value of mQ. The values of

the standard model parameters used in our calculations are found in Appendix A.

There are uncertainties in the predicted NLO cross sections associated with the choices

of the renormalization scale (µR) and the factorization scale (µF) and with the choice of

the PDFs. We elect to use the CTEQ6.6M PDFs [30] for our evaluations. Other sets of

1Our numerical code for the NLO QQinclusive cross section is based on the analytic results in Refs. [26–29].

3

HaLQQ production at the LHC

:s=7 TeV

:s=10 TeV

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Black: Σ H10 TeVL

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Red: Σ H14 TeVL

Σ H10 TeVL

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FIG. 2: (a) NLO QQ pair production cross sections as a function of the quark mass mQat the

LHC. The square symbols denote the results of the exact calculation, while the curves present

results of our phenomenological ﬁt. (b) Ratios of the cross sections at diﬀerent c.m. energies.

(c-e) Theoretical uncertainties of the QQproduction cross section. The band denotes the PDF

uncertainties, while the black solid (red dashed) curve denotes the scale dependence obtained by

varying the renormalization scale by a factor 2 about the central value µ0=mQ.

PDFs are available, notably the MSTW [31] and NNPDF [32] sets, and one might use these

sets in addition to the CTEQ6.6M set to compute a range of predictions. All of these PDF

sets include an analysis of the uncertainties in the PDF determinations associated both with

uncertainties in the data used in the global ﬁts and with the choices of theoretical expressions

used to ﬁt the data. A comparison may be found in Ref. [31]. In this paper, we compute

the PDF uncertainty (at 90% C.L.) from the master formula in Eq. (2.5) in Ref. [30], using

all 44 sets of the CTEQ6.6M package. For heavy quark production at the large masses we

are considering, the PDF uncertainties arise primarily from the valence quark PDF where

diﬀerent PDF sets tend to agree better than they do for gluon distributions.

In Fig. 2(c)-2(e) we display the uncertainties associated with the choices of PDFs and the

4

renormalization/factorization scale. 2The uncertainties are portrayed relative to the cross

section with the best-ﬁt PDF by the bands in Fig. 2. The PDF uncertainties are relatively

large because the mass of the heavy quark requires that the PDFs be sampled in regions

where they are relatively unconstrained. The PDF uncertainties decrease with increasing

c.m. energy.

The uncertainties in the NLO cross section associated with the renormalization scale

(µR) and factorization scale (µF) are shown in Fig. 2. These uncertainties can be considered

as an estimate of the size of unknown higher-order contributions. In this study, we set

µ=µR=µFand vary it around the central value of µ0=mQ, where mQis the mass of the

heavy quark. Typically, a factor of 2 is used as a rule of thumb, and we display curves with

µ= 2µ0and µ=µ0/2. In Figs. 2(c)-2(e) we plot 1 + σ(µi)/σ(µ0) as a function of mQ. The

cross sections vary between about −10% for µ= 2µ0and +10% ∼15% for µ=µ0/2. We

note that the scale dependence at the three energies is insensitive to mQ. It is comparable

to the PDF uncertainties in the region of relatively small mQbut is much smaller than the

PDF uncertainties in the region of large mQ.

We ﬁnd that our calculated cross sections can be ﬁtted well by a simple three parameter

analytic expression:

log σ(mQ, µ)

pb =A(µ)mQ

TeV −1+B(µ) + C(µ)mQ

TeV ,(1)

where the units of σand mQare picobarn (pb) and TeV. The variation with respect to the

reference cross section (corresponding to µ=mQ) gives the NLO scale dependence ∆µ(mQ).

The values of ∆µ, and coeﬃcients A,B,Care listed in Table I. Our results are presented

in the form

σ=σ(µ0)∆σ(µ0/2)

∆σ(2µ0),(2)

where σ(µ0) is our prediction at the scale µ0. The cross sections at the scale µ0/2 and 2µ0

read as σ(µ0/2) = σ(µ0) + ∆σ(µ0/2) and σ(2µ0) = σ(µ0) + ∆σ(2µ0), respectively.

As is evident in Fig. 2(a), the ﬁtting function and parameters in Table I provide an

excellent representation of the calculated cross sections over the range of heavy quark masses

2For the scale dependence, ∆σ/σ0is deﬁned as (σ(µi)−σ(µ0))/σ(µ0), while for the PDF dependence

∆σis deﬁned as ∆σ= 1/4Pn

i(σ+

i−σ−

i)2, where where σ±

idenotes the upper (lower) cross section for

the ith eigenvector of the PDF sets and n= 22 for CTEQ6.6M PDF sets. The PDF uncertainties are

then plotted as 1 ±∆σ/σ0where σ0stands for the cross section of the averaged PDF set. This deﬁnition

applies to all other ﬁgures with cross section uncertainties.

5

Black : s=7 TeV

Red : s=10 TeV

Blue: s=14 TeV

Solid : qq

Dashed : gg

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FIG. 3: Fractions of the q¯qand gg initial state contributions to the NLO total cross sections for

QQ pair production at the LHC.

shown in the ﬁgure. We caution that the expression should not be used outside this range.

One could extend the region of applicability of a ﬁt to lower mass, say down to the top

quark mass, by simply increasing the number of terms in the polynomial expression. In

Appendix B we extend our ﬁt of the cross sections for heavy quark pair production to cover

the wider mass range, 250 to 700 GeV. We also present a tabulation showing our exact NLO

calculation and our ﬁtted results. The numerical comparison shows that our ﬁtted values

agree with the exact calculations well within 1%. Interested readers may contact the authors

for accurate ﬁtting functions that cover the mass range from the top quark mass to 4 TeV.

One reason more terms may be needed in a polynomial expression is that processes with

initial state gluons play an important role in pair production of relatively light quarks. As

shown in Fig. 3, at √s= 7 TeV, the gluon initial state contribution dominates over the

quark initial state when mQ<550 GeV; at √s= 10 TeV when mQ<750 GeV; while at

√s= 14 TeV when mQ<1 TeV. For relatively light quark production, say mQ<300 GeV,

1/y2and 1/y3terms (y≡mQ/TeV) are required to ﬁt the total cross section. In Appendix C,

we explore the apparent process independence of the remarkably simple analytic expression

Eq. 1 and discuss its range of applicability in the heavy fermion mass.

In view of the interest in the top quark as the heaviest of the known quarks, we provide

NLO predictions of the top quark pair production cross sections in Table II at three LHC

energies for a few values of mt. Our result at 14 TeV is consistent with Ref. [33].

6

TABLE I: Fitting parameters (A, B, C) of the parametric formula [Eq. 1] for the NLO total

cross sections (pb) for heavy quark production at the LHC. The central values are given for the

scale choice µR=µF=µ0=mQ. The superscript and subscript denote the deviations in the

coeﬃcients from the central value when the scale is µ0/2 (superscript) or 2µ0(subscript). We use

the CTEQ6.6M PDFs.

Process Parameter √s= 7 TeV √s= 10 TeV √s= 14 TeV

A0.84569+0.04779

−0.03441 1.76661+0.03079

−0.01912 2.03833+0.02345

−0.01292

qq/gg →Q¯

QB1.48655−0.05339

−0.02244 −0.39194+0.00546

−0.06751 −0.45930+0.03295

−0.08482

C−8.87048+0.09955

−0.08488 −5.82142+0.04314

−0.03726 −4.45853+0.01784

−0.01642

A0.72823+0.00130

−0.01262 0.72901−0.00419

+0.00981 0.69166−0.00600

+0.01911

qb →q′TB1.66318−0.03563

+0.00630 2.28354−0.01591

+0.00541 2.94279−0.01395

−0.01098

C−3.38334+0.05251

−0.04100 −2.57503+0.03980

−0.03535 −2.06090+0.03918

−0.02618

A0.77742+0.00580

−0.00158 0.78617+0.00133

+0.00520 0.75829−0.00266

+0.00743

q¯

b→q′¯

TB0.86229−0.04447

+0.03096 1.53387−0.03343

+0.01617 2.22983−0.01720

+0.01447

C−3.57353+0.05666

−0.05169 −2.72791+0.04759

−0.03900 −2.18287+0.03754

−0.03617

A1.37193−0.00541

+0.01703 1.41641+0.00152

+0.00749 1.46925−0.00453

+0.00596

q¯

q′→T¯

bB−3.44671+0.03710

−0.05685 −3.45805+0.01625

−0.02949 −3.33539+0.02351

−0.02316

C−5.22956+0.02415

−0.01643 −3.81301+0.02269

−0.01891 −3.00593+0.01228

−0.01352

A1.79403−0.00423

+0.02359 1.77007−0.00699

+0.00967 1.68940−0.00650

+0.00815

q¯

q′→¯

T b B−5.40590+0.02628

−0.06414 −5.03094+0.03140

−0.03479 −4.44604+0.02719

−0.02747

C−4.81694+0.02980

−0.01401 −3.67831+0.01489

−0.01460 −3.08803+0.00993

−0.01088

A0.09505−0.00087

+0.00202 0.98804+0.01201

−0.00090 1.07010−0.00263

+0.00304

q¯

q′→T¯

BB−1.08915+0.03084

−0.03419 −2.21256−0.01419

−0.00969 −2.20446+0.01778

−0.01452

C−10.4782+0.04744

−0.04719 −4.66219+0.04337

−0.03293 −3.75217+0.01638

−0.01956

A0.65738+0.01149

−0.00549 1.15500−0.00182

+0.00280 1.18663−0.00354

+0.00487

q¯

q′→¯

T B B−4.02175−0.01027

−0.00550 −5.09831+0.02561

−0.02683 −4.73348+0.02658

−0.02800

C−9.05880+0.07512

−0.06865 −6.08544+0.02944

−0.03100 −4.91903+0.01753

−0.01835

7

TABLE II: NLO top quark pair production cross sections (pb) at the LHC with three diﬀerent

c.m. energies based on the CTEQ6.6M PDFs. The central values are given for the scale choice

µR=µF=µ0=mt. The superscript and subscript denote the deviations of the coeﬃcients from

the central value when the scale is µ0/2 (superscript) or 2µ0(subscript).

mt(GeV) √s= 7 TeV √s= 10 TeV √s= 14 TeV

171 157.4+18.9

−20.3396.5+47.2

−48.4877.2+103.1

−101.4

173 148.2+17.7

−19.2374.5+44.4

−45.8830.9+93.9

−94.1

175 139.6+16.5

−18.1354.0+41.6

−43.4787.5+91.3

−91.5

177 131.6+15.6

−17.0335.0+39.2

−41.1747.2+86.4

−86.9

III. HEAVY QUARK PRODUCTION THROUGH THE ELECTROWEAK IN-

TERACTION

Heavy quark production occurs also through electroweak interactions. One example is

the heavy top quark partner (T) in the Little Higgs Model [14, 15], which is responsible for

canceling large quantum corrections to the Higgs boson mass from the SM top quark loop.

In addition to its pair production through QCD interactions, the Tquark can be produced

singly in association with a light quark through the processes

qb →q′T, and q¯

q′→T b.

It may also be produced in association with a heavy Bquark via the s-channel subprocess

q¯

q′→W∗→T B.

We consider ﬁrst single Tquark production in association with a SM bquark via an

s-channel subprocess, illustrated at leading order in Fig. 1(b). This subprocess involves a

weak interaction vertex W-T-band a quark mixing matrix VT b . For simplicity, we assume

that the coupling of W-T-bis same as the SM W-t-bcoupling and that |VT b|= 1. Our results

may be rescaled for other choices. We calculate the T¯

band T b results separately since the

pp initial state at the LHC is not a CP eigenstate. The renormalization and factorization

scales are chosen to be µR=µF=pQ2

Vwhere QVdenotes the four-momentum of the W

boson in the propagator.

The square symbols in Fig. 4 show the predicted production cross sections for T¯

band T b

at the LHC for three c.m. energies. The curves are our ﬁt with the parametrization in Eq. 1.

8

HaLq+q¢W+T+b

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