Next-to-Leading Order Cross Sections for New Heavy Fermion Production at Hadron Colliders
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
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
High Energy Physics Division, Argonne National
Laboratory, Argonne, Illinois 60439, USA and
Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA
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: email@example.com
†Electronic address: firstname.lastname@example.org
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
(a) (b) (c)
(d) (e) (f)
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.
heavy fermion production cross sections, and their dependence on the mass of the heavy
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) , 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 , 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
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  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].
HaLQQ production at the LHC
Black: Σ H10 TeVL
Σ H7 TeVL
Red: Σ H14 TeVL
Σ H10 TeVL
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  and NNPDF  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. . In this paper, we compute
the PDF uncertainty (at 90% C.L.) from the master formula in Eq. (2.5) in Ref. , 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
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
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
log σ(mQ, µ)
TeV −1+B(µ) + C(µ)mQ
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
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)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.
Black : s=7 TeV
Red : s=10 TeV
Blue: s=14 TeV
Solid : qq
Dashed : gg
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. .
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
T b B−5.40590+0.02628
T B B−4.02175−0.01027
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
III. HEAVY QUARK PRODUCTION THROUGH THE ELECTROWEAK IN-
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¯
It may also be produced in association with a heavy Bquark via the s-channel subprocess
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.