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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
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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
¯ q
N
¯N
q
¯ q
E−
E+
Z
Z/γ
(a)(b)(c)
(d)(e)(f)
W
g
W
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.
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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 b prime (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 Q can be pair produced via the strong
interaction processes,
q¯ q → g → QQ
andgg → QQ,
where g denotes a gluon. In this work we assume the coupling g-Q-¯Q is 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.1
The 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 14TeV
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 QQ inclusive cross section is based on the analytic results in Refs. [26–29].
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?a? QQ production at the LHC
? :
s ?7 TeV
? :
s ?10 TeV
? :
s ?14 TeV
0.6 0.8 1.01.2
mQ?TeV?
1.41.6 1.82.0
10?4
0.001
0.01
0.1
1
10
Σ?pb?
?b?
Black:Σ??10?TeV?
Σ??7?TeV?
Red:Σ??14?TeV?
Σ??10?TeV?
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
mQ?TeV?
0
10
20
30
40
50
60
R?Σi?Σj?
?c? LHC 7TeV
Μ?Μ0?2
Μ?2Μ0
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
mQ?TeV?
0.0
0.5
1.0
1.5
2.0
1??Σ?Σ0
?d? LHC 10TeV
Μ?Μ0?2
Μ?2Μ0
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
mQ?TeV?
0.6
0.8
1.0
1.2
1.4
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?e? LHC 14TeV
Μ?Μ0?2
Μ?2Μ0
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
mQ?TeV?
0.6
0.8
1.0
1.2
1.4
1??Σ?Σ0
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 QQ production 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
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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 σ and 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, C are 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/σ0 is defined as (σ(µi) − σ(µ0))/σ(µ0), while for the PDF dependence
∆σ is defined as ∆σ = 1/4?n
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.
i(σ+
i− σ−
i)2, where where σ±
idenotes the upper (lower) cross section for
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Black :
s ?7 TeV Red :
s ?10 TeV Blue:
s ?14 TeV
Solid : qq
Dashed : gg
0.51.0
mQ?TeV?
1.52.0
0.0
0.2
0.4
0.6
0.8
1.0
Fraction
FIG. 3: Fractions of the q¯ q and 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 4TeV.
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 = 7TeV, the gluon initial state contribution dominates over the
quark initial state when mQ< 550GeV; at√s = 10TeV when mQ< 750GeV; while at
√s = 14TeV when mQ< 1TeV. For relatively light quark production, say mQ< 300GeV,
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 14TeV is consistent with Ref. [33].
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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.
ProcessParameter
√s = 7TeV
√s = 10TeV
√s = 14TeV
A
0.84569+0.04779
−0.03441
1.76661+0.03079
−0.01912
2.03833+0.02345
−0.01292
qq/gg → Q¯Q
B
1.48655−0.05339
−0.02244
−0.39194+0.00546
−5.82142+0.04314
0.72901−0.00419
−0.06751
−0.45930+0.03295
−4.45853+0.01784
0.69166−0.00600
−0.08482
C
−8.87048+0.09955
0.72823+0.00130
−0.08488
−0.03726
−0.01642
A
−0.01262+0.00981+0.01911
qb → q′T
B
1.66318−0.03563
+0.00630
2.28354−0.01591
+0.00541
2.94279−0.01395
−0.01098
C
−3.38334+0.05251
0.77742+0.00580
−0.04100
−2.57503+0.03980
0.78617+0.00133
−0.03535
−2.06090+0.03918
0.75829−0.00266
−0.02618
A
−0.00158+0.00520 +0.00743
q¯b → q′¯T
B
0.86229−0.04447
+0.03096
1.53387−0.03343
+0.01617
2.22983−0.01720
+0.01447
C
−3.57353+0.05666
1.37193−0.00541
−0.05169
−2.72791+0.04759
1.41641+0.00152
−0.03900
−2.18287+0.03754
1.46925−0.00453
−0.03617
A
+0.01703 +0.00749 +0.00596
q¯q′→ T¯b
B
−3.44671+0.03710
−5.22956+0.02415
1.79403−0.00423
−0.05685
−3.45805+0.01625
−3.81301+0.02269
1.77007−0.00699
−0.02949
−3.33539+0.02351
−3.00593+0.01228
1.68940−0.00650
−0.02316
C
−0.01643
−0.01891
−0.01352
A
+0.02359
+0.00967
+0.00815
q¯q′→¯Tb
B
−5.40590+0.02628
−4.81694+0.02980
0.09505−0.00087
−0.06414
−5.03094+0.03140
−3.67831+0.01489
0.98804+0.01201
−0.03479
−4.44604+0.02719
−3.08803+0.00993
1.07010−0.00263
−0.02747
C
−0.01401
−0.01460
−0.01088
A
+0.00202
−0.00090 +0.00304
q¯q′→ T¯B
B
−1.08915+0.03084
−10.4782+0.04744
0.65738+0.01149
−0.03419
−2.21256−0.01419
−4.66219+0.04337
1.15500−0.00182
−0.00969
−2.20446+0.01778
−3.75217+0.01638
1.18663−0.00354
−0.01452
C
−0.04719
−0.03293
−0.01956
A
−0.00549 +0.00280+0.00487
q¯q′→¯TB
B
−4.02175−0.01027
−9.05880+0.07512
−0.00550
−5.09831+0.02561
−6.08544+0.02944
−0.02683
−4.73348+0.02658
−4.91903+0.01753
−0.02800
C
−0.06865
−0.03100
−0.01835
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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 = 7TeV√s = 10TeV√s = 14TeV
171157.4+18.9
−20.3
396.5+47.2
−48.4
877.2+103.1
−101.4
173
148.2+17.7
−19.2
374.5+44.4
−45.8
830.9+93.9
−94.1
175
139.6+16.5
−18.1
354.0+41.6
−43.4
787.5+91.3
−91.5
177
131.6+15.6
−17.0
335.0+39.2
−41.1
747.2+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 T quark can be produced
singly in association with a light quark through the processes
qb → q′T, andq¯q′→ Tb.
It may also be produced in association with a heavy B quark via the s-channel subprocess
q¯q′→ W∗→ TB.
We consider first single T quark production in association with a SM b quark via an
s-channel subprocess, illustrated at leading order in Fig. 1(b). This subprocess involves a
weak interaction vertex W-T-b and a quark mixing matrix VTb. For simplicity, we assume
that the coupling of W-T-b is same as the SM W-t-b coupling and that |VTb| = 1. Our results
may be rescaled for other choices. We calculate the T¯b and Tb 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=
?Q2
Vwhere QV denotes the four-momentum of the W
boson in the propagator.
The square symbols in Fig. 4 show the predicted production cross sections for T¯b and Tb
at the LHC for three c.m. energies. The curves are our fit with the parametrization in Eq. 1.
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?a? q?q??W??T?b
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s ?7 TeV
s ?10 TeV
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s ?14 TeV
0.4 0.60.8 1.01.21.4
10?4
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?b? q?q??W??T?b
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s ?7 TeV
? :
? :
s ?10 TeV
s ?14 TeV
0.40.6 0.81.0 1.21.4
10?5
10?4
0.001
0.01
mT?TeV?
Σ?pb?
?c? LHC 7TeV
Μ?Μ0?2
Μ?2Μ0
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
mT?TeV?
0.6
0.8
1.0
1.2
1.4
1??Σ?Σ0
?d? LHC 10TeV
Μ?Μ0?2
Μ?2Μ0
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
mT?TeV?
0.8
0.9
1.0
1.1
1.2
1??Σ?Σ0
?e? LHC 14TeV
Μ?Μ0?2
Μ?2Μ0
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
mT?TeV?
0.8
0.9
1.0
1.1
1.2
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FIG. 4: NLO production cross sections for (a) T¯b and (b)¯Tb at the LHC as a function of mT,
and (c-e) PDF and scale uncertainties of the T¯b production cross section. The Tb production cross
section exhibits similar uncertainties.
We note that this simple parametrization fits single T production via the s-channel process
very well for the three LHC energies. The fitting parameters are presented in Table I. The
cross section for single T production is larger than the cross section for single T production
by almost a factor 2 ∼ 3. This is due to the difference in parton densities of the colliding
protons. While in both cases the antiquark is from the quark sea of one of the incoming
protons, the probability that it collides with an up quark from the other proton is higher
than the probability for a collision with a down quark. Since T¯b and Tb production exhibit
similar uncertainties, we present only the results for T¯b production in Figs. 4(c-e). We note
the scale dependence at the three energies is not very sensitive to mT, and it is much smaller
than the PDF uncertainties in the large mT region. An increase in the c.m. energy reduces
the PDF uncertainties in the large mT region sizably.
In the “sequential” fourth generation model, the heavy T and B quarks form an isodou-
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?a? q?q??W??T?B
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s ?7 TeV
s ?10 TeV
s ?14 TeV
0.5 0.6 0.7
mQ?TeV?
0.8 0.91.0
1.00
0.50
0.20
0.10
0.05
0.02
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2.00
5.00
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s ?7 TeV
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s ?10 TeV
? :
s ?14 TeV
0.5 0.60.7
mQ?TeV?
0.80.91.0
1.00
0.50
0.20
0.10
0.05
0.02
0.01
2.00
5.00
Σ?fb?
?c? LHC 7TeV
Μ?Μ0?2
Μ?2Μ0
0.50.6 0.7
mQ?TeV?
0.8 0.91.0
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1??Σ?Σ0
?d? LHC 10TeV
Μ?Μ0?2
Μ?2Μ0
0.5 0.60.7
mQ?TeV?
0.8 0.91.0
0.8
0.9
1.0
1.1
1.2
1??Σ?Σ0
?e? LHC 14TeV
Μ?Μ0?2
Μ?2Μ0
0.5 0.60.7
mQ?TeV?
0.80.9 1.0
0.8
0.9
1.0
1.1
1.2
1??Σ?Σ0
FIG. 5: NLO production cross sections for (a) TB and (b) TB at the LHC as a function of
mQ= mT= mB, and (c-e) PDF and scale uncertainties of the TB production cross section. The
TB production cross section exhibits similar uncertainties.
blet, and their interaction with the W boson is identical to the W-t-b interaction in the SM
with the substitution of Vtb→ VTB= 1. For simplicity, we take the T and B quarks to be
degenerate, i.e. mT = mB= mQ. The T and B quarks can be produced together via the
Drell-Yan process (with an s-channel W∗). In Fig. 5 we show our calculated cross sections
and the fitted curves based on the parametrization in Eq. 1. The parametrization works
well at√s = 10TeV and 14TeV but less adequately at 7TeV. The inclusion of more terms
in the polynomial would improve the fit. The fitting parameters are listed in Table I.
Last, we consider single T quark production via a t-channel exchange diagram, illustrated
at leading order in Fig. 1(c). The production cross section is larger than in the s-channel
case because it is not subject to 1/s suppression. In this calculation we adopt the so-called
double deep inelastic-scattering scale proposed in Ref. [34]. Color conservation enforces a
natural factorization of the scales. The fermion line in Fig. 1(c) that does not include a heavy
quark probes a proton with the deep inelastic-scattering scale
?Q2
V, which is identical to
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?
?
?a? q?b?q??T
? :
s ?7 TeV
? :
s ?10 TeV
? :
s ?14 TeV
0.60.81.0 1.2
mT?TeV?
1.41.6 1.82.0
1.00
0.50
5.00
0.10
0.05
10.00
50.00
0.01
Σ?pb?
?b? T
Black:Σ??10?TeV?
Σ??7?TeV?
Red:Σ??14?TeV?
Σ??10?TeV?
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
mT?TeV?
0
2
4
6
8
10
R?Σi?Σj?
?
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?c? q?b?q??T production at the LHC
? :
s ?7 TeV
? :
s ?10 TeV
? :
s ?14 TeV
0.60.8 1.01.2
mT?TeV?
1.41.6 1.8 2.0
1.000
0.500
5.000
0.100
0.050
0.010
0.005
0.001
Σ?pb?
?d? T
Black:Σ??10?TeV?
Σ??7?TeV?
Red:Σ??14?TeV?
Σ??10?TeV?
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
mT?TeV?
0
2
4
6
8
10
12
R?Σi?Σj?
?e? LHC 7TeV
Μ?Μ0?2
Μ?2Μ0
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
mT?TeV?
0.0
0.5
1.0
1.5
2.0
1??Σ?Σ0
?f? LHC 10TeV
Μ?Μ0?2
Μ?2Μ0
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
mT?TeV?
0.6
0.8
1.0
1.2
1.4
1??Σ?Σ0
?g? LHC 14TeV
Μ?Μ0?2
Μ?2Μ0
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
mT?TeV?
0.6
0.8
1.0
1.2
1.4
1??Σ?Σ0
FIG. 6: Single heavy (a) T production and (c) T production via the t-channel subprocess as a
function of mT. The square symbols denote the results of the exact calculation while the curves
present results of our phenomenological fit. (b) and (d) Ratios of the cross sections at different
c.m. energies. (e-g) Theoretical uncertainties of the single T production 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= mT.
the virtuality of the W boson through NLO. The fermion line that connects to a heavy
quark sees the deep inelastic-scattering scale for massive quarks of
?
Q2
V+ m2
Q.
Our simple parametrization also works well for the t-channel results as may be seen in
Figs. 6(a) and 6(c). The fitting parameters are found in Table I. Again, an increase in
the c.m. energy from 7 to 14TeV can enhance the total cross section markedly, depending
11
Page 12
on the value of mT; see Figs. 6(b) and 6(d). The PDF and scale uncertainties are shown
in Figs. 6(e)-6(g). Since single T and single T production exhibit similar uncertainties, we
present only the results for single T production. We note the scale dependence at the three
energies is not sensitive to mT. The cross sections vary between ±2% for mT∼ 500GeV to
±7% for mT∼ 2TeV. The scale dependence is much smaller than the PDF uncertainties in
the large mT region.
IV.EXOTIC LEPTON PRODUCTION
Several variants of the seesaw mechanism have been proposed to explain light neutrino
masses. Positive signals could be observed at the LHC in the case that seesaw messengers
exist at the TeV scale or below. Three types of tree-level seesaw models generate light
neutrino Majorana masses via introduction of (1) a right-handed neutrino singlet, (2) a
complex scalar triplet ∆ with hypercharge Y = 1, and (3) a lepton triplet Σ with Y = 0. In
these seesaw models, neutrinos achieve their masses via a lepton number violating operator
at the scale Λ. The scale is not necessarily very high, and it might be around a TeV. Searches
for these exotic leptons and, if observed, measurement of their properties would verify and
even distinguish different seesaw mechanisms [35]. For this purpose, an accurate calculation
including higher-order QCD corrections is in order.
TABLE III: Coupling strengths of the gauge vertices in heavy lepton pair production for differ-
ent models, where g denotes the usual weak coupling strength. Note that all the couplings are
vectorlike.
WENZEEγEE ZNN
Majorana triplet
ggcW
e···
···
g
2cW
Dirac triplet
g
g
gcW
e
Lepton isodoublet
√2
g
cW(−1
2+ s2
W)
e
Charge singlet
······e···
12
Page 13
We examine the following three possibilities for exotic lepton production:
q¯ q → γ/Z → E+E−,
q¯ q → Z → NN,
q¯q′→ W → EN,
where E(N) denotes the new heavy charged lepton (heavy neutrino). The interactions of
gauge bosons and heavy leptons are summarized in Table III for four interesting NP models:
(1) a Majorana lepton triplet with Y = 0, (2) a Dirac lepton triplet with Y = 0, (3) a lepton
isodoublet, (4) a charge singlet. The motivation for these models and more details can be
found in Ref. [36]. It is worth mentioning the following properties:
• All the interactions given in the Table III are vectorlike.
• There are no ZNN interactions in the Majorana/Dirac lepton triplet model because
the triplet has zero hypercharge, and NN pairs are not produced.
• A Dirac lepton triplet is formed by two degenerate Majorana triplets with opposite
CP parities. As shown in Ref. [37], the heavy lepton fields can be redefined in such
a way that the Lagrangian is written in terms of two charged leptons E−
1, E+
2and a
Dirac neutrino N. As a consequence of the presence of two charged fermions instead of
only one, the total heavy lepton production cross section is twice that for a Majorana
triplet. In this work, however, we consider only one flavor of charged lepton in both
the Majorana triplet and the Dirac triplet.
To make our results more useful, rather than focusing on a specific NP model, we consider
the following effective interaction in our numerical calculation,
L = −g
cW
?EγµE + NγµN?Zµ−
g
√2
?EγµNW−
µ+ NγµEW+
µ
?+ eEγµEAµ,(3)
where the symbol g stands for the usual weak coupling strength. Numerical results for the
models listed in Table III can be easily derived from ours by simply rescaling the coupling
strengths, except for the process q¯ q → γ/Z → E+E−where there are interference effects of
the photon and Z boson contributions. However, since the threshold for two heavy charged
leptons is much higher than the Z boson mass, the interference effects are subleading. We
present separately the NLO cross sections for q¯ q → γ → E+E−and q¯ q → Z → E+E−; the
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Page 14
?
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?a? E?N production
? :
? :
? :
s ?7 TeV
s ?10 TeV
s ?14 TeV
0.30.40.50.6
mL?TeV?
0.70.80.91.0
0.01
0.1
1
10
100
Σ?fb?
?
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?b? E?N production
0.3 0.40.5 0.6
mL?TeV?
0.7 0.80.91.0
0.01
0.1
1
10
Σ?fb?
?c? LHC 7TeV
Μ?Μ0?2
Μ?2Μ0
0.50.6 0.7
mL?TeV?
0.80.9 1.0
0.6
0.8
1.0
1.2
1.4
1??Σ?Σ0
?d? LHC 10TeV
Μ?Μ0?2
Μ?2Μ0
0.5 0.60.7
mL?TeV?
0.8 0.91.0
0.8
0.9
1.0
1.1
1.2
1??Σ?Σ0
?e? LHC 14TeV
Μ?Μ0?2
Μ?2Μ0
0.50.6 0.7
mL?TeV?
0.80.9 1.0
0.8
0.9
1.0
1.1
1.2
1??Σ?Σ0
FIG. 7: NLO production cross sections for (a) E+N and (b) E−N production at the LHC as a
function of mL = mE = mN, and (c-e) PDF and scale uncertainties of E+N production. The
E−N production cross section exhibits similar uncertainties.
couplings involved in the two processes are overall factors. For all the s-channel processes
listed above, the renormalization and factorization scales are chosen to be µ =?Q2
QV denotes the four-momentum of the gauge boson in the propagator. For simplicity, we
Vwhere
take the E and N leptons to be degenerate, i.e. mE= mN= mL.
In Figs. 7(a) and 7(b) we display the NLO cross sections for the q¯q′→ W+→ E+N and
q¯q′→ W−→ E−N processes, respectively. Both processes exhibit similar scale and PDF
uncertainties as shown in Figs. 7 (c)-7(e). The fitting parameters are given in Table IV.
We also plot the NLO cross sections for the q¯ q → γ → E+E−and the q¯ q → Z → E+E−
processes in Fig. 8. Both processes exhibit similar scale and PDF uncertainties as shown in
Fig. 8(c)-8(e). Note that the cross sections for q¯ q → Z → NN are the same as those for
q¯ q → Z → E+E−owing to the same assignment of couplings in Eq. 3; therefore, they are
not shown here.
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?a? qq?Γ?E?E?
? :
? :
? :
s ?7 TeV
s ?10 TeV
s ?14 TeV
0.30.40.50.6
mE?TeV?
0.70.80.91.0
0.001
0.01
0.1
1
10
Σ?fb?
?
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?
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?
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?b? qq?Z?E?E?
0.30.40.5 0.6
mE?TeV?
0.70.8 0.91.0
0.01
0.1
1
10
100
Σ?fb?
?c? LHC 7TeV
Μ?Μ0?2
Μ?2Μ0
0.5 0.60.7
mE?TeV?
0.8 0.9 1.0
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1??Σ?Σ0
?d? LHC 10TeV
Μ?Μ0?2
Μ?2Μ0
0.5 0.60.7
mE?TeV?
0.8 0.91.0
0.8
0.9
1.0
1.1
1.2
1??Σ?Σ0
?e? LHC 14TeV
Μ?Μ0?2
Μ?2Μ0
0.50.6 0.7
mE?TeV?
0.80.9 1.0
0.8
0.9
1.0
1.1
1.2
1??Σ?Σ0
FIG. 8: NLO production cross sections for (a) E+E−production via an intermediate virtual
photon and (b) via an intermediate Z boson at the LHC as a function of mE, and (c-e) PDF and
scale uncertainties of E+E−production via an intermediate virtual photon. Production via an
intermediate Z boson exhibits similar uncertainties.
V.SUMMARY
We present NLO predictions for the cross sections for new heavy quark and lepton pro-
duction at three Large Hadron Collider energies. We treat the cases of pair production of
heavy quarks via strong interactions, single heavy quark production via electroweak interac-
tions, and the production of exotic 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 assist in experi-
mental simulations and in phenomenological studies of new physics models containing new
heavy quarks and/or leptons.
Acknowledgments. E. L. B. is supported at Argonne by the U. S. Department of Energy
under Contract No. DE-AC02-06CH11357. Q. H. C. is supported in part by the Argonne
15
Page 16
TABLE IV: The fitting parameters (A, B, C) in the parametric formula [Eq. 1] for the NLO total
cross sections (pb) for heavy lepton production at the LHC. The central values are given for the
scale choice µR= µF = µ0= mE. The superscript and subscript denote the deviations of the
coefficients from the central value when the scale is µ0/2 (superscript) or 2µ0(subscript). We use
the CTEQ6.6M PDFs.
ProcessParameter
√s = 7TeV
√s = 10TeV
√s = 14TeV
A
0.71010+0.00073
+0.00943
0.77778−0.00278
+0.00406
0.81919−0.00250
+0.00335
q¯q′→ E+N
B
−2.90505+0.02141
−9.41140+0.05659
0.99258−0.00287
−0.05227
−3.16717+0.02513
−6.66323+0.03454
0.98794−0.00431
−0.02697
−3.11747+0.02228
−5.20543+0.02221
0.94199−0.00381
−0.02179
C
−0.09655
−0.03572
−0.02455
A
+0.00593 +0.00560 +0.00486
q¯q′→ E−¯ N
B
−5.03622+0.02662
−8.53154+0.05327
0.90798−0.00402
−0.03759
−4.74631+0.03171
−6.43970+0.02594
0.89357−0.00376
−0.03380
−4.18717+0.02669
−5.39397+0.01772
0.86080−0.00328
−0.02725
C
−0.04582
−0.02674
−0.01906
A
+0.00298 +0.00524+0.00425
q¯ q → γ → E+E−
B
−5.74818+0.03299
−8.18368+0.04472
0.94114−0.00508
−0.02533
−5.46629+0.03040
−6.23857+0.02602
0.90624−0.00414
−0.03321
−5.01876+0.02561
−5.19925+0.01799
0.88538−0.00339
−0.02570
C
−0.05541
−0.02566
−0.01941
A
+0.00760 +0.00554+0.00434
q¯ q → Z → E+E−
B
−3.56798+0.03728
−8.45564+0.04295
−0.04573
−3.22104+0.03146
−6.44243+0.02677
−0.03400
−2.82716+0.02556
−5.28945+0.01899
−0.02557
C
−0.03792
−0.02707
−0.02084
National Laboratory and University of Chicago Joint Theory Institute (JTI) Grant 03921-
07-137, and by the U.S. Department of Energy under Grants No. DE-AC02-06CH11357 and
No. DE-FG02-90ER40560. Q. H. C. is grateful to C.-P. Yuan for useful discussions. E. L. B.
thanks the Galileo Galilei Institute for Theoretical Physics in Florence for hospitality, and
the Istituto Nazionale di Fisica Nucleare (INFN) for partial support during the final stages
of this work.
Note added: While finalizing the write-up of this work, we became aware of a recent
paper [38] in which the t channel production of single t′is calculated at the LHC. Our
results agree with those in Ref. [38].
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Appendix A: Standard model parameters
For our numerical evaluations, we choose the following set of SM input parameters:
GF= 1.16637 × 10−5GeV−2,
mZ= 91.1875GeV,
α = 1/137.0359895,
αs(mZ) = 0.1186,
mt= 173.1GeV,sin2θeff
W = 0.2314.
Following Ref. [39], we derive the W boson mass as mW= 80.385GeV. Thus, the square of
the weak gauge coupling is g2= 4√2m2
WGF.
Appendix B: Numerical fit of the heavy quark pair production cross section
In this appendix we use heavy quark pair production via the QCD interaction as a sample
illustration of how one may extend the region of applicability of a fit to a wider mass range
simply by increasing the number of terms in the polynomial expression. We treat heavy
quark production with masses within the range 250 GeV and 700 GeV. We choose the
following formula to fit the cross section:
log
?σ(mQ,µ)
pb
?
=A
x2+B
x+ C + Dx + Ex2,(B1)
where x = mQ/TeV. The fitting functions at the LHC at three energies are given as follows:
7 TeV : 1.33969x2− 10.4918x + 1.36903 + 1.32168/x − 0.0684983/x2,
10 TeV : 1.87135x2− 9.82576x + 2.34236 + 1.28886/x − 0.0682711/x2,
14 TeV : 2.16753x2− 9.46830x + 3.33295 + 1.22263/x − 0.0647251/x2,(B2)
where the renormalization and factorization scales are chosen as µR = µF = µ0 = mQ.
Table V displays the exact and fitted values of the NLO cross section (pb). The numerical
values of the exact and fitted cross sections agree very well.
Appendix C: Influence of parton luminosities on the heavy fermion mass depen-
dence
We comment in this appendix on the origins of the remarkably simple analytic expres-
sion Eq. 1 and on its range of applicability in the heavy fermion mass. The expression
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TABLE V: The exact and fitted values of the NLO cross section (pb) for heavy quark pair pro-
duction at the LHC at three c.m. energies, where the scale is chosen as µR= µF= µ0= mQ.
mQ
7TeV10TeV 14TeV
(GeV) Exact Fit Exact Fit
Exact Fit
25020.51 20.50 58.34 58.32 142.10 142.06
3007.289 7.291 22.21 22.21 57.03 57.04
3502.940 2.939 14.38 14.38 25.78 25.78
4001.301 1.301 9.565 9.561 12.74 12.74
4500.617 0.618 2.285 2.286 6.742 6.743
5000.310 0.310 1.224 1.224 3.771 3.770
550 0.162 0.162 0.685 0.685 2.203 2.204
6000.088 0.088 0.398 0.398 1.336 1.337
6500.049 0.049 0.239 0.239 0.837 0.837
7000.028 0.028 0.147 0.147 0.539 0.539
works well for strong pair production of heavy quarks, as well as for electroweak single
heavy fermion production and electroweak pair production of heavy fermions. This appar-
ent process independence suggests that the heavy fermion mass dependence of the parton
luminosities [40–42] is playing a dominant role.
The quark-antiquark and gluon-gluon contributions to the cross section for heavy quark
pair production at hadron colliders can be written as
σq¯ q(s) =
?
q=u,d
?1
τmin
?1
τmin
dτ
?1
τ
dxa
xa
?
fq/P(xa,µ0)f¯ q/P
?τ
xa,µ0
?
+ (u ↔ ¯ u)
?1
ˆ s
?ˆ sˆ σq¯ q→Q¯Q(ˆ s)?,
σgg(s) =dτ
?1
τ
dxa
xa
fg/P(xa,µ0)fg/P
?τ
xa,µ0
?1
ˆ s
?ˆ sˆ σq¯ q→Q¯ Q(ˆ s)?,(C1)
where ˆ σ is the operative parton-level cross section, τmin = 4m2
Q/s, and ˆ s = τs with s
being the square of the c.m. energy of LHC. In order to represent a process independent
situation, we approximate the (dimensionless) square brackets by one. As a result, the only
remaining effect of the heavy quarks appears in the lower limit of integration τmin. We
choose µR= µF= mQand use the CTEQ6.6M average PDF set.
Using Eq. C1, we generate “cross sections” as a function of mQ, and we then fit the
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?a? qq initial state
? :
? :
? :
s ?7 TeV
s ?10 TeV
s ?14 TeV
0.6 0.81.0 1.21.41.6 1.82.0
1.0
0.5
5.0
0.1
10.0
50.0
100.0
500.0
mQ?TeV?
Σ?pb?
?
?
?
?
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?
?
?b? gg initial state
? :
? :
? :
s ?7 TeV
s ?10 TeV
s ?14 TeV
0.6 0.81.01.2 1.41.6 1.82.0
0.01
0.1
1
10
100
1000
mQ?TeV?
Σ?pb?
FIG. 9: Fit to the trial scattering cross sections in the range of 0.5 < mQ< 2.0 TeV: (a) for the
quark-antiquark annihilation channel, and (b) for the gluon fusion channel.
results with Eq. 1. The agreement is excellent for the q¯ q and gg cases over the range of
heavy quark masses shown in Fig. 9, 0.5 < mQ< 2.0 TeV. The square symbols in the figure
are the predicted cross sections, and the solid curves represents the fits.
If we try to extend our study to smaller masses than shown in Fig. 9 we find that the
simple expression Eq. 1 is no longer adequate, as shown in Fig. 10(a). When τminis too
small, we sample parton densities in the region of small Bjorken x where they grow rapidly
with decreasing x. More terms in the polynomial are needed for a better fit. We can fit the
gluon fusion cross section well with the addition of two more terms; see Fig. 10(b)
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?a? gg:1
y?1?y
y ?
mQ
TeV
0.00.5 1.01.5 2.0
0.001
0.1
10
1000
mQ?TeV?
Σ?pb?
?
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?b? gg:1
y3?1
y2?1
y?1?y
0.0 0.51.0 1.52.0
0.001
0.1
10
1000
mQ?TeV?
Σ?pb?
FIG. 10: Fit to the trial scattering cross section for the gg channel extended into the region of
small mQ.
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