Precise QCD predictions on top quark pair production mediated by massive color octet vector boson at hadron colliders
ABSTRACT We present a theoretical framework for systematically calculating
next-to-leading order (NLO) QCD effects to various experimental observables in
models with massive COVB in a model independent way at hadron colliders.
Specifically, we show the numerical results for the NLO QCD corrections to
total cross sections, invariant mass distribution and AFB of top quark pairs
production mediated by a massive COVB in both the fixed scale (top quark mass)
scheme and the dynamical scale (top pair invariant mass) scheme. Our results
show that the NLO QCD calculations in the dynamical scale scheme is more
reasonable than the fixed scheme and the naive estimate of the NLO effects by
simple rescaling of the LO results with the SM NLO K-factor is not appropriate.
arXiv:1201.0672v1 [hep-ph] 3 Jan 2012
Precise QCD predictions on top quark pair production
mediated by massive color octet vector boson at hadron colliders
Hua Xing Zhu,1Chong Sheng Li,1,2, ∗Ding Yu Shao,1Jian Wang,1and C.-P. Yuan2,3, †
1Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China
2Center for High Energy Physics, Peking University, Beijing 100871, China
3Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
We present a theoretical framework for systematically calculating the next-to-leading order (NLO)
QCD correction to the production of top quark pairs at hadron colliders, in a class of models with
massive color octet vector bosons. We find that NLO corrections can decrease new physics cross
section by about 50%, and top quark forward-backward asymmetry by 1 − 2%. The uncertainty in
predicting the invariant mass distribution is reduced at NLO, and the shape of resonance is strongly
modified by NLO corrections when taking the renormalization and factorization scales as the scale
of top quark mass.
In many extensions of the Standard Model (SM), the
massive Color Octet Vector Bosons (COVB) are neces-
sarily engaged at the TeV scale. For example, in the
top-color , warped (RS) or universal extra dimen-
sions [2, 3], technicolor  and chiral color models .
In all these cases, the COVBs could have large impact
on top quarks production, which are being copiously
produced at the CERN Large Hadron Collider (LHC).
With a large sample of t¯t data, CDF collaboration at
the Tevatron has recently reported an observation of a
large Forward-Backward asymmetry (AFB) in t¯t produc-
FB= 0.158 ± 0.075, to be compared with the SM
prediction 0.058±0.009. The disagreement is more signif-
icant in the region of large t¯t invariant mass, where CDF
reported AFB(mt¯ t> 450GeV) = 0.475 ± 0.114, and the
SM gives 0.088±0.013 . This leads to more than 3.4σ
deviation from the SM prediction , which resulted in
extensive studies on this observable in various models be-
yond the SM. Among these, models with massive COVB
are in particular attractive, c.f., Ref.  and references
therein. A representative is the RS model . However,
those studies were all carried out at the leading order
(LO) accuracy, and it’s not clear whether the next-to-
leading order (NLO) QCD corrections would change the
The existence of such massive COVB will induce a res-
onant peak in the t¯t invariant mass distribution [10–13].
There have been substantial efforts in searching for nar-
row width resonant in t¯t production at the Tevatron and
LHC. The recent analysis by CDF using 4.8 fb−1inte-
grated luminosity sees no evidence of resonant production
of t¯t pairs in the lepton+jets channel, and a model with
leptophobic Z′is excluded for mZ′ < 900 GeV . Sim-
ilar searches, but in semileptonic channel, for Z′with a
narrow width have also been performed by the CMS and
ATLAS collaborations at the LHC [15–18].
It is well known that QCD effects play an important
role in t¯t production. The NLO QCD corrections to the
SM t¯t production have been known for a long time [19–
21]. It was found that the NLO corrections significantly
enhance the t¯t total cross sections and lead to large AFB,
although is still smaller than the current data. In the case
of massive COVB, the QCD gauge interaction is uniquely
determined by its color content, resembling a SM gluon.
Therefore, it’s reasonable to expect that QCD will also
have significant effects on the processes mediated by the
massive COVB, at least at an energy scale comparable
to the mass of COVB. This has motivated the model
dependent calculation of COVB production by gluon fu-
sion  and the model independent (using dimension-
six operators) calculation of t¯t production mediated by
COVB . However, a complete NLO analysis of the
QCD effects to models with massive COVB in the reso-
nant t¯t region is still absent.1In this letter, we present
a model independent complete NLO QCD corrections to
top quark pair production mediated by a general mas-
sive COVB, and show a detailed numerical analysis of
top quark pair production, including invariant mass dis-
tribution and AFBat the NLO level. We also show that
the NLO corrections significantly stabilize the renormal-
ization and factorization scale dependence, as compared
to the LO results.
Below, we briefly outline our approach to systemati-
cally calculate the NLO QCD effects to processes with
COVB production. We consider a model independent
massive COVB originated from a broken SU(3) gauge
group. The effective Lagrangian in the unitary gauge
can be written as
where a = 1,..,8 is the broken SU(3) “color” index, and
µ = 0,..,3 is a Lorentz index. Fµν = Fa
strength tensor, where Tais the conventional Gell-mann
matrix with the normalization Tr[TaTb] =
µνTais the field
1While this work was completed, Ref.  appeared, which cal-
culates the singly production of color octet vector boson in the
narrow width approximation.
mass of the COVB is denoted by MG. The QCD color
interaction between COVB and SM gluon can be eas-
ily implemented by changing the ordinary derivative into
where g1 is the coupling constant of QCD. The La-
grangian in Eq. (1), after the replacement in Eq. (2), is
already invariant under the conventional SU(3)ctrans-
formation.If desired, NLO QCD calculation can be
done with the Feynman rule derived from the above La-
grangian, where unitary gauge is chosen for the broken
SU(3) gauge symmetry. However, it is well known that
loop calculation in unitary gauge is inconvenient, because
of the bad ultra-violet (UV) behavior of the propagator
in unitary gauge. Instead, we choose to carry out the
calculation in the conventional ’t Hooft-Feynman gauge
for the broken gauge group. For this purpose, we need to
separate the longitudinal component (the would-be Gold-
stone boson) of the COVB from Eq. (1). This is done by
modifying the mass term in Eq. (1) as follows,
Gµ(x) → U(x)†
where we have introduced the π field (the would-be Gold-
stone field) via U(x) = eiπa(x)Ta/f, and the symmetry
breaking scale f = MG/g2, with g2 being the coupling
constant of the broken SU(3) gauge symmetry. It’s easy
to check that the mass term is now invariant under the
Gµ(x) → V (x)
U(x) → V (x)U(x), (5)
where V (x) is a finite gauge transformation V (x) =
Similar to the unitary gauge, the in-
teraction between Gµ, π and QCD gluon is obtained
by changing the ordinary derivative to covariant deriva-
tive, as in Eq. (2).
Expanding the mass term of the
COVB in Eq. (1), we find that there is kinetic mixing
between the COVB and the would-be Goldstone boson:
troducing the gauge fixing term:
µ(x). This mixing can be canceled by in-
µ(x) + MGπa(x))2, (6)
which is similar to the gauge fixing term for the SM QCD
Lagrangian in ’t Hooft-Feynman gauge. The ghost La-
grangian is given by
Lg= ¯ ui(x)
where i = 1 for the SM QCD ghost, and i = 2 for the
broken SU(3) ghost. θi(x) are the infinitesimal gauge
transformation parameters for the corresponding gauge
group. All the Feynman rules determined by gauge sym-
metry can then be derived. We have checked that this
set of Feynman rules is in agreement with that derived
in Ref.  for the case of Randall-Sundrum model .
Before continuing, we should define the precise mean-
ing of the NLO QCD corrections in this paper.
the LO, there are three parts of contributions to the
t¯t cross section in models with massive COVB: the
squared SM amplitudes, the interference between the new
physics (NP) amplitudes and the SM amplitudes, and the
squared NP amplitudes. The NLO QCD corrections in
this paper refer to the O(αs) corrections to these three
parts separately. All the relevant Feynman rules can be
derived from the effective Lagrangian given above. The
results obtained in this way reflect the model indepen-
dent corrections from QCD interaction. As a final com-
ment, we note that our NLO QCD corrections are the
non-Abelian analogy of the QED corrections to the W±
boson production process in hadron collision , and
the neglected contributions in our calculation are simi-
lar to the genuine weak corrections there , which can
be mostly absorbed into the redefinition of boson and
fermion couplings at the LO.
Extending the approach shown in our paper , we
calculate the one-loop renormalized helicity amplitudes,
with the unstable particle treated in the complex mass
scheme . Loop integrals with complex arguments
appear in the one-loop amplitudes are evaluated with
ONELOOP . Real emission matrix elements are gen-
erated by a modified version of MADGRAPH . Soft
and (or) collinear singularities of real corrections are
dealt with by the dipole method , implemented in
the MADDIPOLE package . Throughout our calcu-
lation, the pole mass of top quark is chosen as mt= 173.1
GeV, and αs(MZ) = 0.118.
obtained using the CTEQ6L parton distribution func-
tions (PDFs)  with one-loop running of αs, while NLO
cross sections are obtained using the CTEQ6M PDFs
with two-loop running of αs. In the numerical calcula-
tion below we present the results for a benchmark axial-
gluon model . As an example, the coupling between
the massive COVB and quarks in these model are chosen
LO cross sections are
vq(mt) = 0,
vt(mt) = 0,
aq(mt) = 1.5,
at(mt) = −1.5,
where g1vq(t)and g1aq(t)are the vector and axial coupling
of the light quark (top quark), defined at the scale mt.
The evolution of vq,t(µ) under the change of scale is given
by v(µ)q,t = v(µ0)q,t(αs(µ)/αs(µ0))15/(2β0), where
β0= 23/3 is the LO QCD beta function for Nc= 3 and
nf = 5. The evolution equation of aq,t(µ) has the same
form as vq,t(µ). Unless specified, the mass of the COVB
is chosen as 1.5 TeV.
First we define the NP cross section, σNP, as the dif-
ference between t¯t cross section in a model with a new
massive COVB, and the SM:
σNP= σSM+NP− σSM. (10)
The SM cross sections, including both the q¯ q- and gg-
channel, are calculated with the program MCFM .
1500 1550 1600 1650 1700 17501800 1850 1900 1950 2000
Fixed Scale LO
Fixed Scale NLO
15001550 1600 1650 1700 1750 1800 1850 1900 1950 2000
Dynamical Scale LO
Dynamical Scale NLO
two different benchmark schemes. The black (red) bands are
the LO (NLO) uncertainties, estimated by varying the scales
around their default values by a factor of two within each
scheme. The blue dashed lines are the naive estimates of the
NLO effects by simple rescaling of the LO results with the
σNP, defined in Eq. (10), as functions of MG for
In Fig. 1, we plot the NP contribution to the cross
section at the LHC with√s = 7 TeV, as a function of
MG. The bands reflect the scale uncertainties estimated
by varying the renormalization and factorization scales
around their default values by a factor of two. We present
the results in two benchmark schemes, namely the fixed
scale scheme, where the scales are fixed to be mt, and the
dynamical scale scheme, where the scales are set to be the
invariant mass of the top quark pair mt¯ t. We find that
the NLO QCD effects are much larger (by about 50%)
in the fixed scale scheme than the dynamical scheme for
our choice of parameters, and the naive estimate of the
NLO effects by simple rescaling of the LO results with
the SM NLO K-factor is not appropriate. It is also clear
that the inclusion of NLO QCD effects strongly reduces
the theoretical uncertainty of the NLO cross section in
either scheme, comparing with the LO ones, as expected.
Fig. 2 gives the LO and NLO invariant mass distribu-
tion of the top quark pair at the LHC with√s = 7 TeV,
including contribution from the SM LO and NLO top
quark pair production. It can be seen that in the fixed
scale scheme, NLO QCD corrections significantly change
the shape of the LO curve, leading to the reduction of
SM QCD NLO + NP LO
SM QCD NLO + NP NLO
0 200400600800 1000 120014001600 18002000
FIG. 2. LO (dashed lines) and NLO (solid lines) invariant
mass distribution of the top quark pair at the LHC with√s =
7 TeV. Also shown are the differences between two different
schemes at the LO and NLO.
the width of the resonance, which is important for accu-
rate extraction of the mass and width of resonance from
t¯t invariant mass distribution experimentally. The NLO
width can be expressed analytically as
From Eq. (11), it’s obvious that the width of COVB is
reduced at the NLO when µ = mt, and the large log-
arithmic contribution can be canceled by a dynamical
scale choice µ = mt¯ t. This is confirmed in Fig. 2, where
the NLO results for the dynamical scale scheme has rela-
tively small corrections comparing with the LO one. The
predictions in the two scheme at the NLO level are close
to each other, while at the LO level they show large differ-
ence, as shown in the lower panel of Fig. 2. The difference
between the two schemes reflects the uncertainty of the
theoretical prediction. Hence, the NLO result leads to a
smaller theoretical uncertainty in mt¯ tdistribution, which
could improve the accuracy of extracting the theory pa-
rameters of NP models from comparing to experimental
data. We also note that similar conclusion holds in other
cases, e.g., KK gluon in RS model.
Fig. 3 shows the LO and NLO contribution to the AFB
as a function of the mass of the COVB at the Tevatron
with√s = 1.96 TeV in the center of mass frame of the t¯t
pair. The results are given for both the total asymmetry
and the asymmetry in the large invariant mass region,
mt¯ t > 450 GeV, respectively. As we know, the NLO
QCD corrections in the SM enhance the AFBcomparing
with the LO results , but the NP contributions at the
NLO level only reduce the AFBby 1 − 2%. Similar be-
haviors are also found for vector-like coupling (not shown
in this paper). Therefore the LO predictions for AFBare
stable against the NLO QCD corrections. This increases
the confidence on the massive COVB explanation of the
Tevatron AFB anomaly. If this anomaly can be further
confirmed, it could be a first hint of new physics signal.
150015501600 1650 170017501800 185019001950 2000
NP LO + SM NLO QCD
NP NLO + SM NLO QCD
FIG. 3. NP contributions to the AFB in the t¯t center of mass
frame as a function of the mass of the COVB at the Tevatron
with√s = 1.96 TeV. In the upper lines only the AFB in the
large invariant mass region, mt¯ t> 450 GeV, are plotted, while
in the bottom total AFB are shown.
As a further application of our results, we plot in Fig. 4
the NP cross section, σNP, for a specific RS model consid-
ered in the Ref. . Also plotted are the experimental
exclusion limit extracted from the same paper. Our exact
theoretical predictions are given for three different scale
choices, µ = mt, µ = MG and µ = mt¯ t, for both the
LO and NLO. It can be seen that while the NLO QCD
effects are moderate in the µ = mt¯ tcase, they are large
in both the µ = mtand µ = MGcases. We also plot in
Fig. 4 the LO results (the black solid line) with only the
contributions from NP squared amplitudes, i.e., without
the interference with the SM amplitudes. It’s clear in
Fig. 4 that considering the NP squared amplitudes con-
tributions alone obviously underestimates the NP cross
section, and including the interference contributions is
necessary for reliable predictions. We note that while
the mass limit for KK gluon is very different at the LO
for the three kinds of scale choices, their differences are
significantly reduced at the NLO. Hence, the NLO results
can be used for precise extraction of mass limit for KK
In conclusion, we have presented a theoretical frame-
work for systematically calculating the NLO QCD effects
to various experimental observables predicted by mod-
400600800 10001200 14001600
LO NP squared amplitude contributions (
FIG. 4. LO and NLO predictions for σNP in a specific RS
model . The expected and observed limits on cross section
are extracted from the Ref. .
els with massive COVB, in a model independent way.
Specifically, we have showed the numerical results of the
NLO QCD corrections to the total cross sections, invari-
ant mass distribution and the AFBof the top quark pairs
produced by mediating a massive COVB. Our results
show that, for our choice of parameters, the NLO correc-
tions of the NP cross section are much larger in the fixed
scale scheme than in the dynamical scale scheme, and
the naive estimate of the NLO effects by simple rescal-
ing of the LO results with the SM NLO K-factor is not
appropriate. We have also showed that the NLO QCD
corrections only reduce the AFBby 1−2%, as compared
to the LO prediction. This result leads to increased con-
fidence on the massive COVB models explanations of the
top quark AFBobserved at the Tevatron by CDF  and
D0 collaborations . Moreover, for the invariant mass
distribution, we find that the NLO QCD corrections re-
duce the width of the resonant particle in the fixed scale
scheme (with µ ∼ mt), and therefore change the shape of
the distribution, and the effects of the NLO QCD correc-
tions can be suppressed in the dynamical scale scheme.
The difference in the results using the fixed scale and
dynamical scale schemes indicates the size of theoretical
uncertainty in our predictions, and we have showed that
the uncertainty is reduced by the inclusion of NLO QCD
We would like to thank Liang Dai and Jun Gao for col-
laboration on early stage of this project, and Qing-Hong
Cao, Michele Petteni, Bernd Stelzer, and Jing Shu for
helpful discussion. This work was supported in part by
the National Natural Science Foundation of China, under
Grants No.11021092 and No.10975004. C.P.Y acknowl-
edges the support of the U.S. National Science Founda-
tion under Grand No. PHY-0855561.
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