Higher-order soft gluon corrections in single top quark production at the LHC

Abstract

I present a calculation of soft-gluon corrections to single top quark production in pp collisions at the LHC via the Standard Model partonic processes in the t and s channels and associated top quark and W boson production. Higher-order soft-gluon corrections through next-to-next-to-next-to-leading order (NNNLO) are calculated at next-to-leading logarithmic (NLL) accuracy. The soft-gluon corrections in the s channel and in tW production are large and dominant, while in the t channel they are not a good approximation of the complete QCD corrections.

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arXiv:hep-ph/0701080v1 10 Jan 2007
Higher-order soft gluon corrections in single top quark
production at the LHC
Nikolaos Kidonakis
Kennesaw State University, Physics #1202,
1000 Chastain Rd., Kennesaw, GA 30144-5591
Abstract
I present a calculation of soft-gluon corrections to single top quark production in
pp collisions at the LHC via the Standard Model partonic processes in the t and s
channels and associated top quark and W boson production. Higher-order soft-gluon
corrections through next-to-next-to-next-to-leading order (NNNLO) are calculated at
next-to-leading logarithmic (NLL) accuracy. The soft-gluon corrections in the s channel
and in tW production are large and dominant, while in the t channel they are not a
good approximation of the complete QCD corrections.
1 Introduction
The production of top quarks in hadronic collisions can proceed via strong interaction pro-
cesses involving top-antitop pair production, or via electroweak processes involving the pro-
duction of a single top (or antitop) quark. The top quark was first discovered via t¯t pro-
duction at the Fermilab Tevatron by the CDF and D0 Collaborations in 1995 [1, 2]. The
observation of single top quark events has been more elusive but there has been recent
evidence of such events by the D0 Collaboration [3] with a cross section consistent with the-
oretical expectations [4]. Single top quark production is an important production mode for
further understanding of electroweak theory, measurement of the VtbCKM matrix element,
and the search for new physics (see e.g. [5, 6, 7, 8, 9]), so it is crucial to have accurate
predictions for the cross sections.
The Large Hadron Collider (LHC) at CERN has good potential for observation of single
top quark events. The production of single top quarks in proton-proton collisions can proceed
through three distinct partonic processes. The t-channel processes (qb → q′t and ¯ qb → ¯ q′t,
Fig. 1a) involve the exchange of a space-like W boson, the s-channel processes (q¯ q′→¯bt,
Fig. 1b) proceed via the exchange of a time-like W boson, and associated tW production
(bg → tW−, Fig. 1c) involves the production of a top quark in association with a W boson.
At the LHC the t-channel process is numerically the largest, the s-channel process is much
smaller, and associated tW production is intermediate in magnitude.
The next-to-leading order (NLO) QCD corrections to single top quark production have
been calculated in Refs. [10, 11, 12, 13, 14]. In this paper we calculate the contribution
of soft-gluon corrections to the cross section at the LHC beyond NLO. These corrections
are normally dominant near partonic threshold for the production of a specified final state
[15, 16, 17] and they arise from incomplete cancellations of infrared divergences between
virtual diagrams and real diagrams with soft (i.e. low-energy) gluons. For the partonic
process f1+ f2 → t + X the kinematical invariants are s = (p1+ p2)2, t = (p1− pt)2,
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b
q (¯ q)
W
t
q′(¯ q′)
(a)
¯ q′
q
W
t
¯b
(b)
g
b
b
W
t
(c)
Figure 1: Leading-order t-channel (a), s-channel (b), and associated tW production (c)
diagrams for single top quark production.
u = (p2− pt)2, s4= s + t + u − m2
energy to produce the final state, s4→ 0. The threshold corrections then take the form of
logarithmic plus distributions, [lnl(s4/m2
corrections. These threshold corrections exponentiate; the resummation is carried out in
moment space and it follows from the refactorization of the cross section into hard, soft,
and jet functions that describe, respectively, the hard scattering, noncollinear soft gluon
emission, and collinear gluon emission from the partons in the scattering [15, 16]. To obtain
a physical cross section, the moment-space result must be inverted to momentum space
thus necessitating a prescription to handle the Landau singularity. To avoid prescription
ambiguities [17] we provide fixed-order expansions of the resummed cross section, as has
been done for many other processes [4, 8, 17, 18, 19, 20] (for a review see [21]).
In the n-th order corrections in the strong coupling, αs, the leading logarithms (LL) are
those with l = 2n−1 while the next-to-leading logarithms (NLL) are those with l = 2n−2.
We calculate NLO, NNLO, and NNNLO soft-gluon threshold corrections at NLL accuracy,
i.e. at each order including both leading and next-to-leading logarithms. We denote these
corrections as NLO-NLL, NNLO-NLL, and NNNLO-NLL, respectively. More details are
given in Ref. [4], where single top quark production at the Tevatron was discussed.
A calculation of threshold corrections is only meaningful if these corrections dominate
the perturbative expansion, i.e. if they are a good approximation to the complete QCD
corrections. This can be determined by comparing the NLO soft-gluon corrections with the
complete NLO corrections. At Tevatron energies both top-antitop pair [17, 18] and single top
quark [4] production satisfy this criterion. Since the LHC energy is much bigger it cannot be
assumed that the same holds there. Moreover, since the kinematics and color flows are quite
different for each channel, the validity of the threshold approximation must be examined for
each channel seperately. Indeed, a study of the corrections shows that while the threshold
approximation works well for the s channel and for tW production, it is not valid for the t
channel at the LHC. Therefore for the t channel we only update the NLO cross section.
In Section 2 we present numerical results for single top quark production via the t channel
at the LHC at NLO. Results including NNLO-NLL and NNNLO-NLL soft gluon corrections
are provided for the s channel in Section 3 and for associated tW production in Section
4. We note that in the t and s channels the cross section for single antitop production is
different from that for single top production and we provide results for the antitop quark as
t− m2
X. Near threshold, i.e. when we have just enough
t)/s4]+, where l ≤ 2n − 1 for the n-th order QCD
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well. We use standard electroweak parameters [22] throughout.
2 Single top quark production via the t channel at the
LHC
We begin with the t channel. The dominant processes (with percentage contribution to the
cross section) are ub → dt (68.1% of the leading order cross section) and¯db → ¯ ut (11.5%).
Additional processes involving only quarks are cb → st (6.5%) and Cabibbo-suppressed
processes (ub → st, cb → dt, us → dt, and further suppressed processes). Additional pro-
cesses involving antiquarks and quarks are ¯ sb → ¯ ct (8.3%) and Cabibbo-suppressed processes
(¯db → ¯ ct, ¯ sb → ¯ ut,¯ds → ¯ ut, and further suppressed modes).
A calculation of the NLO soft-gluon corrections shows that they are not a good ap-
proximation to the full NLO QCD corrections in the t channel. In fact they are large and
negative while the exact NLO corrections [13] are small. Therefore for the t channel we only
update the NLO results using the MRST 2004 parton distribution functions (pdf) [23] and
we provide uncertainties for the cross section.
The NLO cross section for a top quark mass mt= 175 GeV is σt−channel
146 ± 4 ± 3 pb, where the first uncertainty is from variation of the factorization and renor-
malization scales, µFand µR, between mt/2 and 2mt, and the second uncertainty is from the
pdf [24]. Adding these uncertainties in quadrature gives σt−channel
pb.
The NLO cross section using the most recent value for the top quark mass, mt= 171.4±
2.1 GeV [25], is σt−channel
top
(mt= 171.4 ± 2.1GeV) = 150 ± 5 ± 2.5 ± 3 pb, where the first
uncertainty is due to the scale dependence, the second is due to the mass (±2.1 GeV),
and the third is due to the pdf. Adding these uncertainties in quadrature we find that
σt−channel
top
(mt= 171.4 ± 2.1GeV) = 150 ± 6 pb.
The cross section for the production of an antitop quark at the LHC is different from that
for a top quark. The corresponding results for the two choices of mass are σt−channel
175GeV) = 89±3±2 pb = 89±4 pb and σt−channel
pb = 92 ± 4 pb.
top
(mt= 175GeV) =
top
(mt= 175GeV) = 146±5
antitop
(mt=
antitop
(mt= 171.4±2.1GeV) = 92±3±1.5±2
3 Single top quark production via the s channel at the
LHC
We continue with the s channel. The dominant process is u¯d →¯bt (85.0% of the leading-
order cross section). Additional processes are c¯ s →¯bt (10.2%) and Cabibbo-suppressed
processes (u¯ s →¯bt, c¯d →¯bt, u¯d → ¯ st, and further suppressed modes).
In Table 1 we give results for the LO cross section and for the approximate NLO, NNLO,
and NNNLO cross sections which include the threshold soft-gluon corrections at NLL ac-
curacy. The soft-gluon corrections are relatively large for this channel; this is also true for
the exact NLO corrections [13]. We also note that the approximate NLO cross section is
only about 4% larger than the exact NLO result, showing that the threshold corrections
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s channel top
mt= 170
mt= 172
mt= 175
LO
5.40
5.18
4.87
NLO approx
7.53
7.23
6.79
NNLO approx
8.08
7.76
7.29
NNNLO approx
8.33
8.00
7.52
Table 1: The leading-order and approximate higher-order cross sections for top quark pro-
duction in the s channel in pb for pp collisions with
175 GeV. We use the MRST2004 NNLO pdf [23] and we set µF= µR= mt.
√S = 14 TeV and mt= 170, 172, and
160
165
170
175
180
mt (GeV)
0
2
4
6
8
10
12
14
σ (pb)
LO
NLO approx
NNLO approx
NNNLO approx
Single top at the LHC s-channel S1/2=14 TeV µ=mt
160
165
170
175
180
mt (GeV)
1
1.2
1.4
1.6
1.8
2
K factor
NLO approx / LO
NNLO approx / LO
NNNLO approx / LO
Single top at the LHC s-channel S1/2=14 TeV µ=mt
Figure 2: Left: The cross section for single top production at the LHC in the s channel.
Here µ = µF= µR= mt. Right: the K factors.
are dominant and provide the bulk of the QCD corrections and, thus, that the threshold
approximation works quite well.
As shown in Table 1, the NNNLO approximate cross section for mt= 175 GeV is 7.52 pb.
After matching to the exact NLO cross section [13], we find that the matched cross section
(i.e. exact NLO plus NNLO-NLL and NNNLO-NLL threshold corrections) is σs−channel
175GeV) = 7.23+0.53
the second is due to the pdf. Adding these uncertainties in quadrature we find σs−channel
175GeV) = 7.23+0.55
The NNNLO approximate cross section for mt= 171.4 GeV is 8.10 pb. If we match this to
the exact NLO cross section, then we find σs−channel
top
0.14 pb, where the first uncertainty is due to scale dependence, the second is due to the mass
(±2.1 GeV), and the third is the pdf uncertainty. Adding these uncertainties in quadrature
we find that σs−channel
top
(mt= 171.4 ± 2.1GeV) = 7.80+0.70
In Fig. 2 we plot the cross section for single top quark production in the s channel at
the LHC with
µ = mt. On the left-hand side we plot the LO and approximate higher-order cross sections
as functions of the top quark mass. On the right-hand side we show the K factors, defined as
top
(mt=
−0.45±0.13 pb, where the first uncertainty is due to the scale dependence and
top
(mt=
−0.47pb.
(mt= 171.4±2.1GeV) = 7.80+0.58
−0.48
+0.36
−0.33±
−0.60pb.
√S = 14 TeV setting both the factorization and renormalization scales to
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the ratios of the higher-order cross sections to the LO cross section. The K factors are quite
large, thus showing that the corrections provide a big enhancement to the cross section, and
are fairly constant over the top-quark mass range shown. As seen in the plot, the NLO-
NLL corrections provide a 40% increase of the LO cross section, the NNLO-NLL corrections
provide an additional 10%, and the NNNLO-NLL corrections a further 5%.
s channel antitop
mt= 170
mt= 172
mt= 175
LO
3.44
3.30
3.08
NLO approx
4.22
4.05
3.78
NNLO approx
4.14
3.96
3.71
NNNLO approx
4.09
3.92
3.67
Table 2: The leading-order and approximate higher-order cross sections for antitop quark
production in the s channel in pb for pp collisions with
and 175 GeV. We use the MRST2004 NNLO pdf [23] and we set µF= µR= mt.
√S = 14 TeV and mt= 170, 172,
The cross section for single antitop quark production in the s channel at the LHC is
different from that for single top quark production. In Table 2 we give results for antitop
quark production at the LHC. The approximate NLO cross section is only about 10% smaller
than the exact NLO result [13], showing again that the threshold corrections provide the
bulk of the QCD corrections. The soft-gluon corrections beyond NLO are small and negative.
As shown in Table 2, the NNNLO approximate cross section for mt= 175 GeV is 3.67
pb. After matching to the exact NLO cross section [13] we find σs−channel
4.03+0.10
due to the pdf. Adding these uncertainties in quadrature we find σs−channel
4.03+0.14
The NNNLO approximate cross section for mt= 171.4 GeV is 3.96 pb. If we match this
to the exact NLO cross section, we find σs−channel
antitop
0.11 pb, where the first uncertainty is due to scale dependence, the second is due to the mass
(±2.1 GeV), and the third is the pdf uncertainty. Adding these uncertainties in quadrature
we find that σs−channel
antitop
(mt= 171.4 ± 2.1GeV) = 4.35 ± 0.26 pb.
In Fig. 3 we plot the cross section for single antitop quark production in the s channel
at the LHC with
µ = mt. On the left-hand side we plot the LO and approximate higher-order cross sections
versus the top quark mass. The K factors are shown on the right. They are not as big
as for single top production in this channel and in fact the NNLO-NLL and NNNLO-NLL
corrections are small and negative. As seen in the plot, the NLO corrections provide a 23%
increase of the LO cross section, the NNLO corrections provide an additional -3%, and the
NNNLO corrections a further -1.5%.
antitop
(mt= 175GeV) =
−0.12±0.10 pb, where the first uncertainty is due to the scale dependence and the second is
antitop
(mt= 175GeV) =
−0.16pb.
(mt= 171.4 ± 2.1GeV) = 4.35+0.11
−0.13
+0.21
−0.19±
√S = 14 TeV setting both the factorization and renormalization scales to
4 Associated tW production at the LHC
Associated tW production proceeds via bg → tW−(Cabibbo-suppressed contributions from
sg → tW−and dg → tW−are negligible).
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