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我们提出了通过胶子-胶子融合生产一对壳上Z玻色子的结果。 这个过程既通过希格斯玻色子的产生和衰变发生,也通过连续产生,Z玻色子耦合到无质量夸克或夸克的循环中。 我们计算两个过程的干扰及其对横截面的贡献,直至并包括阶数O(αs 3)。 通过振幅的两环贡献都是解析已知的,除了通过质量m的前夸克的环产生的连续体。 后一种贡献对于两个Z玻色子的不变质量(以其轻子衰变产物的质量m 4 l衡量)很重要,因为纵向玻色子的贡献使m 4 l≥2 m。 我们检查了涉及夸克的所有虚拟幅度的贡献,因为关于重夸克极限的扩展与保形映射和Padé逼近相结合。 与已知的分析结果进行比较,可以让我们评估重夸克及其扩展的有效性。 我们给出了针对这种干扰(包括真实和虚拟辐射)的NLO校正的结果。
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JHEP08(2016)011
Published for SISSA by Springer
Received: May 20, 2016
Accepted: July 19, 2016
Published: August 1, 2016
Two loop correction to interference in gg → ZZ
John M. Campbell,
a
R. Keith Ellis,
b
Michal Czakon
c
and Sebastian Kirchner
c
a
Fermilab,
Batavia, IL 60510, U.S.A.
b
IPPP, University of Durham,
South Road, Durham DH1 3LE, U.K.
c
Institut f¨ur Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University,
D-52056 Aachen, Germany
E-mail: johnmc@fnal.gov, keith.ellis@durham.ac.uk,
mczakon@physik.rwth-aachen.de, kirchner@physik.rwth-aachen.de
Abstract: We present results for the production of a pair of on-shell Z bosons via gluon-
gluon fusion. This process occurs both through the production and decay of the Higgs
boson, and through continuum production where the Z boson couples to a loop of massless
quarks or to a massive quark. We calculate the interference of the two processes and its con-
tribution to the cross section up to and including order O(α
3
s
). The two-loop contributions
to the amplitude are all known analytically, except for the continuum production through
loops of top quarks of mass m. The latter contribution is important for the invariant mass
of the two Z bosons, (as measured by the mass of their leptonic decay products, m
4l
), in a
regime where m
4l
≥ 2m because of the contributions of longitudinal bosons. We examine
all the contributions to the virtual amplitude involving top quarks, as expansions about
the heavy top quark limit combined with a conformal mapping and Pad´e approximants.
Comparison with the analytic results, where known, allows us to assess the validity of the
heavy quark expansion, and it extensions. We give results for the NLO corrections to this
interference, including both real and virtual radiation.
Keywords: NLO Computations, QCD Phenomenology
ArXiv ePrint: 1605.01380
Open Access,
c
The Authors.
Article funded by SCOAP
3
.
doi:10.1007/JHEP08(2016)011
JHEP08(2016)011
Contents
1 Introduction 1
2 Higgs production in gluon-gluon fusion and decay to ZZ 4
2.1 Preliminaries 4
2.2 Large-mass expansion and improvements 7
2.2.1 Rescaling with exact leading-order result 9
2.2.2 Conformal mapping and Pad´e approximants 10
2.2.3 Comparison of LME with full result 11
3 Virtual corrections to SM ZZ production via massive quark loops 13
3.1 Preliminaries 14
3.2 Projected exact result at one loop 15
3.3 Large-mass expansion at one loop 16
3.4 Large-mass expansion at two loops 18
3.4.1 Non-anomalous diagrams 19
3.4.2 Anomalous diagrams 21
3.5 Visualisation of large-mass expansion results for gg → ZZ 24
4 Real corrections to SM ZZ production 26
5 Results 28
6 Conclusions 32
A Definition of scalar integrals 34
B Scale dependence of the finite remainder 34
1 Introduction
The production of four charged leptons is a process of great importance at the LHC. It was
one of the discovery channels of the Higgs boson at the LHC. It also provides fundamental
tests of the gauge structure of the electroweak theory through the high-energy behaviour.
Four charged leptons are predominantly produced by quark anti-quark annihilation; the
mediation is by photons or Z bosons dependent on the mass of the four leptons, m
4l
.
A smaller contribution, which however grows with energy is provided by gluon-gluon
fusion. The Higgs boson is of course produced in this channel; in the Standard Model
(SM) this occurs predominantly through the mediation of a loop of top quarks. As pointed
– 1 –
JHEP08(2016)011
Figure 1. Representative diagrams for the ZZ production. In the following we will suppress the
Z-decays to leptons.
out by Kauer and Passarino [1], despite the narrow width of the Higgs boson, the Higgs-
mediated diagram gives a significant contribution for m
4l
> m
H
. If we examine the tail of
the Higgs-mediated diagrams there are three phenomena occurring:
• The opening of the threshold for the production of real on-shell Z bosons, m
4l
> 2m
Z
.
• The region m
4l
= 2m, (m is the top quark mass) where the loop diagrams develop
an imaginary part.
• The large m
4l
region, m
4l
> 2m, where the destructive interference between the
Higgs-mediated diagrams leading to Z bosons and the continuum production of on-
shell Z bosons is most important.
A feature of this tail is that it depends on the couplings of the Higgs boson to the
initial and final state particles but not on the width of the Higgs boson. Assuming the
couplings of the on- and off-peak Higgs-mediated amplitudes are the same, it has been
proposed to use this property to derive upper bounds on the width of the Higgs boson [2].
Note that models with different on- and off-peak couplings can be constructed [3].
In the following we shall refer to the production of the bosons V
1
, V
2
. Gluon-gluon fu-
sion first contributes to the cross section for electroweak gauge boson production pp → V
1
V
2
as shown in figure 1(c)-(e) at O(α
2
S
), which is the next-to-next-to-leading-order (NNLO)
with respect to the leading-order (LO) QCD process shown in figure 1(a); no two-loop
gg → V
1
V
2
amplitudes participate in this order in perturbation theory.
In the context of the Higgs boson width, however, the interference between the Higgs-
mediated Z boson pair-production and the Standard Model continuum at next-to-leading-
order (NLO) QCD already requires knowledge of the one- and two-loop gg → (H →)V
1
V
2
amplitudes. The requirement for more precise estimates to the Higgs boson width were
– 2 –
JHEP08(2016)011
emphasised in [4–6]. Signal-background interference effects beyond the leading order have
been considered in ref. [7] for the process gg → H → W
+
W
−
for the case of a heavy
Higgs boson.
In this work we will limit ourselves to the Z boson pair final state, due to its importance
at the LHC. At LO [8] and NLO [9–12] the amplitudes for single Higgs boson production
have been known for quite some time. At LO, the amplitude for the SM continuum
gg → ZZ process occurs via massless and massive fermion loops and results are available
in each case [13–16].
The situation, however, is different for the NLO continuum process, although vast
progress in terms of two-loop amplitudes has been made [17–22]. Recently two-loop gg →
ZZ amplitudes
1
via massless quarks became available [21, 22]. The complete computation
of two-loop amplitudes with massive internal quark loops, on the other hand, is commonly
assumed to be just beyond present technical capabilities. Although the contribution of the
top quark loops to these diagrams is smaller than the contribution of the light quarks in the
region just above the Z-pair threshold, in the high m
4l
region the amplitude is dominated
by the contributions of longitudinal Z bosons that couple to the top quark loops. Recently
a first heavy top quark approximation for the two-loop gg → ZZ amplitude with internal
top quarks was published [6]. In that work only the leading term in the s/m
2
expansion
was considered. In that approximation, the vector-coupling of the Z boson to the top
quark does not contribute. In addition an approximate treatment of this process at higher
orders, based on soft gluon resummation, was presented in ref. [23].
In the present work we will push this analysis further. We start by presenting our
results for the LO and NLO Higgs-mediated ZZ production in terms of the s/m
2
expansion
in section 2, despite the fact that the full result is known. This part is required for the
later interference with the SM continuum. Furthermore, it is well suited to introduce our
notation in section 2.1 and to assess the validity of the approximation methods with respect
to the exact known (N)LO amplitudes in section 2.2.
The results for the LO and virtual NLO contributions to the SM continuum with
massive quark loops will be given in section 3 as a large-mass expansion (LME) with
terms up to (s/m
2
)
6
. We will limit our discussion to the interference between the Higgs-
mediated term and the continuum term. Similar to [6] we will consider on-shell Z bosons
in the final state. A theoretical predictions for off-shell Z bosons would be optimal, but
in order to reduce the number of scales in the problem, we restrict ourselves to on-shell Z
bosons. Since we are primarily interested in the high-mass behaviour this is an appropriate
approximation. A limited number of scales is beneficial when we consider the extension of
our approach to a full calculation. In section 4 we summarize our treatment of the real
radiation contribution, which makes use of results already presented in ref. [16].
The results of our calculation, including loops of both massless and massive quarks,
will be presented in section 5. We will compare the effects of the NLO corrections to the
interference contribution with the corresponding corrections to the Higgs diagrams alone.
1
Actually, the results in [21] and [22] allow for arbitrary off-shell electroweak gauge bosons in the fi-
nal state.
– 3 –
JHEP08(2016)011
In addition, we will discuss the impact of our results on analyses of the off-shell region that
aim to bound the Higgs boson width.
All expansion results from section 2 and section 3.4.1 are provided via ancillary files
on arXiv as FORM and Mathematica readable code.
2 Higgs production in gluon-gluon fusion and decay to ZZ
In this section we give a detailed discussion of single Higgs boson production at LO and
NLO QCD and its subsequent decay to a pair of on-shell Z bosons. As mentioned earlier
the LO and NLO amplitudes for single Higgs boson production have been known for a long
time; either approximate results in terms of Taylor expansions in the inverse of the top
quark mass s/m
2
[8, 12, 24–28] or results keeping the exact top mass dependence [12, 29].
It is understood that, whenever feasible and available, the exact results for LO and
NLO amplitudes are used. However, we are mainly interested in approximations to the
interference contributions Re
A
LO
B
(N)LO
, where A denotes the Higgs-mediated and B
the SM continuum amplitude. Since no exact results are available for B
NLO
we will use
the, so-called, large-mass expansion [30] as an approximation of the SM continuum. Hence,
for consistency, we also perform the expansion of the Higgs-mediated amplitude A to
high powers in s/m
2
. Expansion of the two-loop Higgs-mediated amplitude A
NLO
and its
comparison to available results from the literature provides moreover a helpful check of our
expansion routines due to the general structure of the LME.
Furthermore, the large-mass expansion in powers of s/m
2
is formally only valid below
the threshold of top quark pair-production, as m is assumed to be much larger than any
other scale in the problem, e.g. s m
2
. As extensively discussed in literature the naive
LME can be drastically improved at (and even far above) threshold by taking the next mass
threshold into account, see ref. [30] and references within, or by rescaling the approximated
NLO result by the exact LO result, see e.g. refs. [31, 32]. We will address this issue in
section 2.2.3 and try to draw conclusions for the SM continuum.
2.1 Preliminaries
The amplitudes for single Higgs boson production
g(p
1
, α, A) + g(p
2
, β, B) → H(p
1
+ p
2
), s = (p
1
+ p
2
)
2
, (2.1)
are illustrated in figure 2 for the one-loop and two-loop case. The largest contribution is
due to the internal massive top quark loop; in the following we will ignore the contribution
of other quarks for the Higgs production process.
The gg → H amplitude, with color (Lorentz) indices A, B(α, β) for the initial state
gluons, can be written as
A
0,AB
αβ
(α
0
S
, m
0
, µ, )
E
= −iδ
AB
g
W
2m
W
4
3
(g
αβ
p
1
· p
2
− p
1,β
p
2,α
)
A
0
(α
0
S
, m
0
, µ, )
, (2.2)
such that the reduced matrix element
A
0
(α
0
S
, m
0
, µ, )
is dimensionless and can be ex-
pressed as a function of µ
2
/s and r
t
= m
2
/s. The bare on-shell amplitudes admit the
– 4 –
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