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我们在标准模型有效场理论的上下文中检查了三线性希格斯耦合β的约束,这些约束源自关联(V h)和矢量玻色子融合(VBF)希格斯在pp碰撞中的产生。 计算并插入因维度6运算符O 6 = A ˆ(H H H)3而产生的pp Vh和pp j jjh的1回路贡献。 »对希格斯玻色子的部分衰减宽度的$$ \ mathcal {O} \ left(\ lambda \ right)$$校正。 使用下一个至下一个领先的QCD预测,我们分析Vh和VBF希格斯产生中的信号强度的变化的当前和即将进行的测量的灵敏度。 我们显示,将来的大型强子对撞机运行可能能够探测到η的修饰,其敏感性与从两次希格斯产量的测定中预期的敏感性相似。 还研究了差分Vh和VBF希格斯分布对修改的h 3耦合的敏感性。
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JHEP07(2017)083
Published for SISSA by Springer
Received: October 27, 2016
Revised: May 13, 2017
Accepted: June 27, 2017
Published: July 17, 2017
Constraints on the trilinear Higgs coupling from
vector boson fusion and associated Higgs production
at the LHC
Wojciech Bizo´n,
a
Martin Gorbahn,
b
Ulrich Haisch
a,c
and Giulia Zanderighi
a,c
a
Rudolf Peierls Centre for Theoretical Physics, University of Oxford,
OX1 3NP Oxford, U.K.
b
Department of Mathematical Sciences, University of Liverpool,
L69 7ZL Liverpool, U.K.
c
CERN, Theoretical Physics Department,
CH-1211 Geneva 23, Switzerland
E-mail: Wojciech.Bizon@physics.ox.ac.uk,
Martin.Gorbahn@liverpool.ac.uk, Ulrich.Haisch@physics.ox.ac.uk,
Giulia.Zanderighi@physics.ox.ac.uk
Abstract: We examine the constraints on the trilinear Higgs coupling λ that originate
from associated (V h) and vector boson fusion (VBF) Higgs production in pp collisions in the
context of the Standard Model effective field theory. The 1-loop contributions to pp → V h
and pp → jjh that stem from insertions of the dimension-6 operator O
6
= −λ
H
†
H
3
are calculated and combined with the O(λ) corrections to the partial decay widths of
the Higgs boson. Employing next-to-next-to-leading order QCD predictions, we analyse
the sensitivity of current and forthcoming measurements of the signal strengths in V h
and VBF Higgs production to changes in λ. We show that future LHC runs may be able
to probe modifications of λ with a sensitivity similar to the one that is expected to arise
from determinations of double-Higgs production. The sensitivity of differential V h and
VBF Higgs distributions to a modified h
3
coupling is also studied.
Keywords: Effective Field Theories, Higgs Physics, Perturbative QCD
ArXiv ePrint: 1610.05771
Open Access,
c
The Authors.
Article funded by SCOAP
3
.
https://doi.org/10.1007/JHEP07(2017)083

JHEP07(2017)083
Contents
1 Introduction 1
2 Preliminaries 3
3 Corrections to the V V h vertex 4
4 Corrections to the Higgs partial decay widths 5
5 Description of the V h calculation 7
6 Description of the VBF Higgs calculation 9
7 Numerical results 11
7.1 Modifications of the Higgs production cross sections 11
7.2 Modifications of the Higgs decays 12
7.3 Modifications of the V h and VBF Higgs distributions 13
7.4 Constraints on ¯c
6
from double-Higgs production 16
7.5 Constraints on ¯c
6
from V h and VBF Higgs production 16
8 Conclusions 18
1 Introduction
Within the Standard Model (SM), the mass and the self-interactions of the Higgs field h
are parametrised by the potential
L
SM
⊃ −V
SM
= −
m
2
h
2
h
2
− λvh
3
−
κ
4
h
4
, (1.1)
where v = 246.22 GeV denotes the Higgs vacuum expectation value and
λ = κ =
m
2
h
2v
2
. (1.2)
The LHC measurement of the Higgs-boson mass m
h
= 125.09 GeV [1] determined the
first term in (1.1), but the h
3
and h
4
couplings, and in particular the SM relation (1.2)
have not been tested. Trying to constrain the Higgs self-couplings and thereby exploring
the mechanism of electroweak symmetry breaking (EWSB) is hence an important goal
of forthcoming LHC runs and other future high-energy colliders such as a hadron-hadron
Future Circular Collider or a Circular Electron-Positron Collider.
One way to constrain the coefficients λ and κ in (1.1) consists in measuring double-
Higgs and triple-Higgs production. Since the cross section for pp → 3h production is of
– 1 –

JHEP07(2017)083
O(0.1 fb) at 14 TeV centre-of-mass energy (
√
s) even the high-luminosity option of the
LHC (HL-LHC) will only be able to set very loose bounds on the Higgs quartic. The
prospect to observe double-Higgs production at the HL-LHC is considerably better be-
cause at 14 TeV the pp → hh production cross section amounts to O(35 fb) [2–9]. Mea-
suring double-Higgs production at the HL-LHC however still remains challenging (see for
instance [10–27]) and as a result even with the full data set of 3 ab
−1
only an O(1) deter-
mination of the trilinear Higgs coupling seems possible under optimistic assumptions.
A second possibility consists in studying the effects that a modification of λ has at loop
level in single-Higgs production. In fact, such indirect probes of the h
3
coupling have been
first proposed in the context precision studies of e
+
e
−
→ hZ [28, 29] and subsequently
extended to observables accessible at hadronic machines such as the LHC [30, 31]. For
both types of colliders it has been shown that future determination of λ via loop effects
are complementary to the direct HL-LHC determination through pp → hh, since these
probes can provide competitive constraints under the simplified assumption that new-
physics effects dominantly modify the h
3
coupling.
This paper is a sequel to the article [30], in which two of us have calculated the O(λ)
corrections to the gg → h and h → γγ processes that arise at the 2-loop level within the
SM effective field theory (SMEFT). The discussion in the present paper focuses instead
on associated (V h) and vector boson fusion (VBF) Higgs production. Specifically, we
compute the 1-loop contributions to the pp → V h and pp → jjh amplitudes that result
from insertions of the effective operator O
6
= −λ
H
†
H
3
. Combining these contributions
with the O(λ) corrections to the partial decay widths of the Higgs boson, we analyse the
sensitivity of present and future LHC measurements of the V h and VBF Higgs processes
to shifts in the trilinear Higgs interactions. In order to obtain high-precision predictions
for the V h and VBF Higgs cross sections we include QCD corrections up to next-to-next-
to-leading order (NNLO) in our study. We find that HL-LHC measurements of the V h
and VBF signal strengths may allow to set bounds on the Wilson coefficient of O
6
that are
comparable to the limits that are expected to arise from HL-LHC determinations of pp →
hh. By studying differential distributions it may even be possible to improve the obtained
constraints. We present NNLO predictions for the V h and VBF Higgs distributions that
are most sensitive to the shifts in the trilinear Higgs interactions. Our analysis shows that
measurements of the spectra in V h production provide sensitivity to the relative sign of
the Wilson coefficient of O
6
. The discriminating power in VBF Higgs production is less
pronounced compared to the V h channels. A similar investigation of the V h and VBF Higgs
processes in an anomalous coupling approach was presented in [31]. Whenever indicated
we will highlight the similarities and differences between this and our work.
The article at hand is structured in the following way. In section 2 we introduce the
effective interactions relevant for the computations performed in our paper. The results
of our loop calculations of the V V h vertex and the partial decay widths of the Higgs
boson are presented in section 3 and 4, respectively. The computations of the vector boson
mediated Higgs cross sections and distributions are described in section 5 and 6. Our
numerical analysis is presented in section 7. Both LHC Run I and HL-LHC constraints on
the trilinear Higgs coupling are considered. Section 8 contains our conclusions.
– 2 –

JHEP07(2017)083
2 Preliminaries
Physics beyond the SM (BSM) can be described in a model-independent fashion by sup-
plementing the SM Lagrangian L
SM
by effective operators O
k
of mass dimension six. In
our article, we will consider the following Lagrangian
L = L
SM
+
X
k=6,H
¯c
k
v
2
O
k
, (2.1)
where
O
6
= −λ
H
†
H
3
, O
H
=
1
2
∂
µ
H
†
H
∂
µ
H
†
H
, (2.2)
with λ defined as in (1.2) and H denoting the SM Higgs doublet. The dimension-6 operators
introduced in (2.2) modify the trilinear Higgs coupling. Upon canonical normalisation of
the Higgs kinetic term, one finds
L ⊃ −λc
3
vh
3
= −λ
1 + ¯c
6
−
3¯c
H
2
vh
3
, (2.3)
where the Wilson coefficients ¯c
6
and ¯c
H
as well as the trilinear Higgs coupling λ are all
understood to be evaluated at the weak scale hereafter denoted by µ
w
.
It is important to realise that the indirect probes of the trilinear Higgs coupling con-
sidered in our work measure c
3
, i.e. the coefficient multiplying the interaction term −λvh
3
in the effective Higgs potential after EWSB. Relating the coefficient c
3
to any underlying
theory, such as for instance the SMEFT, necessarily involves model assumptions. In the
following we will focus our attention on BSM scenarios where the Wilson coefficient ¯c
6
rep-
resents the only relevant modification of the h
3
vertex. Corrections due to ¯c
H
are on the
other hand ignored. Such effects will cause a universal shift in all Higgs-boson couplings
at tree level and also induce logarithmically-enhanced contributions to the oblique param-
eters S and T at the 1-loop level [32]. The Wilson coefficient ¯c
H
can therefore be probed
by means other than V h or VBF Higgs production that are the focal point of the present
work. We also do not consider effects of dimension-8 operators such as −λ¯c
8
/v
4
H
†
H
4
.
1
Under these model assumptions one obtains the simple relation
c
3
= 1 + ¯c
6
, (2.4)
which allows one to parameterise modifications of the h
3
vertex in terms of the Wilson
coefficient ¯c
6
. In our article we will use this parameterisation, but emphasise that all
formulas and results presented in the following sections can be translated to an anomalous
coupling approach by simply replacing ¯c
6
with c
3
− 1. In fact, we have verified that to
the perturbative order considered here and in [31] the calculations of V h and VBF Higgs
production in the SMEFT and the anomalous coupling framework agree exactly if the
relation (2.4) is taken into account.
1
The effects of −λ¯c
8
/v
4
H
†
H
4
could be easily incorporated in our analysis by shifting the coefficient c
3
introduced in (2.3) by 2¯c
8
[30].
– 3 –
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