#### Electroweak Higgs boson production in the standard model effective field theory beyond leading order in QCD

Eur. Phys. J. C
Electroweak Higgs boson production in the standard model effective field theory beyond leading order in QCD
Céline Degrande 2
Benjamin Fuks 0 1
Kentarou Mawatari 5 6
Ken Mimasu 3 4
Verónica Sanz 4
0 CNRS, UMR 7589, LPTHE , 75005 Paris , France
1 UPMC Univ. Paris 06 , Sorbonne Universités, UMR 7589, LPTHE, 75005 Paris , France
2 CERN, Theory Division , Geneva 23 CH-1211 , Switzerland
3 Centre for Cosmology , Particle Physics and Phenomenology (CP3) , Université catholique de Louvain , Chemin du Cyclotron, 2, B-1348 Louvain-la-Neuve , Belgium
4 Department of Physics and Astronomy, University of Sussex , Brighton BN1 9QH , UK
5 Theoretische Natuurkunde and IIHE/ELEM, International Solvay Institutes, Vrije Universiteit Brussel , Pleinlaan 2, 1050 Brussels , Belgium
6 Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes , CNRS/IN2P3, 53 Avenue des Martyrs, 38026 Grenoble , France
We study the impact of dimension-six operators of the standard model effective field theory relevant for vector-boson fusion and associated Higgs boson production at the LHC. We present predictions at the next-to-leading order accuracy in QCD that include matching to parton showers and that rely on fully automated simulations. We show the importance of the subsequent reduction of the theoretical uncertainties in improving the possible discrimination between effective field theory and standard model results, and we demonstrate that the range of the Wilson coefficient values allowed by a global fit to LEP and LHC Run I data can be further constrained by LHC Run II future results.
1 Introduction
The LHC Run I and early Run II data have not yet put forward
any strong evidence of physics beyond the standard model
(SM) and limits on new states have instead been pushed to
higher and higher energies. As a consequence, the effective
field theory (EFT) extension of the SM (SMEFT) has become
increasingly relevant. The SMEFT is built from the SM
symmetries and degrees of freedom (including the Higgs sector)
by adding new operators of dimension higher than four to
the SM Lagrangian. Being a tool to parameterise the search
for new anomalous interactions, it is fully complementary
to direct searches for new particles. Interpreting data in the
context of the SMEFT hence allows us to be sensitive to new
physics beyond the current energy reach of the LHC in a
model-independent way.
The formulation of the effective Lagrangian restricted to
operators of dimension of at most six relies on the
definition of a complete and non-redundant operator basis [
1–
3
] and should additionally include the translations among
the possible choices [
4
]. This has been intensively discussed
and will be soon reported by the Higgs cross section
working group [
5
]. Moreover, since we try to observe small
deviations from the SM, precise theoretical predictions are
required both in the SM and in the SMEFT framework.
The accumulation of LHC data and the subsequent
precision obtained indeed call for a similar accuracy on the
theoretical side, which demands the inclusion of higher-order
corrections.
What we present in this paper is a part of the
current theoretical activities aiming for precision predictions
for electroweak Higgs-boson production at the LHC, i.e.
Higgs boson production in association with a weak boson
(VH) and via vector-boson fusion (VBF). In this
context, NLO+PS (next-to-leading order plus parton-shower)
matched predictions for VH and VBF production in the SM
have been released both in the MC@NLO [
6–8
] and
Powheg [
9–11
] frameworks and merged NLO samples
describing VH production including up to one additional jet have
been generated in both the Powheg [
12
] and Sherpa [
13
]
platforms. NLO QCD corrections along with the
inclusion of anomalous interactions have been further
investi
2 Theoretical framework
2.1 Model description
In the SM of particle physics, the elementary particles and
their interactions are described by a quantum field theory
based on the SU (3)C × SU (2)L × U (1)Y gauge symmetry.
The vector fields mediating the gauge interactions lie in the
adjoint representation of the relevant gauge group,
SU (3)C → G = (8, 1, 0),
SU (2)L → W = (1, 3, 0),
U (1)Y → B = (1, 1, 0),
gated for VH [
14
] and VBF [
15
] Higgs boson
production, and matched to parton showers in the Higgs
characterisation framework [
16,17
]. Finally, electroweak
corrections as well as anomalous coupling effects for VH
production have been included in the Hawk program [18]
that also contains NLO QCD contributions. In contrast,
fixed-order predictions are known to a higher accuracy
for both VH and VBF SM Higgs production processes
[
19–22
].
In the SMEFT framework (in contrast to the
anomalous coupling approach), the VH process has been studied
at the NLO+PS accuracy within the Powheg–Box
framework [
23
], where a subset of the 59 independent
dimensionsix operators was taken into account. In this paper,
similarly, we consider five operators which are relevant for VH
production. Firstly, we independently provide NLO+PS
predictions for the VH process by using a different framework
via a joint use of FeynRules [
24
], NloCT [
25
] and
MadGraph5_aMC@NLO (MG5_aMC) [
26
] programs. This
approach provides a fully automatic procedure linking the
model Lagrangian to event generation matched to parton
showers at NLO. Our work hence not only independently
validates the previous results obtained with Powheg- Box
but also includes the additional benefits stemming from the
flexibility of the FeynRules program. As a result, one can
exploit generators like MG5_aMC for simulating any desired
process at the NLO+PS accuracy (i.e. the VBF process in our
case) for which the same operators play a role. This is the
second part of our paper, which presents the first SMEFT
results for this process. Although we only present results
for a couple of benchmark scenarios motivated by global
fit results, our predictions can be straightforwardly
generalised to any scenario by using our public Universal
FeynRules Output (UFO) model [
27
] within MG5_aMC [
28–
30
].
We emphasise that, following some recent results in the
t t¯H channel [
31
], this work represents a step towards a
complete SMEFT operator basis implementation for Higgs
physics at the NLO QCD accuracy, which will be beneficial
to both the theoretical and experimental communities.
In Sect. 2 we provide the necessary theoretical
ingredients to calculate NLO-QCD corrections for VH and VBF
Higgs production in the SMEFT. We also discuss current
constraints on the Wilson coefficients originating from a LEP
and LHC Run I global fit analysis with which we inform
our benchmark point selection. In Sect. 3 we describe our
setup for NLO computations matched to parton showers.
We present our numerical results in Sects. 4 and 5, and also
assess the validity of the EFT given the current constraints
on the Wilson coefficients. We assess the future LHC reach
in Sect. 6, before concluding in Sect. 7. Practical information
for event simulation and model validations are provided in the
appendix.
(1)
(2)
(3)
(4)
where the notations for the representation refer to the full
SM symmetry group. The chiral content of the theory is
defined by three generations of left-handed and right-handed
quark (Q L , u R and dR ) and lepton (L L and eR ) fields
whose representation under the SM gauge group is given
by
Q L =
u R =
L L =
u L
dL
3, 1,
νL
L
=
=
3, 2,
2
3 , dR =
where the components of the doublet are given in terms of
the physical Higgs field h shifted by its vacuum expectation
value v and the Goldstone bosons G± and G0 that are eaten
by the weak bosons to give them their longitudinal degree of
freedom.
In the EFT framework, new physics is expected to appear
at a scale large enough so that the new degrees of
freedom can be integrated out. As a result, the SM Lagrangian
LSM is supplemented by higher-dimensional operators Oi
parameterising all effects beyond the SM,
L = LSM +
∞
n=1 i
c¯ni
n Oni .
Restricting ourselves to operators of dimension six, the most
general gauge-invariant Lagrangian L has been known for a
long time [
32–34
] and can be expressed in a suitable form by
choosing a convenient basis of independent operators Oi [
1–
3
]. In this work, we focus on five specific, bosonic
operators,1 which are relevant to the VH and VBF processes,
taken from the strongly interacting light Higgs (SILH) basis
[
2,36,37
],2
g 2
L = LSM + 4 2 c¯BB
i g
+ 2 2 c¯W
i g
c
+ 2 2 ¯B
i g
+
+
i g
2 c¯HW Dμ
2 c¯H B Dμ
†
Bμν Bμν
←→
†T2k D μ
Dν W k,μν
†←→Dμ
∂ν Bμν
†T2k Dν
W k,μν
† Dν
Bμν .
The Wilson coefficients c¯ are free parameters, T2k are the
generators of SU (2) (with Tr(T2k T2l ) = δkl /2) in the
fundamental representation and the Hermitian derivative operators
are defined by
†←→Dμ
←→
†T2k D μ
In our conventions, the gauge-covariant derivatives and the
gauge field strength tensors read
Wμkν = ∂μWνk − ∂ν Wμk + g i j k W μiWνj ,
Bμν = ∂μ Bν − ∂ν Bμ,
Dρ Wμkν = ∂ρ Wμkν + g i j k Wρi Wμjν ,
1
Dμ = ∂μ − i gT2k Wμk − 2 i g Bμ ,
where i j k are the structure constants of SU (2). In addition, g
and g denote the coupling constants of SU (2)L and U (1)Y ,
respectively.
After the breaking of the electroweak symmetry down
to electromagnetism, the weak and hypercharge gauge
eigenstates mix to the physical W -boson, Z -boson and the
photon A,
1
Wμ± = √ (Wμ1 ∓ i Wμ2),
2
Zμ
Aμ
cˆW −sˆW
= sˆW cˆW
Wμ3 .
Bμ
1 The relevant fermionic operators are also considered in, e.g., [
35
].
2 Although the W -boson mass mW and v are usually used as expansion
parameters in this basis, our model explicitly uses a cutoff scale . For
all our numerical results, we set = mW . We also point out a relative
factor 2 difference in our definition of OW and OHW with respect to
Refs. [
2,36,37
].
(5)
(6)
(7)
(8)
We have introduced in this expression the sine and cosine of
the Weinberg mixing angle sˆW ≡ sin θˆW and cˆW ≡ cos θˆW ,
which diagonalise the neutral electroweak gauge boson mass
matrix. The higher-dimensional operators of Eq. (5) induce
a modification of the gauge boson kinetic terms that become,
in the mass basis and after integration by parts,
1
Lkin = − 2 1 −
We have made use here of the freedom related to the removal
of the photon and Z -boson mixing terms induced by the
higher-order operators. This mixing can indeed be absorbed
either in a photon field redefinition, or in a Z -boson field
redefinition, or in both (as in Eq. (10)). In order to minimise
the modification of the weak interactions with respect to the
SM, we additionally redefine the weak and hypercharge
coupling constants
e
g → s
ˆW
1 − e82sˆvW22c¯W2 , g → cˆW
e
e2v2c¯BB .
1 − 4cˆW2 2
(11)
As a result of this choice, the relations between the
measured values for the electroweak input and all internal
electroweak parameters are simplified. The Z -boson mass m Z is
now given by
the Z -boson decay data. Those constraints can nonetheless
be modified if other dimension-six operators are added to the
Lagrangian of Eq. (5).
In unitary gauge and rotating all field to the mass basis, all
three-point interactions involving a single (physical) Higgs
boson and a pair of electroweak gauge bosons are given by
Lhvv = − 4 ghγ γ Aμν Aμν h − 41 gh(1zz) Zμν Z μν h
1
− gh(2zz) Zν ∂μ Z μν h + 21 gh(3zz) Zμ Z μh
− 21 gh(1zγ) Zμν Aμν h − gh(2zγ) Zν ∂μ Aμν h
− 21 gh(1w)w Wμ+ν W −μν h − gh(2w)w [Wν+∂μW −μν h + h.c.]
+ gh(3w)w Wμ+W −μh,
(16)
where integration by parts has been used to reduce the number
of independent Lorentz structures. Table 1 shows the relation
between the couplings in Eq. (16) and the Wilson coefficients
in Eq. (5). As a reference, we also compare our conventions
to those of the previous SILH Lagrangian implementation
of Ref. [
37
] and of the Higgs characterisation Lagrangian of
Ref. [
16
].
2.2 Constraints from global fits of LEP and LHC Run I data
In this section we summarise the current bounds on the
Wilson coefficients associated with the effective operators under
consideration.
We start from the results of previous works [
38,39
], where
a global fit to LEP and LHC Run I data has been performed.
The results imply constraints on several linear combinations
of the c¯ coefficients appearing in Eq. (5) that we present in
ev
m Z = 2sˆW cˆW
e2v2
1 + 8cˆW2 2 (cˆW2 c¯W + 2c¯B ) ,
while the photon stays massless and the expression of the
W -boson mass mW is unchanged respect to the SM one.
We define the electroweak sector of the theory in terms of
the Fermi coupling constant GF as extracted from the muon
decay data, the measured Z -boson mass m Z and the
electromagnetic coupling constant α in the low-energy limit of the
Compton scattering. The vacuum expectation value of the
Higgs field can therefore be derived from the Fermi constant
as in the SM, v2 = 1/(√2GF) . After the field redefinitions
of Eq. (10), the electromagnetic interactions of the fermions
to the photon field turn out to be solely modified by the OW
operator, so that the electromagnetic coupling constant e is
related to the input parameter α as
e =
√
4π α 1 +
π αv2c¯W .
2 2
Furthermore, the shift in the cosine of the Weinberg mixing
angle cos θˆW can be derived, at first order in 1/ 2, from the
Z -boson mass relation of Eq. (12) along with Eq. (13),
c2 2
ˆW = c˜W −
2π αs˜W2 v2
2
c˜2W
c˜W2 c¯W + c¯B ,
with c˜2W ≡ cos 2θ˜W , s˜W ≡ sin θ˜W and
c2 1
˜W ≡ cos2 θ˜W = 2
1 +
1 −
4π αv2
m2Z
.
As a consequence, the c¯W and c¯B parameters are
constrained by the measurement of the W -boson mass and by
Table 2, each limit having been obtained by marginalising
over all other coefficients. Leading-order (LO) theoretical
predictions have been used and in addition, the
modifications of the electroweak parameters computed in Eq. (13)
and Eq. (14) have not been considered for LHC predictions.
We have nevertheless checked that the corresponding effects
are small compared with the LHC Run I sensitivity, as also
noted by the ATLAS collaboration [
40
].
In many classes of SM extensions (featuring in
particular an extended Higgs sector), certain relations among the
coefficients appear. For instance, it is common that matching
conditions such that gh(2w)w ∝ c¯H W + c¯W = 0 appear [
41
]. In
this case, the global fit generates the more stringent constraint
c¯H W = −c¯W = [0.0008, 0.04] when one sets the effective
scale to = mW [
38
].
2.3 Benchmark points
For both production processes of interest, we consider two
benchmark scenarios in the Wilson coefficient parameter
space. These two points are selected to be compatible with
the global fit results discussed in Sect. 2.2.
We first make use of the fact that, as seen in Table 2,
electroweak precision observables strongly constrain a
particular linear combination of the c¯W and c¯B Wilson
coefficients beyond a precision than can be hoped for at the
LHC. We therefore impose c¯B = −c¯W /2, which in turn
leads to an allowed range (setting = mW ) for c¯W of
[−0.035, 0.005], as obtained from the second constraint on
these two parameters. In order to highlight the impact of the
two new Lorentz structures appearing in the interaction
vertices of the Lagrangian of Eq. (16), we allow for non-zero
values for both the c¯H W and c¯W coefficients.
Our benchmark scenarios are defined in Table 3. In the first
setup, we only switch on the OHW operator (which induces
new physics contributions to both the gh(1v)v and gh(2v)v
structures). With the second point, we additionally fix c¯W to an
equal and opposite value relying on the constraint relation
brought up in Sect. 2.2. This allows for turning on solely the
gh(1v)v coupling (see Table 1).
3 Setup for NLO+PS simulations
Our numerical results are derived at the NLO accuracy in
QCD thanks to a joint use of the FeynRules/NloCT and
MG5_aMC packages. The EFT Lagrangian of Eq. (5) has
been implemented in FeynRules [
24
], while the
computation of the ultraviolet counterterms and the rational R2
terms necessary for numerical loop-integral evaluation has
been done by NloCT [
25
] that relies on FeynArts [
42
].
The model information is then provided to MG5_aMC [
26
]
in the UFO format [
29
]. Within MG5_aMC, loop-diagram
contributions are numerically evaluated [
43
] and combined
with the real emission pieces within the FKS subtraction
scheme [
44,45
]. Short-distance events are finally matched to
parton showers according to the MC@NLO prescription [6].
We generate events for 13 TeV LHC collisions using the
LO and NLO NNPDF2.3 set of parton densities [
46
] for LO
and NLO simulations, respectively. Events are then showered
and hadronised within the Pythia8 infrastructure [
47
], which
is also used for handling Higgs-boson decays. This latter
step relies on eHdecay [
48
] that computes all branching
fractions of the Higgs boson into the relevant final states to
the first order in the Wilson coefficients. This procedure has
the advantage of providing a correct normalisation for the
production rates that includes all effects originating from the
EFT operators. For the two adopted benchmark points, the
deviations from the SM branching ratios are found to be very
small.
Event reconstruction and analysis are performed using the
MadAnalysis5 [
49
] framework, which makes use of all jet
algorithms implemented in the FastJet program [
50
]. Jets
are defined using the anti-kT algorithm [
51
] with a radius
parameter of 0.4.
Theoretical uncertainties due to renormalisation (μR ) and
factorisation (μF) scale variations are accounted for thanks
to the reweighting features of MG5_aMC [
52
]. At the event
generation stage, nine alternative weights are stored for each
event, corresponding to the independent variation of the two
scales by a factor of two up and down with respect to a
central scale μ0. Since the parton shower is unitary, this could
be used to reweight the events after showering and
reconstruction, saving a great deal of computational time and
storage. The scale variation uncertainty is taken to be the largest
difference between the central scale and the alternative scale
choice predictions. We use as a central scale μ0 = HT /2 and
m W for VH and VBF processes, respectively, where HT is
defined at the parton-level as the scalar sum of the transverse
momentum of all visible final-state particles and the missing
transverse energy. We refer the reader to the appendix for
further technical details on event generation.
4 Higgs production in association with a vector boson
Higgs-boson production in association with a vector boson is
an excellent probe for new physics, as the momentum
transfer in the process is directly sensitive to the Lorentz structure
appearing in the interaction vertices [
53
]. The use of
differential information at the LHC Run I has therefore enhanced the
sensitivity of Higgs data to possible new physics effects [
39
].
Those Run I studies have, however, relied on predictions
evaluated at the LO accuracy in QCD but, with the improved
capabilities of the LHC Run II, NLO QCD effects become
more relevant and more precise predictions are in order.
To showcase our NLO simulation setup for associated VH
production, we study various differential distributions in the
p p →
H W +
→ b b¯ + + E/
(17)
channel, where E/ stands for the final-state missing energy.
We impose the requirement that both b-jets and leptons have
a pseudorapidity, η, and a transverse momentum, pT ,
satisfying |η| < 2.5 and pT > 25 GeV, respectively, while
nonb-tagged jets are instead allowed to be more forward, with
|η| < 4, for the same pT requirement. We select events by
demanding the presence of one lepton and two b-jets based on
truth-level hadronic information, a b-tagged jet being defined
by the presence of a b-hadron within a cone of radius R = 0.4
centred on the jet direction.
In Fig. 1, we present the transverse momentum spectrum
of the bb¯ system (upper left), of the leading (upper centre)
and next-to-leading (upper right) b-jets, of the lepton
(middle left) and of the leading jet (middle centre). We then focus
on the distribution in pseudorapidity for the bb¯ system
(middle right), in the transverse mass of the W -boson and Higgs
boson (lower left) and of the W -boson, Higgs boson and
leading-jet system (lower centre) and in the total transverse
energy (lower right). In each subfigure, the results are shown
both at the LO+PS and NLO+PS accuracies, together with
uncertainties related to scale variation.
For each studied observable, we investigate in the first two
lower bands of each subfigure the relative difference between
the predictions in the SM and in the case of both considered
benchmark points A and B,
− 1 for
i = A, B.
(18)
i σi
δS M = σS M
In the last band, we additionally show differential K -factors
defined as the binned ratio of NLO to LO predictions taking
only the total NLO uncertainty into account.
The predictions are found to be stable under radiative
corrections, as expected for any process with a Drell–Yan-like
topology. The obtained K -factors are indeed relatively flat
and independent of the EFT parameters, with the exception
of the observables that rely on the leading-jet kinematics
which turn out to be much harder at NLO. Hard QCD
radiation contributions originating from the matrix element are
in this case included, in contrast to the LO setup where QCD
radiation is only described by the parton shower and thus
modelled in the soft-collinear kinematical limit.
LO predictions are found to be inaccurate and do not
overlap with the NLO results even after considering scale
variation uncertainties. This is particularly true at high
transverse momentum pT , transverse mass MT and total
transverse energy ET . This behaviour is once again expected for
a Drell–Yan-like process that does not depend on αS at fixed
LO. If one were to use the difference between the LO and
NLO results as an error estimate for the LO predictions and
the scale variation only for the NLO, then the reduction of
the theory error would be better reflected by Fig. 1. The
δiS M ratios also remain stable with respect to QCD
corrections, except at very high energies for the benchmark point
A where small differences appear between the LO and NLO
predictions. These would, by construction, be covered by the
aforementioned improved definition of the LO theoretical
uncertainties.
All distributions strongly depend on the value of the
EFT Wilson coefficients. For the adopted scenario A, large
enhancements are observed in the tails of the pT , MT and ET
distributions, which correspond to a centrally produced bb¯
system (with a small pseudorapidity). In contrast, event rates
are only rescaled by about 15–20% with respect to the SM
for the scenario B. This originates from the gh(2v)v coupling
that vanishes in this scenario, so that only the gh(1v)v coupling
drives the EFT behaviour in the high-energy tails. However,
this latter coupling is known to yield a smaller impact than
the gh(2v)v coupling [
17, 23
] and it is therefore the presence of
the gh(2v)v interaction vertex in scenario A that leads to the
large observed deviations. This constitutes a very
promising avenue for setting limits on EFT parameters from W H
studies and similar behaviour can be observed for Z H
production, where the c¯H B and c¯B B coefficients additionally
play a role. In this case, the gluon fusion initiated
contribution should, however, also be considered, as discussed in
Refs. [
23, 54
].
While such large enhancements can be exploited to obtain
powerful constraints on the SMEFT Wilson coefficients, they
do raise the question of the validity of the EFT approach at
large momentum transfer [
39, 55–57
]. This question could
400 500
MWHj [GeV ]
T
W+H: H→ bb¯, W → l+ν
LHC 13 TeV
W+H: H→ bb¯, W → l+ν
LHC 13 TeV
]V 10−1
e
G
/
b
[f |SM + EFT|2
dσ HdpT10−2 |c¯SHMW|=2+0.I0N3T, c¯W = c¯B = 0.
c¯HW = −c¯W = 0.03, c¯B = 0.015
120
δEFT(%) 60
0
Higgs pT
W+H: H→ bb¯, W → l+ν
LHC 13 TeV, LO+PS
Leading b-jet pT
W+H: H→ bb¯, W → l+ν
LHC 13 TeV, LO+PS
]V 10−1
e
G
/
b
f
dσ[1bpdT10−2
10−3
120
δEFT(%) 60
0
|SM + EFT|2
|SM|2 + INT
c¯HW = 0.03, c¯W = c¯B = 0.
c¯HW = −c¯W = 0.03, c¯B = 0.015
]V 10−1
e
G
/
b
dσ[fdpT10−2
10−3
120
δEFT(%) 60
0
Lepton pT
W+H: H→ bb¯, W → l+ν
LHC 13 TeV, LO+PS
|SM + EFT|2
|SM|2 + INT
c¯HW = 0.03, c¯W = c¯B = 0.
c¯HW = −c¯W = 0.03, c¯B = 0.015
50
100
be addressed with the use of dedicated benchmark models to
compare the breakdown of the EFT framework against
wellmotivated ultraviolet-complete models [
41, 58
]. At a more
simplistic level we can also make use of the MG5_aMC
ability to select only interference contributions (at LO) to
assess the impact of the squared EFT terms given our
benchmark choices (technical details are described in Appendix A).
Figure 2 shows a selection of distributions, overlaying
predictions with and without this squared term. We observe
significant differences between the two choices which are
greater than the scale uncertainty of the predictions.
Depending on the observable, these can range from 40 to 100% on
the interference-only prediction for the benchmark scenario
A, while they are much milder for the benchmark scenario B.
This suggests that current sensitivities on this region of the
Wilson coefficient parameter space may not yet lend
themselves to an EFT interpretation within the validity of the
framework. A reduction of the production rate from the SM
value, as seen for benchmark scenario B, moreover indicates
the dominance of the interference term between the SM and
EFT contributions given that the squared terms are
positivedefinite (Fig. 3).
5 Higgs production via vector boson fusion
Another powerful probe of anomalous higher-derivative
interactions between weak and Higgs bosons consists of the
VBF Higgs production mode where it is produced in
association with two forward jets,
p p → ( H
→ γ γ ) + j j.
(19)
Our event selection requires the presence of at least two jets
with a pseudorapidity |η| < 4.5 and a transverse momentum
pT > 25 GeV, and we additionally impose the requirement
that the Higgs boson decays into a pair of photons with a
pseudorapidity satisfying |η| < 2.5 and a transverse momentum
pT > 20 GeV. We moreover include a standard VBF
selection on the invariant mass M j j and pseudorapidity separation
η j j of the pair of forward jets,
M j j > 500 GeV
and
η j j > 3.
(20)
Several kinematical observables are sensitive to the
momentum flow in the VBF process, for which EFT contributions
deviate from the SM prediction. We consider in Fig. 4 the
distribution in the transverse momentum of the diphoton system
(upper left), in the pT of the leading (upper centre) and
subleading (upper right) jets, in the invariant mass of the dijet
system (lower left), as well as in its pseudorapidity (lower
centre) and azimuthal angular (lower right) separations. The
consistent definition of scale uncertainties that are possible
with the NLO predictions helps to quantify the
discriminatory power between the new physics benchmarks and the
SM. Similarly to the VH process, the NLO corrections are
independent of the EFT parameters and cannot be completely
described by an overall K -factor.
In contrast to the VH process, we observe a depletion
of the production rate for both benchmark scenarios, which
mainly impacts the high-energy tails of the differential
distributions. This indicates that our predictions may be safer
with respect to the validity of the EFT, as it implies that
the interference term dominates over the EFT squared one.
In particular, the effects for the benchmark scenario B are
more pronounced with respect to the SM compared to the
V H production case and show some different shape
deformations. This illustrates the complementarity between the
VH and VBF processes in disentangling the possible EFT
sources for any potential deviation. Although the
correlations between the forward jets as well as between the jets and
the Higgs boson are known to be sensitive to new physics
effects [
59, 60
], those are less sensitive than the
individual Higgs and jet pT distributions for our two benchmark
scenarios.
]
V
e
G
/
b
[f
VBF: H → γγ
LHC 13 TeV
100
pTj2 [GeV ]
Di-jet azimuthal difference
We also repeat the simple EFT validity analysis performed
for the V H case and assess the impact of the EFT squared
terms at LO, as shown in Fig. 4. As suggested by the
depletion effect of the EFT operators in the high energy bins of
the differential distributions, the squared terms appear much
more under control in this process compared to the V H case.
Within the ranges of our predictions the impact of the squared
term is again most pronounced for the benchmark A,
reaching at most 5–12% , while for benchmark B their effect is
much smaller.
6 Future LHC reach
Before concluding, we attempt to estimate the reach of the
LHC Run II with respect to the Wilson coefficients
considered in our benchmark scenarios. Our results so far suggest to
make use of the high energy tails of differential distributions
as handles for new physics. For concreteness, we focus on
the associated production process
p p → H W ± → b b¯ ± + E/ T
which has already been searched for at both LHC Run I [
61
]
and II [
62
]. In both analyses, a large number of signal and
control regions are defined according to the lepton and
additional jet multiplicities, as well as to the vector boson
transverse momentum pTV . These are combined in a global fit to
obtain the corresponding SM Higgs signal strength. In this
fitting procedure, several dominant components of the
background, namely t t¯ and W -boson production in association
with heavy-flavour jets, are left free to float. We consider
as signal regions the pTV overflow bin in the single lepton
channel for both the 0-jet and 1-jet categories. The Run II
study, however, makes use of multivariate methods in the
event selection process that are difficult to reproduce a task
that definitely lies beyond the scope of the simple estimate
we are intending to derive. We therefore choose to consider
only the cut-based signal selection procedure employed in
the Run I analysis and then project the results for various
Run II integrated luminosities.
6.1 Signal prediction and background estimation
In order to estimate the number of background events in the
single lepton signal regions of a possible cut-based, LHC
Run II analysis, we extrapolate the results of the
corresponding Run I analysis. We first consider the dominating t t¯
contribution which arises from semi-leptonic top-antitop decays
and which makes up 54 and 85% of the total background in
the 0-jet and 1-jet categories, respectively. As a crude
estimate for the corresponding 13 TeV yields, we compute a
transfer factor ftir (with i = 0, 1 for the 0-jet and 1-jet
categories) by generating large statistics of SM semi-leptonic t t¯
events at centre-of-mass energies of 8 and 13 TeV on which
we apply the kinematic selection of the Run I analysis
summarised in Appendix B. The transfer factor ftir is defined as
the ratio of the two fiducial cross sections and we deduce the
Run II analysis background contributions by multiplying the
i
8 TeV SM expectation σbkg inferred from the Run I
background event counts Nbikg assuming 25 fb−1 of 8 TeV data.
These should not depend much on the actual composition of
7 and 8 TeV data analysed, particularly in the high transverse
momentum overflow bin which is dominated by 8 TeV data.
Table 4 summarises the information obtained and used in
the subsequent analysis. Our theoretical predictions for the
t t¯ contributions to the 0- and 1-jet signal regions at 8 TeV,
σ overf., lie within a factor 2 of the cross-sections inferred from
8
the post-selection, fitted background decomposition
presented in Table 5 of Ref. [
61
]. Due to the multi-variate nature
of the recent Run II analysis, its fitted background yields
cannot be used to validate the results of our projection, which
rather represents the scenario in which a cut-based analysis
similar to the Run I counterpart were performed at 13 TeV.
The signal predictions have been generated using the
previously described UFO implementation, and both the t t¯ and
W H contributions have been simulated at the NLO accuracy
in QCD as described in previous sections, the fixed-order
results being matched with Pythia 8 for both handling the
top decays and the parton showering. A grid of points
spanning the allowed region of the (c¯HW , c¯W ) parameter space,
including the SM prediction, was simulated and the generated
events were passed through the same kinematic selection of
Appendix B. Following the previous discussion on the
existing constraints, we assume the simplification c¯W = −c¯B /2.
Such a relation would not be retained in a complete global fit
including, e.g., LEP data. However, it is instructive to follow
this simplified path as it assesses the sensitivity of the LHC
to the direction in the (c¯W , c¯B ) plane that is orthogonal, and
thus complementary, to the one tightly constrained by
precision measurements at the Z -pole. We have derived
leastsquares-fitted quadratic polynomial forms for the 0- and 1-jet
overflow bin cross sections in the two-dimensional
parameter plane σ Wi H (c¯HW , c¯W ). Our results, moreover, embed a
0-jet bin, pTW > 200 GeV
1-jet bin, pTW > 200 GeV
Combined, pTW > 200 GeV
Fig. 5 95% confidence intervals in the (c¯HW , c¯W ) plane depicting the
projected reach at the LHC Run II extracted from data in the pTW
overflow bin of the corresponding Run I analysis performed in Ref. [
61
].
We consider three different choices for the integrated luminosities, and
the dashed lines indicate the previously obtained marginalised limits on
the Wilson coefficients from the global fit of Refs. [
38,39
]
b-tagging efficiency of 70%, and more information (in
particular on the explicit coefficients of the fits) is given in
Appendix B.
6.2 Results
Our results have been derived from the fitted functional forms
for the signal cross sections in combination with the
projected background yields. We have performed a χ 2 analysis
to extract 95% confidence intervals assuming L = 30, 300
and 3000 fb−1 of integrated luminosities of 13 TeV proton–
proton collisions. Denoting by Bi and Si the event counts
in the signal region in the background-only and
signal-plusbackground hypotheses, respectively, we have
Bi = L( ftirσbikg + σ Wi H (0, 0));
Si = L( ftirσbikg + σ Wi H (c¯H W , c¯W ));
χ 2
=
i
i
( Bi − Si )2
Bi
,
L σ Wi H (0, 0) − σ Wi H (c¯H W , c¯W )
ftirσbikg + σ Wi H (0, 0)
2
.
The 95% confidence intervals are obtained at the boundary
of χ 2 = 5.99 which equates to the corresponding p-value
for a χ 2 distrbution with two degrees of freedom. Figure 5
depicts these confidence intervals for the 0- and 1-jet bin
separately as well as their combination, for the three integrated
luminosity points. For comparison, the marginalised single
parameter exclusion regions established in Table 2 and the
benchmark discussion of Sect. 2.3 are indicated.
This simplified projection shows that this type of analysis
is likely to substantially improve the existing limits on these
Wilson coefficients in combination with existing data. Since
(21)
the 1-jet category suffers both from a larger background and
smaller signal contribution, its relative impact on the
overall reach is small. The blind direction associated with this
measurement lies very close to the c¯H W = −c¯W line,
corresponding to the benchmark choice B of the earlier sections.
This is consistent with the very mild expected impact of this
particular new physics scenario in the high energy tails of the
differential distributions for the pp → W + H process (see
Sect. 4). Nevertheless, our results suggest that in the general
case, taking the full integrated luminosity of LHC Run II
will individually allow to constrain Wilson coefficients with
a precision of a few per-mille and the results presented in
Sect. 5 indicate that combining VBF and WH studies may
break this degeneracy.
7 Conclusions
We have presented FeynRules and UFO
implementations of dimension-six SMEFT operators affecting
electroweak Higgs-boson production, which can be used for
NLO(QCD)+PS accurate Monte Carlo event generation
within the MG5_aMC framework. We have considered five
SILH basis operators and have accounted for all field
redefinitions that are necessary to canonically normalise the theory.
Moreover, the ensuing modifications of both the gauge
couplings and the relationships between the electroweak input
and the derived parameters have also been included. We
have showcased the strength of our approach by
simulating both associated VH and VBF Higgs-boson production
at the 13 TeV LHC, selecting a pair of benchmark scenarios
informed both by recent limits from global fits to the LEP and
LHC Run I data and by theoretical motivations originating
from integrating out certain popular ultraviolet realisations.
We have found that EFT predictions and deviations from
the SM are stable under higher-order corrections. Overall, we
have also observed a significant reduction of the theoretical
errors, which would have an impact on the future
measurements aiming to unravel dimension-six operator
contributions.
Furthermore, as a test for the validity of the EFT approach,
we have proposed to compare distributions that either include
the full matrix element (embedding all SM and new physics
contributions) or account solely for the interference of the
SM component with the new physics component. For our
benchmark choices that saturate current experimental
limits, the differences were observed to be large in the
kinematic extremes of some of our distributions, particularly for
V H production. This points to the possibility that the EFT
description is breaking down in these regions of the
parameter space and that the most precise measurements undertaken
at the LHC Run II may be required to probe the EFT (while
staying in its region of validity).
When comparing results for the VH and VBF channels,
we have found that both Higgs-boson production modes are
sensitive to new physics, but the VH one seems to have a
better handle on g(1)-type (Vμν V μν h) structure, since several
key distributions display deviations that may be more
easily distinguished from the background. Although the QCD
K -factors have been observed not to depend on the EFT
parameters, the reduced theoretical uncertainties are crucial
for disentangling a non-vanishing contribution of the
g(1)type structure to the predictions from the SM. This has been
singled out in our study of the benchmark point B.
Moreover, our results exhibit an interesting complementarity of
the two Higgs production channels, since the interference
pattern between the SM and the SMEFT contributions is quite
different and benchmark-dependent.
In order to estimate the reach that might be possible at
LHC Run II, we have performed a simplified analysis
projecting the Run I SM background expectations in a search
for W H associated production and combining this
information with LHC Run II signal predictions obtained using our
implementation. Using the overflow bin of the reconstructed
W -boson transverse momentum distribution in the single
lepton channel as a probe for EFT effects suggests that the LHC
Run II will significantly improve the current limits obtained
from global fits. Clearly both the V H and VBF processes
deserve further investigation including detector effects and
an analysis strategy to reject the SM backgrounds. In this
case, the new physics contributions to the SM background
processes should also correctly be accounted for, since
effective operators affecting electroweak Higgs-boson production
also impact the normalisation of the triple gauge-boson
interactions both directly and indirectly via the aforementioned
field redefinitions [
23,38,41,63,64
].
Finally, our work has demonstrated a proof-of-concept
for automated NLO+PS simulations in the SMEFT
framework. To this aim, we have limited ourselves to a small set of
dimension-six operators and a pair of benchmark points. This
is characterised as a first step towards a complete operator
basis implementation, with which the renormalisation group
running of the Wilson coefficients [
31,65–69
] could also be
supplemented in the future.
Acknowledgements We would like to express a special thanks to Fabio
Maltoni for useful discussions. We moreover acknowledge the
organisers of the 2015 ‘Les Houches–Physics at TeV colliders’ workshop
and the Mainz Institute for Theoretical Physics for their hospitality and
support during the completion of this work. CD is a Durham
International Junior Research Fellow. BF and KMa have been supported by the
Theory-LHC-France initiative of the CNRS (INP/IN2P3) and KMi and
VS by the Science and Technology Facilities Council (Grant number
ST/J000477/1).
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
A Simulation in MadGraph5_aMC@NLO
A.1 Technical details
Our HEL@NLO UFO model can be downloaded from the
FeynRules model database [
27
]. It can be used for
generating events at the NLO accuracy in QCD using software such
as MG5_aMC via to the automated procedure detailed in
Sect. 3. Event generation for W + H production is achieved
by typing in the MG5_aMC interpreter
import model HELatNLO
generate p p > h ve e+ [QCD]
output
launch
Since the usual decay syntax of MG5_aMC is not available
for NLO event generation, we directly request the presence of
the W -boson decay products in the final state. An alternative
way would require one to simulate the production of a Higgs
boson in association with an on-shell W -boson that is
subsequently decayed within the MadSpin infrastructure [
70,71
]
before invoking the parton showering. On the other hand,
VBF Higgs-boson production is achieved by typing in the
generation command
generate p p > h j j $$ w+ w- z a
QCD=0 [QCD]
Higgs pT
W+H: H→ bb¯, W → l+ν
LHC 13 TeV, NLO+PS
Lepton pT
W+H: H→ bb¯, W → l+ν
LHC 13 TeV, NLO+PS
]
eV 10−1
G
/
b
[f
dσHdpT
10−2
20
δ(%) 100
-10
-20
MG5 aMC
POWHEG/MCFM
c¯HW = 0.03, c¯W = c¯B = 0.
c¯HW = −c¯W = 0.03, c¯B = 0.015
]V 10−1
e
G
/
b
dσ[fdpT10−2 MG5 aMC
POWHEG/MCFM
c¯HW = 0.03, c¯W = c¯B = 0.
c¯HW = −c¯W = 0.03, c¯B = 0.015
Leading b-jet pT
W+H: H→ bb¯, W → l+ν
LHC 13 TeV, NLO+PS
MG5 aMC
POWHEG/MCFM
c¯HW = 0.03, c¯W = c¯B = 0.
c¯HW = −c¯W = 0.03, c¯B = 0.015
50
100
The removal of the s-channel gauge boson contributions (via
the $$ syntax) avoids the generation of VH topologies where
the gauge boson decays hadronically.
In the parameter card, users may set values for the five
Wilson coefficients defined in Eq. (5) as well as for the cutoff
scale . The modifications to the electroweak parameters in
terms of the input in the (GF, m Z , α) scheme are taken into
account, as well as the shifts induced by the field redefinitions
discussed in Sect. 2.
At the FeynRules level, the final Lagrangian
involving all of the redefined fields and parameters is expanded
up to O(1/ 2). From the point at which this truncation
occurs, all subsequent performed calculations will
necessarily induce some O(1/ n) (with n > 2) dependence from,
e.g., higher powers in the electroweak couplings.
Furthermore, MG5_aMC constructs its matrix elements by
squaring helicity amplitudes, so that the squared EFT contribution
terms that are formally of O(1/ 4) are by default retained
on top of the leading interference terms with the SM of
O(1/ 2). A positive definite cross section is ensured at the
price of including higher-order terms. It is therefore
important to remain in the regime where the EFT expansion can be
trusted in that higher-order contributions are sufficiently
suppressed. Departure from this safe zone may be reflected by
a rapid growth of amplitudes with energy leading to extreme
deviations in the tails of the distributions, as well as by
discrepancies between independent Monte Carlo setups
performing their truncation in different ways. Alternatively, this
issue could be investigated thanks to some recent
developments in MG5_aMC that allow the user to specify an order
for the squared matrix element calculation, like QED∧2 or
QCD∧2. However, this feature is currently only available for
LO computations. For including the EFT effects in the
simulation, the coupling order parameter NP can hence be set
either to NP=1 to retain the full amplitude squared or to
NP∧2 <= 1 to throw away the aforementioned EFT squared
terms, e.g.
generatepp > hvee + NP2 <= 1
Finally, due to the details of the implementation, we advise
users seeking to recover the SM limit to avoid setting the
Wilson coefficients to zero. It is preferable to set them either
to very small non-zero values or to use restrictions. The cutoff
scale parameter NPl may also be fixed alternatively to a very
large value.
A.2 Comparison with the MCFM/POWHEG-BOX
implementation
We have verified that our results in the W H channel are
compatible with those stemming from an alternative existing
implementation, which is based on MCFM [
72–74
] and the
Powheg- Box framework [
75,76
] and that has been
introduced in Ref. [23]. We have scrutinised several differential
distributions for both our benchmark points in the Powheg
and MG5_aMC framework, as presented in Fig. 6. A good
consistency has been found up to statistical uncertainties and
despite the difference in the dynamical scale choice which is
taken as HT /2 in MG5_aMC and the invariant mass of the
Higgs-vector boson system in the Powheg- Box
implementation.
B Kinematic selection of ATLAS-CONF-2013-079 and signal fit
We summarise here the kinematic selection through which
the t t¯ and signal W H events have been passed in order to
determine the background transfer factor and signal
efficiencies of the analysis performed in Ref. [
61
]. The signal region
used is the pW > 200 GeV overflow bin in the 0- and 1-jet
T
categories of the single-lepton channel. The kinematic
selection for this channel applied to our event samples after parton
shower is as follows:
• We require the final state to contain exactly one lepton
with |η| < 2.47 and ET > 25 GeV.
• Jets are reconstructed by means of the anti-kT jet
algorithm with a radius parameter R = 0.4, and we discard
the jet candidates for which the conditions |η| < 4.5 and
pT > 20 GeV are not satisfied.
• We require exactly two b-jets with a pseudorapidity
|η| < 2.5, the hardest one being further imposed to have
a transverse momentum pT > 45 GeV.
Not more than one additional jet in the |η| < 2.5 region
is allowed and any event with a jet with pT > 30 GeV in
the |η| > 2.5 region is also rejected. Events are then split
into the 0- and 1-jet categories based on the presence of an
extra, non-forward jet softer than the two b-jets. The
vector boson transverse momentum pW is defined as the
vecT
tor sum of the transverse momentum of the lepton and the
missing transverse energy. Additionally, in the pTW > 200
GeV overflow bin, a cut of 50 GeV on the missing energy
is imposed as well as a cut on the distance between the two
b-jets, Rbb ≡ ηb2b + φb2b < 1.4. A flat b-tagging
efficiency of 70% is additionally assumed.
34 signal samples have been simulated with the Wilson
coefficients being taken in the ranges −0.02 < c¯H W < 0.03
and −0.03 < c¯W < 0.01, the total rates being rescaled by
the Higgs branching fraction to bb¯ obtained with
eHDECAY. These events have then been passed through the above
selection in order to determine the functional forms for the
0- and 1-jet signal cross sections in pTW overflow bin, σ 0 and
σ 1. A least-squares fit yields
σW H = 1.135(1 + 71 c¯H W + 79.5 c¯W + 228 c¯2H W
0
+ 253 c¯W2 + 4711 c¯H W c¯H W ) fb,
σW H = 0.607(0.9 + 66.8 c¯H W + 74.6 c¯W + 232 c¯2H W
1
+ 243 c¯W2 + 4621 c¯H W c¯H W ) fb.
(22)
(23)
The fit coefficients are within 5–10% of one another, which is
to be expected given that our signal process receives
contributions from an electroweak vertex and should not be directly
sensitive to additional QCD radiation.
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