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JHEP07(2017)081

Published for SISSA by Springer

Received: April 12, 2017

Revised: June 5, 2017

Accepted: June 20, 2017

Published: July 17, 2017

NLO electroweak corrections in general scalar singlet

models

Raul Costa,

a,b

Marco O.P. Sampaio

and Rui Santos

c,d,e

CERN,

Geneva, Switzerland

Departamento de F´ısica da Universidade de Aveiro

and CIDMA (Center for Research & Development in Mathematics and Applications),

Campus de Santiago, 3810-183 Aveiro, Portugal

Instituto Superior de Engenharia de Lisboa — ISEL,

1959-007 Lisboa, Portugal

Centro de F´ısica Te´orica e Computacional, Universidade de Lisboa,

1649-003 Lisboa, Portugal

LIP, Departamento de F´ısica, Universidade do Minho,

4710-057 Braga, Portugal

E-mail: raul.costa@cern.ch, msampaio@ua.pt, rasantos@fc.ul.pt

Abstract: If no new physics signals are found, in the coming years, at the Large Hadron

Collider Run-2, an increase in precision of the Higgs couplings measurements will shift the

discussion to the eﬀects of higher order corrections. In Beyond the Standard Model (BSM)

theories this may become the only tool to probe new physics. Extensions of the Standard

Model (SM) with several scalar singlets may address several of its problems, namely to

explain dark matter, the matter-antimatter asymmetry, or to improve the stability of the

SM up to the Planck scale. In this work we propose a general framework to calculate one

loop-corrections to the propagators and to the scalar ﬁeld vacuum expectation values of

BSM models with an arbitrary number of scalar singlets. We then apply our method to

a real and to a complex scalar singlet models. We assess the importance of the one-loop

radiative corrections ﬁrst by computing them for a tree level mixing sum constraint, and

then for the main Higgs production process gg → H. We conclude that, for the currently

allowed parameter space of these models, the corrections can be at most a few percent.

Notably, a non-zero correction can survive when dark matter is present, in the SM-like

limit of the Higgs couplings to other SM particles.

Keywords: Beyond Standard Model, Higgs Physics, Spontaneous Symmetry Breaking

ArXiv ePrint: 1704.02327

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP07(2017)081

JHEP07(2017)081

radiative corrections. In fact, if we are faced with a situation where no direct hint of new

physics is found, manifestations of BSM physics can only appear through deviations in the

measured Higgs couplings.

In this work we will focus on extensions of the SM where an arbitrary number of singlets

is added to the SM ﬁeld content. These are the simplest extensions of the scalar sector

that

introduce a dark matter candidate [2–17]. These models also allow for a strong ﬁrst-order

phase transition during the era of Electroweak Symmetry Breaking (EWSB) if the extension

comprises at least two singlets [18–22]. Hence, at least two of the outstanding problems of

the SM can be solved within the framework of these models, namely a candidate for dark

matter and a solution to the matter antimatter asymmetry via electroweak baryogenesis. It

should be noted that extensions with only one dark scalar singlet are basically excluded by

the latest LUX results [23] when combined with the requirement that the dark matter relic

density of the model matches the one obtained from the Cosmic Microwave background

data. We have veriﬁed that this is not the case when at least two new singlets are added

to the SM, one of them being dark and the other mixing with the SM-like Higgs.

If the LHC indeed does not ﬁnd strong signs of new physics, such as new particle

states, the scale for such new physics may be as large as the GUT or the Planck scales.

This energy is unattainable by any current or planned collider experiments so we may have

to work in a framework that is a good description of the fundamental interactions up to

some high energy scale. Thus, any eﬀective description that improves theoretical problems

of the SM is an interesting candidate. In a previous work we have shown that the complex

singlet extension of the SM also improves the stability of the SM. In fact, the presence of

a heavier scalar state, which has to be heavier than about 140 GeV, can stabilise the SM

up to the Planck scale [24].

In this article we focus on the issue of determining electroweak (EW) radiative correc-

tions in general scalar SM extensions, with emphasis on the scalar singlet models frame-

work. Our main goal is to ﬁnd a general set of expressions that allow us to obtain next to

leading order (NLO) electroweak corrections to the parameters of a model with any number

of scalar singlet ﬁelds. We will go beyond the eﬀective potential approach recently studied

in [25], which is valid only when the new degrees of freedom are heavy. Thus, though we

formulate our results to connect to that limit, they are valid for any external momentum

scale (contrarily to the eﬀective potential approximation which is valid for small external

momenta). In our framework we obtain a set of conditions consistently truncated in an

expansion in powers of ~ which, once a number of consistent independent input parame-

ters are chosen, deliver the NLO EW corrections for the remaining parameters. A special

attention is given to the treatment of tadpoles and propagators and we provide a generic

strategy to easily transform between diﬀerent schemes. In connection with the eﬀective

potential approximation we also discuss, on general grounds, the issue of infrared diver-

gences. We then apply our method to the real scalar singlet extension (RxSM) and to the

complex scalar singlet extension (CxSM) of the SM. However, we note that the method

is ready to be applied to SM extensions with an arbitrary number of singlet ﬁelds and

For a recent review on scalar extensions of the SM see [1].

– 2 –

JHEP07(2017)081

that many of our formulas are also useful for other scalar extensions of the SM. In par-

ticular, our approach is especially suited for the automation of the computation of higher

order corrections in general purpose numerical tools to scan the parameter space of scalar

extensions of the SM [26, 27].

Higher order corrections to real singlet extensions of the SM have been performed

in [28–30]. The corrections to the SM-like Higgs coupling to fermions and gauge bosons

was shown to be of the order of 1% [28]. Furthermore the corrections were maximal in the

decoupling limit where the model becomes indistinguishable from the SM. Electroweak

corrections to the decay H → hh were performed in [29]. With the main theoretical

and experimental constraints taken into account, it was shown that corrections to the

triple scalar vertex (Hhh) are of the order of a few percent. In [30] NLO corrections to the

electroweak precision parameter ∆r were computed and confronted with the W-boson mass

measurement. Calculations of higher order corrections in the complex singlet extension of

the SM are still not available. With this work we will not only present a set of equations

to renormalise the parameters of the theory at one loop but we will then also use them

to calculate the electroweak corrections to Higgs production via gluon fusion. This last

calculation is performed near the decoupling limit with the main purpose to understand

the contributions of the triple scalar couplings of the various scalars running in the loops

at NLO. Clearly, with all the SM-like Higgs coupling close to the SM ones, the only large

eﬀects in the radiative corrections would have to come from such scalar-scalar interactions.

The numerical analysis in our examples will be performed for three particular cases: the

broken RxSM, with a new Higgs boson mixing with the SM-like one, and the broken and

symmetric CxSM with, respectively, three mixing Higgs bosons, and two mixing Higgs

bosons and a dark matter scalar. We will ﬁnd that, consistently with earlier calculations

for the NLO corrections to the decays, the corrections are very small, of the order of a few

percent, also for production. Nevertheless, we will ﬁnd that the presence of a dark matter

particle can enhance the corrections, even very close to the SM-like limit, compared with

the other models (though still in the few percent order).

The smallness of the electroweak corrections in the real singlet models calls for prudence

in the claims of measurable diﬀerences relative to SM Higgs couplings. The interference

eﬀects for this kind of BSM scenarios was ﬁrst addressed in [31] for the real singlet model,

showing that interference eﬀects to gg → h

∗

, H

(∗)

→ ZZ → 4l can be important away

from the non-SM scalar (H) peak region. Although the interference eﬀects can be of up to

order O(1) for the integrated cross sections for the 8 TeV LHC [32], judicious kinematical

cuts can reduce the interference eﬀects to O(10%). Interference eﬀects at NLO QCD were

discussed in [33] for the process gg → h

∗

, H

(∗)

→ hh. It was shown that the double Higgs

invariant mass can increase by up to 20% or decrease by up to 30% depending of the heavier

Higgs mass. More importantly, interference eﬀects can signiﬁcantly distort the kinematic

distribution around the resonant peak of the heavy Higgs. Recently the eﬀects of higher

order operators in the real singlet model [34] again showed that large cancellations can

occur due to interference eﬀects between the two sectors. In conclusion, if a signiﬁcant

deviation is found in Higgs couplings, the radiative corrections have to be combined with

the interference for a proper interpretation of the results.

– 3 –

JHEP07(2017)081

The structure of the paper is as follows. In the ﬁrst two sections we start by deﬁning

our strategy. We present the Lagrangians and ﬁx the notation in section 2 and then, in

section 3, we obtain our master linear system that, given a choice of input parameters,

provides as output the remaining renormalised parameters at NLO EW. The issue of

infrared divergences in connection with the eﬀective potential approximation is discussed

in sections 3.1 and 3.2. In section 4 we apply the procedure ﬁrst to a general class of scalar

singlet extensions of the SM and then specialise to the RxSM and to the CxSM, section 4.4,

for which we provide a numerical analysis in section 4.5. Our conclusions are summed up

in section 5 and several useful formulae/derivations are provided in the appendices.

2 Deﬁnitions and notation

To deﬁne a general four dimensional gauged Quantum Field Theory (QFT) Lagrangian

we use the notation of [35] with a few adaptations [24, 25]. We assume a decomposition

of a general renormalisable Lagrangian where the gauge basis ﬁelds are such that: i) all

scalar ﬁeld multiplets are decomposed as N

canonically normalised real scalar ﬁelds, Φ

(i = 1, . . . , N

), ii) all fermion multiplets are decomposed as a set of N

1/2

two-component

Weyl fermions, Ψ

(I = 1, . . . , N

1/2

) and iii) there are N

gauge bosons in the adjoint

representation of the gauge group, i.e. A

(a = 1, . . . , N

). We adopt the Einstein con-

vention where repeated indices which are one up (superscript) and one down (subscript)

are summed over. If the repeated indices are both down or both up they are not summed

over. All (non-spacetime) latin indices are assumed to be in Euclidean space — they are

lowered and raised with the identity matrix. The gauge basis interaction Lagrangian (i.e.

suppressing kinetic terms) is then composed of the following terms:

−L

= L

ijk

ijkl

−L

IJk

+ c.c. (2.1)

−L

abij

aµ

+ G

aij

aµ

∂

−L

F G

= −G

a J

aµ

†I

¯σ

−L

= −G

abc

aµ

bν

∂

abe

cµ

dν

− G

abc

aµ

∂

¯ω

where the ghost ﬁelds are represented by ω

and c.c. denotes complex conjugation. We call

this the L-basis following the nomenclature in [24, 25], where the pure scalar, fermionic

and gauge interaction coupling tensors are denoted, respectively by {L

...

, Y

...

, G

...

} with

. . . replaced by suitable sets of indices. Note that for a simple gauge group G

abc

is given

by G

abc

= g f

abc

with g the gauge coupling constant and f

abc

the structure constants

of the gauge group. For a direct product group we can still encode all information in

abc

by requiring a block structure. This can be represented using sub-ranges for the

indices a

= 1, . . . , n

, a

= n

+ 1, . . . , n

, etc. . . if components that have indices not

all in the same sub-range are zero. More concretely we would have G

= g

, etc. and, for example, G

= 0.

– 4 –

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