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Eur. Phys. J. C (2017) 77:176

DOI 10.1140/epjc/s10052-017-4745-5

Regular Article - Theoretical Physics

Higgs EFT for 2HDM and beyond

Hermès Bélusca-Maïto

1,a

, Adam Falkowski

1,b

, Duarte Fontes

2,c

, Jorge C. Romão

2,d

, João P. Silva

2,e

Laboratoire de Physique Théorique, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France

CFTP, Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049 Lisbon, Portugal

Received: 16 December 2016 / Accepted: 9 March 2017 / Published online: 21 March 2017

Abstract We discuss the validity of the Standard Model

Effective Field Theory (SM EFT) as the low-energy effective

theory for the two-Higgs-doublet Model (2HDM). Using the

up-to-date Higgs signal strength measurements at the LHC,

one can obtain a likelihood function for the Wilson coef-

ﬁcients of dimension-6 operators in the EFT Lagrangian.

Given the matching between the 2HDM and the EFT, the

constraints on the Wilson coefﬁcients can be translated into

constraints on the parameters of the 2HDM Lagrangian. We

discuss under which conditions such a procedure correctly

reproduces the true limits on the 2HDM. Finally, we employ

the SM EFT to identify the pattern of the Higgs boson cou-

plings that are needed to improve the ﬁt to the current Higgs

data. To this end, one needs, simultaneously, to increase the

top Yukawa coupling, decrease the bottom Yukawa coupling,

and induce a new contact interaction of the Higgs boson with

gluons. We comment on how these modiﬁcations can be real-

ized in the 2HDM extended by new colored particles.

1 Introduction

Effective ﬁeld theories (EFTs) allow one to describe the low-

energy dynamics of a wide class of quantum theories [1–3].

The idea is to keep only the subset of light degrees of free-

dom, while discarding the heavy ones that cannot be pro-

duced on-shell in the relevant experimental setting. Virtual

effects of the heavy particles on low-energy observables are

represented by an inﬁnite series of operators constructed out

of the light ﬁelds.

In the context of the LHC experiments, the light degrees of

freedom are those of the Standard Model (SM), and the heavy

e-mail: hermes.belusca@th.u-psud.fr

e-mail: afalkows017@gmail.com

e-mail: duartefontes@ist.utl.pt

e-mail: jorge.romao@tecnico.ulisboa.pt

e-mail: jpsilva@cftp.ist.utl.pt

ones correspond to hypothetical new particles. The low-

energy effective description of such a framework is called

the SM EFT; see e.g. [4–9] for reviews. The SM EFT allows

for a uniﬁed description of many possible signals of physics

beyond the SM (BSM), assuming the new particles are too

heavy to be directly produced. This model-independence is a

great asset, given we currently have little clue about the more

complete theory underlying the SM. Another strength of this

approach is that constraints on the EFT parameters can eas-

ily be translated into constraints on masses and couplings in

speciﬁc BSM constructions. Thus, once experimental results

are interpreted in the EFT language, there is no need to re-

interpret them in the context of every possible model out

there.

A less appealing feature of EFTs is that the Lagrangian

contains an inﬁnite number of interaction terms and param-

eters, in contrast to renormalizable theories. In the SM EFT,

these terms are organized in an expansion

eff

= L



(6)



(6)



(8)



(8)

+···, (1.1)

where L

is the SM Lagrangian,  is the mass scale of BSM

physics, each O

(D)

is an SU(3) × SU(2) × U(1) invariant

operator of canonical dimension D, and the parameters c

(D)

are called the Wilson coefﬁcients. Terms with odd D are

absent assuming baryon and lepton number conservation.

In practice, the series in Eq. (1.1) must be truncated, such

that one works with a ﬁnite set of parameters. In most appli-

cations of the SM EFT, terms with D ≥ 8 are neglected.

This corresponds to taking into account the BSM effects

that scale as O(m

/

), and neglecting those suppressed by

higher powers of . It is important to discuss the validity of

such a procedure for a given experimental setting [10]. More

precisely, the questions are: (1) whether the truncated EFT

gives a faithful description of the low-energy phenomenol-

ogy of the underlying BSM model, and (2) to what extent

experimental constraints on the D = 6 Wilson coefﬁcients

123

176 Page 2 of 14 Eur. Phys. J. C (2017) 77 :176

are affected by the neglected higher-dimensional operators.

Generically, in the context of LHC Higgs studies the trun-

cation is justiﬁed if  is much larger than the electroweak

scale. But, to address the validity issue more quantitatively

and identify exceptional situations, it is useful to turn to con-

crete models and compare the description on physical observ-

ables in the full BSM theory with that in the corresponding

low-energy EFT. Such an exercise provides valuable lessons

about the validity range and limitations of the SM EFT.

In this paper we perform that exercise for the Z

symmetric CP-conserving two-Higgs-doublet model

(2HDM). We compare the performance of the full model and

its low-energy EFT truncated at D = 6 to describe the Higgs

signal strength measurements at the LHC. To this end, we

ﬁrst update the tree-level constraints on the 2HDM parame-

ter space using the latest Higgs data from Run-1 and Run-2 of

the LHC. We use the same data to derive leading-order con-

straints on the parameters of the SM EFT. Given the match-

ing between the EFT and the 2HDM parameters [11–13], the

EFT constraints can be subsequently recast as constraints on

the parameter space of the 2HDM. By comparing the direct

and the EFT approaches, we identify the validity range of the

EFT framework where it provides an adequate description of

the impact of 2HDM particles on the LHC Higgs data.

We also remark that neither the SM nor the 2HDM pro-

vides a very good ﬁt to the Higgs data, mostly due to some ten-

sion with the measured rate of the t

th production and h → b

decays. If the current experimental hints of an enhanced t

and suppressed h → b

b are conﬁrmed by the future LHC

data, the 2HDM alone will not be enough to explain these.

Here the EFT approach proves to be very useful in suggest-

ing extensions of the 2HDM that better ﬁt the current Higgs

data. In particular, we show that a good ﬁt requires simulta-

neous modiﬁcations of the EFT parameters controlling the

top and bottom Yukawa couplings and the contact interac-

tion of the Higgs boson with gluons. We show how these

modiﬁcations can be realized in the 2HDM extended by new

colored particles coupled to the Higgs.

This paper is organized as follows. In Sect. 2 we review

the 2HDM and its low-energy EFT. In Sect. 3 we compare

the direct and the EFT constraints on the parameter space

imposed by the Higgs measurements. In Sect. 4 we discuss

how to improve the ﬁt to the LHC Higgs data by extending

the 2HDM with new colored states coupled to the Higgs.

2 Formalism

2.1 CP-conserving 2HDM

We start by reviewing the (non-supersymmetric) 2HDM

[14–16], closely following the formalism and notation of

Ref. [17]. We consider two Higgs doublets 

and 

, both

transforming as (1, 2)

1/2

under the SM gauge group. Both

doublets may develop a vacuum expectation value (VEV)

parametrized as 

=

√

, with v

= v cos β ≡ vc

= v sin β ≡ vs

, and v = 246.2 GeV. We assume that all

parameters in the scalar potential are real, which implies the

Higgs sector preserves the CP symmetry at the leading order.

Furthermore, we assume that the Lagrangian is invari-

ant under a discrete Z

symmetry, under which the doublets

transform as 

→+

and 

→−

. This symmetry is

allowed to be broken only softly, that is to say, only by mass

parameters in the Lagrangian. The Z

symmetry constrains

the possible form of Yukawa interactions. There are four pos-

sible classes of 2HDM, depending on how the SM fermions

transform under the Z

symmetry. They are summarized in

the following table:

Type-I Type-II -speciﬁc

(Type-X)

Flipped

(Type-Y)

Up-type 



Down-type 



Leptons 



It is often more convenient to work with linear combina-

tions of 

and 

deﬁned by the rotation







−s







. (2.1)

It follows that H

=

√

, H

=0. Note that H

and H

unlike 

, are not eigenstates of the Z

symmetry. The linear

combinations H

deﬁne the so-called Higgs basis [18], while

the original doublet 

are referred to as the Z

basis.

In the Higgs basis, the scalar potential takes the form

V (H

, H

) = Y

+ Y

+ (Y

†

+ h.c.) +

+ Z

†

)(H

†

)



†

)

+ (Z

)(H

†

) + h.c.



, (2.2)

where the parameters Y

and Z

are all real. The Z

symmetry

is manifested by the fact that only 5 of the Z

are independent,

as they satisfy 2 relations:

− Z

1 − 2s

+ Z

345

− Z

1 − 2s

−

1 − 2s

− Z

(2.3)

where Z

345

≡ Z

+ Z

. The Yukawa couplings are

given by

123

Eur. Phys. J. C (2017) 77 :176 Page 3 of 14 176

Yukawa

=−

†

− H

†

− H

†



−

tan β

†

−

tan β

†

−

tan β

†



+ h.c., (2.4)

where

= iσ

∗

, and the coefﬁcients of the H

Yukawa

couplings are summarized in the table below:

Type-I Type-II Type-X Type-Y

11 1 1

1 −tan

β 1 −tan

1 −tan

β −tan

β 1

In the Higgs basis, the doublets can be parametrized as



−ıG

√

(v + s

β−α

h + c

β−α

+ ıG

)





√

β−α

h − s

β−α

+ ıA)



(2.5)

where G

and G

are the Goldstone bosons eaten by W

and Z, while H

and A are the charged scalar and neutral

pseudo-scalar eigenstates. The two neutral scalars h, H

are

mass eigenstates, while the parameter c

β−α

≡ cos(β − α)

determines their embedding in the two doublets H

In the

following we will identify h with the 125 GeV Higgs boson.

The equations of motion for H

and H

imply the vacuum

relations

=−

, Y

=−

. (2.6)

The masses of the charged scalar and the pseudo-scalar are

given by

= Y

, m

= Y

+ Z

− Z

. (2.7)

The mixing angle is related to the parameters of the potential

tan(2(β −α)) ≡−

β−α

1 − 2c

β−α

+ Z

345

/2 − Z

(2.8)

The masses of the neutral scalars can be written as

= v



β−α



The angle α can be deﬁned as the rotation angle connecting the compo-

nents of the original Higgs doublets 

and 

to the mass eigenstates.

β−α

+ Z

345

β−α

/2 − Z

β−α

1 − 2c

β−α

. (2.9)

Finally, the couplings of the CP-even scalar, h, to the elec-

troweak gauge bosons are given by

hV V

(2m

μ,−

+ m

)



1 − c

β−α

(2.10)

and to the fermions by

hf f

=−







1 − c

β−α

+ η

β−α

tan β



. (2.11)

By convention, the sign of the h couplings to WW and ZZ

is ﬁxed to be positive (this can always be achieved, without

loss of generality, by redeﬁning the Higgs boson ﬁeld as

h →−h). On the other hand, the sign of the h couplings

to a fermion may be positive or negative, depending on the

value of c

β−α

and tan β.Thealignment limit is deﬁned by

β−α

→ 0, that is to say, when h has SM couplings. There

is a strong evidence, both from Higgs and from electroweak

precision measurements, that the couplings of the 125 GeV

boson to W and Z bosons are very close to those predicted

by the SM. Therefore the 2HDM has to be near the alignment

limit to be phenomenologically viable. From Eq. (2.8), the

condition for alignment is

||Y

+ Z

345

/2 − Z

|. (2.12)

One way to satisfy this is by making Y

large, Y

 v

which is called the decoupling limit because then H

, A and

become heavy. Another way to ensure alignment is to

take |Z

|small enough, |Z

|1. If the condition Eq. (2.12)

is satisﬁed with Y

 v

then we speak of alignment without

decoupling.

2.2 Low-energy EFT

For Y

≡ 

 v

and Y

∼ Y

∼ v,Eqs.(2.7) and (2.9)

imply m

∼ m

∼ , and the spectrum below

the scale v is that of the SM. Consequently, we can describe

Higgs production and decays at the LHC in the framework

of the so-called SM EFT, where the heavy particles are inte-

grated out, and their effects are represented by operators with

canonical dimensions D > 4 added to the SM. Below we dis-

cuss the Lagrangian of the low-energy effective theory for the

2HDM, treating 1/ as the expansion parameter. We ﬁrst

review the known results concerning the D = 6 operators

in the EFT with tree-level matching [11,12]. This is enough

for the purpose of this paper, in which the main focus is

the accuracy of the EFT to describe the current LHC Higgs

measurements. Matching beyond D = 6 and tree level was

discussed in Refs. [11,13,19,20], and we will come back to

it in an upcoming publication [21].

123

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