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arXiv:0711.4022v2 [hep-ph] 10 Apr 2008

UWThPh-2007-28

A precision constraint on multi-Higgs-doublet models

W. Grimus,(1)∗L. Lavoura,(2)†O.M. Ogreid,(3)‡and P. Osland(4)§

(1)Fakult¨ at f¨ ur Physik, Universit¨ at Wien

Boltzmanngasse 5, 1090 Wien, Austria

(2)Universidade T´ ecnica de Lisboa and Centro de F´ ısica Te´ orica de Part´ ıculas

Instituto Superior T´ ecnico, 1049-001 Lisboa, Portugal

(3)Bergen University College, Bergen, Norway

(4)Department of Physics and Technology, University of Bergen

Postboks 7803, N-5020 Bergen, Norway

10 April 2008

Abstract

We derive a general expression for ∆ρ (or, equivalently, for the oblique parameter

T) in the SU(2)×U(1) electroweak model with an arbitrary number of scalar SU(2)

doublets, with hypercharge ±1/2, and an arbitrary number of scalar SU(2) singlets.

The experimental bound on ∆ρ constitutes a strong constraint on the masses and

mixings of the scalar particles in that model.

∗E-mail: walter.grimus@univie.ac.at

†E-mail: balio@cftp.ist.utl.pt

‡E-mail: omo@hib.no

§E-mail: per.osland@ift.uib.no

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1 Introduction

In the Standard Model (SM), the parameter

ρ =

m2

W

m2

Zcos2θW, (1)

where mWand mZare the masses of the W±and Z0gauge bosons, respectively, and θW

is the weak mixing angle, gives the relative strength of the neutral-current and charged-

current interactions in four-fermion processes at zero momentum transfer [1]. At tree

level ρ is equal to one, and it remains one even if additional scalar SU(2) doublets, with

hypercharge ±1/2, are added to the SM.1At one-loop level, the vacuum-polarization

effects, which are sensitive to any field that couples either to the W±or to the Z0,

produce the vacuum-polarization tensors (V = W,Z)

Πµν

V V(q) = gµνAV V

?q2?+ qµqνBV V

?q2?, (2)

where qµis the four-momentum of the gauge boson. Then, deviations of ρ from unity

arise, which are determined by the self-energy difference [1, 2]

AWW(0)

m2

W

−AZZ(0)

m2

Z

. (3)

The precise measurement [3], at LEP, of the W±and Z0self-energies is in striking agree-

ment with the SM predictions [4] and provides a strong constraint on extended electroweak

models. For instance, one can constrain the two-Higgs-doublet model (2HDM) in this

way [5, 6].

In this paper we are interested in the contributions to the ρ parameter generated by

an extension of the SM. Therefore, we define a ∆ρ which refers to the non-SM part of the

quantity (3):

∆ρ =

?AWW(0)

m2

W

−AZZ(0)

m2

Z

?

SM extension

−

?AWW(0)

m2

W

−AZZ(0)

m2

Z

?

SM

. (4)

The SM contributions to the quantity (3) are known up to the leading terms at three-loop

level [7]. However, the consistent SM subtraction in equation (4) only requires the one-

loop SM result. In the same vein, we are allowed to make the replacement m2

in equation (4), writing instead

Z= m2

W/c2

W

∆ρ =

?AWW(0) − c2

WAZZ(0)

m2

W

?

SM extension

−

?AWW(0) − c2

WAZZ(0)

m2

W

?

SM

. (5)

Here and in the following, we use the abbreviations cW= cosθW, sW= sinθW.

At one loop, the contributions of new physics to the self-energies constitute intrinsically

divergent Feynman diagrams, but the divergent parts cancel out among different diagrams,

1Other scalar SU(2) × U(1) representations are also allowed, as long as they have vanishing vacuum

expectation values.

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between AWW(0) and c2

contributions laid out in equation (5). If the new-physics model is renormalizable, then

∆ρ is finite. The cancellations finally leave either a quadratic or a logarithmic dependence

of ∆ρ on the masses of the new-physics particles. The pronounced effects of large masses

is what renders the parameter ∆ρ so interesting for probing physics beyond the Standard

Model.

The functions AV V(q2) contain more information about new physics than the one just

provided by ∆ρ. In fact, for new physics much above the electroweak scale, a detailed

analysis of the so-called “oblique corrections” lead to the identification of three relevant

observables, which were called S, T and U in [8] and ǫ1, ǫ2and ǫ3in [9].2While these two

sets of observables differ in their precise definitions, the quantity of interest in this paper

is simply

∆ρ = αT = ǫ1,

WAZZ(0), and also, eventually, through the subtraction of the SM

(6)

where α = e2/(4π) = g2s2

It is not straightforward to obtain a bound on ∆ρ from electroweak precision data.

One possibility is to add the oblique parameters to the SM parameter set and perform

fits to the data. However, since the SM Higgs-boson loops themselves resemble oblique

effects, one cannot determine the SM Higgs-boson mass mhsimultaneously with S and

T [4]. To get a feeling for the order of magnitude allowed for ∆ρ, we quote the number

W/(4π) is the fine-structure constant.

T = −0.03 ± 0.09 (+0.09), (7)

which was obtained in [4] by fixing U = 0. For the mean value of T, the Higgs-boson

mass mh= 117 GeV was assumed; the mean value in parentheses is for mh= 300 GeV.

Equation (7) translates into ∆ρ = −0.0002 ± 0.0007 (+0.0007).

There is a vast literature on the 2HDM—see [11] for a review, [12] for the renormal-

ization of the model, [13, 14] for the possibility of having a light pseudoscalar compatible

with all experimental constraints, and [15, 16], and the references therein, for other var-

ious recent works. However, just as the 2HDM may differ significantly from the SM, a

general multi-Higgs-doublet model may be quite different from its minimal version with

only two Higgs doublets [17]. Three or more Higgs doublets frequently appear in models

with family symmetries through which one wants to explain various features of the fermion

masses and mixings; for some examples in the lepton sector see the reviews in [18].

In this paper we present a calculation of ∆ρ in an extension of the SM with an arbitrary

number of Higgs doublets and also, in addition, arbitrary numbers of neutral and charged

scalar SU(2) singlets. Our results can be used to check the compatibility of the scalar

sector of multi-Higgs models with the constraints resulting from the electroweak precision

experiments.

Recently, there has been some interest in “dark” scalars [19, 20]. These are scalars that

have no Yukawa couplings, and are thus decoupled from ordinary matter. Furthermore,

they have no vacuum expectation values (VEVs) and therefore display truncated couplings

to the gauge bosons. However, they would have quadrilinear vector–vector–scalar–scalar

2For new physics at a mass scale comparable to the electroweak scale three more such “oblique pa-

rameters” have been identified in [10].

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and trilinear vector–scalar–scalar (but no vector–vector–scalar) couplings, and would thus

also contribute to, and be constrained by, ∆ρ.

The plan of the paper is as follows. In Section 2 we present a description of our

extension of the SM and the final result of the calculation of ∆ρ; this section is self-

consistent and the result can be used without need to consult the rest of the paper. The

details of the calculation are laid out in Section 3. The application of our ∆ρ formula to

the general 2HDM is given in Section 4. The summary of our study is found in Section 5.

2 The model and the result for ∆ρ

2.1 The model

We consider an SU(2) × U(1) electroweak model in which the scalar sector includes nd

SU(2) doublets with hypercharge 1/2,3

φk=

?ϕ+

k

ϕ0

k

?

,k = 1,2,...,nd. (8)

Moreover, we allow the model to include an arbitrary number and variety of SU(2)-singlet

scalars; in particular, nccomplex SU(2) singlets with hypercharge 1,

χ+

j,j = 1,2,...,nc

(9)

and nnreal SU(2) singlets with hypercharge 0,

χ0

l,l = 1,2,...,nn. (10)

In general, our model may include other scalar fields, singlet under the gauge SU(2), with

different electric charges.

The neutral fields are allowed to have vacuum expectation values (VEVs). Thus,

?0??ϕ0

k

??0?

??0?

=

vk

√2, (11)

?0??χ0

l

= ul, (12)

the vkbeing in general complex. (The ulare real since the χ0

usual v =??nd

fields around their VEVs,

lare real fields.) We define as

k=1|vk|2?1/2≃ 246GeV. Then, the masses of the W±and Z0gauge bosons

are, at tree level, mW = gv/2 and mZ= mW/cW, respectively.4We expand the neutral

ϕ0

k

=

1

√2

?vk+ ϕ0

k

′?, (13)

3Equivalently, we may consider the model to contain SU(2) doublets with hypercharge −1/2, since

˜φk≡ iτ2φ∗

k=

?

ϕ0

−ϕ−

k

∗

k

?

is also a doublet of SU(2).

4Since the neutral singlet fields carry no hypercharge, their VEVs uldo not contribute to the masses

of the gauge bosons.

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χ0

l

= ul+ χ0

l

′. (14)

Altogether, there are n = nd+ nc complex scalar fields with electric charge 1 and

m = 2nd+ nnreal scalar fields with electric charge 0. The mass matrices of all these

scalar fields will in general lead to their mixing. The physical (mass-eigenstate) charged

and neutral scalar fields will be called S+

respectively. Note that the fields S0

µbto denote the mass of S0

a(a = 1,2,...,n) and S0

bare real. We use mato denote the mass of S±

b. We have

b(b = 1,2,...,m),

aand

ϕ+

k

=

n

?

n

?

m

?

m

?

a=1

UkaS+

a, (15)

χ+

j

=

a=1

TjaS+

a, (16)

ϕ0

k

′

=

b=1

VkbS0

b, (17)

χ0

l

′

=

b=1

RlbS0

b, (18)

the matrices U, T, V and R having dimensions nd× n, nc× n, nd× m and nn× m,

respectively. The matrix R is real, the other three are complex. The matrix

˜U ≡

?

U

T

?

(19)

is n × n unitary; it is the matrix which diagonalizes the (Hermitian) mass matrix of the

charged scalars. The real matrix

˜V ≡

ReV

ImV

R

(20)

is m×m orthogonal; it diagonalizes the (symmetric) mass matrix of the real components

of the neutral-scalar fields.5

There are in the spontaneously broken SU(2)×U(1) theory three unphysical Goldstone

bosons, G±and G0. For definiteness we assign to them the indices a = 1 and b = 1,

respectively:

S±

1

S0

= G±,

= G0.

(21)

(22)

1

Thus, only the S±

to true particles. In the general ’t Hooft gauge that we shall use in our computation, the

masses of G±and G0are arbitrary and unphysical, and they cannot appear in the final

result for ∆ρ.

awith a ≥ 2 are physical and, similarly, only the S0

bwith b ≥ 2 correspond

5Our treatment of the mixing of scalars is inspired by [21].

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