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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

1

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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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2.2 The result

As we shall demonstrate in the next section, the value of ∆ρ in the model outlined above

is

?

a=2

b=2

m−1

?

n−1

?

m

?

−3?F?m2

where ma, ma′ denote the masses of the charged scalars and µb, µb′ denote the masses

of the neutral scalars. The term (23b) contains a sum over all pairs of different physical

neutral scalar particles S0

of different charged scalars, excluding the Goldstone bosons G±, i.e. 2 ≤ a < a′≤ n. The

term (23e) consists of the subtraction, from the rest of ∆ρ, of the SM result—mhis the

mass of the sole SM physical neutral scalar, the so-called Higgs particle.

In equation (23), the function F of two non-negative arguments x and y is

x + y

2

0

∆ρ =

g2

64π2m2

W

n

?

?Im?V†V?

?

?Im?V†V?

Z,m2

h

?− F?m2

m

?

???U†V?

ab

??2F?m2

?2F?µ2

??2F?m2

?2?F?m2

W,m2

a,µ2

b

?

(23a)

−

b=2

m

?

b′=b+1

bb′

b,µ2

b′?

(23b)

−2

a=2

n

a′=a+1

???U†U?

aa′

a,m2

a′?

(23c)

+3

b=2

1b

Z,µ2

b

?− F?m2

???,

W,µ2

b

??

(23d)

h

(23e)

band S0

b′; similarly, the term (23c) contains a sum over all pairs

F (x,y) ≡

−

xy

x − ylnx

y

⇐ x ?= y,

⇐ x = y.

(24)

This is a non-negative function, symmetrical under the interchange of its two arguments,

and vanishing if and only if those two arguments are equal. This function has the impor-

tant property that it grows linearly with max(x,y), i.e. quadratically with the heaviest-

scalar mass, when that mass becomes very large. Unless there are cancellations, this leads

to a quadratic divergence of ∆ρ for very heavy scalars (Higgs bosons).

If there are in the model any SU(2)-singlet scalars with electric charge other than 0

or ±1, then the existence of those scalars does not contribute to ∆ρ, they do not modify

equation (23), at one-loop level, in any way.

A simplification occurs when there are in the model no SU(2)-singlet charged scalars

χ+

term (23c) vanishes.

When there are in the model no SU(2)-singlet neutral scalars χ0

R, hence Re?V†V?

and (23d) one may write

bb′

Thus, in an nd-Higgs-doublet model without any scalar singlets, one has simply

j. In that case, there is no matrix T, hence the matrix U is unitary by itself, and the

l, there is no matrix

bb′=?ReVTReV + ImVTImV?

bb′= δbb′. Then, in the terms (23b)

?2.

???V†V?

g2

64π2m2

??2instead of?Im?V†V?

?nd

a=2

b=2

bb′

∆ρ =

W

?

2nd

?

???U†V?

6

ab

??2F?m2

a,µ2

b

?

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−

2nd−1

?

2nd

?

−3?F?m2

b=2

2nd

?

b′=b+1

???V†V?

??2?F?m2

h

?− F?m2

bb′

??2F?µ2

Z,µ2

b,µ2

b′?

+3

b=2

???V†V?

1b

b

?− F?m2

h

???.

W,µ2

b

??

Z,m2

W,m2

(25)

Our general results have been checked to be consistent with specific results for ∆ρ in a

few models. These include the results for both the CP conserving version [5, 13] and the

CP non-conserving version [16] of the 2HDM.6It has also been checked against a model

containing one doublet and one scalar singlet [23].

3 Derivation of the result

This section contains the derivation of equation (23). It may be skipped by those who

are not interested in the details of that derivation.

3.1The Lagrangian

We use the conventions of [22]. The covariant derivative of the doublets is

Dµφk=

∂µϕ+

k− i

g

√2W+

∂µϕ0

µϕ0

k+ ig (s2

g

√2W−

W− c2

2cW

µϕ+

W)

Zµϕ+

k+ ieAµϕ+

k

k− i

k+ i

g

2cW

Zµϕ0

k

(26)

and the covariant derivative of the charged singlets is

Dµχ+

j= ∂µχ+

j+ igs2

W

cW

Zµχ+

j+ ieAµχ+

j. (27)

The covariant derivative of the neutral singlets is, of course, just identical with their

ordinary derivative. We use the unitarity of˜U in equation (19), in particular

?T†T?

a′a= δa′a−?U†U?

a′a.(28)

We also use the orthogonality of˜V in equation (20) to arrive at the gauge-kinetic La-

grangian

nd

?

n

?

k=1

(Dµφk)†(Dµφk) +

nc

?

j=1

?Dµχ−

m

?

j

??Dµχ+

??∂µS0

j

?+1

?

2

nn

?

l=1

?∂µχ0

l

??∂µχ0

l

?

=

a=1

?∂µS−

a

??∂µS+

a

?+1

2

b=1

?∂µS0

bb

(29a)

6There is some discrepancy between our result and the one presented in Section 4 of [11].

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+m2

WWµ−W+

µ+ m2

Z

ZµZµ

2

(29b)

+imW

n

?

?

a=1

?W−

Im?ω†V?

WmZZµ?

?S+

n

?

m−1

?

m

?

baW−

µ

?S0

mWW+

µ

?ω†U?

a∂µS+

a− W+

µ

?U†ω?

a∂µS−

a

?

(29c)

+mZZµ

m

b=1

b∂µS0

b

(29d)

−?emWAµ+ gs2

+ieAµ

?

+i

2cW

n

?

a∂µS+

a=1

??ω†U?

?

aW−

µS+

a+?U†ω?

aW+

µS−

a

?

(29e)

n

a=1

a∂µS−

a− S−

a

(29f)

g

Zµ

a,a′=1

?2s2

m

?

??U†V?

Wδaa′ −?U†U?

Im?V†V?

abW+

µ

a′a

??S+

a∂µS−

a′ − S−

a′∂µS+

a

?

(29g)

+

g

2cW

Zµ

b=1

b′=b+1

bb′

?S0

b∂µS0

b′ − S0

b′∂µS0

b

?

(29h)

+ig

2

n

?

a=1

b=1

?S−

a∂µS0

ZµZµ

2

a∂µS0

b− S0

b∂µS−

a

?

+?V†U?

+g

b∂µS+

a− S+

b

m

??

(29i)

?

µWµ−+mZ

cW

?

?

??U†V?

m

?

b=1

S0

bRe?ω†V?

abW+

b

(29j)

−

?eg

?g2

2Aµ+g2s2

W

2cW

Zµ

?

n

?

ZµZµ

2

a=1

m

?

b=1

S0

b

µS−

a+?V†U?

b′S0

b

baW−

µS+

a

?

(29k)

+

4Wµ−W+

µ+

g2

4c2

W

?

b,b′=1

?V†V?

b′bS0

(29l)

+g2

2Wµ−W+

µ

n

?

S−

a,a′=1

n

?

n

?

n

?

?U†U?

aS+

a

a′aS−

a′S+

a

(29m)

+2e2AµAµ

2

a=1

(29n)

+eg

cW

AµZµ

a,a′=1

?2s2

Wδaa′ −?U†U?

?4s4

a′a

?S−

a′S+

a

(29o)

+g2

2c2

W

ZµZµ

2

a,a′=1

Wδaa′ +?1 − 4s2

W

??U†U?

a′a

?S−

a′S+

a.(29p)

In lines (29c)–(29e) and (29j) we have used an nd-vector ω defined by ωk ≡ vk/v. By

identifying lines (29c) and (29d) with the usual terms [22] mixing the W±and Z0gauge

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bosons with the G±and G0Goldstone bosons, respectively,

imW

?W−

µ∂µG+− W+

µ∂µG−?+ mZZµ∂µG0,

we conclude that the components of the Goldstone bosons are given by [21]

Uk1=vk

Vk1= ivk

v,

v,

henceTj1= 0, (30)

henceRl1= 0. (31)

Therefore, we may rewrite line (29e) as

−?emWAµ+ gs2

WmZZµ??W−

µG++ W+

µG−?

(32)

and line (29j) as

− g

?

mWW+

µWµ−+mZ

cW

ZµZµ

2

? m

?

b=2

S0

bIm?V†V?

1b.(33)

The sum starts at b = 2 because Im?V†V?

than 0 or ±1, then those scalars do not mix with components of the doublets. Their

covariant derivative is

11= 0.

If there are in the theory any SU(2)-singlet scalars S±Qwith electric charge ±Q other

DµS+Q= ∂µS+Q+ igs2

WQ

cW

ZµS+Q+ ieQAµS+Q.(34)

This yields, in particular, the following two interaction terms in the Lagrangian:

L = ··· +igs2

WQ

cW

Zµ

?S+Q∂µS−Q− S−Q∂µS+Q?

?2

(35a)

+

?gs2

WQ

cW

ZµZµS−QS+Q. (35b)

3.2The Feynman diagrams

In our model, in the computation of the vacuum polarizations of the gauge bosons W±

and Z0there are four types of Feynman diagrams involving scalar fields:

Type (a) diagrams: A scalar branches off from the gauge-boson line and loops back to the

same point in that gauge-boson line—see figure 1(a). When the scalar is neutral,

the relevant interaction terms in the Lagrangian are the ones in line (29l), for b′= b;

but then the contribution to ∆ρ vanishes, since one obtains Πµν

the scalar is charged, the relevant terms in the Lagrangian are those in line (29m)

for Πµν

WW= c2

WΠµν

ZZ. When

WWand line (29p) for Πµν

ZZ, in both cases for a′= a.

Type (b) diagrams: The gauge-boson line splits into two scalar lines which later reunite to

form a new gauge-boson line—see figure 1(b). The relevant terms in the Lagrangian

are those in line (29i) for Πµν

WW, and those in lines (29g) and (29h) for Πµν

ZZ.

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(a)

(b)

(c)

Figure 1: Three types of Feynman diagrams occurring in the calculation of the vacuum

polarizations.

Figure 2: Tadpole diagrams which do not contribute to ∆ρ.

Type (c) diagrams: A neutral scalar branches off from the gauge-boson line and loops to

a later point in that gauge-boson line—see figure 1(c). The interaction terms in the

Lagrangian responsible for these Feynman diagrams are those in expression (33).

Type (d) diagrams: A neutral scalar branches off, with zero momentum, from the gauge-

boson line, and produces a loop of some stuff—see figure 2. These “tadpole” Feyn-

man diagrams originate from the interaction terms in expression (33). They yield

a vanishing contribution to ∆ρ since one obtains Πµν

omit the tadpole diagrams altogether.

WW= c2

WΠµν

ZZ. Hence we may

3.3 Computation of the loop diagrams

We use dimensional regularization in the computation of the Feynman diagrams. The

dimension of space–time is d. An unphysical mass µ is used to keep the dimension of each

integral unchanged when d varies. We define the divergent quantity

div ≡

2

4 − d− γ + 1 + ln?4πµ2?,

In the computation of type (a) Feynman diagrams the where γ is Euler’s constant.

relevant momentum integral is

µ4−d

?

ddk

(2π)d

gµν

k2− A + iε=igµν

16π2A(div − lnA), (36)

where A is the mass squared of the scalar particle in the loop. In order to compute the

type (b) and type (c) Feynman diagrams we need first to introduce a Feynman parameter

x, which is later integrated over from x = 0 to x = 1. For type (b) diagrams we have

µ4−d

?

ddk

(2π)d

?1

0

dx

4kµkν

[k2− Ax − B (1 − x) + iε]2

=

igµν

16π2[A(div − lnA)

10

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+B (div − lnB) + F (A,B)],(37)

where A and B are the masses squared of the scalars in the loop, and the four-momentum

qµof the external gauge-boson line is taken to obey q2= 0. Notice the presence of terms

of the form A(div − lnA) in both diagrams of types (a) and (b); we shall soon see that

those terms cancel out in the computation of ∆ρ, leaving only the F functions from the

type (b) diagrams. For type (c) diagrams the relevant integral is

µ4−d

?

ddk

(2π)d

?1

0

dx

gµν

[k2− Ax − B (1 − x) + iε]2

=

igµν

16π2

+F (A,B)].

1

A

?

A(div − lnA) −A + B

2

(38)

This integral is symmetric under the interchange of A and B; equation (38) presents a

seemingly asymmetric form, but it is in fact symmetric. The reason for expressing the

integral in this way is that, due to cancellations, only the terms F (A,B) survive in the

end.

3.4The contributions to ∆ρ from diagrams of types (a) and (b)

Using (29m) and (36), we see that the contribution to AWW(q2) of type (a) Feynman

diagrams with charged scalars in the loop is

A(a)

WW

?q2?= −

g2

32π2

n

?

a=1

?U†U?

aam2

a

?div − lnm2

a

?. (39)

In the same way, using (29p),

A(a)

ZZ

?q2?= −

g2

32π2c2

W

n

?

a=1

?4s4

W+?1 − 4s2

W

??U†U?

aa

?m2

a

?div − lnm2

a

?. (40)

Proceeding to the type (b) Feynman diagrams, from (29i) and (37) we find that

A(b)

WW(0) =

g2

64π2

n

?

?div − lnµ2

64π2

a=1

m

?

b=1

?U†V?

b

?+ F?m2

?U†U?

?V†V?

m

?

ab

?V†U?

ba

?m2

??

a

?div − lnm2

a

?

+µ2

b

a,µ2

b

=

g2

?

2

n

?

a=1

aam2

a

?div − lnm2

?

??2F?m2

a

?

(41a)

+

m

?

n

?

b=1

bbµ2

b

?div − lnµ2

b

(41b)

+

a=1

b=1

???U†V?

ab

a,µ2

b

?

?

.(41c)

We have used

n

?

a=1

?U†V?

ab

?V†U?

11

ba=?V†V?

bb,(42)

Page 12

which follows from the unitarity of˜U, i.e. from [21]

UU†= 1nd×nd. (43)

We have also used

m

?

b=1

?U†V?

ab

?V†U?

ba= 2?U†U?

aa, (44)

which follows from the orthogonality of˜V , i.e. from [21]

ReV ReVT

ReV ImVT

= ImV ImVT

= ImV ReVT

= 1nd×nd,

= 0nd×nd.

(45)

Considering now the self-energy of the Z0boson, we find

A(b)

ZZ(0) =

g2

64π2c2

W

?

n

?

a,a′=1

?2s2

Wδaa′ −?U†U?

a

?+ m2

?Im?V†V?

?div − lnµ2

64π2c2

W

a=1

a′=a+1

n

?

m−1

?

m

?

a′a

??2s2

a′?+ F?m2

Wδaa′ −?U†U?

aa′

?

×?m2

+

?

×?µ2

g2

a

?div − lnm2

?

a′

?div − lnm2

a,m2

a′??

m−1

b=1

m

b′=b+1

bb′

?2

?div − lnµ2

???U†U?

??U†U?

?2F?µ2

?div − lnµ2

bb

?+ µ2

?

b′

b′?+ F?µ2

??2F?m2

?m2

b,µ2

b,µ2

b′???

=

?

2

n−1

?

n

aa′

a,m2

a′?

(46a)

+2

a=1

?4s4

m

?

W+?1 − 4s2

?Im?V†V?

?V†V?

W

aa

a

?div − lnm2

a

?

(46b)

+

b=1b′=b+1

bb′

b′?

(46c)

+

b=1

bbµ2

bb

??

.(46d)

We have used

m

?

b′=1

?Im?V†V?

bb′

?2=?V†V?

bb,(47)

which follows from equations (45).

Putting everything together, we see that

the A(a)

WW(q2) of equation (39) cancels out the line (41a) of A(b)

WW(0);

the A(a)

ZZ(q2) of equation (40) cancels out the line (46b) of A(b)

ZZ(0);

the line (41b) of A(b)

c2

WW(0) cancels out the line (46d) of A(b)

ZZ(0) in the subtraction AWW−

WAZZ.

12

Page 13

In this way we finally obtain

A(a+b)

WW(0) − c2

WA(a+b)

ZZ

(0) =

g2

64π2

?

n

?

?

m

?

a=1

m

?

b=1

???U†V?

???U†U?

?Im?V†V?

ab

??2F?m2

??2F?m2

?2F?µ2

a,µ2

b

?

(48a)

−2

n−1

?

m−1

?

a=1

n

a′=a+1

aa′

a,m2

a′?

(48b)

−

b=1

b′=b+1

bb′

b,µ2

b′?

?

. (48c)

The positive term (48a) originates from A(b)

come from A(b)

If there are in the electroweak theory any scalar SU(2) singlets with electric charges

other than 0 or ±1, then the relevant terms in the Lagrangian are those in equation (35).

The term (35b) generates a type (a) Feynman diagram which exactly cancels the type (b)

Feynman diagram generated by the term (35a).7Thus, scalar SU(2) singlets with electric

charge different from 0 and ±1 do not affect ∆ρ at all.

The sums in equation (48) include contributions from the Goldstone bosons G±= S±

and G0= S0

which are arbitrary in a ’t Hooft gauge. The terms which depend on those masses are,

explicitly,

WWwhile the negative terms (48b) and (48c)

ZZ.

1

1. These Goldstone bosons have unphysical masses m1and µ1, respectively,

???U†V?

+

?

+

?

−2

11

??2F?m2

???U†V?

???U†V?

?

m

?

1,µ2

1

?

(49a)

m

b=2

n

1b

??2F?m2

??2F?m2

??2F?m2

?2F?µ2

1,µ2

b

?

?

(49b)

a=2

a1

a,µ2

1

(49c)

n

a=2

???U†U?

?Im?V†V?

1a

1,m2

a

?

(49d)

−

b=2

1b

1,µ2

b

?. (49e)

One may eliminate some of these terms by using equations (30) and (31).

?U†U?

?U†V?

?U†V?

since Re?V†V?

7This cancellation is analogous to the one between equation (40) and line (46b).

Indeed,

a1=

1a= −?T†T?

11= i. In the term (49b) one may write

1a= 0 when a ?= 1, because Tj1 = 0 for any j; also,

a1= 0 for a ?= 1. Therefore, the terms (49c) and (49d) vanish. In the term (49a),

?U†V?

i?U†U?

1b= i?V†V?

1b= −Im?V†V?

1b⇐ b ?= 1,(50)

1b=

?ReVTReV + ImVTImV?

1b= −?RTR?

1b= 0. In this way, the

13

Page 14

terms (49) are reduced to

F?m2

1,µ2

1,µ2

1

?+

m

?

b=2

?Im?V†V?

1b

?2?F?m2

1,µ2

b

?− F?µ2

1,µ2

b

??. (51)

The term F (m2

it is eliminated when one subtracts the SM result from the Multi-Higgs-doublet-model

one. The other terms in the expression (51) are cancelled out by the diagrams of type (c),

as we shall see next.

1) is independent of the number of scalar doublets and singlets, hence

3.5The contributions to ∆ρ from diagrams of type (c)

To compensate for the unphysical masses of the Goldstone bosons, the propagators of

gauge bosons W±and Z0with four-momentum kµare, in a ’t Hooft gauge,

−kµkν

m2

−kµkν

m2

W

i

k2− m2

i

k2− µ2

1

+

?

?

−gµν+kµkν

−gµν+kµkν

m2

W

?

?

i

k2− m2

i

k2− m2

W

,(52)

Z

1

+

m2

ZZ

, (53)

respectively, i.e. they contain a piece with a pole on the unphysical masses squared m2

and µ2

Using these propagators to compute the type (c) Feynman diagrams, one obtains

1

1, respectively.

A(c)

WW(0) =

g2

64π2

m

?

W+ µ2

m

?

Z+ µ2

b=2

?Im?V†V?

b

?− F?m2

?Im?V†V?

b

?− F?µ2

1b

?2?−m2

1,µ2

1

?div − lnm2

?− 3F?m2

?2?−µ2

1,µ2

b

?− 3F?m2

1

?− 3m2

??,

1

?− 3m2

??.

W

?div − lnm2

W

?

+2?m2

g2

64π2c2

b

W,µ2

b

(54)

A(c)

ZZ(0) =

W

b=2

1b

1

?div − lnµ2

Z,µ2

Z

?div − lnm2

Z

?

+2?m2

b

(55)

The factors 3 originate in a partial cancellation between the contributions from the pieces

−gµνand kµkν/m2

four times larger than, and with opposite sign relative to, the latter one, cf. equations (37)

and (38). Performing the subtraction relevant for ∆ρ, one obtains

Vin the propagator of the gauge boson V , the former contribution being

A(c)

WW(0) − c2

WA(c)

ZZ(0) =

g2

64π2

m

?

1

?div − lnm2

−F?m2

b=2

?Im?V†V?

?div − lnm2

W− m2

1,µ2

b

?+ F?µ2

b

1b

?2

×?−m2

+2?m2

−3F?m2

1

?+ µ2

1

?div − lnµ2

Z

1

?

(56a)

(56b)

(56c)

(56d)

(56e)

−3m2

WW

?+ 3m2

1,µ2

b

Z,µ2

?div − lnm2

Z

?

Z

?

?

W,µ2

?+ 3F?m2

b

??.

14

Page 15

The terms (56a)–(56c) are independent of the number of scalar doublets. They disappear

when one subtracts the Standard-Model result from the multi-Higgs-doublet-model one,

since

m

?

The terms (56d), which involve the masses of the Goldstone bosons, cancel out the terms

in (51), except the first one, which is cancelled by the subtraction of the SM result.

We have thus finished the derivation of equation (23) for ∆ρ.

b=2

?Im?V†V?

1b

?2=?V†V?

11= 1. (57)

4 The 2HDM and the Zee model

In this section we give, as examples of the application of our general formulae, the expres-

sions for ∆ρ in the 2HDM and also in the model of Zee [24] for the radiative generation

of neutrino masses, which has one singly charged SU(2) singlet together with the two

doublets.

In the study of the 2HDM it is convenient to use the so-called “Higgs basis,” in which

only the first Higgs doublet has a vacuum expectation value. In that basis,

φ1=

?

G+

(v + H + iG0)?√2

1and G0≡ S0

scalar, which has mass m2. Thus, the matrix U, which connects the charged components

of φ1and φ2to the eigenstates of mass, is in the Higgs basis of the 2HDM equal to the unit

matrix. On the other hand, H, R and I, which are real fields, must be rotated through

a 3 × 3 orthogonal matrix O to obtain the three physical neutral fields S0

Without lack of generality we choose detO = +1. Thus, the 2 × 4 matrix V , defined

through

?

,φ2=

?

S+

2

(R + iI)?√2

?

.(58)

Here, G+≡ S+

1are the Goldstone bosons, while S+

2is the physical charged

2,3,4:

H

R

I

= O

S0

S0

S0

2

3

4

. (59)

?

H + iG0

R + iI

?

= V

G0

S0

S0

S0

2

3

4

, (60)

is

V =

?

i

0 O21+ iO31 O22+ iO32 O23+ iO33

O11

O12

O13

?

.(61)

Therefore,

V†V =

1−iO11 −iO12 −iO13

1 iO13

iO12 −iO13

iO13

iO12

iO11

−iO12

iO11

1

1

−iO11

. (62)

15

Page 16

The value of ∆ρ in the 2HDM is therefore, using our formula in equation (25),

∆ρ =

g2

64π2m2

W

?

2,µ2

4

?

b=2

?1 − O2

?− O2

?F?m2

1b−1

?F?m2

2,µ2

2,µ2

b

?

−O2

13F?µ2

?

3

12F?µ2

Z,µ2

4

?− O2

W,µ2

11F?µ2

b

?− F?m2

3,µ2

4

?

+3

4

b=2

O2

1b−1

b

?− F?m2

2,3,4, respectively, while mhis the mass of the Higgs

Z,m2

h

?+ F?m2

W,m2

h

??

?

, (63)

where µ2,3,4denote the the masses of S0

boson of the SM. Equation (63) reproduces, in a somewhat simplified form, the result for

∆ρ in the 2HDM previously given in [16].

A special case of the 2HDM is the model with one “dark” scalar doublet. This means

that a second doublet is added to the SM, but that doublet has no VEV and it does not

mix with the standard Higgs doublet [19]. We should then identify H with the usual

Higgs particle. Thus, O11= 1 and µ2= mh. Equation (63) then simplifies to [25, 20]

∆ρ =

g2

64π2m2

W

?

4

?

b=3

F?m2

2,µ2

b

?− F?µ2

3,µ2

4

??

. (64)

This quantity is small if the three masses m2, µ3and µ4are close together. Notice that in

this case of a “dark” scalar doublet there are no vector–vector–scalar couplings involving

the additional doublet, hence ∆ρ stems exclusively from type (a) and type (b) Feynman

diagrams.

In the model of Zee there is, besides the two scalar SU(2) doublets

φ1=

?

G+

(v + H + iG0)?√2

?

,φ2=

?

H+

(R + iI)?√2

?

, (65)

also one scalar SU(2) singlet χ+with unit electric charge. Therefore there is a 2 × 2

unitary matrix K such that

?H+

χ+

?

= K

?S+

2

S+

3

?

, (66)

where S+

tively. So, now the matrix U of equation (15) is

2and S+

3are the physical charged scalars, which have masses m2and m3, respec-

U =

?100

0 K11 K12

?

, (67)

so that

U†U =

1

0

0 K11K∗

00

|K11|2

K∗

|K12|2

11K12

12

.(68)

16

Page 17

Equations (61) and (62) retain their validity, and

U†V =

i

0 K∗

0 K∗

O11

O12

O13

11(O21+ iO31) K∗

12(O21+ iO31) K∗

11(O22+ iO32) K∗

12(O22+ iO32) K∗

11(O23+ iO33)

12(O23+ iO33)

.(69)

Therefore, using our general formula (23) for ∆ρ, we see that, in the model of Zee,

∆ρ =

g2

64π2m2

W

?

4

?

b=2

?1 − O2

2,m2

1b−1

?

3

?

a=2

|K1a−1|2F?m2

a,µ2

b

?

−2|K11K12|2F?m2

−O2

4

?

3

?

13F?µ2

2,µ2

3

?− O2

?F?m2

12F?µ2

Z,µ2

2,µ2

4

?− O2

W,µ2

11F?µ2

b

?− F?m2

3,µ2

4

?

+3

b=2

O2

1b−1b

?− F?m2

Z,m2

h

?+ F?m2

W,m2

h

??

?

. (70)

5 Summary

In this paper we have derived the formula for the parameter ∆ρ, as defined in equation (4),

in an extension of the Standard Model characterized by an arbitrary number of scalar

SU(2) doublets (with hypercharge ±1/2) and singlets (with arbitrary hypercharges). Our

formalism is completely general, using only the masses of the scalars and their mixing

matrices, which ensures that our formulae are always applicable. The computation has

been carried out in a general Rξ gauge, thereby demonstrating that the final result is

independent of the masses of the unphysical scalars. We have also explicitly demonstrated

that all infinities cancel out in the final result for ∆ρ. In order to ease the consultation of

this paper, the formulae for ∆ρ given in Section 2 have been completely separated from

their derivation presented in Section 3. Our results can be applied either to check the

viability of a model or to constrain its parameter space, by comparing the ∆ρ, calculated

in that model, with numerical bounds on ∆ρ obtained from a fit to precision data—

for instance, the bound (7) found in [4]. As an illustration of our general formulae, in

Section 4 we have worked out the specific cases of the two-Higgs-doublet model, with and

without one extra charged scalar singlet.

Acknowledgements:

feld for helpful discussions. The work of L.L. was supported by the Portuguese Funda¸ c˜ ao

para a Ciˆ encia e a Tecnologia through the project U777–Plurianual. W.G. and L.L.

acknowledge support from EU under the MRTN-CT-2006-035505 network programme.

W.G. thanks S. Dittmaier, W. Hollik, M. Krawczyk and H. Neu-

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