arXiv:0711.4022v2 [hep-ph] 10 Apr 2008
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
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.
In the Standard Model (SM), the parameter
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 . 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)
V V(q) = gµνAV V
?q2?+ qµqνBV V
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]
The precise measurement , at LEP, of the W±and Z0self-energies is in striking agree-
ment with the SM predictions  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
The SM contributions to the quantity (3) are known up to the leading terms at three-loop
level . 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
?AWW(0) − c2
?AWW(0) − c2
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
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
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  and ǫ1, ǫ2and ǫ3in .2While these two
sets of observables differ in their precise definitions, the quantity of interest in this paper
∆ρ = αT = ǫ1,
WAZZ(0), and also, eventually, through the subtraction of the SM
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 . 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  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  for a review,  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 . 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 .
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
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
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rameters” have been identified in .
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