#### Limiting two-Higgs-doublet models

Alessandro Broggio
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Eung Jin Chun
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Massimo Passera
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Ketan M. Patel
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Sudhir K. Vempati
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CH-
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Villigen
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Switzerland
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I-
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Padova
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Italy
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Open Access
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c The Authors.
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Bangalore 560012
,
India
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Centre for High Energy Physics, Indian Institute of Science
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Seoul 130-722
,
Korea
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Korea Institute for Advanced Study
4
INFN - Sezione di Padova
5
Paul Scherrer Institut
on the parameter space relevant to explain the present muon g 2 anomaly, a, in four different types of models, type I, II, lepton specific (or X) and flipped (or Y). We show that the strong constraints provided by the electroweak precision data on the mass of the pseudoscalar Higgs, whose contribution may account for a, are evaded in regions where the charged scalar is degenerate with the heavy neutral one and the mixing angles and satisfy the Standard Model limit /2. We combine theoretical constraints from vacuum stability and perturbativity with direct and indirect bounds arising from collider and B physics. Possible future constraints from the electron g 2 are also considered. If the 126 GeV resonance discovered at the LHC is interpreted as the light CP-even Higgs boson of the 2HDM, we find that only models of type X can satisfy all the considered theoretical and experimental constraints.
1 Introduction 2 3 4
Electroweak constraints
Theoretical constraints on the splitting MA-MH+
Constraints from the muon g 2
Constraints from the electron g 2
The ATLAS and CMS Collaborations at the LHC [1, 2] found a neutral boson with a
mass of about 126 GeV which confirms the Brout-Englert-Higgs mechanism. It is now of
imminent interest to check whether this new boson is the unique one following exactly
the Standard Model (SM) prediction, or if there are other bosons participating in the
electroweak (EW) symmetry breaking. One of the simplest way to extend the SM is to
consider two Higgs doublets participating in the EW symmetry breaking instead of the
standard single one. There are in fact several theoretical and experimental reasons to go
beyond the SM and look forward to non-standard signals at the next run of the LHC and
at future collider experiments. For reviews on two-Higgs-doublet models, see [3, 4].
A major constraint to construct models with two Higgs doublets (2HDMs) arises from
flavour changing neutral currents, which are typically ubiquitous in these models.
Requiring Natural Flavour Conservation (NFC) restricts the models to four different classes
which differ by the manner in which the Higgs doublets couple to fermions [46]. They
are organized via discrete symmetries like Z2 under which different matter sectors, such as
right-handed leptons or left-handed quarks, have different charge assignments. These
models are labeled as type I, II, lepton-specific (or X) and flipped (or Y). Normalizing the
Yukawa couplings of the neutral bosons in such a way that the explicit Yukawa interaction
terms in the Lagrangian are given by (yf) mf ff for the CP-even scalars = h, H (lighter
basis, the yfh,H,A factors are summarized in table I for each of these four types of 2HDMs
the diagonalization angle of the two CP-even Higgs bosons (v = pv12 + v22 = 246 GeV).
However it should be noted that in addition to these models, NFC can also occur in models
with alignment, as in ref. [7]. In this class of models, more general sets of relations are
imposed on the field content using discrete symmetries similar to Z2, which still conserve
flavour but allow for CP violation. A class of 2HDMs also exists where one of the Higgs
doublets does not participate in the dynamics and remains inert [8, 9]. Finally, in the
so-called type III models both up and down fermions couple to both Higgs doublets. A
detailed analysis of flavour and CP violation in type III models can be found in [10] and
in type II and Ymodels.
One of the possible experimental indications for new physics is the measurement of the
out favourable extensions of the SM. In this paper we will study if such a deviation can
two-loop diagrams [1115]. However, a light pseudoscalar may be in conflict with a heavy
charged scalar whose mass is strongly constrained by direct and indirect searches. In fact,
the general 2HDM lower bound on the mass of the charged scalar H from direct searches
at LEP2 is MH >79 GeV [16], and even stronger indirect bounds can be set from B-physics
In 2HDMs, the observed 126 GeV resonance can be identified with any of the two
CP-even Higgs bosons.1 In the present paper we identified this resonance with the lightest
CP-even scalar h. This interpretation is possible in all four 2HDMs types considered here.
neutral Higgs h with the gauge bosons and fermions attain the SM values. In fact, the
measured signal strengths and production cross section of such a particle are in very good
agreement with the corresponding SM predictions [1834].
searches at colliders for the Higgs bosons h, H, A and H, B-physics observables, EW
precision measurements and theoretical considerations of vacuum stability and perturbativity.
The question then arises: which of these models are preferred by the present set of direct
and indirect constraints? In this work we addressed this question concentrating on the
1In this paper, we work in the CP-conserving case i.e, we assume all the parameters to be real. The
CP-violating case (see [4] for a review) is interesting in its own right as it can significantly modify the
phenomenology (see for example ref. [17] and references therein). We will leave the CP-violating case for a
four models described in table 1. Our analysis shows that only models of Type X (lepton
specific) survive all these constraints.
The paper is organised as follows. In section 2 we present a detailed analysis of
the EW constraints on the masses of the pseudoscalar boson A, charged scalar H, and
additional neutral heavy scalar H. We study radiative corrections in the 2HDMs and, in
particular, the impact of the precise measurements of the W boson mass MW and the
hierarchy between A and H is allowed by the Higgs measurements at the LHC and by
the theoretical constraints on vacuum stability and perturbativity, which is discussed in
section 3. In section 4 we present the additional contributions of the 2HDMs to the muon
drawn in section 6.
Electroweak constraints
In this section we analyze the constraints arising from EW precision observables on 2HDMs.
As it was shown for the first time in [36], in the SM the W mass can be computed
perturbatively by means of the following relation
M W2 =
1
form factor Z ll evaluated at q2 = MZ2 .
where sin2W = 1 M W2 /MZ2 [36] and kl(q2) = 1 + kl(q2) is the real part of the vertex
where the tilded quantities indicate the additional 2HDM contributions not contained in the
SM prediction. These additional corrections depend only on the particles and parameters of
the extended Higgs sector which are not present in the SM part. The radiative corrections
QCD corrections [38, 39] (for a review of these corrections we refer the reader to [40]). For
our purposes, this level of accuracy in the SM part is not needed, and in our codes [35] we
implemented the full one-loop SM result plus the leading two-loop contributions of [41
this contribution and found agreement with the previous results. The additional 2HDM
of [44]. For convenience, the calculation was carried out in the MS scheme and then
translated to the on-shell scheme by means of the relations derived in [37, 45]. The analytic
where a virtual Higgs is attached to an external fermion line, since they are suppressed by
factors of O(Mf /MW ). As a result, no new contributions to vertex and box diagrams are
present with respect to the SM ones. All the additional diagrams fall in the class of bosonic
constraints do not depend on the way fermions couple to the Higgs bosons and, therefore,
all four types of 2HDMs discussed in this paper share the same EW constraints.
The 2HDM predictions for MW and sin2eleffpt depend on the Z boson mass MZ =
light quarks, h(5a)d(MZ2 ) = 0.02763 (14) [47], the masses of the neutral Higgs bosons
the LHC results on Higgs boson searches [1834]. To analyze the constraints on 2HDMs
M W2HDM M WEXP !2
M WEXP = 80.385 0.015 GeV,
The results of our analysis are displayed in figure 1, where we chose three different
boson coupling = /2, Mh = 126 GeV, and we set MZ , mt, s(MZ ) and h(5a)d(MZ2 )
to their experimental central values. The green, yellow and gray regions of the plane MA vs.
2.3, 6.2, 11.8, respectively, which are the critical values corresponding to the 68.3, 95.4, and
Note that in the case of a large splitting between MH and MH , MA is required to be
almost degenerate with MH in order to satisfy the EW constraints.
This point has
already been remarked upon in [23, 29] (see also [51] for alternative conditions to satisfy
the EW constraints). In addition, we observe that all values of MA are allowed when MH
and MH are almost degenerate. This useful result will be used in section 4.
Theoretical constraints on the splitting MA-MH+
Although, as shown in the previous section, any value of MA is allowed by the EW precision
constrained by theoretical considerations of vacuum stability and perturbativity. Since
discrepancy), it is important to check how small MA is allowed to be. In this section we
study such constraints in a semi-analytical way.
The CP-conserving 2HDM with softly broken Z2 symmetry is parametrized by seven
V = m121|1|2 + m222|2|2 m122(12 + 21)
where the Higgs vacuum expectation values are given by h1,2i = 12 (0, v1,2)T . The masses
and perturbativity conditions put bounds on these parameters and correlate the masses
of different neutral and charged scalars. For example, the vacuum stability condition
and the requirement of global minimum is imposed by the condition [52]
m122(m121 m222p1/2)(tan (1/2)1/4) > 0 ,
perturbativity criterion, we will consider three different values for the maximum couplings
to see their impact on the allowed mass spectrum. A large separation between any two
scalar masses in 2HDM is controlled by the above constraints.
MA2 =
M H2 = MA2 +
conditions (3.2), (3.3) and (3.4) on the above equations. As can be seen from eq. (3.5),
mentioned above, one gets the regions allowed in MA-M plane as shown in figure 2.
can clearly see that for a light pseudoscalar with MA . 100 GeV the charged Higgs boson
mass gets an upper bound of MH . 200 GeV. Also, figure 2 shows the presence of lower
the implications of these correlations in the following sections.
MH MH = {20,40, ,230,}4G}eVanadndvanmisahxin=g 4M..
Constraints from the muon g 2
contributions: aSM = aQED + aEW + aH. The QED prediction, computed up to five loops,
aEW = 153.6 (1.0) 1011 [5456]. The latest calculations of the hadronic leading order
way to improve its evaluation [6164]. Very recently, also the next-to-next-to leading order
hadronic corrections have been determined: insertions of hadronic vacuum polarizations
were computed to be aHNNLO(vp) = 12.4 (1) 1011 [65], while hadronic light-by-light
aHLO = 6903 (53) 1011 of [58] (which roughly coincides with the average of the three
116 (39) 1011 of [57] and the rest of the other SM contributions, we obtain
is, therefore, a aEXP aSM = +262 (85) 1011, i.e. 3.1 (all errors were added
and charged Higgs bosons are [7375]
where j = {h, H, A, H}, rj = m2/Mj2, and
fh,H (r) =
fA(r) =
fH (r) =
x2(2 x)
1 x + rx2
x
1 x + rx2
dx x(1 x)
1 (1 x)r
in the additional 2HDM contribution (in any case, this contribution is negligible: setting
Mh = 126 GeV and y
h = 1 we obtain 2 1014). The formulae in eqs. (4.3)(4.6) show
fh,H (r) = ln r 7/6 + O(r),
fA(r) = + ln r + 11/6 + O(r),
fH (r) = 1/6 + O(r),
showing that in this limit fH (r) is suppressed with respect to fh,H,A(r).
The one-loop results in eqs. (4.7)(4.9) also show that, in the limit r
roughly scales with the fourth power of the muon mass. For this reason, two-loop effects
may become relevant if one can avoid the suppression induced by these large powers of the
to the muon g2 is [11, 12, 15, 54]
X Nfc Qf2 yi yfi rfi gi(rfi ),
where i = {h, H, A}, rfi = m2 /Mi2, and mf , Qf and Nfc are the mass, electric charge and
f
number of color degrees of freedom of the fermion f in the loop. The functions gi(r) are
gi(r) =
x(1 x) r
x(1 x)
fh = 1 and summing over top, bottom and tau lepton loops we obtain
become larger than the one-loop ones. Moreover, the signs of the two-loop functions gh,H
(negative) and gA (positive) for the CP-even and CP-odd contributions are opposite to
those of the functions fh,H (positive) and fA (negative) at one-loop. In type II models
with MA > 3 GeV can therefore generate a sizeable positive contribution which can account
A similar conclusion is valid for the pseudoscalar
contribution in type X models [78]. In fact, we notice from the pseudoscalar Yukawa
couplings in table 1 that the contribution of the tau lepton loop is enhanced by a factor
models of type I and Y.
chosen instead (in fact, a lower MH induces a slightly larger negative scalar contribution
experimentally [76] (see also [77]).
line corresponds to MA = Mh/2 (see text for an explanation).
for the down-type quark contribution. It is important to note that, on the contrary,
In 2HDMs of type II (and Y) a very stringent limit can be set on MH from the flavour
is much stronger than the model-independent one obtained at LEP, MH > 79.3 GeV at
95% CL [16, 83]. This strong constraint MH > 380 GeV, combined with the theoretical
requirements shown in figure 2, leads to MA & 300 GeV. In turn, this lower bound on MA
type II models are strongly disfavoured by these combined constraints. On the other hand,
no such strong flavour bounds on MH exist in type X models [4, 81]. These models are
large branching fraction even in the SM decoupling limit [78]. It will be thus interesting
to perform a dedicated analysis for such a process by considering the further decays of
light A in the type-X 2HDM. We do not perform such analysis here and leave it for future
studies. However, we would like to emphasize that any limits arising from this process can
still be avoided by considering the region MA > Mh/2. From figure 3, we see that even if
MA > Mh/2, there is still sufficient parameter space left which can provide an explanation
Constraints from the electron g 2
to be relevant; with this assumption, the measurement of ae is equated with the SM
However, as discussed in [84], in the last few years the situation has been changing thanks
to several theoretical [85] and experimental [86] advancements in the determination of ae
dominating, has been significantly reduced, and one can start to view ae as a probe of
physics beyond the SM.
aEXP aeSM = 10.5(8.1)1013, i.e. 1.3 standard deviations, thus providing a beautiful test
e
ae = 10.5 1013, one still gets a negative value, which cannot be accounted for in
the region of parameter space shown in figure 3. (Increasing the present central value by
1.4 one gets +0.8 1013, which is the input used to draw the dotted lines in figure 3.)
In recent times there has been renewed interest in the phenomenology of models with two
Higgs doublets. Most of the focus has been on four possible variations of them, namely,
type I, II, X (or lepton specific) and Y (flipped). In this work we presented a detailed
phenomenological analysis with the aim of challenging these four models.
constraints from electroweak precision tests, vacuum stability and perturbativity, direct
In these models, all the Higgses couple similarly to the gauge bosons, but differently to
the fermions. Therefore, the electroweak constraints (along with the perturbativity and
vacuum stability ones) are common to all of them, while the rest of the constraints vary
from model to model. Using a stringent set of precision electroweak measurements we
searches, all values of MA are allowed when MH and MH are almost degenerate. We
considered a CP-conserving scenario where the 126 GeV resonance discovered at the LHC
has been identified with the lightest CP-even boson h.
The 2HDM predictions for observables which depend on fermion couplings are expected
is the key distinguisher between the various types. A light pseudoscalar with couplings
bound on the mass of the charged scalar which, taken together with the perturbativity and
vacuum stability constraints, was shown to leave hardly any space for a light pseudoscalar.
because both up and down type quarks couple to the same Higgs doublet in these models.
Therefore, we showed that type X models are the only ones which can accommodate the
further significant bounds on 2HDMs.
even if this parameter region could be elusive because the productions of the additional
leading search channels for the extra bosons would then be pair or associated productions
A, H l+l, which can be readily tested at the next run of the LHC [4, 89].
We would like to thank G. Degrassi, T. Dorigo, A. Ferroglia, P. Paradisi, A. Sirlin and
G. Venanzoni for very useful discussions. MP and KMP also thank the Department of
Physics and Astronomy of the University of Padova for its support. Their work was
supported in part by the PRIN 2010-11 of the Italian MIUR and by the European Program
INVISIBLES (PITN-GA-2011-289442). MP and SKV thank the hospitality of KIAS, Seoul,
where this work started. SKV also thanks DST, Govt. of India, for the support through
the project SR/S2/RJN-25/2008.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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