1 Introduction


Lepton-Specific Two Higgs Doublet Model

as a Solution of Muon Anomaly

Tomohiro Abe, Ryosuke Sato and Kei Yagyu

Institute of Particle and Nuclear Studies,

High Energy Accelerator Research Organization (KEK),

Tsukuba 305-0801, Japan

School of Physics and Astronomy, University of Southampton,

Southampton, SO17 1BJ, United Kingdom

1 Introduction

The anomalous magnetic moment of the muon , so-called muon , is a very precisely measured observable. The latest measurement of by the E821 collaboration [1] gives


As it has been well known that there is a discrepancy between the experimental value and the prediction of the standard model (SM). According to the calculation evaluated in Refs. [2, 3]

the discrepancy is more than the 3 level, which can be considered as an indirect evidence of the existence of a new physics model. This discrepancy will be further probed at Fermilab [4] and J-PARC [5] in the near future. Since the size of the deviation is the same order as the electroweak contribution [6], we expect that new physics exists at the electroweak scale if the strength of new interactions is as large as that of the weak interaction. In such a new physics scenario, new particles are expected to be light enough to be directly discovered at the LHC. Therefore, it is quite interesting to consider models beyond the SM as a solution of the muon anomaly.

Among various models which can explain the anomaly (for a review, e.g., see Ref. [7]), two Higgs doublet models (2HDMs) give simple solutions. In 2HDMs, there are extra Higgs bosons (, , and ) in addition to the SM-like Higgs boson (), and they can give new contributions to . Usually, a softly-broken discrete symmetry is imposed [8] to avoid flavor changing neutral current (FCNC) processes at the tree level. Under the symmetry, four independent types of Yukawa interactions are allowed depending on the assignment of the charge to the SM fermion [9, 10], which are called as Type-I, Type-II, Type-X (or lepton specific) and Type-Y (or flipped) [11]. In all the types of Yukawa interactions, the lepton couplings to the extra Higgs bosons can be sizable enough to explain . In the Type-I and Type-Y 2HDMs, however, the top Yukawa coupling also becomes large together with the enhancement of the lepton couplings. This is disfavored from the view point of perturbativity. Thus, the Type-II and Type-X 2HDMs are suitable to solve the muon anomaly.

The muon has been calculated in a number of papers within 2HDMs [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. In the early 2000s, this was calculated at the one-loop level in the Type-II 2HDM in Ref. [12]. After that, it was pointed out in Refs. [13, 14] that the two-loop Barr-Zee type diagrams [23, 24] give a significant contribution to if a mass of is GeV and if there is large or couplings. In Ref. [19], the implication on collider signatures was discussed in the Type-X 2HDM, namely, the process can be important in the favored parameter region by . After the discovery of the Higgs boson at the LHC [25, 26], the muon has been reanalyzed by taking into account the Higgs boson search data in addition to the previous experimental constraints [20, 21, 22]. Furthermore, the recent observation of at the LHC [27] gives a new constraint on the parameter space of 2HDMs [21].

The difference between the Type-II and Type-X 2HDMs is the quark couplings to the extra Higgs bosons. In the Type-II 2HDM, both the lepton and down-type quark couplings are enhanced simultaneously, and thus the model is severely constrained by flavor physics and direct searches of the extra Higgs bosons. On the other hand, in the Type-X 2HDM, the quark couplings to the extra Higgs bosons are suppressed when the lepton couplings are enhanced. Thus, the constraints are weaker than those in the Type-II 2HDM. In fact, it was clarified in Refs. [20, 21] that only the Type-X 2HDM can solve the muon anomaly with satisfying the current experimental data111 In addition to the muon anomaly, there are several other motivations for this model. For example, see Refs. [28, 29]. .

Another important constraint comes from the lepton flavor physics. In the Type-X 2HDM, the constraint from the leptonic decay [30, 31, 11, 32] gives a severe constraint on the parameter space favored to explain the anomaly because of the lepton coupling enhancements. However, this important constraint has not been included in the previous analyses. Therefore, in this paper, we calculate the leptonic decay and the vertex at the one-loop level in the Type-X 2HDM in order to compare the precise experimental measurements. We then investigate the favored parameter region by under these constraints in addition to those already known. Furthermore, we evaluate the running of the scalar quartic couplings by renormalization group equations (RGEs), and require that the couplings do not become too large up to a certain energy scale, for example 10 TeV. We find that extra Higgs boson loop contributions can reduce the discrepancy in to be 2 level, but not less than 1 level. We then study the collider phenomenology in the favored parameter region.

This paper is organized as follows. In Sec. 2, we define the Lagrangian of the 2HDM, and derive the Higgs boson couplings with the gauge bosons and the fermions. In Sec. 3, we discuss constraints from direct searches for the extra Higgs bosons at LEP II and the LHC Run-I, electroweak precision observables, the decay of , the leptonic decay, and the triviality bound. In Sec. 4, we show the favored parameter regions by the muon anomaly. In Sec. 5, we discuss the collider phenomenology of the extra Higgs bosons at the LHC, the deviations in the SM-like Higgs boson couplings, and the decay branching fractions of . We also discuss the exotic decay mode: . Conclusion is given in Sec. 6. In Appendix, we present the expressions for the decay rates of extra Higgs bosons and those for the parton level cross sections for the production of extra Higgs bosons at the LHC.

2 The 2HDMs

Table 1: Assignment of the parity and factors in Eq. (22) in each type of the Yukawa interactions.

In this section, we define the Lagrangian of the 2HDM, in which the Higgs sector is composed of two doublet scalar fields and . To avoid the tree level FCNC, we impose a symmetry in the Higgs sector which can be softly-broken in general. Under the parity, four types of Yukawa interactions are defined depending on the assignment of charge as listed in Table 1.

The most general Higgs potential with the softly-broken parity is given as


Throughout the paper, we consider the CP-conserving case of the Higgs sector for simplicity, so that the imaginary parts of and are assumed to be zero. The Higgs fields are parametrized as


where and are the VEVs of the Higgs doublets which are related to the Fermi constant by . The ratio of the two VEVs is parametrized by .

The mass eigenstates of the scalar bosons are expressed by introducing the mixing angles and as


where and are the Nambu-Goldstone bosons which are absorbed by the and bosons as the longitudinal component, respectively.

The squared masses for the physical Higgs bosons are given by


where describes the breaking scale of the symmetry, and are given by


where . The mixing angle is also expressed in terms of as


All the quartic coupling constants in the Higgs potential can be rewritten in terms of the physical parameters as


where and .

A size of some combinations of ’s in the Higgs potential is constrained by taking into account perturbative unitarity [33, 34, 35, 36] and vacuum stability [37, 38]. Through Eq. (12), such a constraint can be translated into a bound on the physical parameters; e.g., the masses of the scalar bosons. First, the condition for vacuum stability; i.e., the requirement for bounded from below in any direction of the Higgs potential with large scalar fields, is given by [37, 38]


Second, the perturbative unitarity bound is obtained by requiring that all the eigenvalues of the -wave amplitude matrix for the elastic scatterings of two body boson states are satisfied as


All the independent eigenvalues were derived in Refs. [34, 35, 36] as


The Yukawa interaction terms are given by


where . In Eq. (21), , and are either or depending on the type of Yukawa interaction. In the mass eigenstates of the scalar bosons, the interaction terms are expressed as


where for (), and is the Cabibbo-Kobayashi-Maskawa matrix element. The and factors are defined by


The factors in Eqs. (22) and (23) are given in Table 1.

From the kinetic terms of the scalar fields, the ratios of the coupling constant among the CP-even scalars and gauge bosons are extracted as


As it is seen in Eqs. (22), (23) and (24), in the limit of , both and couplings become the same as those in the SM, so that we can call this limit as the SM-like limit.

3 Constraints on the Type-X 2HDM

Figure 1: One-loop (left) and two-loop Barr-Zee (right) diagrams which give corrections to the muon .

In the 2HDMs, the one-loop diagrams and the two-loop Barr-Zee type diagrams shown in Fig. 1 give dominant contributions to the muon . It has been known that the Barr-Zee type diagrams give a sizable positive contribution to in the case of a large coupling and a small as pointed it out in Refs. [13, 14]. In the Type-X 2HDMs, a large can be realized by taking since as shown in Table 1. Typically, when and GeV, the muon anomaly can be explained in the Type-X 2HDM [20]. In this section, we focus on the Type-X 2HDM with the large and small scenario to explain the anomaly, and we discuss important experimental constraints in this situation.

3.1 Direct searches for the extra Higgs bosons

There has been no signal of the extra Higgs bosons at any collider experiments. This gives lower limits on the masses of the extra Higgs bosons depending on the magnitude of couplings with SM particles. We first summarize the current bounds from the LEP II experiment, and we also review those from the LHC Run-I.

3.1.1 Lep Ii

There are constraints on the masses of the extra Higgs bosons from the direct production at the LEP II experiment with the maximal collision energy to be about 200 GeV. From the pair production process the lower bound was obtained by at 95 % C.L. [39] under the assumption of which is realized by , in the Type-X 2HDM.

From the pair production of the neutral Higgs bosons , the lower bound for the sum of and is given to be about 190-195 GeV for  [40] under the assumption of which is realized by and in the Type-X 2HDM.

The searches for and from the bremsstrahlung process have also been performed for the range of . This process gives an upper bound on for a fixed value of . For example, and are respectively excluded at 95% C.L. for () to be 30 GeV and 15 GeV [41] with the case of Br()Br()=1.

We note that the branching fractions for the extra Higgs bosons into a fermion pair can be reduced when there is a non-zero mass splitting among them. For example, and open in the case of and , respectively. There also happen the inverse processes like and as long as they are kinematically allowed. In such decay modes associated with a gauge boson, the bounds on masses on the extra Higgs bosons can be weaker than those given in the above.

3.1.2 LHC Run-I

At the LHC, extra Higgs boson searches have been performed in various channels. In the most of channels, an enhancement of the Yukawa couplings of the extra Higgs bosons becomes important to obtain a bound on their masses or coupling constants. However, in the Type-X 2HDM, the couplings of the neutral extra Higgs bosons to the quarks are suppressed by . Thus, the processes such as and do not set a limit on the masses in a large case.

Similar to the neutral Higgs boson productions, the cross section of the production such as is also suppressed by in the Type-X 2HDM. If , the top decay can be used to constrain . From the process with , the upper limit on BRBR has been driven to be between 0.23% and 1.3% at 95% C.L. for in the range of 80 GeV to 160 GeV [42]. This gives the bounds, for example, and 15 for and 150 GeV at 95% C.L. in the Type-X 2HDM using 0.23% of the product of the branching fractions.

Apart from the production processes via Yukawa couplings, one must take care of the decay in the case of . In the Type-X 2HDM, this typically gives the four final state, because the decay can be the main decay mode as explained in Sec. 3.1.1. In Ref. [43], the upper bound on is given to be about for and - for . In the 2HDMs, the branching fraction is determined by the dimensionless coupling defined as the coefficient of the vertex in the Lagrangian; i.e., which is given by


The partial decay width of is then expressed by


where MeV is the total decay width of the SM Higgs boson for  [44]. Therefore, to satisfy Br(), is required. We can simply take by setting an appropriate value of from Eq. (25) as


In the case of , and , we obtain


From the above expressions, we find that the SM-like behavior of is realized by taking , because of .

3.2 Electroweak precision observables

The extra Higgs bosons can modify the electroweak precision observables from the SM prediction via the loop effects. Such an effect can be used as an indirect search for the extra Higgs bosons and also used to constrain parameter space in the 2HDM. In this subsection, we discuss the constraints from the oblique parameters and the boson decay.

3.2.1 Oblique parameters

The electroweak oblique , and parameters are introduced by Peskin and Takeuchi [45] which parametrize new physics effects on the gauge boson two point functions. These parameters are calculated in 2HDMs in Refs. [46, 47, 48, 49, 50, 51, 52]. In the SM-like limit , these parameters are given to be the same formulae as those given in the inert doublet model [53]. For the case of , the contribution from the additional scalar bosons is given by


We also find that is the same order as in our setup for large regime. If we take and , the Higgs potential respects the custodial symmetry [48, 50], which makes . The and parameters driven by the Gfitter group [54] are


with the reference values of GeV and GeV. The prediction of parameter is inside the error of the measured value, and the parameter constrains on the mass splitting . Hence we take to avoid the constraint on the oblique parameters in the following analysis.

3.2.2 boson decay

The property of the boson such as the mass, the total width, and the decay branching ratios were precisely measured at the LEP experiment. If new physics particles modify such a precisely measured quantity, their masses and/or couplings are severely constrained.

In our scenario, the vertex can be significantly deviated from the SM prediction by loop effects of the extra Higgs bosons, because they strongly interact with charged leptons in the large case. In order to discuss how the modified vertex affects the observables, we define the effective vertex as


where and being the weak mixing angle. Although there are several definitions for , we here use the on-shell definition [55] of it which is determined by using and boson masses, i.e., . The effective vector coupling and axial vector coupling can be separately written by the contributions from the tree level and from the one-loop level as


where the tree level contributions are expressed as


with being the electric charge of . The loop contributions and are composed of the counter term and the one particle irreducible (1PI) vertex correction diagram:


After imposing the on-shell renormalization conditions, the counter term contribution is expressed by [56]


where are the 1PI diagram contributions to the fermion two point functions defined as


In the SM-like limit , the deviation in and purely comes from the extra Higgs boson loop diagrams. In this case for , we obtain


where the loop functions are given as


In the above expressions, we neglect the mass of the tau lepton in the loop functions. We note that the function is invariant under the interchange of , so that and does not change the value under .

Let us apply the modified vertex to the leptonic partial decay width of the boson :


We define the ratio of the partial width of to that of as


The deviation in the ratio from the SM predictions are then given by


The SM prediction is given by


The measured values of the leptonic decay width and are given by [60]


We find that is excluded for  (50) GeV when  GeV. The bound becomes weaker for the larger . We will combine the constraint from the decay with the muon result in Sec. 4.

3.3 Flavor experiments

Effects of the extra Higgs bosons can appear in various observables measured at flavor experiments. Therefore, similar to the electroweak precision measurements, flavor measurements can be used to constrain the parameter space in the 2HDM. In this subsection, we discuss and the leptonic decay of .


It was pointed out in Ref. [57] that the branching fraction of the process in the Type-II 2HDM is enhanced by the factor which comes from the box type and the penguin type diagrams with extra Higgs boson mediation. The origin of the dependence is that both the charged lepton and down-type quark couplings to the extra Higgs bosons are proportional to in the Type-II 2HDM. On the other hand, in the Type-X 2HDM, the lepton couplings are enhanced by , while the quark couplings are suppressed by . Thus, the dependence does not appear in the branching fraction of , and the additional leading contribution is almost independent on for large . Although the deviation from the SM becomes mild as compared to the case of Type-II, we check because the light CP-odd Higgs boson could give a sizable contribution, which is required to explain muon anomaly.

For the calculation of in the 2HDM, we use the formulae given in Ref. [58]. The observed branching fraction at the LHC is [27]. We show the constraint from on the parameter space of 2HDM in Sec. 4.

3.3.2 Leptonic decay at the tree level

Figure 2: The ratio of the decay rate (upper left and upper right) and the Michel parameters (lower left) and (lower right) as a function of in the leptonic tau decay. In each panel, outside regions from the dotted lines are excluded at the 95 % C.L.

In the SM, the leptonic decay is caused by the boson exchange diagram at tree level. In the 2HDM, the mediated diagram also contributes to the leptonic decay. The effect of contribution on the partial decay width of was calculated in Refs. [31, 11], and that on the Michel parameters, which is defined just below, in Ref. [32].

The differential decay rate of is given in terms of the Michel parameters ( and ) and defined in Eqs. (52) and (53) as [59]


where , and with being the muon energy. is the polarization of the tau, and is the angle between the polarization and the momentum direction of the muon. The functions and are defined as


By using , we find222We find these expressions are inconsistent with Ref. [32]


We see that and are equal to the SM values at the tree level. The observed Michel parameters of the decay are and [60]. The ratio of the decay rate in the 2HDM to that in the SM prediction is given as [31, 11]