Dielectron and Diphoton channels in 2HDM

It was observed recently by Randall and Wise that, any particle in the Standard Model cannot decay to $e^+e^-$ and $\gamma \gamma$ final states with comparably measurable branching ratios. This is also true for most extensions of the Standard Model, with the Randall-Sundrum model as an outstanding exception. In this paper, we show that two-Higgs-Doublet-Models (2HDM) yield another possible exception if certain parameters are properly chosen. In addition, we have checked that this model survives the tests of low energy processes, including the anomalous magnetic moment and electric dipole moment of leptons, lepton-flavor-violating decays $\mu^- \to e^- \gamma$ and $e^- e^+ e^-$.


I. INTRODUCTION
It was observed recently that [1], no resonance can decay to e + e − and γγ with comparably First, let us recapitulate the arguments [1] why e + e − and γγ channels do not appear simultaneously in the SM. Due to conservation of angular momentum, fermionic resonance can decay to neither e + e − nor γγ final states. Furthermore, Landau-Yang theorem [3] prohibits a spin-1 resonance decaying into two photons, which follows from general assumptions of Lorentz invariance, gauge invariance, and Bose symmetry of the photons. In principle, a spin-0 resonance can decay to e + e − either through the electroweak gauge interaction or the Yukawa interaction. However, the gauge channel is helicity suppressed while the Yukawa channel is suppressed by the ratio of electron mass to the vacuum expectation value (VEV) of the SM Higgs. For a resonance of spin-2 or higher spin, e + e − channel is forbidden at tree level simply because the resonance can not couple directly to a virtual spin-1 gauge boson.
The arguments hold also true for the case of dimuon and diphoton signals, if the muon mass is much smaller than the energy scale of the resonance.
Thus comparably measurable dielectron and diphoton signals clearly imply that leptons must have direct interaction with either a spin-2 (or higher spin) boson or a new scalar beyond the SM Higgs. The former possibility has been examined in [1] where one gets the KK modes of RS graviton as a concrete example. This is actually not hard to understand, as gravity couples universally to energy-momentum tensor of any fields, including electrons and photons. In this letter, we will instead focus on the latter case, by considering models with new scalar fields that can decay into dielectron and diphoton with comparably measurable rates. In particular, we will concentrate on extensions of the SM with an extra Higgs doublet, which is usually dubbed as the two Higgs doublet models (2HDM).
In 2HDM, there are eight degrees of freedom in the scalar sector to start with. After spontaneously symmetry breaking, five scalars will remain but two of them are charged.
Among the three neutral ones, one resembles the SM Higgs and the additional two could play the role of new resonances. These neutral scalars couple to electron directly through Yukawa interactions. To avoid constraints on the Yukawa couplings due to the light electron mass compared to the electroweak scale, the VEV of the additional Higgs doublet must be (almost) zero. This type of model is usually referred to as type III 2HDM, in which the neutral scalars can decay to e + e − at tree level with significant rate. Of course, care should be taken to avoid large FCNC and CP violation effects. On the other hand, the γγ decay is loop suppressed in these models and smaller than that in the SM due to the absence of contributions from gauge particles. To make a sizable diphoton decay rate, some mechanism for enhancement is needed for this loop level process. In the letter, we will see that such enhancement is still feasible within current experimental limits. Putting all things together, the extra scalars can be made to decay into both e + e − and γγ final states with comparably measurable branching ratios. As one would have expected, here one has to choose parameters properly and certain level of fine tuning is necessary. This paper is organized as follows: In Section II, we will show how to get significantly comparable branching ratios for e + e − and γγ channels in 2HDM. Phenomenological constraints on these models are presented in Section III, including the anomalous magnetic

II. COMPARABLE BRANCHING RATIOS IN TYPE III 2HDM
In Type III 2HDM, one scalar doublet resembles the role of the SM Higgs boson, which will be denoted as φ 1 . We denoted the additional doublet as φ 2 . The general interaction between φ 2 and fermions reads with left (right) handed projection L(R) = (1 ∓ γ 5 )/2,Q Li (l Li ) are left handed quark (lepton) doublets, U Rj , D Rj and E Rj are right handed up quark, down quark and lepton singlets, respectively, with family indices i, j. In general, all Yukawa matrices ξ U,D,E ij are non-diagonal and complex 1 . This may result in FCNC and CP violating effects, which will be addressed in the next section. The general Higgs potential [4] which spontaneously breaks where the λ i 's are all real parameters.
In type I and type II 2HDM, the non-zero VEVs of φ 1 and φ 2 make contributions to fermion masses proportional to the corresponding Yukawa couplings in (1). The branching ratio of the e + e − final state from scalar particles is severely suppressed by the tiny electron mass. In type III 2HDM, those Yukawa couplings are not confined by such constraints, since φ 2 has a (almost) zero VEV. One might hope to find viable models with comparably measurable branching ratios for e + e − and γγ channels.
Only φ 1 has a non-zero VEV while φ 2 does not in type III 2HDM. That is, We can thus choose a gauge such that In this gauge, the potential in (2) can be re-expressed as, The internal lines represent either leptons or quarks.
One sees that H 1 , H 2 and h are mass eigenstates, i.e., there is no mixing between the Higgs bosons. To be definite, H 1 is always assumed to be lighter than H 2 , without losing any generality. Now we will see how to make H 1 decaying into both e + e − and γγ channels with comparable and significant rates.
It is clear from (1) that H 1 decays into e + e − at the tree level. By ignoring the electron mass, the decay width is where M H 1 is the mass of H 1 and ξ E ee is the Yukawa coupling between H 1 and electron. Meanwhile, H 1 decays to γγ through triangle fermion loop (see Fig. (1)). Notice that the triangle gauge boson loops, which are dominant in the SM, are absent here. This is because that the term (D µ φ 2 ) † (D µ φ 2 ) only results in a four-point vertex φ + 2 φ 2 B µ B µ due to the vanishing vev of φ 2 . Similarly the scalar loops do not appear here either.
with F µν = 1 2 ǫ µνρσ F ρσ . There are both CP even amplitude A + and CP odd amplitude A − , as the Yukawa couplings are complex, where t j = M 2 j /M 2 H 1 , F = E, U or D for the virtual fermions to be charged leptons, up quarks or down quarks. Q j are their charges. The decay width of H 1 → γγ is thus, which agree with results in [5].
Now we are in a position to make H 1 decays into e + e − and γγ with comparable and significant rate. To enhance the loop-suppressed γγ channel, the Yukawa couplings of H 1 to heavy leptons and/or quarks should be much larger than that of H 1 to electron. Let us first consider case of a quark-phobic φ 2 . We need then a heavy lepton to have a large coupling with φ 2 , which we may, for example, choose it to be the τ lepton. Taking the mass of H 1 to be a few hundred GeV, |ξ E τ τ |/|ξ E ee | > 10 3 is required to make the branching ratios for e + e − and γγ comparable. However with this parameter set, the e + e − channel would be too rare to be observable: So a quark-phobic φ 2 will not work.
At first glance, the same would happen if φ 2 has large couplings to quarks. Naively, Br(H 1 → e + e − ) ≪ Br(H 1 → qq) would make e + e − final state too scarce to be observed.
Fortunately, if the mass of H 1 is smaller than 340 GeV, roughly twice of top quark mass, H 1 cannot decay to top quark pairs. In this case, one can make the e + e − and γγ decay channels comparable and observable, by having ξ U tt ≫ ξ E ee and keeping the Yukawa couplings between φ 2 and the other quarks small. To get Γ(H 1 → e + e − )/Γ(H 1 → γγ) ≃ 1, one needs the ratio of Yukawa couplings to be ξ U tt /ξ E ee ≃ 1800. To make e + e − and γγ decay channels observable, we may assume universal coupling of H 1 to all fermions except the top quark 2 . With this parameter set, the main decay channel will be H 1 → gg. It is then easy to estimate the branching ratios: which is not too small and should be measurable. Therefore with such a specific parameter set, it would be possible for Type III 2HDM to produce significant and comparable branching ratios for e + e − and γγ decays.
In this parameter set, the production of H 1 at hadron collider is mainly through gluon fusion, which is similar to that of the SM Higgs. Therefore, to copiously produce H 1 at an observable rate, one would expect ξ U tt to be of order one. Generically, one needs ξ E ee to be no less than 10 −4 . In the following, we will discuss whether this is possible concerning phenomenological constraints from low energy processes.

III. PHENOMENOLOGICAL CONSTRAINTS AND IMPLICATIONS
With extra not-so-heavy Higgs particles, the type III 2HDM has rich phenomenological implications at low energy. For example, it may contribute to the anomalous magnetic moments of leptons, which have been measured to very high precision. Complex phases of the Yukawa couplings of H 1 to leptons may contribute to leptonic EDMs at one-loop level.
Finally, non-diagonal Yukawa couplings of H 1 to leptons would lead to LFV processes, such as µ → eγ and µ − → e − e + e − . Non-diagonal Yukawa couplings of H 1 to quarks might produce large hadronic FCNC processes and complex phases may induce CP violating effects in hadrons, which are both tightly constrained experimentally. However these couplings do not affect e + e − and γγ signals at all, therefore we will simply assume that their effects can be tuned away and not consider them further.
The complex Yukawa matrix for leptonic couplings ξ E ij contains 18 real parameters, which are obviously too many. So we will consider two extreme scenarios, just for illustration, in numerical discussions: Scenario I (S1): the leptonic Yukawa matrix is diagonal with a universal CP phase π/6, i.e. ξ E ij = |ξ ij |e iθ ij , with |ξ ij | = 0(for i = j) and θ ii = π/6(for i = e, µ, τ ). Scenario II (S2) all leptonic Yukawa couplings are assumed to have universal magnitude with CP phase π/6, namely ξ E ij = ξe iπ/6 . Actually, numerical analyses in these two scenarios yield rather reasonable estimates on constraints on the general Yukawa matrix. The extra scalars in 2HDM may contribute to the anomalous magnetic moment at one-loop level, as shown in Fig. (2). We will assume a 2HDM e < 10 × 10 −12 which is slightly larger than the theoretical error, to get a restriction on the corresponding Yukawa couplings. For muon, there is some discrepancy between experiments and the SM calculation [6]: a exp µ − a SM µ = (292 ± 63 ± 58) × 10 −11 , so we assume an upper limit a 2HDM µ < 292 × 10 −11 .
In S1, the expression (12) is proportional to M 2 e (M 2 µ ) for electron (muon) anomalous magnetic moment. That is why, though the upper limit on a 2HDM µ is about 300 times larger than that of a 2HDM e , the constraint ξ µµ < 0.64 from a 2HDM µ < 292 × 10 −11 is more severe than the limit ξ ee < 5.9 from a 2HDM e < 10 × 10 −12 .
In S2, the expression (12) is dominated by M e M τ (M µ M τ ) term for the electron (muon) anomalous magnetic moment. In this scenario, the electron g − 2 leads to ξ < 0.18, which is slightly smaller than the restriction ξ < 0.21 from the muon g − 2.

B. EDM
The electron and muon EDMs are extremely small in the SM, which arise at three-loop level. But in the present model, the complex Yukawa couplings give rise to non-zero EDMs at one loop level, as shown in Fig.(2). Thus these couplings might be severely constrained by the experimental data. The current experimental measurements cited by PDG2008 [6] are  [8] and d µ < 10 × 10 −19 e cm (95% C.L.) [9]. We will use these upper limits, instead of the PDG numbers, to obtain constraints. The leptonic EDM d i (i = e, µ) has contributions from neutral Higgs bosons H 1 , H 2 and charged one H + . The contribution from the internal H + is suppressed by the neutrino masses, which can be safely dropped.
The contribution from the neutral Higgs bosons is As expected, d i vanishes when the phase of ξ E ij is zero and will not yield any constraints on the Yukawa couplings. If only virtual τ lepton is considered inside the loop and the external lepton mass is neglected, our formula agrees with [11]. The above equation is also consistent with [10].
In S1, the expression (13) is proportional to M e (M µ ) for electron (muon) EDM. It is then straightforward to find ξ ee < 0.013 and ξ µµ < 31 from the experimental upper limits.
In S2, the expression (13) is dominated by M τ term for both d e and d µ . Therefore the constraint ξ < 3.9 × 10 −4 from d e is significantly tighter than the limit ξ < 9.6 from d µ .
The decay amplitude of µ → eγ can be expressed as where A L and A R are, For e − e + e − channel, ignoring the mass of electron, the decay width can be calculated as In S1, these LFV processes are forbidden because of the diagonal lepton Yukawa coupling.

IV. SUMMARY
We have so far found that, type III 2HDM may produce comparably observable e + e − and γγ signals simultaneously from decays of neutral scalars. This provides an interesting alternative to the RS model. To make this happen, certain amount of fine tunings are necessary. First, to enhance the loop suppressed γγ channel to be comparable with e + e − channel, the Yukawa coupling of H 1 to top quark (ξ U tt ) must be a few thousand times larger than those to the rest of fermions. In addition, the mass of H 1 should be smaller than twice of the top quark mass, to prevent H 1 decays predominantly to top-anti-top pair. Under these conditions, H 1 would decay mainly to two gluons, with the branching ratios of e + e − and γγ channels to be about half a percent.
Notice that these requirements determine only the relative strength of different Yukawa couplings, while their absolute sizes would determine the number of events produced in colliders. If the parameters in these models do meet the above requirements, H 1 will be mainly produced via gluon fusion in hadron colliders, analogous to the SM Higgs. For H 1 to be produced roughly at the same rate as the SM Higgs, ξ U tt needs to be of order one, which means that the electron Yukawa coupling ξ E ee should be around few ×10 −4 . Generically, the non-diagonal and complex Yukawa couplings lead to FCNC and CP violating processes. We have thus examined constraints from the anomalous magnetic moment and EDM of leptons, LFV processes µ − → e − γ and e − e + e − . Because of the large number (18 in total) of free parameters in Yukawa interactions, we have chosen two scenarios as illustration for numerical discussions. In the scenario of diagonal Yukawa couplings with an universal CP phase, we find the most stringent constraint ξ E ee < 0.013 from electron EDM, and ξ E µµ < 0.64 from muon g−2 measurement. In the scenario of universal Yukawa couplings, the most stringent constraint ξ < 3.9 × 10 −4 comes again from the electron EDM. In both scenarios, the constraint on ξ E ee is above the requirement from the dielectron and diphoton signals, so the model survives existing phenomenological tests. The same conclusion also holds true for general Yukawa interactions.
So, if resonances below 2M t are found in experiments to decay into e + e − and γγ with comparable rates, more work would be needed to delineate their origins. Under this circumstance, we have just seen that type III 2HDM provides one alternative possibility, in addition to the RS-model. It would be interesting to see whether there exist other possibilities.