Limits on Neutrinophilic Two-Higgs-Doublet Models from Flavor Physics

We derive stringent limits on neutrinophilic two-Higgs-doublet models from low-energy observables after the discovery of the Higgs boson and of the mixing angle $\theta_{13}$. These decays can constrain the plane spanned by $m_{H^\pm}$, the mass of the new charged Higgs, and $v_2$, the vacuum expectation value of the new neutrinophilic scalar doublet. Lepton flavor conserving decays are not able to set meaningful bounds, since they depend strongly on the unknown neutrino absolute mass scale. On the other hand, loop induced lepton flavor violating decays, such as $\mu \to e \gamma$, $\mu \to 3 e $ or $\mu \to e$ in nuclei are currently responsable for the best limits today. If $v_2 \lesssim 1 \, (0.1)$ eV we get $m_{H^\pm} \gtrsim 250 \, (2500)$ GeV at 90% CL. In the foreseen future these limits can improve by at least a factor of 100.


Introduction
The discovery of neutrino oscillations leaves basically no doubts on the fact that neutrinos are massive particles. Still, whether neutrinos are Dirac or Majorana particles is an open question. On a general ground, new physics in the neutral and charged lepton sectors are expected to be connected.
As it is well known, the easiest way to generate neutrino masses is via the addition of at least two right-handed (RH) neutrinos to the Standard Model (SM) particle content, with a Yukawa interaction L ⊃ H L Y ν ν R . Since the ν R are necessarily gauge singlets, no gauge symmetry can forbid a Majorana mass term L ⊃ M R ν R ν R . Light neutrinos are a natural outcome in a seesaw scenario [1], where M R is assumed to be very large. Once the RH neutrinos are integrated out, the Weinberg operator ( H) 2 /Λ is generated [2], producing Majorana neutrinos. In addition, heavy RH neutrinos may also generate the right amount of matter-antimatter asymmetry through leptogenesis [3].
On the contrary, if we insist on light RH neutrinos (i.e. small M R ), small neutrino masses require Y ν ∼ 10 −(12÷13) , roughly 7 orders of magnitude smaller than the electron Yukawa coupling. Moreover, the presence of the RH Majorana mass makes the neutrinos pseudo-Dirac particles, rather than Dirac. In this case, baryogenesis is still possible through neutrinogenesis [4], but the stringent cosmological limits on the number of relativistic degrees of freedom (RDOF) obtained by Planck [5] are a potential shortcoming. The easiest way out is to impose the light ν R to not contribute at all to the RDOF, i.e. to decouple from the plasma well before Big Bang Nucleosynthesis.
The tiny Yukawa couplings in the pseudo-Dirac case are of course a possibility, as they may find an explanation in a theory of flavor (just like any theory of flavor would explain the 5 orders of magnitude difference between the top and the electron Yukawa couplings). However, to get pure Dirac states, an additional symmetry must be imposed to forbid the RH Majorana mass. To avoid charging also some SM state under the same symmetry, this is most easily realized decoupling neutrino masses from the SM Higgs doublet. A second scalar, charged under the additional symmetry, can thus be introduced, whose vacuum expectation value (vev) is responsible for neutrino masses [6]. If the vev happens to be in the eV range, we obtain the correct order of magnitude for the neutrino masses with O(1) Yukawa couplings. Models with such a neutrinophilic doublet and pure Dirac neutrinos were studied in some detail in [7]. Recently, after the discovery of the Higgs boson by the LHC experiments, these minimal Neutrinophilic Two-Higgs-Doublet Models (ν2HDM) were revisited and strong bounds were imposed on their scalar spectrum [8].
We focus here on additional experimental consequences from charged lepton flavor physics on ν2HDM, in particular, after the last neutrino mixing angle, θ 13 , was precisely measured by the reactor experiments Daya Bay [9] and Double Chooz [10]. Note that non-standard interactions in the neutrino sector only involve RH neutrinos in these models, so they are not expected to affect neutrino oscillations.
In Sec. 2 we describe the main features of ν2HDM that are important to understand how they modify lepton flavor physics. In Sec. 3 we calculate the most constraining processes involving charged leptons that are affected by ν2HDM: tree-level flavor conserving leptonic decays, loop induced flavor violating leptonic decays, tree-level flavor violating Z and Higgs decays. We derive current bounds on the model parameters by using the most precise experimental data available today and estimate the improved sensitivity of future experiments to these parameters. In Sec. 4 we present our final conclusions.

A brief description of the model
Let us now describe the class of models we are interested in. We extend the SM particle content to include a second Higgs doublet H 2 , as well as three RH neutrinos ν R . H 2 has the same gauge quantum numbers as the ordinary Higgs H 1 , while the RH neutrinos ν R are gauge singlets. The interactions we are interested in are given by where H 2 = iσ 2 H * 2 is the conjugate of H 2 . The previous lagrangian can be easily obtained requiring H 2 and ν i R to have the same charge under an additional global U (1) X [7], or imposing a Z 2 symmetry [6]. In the following we will not be concerned with the specific realization from which Eq. (2.1) is obtained, focusing only on its consequences on flavor processes. Using the PMSN matrix U to express the gauge eigenstates ν α in terms of the mass eigenstates ν i , ν α = U αi ν i , from Eq. (2.1) we get 2 ) T . Let us stress that if an exact U (1) X symmetry is present also in the scalar potential, A is a strictly massless Goldstone boson. This calls for an explicit symmetry breaking to make the model phenomenologically viable, which in [7] is given by a soft term in the potential. 1 Given its importance for the following sections, let us rewrite the coupling involving the charged Higgs in the neutrino flavor basis: It is clear that apart from the v 2 dependence, the coupling is completely fixed in terms of (known) neutrino parameters. Integrating out the massive charged Higgs boson, we obtain where m αβ = U αi m ν i U * βi and only the non-standard neutrino interaction term is displayed. As already anticipated in the introduction, this term only involves RH neutrinos. Since Eq. (2.4) does not have an analog term in the quark sector, we do not expect RH neutrinos to be produced at a relevant rate by nuclear processes, for instance, in the sun, in such a way that there is no modification of neutrinos propagation through matter.

Experimental Constraints from Charged Lepton processes
The phenomenology of a neutrinophilic charged Higgs in low-energy processes is different from the one of a generic 2HDM, mainly because the couplings with leptons are highly enhanced by a factor v 1 /v 2 1, while the couplings with quarks are highly suppressed by v 2 /v 1 1. As a first consequence, the ν2HDM easily evades the limits coming from hadronic observables such as B → X s γ, mesons mixing and, more importantly, leptonic and semi-leptonic B meson αα as a function of the lightest neutrino mass m 0 obtained scanning over the 1σ range of the oscillation parameters [15]. Blue region: α = e, green region: α = µ, red region: α = τ . The left panel refers to the Normal neutrino mass ordering (NO), the right panel to the Inverted ordering (IO). The gray points are excluded by Planck's limit on the sum of neutrinos masses [5,16].
decays [12][13][14]. On the other hand, leptonic observables (normally suppressed by neutrino masses) may now receive sizable contributions, as we are now going to show. Our main results are summarized in Fig. 4, where we show the current bounds (left panel) and the expected future sensitivities (right panel) on the (m H ± , v 2 ) plane.

Lepton Flavor Conserving Decays
Let us look at the tree level µ and τ leptonic decays. The charged scalar contribution to α → βν ν induces a violation of lepton flavor universality (LFU), which can be effectively encoded in the definition of "flavorful" gauge couplings g α [17]. Experimentally, they can be measured from the τ and µ lifetimes. The total decay width for α → β νν in the presence of a charged Higgs boson can be written as Γ Although from Eq. (3.1) it may look like it is possible to use the experimental results on g µ /g e and g µ /g τ to extract bounds on ρ, we stress that (i) the experimental data are well compatible with lepton flavor universality at 1 σ, 2 and (ii) although the differences m 2 αα − m 2 ββ are independent of the value of the lightest neutrino mass m 0 , the individual values of m 2 ee and m 2 τ τ depend crucially on m 0 , as shown in Fig. 1. From this plot, it is clear that the flavorful couplings reach very small values for m 2 0 ∆m 2 ij , making them always compatible with the currents bounds on LFU [17]. Since the absolute neutrino mass scale is still unknown, no information can be extracted from these observables.

Lepton Flavor Violating Decays
We will study here loop induced lepton flavor violating (LFV) processes that can currently, or in the near future, be constrained by data, and the corresponding consequences on the allowed values of the ν2HDM parameters.

3.2.1
α → β γ Let us start with loop induced processes, for which strong experimental constraints are available, at least in the µ → eγ channel. For a generic process α → β γ, the scalar mediated branching ratio reads [18] The strongest experimental bound on this type of process comes from the MEG-2 upper limit BR(µ → eγ) < 5.7 × 10 −13 [22], while weaker bounds on the other channels are obtained by the BaBar Collaboration, BR(τ → eγ) < 3.3 × 10 −8 and BR(τ → µγ) < 4.4 × 10 −8 [23]. In terms of ρ (defined below Eq. (3.1)), we get the 90% C.L. bounds 3 This is the best limit at present on the parameters v 2 and m H ± , and it already implies that, insisting on v 2 1 eV, we must have m ± H 250 GeV. With the future improvement on the MEG expected sensitivity, BR(µ → eγ) ∼ 5 × 10 −14 [24,25], the corresponding bound on ρ can be improved by about one order of magnitude to ρ 0.4 eV −2 . The limits imposed by the MEG bound on the (m H ± , v 2 ) plane are shown in Fig. 4, blue line, for the current result (left panel), as well as for the expected future sensitivity (right panel).
We would like to point out that in case a positive sign of µ → eγ is observed in the near future, ν2HDM predicts a relation between BR(µ → eγ) and BR(τ → eγ, µγ) which is actually sensitive to δ CP . We show in Fig. 2 the ratio of these branching ratios as a funtion of δ CP . Although it is very unlikely to be able to experimentally probe BR(τ → eγ, µγ) down to 10 −14 in the near future, from Fig. 2 we see that a limit on BR(µ → eγ) would also set a universality assumption [21].

3.2.2
α → 3 β We now turn our attention to processes involving three charged leptons in the final state. These processes can be described by the loop induced effective lagrangian L eff = e m α 2 A D¯ β σ µν α F µν + eA N D¯ β γ µ P L α A µ + e 2 B(¯ α γ µ P L β )(¯ β γ µ P L β ) + h.c., (3.4) where m α is the mass of the charged lepton in the initial state, and A (N )D and B are Wilson coefficients associated to γ-penguin diagrams and charged Higgs boxes, respectively. Neglecting neutrino masses and imposing m β /m α 1, we derived the following expressions, (3.5) (3.6) where q 2 is the photon squared momentum in the penguin-like diagrams 4 . We neglect the Z-penguin diagrams, since they are suppressed by m β and by the Z boson mass.
Our results for the Wilson coefficients are in full agreement with those of Ref. [26], where the impact of the scotogenic model on µ → 3e was studied. Although these models are intrinsically different, one can retrieve the ν2HDM loop functions by replacing the mass of RH neutrinos by the active ones and by matching the Yukawa lagrangians of the two models. The only difference between the two calculations is a new Z-penguin diagram with two neutrinos in the loop, which is only present in the ν2HDM model (see Fig. 3). This diagram does not exist in the scotogenic model, because the Z 2 symmetry forbids the mixing between the active and sterile neutrinos. However, this diagram is additionally suppressed by the active neutrino mass, giving a negligible contribution.
In terms of the coefficients defined above, the branching ratio reads where the sub-leading terms in m β /m α have been neglected. Notice that the box terms have a different dependence on v 2 and therefore this expression cannot be expressed only in terms of ρ. Moreover, from Eqs. For relatively large values of v 2 , in the region where the penguin diagrams dominate, we can use the current experimental limit, BR(µ → e − e − e + ) < 1 × 10 −12 [27] to directly put the bound ρ 22 eV −2 . This is not possible for small v 2 , in the region where the box dominates, since the corresponding Wilson coefficient cannot be expressed in terms of ρ. The total bound is shown in Fig. 4, green line. The current experimental limit (left panel), is stronger than the limit derived from µ → eγ for v 2 0.01 eV. The situation will change with the future Mu3e experiment, which aims to reach an ultimate sensitivity of BR(µ → e − e − e + ) ∼ 1 × 10 −16 [28]. As can be seen from Fig. 4, right panel, the future sensitivity on µ → 3e is expected to be stronger than the one from µ → eγ.

µ → e in nuclei
The µ−e conversion in nuclei is another LFV process that appears in ν2HDM. It is important to note that the experimental collaborations have announced great future sensitivities, making this a relevant bound for different neutrino mass models. In our framework, the dominant contributions are only the γ-penguins, since the Z-penguins are suppressed by the electron and Z boson masses, while box diagrams and scalar penguins are suppressed by the tiny  Table 2. Present bound and future sensitivity on the µ − e nuclear conversion rate [35].
coupling of the neutrinophilic scalars with quarks. Keeping only the dominant contributions, the conversion rate is given by [26,29,30] CR(µ − e, nucleus) = where p e , E e ≈ m µ are the electron momentum and energy, respectively, which are approximately equal to the muon mass; Z, N are the number of protons and neutrons in the nucleus, Z eff is the effective atomic charge and F p is the nuclear matrix element, given in Table 1 for the nuclei we are considering in our study. Notice that the conversion rate is normalized to the muon capture rate Γ capt . The coefficients g (0,1) LV are given by [29,30] g (0) We stress that we consider only vector couplings, since only the γ-penguins are relevant for this process. Also, we should note that only valence quarks will be relevant for our purpose, because the sea quarks, like the strange quark, interact effectively only through the scalar part [29]. The coupling g LV q is given by where Q q is the quark charge. For completeness, we quote here the values of the coefficients G (q,p) V [29]: From the present bounds on the µ − e conversion rate, Table 2, we get ρ 30 eV −2 for titanium (Ti), and ρ 13.5 eV −2 for gold (Au), while from the future expected sensitivities in aluminium (Al) and titanium we get ρ 0.020 eV −2 and ρ 0.015 eV −2 , respectively. We stress that µ − e conversion in nuclei will be, in the future, the most sensitive process to probe ν2HDM, if the experiments reach the announced sensitivities. In Figure 4 we see how µ − e conversion sets limits on the (m H ± , v 2 ) plane using the present results (left panel) and

Z → α β
Besides their impact on charged lepton decays, neutrinophilic scalars can give rise to LFV Z boson decays. In our framework, the additional one-loop diagrams contributing to this process are shown in Fig. 3, where α and β are charged leptons of different flavors. The effective Hamiltonian can be written as Neglecting fermion masses, the Wilson coefficient C V is given by In the x Z 1 limit this expression can be further simplified to The Wilson coefficient C V directly enters in the expression for the Z → α β decay width, which in the limit of vanishing lepton masses is given by where m Z and Γ Z are the Z mass and total decay width and we took into account that . On the experimental side, the most constraining bound comes from the ATLAS upper limit BR(Z → e ± µ ∓ ) < 7.5 × 10 −7 [36]. The channels with a τ lepton in the final state were only studied at LEP and have weaker experimental limits, BR(Z → e ± τ ∓ ) < 9.8 × 10 −6 and BR(Z → µ ± τ ∓ ) < 1.2 × 10 −5 [37].
Using ATLAS current limit on Z → e ± µ ∓ we get an upper bound ρ 3.5 × 10 3 eV −2 , much weaker than any of the bounds presented so far. Even considering the expected sensitivity at a future electron-positron collider operating at the Z pole (TLEP), BR(Z → e ± µ ∓ ) ∼ 10 −13 [35], the situation is not going to improve much. Instead, if we consider the current bounds on the parameter ρ coming from µ → eγ, then we can predict BR(Z → e ± µ ∓ ) 10 −17 and BR(Z → µ ± τ ∓ ) 10 −16 .

h → α β
In this section we briefly discuss the LFV process h → α β , where h denotes the SM-like Higgs. The effective Hamiltonian describing this decay can be written as (3.17) Assuming m β /m α 1, the Wilson coefficient C L reads with y H = m 2 h /m 2 H ± . The coupling g hH + H − is defined as the trilinear coupling hH + H − and depends on the particular realization of the scalar sector. In the asymptotic limit, (3.20) The decay rate (3.21) In order to make predictions for LFV Higgs decays, one needs to consider a specific realization of the scalar potential and scan over the parameter space allowed by theoretical and phenomenological constraints. Here, we consider the model proposed in Ref. [7] which, as discussed in [8] is the only minimal realization of the ν2HDM still consistent with electroweak precision measurements. Scanning over the allowed parameter space of this model (see [8] for details), we obtain a prediction for Γ(h → µτ ), the largest LFV Higgs decay, as a function of BR(µ → eγ), as shown in Fig. 5. We see that due to the stringent limit imposed by MEG-2, Γ(h → µτ ) cannot exceed 10 −9 MeV, so this model cannot possibly explain a branching ratio as high as BR(H → µτ ) = (0.84 +0.39 −0.37 ) % as measured by CMS [38]. Unfortunately, such a small branching ratio is also completely out of the reach of the LHC or even of any foreseen future Higgs precision experiment.

Conclusions
We have focused our study on the neutrinophilic two-Higgs-doublets model scenario, ν2HDM, in which a neutrinophilic Higgs doublet is responsible for neutrino masses through a tiny vev v 2 1 eV and in which neutrinos are Dirac particles. Interestingly, in this scenario indirect limits are much more effective in constraining the parameter space than direct collider searches. This is due to the fact that the new scalars basically only couple to leptons and gauge bosons, in such a way that only the quite weak direct limit from LEP applies, m H ± 80 GeV [39].
Indirect limits can come either from lepton flavor conserving or from lepton flavor changing processes. The important point is that such flavor effects are controlled by the effective neutrino mass m 2 αβ , defined below Eq. (3.1), so that they can be well predicted now that, thanks to the measurement of the last mixing angle θ 13 , we are entering in the era of precision neutrino physics. This allows to put stringent limits on two of the unknown parameters of the ν2HDM, namely the mass of the neutrinophilic charged Higgs boson, m H ± , and the vev of the neutrinophilic doublet, v 2 . Let us stress that, although the neutrinophilic charged Higgs boson modifies the neutrino propagation in matter, this modification affects only the right-handed neutrinos, which are not produced with a relevant rate in the sun.
In this paper we have investigated limits coming from µ → eγ, τ → µ(e)γ, µ → 3e, µ → e in nuclei, lepton flavor violating Z and Higgs decays. Other limits, like the one coming from Big Bang Nucleosynthesis, turn out to be weaker (see Appendix A). Our main results are summarized in Fig. 4. On the left panel we show the current bounds, which are dominated by µ → eγ for v 2 0.01 eV and by µ → eee for v 2 0.01 eV. For example, in the region dominated by µ → eγ, we get a lower bound of m H ± 250 (2500) GeV for v 2 1 (0.1) eV at 90% CL, while in the region dominated by µ → 3e the lower bound on the charged Higgs boson mass is worse than (30 − 40) TeV. Since the philosophy of the ν2HDM is to allow for O(1) Yukawa couplings in the neutrino sector, we do not expect the v 2 0.01 eV region to be particularly relevant. In the right panel of Fig. 4 we instead show the future sensitivity, in which the limits could be largely dominated by µ − e conversion in nuclei. In this case, if nothing is observed, we expect the lower bound for v 2 = 1 eV to get as stringent as m H ± 2 TeV, and the one for v 2 = 0.1 eV to become m H ± 20 TeV.
Let us conclude mentioning that this model predicts lepton flavor violating Higgs decays, which is an interesting possibility in light of the recently CMS observation of an excess in the h → µτ channel [38]. Unfortunately, the region compatible with the observed value for the branching ratio is already excluded by the µ → eγ limit, Fig. 5, in such a way that such an excess cannot be observed in the ν2HDM framework. Similarly, BR(Z → α β ), with α = β , is constrained to be 10 −16 by the experimental limit on µ → eγ, making this observable beyond the reach of the current experiments.