Vacuum stability in neutrinophilic Higgs doublet model

A neutrinophilic Higgs model has tiny vacuum expectation value (VEV), which can naturally explain tiny masses of neutrinos. There is a large energy scale hierarchy between a VEV of the neutrinophilic Higgs doublet and that of usual standard model-like Higgs doublet. In this paper we at first analyze vacuum structures of Higgs potential in both supersymmetry (SUSY) and non-SUSY neutrinophilic Higgs models, and next investigate a stability of this VEV hierarchy against radiative corrections. We will show that the VEV hierarchy is stable against radiative corrections in both Dirac neutrino and Majorana neutrino scenarios in both SUSY and non-SUSY neutrinophilic Higgs doublet models.


Introduction
The recent neutrino oscillation experiments gradually reveal a structure of lepton sector [1,2]. However, from the theoretical point of view, smallness of neutrino mass is still a mystery and it is one of the most important clues to find new physics beyond the standard model (SM). A lot of ideas have been suggested to explain the smallness of neutrino masses comparing to those of quarks and charged leptons. How about considering a possibility that the smallness of the neutrino masses is originating from an extra Higgs doublet with a tiny vacuum expectation value (VEV). This idea is that neutrino masses are much smaller than other fermions because the origin of them comes from different VEV of different Higgs doublet, and then we do not need extremely tiny neutrino Yukawa coupling constants. This kind of model is so-called neutrinophilic Higgs doublet model [3]- [13], where a neutrinophilic Higgs take a VEV of O(0.1) eV in Dirac neutrino scenario [5,6,8,9], while a VEV of O(1) MeV in Majorana neutrino scenario with TeV-scale seesaw [3,4,7,10,11,12,13]. The non-supersymmetric (non-SUSY) neutrinophilic Higgs doublet model is sometimes called νTHDM. The (collider) phenomenology in νTHDM is interesting, since a charged Higgs boson is almost originated from the extra neutrinophilic Higgs doublet and its couplings to neutrinos are not small. The characteristic signals of the νTHDM could be detected at LHC and ILC experiments [9,11]. Not small neutrino Yukawa couplings in the νTHDM can also make low energy thermal leptogenesis work [12]. The SUSY version of neutrinophilic Higgs doublet model have been suggested in Refs. [12,13], where a thermal leptogenesis in a low energy scale works without gravitino problem [12,13] Anyhow, a neutrinophilic Higgs model has tiny VEV, and there is a large energy scale hierarchy between a VEV of the neutrinophilic Higgs doublet and that of usual SM-like Higgs doublet. In this paper, we at first analyze vacuum structures of Higgs potential in both SUSY and non-SUSY neutrinophilic Higgs models, and next investigate a stability of this VEV hierarchy against radiative corrections. We will show that the VEV hierarchy is stable against radiative corrections in both Dirac neutrino and Majorana neutrino scenarios in both SUSY and non-SUSY neutrinophilic Higgs doublet models.

νTHDM
Let us analyze vacuum structures of Higgs potential in non-SUSY neutrinophilic Higgs model, i.e., νTHDM at first, and next investigate a stability of this VEV hierarchy against radiative corrections.

Vacuum structure in tree-level potential
We here overview the νTHDM, where we introduce a neutrinophilic Higgs doublet Φ ν and Z 2 -parity as follows.
1. |v 2 | |v 1 | case: This vacuum is what the νTHDM wants to realize. The magnitudes of VEVs are given by and a potential height at the vacuum is given by 3), and thus, The case of v 2 where a potential height is given by 3. |v 1 | ∼ |v 2 | case: Neglecting tiny parameter m 2 3 , the stationary conditions Eqs.(2.3) and Then, VEVs are given by and the potential height at the vacuum is estimated as Notice that the νTHDM wants to realize the vacuum in Eq.(2.5) so that this vacuum at This is a necessary condition for v 1 v 2 to be the global minimum, and an additional condition − 2λ 2 makes the vacuum true global minimum. For the potential to be bounded from below [10,16], quartic terms must satisfy These are the conditions of bounded below of the Higgs potential. We can show that a case ofλ < 0 cannot satisfy the global minimum condition. Therefore, only a case ofλ > 0 can satisfy the global minimum condition. Thus, in order for the desirable vacuum v 1 v 2 to be the global minimum, a condition is needed. Next, let us estimate a curvature (mass squared) at each vacuum, which is given by Then, the eigenvalue equation (eigenvalue: x) is given by and we can estimate the curvature for above three cases.
The squared masses of the charged Higgs and of the pseudoscalar must be also positive. These conditions are equivalent to Summarizing conditions for the vacuum we want, at first,λ 2 − λ 2 λ 2 must be positive for the vacua of |v 1 | |v 2 | and |v 1 | |v 2 | to be lower than that of |v 1 | ∼ |v 2 |, and − makes the vacuum of |v 1 | |v 2 | the global minimum. Note thatλ must be also positive to be consistent with the conditions of the potential bounded from below. Next, positive curvature conditions are m 2 2 > 0 orλm 2 1 + λ 1 m 2 2 > 0 with m 2 2 < 0. Finally, positive curvature of the charged Higgs and the pseudoscalar components require m 2 2 + λ 3 v 2 1 > 0 and m 2 2 + (λ 3 + λ 4 − λ 5 )v 2 1 > 0 at |v 1 | |v 2 |. In Table 1, we show which vacuum becomes the global minimum depending on signs of m 2 2 ,λ, andλ 2 − λ 1 λ 2 . Can a "local minimum" at |v 2 | |v 1 | in (2), (4) and (5) be our vacuum? It might be possible if a life time of the local minimum is long enough. There is a transition process from the local minimum at |v 2 | |v 1 | to the global minimum at |v 1 | ∼ |v 2 |. Its transition probability of tunneling rate suggests the life time is much shorter than an age of our universe, since a "distance" and a "height" of wall between the local and global minimums are both O(100) GeV with O(1) couplings of λ i in Higgs potential. So, unfortunately, the local minimum cannot be our vacuum. Therefore, in the νTHDM, we must use the suitable parameter setup as (1) Before closing this subsection, we comment on recent analyzes of vacuum structure in general THDM. For example, in Ref. [16], they investigated the vacuum instability of charge and/or CP breakings at tree level. As for so-called Inert Doublet Model (IDM) [17], it has exact Z 2 -symmetry with m 2 3 = 0. This Inert Doublet does not couple with any matter fermions, which is crucial difference from our model.

Stability against radiative corrections
Now we are in a position to investigate the stability of the VEV hierarchy |v 2 | |v 1 | against radiative corrections. First of all, we should remind that the small magnitude of |m 2 3 | plays a crucial role for generating the tiny VEV of |v 2 |( |v 1 |). Its smallness is guaranteed against radiative corrections, since it is the "soft" breaking mass parameter of the Z 2 -symmetry. As noted in Ref. [8], the radiative correction to this parameter is expected to be logarithmic. For analyses of the vacuum stability, we should use Coleman-Weinberg type 1-loop effective potential [18], and analyze the stability of the VEV hierarchy. This 1-loop effective potential contains infinite number of irrelevant operators with zero-momentum Higgs fields in the external lines, and is calculated by a summation of them. However, for the investigation of stability of the VEV hierarchy, it is enough for us to pick up only diagrams which have external lines of mixture of Φ and Φ ν . Furthermore, we should notice that, when one Φ ν is added in the external lines, a coefficient of the effective operator should have suppression factor, |v 2 /m 1,2 |. Thus, we investigate diagrams which have only one Φ ν in the external lines.
At first, we focus on marginal operators in the effective potential. The most dangerous marginal operator for the instability of the VEV hierarchy is λ 6 |Φ 2 |(Φ † Φ ν ) (+h.c.), which is induced from diagrams in Fig.1 (a) and (b). It is because this operator breaks Z 2 -parity and induces linear term of v 2 , which might possibly destroy the VEV hierarchy. Here we note that Fig.1 (a) and (b) are only 1-loop diagrams which induce λ 6 |Φ 2 |(Φ † Φ ν ) (+h.c.). Neither lepton nor quark 1-loop diagrams contribute λ 6 due to the Z 2 -parity, since one additional external Φ ν needs one additional right-handed neutrino propagator inside a loop which requires one more Φ ν . Fig. 1 (c) and (d) induce another Z 2 -parity violating operator,   c)) is expected to dominate (b) ((d)) because of |λ i | 2 g 4 2 , so λ 6 and λ 7 are estimated as Taking into account all irrelevant operators which have only one φ ν in the effective operator, correction for |λ 6 | might be of order 3 This correction contributes the stationary condition of v 2 in Eq.(2.4), and modifies it as Remind again that tiny VEV of |v 2 |( |v 1 |) is originated from tiny term of m 2 3 v 1 . Thus, an induced term from the radiative correction of λ 6 2 v 3 1 must be smaller than m 2 3 v 1 to preserve the VEV hierarchy. Actually, the ratio of them is estimated as at most. This means that the order of |v 2 | is not changed but its factor might be modified about 0.8 (2) by the radiative corrections in Majorana (Dirac) neutrino scenario. This magnitude comes from a maximal (may be over-) estimation, and anyhow, the orders of VEVs are not changed. (Actually, this modification becomes much smaller about O(1)%, if we use Higgs self-couplings of O(0.1).) Thus, the VEV hierarchy itself is stable against radiative corrections. As for higher-loop effects, they are at least suppressed by an additional loop-factor 1 16π 2 , and we cannot find any diagrams which have larger contribution than above diagrams. Therefore, the VEV hierarchy itself is stable against radiative corrections, and we can conclude radiative corrections do not destroy the VEV hierarchy in both Dirac and Majorana neutrino scenarios.

SUSY neutrinophilic Higgs doublet model
In this section, we analyze vacuum structures of Higgs potential in the SUSY neutrinophilic Higgs doublet model at first, and next investigate a stability of this VEV hierarchy against radiative corrections.

Vacuum structure in tree-level potential
The SUSY neutrinophilic Higgs doublet model has four Higgs doublets [12,13], and the superpotential is given by 8π 2 |B ρ m 2 | at most, where m is a Higgs mass in a loop. Notice that neither (s)lepton nor (s)quark contribute λ at 1-loop level due to the Z 2 -parity similarly in non-SUSY νTHDM. It is because one additional external H ν needs one additional right-handed neutrino propagator inside a loop, which requires one more external H ν . Anyhow, this term modifies the stationary condition of v ν in Eq.(3.35) as Taking into account all irrelevant operators which have only one H ν in the effective operator, correction for |λ | might be of order Remind that tiny VEV of v ν is originated from the small mass parameters ofB ρ as in Eq.(3.36). Thus, in order to preserve the VEV hierarchy, | λ 2 v u v 2 d | must be smaller than |B ρ v d | in Eq.(3.43). And, this ratio is estimated as This value is too small to influence the stationary conditions in both Dirac and Majorana neutrino scenarios. We can also show that higher-loop diagrams induce smaller corrections due to the loop suppression factors. Therefore, we can conclude that the potential is stable against radiative corrections in SUSY neutrinophilic Higgs doublet model.

Summary
A neutrinophilic Higgs model has tiny VEV, which can naturally explain tiny masses of neutrinos. There is a large energy scale hierarchy between a VEV of the neutrinophilic Higgs doublet and that of usual SM-like Higgs doublet. In this paper, we have analyzed vacuum structures of Higgs potential in both SUSY and non-SUSY neutrinophilic Higgs models, and next investigated a stability of this VEV hierarchy against radiative corrections. We have shown that the VEV hierarchy is stable against radiative corrections in both Dirac neutrino and Majorana neutrino scenarios in both SUSY and non-SUSY neutrinophilic Higgs doublet models.

Note added
After preparing our submission of this paper, we notice a paper [19], where authors also analyzed the vacuum stability against radiative corrections in the non-SUSY νTHDM with Dirac neutrino scenario. Their results are consistent with ours. They calculated the 1-loop effective potential and the quantum corrections to VEV hierarchy. On the other hand, we estimated the most dangerous contributions to the VEV hierarchy and confirmed the stability also in SUSY and Majorana cases.