Dependence of the leptonic decays of H^- on the neutrino mixing angles theta_{13} and theta_{23} in models with neutrinophilic charged scalars

In the Higgs Triplet Model and the neutrinophilic Two-Higgs-Doublet Model the observed neutrinos obtain mass from a vacuum expectation value which is much smaller than the vacuum expectation value of the Higgs boson in the Standard Model. Both models contain a singly charged Higgs boson (H^-) whose Yukawa coupling is directly related to the neutrino mass (i.e. a"neutrinophilic charged Higgs"). The partial decay widths of H^- into a charged lepton and a neutrino (H^- to l^- nu) depend identically on the neutrino masses and mixings in the two models. We quantify the impact of the recent measurement of sin^2(2theta_{13}), which plays a crucial role in determining the magnitude of the branching ratio of H^- to e^- nu for the case of a normal neutrino mass ordering if the lightest neutrino mass m_0<10^{-3} eV. We also discuss the sizeable dependence of H^- to mu^- nu and H^- to tau^- nu on sin^2(theta_{23}), which would enable information to be obtained on sin^2(theta_{23}) and the sign of \Delta m^2_{31} if these decays are measured. Such information would help neutrino oscillation experiments to determine the CP-violating phase \delta.


I. INTRODUCTION
The ATLAS [1] and CMS [2] experiments at the CERN Large Hadron Collider (LHC) have discovered a new boson with a mass of approximately 125 GeV. The measurements of its branching ratios (BRs) are consistent (within experimental error) with those predicted by the Higgs boson [3] of the Standard Model (SM). Current LHC data [4] also suggests that the new particle's spin and parity are compatible with the values expected for the SM Higgs boson. It is now widely believed that this discovery corresponds to a fundamental scalar particle with a vacuum expectation value (vev) i.e. it is a species of Higgs boson. Consequently, there is increased motivation to search for additional scalars which would belong to an extension of the SM with a non-minimal Higgs sector. Such models might also provide a mechanism for the generation of neutrino mass. Although the solitary Higgs boson in the SM can provide a Dirac mass term for the observed neutrinos by assuming the existence of three generations of right-handed neutrinos, such a mechanism would not be testable at the LHC. Extensions of the Higgs sector of the SM may involve an additional SU(2) L -multiplet of scalar fields whose vev solely provides neutrino masses. We refer to these scalar fields as "neutrinophilic scalars". In this paper we will consider two such models which are potentially testable because they predict neutrinophilic charged scalars (H ± ) which might be light enough to be discovered at the LHC.
Neutrinos may obtain a Majorana mass via the vev of a neutral Higgs boson in an isospin triplet representation [5][6][7][8][9]. A particularly simple implementation of this mechanism of neutrino mass generation is the "Higgs Triplet Model" (HTM) in which the SM Lagrangian is augmented solely by an SU(2) L -triplet of scalar particles (denoted by ∆) with hypercharge Y = 2 [5,8,9]. In the HTM there are three electrically neutral Higgs scalars: h 0 and H 0 are CP-even, and A 0 is CP-odd. These scalar eigenstates are mixtures of the doublet and triplet neutral fields, but the mixing angle is very small in most of the parameter space of the HTM because of the hierarchy of the vevs, v ∆ ≪ v, where v(= 246 GeV) is the vev of the neutral doublet field, and v ∆ is the vev of the triplet field. There are also electrically charged scalars: a doubly charged scalar (H ±± ) and a singly charged scalar (H ± ).
The Higgs sector of the SM may be extended with a second SU(2) L -doublet scalar field of hypercharge Y = 1 (denoted by Φ ν ) which has a Yukawa interaction only with right-handed neutrinos. The phenomenology is discussed in Ref. [10] for the case where the right-handed neutrinos also have their Majorana mass terms [11]. If right-handed neutrinos do not have Majorana masses [12][13][14] then the neutrinos are Dirac fermions, and their mass matrix (m D ) iℓ is solely given by a product of new Yukawa coupling matrix (y ν ) iℓ and the vev v ν of the second scalar doublet. The vev v ν is generated via spontaneous breaking of a global symmetry in Ref. [12] while it is obtained via soft-breaking of a global symmetry in Refs. [13,14]. We refer to the model of Dirac neutrinos in Refs. [13,14] as the "neutrinophilic Two Higgs Doublet Model" (ν2HDM). Like the HTM, the ν2HDM also predicts three electrically neutral Higgs scalars (two being CP-even, and one being CP-odd), as well as a singly charged scalar.
In the context of both the HTM and the ν2HDM the simplest candidate for the observed boson at ∼ 125 GeV would be the lightest CP-even h 0 . This scalar eigenstate has BRs which are very similar to those of the SM Higgs boson in most of the parameter space of the two models with v ∆ , v ν ≪ v. At present, the measured BRs of the 125 GeV boson are fully consistent with those of the Higgs boson of the SM. The current experimental errors allow deviations from the BRs of the SM Higgs boson of the order of 20% to 30%. The decay channel to two photons is sensitive to the virtual effects of H ± and H ±± [15][16][17][18][19][20], and the measurement of this decay now constrains the parameters of the scalar potentials in the above models, especially the mass of H ±± and the trilinear coupling h 0 H ++ H −− in the HTM. The result of the ATLAS experiment [21] with all the data taken at √ s = 7 TeV and √ s = 8 TeV is R γγ = 1.65 ± 0.24(stat) +0.25 −0.18 (syst), where R γγ = 1 for the SM Higgs boson. The CMS experiment measures R γγ = 0.78 ± 0.27 with a Multi-Variate-Analysis and R γγ = 1.11 ± 0.31 with a cut-based analysis [22]. If future measurements show a statistically significant deviation from R γγ = 1, then this result could be readily explained by the presence of charged scalars.
The HTM and the ν2HDM provide identical dependences of the partial decay widths for H ±± → ℓ ± ν on the six neutrino oscillation parameters and the unknown mass of the lightest neutrino, where the main uncertainty comes from the latter parameter. Quantitative studies were performed in the context of the HTM in Ref. [23], and in the ν2HDM in Refs. [13,14]. In the HTM (in which the neutrinos are Majorana particles) the prediction for BR(H ± → ℓ ± ν) is of particular importance because its value does not depend on the two unknown Majorana phases in the neutrino mass matrix. This result is in contrast to the prediction for BR(H ±± → ℓ ± ℓ ± ) in the HTM, which does depend on the values of the Majorana phases and thus such BRs have more uncertainty. Consequently, if a H ±± and H ± were discovered at the LHC, a measurement of BR(H ± → ℓ ± ν) would provide a more robust means of determining whether the mass of the neutrinos arose solely from a triplet vev v ∆ (which is the case in the HTM), or from a combination of mechanisms which may or may not include a triplet vev.
In this work we study the dependence of BR(H ± → ℓ ± ν) on the neutrino oscillation parameters, in particular the mixing angles sin 2 2θ 13 and sin 2 θ 23 of U MNS . Previous studies [13,14,23] considered the dependence of BR(H ± → ℓ ± ν) on these parameters by scanning over their allowed ranges and presenting the results as scatter plots. The aim of the present work is to clarify the effect of varying each of these mixing angles individually, with special attention given to the impact of the recent measurement of sin 2 2θ 13 . We also pay attention to the dependence on sin 2 θ 23 , whose uncertainty (whether sin 2 θ 23 > 0.5 or sin 2 θ 23 < 0.5) might be the main hindrance in the determination of the CP-violating phase δ in neutrino oscillation experiments.
Our work is organised as follows. In section II we briefly introduce the HTM and the ν2HDM, and discuss the ongoing measurements of the neutrino oscillation parameters. In section III we present our numerical results for BR(H ± → ℓ ± ν). Conclusions are given in section IV.

II. THE HIGGS TRIPLET MODEL AND NEUTRINOPHILIC 2HDM
The HTM and the ν2HDM are models with a non-minimal Higgs sector in which the observed neutrinos obtain mass as a product of a Yukawa coupling and the vev of a new scalar field. The two models predict the same specific relationship between the neutrino parameters and the partial widths of the decay channels H ± → ℓ ± ν. In this section we briefly introduce both models, and then summarise the current experimental status of the measurements of the neutrino oscillation parameters.

A. HTM
In the HTM [5,8,9] a Y = 2 complex SU(2) L isospin triplet of scalar fields, T = (T 1 , T 2 , T 3 ), is added to the SM Lagrangian. This model has the virtue of providing Majorana masses for the observed neutrinos without the introduction of SU(2) L singlet neutrinos. The following SU(2) L ⊗ U(1) gauge-invariant Yukawa interaction is introduced: Here h ℓℓ ′ (ℓ, ℓ ′ = e, µ, τ ) is a complex and symmetric coupling, C is the Dirac charge conjugation operator, σ i (i = 1-3) are the Pauli matrices, L ℓ = (ν ℓL , ℓ L ) T is a left-handed lepton doublet, and ∆ is a 2 × 2 representation of the Y = 2 complex triplet fields: where arises from the minimisation of the scalar potential and leads to the following mass matrix for Majorana neutrinos: The most general SU(2) L ⊗ U(1) Y invariant form of the scalar potential is given in Refs. [8,24,25] and a detailed study of the theoretical constraints on its parameters has been performed in Ref. [26]. The conservation of lepton number is broken by two units due to a soft-breaking term µΦ T iσ 2 ∆ † Φ (here µ is a dimensional coupling constant), which gives rise to v ∆ and thus neutrino masses. This soft-breaking term might be suppressed by a radiative mechanism [27].
A distinctive signal of the HTM would be the observation of H ±± , whose mass (m H ±± ) may be of the order of the electroweak scale. Such particles could be produced with sizeable rates at hadron colliders through the processes qq → γ * , Z * → H ++ H −− [44][45][46][47][48] and [40,44,49]. Direct searches in these channels have been carried out by the ATLAS [50] and CMS collaborations [51], using about 5 fb −1 of data at √ s = 7 TeV.
If m H ±± > m H ± then the decay H ±± → H ± W ± * can be dominant, even for relatively small mass splittings m H ±± − m H ± . At present there has been no direct search in this channel.
In this work we will study in detail the branching ratios of the leptonic decays of the singly charged Higgs, Ref. [23]). In order to avoid decays of the form H ± → H 0 W * [23,25,40,54,55] and H ± → H ±± W * [56] (which can be dominant in the HTM) we assume m H ± ≃ m H 0 ≃ m H ±± . Since the vertex H ± tb is suppressed by v ∆ the decay width for t → H ± b with m H ± < m t − m b is negligible, and thus searches at the LHC in this channel will have no sensitivity. There were searches at the CERN LEP experiment for e + e − → H + H − with H ± → τ ± ν, in which the limit m H ± ∼ > 90 GeV was derived [57]. For the decay channels H ± → e ± ν and H ± → µ ± ν, the limits from explicit searches at LEP for sleptonsl in supersymmetric models can be applied [58] (i.e. searches for e + e − →l +l− with ℓ ± → ℓ ± χ 0 1 for ℓ ± = e ± , µ ± , where χ 0 1 is the lightest neutralino which appears as missing energy). Again, these limits can be satisfied by m H ± ∼ > 90 GeV.
Previous studies of BR ℓν in the HTM have been performed in Ref. [23]. The partial width of H ± → ℓ ± ν i is determined from eq. (1) and is proportional to After summing over the three mass eigenstates of neutrinos, the summed partial width Γ(H ± → ℓ ± ν) is given by Note that the summation ensures that the dependence on the Majorana phases vanishes, unlike the case for Γ(H ±± → ℓ ± ℓ ± ), and this notable result was first pointed out in Ref. [23].
Explicit forms of (m † L m L ) ℓℓ are given by The effect of the CP-violating phase δ is negligible [59] because it appears with s 13 ∆m 2 21 , which is much smaller than |∆m 2 31 | (with an O(1) coefficient) in the second term in the right hand-side of eqs. (9)- (12).
In the HTM the production process q ′ q → W → H ±± H ∓ affords the best detection prospects for H ± → ℓ ± ν for a given m H ± . This mode (with m H ±± = m H ± ) has already been taken into account in the search for H ±± by the CMS collaboration in Ref. [51]. To our knowledge there has not been a dedicated search for qq → γ, Z → H + H − at the LHC, and we are not aware of a simulation of the detection prospects for √ 8 TeV and L ≃ 20 fb −1 .

B. ν2HDM
In the ν2HDM, the SM is extended with three right-handed gauge singlet fermions ν iR and a second scalar SU(2) L -doublet Φ ν = (φ + ν , φ 0 ν ) T , which is in the same representation as Φ under the SM gauge group. A global U(1) symmetry is imposed, under which Φ ν and the three ν iR have charge +1 and all the other fields are uncharged [13]. The following Yukawa interaction is added to that of the SM: where (y ν ) iℓ is the 3 × 3 matrix of Yukawa coupling constants for neutrinos. Note that the U(1) symmetry forbids Majorana mass terms 1 2 m iR (ν iR ) T C ν iR . If the global U(1) symmetry is softly broken only by m 2 12 Φ † Φ ν [13], there arises a vev v ν ≡ √ 2 φ 0 ν and lepton number is conserved. 1 The smallness of the neutrino masses can be naturally understood if the soft-breaking term is generated at the loop level [60,61]. Then, ν ℓL and ν iR become three Dirac neutrinos whose mass matrix is simply given by One can take ν iR as the right-handed components of the mass eigenstates ν i without loss of generality, which leads to the following expression: The charged Higgs H − in the ν2HDM decays into ℓ L ν R while H − in the HTM decays into ℓ L ν L . The partial decay widths for H ± → ℓ ± ν (summed over all the neutrino species) are calculated as (4) and (15), and explicit expressions are presented in eqs. (7)- (12). Thus, the dependence of Γ(H ± → ℓ ± ν) on the neutrino parameters is identical in both the HTM and the ν2HDM. Previous studies of BR ℓν in the ν2HDM have been performed in Ref. [13]. Detection prospect of H ± at LHC is discussed in Ref. [14].
In the ν2HDM, the cross section for qq → γ, Z → H + H − is larger than that in the HTM by a factor of 2.7. This is a consequence of the different isospin of H ± (I 3 = 0) in the HTM and H ± (I 3 = ±1/2) in the ν2HDM. Hence the detection prospects in the channel qq → γ, Z → H + H − are significantly better in the ν2HDM than in the HTM, as emphasised in Ref. [14].

C. Neutrino oscillation parameters
As shown above, the decay widths of H ± → ℓ ± ν depend on the neutrino parameters. Neutrino oscillation experiments involving solar [28], atmospheric [29], accelerator [30][31][32], and reactor neutrinos [33][34][35][36][37][38] are sensitive to the mass-squared differences and the mixing angles, and give the following preferred values and ranges: where θ 12 , θ 13 < π/4. We use these values in our numerical analysis unless otherwise mentioned. Varying |∆m 2 31 |, ∆m 2 21 and sin 2 2θ 12 within their allowed ranges only causes a very small error in BR ℓν e.g. the value of BR eν in our analysis with m 1 ≃ 0 (∆m 2 31 > 0) has about a 10 % error in total from varying them, while the effect of varying sin 2 2θ 13 (which we will study in detail) is much larger. Information on the mass m 0 of the lightest neutrino and the Majorana phases cannot be obtained from neutrino oscillation experiments. This is because the oscillation probabilities are independent of these parameters, not only in vacuum but also in matter. If m 0 > ∼ 0.2 eV, a future 3 H beta decay experiment [62] can measure m 0 . Experiments which seek neutrinoless double beta decay (See e.g., Ref. [63] for a review) are only sensitive to a combination of neutrino masses and phases when neutrinos are Majorana fermions.
The value of δ in completely unknown. Measurement of δ is a main goal of oscillation experiments with accelerator neutrinos [64][65][66][67]. The measurement uses appearance modes (e.g. ν µ → ν e ) whose dominant terms are controlled by s 2 23 and sin 2 (2θ 13 ). Since this CP-violating parameter is extracted by comparing measurements with a neutrino beam and an antineutrino beam (not on the anti-Earth), the measurement of δ is affected by the sign of ∆m 2 31 due to the effect of the Earth's matter on the oscillations. Since the sign of ∆m 2 31 is also undetermined at present, distinct neutrino mass spectrums are possible. The case with ∆m 2 31 > 0 is referred to as Normal mass ordering (NO) where m 1 < m 2 < m 3 and the case with ∆m 2 31 < 0 is known as Inverted mass ordering (IO) where m 3 < m 1 < m 2 . The sign of ∆m 2 31 can be determined by long baseline oscillation measurements (e.g. in the NOvA experiment [65]) and precise measurements of the oscillations of atmospheric neutrinos (e.g. with the Hyper-Kamiokande [66]).
An important recent result is the knowledge that the small mixing angle θ 13 is now known to be significantly different from zero. The nonzero value of θ 13 makes the measurement of leptonic CP-violation possible (which depends on s 13 sin δ, as can be seen from Eq. (5)) at future experiments. Reactor experiments probe the probability of the disappearance of antielectron neutrinos (ν e ), a process which is sensitive to sin 2 2θ 13 . The Daya Bay collaboration has obtained the value sin 2 2θ 13 = 0.089 ± 0.010 ± 0.005 [34]; the RENO collaboration has obtained sin 2 2θ 13 = 0.113 ± 0.013 ± 0.014 [35]; the Double Chooz collaboration has obtained sin 2 2θ 13 = 0.109 ± 0.030 ± 0.025 (with a Gadolinium analysis) [36] and sin 2 2θ 13 = 0.097 ± 0.034 ± 0.034 (with an analysis which captures neutrons on hydrogen) [37]. Long baseline experiments search for the appearance of ν e from a beam of ν µ , and this process is sensitive to the combination s 2 23 sin 2 2θ 13 . Assuming θ 23 = π/4, T2K has obtained sin 2 2θ 13 = 0.088 +0.049 −0.034 [32] (See also a preliminary update [68]). The NOvA experiment [69] will also measure s 2 23 sin 2 2θ 13 . The ultimate precision is expected to be about 0.005 at 68% confidence level (c.l.) at the Daya Bay (see e.g. Ref. [70]).
The mixing angle sin θ 23 is known to be almost maximal. The currently preferred 2σ range is 0.4 < ∼ s 2 23 < ∼ 0.6 [29]. Long baseline experiments [64,65] will further improve the precision in the determination of s 2 23 by studying the survival probability of ν µ , which is proportional to sin 2 2θ 23 . However, if θ 23 deviates enough from π/4, there are two possible values of θ 23 which give the same value of sin 2 2θ 23 . For example, s 2 23 = 0.4 and 0.6 are obtained for sin 2 2θ 23 = 0.96; this gives about ±20% uncertainty in appearance probabilities (e.g. ν µ → ν e ) which will be used to measure δ. The ambiguity (the "octant degeneracy") on whether s 2 23 > 0.5 or not can be resolved by precise measurement of the atmospheric neutrino (e.g. with the Hyper-Kamiokande experiment [66]). It is also possible to resolve the ambiguity by e.g. utilising the complementarity of reactor and long baseline experiments [71].

III. BR(H ± → ℓ ± ν) IN THE HTM AND IN THE NEUTRINOPHILIC 2HDM
The dependence of BR ℓν on the neutrino parameters has been studied in Ref. [23] in the context of the HTM, and in Ref. [13] in the context of the ν2HDM. Both studies are in agreement, and present BR ℓν as functions of the lightest neutrino mass (m 0 ) for both orderings of neutrino masses. In Refs. [13,23] the BRs of the leptonic decay channels of H ± were displayed as scatter plots in which the neutrino mixing angles and mass differences were varied over the allowed intervals. From those studies it is not clear which values of s 2 23 and sin 2 2θ 13 correspond to the upper and lower limits of the allowed regions of the BRs. Correlations of BRs with respect to the neutrino parameters are not also clear. In this work we clarify the effect of varying s 2 23 and sin 2 2θ 13 individually, and quantify the impact of the recent measurement of sin 2 2θ 13 on the leptonic BRs of H ± . Where comparison is possible our results are in agreement with those in Refs. [13,23]. As explained earlier, we only consider the parameter space of v ∆ (v ν ) < 0.1 MeV for which ℓ BR ℓν ∼ 1. Thus the dependence of BR ℓν on m H ± drops out, but for definiteness we fix m H ± = 500 GeV (200 GeV) in the HTM (ν2HDM). Since we assume m H ± = m H ±± in the HTM, the choice of m H ± = 500 GeV is necessary in order to comfortably satisfy current limits on m H ±± from direct searches for H ±± → ℓ ± ℓ ± at the LHC [50,51]. We also fix v ∆ (v ν ) = 1000 eV. Although BR ℓν is not sensitive to the exact value of v ∆ (v ν ) for v ∆ (v ν ) < 0.1 MeV, contributions of the scalars H ±± (in the HTM) and H ± to lepton-flavour-violating decays such as µ → eee and τ → ℓℓℓ (in the HTM), and µ → eγ are sensitive to the value [13,25,72]. Constraints from these decays are satisfied for v ∆ ∼ > 1000 eV, and so we fix v ∆ (v ν ) = 1000 eV.
In Fig. 1 (upper panel)  The values sin 2 2θ 13 = 0.11, 0.09, and 0.07 correspond to s 2 13 = 0.028, 0.023, and 0.018, respectively. The maximal mixing s 2 23 = 0.5 is used, and all other neutrino parameters are fixed as in eq. (18). For this choice of s 2 23 , one has BR µν ≈ BR τ ν , a result which is due to an approximate µ-τ exchange symmetry of U MNS . Since BR µν and BR τ ν are not very sensitive to sin 2 2θ 13 , we used only sin 2 2θ 13 = 0.09 for BR µν . They are much more sensitive to s 23 . For m 0 < 10 −2 eV it can be seen that BR eν is very sensitive to the value of sin 2 2θ 13 . This can be understood from the explicit expression for (m † L m L ) ee in eq. (7), in which the term s 2 13 ∆m 2 31 can be the dominant one for s 2 13 > ∼ 0.01 and m 1 < ∼ 5 × 10 −3 eV. The shaded region in Fig. 1 (upper panel) between the curves for sin 2 2θ 13 = 0.07 and 0.11 corresponds to the allowed region of BR eν at about 95% c.l. The lowest value is BR eν ≃ 2.7%, and is obtained for sin 2 θ 13 = 0.07 and m 0 < 10 −3 eV. It is notable that this minimum BR eν is considerably larger than the value BR eν = 1% which is obtained for the (now strongly disfavoured) case of sin 2 2θ 13 = 0. Hence the measurement of sin 2 2θ 13 has now disfavoured the parameter space of 1% < BR eν < 2.7%, and the minimum value of BR eν is now three times larger than before for the case of the normal mass ordering. This result improves the detection prospects of the channel H ± → e ± ν at the LHC, and will be discussed in more detail below.
In Fig. 1 (lower panel) we show the m 0 -dependence of BR eν with a light (red) line and BR µν with dark (blue) lines for the case of the normal mass ordering, but this time we fix sin 2 2θ 13 = 0.09 and consider three different values of s 2 23 : Note that s 2 23 does not appear in the expression for (m † L m L ) ee in eq. (7) and so BR eν is completely insensitive to the value of s 2 23 . Therefore in Fig .1 (lower panel) we plot BR eν for s 2 23 = 0.5 only. In contrast, BR µν and BR τ ν are quite sensitive to s 2 23 e.g. for m 0 < 10 −2 eV, where BR µν takes the values ≃ 57%, ≃ 48% and ≃ 39% for s 2 23 = 0.6, 0.5 and 0.4, respectively. Note that BR τ ν for s 2 23 = 0.6, 0.5 and 0.4 are almost given by dot-dashed, solid and dashed curves of BR µν , respectively. The case of s 2 23 > 0.5 leads to BR µν > BR τ ν , while s 2 23 < 0.5 leads to BR µν < BR τ ν . If BR µν > 48 % is measured then this would require s 2 23 > 0.5 and m 0 = m 1 < ∼ 10 −2 eV. We now discuss the case of the inverted mass ordering where m 0 = m 3 . Figure 2 (upper panel) is the analogue of Fig .1 (upper panel), and considers only two values of sin 2 2θ 13 : sin 2 2θ 13 = 0.11 (approximately the 95% c.l. upper limit, a dashed line) and sin 2 2θ 13 = 0 (which is now excluded, a dotted line). Since the dominant contribution of θ 13 to the BRs comes from the combination c 2 13 ∆m 2 13 in eqs. (8), (10), and (12), one has the result that the BRs deviate by only a couple of percent when sin 2 2θ 13 is varied. Figure 2 (lower panel) is the analogue of Fig. 1 (lower panel), again fixing sin 2 2θ 13 = 0.09 and considering three different values of s 2 23 (=0.4, 0.5 and 0.6). Again one sees that the difference between BR µν and BR τ ν is determined by the deviation from maximal mixing for s 2 23 . However, one has the result that s 2 23 > 0.5 leads to BR τ ν > BR µν while s 2 23 < 0.5 leads to BR τ ν < BR µν , which are opposite behaviours to those for the normal mass ordering. This result was not explicitly pointed out in Refs. [13,23], and is due to the fact that dominant contributions of s 2 23 to (m † L m L ) µµ and (m † L m L ) τ τ come with ∆m 2 31 , whose sign is flipped depending on the neutrino mass ordering.
We now study the numerical value of the ratio of BR eν and BR µν as a function of m 0 , for various values of sin 2 2θ 13 and s 2 23 . The ratio does not change even if other decay channels (such as H ± → W ± Z for v ∆ (v ν ) ∼ > 0.1 MeV and H ± → W ± H 0 for m H ± > m H 0 ) have significant BRs. We note that the cross section for qq → H + H − depends on m H ± , and approximate information on m H ± can be obtained from the M T 2 distribution of the signal, as shown in Ref. [14]. However, given the sizeable uncertainty in the extraction of m H ± we propose to use the ratio of BR eν and BR µν in which this uncertainty essentially cancels out, thus enabling a more precise determination of the neutrino parameters. We note that there was no (explicit) quantitative study of this ratio in Refs. [13,14,23], although a qualitative discussion was given in Ref. [14]. In Fig. 3 (upper panel) we show BR eν /BR µν for a normal mass ordering. The central grey region corresponds to s 2 23 = 0.5 and 0.07 < sin 2 2θ 13 < 0.11. The dashed line (dot-dashed line) corresponds to the largest (smallest) value of BR eν /BR µν for a given m 0 , and is obtained for s 2 23 = 0.4(0.6) and sin 2 2θ 13 = 0.11(0.07). As expected, one can see that sin 2 2θ 13 causes the most uncertainty in BR eν /BR µν for smaller values of m 0 , while s 2 23 gives the most uncertainty for larger m 0 . Since the ratio changes monotonically in a wide range (0.05 < ∼ BR eν /BR µν < ∼ 0.9) with respect to m 0 , a measurement of BR eν /BR µν would determine the value of m 0 , which might be more difficult to obtain from H ±± decays alone due to the additional uncertainty from the Majorana phases. For example, BR eν /BR µν ≃ 0.3 means m 0 ≃ 0.02 eV and ∆m 2 31 > 0. In Fig. 3 (lower panel) we show BR µν /BR eν (i.e. the inverse of the ratio plotted in the upper panel of Fig. 3) for an inverted mass ordering. Again, the central grey region corresponds to s 2 23 = 0.5 and 0.07 < sin 2 2θ 13 < 0.11. As expected, varying sin 2 2θ 13 has very little effect on the ratio BR µν /BR eν for an inverted mass ordering. The maximum (minimum) value of BR µν /BR eν with a fixed m 0 is again obtained for s 2 23 = 0.4(0.6) and sin 2 2θ 13 = 0.11(0.07). Hence a precise measurement of this ratio would provide simultaneous information on s 2 23 , m 0 and the neutrino mass ordering. For example, BR µν /BR eν < 0.5 indicates s 2 23 > 0.5, m 0 < ∼ 0.01 eV, and ∆m 2 31 < 0. If the ratio in a range 0.65 -0.9 is observed one can obtain a lower bound on m 0 (= m 3 ).
We now discuss the phenomenology of H ± at the LHC by applying the above results to the phenomenological discussion already given in Ref. [14]. In the ν2HDM the main production process of H ± is via qq → γ, Z → H + H − . A simulation of the detection prospects of this process has been performed in Ref. [14], in which the signatures H + H − → e + e − νν, e ± µ ∓ νν and µ ± µ ∓ νν were studied. Detection prospects are best for the case of an inverted neutrino mass ordering, because the sum of BR eν and BR µν is always above 60%, while for the case of a normal mass ordering this sum of BRs can drop as low as 40%. By combining results for all three channels (e + e − νν, e ± µ ∓ νν and µ + µ − νν), detection at the 5σ level in the ν2HDM for any choice of mass spectrum and mixing parameters was shown to be possible for m H ± = 100 GeV (300 GeV) with between 20 fb −1 and 80 fb −1 (57 fb −1 and 450 fb −1 ) of integrated luminosity at √ s = 14 TeV [14]. Thus a signal could be possible in the early stages of the √ s = 14 TeV run of the LHC. In the HTM, 2.7 times larger integrated luminosity is required because of the different I 3 .
For the region of m 0 < 10 −3 eV where the exact value of sin 2 2θ 13 plays an important role for a normal mass ordering, the first signal of H + H − with m H ± = 100 GeV would come in the channel µ + µ − νν (for which between 10 fb −1 and 80 fb −1 of integrated luminosity would be necessary in the ν2HDM). The small value of BR eν for m 0 < 10 −3 eV ensures that detection of the H + H − → e + e − νν signal would require very large (> 10 4 fb −1 ) integrated luminosities, which are possibly beyond the reach of an upgraded LHC. Reference [14] states that the detection of the channel e ± µ ∓ νν for m H ± = 100 GeV would require integrated luminosities ≃ 650 fb −1 in an optimistic case of sin 2 2θ 13 ≃ 0.12 (a region 0 ≤ sin 2 2θ 13 < ∼ 0.12 is used in Ref. [14]). In a pessimistic case sin 2 2θ 13 = 0, integrated luminosities of a several × 10 3 fb −1 were required in the ν2HDM because of a smaller BR eν . However, as already shown in Fig. 1, the lower bound on BR eν have now been improved by a factor of three by virtue of the recent measurement of sin 2 2θ 13 . The required luminosity to obtain a signal for e ± µ ∓ νν for m 0 < 10 −3 eV has now been reduced to about 1000 fb −1 even in a pessimistic case in the ν2HDM, which is well within the reach of an upgraded LHC. Of course, for m 0 > 10 −3 eV one sees from Fig. 1 that BR eν starts to increase up to its maximum value of BR∼ 30%, and thus signals in all three channels (H + H − → e + e − νν, e ± µ ∓ νν and µ ± µ ∓ νν) would become a possibility with the envisaged integrated luminosities of the LHC.
The exact value of s 2 23 plays a crucial role in determining how much integrated luminosity is required for discovery of H ± , because this parameter has a large effect on BR µν (which is easier to detect) and BR τ ν , unless the neutrinos are quasi-degenerate. If sin 2 2θ 23 ≃ 1 is precisely verified by long baseline experiments in the near future, then such a scenario would act to improve the predictions of BR µν and BR τ ν . Alternatively, if a significant deviation from sin 2 2θ 23 = 1 has been measured by long baseline experiments and H ± of the HTM or ν2HDM has been discovered at the LHC, then a measurement of BR(H ± → µ ± ν)/BR(H ± → e ± ν) could provide information on s 2 23 (and the sign of ∆m 2 31 ) earlier than oscillation experiments, thereby removing the octant degeneracy. Such information would be helpful for CP-violation searches in future oscillation experiments.

IV. CONCLUSIONS
We have studied the branching ratios (BRs) of H ± → e ± ν, H ± → µ ± ν and H ± → τ ± ν in the context of the Higgs Triplet Model and the neutrinophilic Two-Higgs-Doublet Model. We went beyond the analyses of previous papers by quantifying the individual effect of the neutrino mixing angles θ 13 and θ 23 on the above BRs. We showed that the recent measurement of sin 2 2θ 13 = 0.07 -0.11 has important implications for BR(H ± → e ± ν) in the case of a normal neutrino mass ordering. The above measurement of sin 2 2θ 13 rules out (at about 95% c.l.) the previously allowed region of 1% < BR(H ± → e ± ν) < 2.7%, while constraining the BR to lie in the region 2.7% < BR(H ± → e ± ν) < 30%. This ensures that integrated luminosities of about 1000 fb −1 (2700 fb −1 ) should be enough to observe a signal for qq → H + H − → e ± µ ∓ νν in the ν2HDM (HTM) at the upgraded LHC even if m 0 < 10 −3 eV, where BR(H ± → e ± ν) has a minimum value.
We also showed that BR(H ± → µ ± ν) and BR(H ± → τ ± ν) can deviate by up to 20% depending on the value of s 2 23 . For the case of s 2 23 > 0.5 and a normal mass ordering one has the result BR(H ± → µ ± ν) > BR(H ± → τ ± ν), while for s 2 23 < 0.5 one has BR(H ± → µ ± ν) < BR(H ± → τ ± ν). For the case of an inverted neutrino mass ordering one has the converse results. We proposed to use the ratio of BR eν and BR µν in which the uncertainty from m H ± in the production cross section cancels out, thus enabling a more precise determination of the neutrino parameters than for the cases of using BR eν and BR µν alone. Accurate information on s 23 , m 0 and the neutrino mass ordering could then be obtained, some of which might be difficult (m 0 is impossible) to obtain at future neutrino oscillation experiments. Such information would be helpful for CP-violation searches in future oscillation experiments.