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Constraints from muon g-2 and LFV processes in the Higgs Triplet Model

Constraints from the muon anomalous magnetic dipole moment and lepton flavor violating processes are translated into lower bounds on v_Delta*m_H++ in the Higgs Triplet Model by considering correlations through the neutrino mass matrix. The discrepancy of the sign of the contribution to the muon anomalous magnetic dipole moment between the measurement and the prediction in the model is clarified. It is shown that mu to e gamma, tau decays (especially, tau to mu e e), and the muonium conversion can give a more stringent bound on v_Delta*m_H++ than the bound from mu to eee which is expected naively to give the most stringent one.


I. INTRODUCTION
In the standard model of the particle physics (SM), neutrinos are massless particles due to the absence of right-handed neutrinos ν R . The simplest way to give masses to three neutrinos is to add three ν R similarly to other fermions, which corresponds to six additional particles (three ν R and three ν R ) to the SM. In the Higgs triplet model (HTM) [1,2] which we deal with in this article, a complex SU(2) L triplet scalar with the hypercharge Y = 2 is introduced to the SM in order to have neutrino masses. This model can be regarded as one of the simplest extension of the SM because the number of new particles is six in this model also.
The triplet Higgs boson field with hypercharge Y = 2 can be parameterized by where v ∆ is the vacuum expectation value (VEV) of the triplet Higgs boson. The constraint on the rho parameter, ρ 0 = 1.0004 +0.0027 −0.0007 at 2σ CL (page 137 of [3]), gives an upper limit v ∆ /v 0.01 where v = 246 GeV is the VEV of the doublet Higgs field, which corresponds to v ∆ 3 GeV. There is no stringent bound from quark sector on triplet Higgs bosons because they do not couple to quarks. The interaction of the Higgs triplet with lepton doublets L ℓ ≡ (ν ℓL , ℓ L ) T (ℓ = e, µ, τ ) is given by The symmetric matrix h ℓℓ ′ is coupling strength, τ i (i = 1-3) denote the Pauli matrices, and L c ℓ ≡ ν T ℓL C, ℓ T L C with the charge conjugation operator C. The coupling h ℓℓ ′ has a direct relation to the neutrino mass matrix m ℓℓ ′ in the flavor basis through v ∆ as The mass eigenvalues m i are taken to be real positive values. We define ∆m 2 ij ≡ m 2 i − m 2 j and refer to the case of ∆m 2 31 > 0 (∆m 2 31 < 0) as the normal (inverted) mass ordering. Here neutrinos are required to be Majorana particles 1 , and ϕ 1 and ϕ 2 are the Majorana phases [2,4] defined in an interval of [0, 2π). The Maki-Nakagawa-Sakata matrix [5] of the neutrino mixing 2 is parameterized as where s ij ≡ sin θ ij and c ij ≡ cos θ ij , and δ is the Dirac phase. The ranges are chosen as 0 ≤ θ ij ≤ π/2 and 0 ≤ δ < 2π. According to current constraints from neutrino oscillation experiments [7,8,9,10], we use the following values in this article sin 2 2θ 13 < 0.14.
The HTM has seven physical Higgs bosons which are two CP-even neutral bosons h 0 (lighter) and H 0 (heavier), a CP-odd neutral one A 0 , a pair of singly charged bosons H ± , and a pair of doubly charged bosons H ±± . These Higgs bosons contribute to many lepton flavor violating (LFV) processes. Experimental searches for µ →ēee etc. put upper bounds on |h µe ||h ee |/m 2 H ±± etc. (See e.g. [13]), where m H ±± is the mass of H ±± . The couplings h ℓℓ ′ are, however, not free in the HTM because they relate directly to the neutrino mass matrix m ℓℓ ′ as shown in (3). Previous works for dependences of LFV processes on the parameters in m ℓℓ ′ can be found in [6,14,15]. In this article, we consider in detail the correlation of upper bounds on |h * ij h kl |/m 2 H ±± from new physics searches and deal with them as lower bounds on v ∆ m H ±± . 2 We took the definition of the mixing ν ℓ = i U ℓi ν i according to page 517 of [3] although another definition ν ℓ = i U * ℓi ν i is used for example, in [6] and on page 163 of [3]. In latter definition, we need to take complex conjugate in the middle equation of (3).

A. Constraint from the Muon Anomalous Magnetic Dipole Moment
Let us consider first the anomalous magnetic dipole moment (MDM) of muon, a µ ≡ (g − 2)/2. The muon anomalous MDM has been measured very precisely [16] as where the number in parentheses shows 1σ uncertainty. On the other hand, the SM predicts where the hadronic contributions to a SM µ [τ ] and a SM µ [e + e − ] were calculated [17] by using data of hadronic τ decay and e + e − annihilation to hadrons, respectively (See also [18,19,20,21,22]). The deviations of the SM predictions from the experimental result are given by where P L ≡ (1−γ 5 )/2 and ν i represent mass eigenstates of Majorana neutrinos which satisfy conditions ν i = ν c i . The 1-loop contribution of H ± through the triplet Yukawa interaction 3 is calculated as and the H ±± contribution is given by Here we have defined Note that m 2 ℓℓ ′ does not depend on Majorana phases and m 2 ℓℓ is positive definite. Thus a HTM µ ≡ a H ± µ + a H ±± µ is negative definite though ∆a µ is positive. The minus sign of the contributions from Higgs triplets has been known [13,23] but it does not seem to be taken seriously probably because of confusions about the combination (h † h) µµ . Usually the combination seems to be written as (h µµ ) 2 or (h 2 ) µµ , for which sign of a HTM Concerning only on the sign, the 1-loop contribution from H 0 can have the right sign to explain ∆a µ (see [24] for the case in the type II two Higgs doublet model (2HDM-II)).
However, the contribution has a suppression with v 2 ∆ /v 2 in the HTM because ∆ 0 does not couple with charged leptons at the tree level 4 . Although in the Barr-Zee type [25] 2-loop diagrams the right-sign contribution of A 0 can be important in some models like the 2HDM-II [26] and the minimal supersymmetric standard model (MSSM) [27], such a situation does where BR(τ → µν µ ν τ ) = 17%, α = 1/137 stands for the fine structure constant, and G F = 1.17 × 10 −5 GeV −2 denotes the Fermi coupling constant. The effective Lagrangian for the muonium conversion is The formula of the coupling G M M in the HTM and current constraint at 90% CL for that are In Fig. 1, parameters of the neutrino mass matrix are fixed by (5), (6), and the following values as an example: m 1 = 0, sin 2 2θ 13 = 0, ϕ 1 = ϕ 2 = 0. With these values of parameters,   In most of parameter space, the strongest lower bound on v ∆ m H ±± is given by µ →ēee as expected naively from the strong constraint on its branching ratio (20). Figures 2(a) and (b) show contours of the bounds with θ 13 = 0 for the normal and inverted mass ordering, respectively. Note that BR(µ →ēee) does not depend on δ and ϕ 2 for θ 13 = 0. Although the bound on v ∆ m H ±± from µ →ēee is relatively weak for small m 1 in Fig. 2(a), bounds from other LFV processes are weaker than that. It is shown also that ϕ 1 ≃ 0 makes the bound from µ →ēee weak for both of mass orderings. We focus on the case of ϕ 1 = 0 in the next paragraph. In Fig. 2(a)  the same values as ones in Fig. 1. It is seen in Fig. 3(a) that µ →ēee still gives the most stringent bound for ϕ 1 = ϕ 2 = 0 and m 1 0.3 eV although the bound from the muonium conversion gets close to that for large m 1 . If we accept m 1 0.3 eV, the bound from the muonium conversion can be stronger than the bound from µ →ēee. On the other hand, Fig. 3(b) shows that the bound from τ →μee can be more stringent than the one from µ →ēee for m 1 0.06 eV. This is because a parameter set (θ 13 , ϕ 1 , ϕ 2 ) = (0, 0, π) in the region of ∆m 2 ij /m 2 1 ≪ 1 gives and the large difference between experimental constraints (20) and (23) can be compensated for m 1 O(0.1) eV. In Fig. 4 the shaded region shows values of Majorana phases for which the bound from τ →μee becomes more stringent than the one from µ →ēee at m 1 = 0.2 eV for the normal mass ordering. The region is symmetric under a transformation of (ϕ 1 , ϕ 2 − π) → (−ϕ 1 , −ϕ 2 + π) because of |m ℓℓ ′ | = |m * ℓℓ ′ |. Although the bound from µ →ēee is relatively weak for ϕ 1 ≃ 0, the bound is still the most stringent one at around ϕ 2 = 0 because τ →μee is also suppressed. If we take nonzero θ 13 and ignore ∆m 2 21 for (ϕ 1 , ϕ 2 ) = (0, π), eq. (26) is rewritten as while eq. (27) remains valid. Therefore, the shaded region in Fig. 4 at around (ϕ 1 , ϕ 2 ) = (0, π) exists for sin 2 2θ 13 10 −5 . For the inverted mass ordering, the region where τ →μee becomes remarkable is almost same as the one in Figs. 3 and 4 because neutrino masses are almost degenerated in the region. In such a region, we can also expect a signal of τ →μee in future experiments [35,36] with satisfying the current constraint on µ →ēee.

C. Constraints in Cases of BR(µ →ēee) = 0
It has been known that the strong constraint from µ →ēee can be evaded in the cases of m eµ = 0 [14] and m ee = 0 [6]. While it is impossible to have m eµ = 0 with θ 13 = 0, the case of m ee = 0 is possible also for θ 13 = 0 as we mentioned for Fig. 2(a). Such cancellations in the HTM are desired also for experiments [35,36,37] to discover some LFV decays (µ → eγ etc.) [6,14] in the future.
We see that ϕ 1 ≃ 0 is preferred to keep s mgc 13 small for m 1 = 0 and we can confirm s mgc 13 ∝ ∆m 2 21 for ϕ 1 = 0 with eq. (32) in Appendix. It can be found also with eq. (32) for ϕ 1 , ϕ 2 ≪ 1 and ∆m 2 ij = 0 that ϕ 2 ≃ 1 + cos 2θ 12 2 ϕ 1 = 0.68 ϕ 1 (red dotted line in Fig. 6 is preferred to have a small s mgc 13 . Bounds from τ →μee and the muonium conversion can be the most stringent one only for ϕ 1 0.1π. Majorana phases are almost restricted as  ϕ 2 ≃ 0.68 ϕ 1 for the case of a strong constraint from the muonium conversion. It is shown that µ → eγ can be the most stringent bound even for m 1 0.008 eV (cf. Fig. 5) because of large s mgc 13 . At the border to the white region in Fig. 6, we have sin 2 2θ mgc 13 = 0.14. Similarly to Fig. 5, lower bounds on v ∆ m H ±± are shown in Fig. 7 for the inverted mass ordering. Note that ϕ 1 = π can not give m eµ = 0 for the mass ordering because s mgc 13 becomes too large. Very roughly speaking, the results in Figs. 7(a) and (b) are the same as those in Figs. 5(a) and (b), respectively. A difference is that τ →μee can be the most stringent bound at m 3 = 0 while it is not the case for m 1 = 0 in the normal mass ordering. Figure 8 shows the ϕ 1 -dependence for m eµ = 0 with m 3 = 0 where the ϕ 2 -dependence vanishes. For becomes larger than 0.14. We see that µ → eγ can give the most stringent bound even for the inverted mass ordering. For m 3 = 0.2 eV, Fig. 6(b) is almost applicable to see which process gives the most stringent bound. The bound from µ → eγ is the most stringent one except for sin 2 2θ 13 0.04 where the bound from τ →μµµ becomes stronger than that. This is also the case for different values of δ and ϕ 2 .

III. CONCLUSION
In the HTM, it is impossible to have a contribution to the muon anomalous MDM with a plus sign. Therefore, the HTM is qualitatively disfavored by positive ∆a µ [τ ] (∆a µ [e + e − ]) at 1.8σ (3.7σ). If we deal with 2σ bounds to avoid the disagreement with ∆a µ [τ ], v ∆ m H ±± must be greater than 10 3 eV·GeV in most of parameter space of the HTM in order to satisfy a strong constraint on BR(µ →ēee). We found that the bound on v ∆ m H ±± from τ →μee becomes more stringent than that from µ →ēee in a region of m 1 0.06 eV, ϕ 1 0.002π, 0.5π ϕ 2 1.5π, and sin 2 2θ 13 10 −5 . The bound from µ →ēee can be evaded in cases of m eµ = 0 [14] and m ee = 0 [6]. In the case of m eµ = 0 in the normal mass ordering, the strongest bound is given by µ → eγ for m 1 0.01 eV or ϕ 1 0.1π and by τ decays (mainly τ →μee) for m 1 0.01 eV with ϕ 1 0.1π. On the other hand, τ →μee gives the strongest bound in the normal mass ordering except for m 3 ≃ 0 with ϕ 1 ≃ 0.1π where µ → eγ gives the bound. For both of the mass orderings with m eµ = 0, the muonium conversion gives the most stringent bound if Majorana phases satisfy ϕ 2 ≃ 0.68 ϕ 1 for ϕ 1 , ϕ 2 ≪ 1 and m 1 0.1 eV.
In the case of m ee = 0, the strongest bound is obtained from µ → eγ except for the case of sin 2 2θ 13 0.04 where the bound is given by τ →μµµ. By looking over all cases, we see that where we define s mgc 13 ≡ sin θ mgc 13 . Note that s mgc 13 = 0 because it requires m 2 = m 1 . Note also that s mgc 13 is not always acceptable; For example, ϕ 1 = π in the inverted mass ordering gives s mgc 13 > 1. These results are consistent with s mgc 13 and δ mgc used in [6,14] for ϕ 1 , ϕ 2 = 0 or π. Although mixing matrix in [6] is defined as ν ℓ = i U * ℓi ν i in stead of ν ℓ = i U ℓi ν i used in this article, there is no change in formulae of magic values because the difference appears B ≡ (s 4 12 − c 4 12 sin 2 (ϕ 2 − 2δ))∆m 2 31 − s 4 12 ∆m 2 21 ∆m 2 31 , where we define t 13 ≡ s 13 /c 13 .