Study of two-loop neutrino mass generation models

Annals of Physics 365 (2016) 210–222

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Study of two-loop neutrino mass generation models Chao-Qiang Geng a,b,c,∗ , Lu-Hsing Tsai b a

Chongqing University of Posts & Telecommunications, Chongqing, 400065, China

b

Department of Physics, National Tsing Hua University, Hsinchu, Taiwan

c

Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan

article

info

Article history: Received 27 August 2015 Accepted 24 November 2015 Available online 3 December 2015 Keywords: Neutrino mass Majorana neutrino CP phase

abstract We study the models with the Majorana neutrino masses generated radiatively by two-loop diagrams due to the Yukawa ρ ℓ¯ cR ℓR and effective ρ ±± W ∓ W ∓ couplings along with a scalar triplet ∆, where ρ is a doubly charged singlet scalar, ℓR the charged lepton and W the charged gauge boson. A generic feature in these types of models is that the neutrino mass spectrum has to be a normal hierarchy. Furthermore, by using the neutrino oscillation data and comparing with the global fitting result in the literature, we find a unique neutrino mass matrix and predict the Dirac and two Majorana CP phases to be 1.41π , 1.11π and 1.48π , respectively. We also discuss the model parameters constrained by the lepton flavor violating processes and electroweak oblique parameters. In addition, we show that the rate of the neutrinoless double beta decay (0νββ) can be as large as the current experimental bound as it is dominated by the short-range contribution at tree level, whereas the traditional long-range one is negligible. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Although the data from neutrino experiments have implied that at least two neutrinos carry nonzero masses [1–5], the origin of these masses is still a mystery. Apart from the mass generation of

∗

Corresponding author at: Department of Physics, National Tsing Hua University, Hsinchu, Taiwan. E-mail addresses: [email protected] (C.-Q. Geng), [email protected] (L.-H. Tsai).

http://dx.doi.org/10.1016/j.aop.2015.11.010 0003-4916/© 2015 Elsevier Inc. All rights reserved.

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Dirac neutrinos given by the Yukawa couplings with the existence of right-handed neutrinos (νR ), seesaw mechanisms with type-I [6–10], type-II [11–17] and type-III [18] can generate masses for Majorana neutrinos by realizing the Weinberg operator (L¯ cL Φ )(Φ T LL ) at tree-level, where Φ and LL are the doublets of Higgs and left-handed lepton fields, respectively. In these scenarios, either heavy degrees of freedom or tiny coupling constants are required in order to conceive the small neutrino masses. On the other hand, models with the Majorana neutrino masses generated at one-loop [19,20], two-loop [21–24] and higher loop [25–28] diagrams have also been proposed without introducing νR . Due to the loop suppression factors, the strong bounds on the coupling constants and heavy states are relaxed, resulting in a somewhat natural explanation for the smallness of neutrino masses. Among the loop-level mass generation mechanisms, there is a special type of the neutrino models [23,24] in which a doubly charged singlet scalar ρ : (1, 4) and a triplet ∆ : (3, 2)1 under SU(2)L × U(1)Y are introduced to yield the new Yukawa coupling ρ ℓ¯ cR ℓR with the charged lepton ℓR as well as the effective gauge coupling ρ ±± W ∓ W ∓ due to the mixing between ρ ±± and ∆±± , leading to the neutrino masses through two-loop diagrams [23]. As this model is the simplest way to realize the ρ WW coupling, we name it as the minimal two-loop-neutrino model (MTM) [23]. It is interesting to note that ρ ±± W ∓ W ∓ can also be induced from non-renormalizable high-order operators [29–31]. Although MTM can depict neutrino masses at two-loop level, the assumption on the absence of the L¯ c L∆ term makes this model unnatural. To solve this problem, one can simply extend MTM by adding an extra doublet scalar, which together with ∆ carries an odd charge under an Z2 symmetry [32]. We call this model as the doublet two-loop-neutrino model (DTM). On the other hand, ρ ±± W ∓ W ∓ could be granted by inner-loop diagrams, such as those [27,28] with three-loop contributions to neutrino masses, in which the neutral particle in the inner-loops could be a candidate for the stable dark matter. In this study, we will demonstrate that the neutrino mass matrix can be determined in these models by the experimental data. In particular, the neutrino mass spectrum is found to be a normal hierarchy. In addition, the neutrinoless double beta decay (0νββ) is dominated by the short-range contribution at tree level due to the effective coupling of ρ ±± W ∓ W ∓ [23,24,28–30,33,34], instead of the traditional long-range one. However, the neutrino masses in this type of the models are usually oversuppressed as there is not only a two-loop suppression factor, but also a small ratio ml /v with the charged lepton mass ml and vacuum expectation value (VEV) v = 246 GeV of the Higgs field. Furthermore, the lepton flavor violation (LFV) processes could also limit the new Yukawa couplings. To have a large enough neutrino mass, the mixing angle or mass splitting between the two doubly-charged states should be large, which inevitably leads to a significant contribution to the electroweak oblique parameters, especially the T parameter. We will calculate the neutrino masses in detail and check whether there is a tension between these masses and the constraint from the oblique parameter T . This paper is organized as follows. In Section 2, we study the neutrino masses in the two-loop neutrino models. In Section 3, the constraints on the model parameters from lepton flavor violating processes and electroweak oblique parameters are studied. We present the conclusions in Section 4. 2. Two-loop neutrino masses In MTM, we introduce the scalars ρ : (1, 4) and ∆ = (∆++ , ∆+ , ∆0 ) : (3, 2) under SU (2)L × U (1)Y . The relevant terms in the Lagrangian are given by

− L = −µ2Φ (Φ Ď Φ ) + M∆2 (∆Ď ∆) + λΦ (Φ Ď Φ )2 + λ¯ 3 (∆Ď ∆)1 (Φ Ď Φ )1 + λ¯ 4 (∆Ď ∆)3 (Φ Ď Φ )3   κ ∗ 2 Cab c Ď 2 ∗ 2 c ¯ ¯ ¯ ρ(ℓR )a (ℓR )b − µ∆(Φ ) + ρ ∆ + λρ ∆Φ + H.c. , (1) + Yab (LL )a ∆(LL )b + 2

2

√

where Φ = (Φ + , Φ 0 )T with Φ 0 = (ΦR + iΦI )/ 2 is the SM doublet scalar, the indices of a and b represent e, µ and τ , and the subscripts of 1 and 3 in the quartic terms stand for the SU(2) singlet and triplet scalars inside the parentheses, respectively. After the spontaneous symmetry breaking, Φ

1 The convention for the electroweak quantum numbers (I , Y ) with Q = I + Y /2 is used throughout this paper.

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Fig. 1. Diagrams for the neutrino mass generation, where the charged states S ± can be replaced by S1± or S2± when DTM is discussed.

√ 

√

2 Φ 0 , while the neutral component of ∆ also receives a VEV v∆ / 2, acquires a VEV of vΦ = generated via the µ term. Note that by the global fitting result of ρ0 = 1.0000 ± 0.0009 [35], v∆ is constrained to be .5 GeV, so that vΦ ≃ 246 GeV is a good approximation. The κ term in Eq. (1) can produce a mixing term between ρ ±± and ∆±± , resulting in two mass eigenstates P1,2 ¯ to be zero since they have the tree-level and with masses M1,2 , respectively. We will set Yab and λ logarithmic divergent two-loop contributions to neutrino masses, respectively. These two couplings can also be forbidden in a natural way by introducing a new doublet [32] or a singlet scalar [33] with an Z2 symmetry or by replacing ∆ by a higher multiplet, such as ξ : (5, 2) without the discrete symmetry [36]. The scalar mass spectra of MDM are shown in Appendix A.1. We now calculate the neutrino masses from the two-loop diagrams of Fig. 1 in the ’t Hooft– Feynman gauge. The neutrino mass matrix Mν can be written as

(Mν )ℓℓ′ =

1

(16π )

2 2



 2Cℓℓ′ mℓ mℓ′  A(a) + A(b) + A(c ) + A(d) , 2

(2)

v

where the integration results A(a) , A(b) , A(c ) and A(d) , corresponding to the sub-figures (a)–(d) in Fig. 1, are given in Eqs. (B.9)–(B.11) in Appendix B, respectively. Explicitly, we find that the contribution related to A(a) dominates over the other three components. We note that if Mρ is much smaller than M∆ and the mixing angle θ between them is small, this model approximately coincides with the effective theory involving the dimension-7 operator ρ(Dµ Φ )(Dµ Φ )ΦΦ discussed in Ref. [30]. DTM can be viewed as the extension of MTM by introducing a new doublet χ = (χ + , χ 0 )T with √ 0 χ = (χR +iχI )/ 2. This new doublet along with ∆ carries an odd charge under the Z2 symmetry [32]. This discrete symmetry can forbid the tree-level coupling L¯ c L∆ to make the two-loop neutrino mass generation more natural. The relevant part of the Lagrangian is given by

− L = −µ2Φ (Φ Ď Φ ) − µ2χ (χ Ď χ ) + λΦ (Φ Ď Φ )2 + λχ (χ Ď χ )2 + λ4 (Φ Ď χ )(χ Ď Φ ) + M∆2 (∆Ď ∆)  C κ λ5 ab ρ(ℓ¯ cR )a (ℓR )b − µ∆Φ Ď χ Ď + ρ ∗ ∆2 + λρ ∗ ∆Φ χ + (Φ Ď χ )2 + h.c. . (3) + 2

2

2

Since χ does not couple to the SM fermions due to the Z2 symmetry, the √model is similar to the Type-I two-Higgs doublet model [37]. The doublet χ can also have a VEV vχ / 2 = ⟨χ 0 ⟩ due to the negative



2 mass term of χ . We can define the mixing angle sin γ = vΦ / vΦ + vχ2 to characterize the scalar 2 mixing matrix when only scalar doublets are taken into account, where v 2 ≡ vΦ + vχ2 + 2v∆2 with v = 246 GeV. Note that the λ5 term in Eq. (3) breaks the lepton number symmetry explicitly so that the dangerous Majaron can be avoided, while sizable values of λΦ and λχ are needed in order to give the CP even neutral scalar masses and preserve the stability of the scalar potential. The VEV of ∆ in this model is induced by the µ term in Eq. (3), which is proportional to vΦ vχ with fixed values of M∆ and µ. In the model, we have two singly-charged physical states S1± and S2± besides the unphysical Goldstone boson G± , originated from the mixing among Φ ± , χ ± and ∆± . The values

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of the mixing elements between the doublets and ∆ are also proportional to v∆ like MTM. Moreover, the term λρ ∗ ∆Φ χ and its Hermitian conjugate provide another source for the ρ ±± − ∆±± mixing apart from the κ term, with the contribution to θ approximately proportional to sin 2γ . The results on the scalar mass spectra are given in Appendix A.2. The mechanism for the neutrino mass generation in DTM is similar to that in MTM. But, the main coupling related to ρ ±± is from the effective dimension-5 effective operator ρ(χ Φ )2 . The formula for the neutrino mass matrix is given by

(Mν )ℓℓ′ =

  2Cℓℓ′ mℓ mℓ′  s2θ κ λv µ A(a1) + A(a2) + A(a3) + A(b) + A(c ) + A(d) , 2 2 (16π ) v cγ 2 2 2 1

2 2

(4)

where A(ai) and A(j) with i = 1, 2 and 3 and j = b, c and d are defined in Eqs. (B.12)–(B.15), respectively. In Eq. (4), there is a new contribution proportional to λ, which is of O (v∆ /v). Note that the elements 2 of the neutrino mass matrix in MTM are of O (v∆ /v 2 ).

It is crucial that the above types of the two-loop neutrino mass generation, in which ℓ¯ cR ℓR ρ is the only source of the LFV, can lead to an interesting structure for the neutrino mass matrix. The relative sizes among the matrix elements are determined by the combination factors of Cℓℓ′ mℓ mℓ′ . Assuming that each value of Cℓℓ′ is at the same order, there exist interesting hierarchies for the mass matrix elements, given by

(Mν )ee ≪ (Mν )eµ ≪ (Mν )eτ ≪ (Mν )µµ ≪ (Mν )µτ ≪ (Mν )τ τ .

(5)

In particular, (Mν )ee is much less than (Mν )τ τ due to m2e /m2τ ∼ (1/35002 ). In Refs. [38–43], it has been shown that only the normal hierarchy for the neutrino mass spectrum can have the matrix textures in which (Mν )ee together with another matrix element is zero. Clearly, as the mass hierarchies in Eq. (5) naturally realize (Mν )ee ≃ (Mν )eµ ≃ 0, both MTM and DTM predict the normal neutrino mass hierarchy. Recall that in the standard parametrization [35,44], the neutrino mixing matrix VPMNS is given by



VPMNS

c12 c13 = −s12 c23 − c12 s23 s13 eiδ s12 s23 − c12 c23 s13 eiδ



1 × 0 0

0

0 0

eiα21 /2 0

s12 c13 c12 c23 − s12 s23 s13 eiδ −c12 s23 − s12 c23 s13 eiδ

s13 e−iδ s23 c13  c23 c13





eiα31 /2

,

(6)

where sij (cij ) = sin θij (cos θij ) with θij being the mixing angles, δ is the Dirac phase, and α21 and α31 are the two Majorana phases. Note that one of the Majorana phases can be absorbed by the chiral fermion fields if there exists one massless neutrino. For given values of mass square splittings and mixing angles, there are only two solutions for the three CP phases of δ , α21 and α31 , along with the lightest neutrino mass m0 , to satisfy the mass hierarchies in Eq. (5). In particular, by using the central values of the global fitting result for the normal hierarchy mass spectrum, given by [35] sin2 θ12 = 0.308 ± 0.017,





+0.033 sin2 θ23 = 0.437− 0.023 ,

0.26 −5 ∆m221 = 7.54+ eV, −0.22 × 10

0.0020 sin2 θ13 = 0.0234+ −0.0019 ,

∆m232 = (2.39 ± 0.06) × 10−3 eV,

(7) (8)

we find that

(i) : m0 = 5.07 × 10−3 eV,

δ = 0.59 π ,

α21 = 0.89 π ,

α31 = 0.52 π ,

(9)

(ii) : m0 = 5.07 × 10

δ = 1.41 π ,

α21 = 1.11 π ,

α31 = 1.48 π .

(10)

−3

eV,

Note that both solutions in Eqs. (9) and (10) have the same value for m0 but different CP phases. It is interesting to see that the predicted Dirac phase δ = 1.41π in (ii) of Eq. (10) agrees well with that

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given by the global fitting result in Ref. [35]. Taking (ii) in Eq. (10) as the input parameters, the neutrino mass matrix is then given by



0 Mν =  0 1.03ei1.93π

1.03ei1.93π 2.31 ei0.49π  , 2.76 ei0.54π



0 2.40 ei0.56π 2.31 ei0.49π

(11)

in unit of 10−11 GeV. Note that the empty values for (Mν )ee and (Mν )eµ can be placed by some small non-zero values when any of the parameters in (ii) is under slight shifting. 3. Constraints from lepton flavor violation processes and electroweak oblique parameters In both MTM and DTM, as the coupling matrix elements Cab are the only sources of the LFV, the processes of ℓ → ℓ′ ℓ′′ ℓ′′′ (ℓ → ℓ′ γ ) with the tree-level (one-loop) contributions involving ρ ±± could give significant constraints on Cab . However, those on Cee and Ceµ can be ignored since they do not affect the tiny matrix elements (Mν )ee and (Mν )eµ when we discuss the neutrino mass spectrum. Among the current experimental bounds, Br(µ+ → e+ γ ) < 5.7 × 10−13 [45] is the most stringent one to limit of Cab . In particular, we can obtain [28]

 |Ceτ |

cθ2

2

M12

+

s2θ M22



 <

0.336 TeV

2 .

(12)

It is obvious that the largest allowed value of |Ceτ |max from Eq. (12) depends only on M1 since sθ is of order 10−2 . To account for the current experimental data on the neutrino masses as obtained in Eq. (11), the matrix element (Mν )eτ should be around 1.0 × 10−11 GeV. As a result, we can use this value to check whether the mechanism of the neutrino mass generation can work, as shown in Fig. 2. The value of κ is taken to be κ < max(M1 , M2 ), constrained by the perturbativity [46]. In general, a larger allowed value of κ is more possible to give a correct value of (Mν )eτ . To obtain the right values for the neutrino masses, at least one of M1 and M2 should be roughly larger than 2.5 TeV. In Fig. 2(a), (Mν )eτ behaves approximately as an increasing function of M2 due to the weak bound on Cℓℓ′ from the LFV processes, while in Fig. 2(b) it is linearly proportional to M2 . The experimental constraints from the µ → e conversion could also give some hints on M1 . To −13 illustrate the result, we pick out some of the experimental bounds, given by BAu [47], µ→e < 7 × 10 −12 −11 [50]. For MTM and DTM, [49], and BPb BSµ→e < 7 × 10−11 [48], BTi µ→e < 4.6 × 10 µ→e < 4.3 × 10 the dominant contributions come from γ to Z penguin diagrams, which lead to

BAµ→e =

2G2F m5µ  capt

ΓA

2  (p) (n) eAL D(A) + gRV V p (A) + gRV V n (A) ,

(13) (p,n)

capt

where ΓA is the muon capture rate for the nucleus A, the coefficients AL and gRV correspond to the dipole and vector contributions, and D(A) and V p,n (A) are the overlapping functions between e and µ (see Ref. [51] for the details), respectively. Explicitly, we have

√ AL =

e 2



192π 2 Mρ2 GF

l

Cµl Cel , ∗

gRV (q) =

2 s2W MW

2π 2 Mρ2

 Qq

 l

∗

Cµl Cel

5 9

+

1 3

log

m2l Mρ2

 ,

(14)

where Qq is the electric charge of the quark q and gRV (q) is the vector coupling with the quark q, mainly from the γ penguin diagram as the Z diagram is suppressed by the charged lepton masses. Based on (q) (p) (n) the valence quark model, one has the relations between gRV and gRV , given by gRV = 2gRV (u) + gRV (d) (n)

and gRV = 2gRV (d) + gRV (u) [51]. To simplify the problem, we first assume that the coupling constant Ceµ , which cannot be determined by Mν , is negligible, so that we can estimate the µ − e conversion

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215

(a) M1 < M2 .

(b) M1 > M2 . Fig. 2. Plots for (Mν )eτ with (a) M1 < M2 and (b) M1 > M2 .

rate by only calculating the photon penguins that involve τ . By taking Mρ = 1 TeV, Ceτ = 0.33, and Cµτ = 0.0033, we find −13 BAu , µ→e = 3.3 × 10

BSµ→e = 1.8 × 10−13 ,

−13 BPb , µ→e = 2.4 × 10

−13 BAl , µ→e = 1.6 × 10

−13 BTi , µ→e = 2.7 × 10

(15)

which satisfy all the corresponding experimental limits. For this case, the conversion rate for Au can approach to the same order as the upper bound by the experiment. The improvement on the sensitivity of the µ − e conversion [52,53] in the future will either detect the signal or put some more stringent constraint on the models. It is interesting to note that the neutrinoless double beta decay in our models can have a significant different feature from other models with radiative neutrino mass generations. In MTM and DTM, the short-range contributions to the decay dominate over the traditional long-range ones [23,24], with the decay amplitudes proportional to (Mν )ee . It is clear that the long-range parts can be safely neglected due to the small electron mass in (Mν )ee , whereas the short-range ones are proportional only to the Yukawa coupling Cee . As a result, by calculating 0νββ , the upper limit on |Cee | could be derived, despite the fact that it is ignored when discussing the neutrino mass matrix. The half life for 0νββ is given by [54,55] 0νββ

T1/2

= (G01 |ϵ3LLL |2 |M3 |2 )−1 ,

(16)

which leads to [29]

v∆ M12 − M22 , ϵ3LLL = mp (2Cee∗ s2θ ) √ 2 2 2 M1 M2

(17)

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Fig. 3. Allowed regions in M1 − M2 plane for (a) MTM and (b) DTM, where κ = 5 max(M1 , M2 ) and Ceτ = (Ceτ )max are used in MTM and DTM, while sγ = 0.4, λ = 2, λ4 = 1 and λ5 = 1.5 are taken in DTM. Gray areas represent regions without enough neutrino masses, and blue regions located at M1 ≃ M2 are disallowed due to the mixing term between ρ ±± and ∆±± . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

where G01 and |M3 | are the phase space factor and the matrix element for the hadronic sector, respectively, and ϵ3LLL is the coefficient, which is effectively related to the dimension-9 operator (¯uL γµ dL )(¯uL γ µ dL )(¯eR ecR ), defined in Refs. [54,55], and mp is the proton mass. Note that the coefficient in Eq. (17) has no explicit dependence on the electron mass. If one takes M1 = 1 and M2 = 1.5 TeV in MTM, resulting in the maximal value of mixing | sin 2θ| = 0.04, the upper bounds on |Cee | for different target nuclei can be estimated as

|Cee(Ge) | < 0.088,

|Cee(Xe) | < 0.067,

|Cee(Te) | < 0.096,

|Cee(Se) | < 0.36,

|Cee(Nd) | < 0.36, |Cee(Mo) | < 0.13,

(18)

by comparing with experimental upper limits [56–62]. When including the effect of λ in DTM, a larger contribution to sin 2θ could make these upper bounds on |Cee | to be around 20% lower. Combining the typical value of (Mν )eτ and the constraints from the LFV processes, at least one of M1 and M2 should be heavier than around 2.7 TeV in MTM. On the other hand, to get M1 = O (102 ) GeV, M2 needs to be much larger, at least 4 TeV. However, it is more difficult to get the value of M2 less than 1 TeV with a large M1 , which means that to get Mρ = O (100) GeV, at least M∆ & 4 TeV is required. Consequently, it is possible to detect the signals of ρ ±± , mainly through the pair production of ρ ++ ρ −− and subsequently decays with the same sign charged leptons in the final states at the LHC [30,63]. We present the related results in Fig. 3(a). Similar conclusions have been also shown in Fig. 2 and 8.8 of Refs. [33,64], respectively, but they allowed some of the region with M1 ≈ M2 ≈ 1.5 TeV, which is forbidden in this paper. In DTM, the neutrino masses could be lifted up more easily by using a sizable λ as well as sin θ . Moreover, there is a new contribution to the neutrino masses from the dimension-5 operator ρ Φ 2 χ 2 in DTM instead of the dimension-7 one ρ[(Dµ Φ )Φ ]2 in MTM. As an example, if we take sγ = 0.4, λ = 2, λ4 = 1, and λ5 = 1.5, along with the same values of κ and Ceτ in MTM, one finds that M2 . 400 GeV (&2 TeV) for M1 & 550 GeV (.400 GeV). The relevant result is displayed in Fig. 3(b). Finally, we briefly discuss the effects of the electroweak oblique parameters S and T in our models. First of all, in MTM and DTM, we find that the typical value of S is of order 10−3 , which is lower than the current experimental sensitivity. For the T parameter, the mixing between ∆ and Φ gives a logarithmic divergent T due to the non-unity of ρ , but this part could be ignored when v ≪ 5 GeV. In this case, the main contribution to T is given by the mixing between ∆ and ρ , denoted as T(ρ−∆) . It is basically a negative value whose magnitude is limited due to the small mixing angle |sθ | . 0.02. For example, taking M1 = 2 TeV, M2 = 4 TeV, v∆ = 0.5 GeV, and κ = 5M2 , it only leads to T(ρ−∆) = −5 × 10−5 . However, in DTM the mixing angle |sθ | can be enhanced by λ, which is independent of v∆ . Using the input values for the above parameters, and λ = 3, we get T(ρ−∆) = −0.001. Meanwhile, the mixing between Φ and χ can also provide a sizable value to the corresponding parameter T(Φ −χ ) . For example, T(Φ −χ) = 0.04 with sα = sγ = 0.4, λ4 = 1, and λ5 = 1.5, which still fulfills the

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experimental bound −0.02 < ∆T < 0.12 at 1.5 − 1.8 σ confident level [35]. The relevant formulae for T are summarized in Appendix C. 4. Conclusions We have studied the two models of MTM and DTM, in which Majorana neutrino masses are generated radiatively by two-loop diagrams due to the Yukawa ρ ℓ¯ cR ℓR and effective ρ ±± W ∓ W ∓ couplings. We have shown that the lepton violating processes, in particular, µ+ → e+ γ can give stringent constraints on the new Yukawa coupling Cℓℓ′ . By combining with the perturbativity condition for the coupling κ , the light neutrino mass element (Mν )eτ can limit the allowed values of M1 and M2 . In particular, we have found that only M1 and M2 with TeV scale can lead to the correct sizes of the neutrino masses. We have illustrated that the normal neutrino mass hierarchy is a generic feature in these two-loop neutrino mass generation models. Moreover, by using the central values of the neutrino oscillation data and comparing with the global fitting result in the literature [35], we have obtained the unique neutrino mass matrix in Eq. (11) and predicted the Dirac and two Majorana CP phases to be 1.41π , 1.11π and 1.48π , respectively. Finally, we emphasize that the neutrinoless double beta decays can be very large as they are dominated by the short-distance contributions at tree-level, which can be tested in the future experiments and used to constrain the element of Cee . Acknowledgments This work was supported in part by National Center for Theoretical Sciences, National Science Council (Grant No. NSC-101-2112-M-007-006-MY3) and National Tsing Hua University (Grant No. 104N2724E1). Appendix A. Scalar mass spectra in MTM and DTM A.1. MTM The non-self-Hermitian terms in Eq. (1) can be expanded as follows:

∆+ ∆(Φ Ď )2 = ∆++ (Φ + )∗ 2 + 2 √ (Φ + )∗ (Φ 0 )∗ + ∆0 (Φ 0 )∗2 , 2

 ρ ∆ ∗

= 2ρ

2

∗

∆

++

∆ − 0

1 2

(A.1)

 (∆ )

+ 2

.

(A.2)

After obtaining the explicit forms of the scalar potential, we can write down its tadpole conditions

−µ2Φ vΦ + λΦ vΦ3 −

√

2µvΦ v∆ = 0,

(A.3)

v2

2 M∆ v∆ − µ √Φ = 0, 2

(A.4)

which give

µ v2 v∆ = √ Φ2 .

(A.5)

2 M∆

The mixing matrices of CP odd neutral and singly charged states are written as MI2

 =

tβ −tβ

−t β 1



M∆ , 2

where tβ = 2v∆ /vΦ and tβ′ =

2

M± =

√

tβ′2 −tβ′



−tβ′ 1



2 M∆ ,

(A.6)

2v∆ /vΦ . The masses of the neutral CP odd state A and singly charged

states S ± are given by MA = M∆ /cβ2 and MS = M∆ /cβ′ 2 , respectively.

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The doubly-charge mixing matrix is given by



Mρ2

±± M12

±± M12

2 M∆

 ,

(A.7)

where

κv∆

±± M12 = √ . 2

(A.8)

One can easily diagonalize Eq. (A.7), leading to the mixing angle θ ±± 2M12

t2θ =

,

2 Mρ2 − M∆

(A.9)

and eigenvalues of the eigenstates P1,2 M12 = Mρ2 − 2 M22 = M∆ +

s2θ 1 − 2s2θ s2θ 1 − 2s2θ

(M∆2 − Mρ2 ), (M∆2 − Mρ2 ).

(A.10)

A.2. DTM The related operators with χ in Eq. (3) can be written as

∆+ ∆Φ Ď χ Ď = ∆++ Φ +∗ χ +∗ + √ (Φ +∗ χ 0∗ + Φ 0∗ χ +∗ ) + ∆0 Φ 0∗ χ 0∗ , 2   ∆+ + 0 ∗ −− + 0 0 + + ++ 0 0 ρ ∆Φ χ = ρ ∆ Φ χ − √ (Φ χ + χ Φ ) + ∆ Φ χ . 2

(A.11)

The minimization conditions are given by

µ 1 −µ2Φ vΦ + λΦ vΦ3 − (λ4 + λ5 )vΦ vχ2 − √ v∆ vχ = 0,

(A.12)

µ −µ2χ vχ + λχ vχ3 − (λ4 + λ5 )vχ vΦ2 − √ v∆ vΦ = 0,

(A.13)

2 1

vΦ vχ

2

2

2

2 M∆ v∆ − µ √ = 0. 2

(A.14)

 vΦ2 + vχ2 , and sγ = vχ /¯v . The singly charged and CP odd neutral mass matrices are both 3 × 3. In the diagonalization, we use the relation sγ ≫ s′β . The transformation It is convenient to define v¯ =

matrices, VI and V± , of CP odd neutral and singly charged states can be presented by a set of small quantities ϵij and ϵij′ , given as

 VI =

cγ sγ

ϵ31

−s γ cγ

ϵ32

 ϵ13 ϵ23 , 1



cγ

V± =  s γ

ϵ31 ′

−sγ cγ

ϵ32 ′

′  ϵ13 ′  ϵ23 ,

(A.15)

1

where

ϵ32 = ϵ31

2 M∆

c2γ

M∆ − λ5 v¯ 2 s2γ = tβ , 2

tβ ,

(A.16) (A.17)

C.-Q. Geng, L.-H. Tsai / Annals of Physics 365 (2016) 210–222

 ϵ13 = −cγ +  ϵ23 = −sγ −

c2γ

2 M∆



2 M∆



tβ ,

(A.18)

tβ ,

(A.19)

2cγ M∆ − λ5 v¯ 2 2

c2γ

2 2sγ M∆ − λ5 v¯ 2

219

and ′ ϵ32 =

2 M∆

c2γ ′ tβ ,

(A.20)

M∆ − (λ4 + λ5 )¯v 2 s2γ 1 2

2

′ ϵ31 = tβ′ , 

(A.21)

′ ϵ13 = −cγ +

 ϵ23 = −sγ − ′

2 M∆

c2γ



2cγ M∆ − (λ4 + λ5 )¯v 2 1 2

2

2 M∆

c2γ

tβ′ ,

(A.22)

tβ′ .

(A.23)



2 2sγ M∆ − 12 (λ4 + λ5 )¯v 2

2 The mass eigenvalues for two CP odd states are MA21 = λ5 v¯ 2 and MA22 = M∆ , while those for singly

2 charged states MS21 = 21 (λ4 + λ5 )¯v 2 and MS22 = M∆ . Finally, the mixing angle between the doublycharged states is given by

√

2 2κv∆ + λcγ sγ vΦ

t2θ =

2 Mρ2 − M∆

.

(A.24)

For the mixing of CP even neutral states, one can focus on the 2 × 2 mixing matrix between Φ and χ , given by



2 2λΦ vΦ

−(λ4 + λ5 )vΦ vχ

 −(λ4 + λ5 )vΦ vχ . 2λχ vχ2

(A.25)

Rotating the states ΦR = cα h − sα H and χR = sα h + cα H, we obtain two mass eigenvalues Mh and MH to be 2 ′2 2 Mh2 = (2λΦ − (λ4 + λ5 )tγ tα )vΦ cβ cγ ,

 MH2 =

2λΦ + (λ4 + λ5 )

tγ tα

(A.26)

 vΦ2 cβ′2 cγ2 .

(A.27)

If one takes mh = 125.7 GeV [35] as the state found at the LHC [65,66], MH can be obtained as a function of λ4 + λ5 and α . Appendix B. Two-loop neutrino mass B.1. MTM We calculate the neutrino masses by using the ’t Hooft–Feynman Gauge. The relevant vertices ∗ corresponding to the top vertex in Fig. 1(a) are −µ∆−− (Φ − )∗2 and − κ2 ρ ++ (∆+ )2 , which yield (a) − iMℓℓ ′ =

i v∆

 C ′ m m ′   κv  M 2 ∆ ℓℓ ℓ ℓ 0 −4 (I111 − I112 ) (16π 2 )2 v 2 M12 − M22 v 2 cβ2 ′  κv∆  − 4 2 cθ2 (I111 − 2I121 + I221 ) + s2θ (I112 − 2I212 + I222 ) (¯νLc νL ), v

(B.1)

220

C.-Q. Geng, L.-H. Tsai / Annals of Physics 365 (2016) 210–222

with 1

 Iijn =

1−y1



1



dx1 0

0

0

0

1 −x 2

 dx2

dy1

dy2

2 1 − x1

× log(y1 [x1 Mn2 + x2 Mi′2 ] + y2 x1 (1 − x1 )Mj′2 ), ′

(B.2)

′

where we can define M1 = MW and M2 = MS . For Fig. 1(b), we get

−

(b) iMℓℓ′

 √2C ′ m m ′  √2g 2 s s ′ ℓℓ ℓ ℓ (b) W 2θ 2β = (M12 − M22 )(IW − IS(b) )(¯νLc νL ), (16π 2 )2 vΦ 2 2 2 i

(B.3)

with (b)

Ii

1

 =

1−y1

 dy2

1

 dy1

0

0

1 −x 3

 dx3

0



1−x2 −x3

dx2 0

dx1 0

−2y1 (2 − x1 − x2 ) , (x1 + x2 )(1 − x1 − x2 )m2i

(B.4)

where 2 m2i = y1 (x1 M12 + x2 M22 + x3 MW ) + y2 (x1 + x2 )(1 − x1 − x2 )Mi′2 .

(B.5)

(b)

(c )

The amplitude Mll′ should be equal to Mll′ . For Fig. 1d, we have 4 v∆ −i 2gW 2 2 ) − I (d) (MP2 )](¯νLc νL ), √ s2θ (mℓ mℓ′ Cℓℓ′ )[I (d) (MP1 2 2 (16π ) 4 2  1  1−y1  1  1−x2 −4 = dy2 dy1 dx2 dx1 2 ,

(d) −iMℓℓ ′ =

I (d)

0

0

0

m(d)

0

(B.6)

(B.7)

where 2 2 m2(d) = y1 (x1 Mi2 + x2 MW ) + y2 x1 (1 − x1 )MW .

(B.8)

In summary, A(a) , A(b) , A(c ) , and A(d) in Eq. (2) are listed as follows: A(a) = s2β ′ κ

2 M∆

  s′β2 s′β4 ( I − I ) + ( I − I + I − I ) ( I − I ) 111 112 121 122 211 212 221 222 2 ′2 ′4

M12 − M2

cβ

cβ



 + s2β ′ κ cθ2 (I111 − I121 − I211 + I221 ) + s2θ (I112 − I122 − I212 + I222 ) , (b)

(b)

2 A(b) = A(c ) = −2s2β ′ κ MW (IW − IS ),

A(d) = 2s2β ′ κ

4 MW I (d) M12 M22

−

.

(B.9) (B.10) (B.11)

B.2. DTM If we expand the amplitudes A(a1) , A(a2) , and A(a3) up to O (v∆ /v), then the results are given by A(a1) = 0,

A(a2) = 0,

(B.12)



′ ′ ′ ′ 3 A(a3) = sγ cθ2 2cγ3 ϵ31 I111 + 2cγ s2γ ϵ31 (I121 + I211 ) + 2cγ2 ϵ13 (I131 + I311 ) − 2ϵ32 sγ I221

  ′ ′ ′ ′ + 2s2γ ϵ13 (I231 + I321 ) + cγ cθ2 2cγ2 sγ ϵ31 I111 − 2cγ sγ (ϵ31 cγ + ϵ32 sγ )(I121 + I211 )  ′ ′ ′ + 2cγ sγ ϵ13 (I131 + I311 ) + 2cγ s2γ ϵ32 I221 − 2cγ sγ ϵ13 (I231 + I321 )   − s′β cθ2 2cγ3 sγ I111 − 2cγ sγ (cγ2 − s2γ )(I121 + I211 ) − 2cγ s3γ I221 , (B.13)

C.-Q. Geng, L.-H. Tsai / Annals of Physics 365 (2016) 210–222

M12 − M22

A(b) = A(c ) = −2s2θ A(d) = 2sβ ′

s2θ

v

v

(b)

(b)

221

(b)

2 ′ ′ ′ MW (ϵ31 IW − sγ ϵ32 IS1 + ϵ13 IS2 ),

(B.14)

4 (d) MW I

(B.15)

where we have used the same notations as those in Appendix B.1 except M2′ = MS1 and M3′ = MS2 . Appendix C. T parameters in MTM and DTM The T parameter due to the mixing between ρ and ∆ in MTM is given by T(ρ−∆) =

1



2 4π s2W MW



2 s2θ F (M12 , MS2± ) + cθ2 F (M22 , MS2± ) − F (M∆ , MS2± ) − 2cθ2 s2θ F (M12 , M22 ) . (C.1)

The above result in Eq. (C.1) can also be applied to DTM by replacing S ± by S2± . In DTM, since the contribution from the mixing between Φ and χ should also be considered, we find T(Φ −χ) =

1 16π

2 s2W MW

 2 2 (cα−γ − 1)(G(Mh2 , MW ) − G(Mh2 , MZ2 ))

2 + s2α−γ (G(MH2 , MW ) − G(MH2 , MZ2 )) + s2α−γ (F (Mh2 , MS21 ) − F (Mh2 , MA21 ))  2 + cα−γ (F (MH2 , MS21 ) − F (MH2 , MA21 )) + F (MS21 , MA21 ) ,

(C.2)

where the functions F , K , and G are defined by G(x, y) = F (x, y) + 4yK (x, y), F (x, y) =

x+y 2

−

xy x−y

log

x y

(C.3)

,

K (x, y) =

x log x − y log y x−y

.

(C.4)

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