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One-loop renormalization group equations of the neutrino mass matrix in the triplet seesaw model

Within the framework of the standard model plus one heavy Higgs triplet, we derive a full set of one-loop renormalization group equations of the neutrino mass matrix and Higgs couplings in both full and effective theories. The explicit RGEs of neutrino masses, flavor mixing angles and CP-violating phases are also obtained, and their non-trivial running behaviors around the Higgs triplet mass threshold are numerically illustrated.


II. THE TRIPLET SEESAW MODEL
The full Lagrangian of the triplet seesaw model is given by [8]: where the first part represents the SM Lagrangian, and the second part contains the interactions involving the Higgs triplet. The most general form of L ξ is where φ is the SM Higgs doublet withφ = iτ 2 φ * and ∆ is a 2 × 2 representation of the Higgs triplet field [12]: In Eq. (2), f and g are generation indices, and summation over repeated indices is implied. t i are the three dimensional representations of the Pauli matrices andĈ is defined as [13]:Ĉ The covariant derivative D µ reads As already mentioned in the first section, in order to generate tiny light neutrino masses, the mass scale of the Higgs triplet M ξ should be much higher than the typical electroweak scale M Z . In this letter, we take M ξ = 10 10 GeV. In the energy scale µ ≫ M ξ , the full Lagrangian is taken into account. In the low energy limit µ ≪ M ξ , we should use the effective theory by integrating out the heavy triplet field. The effective Lagrangian can be defined by [14] : where Dξ i stands for functional integration over ξ i . Starting from Eq. (7), we can calculate the one-loop level effective Lagrangian by using the steepest-descent method to integrate out the heavy scalar. Keeping only terms of order O(1/M 2 ξ ) through the whole calculation and neglecting all the operators with higher inverse powers of M ξ , we may get the effective operators: where m, n, f and g run over 1, 2, 3. L Higgs has the same form as the Higgs self-coupling in the SM. The four fermion interaction L 4−Fermi may contribute to the lepton flavor violating processes. However, such processes should be strongly suppressed due to the heavy mass of the Higgs triplet. Thus we will neglect this four fermion coupling in the following calculations. L ν−mass is in proportion to the Majorana neutrino mass matrix. In analogy to the Type-I seesaw model, we can also define the effective dimension-5 operator where κ = λ H Y ξ /M ξ . After spontaneous symmetry breaking, neutrinos acquire masses and the neutrino mass matrix is given by Here v ≃ 174 GeV denotes the Higgs vacuum expectation value. Since M ξ ≫ v, the mass scale of neutrinos is then suppressed, and this is the so-called triplet seesaw mechanism.

III. CALCULATIONS OF THE BETA-FUNCTIONS
In our calculations, we always use the dimensional regularization. For the one-loop wavefunction renormalization constants Z above the seesaw scale, we find that For the vertex renormalization constants and the Higgs masses, we obtain The counterterms of the Higgs couplings λ ξ , λ C , λ φ and λ T have also been calculated, By using the counterterms calculated above and the technique described in [11], we obtain the β-functions (β X = µ d dµ X) of Yukawa couplings and λ H : and the anomalous dimensions (γ m = − 1 m µ dm dµ ) of the Higgs masses: It should be noticed that in the triplet seesaw model the mass of the Higgs doublet suffers from the so-called hierarchy problem, which can be prevented in some other supersymmetric models. We also calculate the β-functions of Higgs couplings: The RGEs for the gauge couplings are changed in this model, and we list the results below: Since the Higgs triplet field does not couple with quarks, the RGEs of Yukawa couplings for quarks are the same as those in the SM, and the corresponding results can be found in the literature [15]. By calculating the relevant one-loop diagrams, we obtain the β-function of the effective operator which describes the neutrino masses and mixing at the energy scales below the seesaw scale: It should be mentioned that Eq. (17) has the same form as that in the Type-I seesaw model, only up to a replacement λ → λ + λ † H λ H .

IV. APPLICATIONS
To see the running behaviors of neutrino mixing parameters in the triplet seesaw model, we carry out some numerical analyses by using the β-functions derived above. The lepton flavor mixing matrix, which comes from the mismatch between the diagonalizations of the neutrino mass matrix and the charged lepton mass matrix, is given by with (y e , y µ , y τ ) and (y 1 , y 2 , y 3 ) being the eigenvalues of Y e and Y ξ respectively. In this letter, we adopt the following parameterization [16]: where c ij ≡ cos θ ij and s ij ≡ sin θ ij (for ij = 12, 23 and 13). For the typical choice ∆ 0 ∼ O(0.1 eV), one can estimate that Y ξ ∼ O (1), which means that the relation Y ξ ≫ Y e holds, and the tiny Yukawa coupling Y e in the RGEs of Y ξ and Y e can be safely neglected in the hierarchical mass spectrum case 1 . Then we get the approximate equations of Yukawa couplings: where Using the results above and taking into account the fact that m τ > m µ ≫ m e , we neglect the tiny terms in proportion to y 2 e and arrive at the approximate analytical results of three mixing angles: where ∆ ij = y 2 i − y 2 j with (i, j = 1, 2, 3). Note that, in deriving Eqs. (22)-(24), we have adopted the the parametrization in Eq. (19). Such instructive expressions allow us to do useful analyses of the running behaviors of mixing angles. Due to the hierarchical charged lepton masses, there is in general no enhanced factor compared with the Type-I seesaw model [11]. However, nontrivial running effects may also be acquired from the sizable Yukawa coupling Y ξ . From Eqs. (22)-(24), one can immediately conclude that the corrections to θ 12 and θ 13 should be milder than that to θ 23 since the right-hand sides of Eqs. (22) and (24) are in proportion to either ∆ 21 or θ 13 . In the limit θ 13 → 0 and ∆ 21 → 0, we can see from Eq. (23) thatθ 23 ∝ −∆ 32 . Thus θ 23 will get negative correction in the normal hierarchy case. For illustration, we only show the evolution of θ 23 with different λ H (M ξ ) in Fig. 1. We can see that its running is quite sensitive to λ H and a decrease of several degrees may be acquired from the RGE evolution.
Considering the smallness of θ 13 , the evolution of the Dirac CP-violating phase δ is the same as those of two Majorana phases (ρ, σ) at the leading order of s −1 13 , sin δs 12 c 12 s 23 c 23 This is an interesting feature: once three CP-violating phases are same at certain energy scale, they will keep this equality against the RGE running. We can also see that the small sin θ 13 in the denominator of Eq. (25) dominates the running of CP phases. That means a fixed point [17] should exist for extremely tiny θ 13 . As an example, we plot the evolution of δ with different θ 13 in Fig. 2. Similar results can be obtained for two Majorana phases ρ and σ. By using Eq. (20), we obtain the RGEs of the eigenvalues of Y ξ with i = 1, 2, 3. Note that, for different signs of α ξ , the corrections to y i may be either positive or negative. However, in order to investigate the running of light neutrino masses, one should consider the RGEs of m ξ and λ H simultaneously. In Fig. 3, we present the typical evolution of three light neutrino masses with λ H (M ξ ) = 5 × 10 −5 . We can see that their running effects are appreciable and should not be neglected. A detailed numerical analysis of the triplet seesaw model is worthwhile and the corresponding work will be elaborated elsewhere.

V. SUMMARY
Working in the framework of the SM extended with one heavy Higgs triplet, we have derived a full set of one-loop RGEs for lepton Yukawa and Higgs couplings. Since the triplet seesaw model involves more couplings than the usual Type-I seesaw models, the results are also quite different. Analytical and numerical analyses have been given based on the RGEs we obtained. We find that nontrivial corrections to the mixing parameters can be acquired and they should not be neglected in general. It provides us a possible way to connect the experimental values of lepton flavor mixing parameters with some high energy GUT theories. In conclusion, our formulae are very important for both model building and phenomenological analyses of the triplet seesaw models.