Accurate renormalization group analyses in neutrino sector

We investigate accurate renormalization group analyses in neutrino sector between $\nu$-oscillation and seesaw energy scales. We consider decoupling effects of top quark and Higgs boson on the renormalization group equations of light neutrino mass matrix. Since the decoupling effects are given in the standard model scale and independent of high energy physics, our method can basically apply to any models beyond the standard model. We find that the decoupling effects of Higgs boson are negligible, while those of top quark are not. Particularly, the decoupling effects of top quark affect neutrino mass eigenvalues, which are important for analyzing predictions such as mass squared differences and neutrinoless double beta decay in an underlying theory existing at high energy scale.


Introduction
Neutrino oscillation experiments established that active neutrinos are massive, and the masses are much smaller than the other standard model (SM) fermions. The existence of nonzero neutrino masses is evidence of physics beyond the SM. It is therefore necessary to explain the nonzero and tiny neutrino masses. The seesaw mechanism [1] provides an attractive explanation, and a number of works have been presented in the context of the mechanism. Moreover, recent precision measurements of leptonic mixing angles in the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [2] showed that θ 12 and θ 23 are large, and θ 13 is small but not zero [3,4,5]. These results suggest the mixing angles are much larger than mixing angles of quark sector. Therefore, the nature of the neutrino is a key to study physics beyond the SM.
We can obtain some physical values in arbitrary high energy scale by solving the renormalization group equations (RGEs) and taking the experimental values as boundary conditions. The renormalization group (RG) evolution of the light neutrino mass matrix can be determined by solving the RGE of a coefficient of effective dimension five operator [6]. The RG analyses using the operator are relevant only below the lowest seesaw scale, e.g., the lightest right-handed neutrino mass in type-I seesaw mechanism. However, since the analyses are independent of the models, the analyses are useful for building models in high energy scale, in which the models are the grand unified theory (GUT), and/or have a new symmetry such as a flavor symmetry. In fact, a large number of works respect with the RGEs of the neutrino sector have been presented (e.g., see [7]- [15]). In particular, the RG effects can be large if the neutrino masses are quasi-degenerate [8,12,14,15]. There are also RG analyses in the minimal supersymmetric standard model (MSSM), which can realize the gauge coupling unification and be related to the GUT.
On the other hand, most of the analyses do not consider the decoupling effects of the massive SM particles. When a certain particle is decoupled, contributions from the particle should be subtracted from the RGEs. However, the decoupling effects are independent of the models beyond the SM, since the decoupling effects are of course given in the SM scale. Thus, when we analyze the RG evolution in the MSSM, we should use the subtracted RGEs in the SM scale, while can use the original RGEs in the MSSM scale. This method can basically apply to the other models beyond the SM. In this paper, we consider the RGEs in the SM and the MSSM, and investigate the decoupling effects of top quark and Higgs boson on the light neutrino mass matrix between ν-oscillation and seesaw energy scales. The relevant RGEs of the work will be shown in Appendix.
In our analyses, the light neutrino mass matrix is approximately described only by two parameters. One is an overall factor of the mass matrix, and the other denotes the RG effects of charged lepton Yukawa couplings and affects on the mixing angles. We will show the RG evolution of these parameters in both the SM and MSSM, and find the decoupling effects are negligible for the latter parameter, while not for the former parameter. Moreover, we will find that the effects are almost completely given by top quark decoupling, and the decoupling effects of Higgs boson are negligible. In the MSSM, these fundamental behaviors are the same as in the SM. Besides, when tan β ≃ 1, the RG evolution is similar to the SM results. Next, we will show the RG evolution of the mass squared differences and the mixing angles, in which the results are correspond to the MSSM with tan β = 30. We will find the decoupling effects are negligible for the mixing angles, while not for the mass eigenvalues. These results are important for analyzing predictions such as mass squared differences and neutrinoless double beta decay in an underlying theory existing at high energy scale. We will also discuss the dependence of decoupling effects on mass spectrum of light neutrinos, degeneracy of the masses and CP-phases.

Renormalization Group Evolution of Neutrino Mass Matrix
We consider the extensions of the SM and the MSSM, in which lepton mass terms in low energy scale are effectively given by where Y E , L, E R , and Φ are the Yukawa coupling matrix of charged leptons, left-handed lepton doublets, right-handed charged leptons, and (up-type) Higgs doublet in the SM (the MSSM), respectively. κ is a coefficient of effective dimension five operator. Now an effective light neutrino mass matrix M ν is given by κv 2 , where v is a relevant Higgs vacuum expectation value, that is, v = 174 GeV in the SM and v = 174 × sin β GeV in the MSSM, respectively. On the other hand, the light neutrino mass matrix can also be described by the PMNS matrix U and mass eigenvalues of light neutrinos: where the charged lepton mass matrix is diagonal, and M diag ν is a diagonal matrix, and α, β = e, µ, τ . Then, if the neutrinos are Majorana particles, the mass matrix can be Once one fixes those values at low energy as boundary conditions, one can obtain those values at arbitrary high energy scale by solving the corresponding RGEs. The RGE for κ ≡ M ν /v 2 is given by with t ≡ ln µ (µ is a renormalization scale), where C E = −3/2 in the SM and C E = 1 in the MSSM, respectively. And, at one-loop level, where Y f (f ∈ {E, U, D}) are Yukawa coupling matrices of the charged leptons, up-and down-type quarks, respectively, g i are gauge coupling constants and λ is the Higgs self coupling in the SM. Then, We can write the neutrino mass matrix as M ν (Λ) = R ( I M ν (Λ EW ) I ) at arbitrary high energy scale Λ, where Λ EW is some energy at electroweak scale, R is a flavor blind overall factor, and I is defined by - [11]. I α denote quantum corrections of the charged lepton Yukawa couplings as t EW dt y 2 α with t Λ ≡ ln Λ and t EW ≡ ln Λ EW . Then, the light neutrino mass matrix at arbitrary high energy scale can be written by where r ≡ R/I e . Now we introduce small parameters defined as ǫ τ ≡ I e /I τ − 1 and ǫ µ ≡ I e /I µ − 1. Since ǫ µ ≪ ǫ τ and ǫ µ is numerically almost equal to 0, we can neglect ǫ µ . Thus, Eq. (7) can be well approximated by where we drop the subscript of ǫ τ , that is, ǫ ≡ ǫ τ . To investigate the RG evolution of the mass matrix, all we have to do is calculating r and ǫ at arbitrary energy scale. r is calculated by and ǫ is calculated by The mass eigenvalues and the mixing angles can be extracted from the mass matrix. Note that the mass eigenvalues depend on both r and ǫ, while the mixing angles depend only on ǫ.
In addition to the above discussion, we should consider decoupling effects of the massive SM particles at low energy scale. Among the SM particles the order of their masses

Boundary Conditions
To solve the RGEs, we take the boundary conditions for fermions and bosons as , GeV [16]. α em , θ w , and g 3 are finestructure constant, weak mixing angle, and strong coupling constant, respectively. The experimental values for the mass eigenvalues and the mixing angles in low energy scale are given by the best-fit values [17]: We use these values as the boundary conditions at µ = M Z . In fact, the β-function of κ is zero below µ = M Z . Therefore, our analyses including the decoupling effects can accurately connect ν-oscillation to seesaw energy scale.

RG Evolution of r and ǫ
We show the RG evolution of r and ǫ in this subsection. In our notation, r and ǫ are calculated by Eqs. (9) and (10). In this subsection, we consider the mass spectrum of light neutrinos is the NH and m 1 = 0 eV, and all figures show within M Z ≤ µ ≤ 10 14 GeV.
Since, when we consider the type-I seesaw mechanism, the neutrino Yukawa couplings exceed 1 at higher energy scale than µ = 10 14 GeV, we consider the lower energy scale than µ = 10 14 GeV. And, since we take the boundary conditions of the RGEs at µ = M Z (except for m t ), r = 1 and ǫ = 0 at µ = M Z .
The upper figures of Fig.1 show the RG evolution of r and ǫ in the SM. We can see that the decoupling effects of top quark and Higgs boson are negligible for ǫ, but for r. For r, the difference between including the decoupling effects or not is specifically about 0.8% at µ = 10 14 GeV. In fact, the decoupling effects of Higgs boson are negligible. Thus, the top quark decoupling accounts for the difference, since top quark Yukawa coupling is much larger than the others. The sign inversion of r at log m t ≃ 2.2 just occur due to the top quark decoupling, that is, the sign of β-function of κ is changed when the contributions from top quark are subtracted from the RGEs. On the other hand, since ǫ depends on the integral of charged lepton Yukawa couplings, the decoupling effects are buried in the integrated value, that is, the decoupling effects are negligible.
The lower figures of Fig.1 show the RG evolution of r and ǫ in the MSSM. The gradient of r in high energy scale is positive in the SM, but negative in the MSSM, since top quark Yukawa coupling has positive contribution to the β-function and becomes dominant below µ = O(10 8−9 ) GeV, while gauge couplings have negative contribution and become dominant above µ = O(10 8−9 ) GeV. The gradient of ǫ is negative in the SM, but positive in the MSSM due to the sign of C E . We can see that the values in the MSSM scale depend on tan β. But, the differences between including the decoupling effects or not are almost independent of tan β. The differences are about 1.4% for r at µ = 10 14 GeV, and negligible for ǫ in all energy scale. Figure 2 shows the SUSY threshold dependence of r and ǫ in the MSSM with tan β = 30. The fundamental behaviors are the same as before. We can see that the differences between including the decoupling effects or not are almost independent of the value of SUSY threshold. Figure 3 shows the tan β dependence of r and ǫ in the MSSM with the decoupling effects. When tan β ≃ 1, the RG evolution is similar to the SM results. We can see that the minimal RG effect of r occurs at tan β ≃ 13, and ǫ can be large for large tan β. The reason for ǫ is simply because charged lepton Yukawa couplings are larger as tan β is large. Moreover, top quark Yukawa coupling is smaller, while bottom quark Yukawa coupling is larger. Then, top quark Yukawa coupling accidentally has the minimum at tan β ≃ 13. This is the reason for r. Note that since the RG effects of the mixing angles depend only on ǫ, the mixing angles can be unstable for large ǫ as we will show the next subsection.
In Figs.1 -3, we have considered the mass spectrum of light neutrinos as the NH and m 1 = 0 eV. Note that all figures are the same even if the mass spectrum is the IH, or the lightest neutrino mass is large as 0.05 eV, that is, the masses are quasi-degenerate. When we change the mass spectrum or the absolute neutrino mass, the light neutrino mass matrix (equivalently κ) also changes. But, r, which is proportional to the ratio of κ, does not depend on the magnitude of κ, since the magnitude is canceled in the ratio. ǫ obviously does not depend on the magnitude of κ, since ǫ is calculated by charged lepton Yukawa couplings. Moreover, both r and ǫ are independent of CP-phases, because the arguments of r and ǫ do not change during the RG evolution.
We note that the effective neutrino mass (M ν ) 11 is given by r × (M ν (M Z )) 11 . The amplitude of neutrinoless double beta decay is proportional to (M ν ) 11 . Therefore, we can easily see the RG evolution of the decay amplitude. On the other hand, experiments of the decay can restrict the absolute neutrino mass scale. Since we often consider the neutrino mass scale relates to unknown high energy physics, the RG evolution is important for constructing the models in high energy scale. Similarly, our analyses are useful for the other phenomenological problems, e.g., thermal leptogenesis [18], which is proposed to explain the baryon asymmetry in the universe. In the leptogenesis, the heaviest mass eigenvalue and the absolute neutrino mass are important parameters used to calculate the baryon asymmetry [19]. Since the mass eigenvalues are obtained by r and ǫ, and almost depend on r, the decoupling effects are not negligible. Thus, our results for the neutrino mass might become important for accurate computation in the canonical leptogenesis.

RG Evolution of the Mass Squared Differences
We show the RG evolution of the mass eigenvalues in this subsection and the mixing angles in the next subsection. As mentioned above, the RG evolution of the masses depends on both r and ǫ, while those of the mixing angles depend only on ǫ. In the SM or the MSSM with small tan β, all mixing angles almost stable because of the smallness of ǫ. Thus, we do not consider these cases. From here, all figures correspond to the results in the MSSM with tan β = 30 and SUSY threshold is taken at µ = 1 TeV. Figure 4 shows the RG evolution of the mass squared differences (∆m 2 21 ≡ m 2 2 −m 2 1 and ∆m 2 31 ≡ |m 2 3 − m 2 1 |) with the decoupling effects. The regions between each type of lines can be allowed by arbitrary combination of three CP-phases. For example, the shaded regions are the allowed region for m 1 (or 3) = 0.07 eV. This value of m 1 (or 3) corresponds to the upper bound imposed by Planck 2013 results (Planck + WP + highL + BAO), which is given by i m i ≤ 0.23 eV [20]. We can see that, when m 1 (or 3) becomes large, ∆m 2 21 can drastically vary in high energy scale compared with ∆m 2 31 . The reason can be understood by the RGEs of the mass squared differences, which are written by where C ′ s represent the corresponding coefficients. These RGEs show the feature that the evolution of ∆m 2 21 is more sensitive to the value of m 1 (equivalently the neutrino mass degeneracy) than that of ∆m 2 31 , because of ∆m 2 21 ≪ ∆m 2 31 . Now we note the CP-phase dependences of ∆m 2 21 and ∆m 2 31 . When m 1 = 0 eV in the NH, both ∆m 2 21 and ∆m 2 31 are independent of φ 1 , while when m 3 = 0 eV in the IH, they are independent of |φ 1 − φ 2 |. The reason is because, in the light neutrino mass matrix, the mass eigenvalues are always appeared as m 1 e iφ 1 , m 2 e iφ 2 , m 3 (see Eq. (2)). When m 1 (or 3) = 0 eV, the upper and lower parts of the allowed regions, except ∆m 2 31 in the IH, are taken by δ = 0 and π, respectively. For ∆m 2 31 in the IH, they are taken by δ = π and 0, respectively. The reason why this case is different from the others can be understood by Eq. (12). In the right-hand side of this equation, the sign of term which is proportional to m 2 1 depends on the mass spectrum, since ∆m 2 31 is defined as the absolute value, that is, ∆m 2 31 ≡ |m 2 3 − m 2 1 |. Therefore, ∆m 2 31 in the IH inversely behaves compared with that in the NH. In Particular, the upper and lower bounds are taken by some combinations of CP-phases as Table 1. Figure 5 shows ∆m 2 21 vs. ∆m 2 31 at µ = 10 14 GeV. As seen in Fig.4, the allowed regions are large for large m 1 (or 3) . The gradients of the figures reflect the sign of term which is proportional to m 2 1 in Eq. (12). We can see that the differences between including the Upper bound (0, any, π) / (0, π, π) (0, any, 0) / (0, 0, 0) Lower bound (π, any, 0) / (π, π, 0) (0, any, π) / (π, π, 0) IH ∆m 2 21 ∆m 2

Summary
We have investigated accurate renormalization group analyses in neutrino sector between ν-oscillation and seesaw energy scales. In other words, we have considered the decoupling effects of top quark and Higgs boson on the RGEs of the light neutrino mass matrix. Since the decoupling effects are given in the SM scale and independent of high energy physics, our method can basically apply to any models beyond the SM. Therefore, it is useful to use our method when one constructs the models in high energy scale. The relevant RGEs of the work are shown in Appendix. In our analyses, we have used the effective dimension five operator, then the light neutrino mass matrix is approximately described only with two parameters, that is, r and ǫ. r is the overall factor of the mass matrix, and ǫ denotes the RG effects of charged lepton Yukawa couplings. Using these parameters, the mass eigenvalues depend on both r and ǫ, while the mixing angles depend only on ǫ. We have shown the decoupling effects of top quark and Higgs boson for these parameters. The effects of Higgs boson have been negligible, but those of top quark have been considerable because of the largeness of top quark Yukawa coupling. For r, the differences between including the decoupling effects or not have been about 0.8% in the SM and 1.4% in the MSSM at µ = 10 14 GeV. On the other hand, the differences for ǫ have been negligible in all energy scale, since ǫ depends on the integral of charged lepton Yukawa couplings, and then the decoupling effects are buried in the integrated value. In the MSSM, the differences between including the decoupling effects or not are almost independent of the SUSY threshold and tan β. Besides, when tan β ≃ 1, the RG evolution has been similar to the SM results. These all results have been independent of the mass spectrum of the light neutrinos, the mass degeneracy and all CP-phases. In other words, both r and ǫ do not depend on the absolute neutrino mass scale and all CP-phases.
Next, we have shown the decoupling effects for the mass squared differences and the mixing angles. Once we calculate r and ǫ, we can obtain the mass eigenvalues and the mixing angles by extracting from the light neutrino mass matrix. We have derived the differences between including the decoupling effects or not are about 3.5% (4.0%) for ∆m 2 21 , and 2.9% (2.7%) for ∆m 2 31 at µ = 10 14 GeV in the NH (IH). These magnitudes of the differences have been nearly the same for any CP-phases. Since the mixing angles depend only on ǫ and the differences for ǫ has been negligible, the differences for the mixing angles have been also negligible.
The RG analyses can be applied to some phenomenological problems, e.g. neutrinoless double beta decay or thermal leptogenesis, which were discussed in Ref. [10]. The amplitude of neutrinoless double beta decay is proportional to (M ν ) 11 , which is given by r × (M ν (M Z )) 11 . In the leptogenesis, the heaviest mass eigenvalue and the absolute neu-trino mass are the parameters used to calculate the baryon asymmetry. Thus, accurate RG analyses are important to study these problems. Note that our analyses correct the previous results, and the corrections would be not negligible.

A.3 The RGEs in the MSSM
In the MSSM, we can consider the effective dimension five operator (the coefficient is denoted by κ) in low energy scale. The decoupling effects of the massive SM particle do