hep-ph/9810471 DPNU-98-36 KEK-TH-597 OHSTPY-HEP-T-98-022 Oct. 1998

The renormalization group analysis of the large lepton flavor mixing and the neutrino mass

N. Haba,***E-mail:    N. Okamura,E-mail:    M. Sugiura E-mail:

Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA

Theory Group, KEK, Oho 1-1, Tsukuba 305-0801, Japan

Department of Physics, Nagoya University, Nagoya, 464-8602, Japan

The Superkamiokande experiment suggests the large flavor mixing between and . We show that the mixing angle receives significant corrections from the renormalization group equation (RGE) when both the second and the third generation neutrino masses are larger than . This means that the mixing angle must be small at the decoupling scale of right-handed neutrinos in the model containing a sterile neutrino with the mass spectrum of .

## 1 Introduction

The recent Superkamiokande data suggests the large neutrino flavor mixing between and . According to this experimental result, there have been a lot of theoretical attempts to explain why the large flavor mixing is realized in the lepton sector. One of the interesting approaches is concentrating on the effects of the the renormalization group equation (RGE). The RGE effects cause the enhancement of the neutrino flavor mixing in some situations.

In this paper we analyze the RGE of the neutrino flavor mixing between and in the minimal supersymmetric standard model (MSSM) with right-handed neutrinos. Here we stand the position that the smallness of neutrino masses is explained by the seesaw mechanism. We consider the situation that is much smaller than and , and expect the solar neutrino problem is solved by the oscillation between and a sterile neutrino . This situation is so-called four light neutrino scenario §§§ The four neutrino scenario might also explain the results of the LSND. The LSND results suggest the small mixing between and with . However, the confirmation of this result still awaits future experiments. Recent measurements in the KARMEN detector exclude part of the LSND allowed region. with the mass spectrum of . In this neutrino mass hierarchy, we find that the mixing angle between and receives significant corrections from the renormalization group equation (RGE). In this case the mixing angle at the high energy must be small as long as the mixing at the low energy is maximal. From the view point of the model building, we must find the fundamental theory which induces the small mixing angle at the high energy in the four neutrino scenario with the above mass spectrum.

## 2 The RGE effects of the neutrino flavor mixing

### 2.1 The RGEs of the Yukawa couplings

In this section we show the RGEs of the MSSM with right-handed neutrinos. The superpotential of the MSSM is given by

 W=yuijQiHu¯Uj+ydijQiHd¯Dj+yνijLiHu¯Nj+yeijLiHd¯Ej+μHHuHd+12MRij¯Ni¯Nj, (1)

where the index stands for the generation number . In this paper we neglect Yukawa couplings of the first generation since we consider the case where is much smaller than and . , , , , , and are quark doublet, lepton doublet, right-handed up-sector, right-handed down-sector, right-handed charged lepton, right-handed neutrino and Higgs fields, respectively. is the Majorana mass matrix of the right-handed neutrinos, which is symmetric under the generation indices . is the supersymmetric mass parameter of Higgs particles.

In this model the RGEs of Yukawa couplings are given by

 ddtyu = 1(4π)2[tr(3yuyu†+yνyν†)+3yuyu†+ydyd†−4π(163α3+3α2+1315α1)]yu, ddtyd = 1(4π)2[tr(3ydyd†+yeye†)+3ydyd†+yuyu†−4π(163α3+3α2+715α1)]yd, ddtye = 1(4π)2[tr(3ydyd†+yeye†)+3yeye†+yνyν†−4π(3α2+95α1)]ye, ddtyν = 1(4π)2[tr(3yuyu†+yνyν†)+3yνyν†+yeye†−4π(3α2+35α1)]yν, (2)

where and is a renormalization point, respectively. These equations are available for the energy region of , where denotes the energy scale of the Majorana mass.

Below the scale of , we should take the decoupling effects of heavy neutrinos into account. The effective theory is described without heavy neutrinos. The superpotential of Eq.(1) is modified as

 W=yuijQiHu¯Uj+ydijQiHd¯Dj+yeijLiHd¯Ej−12κijνiνjHuHu. (3)

Here s are the light modes of neutrinos which remain after integrating out the heavy ones. The coupling constant is defined as

 κij=(yνM−1RyνT)ij. (4)

It relates to the mass matrix of the light neutrinos as

 mνij=v2u2κij=v2sin2β2κij, (5)

where

 tanβ≡vuvd,       v2=v2u+v2d, (6)

with and . The value of is given by

 v=MZsin2θW2√απ=245.4(GeV), (7)

with GeV, , and .

In , the RGEs of the Yukawa couplings Eqs.(2) are modified as

 ddtyu = 1(4π)2[tr(3yuyu†)+3yuyu†+ydyd†−4π(163α3+3α2+1315α1)]yu, ddtyd = 1(4π)2[tr(3ydyd†+yeye†)+3ydyd†+yuyu†−4π(163α3+3α2+715α1)]yd, ddtye = 1(4π)2[tr(3ydyd†+yeye†)+3yeye†−4π(3α2+95α1)]ye, (8)

and

 ddtκ=18π2[{tr(3yuyu†)−4π(3α2+35α1)}κ+12{(yeye†)κ+κ(yeye†)T}]. (9)

From Eqs.(8), we can see that the RGEs of quark and charged lepton do not include the neutrino Yukawa couplings contrary to the case of as Eqs.(2). Hence below the running of the Yukawa couplings of quark and charged lepton can be determined independently of that of the neutrinos.

### 2.2 The RGEs of neutrinos in the effective theory

From now on we are concentrating on the RGE effects below the scale of , which are given by Eqs.(8) and (9). Since RGEs of the Yukawa couplings for quark and charged lepton of Eqs.(8) can be solved without information about the neutrino sector as mentioned before, the renormalization point dependences of and are completely determined by the RGEs’ boundary conditions which we take the masses of quark and charged lepton We use GeV, GeV, GeV, GeV, GeV, and GeV at . , the Cabibbo-Kobayashi-Maskawa matrix  and at the weak scale. Then we have only to concentrate on Eq.(9). Here we neglect phases in the flavor mixing matrices of the quark and the lepton sector for simplicity.

For convenience, we take three independent parameters , and instead of [, and ]. Here and are determined from by the following equations:

 κ = (cosθ23sinθ23−sinθ23cosθ23)(κ200κ3)(cosθ23−sinθ23sinθ23cosθ23), (10) δκ2 ≡ κ23−κ22, (11)

where

 κ3=√δκ22(√α+1√α),κ2=√δκ22(√α−1√α), (12)

with

 α≡∣∣∣1+κr1−κrcos2θ23∣∣∣. (13)

By using this relation, the RGE of Eq.(9) can be rewritten into the following three equations Equation (15) was firstly derived by Babu, Leung and Pantaleone in Ref. . :

 ddtκr = −18π2(y2τ−y2μ)κr, (14) ddtsin22θ23 = −18π2sin22θ23(1−sin22θ23)(y2τ−y2μ)1+κr1−κr, (15) ddtδκ2 = 18π2[2{tr(3yuyu†)−4π(3α2+35α1)}+y2τ+y2μ (16) +(y2τ−y2μ)(1+κ2r1−κ2r−12⋅1+κr1−κrsin22θ23)]δκ2.

Both and directly relate to the observed quantities in neutrino oscillation experiments******Since we take a diagonal base of the charged lepton, of this paper is an observable quantity. . The mass squared difference can be written as by using Eq.(5).

As shown before, and are determined without knowing the neutrino Yukawa coupling, thus we can obtain values of and at the weak scale by using only Eqs.(14) and (15). We analyze RGEs of Eqs.(14) and (15) by inputting various values of and for the initial conditions at †††††† As shown later, we do not need to calculate Eq.(16) in our analysis. .

### 2.3 Numerical results of the RGEs

Now we show the numerical results of the RGEs. Figure 1 shows the energy dependence of the values of , which is the coefficients of the RGEs of in Eq.(14) and in Eq.(15). Four lines correspond to the various values of , which we take 5, 20, 35, and 50. The values of do not receive the significant RGE corrections when is small.

Figures 2 show the energy dependence of with . We show the two cases of (a): GeV and (b): GeV with various initial conditions at . From the high energy to the low energy, simply increases. If we take the small , the slope tends to be flat because the value of in Eq.(14) decreases as Fig. 1.

Figures 3 show the energy dependence of the mixing angle with the same values of and as Figs. 2. There are three curves corresponding to the values of , and all of them have the same boundary condition of . From these figures we can see that the mixing angle at the weak scale changes depending on the value of . This difference can be easily understood by comparing Figs. 3 with Figs. 2 and by the existence of the factor in the R.H.S. of the RGE of Eq.(15). Let us see Fig. 2(a) and Fig. 3(a), for example. In the case of , where does not exceed one at all the energy scales, the mixing angle does not receive the significant RGE corrections. It is because the R.H.S. of Eq.(15) does not become so large as to enhance the mixing angle. On the other hand, in the case of exceeds one near the weak scale, which results in the significant enhancement of the mixing angle by the factor . Finally in the case of , exceeds one above the weak scale. Then the mixing angle once becomes maximal at high energy, however, after there it decreases rapidly since the sign of changes.

Figures 4 are the contour plots of the mixing angle at the weak scale. These are obtained by solving Eq.(15) with the various (horizontal axis) and (vertical axis). Here in Figs. 4 is the value at that induces . In the parameter region of , there are no significant RGE corrections and then the mixing angle does not change drastically. The energy dependence of the mixing angle in this case is similar to the solid lines in Figs. 3. Next in the case of , which means , the mixing angle at the weak scale is strongly enhanced near the weak scale, as the dashed lines in Figs. 3. Then the mixing angle at the weak scale becomes maximal independently of the mixing angle at . Finally in the case of , the mixing angle at the weak scale becomes small even for the large mixing angle at , where the energy dependence of the mixing angle is similar to the dotted lines in Figs. 3. From these arguments we can easily see that there is the large enhancement of the mixing angle from the RGE around .

Figures 5 are the contour plots of the heaviest neutrino mass at the weak scale. The horizontal and vertical axes are the same as those of Figs. 4. We determine the masses of neutrino by substituting the parameters in Eq.(12) with the result of Figs. 4 and the experimental value . Since we use the experimental value of instead of evaluating the RGE of Eq.(16), we can determine the masses without any additional input parameters. As becomes large, the region of is limited around . Compared with Figs. 4, it is found that the region where the heaviest mass is larger than corresponds to the region where the mixing angle at the weak scale is always larger than 0.9 despite the small mixing at scale.

Figures 6 and Figures 7 correspond to Figures 4 and Figures 5 with another value of . Compared with the case of , the value of are just shifted to the right in the case of . Around , the value of is slightly larger than that of at the same value of . In general, the smaller becomes, the more approaches one, and the value of around becomes larger. The region where the maximal enhancement of the mixing angle is derived by the RGE exists even in the case of small . The value of around becomes larger corresponding to the smaller value of .

We stress here that the large enhancement factor at the weak scale induced by in Eq.(15) is not the fine-tuning. This factor must be inevitably large with if and are larger than . It is worth noting that the enough enhancement of the mixing angle can be obtained even in the case of neutrino masses. Even in the small , mass is enough to obtain this RGE enhancement.

## 3 Summary and Discussion

In this paper we analyzed the RGE effects of the neutrino flavor mixing between and in the MSSM with right-handed neutrinos. The experimental result of the Superkamiokande suggests the large neutrino flavor mixing between and with eV. Then we found that the mixing angle between and receives significant corrections from the RGE in the case of . In this mass spectrum, the mixing angle at the decoupling scale of right-handed neutrinos must be small when .

Finally we give a brief comment about the LSND experimental result. In the above mass spectrum, both and must be of order 1 eV to explain the LSND result. In this case, the RGE analysis shows that the mixing angle at the decoupling scale of right-handed neutrinos must be negligibly small‡‡‡‡‡‡ Although we have not shown the lines of 1 eV in Figs. 5 and Figs. 7, there exist these lines. They are too small to be drawn apparently in the large . . From the view point of the model building, we must find the fundamental theory which induces the small mixing angle at the high energy scale in the four neutrino scenario with the mass spectrum of .

Acknowledgments

We would like to thank M. Tanimoto for useful discussions. NO would like to thank K. Hagiwara, J. Hisano and Y. Okada for useful discussions and comments. NH would like to thank S. Raby and K. Tobe for helpful discussions and comments. The work of NO is financially supported by the JSPS Research Fellowships for young scientist, No.2996. NH is partially supposed by DOE grant DOE/ER/01545-753.

## References Figure 1: The energy dependence of the values of y2τ−y2μ with various values of tanβ. These values are the coefficients of the RGEs of κr and sin22θ23. Figure 2: The energy dependence of κr with tanβ=50. (a): The initial condition is fixed as κr(MR)=0.8,0.9, and 0.99 at MR=1014 GeV. (b): The initial condition is fixed as κr(MR)=0.8,0.88, and 0.96 at MR=1016 GeV. Figure 3: The energy dependence of sin22θ with tanβ=50. We show two cases of (a): MR=1014 GeV and (b): 1016 GeV. Figure 4: The contour plots of the mixing angle sin22θ23 at the weak scale with tanβ=50. We show two cases of (a):MR=1014 GeV and (b):1016 GeV. The horizontal axes show the value of κr at μ=MR scale, and the vertical axes show the mixing angle at μ=MR The values of sin22θ23 at the weak scale are determined by inputting the initial values of sin22θ23(MR) and κr(MR). Around κc, the RGE effects make the weak scale mixing be large at any initial conditions. Figure 5: The contour plots of the heaviest neutrino mass m3 at the weak scale with tanβ=50. The horizontal and vertical axes are the same as Figs. 4. The values of m3 at the weak scale are determined by inputting the initial values of sin22θ23(MR) and κr(MR), and the experimental value of δm223=1.3×10−3eV2. As the mass becomes heavy, the region of κr(MR) is limited around κc. Figure 6: The contour plots of the mixing angle sin22θ23 at the weak scale with tanβ=35. In this case κc is larger than that of Figs. 4. Figure 7: The contour plots of the heaviest neutrino mass at the weak scale with tanβ=35. The region where m3 is larger than O(0.1eV) corresponds to the region in which the maximal mixing angle is realized at low energy.
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