Leptogenesis and Low energy CP violation, a link

T. Endoh†, S. Kaneko‡, S. K. Kang†§,

T. Morozumi† and M. Tanimoto‡

† Graduate School of Science, Hiroshima University, 739-8526, Japan ‡ Department of Physics, Niigata University, Niigata, 950-2181, Japan § Department of Physics, Seoul National University, Seoul, 151-747, Korea

Abstract. How is CP violation of low energy related to CP violation required from baryon number asymmetry ? We give an example which shows a direct link between CP violation of neutrino oscillation and baryogenesis through leptogenesis.

When the sphaleron process is active, the sum of baryon number and lepton number is not a conserved quantity; \frac{d(B+L)}{dt} = Anomaly ≠ 0. Therefore, the evolution equation of baryon number and lepton number becomes a coupled equation. The present baryon number can be written in terms of the ”initial” lepton number and baryon number as;

B_{now} = \frac{1}{2}(B − L)_{ini.} + \frac{1}{2}(B + L)_{ini.} exp[−\frac{Δt}{τ}] → \frac{1}{2}(B − L)_{ini.}. (1)

Fukugita and Yanagida proposed ”baryogenesis without Grand unification” [1]. Then, B_{ini.} = 0 while the lepton number production is possible because their model includes the heavy Majorana neutrinos and their decays lead to the lepton number asymmetry;

L_{ini} ∼ Γ[N → l^{−}ϕ^{+}] − Γ[N → l^{+}ϕ^{−}].  (2)

The purpose of my talk is give a specific example which shows ”a direct link” between the size and sign of baryon number and CP violation in neutrino oscillatio;

P (ν_{μ} → ν_{e}) − P (ν̄_{μ} → ν̄_{e}) ∼ J = Im(K_{e1}K_{e2}^{∗})(K_{μ1}K_{μ2}^{∗})^{∗},  (3)

where K is MNS matrix.

J_{μ}^{cc} = l̄_{L}γ_{μ}Kν_{L}.  (4)

J ∼ P − P̄ = ΔP is related to MNS matrix. By taking the basis in which the mass matrix for the heavy Majorana neutrinos and charged leptons are real diagonal, the MNS matrix can be obtained through the diagonalization of m_{eff} = −m_{D} \frac{1}{M} m_{D}^{T} . On the other hand the lepton number asymmetry in the same basis is given as;

ϵ_{1} = \frac{Γ[N_{1} → l^{−}ϕ^{+}] − Γ[N_{1} → l^{+}ϕ^{−}]}{Γ[N_{1} → l^{−}ϕ^{+}] + Γ[N_{1} → l^{+}ϕ^{−}]} = −\frac{3M_{1}}{2M_{2}} \frac{Im[(m_{D}^{†}m_{D})_{12}^{2}]}{V^{2}(m_{D}^{†}m_{D})_{11}}, (5)

where V ∼ 1 (TeV) and we take M_{2} ≫ M_{1}. We consider ”the minimal seesaw” which generates L ≠ 0 and ΔP ≠ 0 simultaneously. The minimal model is (3, 2) model with 3 light neutrinos ν_{1}, ν_{2}, ν_{3} and 2 heavy Majorana neutrionos N_{1} and N_{2}.

L = l̄^{i}m_{l}_{i}l^{i} + ν̄^{i}m_{D}_{ij}N_{Rj} + \frac{1}{2} ¯ (N_{Rj})^{c}M_{j}N_{Rj},  (6)

† Talk presented by T. Morozumi, morozumi@hiroshima-u.ac.jp

where i = 1 ∼ 3, j = 1, 2. Dirac mass term 3 × 2 matrix,

m_{D} = ( \begin{matrix}m_{11} & m_{12} \\ m_{21} & m_{22} \\ m_{31} & m_{32} \end{matrix} ) = U_{L} ( \begin{matrix}0 & 0 \\ m_{2} & 0 \\ 0 & m_{3} \end{matrix} ) V_{R},  (7)

where U_{L} (V_{R}) is a 3 × 3 (2 × 2) unitary matrix. The important properties of the model are one light neutrino is exactly massless: det[m_{D} \frac{1}{M} m_{D}^{T}] = 0 and there are three CP violating phases since there are 3 = 6 − 3 imaginary elements in m_{D}. By writing V_{R} as follows;

V_{R} = ( \begin{matrix}c_{R} & s_{R} \\ −s_{R} & c_{R} \end{matrix} ) ( \begin{matrix}exp(−i\frac{γ_{R}}{2} ) & 0 \\ 0 & exp(i\frac{γ_{R}}{2} ) \end{matrix} ) ,  (8)

we can show the leptogenesis is determined by a CP phase γ_{R},

ϵ_{1} ∼ −Im[(m_{D}^{†}m_{D})_{12}^{2}]
∼ −(m_{3}^{2} − m_{2}^{2})^{2}s_{R}^{2}c_{R}^{2} sin 2γ_{R}.  (9)

On the other hand, CP violation in neutrino oscillation, J in Eq.(3), depends on all three CP violating phases because K is determined by the diagonalization as −K^{†}m_{D} \frac{1}{M} m_{D}^{T}K^{∗} and it is sensitive to all CP phases in m_{D}. We give an example for the model in which J is determined by leptogenesis phase γ_{R}. Suppose U_{L} is a real orthogonal matrix [2] as U_{L} = O_{23}O_{12}. MNS matrix K has the following form;

K = O_{23}O_{12} ( \begin{matrix}1 & 0 & 0 \\ 0 & cos θ & sin θ exp[−iϕ] \\ 0 & sin θ exp[iϕ] & cos θ \end{matrix} ) P,  (10)

where P is a diagonal phase matrix which is not relevant for J. θ and ϕ are determined through the diagonalization of −U_{L}^{†}m_{D}V_{R} \frac{1}{M} V_{R}^{T} m_{D}^{T} U_{L}^{∗}. Therefore θ and ϕ do not depend on U_{L} at all. It depends on m_{2}, m_{3}, θ_{R}.γ_{R} besides M_{1} and M_{2}. Those four quantities can be determined from the heavy Majorana decay width, Γ_{1} and Γ_{2} and two light neutrino masses scales Δm_{atm.} and Δm_{solar}. Taking θ_{23} = π/4, θ_{12} = π/4, we have;

K = ( \begin{matrix}\frac{1}{\sqrt{2}} & \frac{cos θ}{\sqrt{2}} & \frac{sin θ exp[−iϕ]}{\sqrt{2}} \\ −\frac{1}{2} & \frac{cos θ−\sqrt{2} sin θ exp[iϕ]}{2} & \frac{sin θ exp[−iϕ]+\sqrt{2} cos θ}{2} \\ \frac{1}{2} & −^{\frac{cos θ+\sqrt{2} sin θ exp[iϕ]}{2}} & \frac{− sin θ exp[−iϕ]+\sqrt{2} cos θ}{2} \end{matrix} ) P. (11)

It is easy to see J ∼ sin ϕ. We can show there is a correlation between ϕ and γ_{R}. If γ_{R} vanishes, m_{eff} becomes a real symmetric matrix. Then, ϕ must vanish in the limit. On the other hand, if γ_{R} does not vanish, the sign of γ_{R} determines the sign of lepton number asymmetry which in turn determines the excess of matter (anti-matter). In our model, the sign of J reflects the sign of γ_{R}. We found the following correlation holds.

B → −B ⇔ L → −L ⇔ γ_{R} → −γ_{R} ⇔ ϕ → −ϕ ⇔ J → −J.  (12)

In Fig.1 and Fig.2, we show the correlation (sin 2γ_{R}, sin ϕ, x + y), where x = \frac{Γ_{1}V^{2}}{M_{1}^{2}} and

y = \frac{Γ_{2}V^{2}}{M_{2}^{2}} , by identifying the two light neutrino masses as \sqrt{Δm_{atm.}^{2}} ∼ 5.5 × 10^{−2} eV and \sqrt{Δm_{solar}^{2}} = (4 ∼ 5) × 10^{−3}(LMA). We take \frac{M_{1}}{M_{2}} = 0.1. The figures are obtained for fixed (x + y) × 10^{−2} (eV) and varying u = x − y.

Figure 1.

Correlation bet. leptogenesis CP phase and low energy CP phase; (sin 2γ_{R}, sin ϕ, x + y). 0.06 < x + y < 0.075 (eV).

Figure 2.

Correlation bet. leptogenesis CP phase and low energy CP phase; (sin 2γ_{R}, sin ϕ, x + y). 0.06 < x + y < 0.066 (eV).

Summary

• leptogenesis phase (γ_{R}) certainly affects the neutrino oscillation CP violation through (ϕ). However, if we measure J only, we can not distinguish the phase for the leptogenesis (ϕ) from the other phases in U_{L} because only a certain combination of them appear. The isolation must be done using some other quantities, double β decay etc [3].

• We show the correlation between CP violating phase for leptogenesis and CP violating phase for neutrino oscillation for a specific choice of U_{L}.

Acknowledgments

The works of T. M. and M. T. are supported by the Grand-in-Aid for Scientific Research of the MEXT, Japan, No.13640290 and No.12047220 respectively. S.K.K is supported by JSPS Invitation Fellowship (No.L02515) and by BK21 program of the Ministry of Education in Korea. We thank H. So, A. Konaka, D. Chang, K. Hagiwara, J. Sato, T. Onogi, Y. Hayato, and O. Yasuda.

References

[1] M. Fukugita and T. Yanagida, Phys. Lett.B174, 45(1986).

[2] G. Branco, T. Morozumi, B. Nobre and M. Rebelo, Nucl. Phys. B 617, 475 (2001).

[3] T. Endoh, S. Kaneko, S. K. Kang, T. Morozumi, and M. Tanimoto, hep-ph 0209020.