Overall feature of CP dependence for neutrino oscillation probability in arbitrary matter profile

We study the CP dependence of neutrino oscillation probability for all channels in arbitrary matter profile within three generations. We show that an oscillation probability for \nu_e \to \nu_\mu can be written in the form P(\nu_e \to \nu_\mu) =A_{e\mu} cos \delta + B_{e\mu} sin \delta + C_{e\mu} without any approximation using the CP phase \delta. This result holds not only in constant matter but also in arbitrary matter. Another probability for \nu_\mu \to \nu_\tau can be written in the form P(\nu_\mu \to \nu_\tau)= A_{\mu\tau} cos \delta + B_{\mu\tau} sin \delta + C_{\mu\tau} + D_{\mu\tau} cos 2\delta + E_{\mu\tau} sin 2\delta. The term which is proportional to sin 2\delta disappear, namely E_{\mu\tau}=0, in symmetric matter. It means that the probability reduces to the same form as in constant matter. As for other channels, probabilities in arbitrary matter are at most the quadratic polynomials of sin \delta and cos \delta as in the above two channels. In symmetric matter, the oscillation probability for each channel reduces to the same form with respect to \delta as that in constant matter.


Introduction
In solar and atmospheric neutrino experiments, the ν e deficit [1] and the ν µ anomaly [2] have been observed. These results strongly suggest the finite mixing angles θ 12 and θ 23 and the finite mass squared differences ∆ 12 and ∆ 23 , where ∆ ij = m 2 i − m 2 j . Within the framework of three generations, there are two more parameters θ 13 and δ to be determined. About θ 13 , only upper bound is obtained from CHOOZ experiment [3] and the information on the CP phase δ is not obtained at all. In order to determine these parameters, several long baseline experiments using artificial neutrino beam will be planned [4], and it is important to study the effect when the neutrino pass through the matter [5]. The main physics goal in these experiments is to measure the value of δ. In this paper, we study the CP dependence of oscillation probability for all channels in arbitrary matter profile.
Before giving our results, let us review the works on CP violation in three neutrino oscillation. At first we introduce the CP-odd asymmetry ∆P CP αβ = P (ν α → ν β ) − P (ν α →ν β ). In disappearance channel, ∆P CP αα is exactly equal to 0 in vacuum and independent of δ. However, ∆P CP αα is not always equal to 0 in matter. This is due to the genuine CP violation and/or fake CP violation from matter effects. In the case of α = e, Kuo and Pantaleone [6] have shown that P (ν e → ν e ) does not depend on δ in the context of solar neutrino problem. Therefore, ∆P CP ee = 0 arises from matter effects 1 . However, in the case of α = µ, P (ν µ → ν µ ) has the CP-odd term in asymmetric matter profile as pointed out by Minakata and Watanabe [7]. We investigate the CP dependence in more detail in this paper.
Let us consider the appearance channels. As the CP-odd asymmetry is proportional to sin δ in vacuum, ∆P CP αβ = 0 means that the discovery of CP violation. However, the situation completely changes when the matter effects are taken into account. Namely, ∆P CP αβ = 0 does not always mean the existence of CP violation [8], because the fake CP violation due to matter effects exists.
Here, it is difficult to separate genuine CP violation due to δ from fake CP violation. One of the methods to solve these problems is to take into account mass hierarchy approximation |∆ 21 | ≪ |∆ 32 |. Actually, some approximate formulae are given by Arafune et al. at low energy region [9] and by Cervera et al. [10] and Freund [11] at high energy region.
Next, we introduce the T-odd asymmetry ∆P T αβ = P (ν α → ν β ) − P (ν β → ν α ). Krastev and Petcov [12] have shown that ∆P T αβ is proportional to sin δ exactly in constant matter. Recently, Naumov [17] , Harrison and Scott [18] have derived the simple identity on the Jarlskog factor J [16] as∆ 12∆23∆31J = ∆ 12 ∆ 23 ∆ 31 J, where quantities with tilde represent those in matter. We can simply understand that ∆P T αβ is proportional to sin δ from this identity. We have studied the matter enhancement ofJ [19] taking advantage of this identity. Furthermore, Parke and Weiler have investigated the matter enhancement of the ∆P T eµ [20]. There are some works on the deviation from constant matter. In long baseline experiments, we need to estimate the validity of constant density approximation because the earth matter density largely changes along to the path of neutrino. The matter profile of the earth is approximately expressed by Preliminary Reference Earth Model(PREM) [21]. Minakata and Nunokawa [22] give the oscillation probability using mass hierarchy and adiabatic approximations. For the distance less than L = 3000 km, the matter density fluctuation is small and the constant density approximation is valid. On the other hand, it has been shown that the fluctuation of the density cannot be ignored for the distance greater than L = 7000km [23,24,25,26]. Furthermore, the constant density approximation is not valid in the case that the matter density profile is different from PREM and the asymmetric part exists. It is pointed out that ∆P T αβ has the term proportional to cos δ in arbitrary matter. [27,28]. We investigate this feature in more detail in this paper.
In previous paper, we have proposed the new method applicable to constant matter. This method is to estimate the product of effective Maki-Nakagawa-Sakata matrix elements [29]Ũ αiŨ * βi without directly calculatingŨ αi . We have shown that the oscillation probability P (ν e → ν µ ) is written in the linear combination 2 of cos δ and sin δ exactly [31] in constant matter. In other channels, for example, It is found that the probability is quadratic polynomial of cos δ, sin δ, and the CP dependence was equal to in vacuum [32].
In this paper, we give the exact CP dependence of oscillation probability for all channels in arbitrary matter profile. For the purpose, we decompose the Hamiltonian H in the form This decomposition plays a key role in our paper. As a result, we obtain the probability for This has the same form with respect to δ as in eq.(1) in constant matter. On the other hand, for ν µ → ν τ , we show that the probability is given by Comparing the equations (4) with (2), the probability in arbitrary matter profile has the term proportional to sin 2δ which does not exist in the probability for constant matter. Furthermore, in the case of symmetric matter profile, we show that this additional term disappears, namely E µτ = 0, and the probability reduces to the same form as in constant matter.

CP Dependence in Arbitrary Matter Profile
In this section, we study the exact CP dependence of neutrino oscillation probability in arbitrary matter profile. The Schrödinger equation for neutrino is where H is the Hamiltonian in matter and ν is flavor eigenstate ν = (ν e , ν µ , ν τ ) T . We introduce the MNS matrix which relates the flavor eigenstate ν α to the mass eigenstate ν i . The MNS matrix U in the standard parametrization is represented as where Γ δ = diag(1, 1, e iδ ) and using the abbreviation s ij = sin θ ij and c ij = cos θ ij . O 13 and O 12 represent 1-3 and 1-2 rotation matrix like O 23 , respectively. By using this relation (6), we can rewrite the H as where a(t) is matter potential defined by a(t) = 2 √ 2G F N (t) e E, and G F , N (t) e , E are respectively Fermi constant, electron number density and neutrino energy. This equation (9) means that the Hamiltonian can be decomposed into two parts. One is 1-2 and 1-3 mixing part which contain matter effects. The other is 2-3 mixing and CP phase δ part which does not contain matter effects. It is noted that this decomposition is garanteed by the relation We can separate CP phase δ from matter effects by taking advantage of this decomposition (9).
Changing ν to ν ′ as the Schrödinger equation (5) is rewritten as where We emphasise that the reduced Hamiltonian H ′ does not contain the 2-3 mixing and CP phase and is real symmetric. For anti-neutrino, we obtain the similar results by the replacements δ → −δ and in a(t) → −a(t). Next, we introduce the time evolution operator S(t) and S ′ (t) which is defined by the solution of the Schrödinger equation The relation between S(t) and S ′ (t) is determined by the transformation (11) and is given by By taking the component of (15), the relation between the time evolution operators for each flavour is given by Here, S αβ represents the transition amplitude for ν β → ν α . S eµ , S eτ and S µτ are obtained from S µe , S τ e and S τ µ respectively by the replacements S ′ αβ → S ′ βα , δ → −δ. Substituting (16)-(21) into the relation the oscillation probabilities in arbitrary matter profile are given by P (ν µ → ν τ ) = A µτ cos δ + B µτ sin δ + C µτ + D µτ cos 2δ + E µτ sin 2δ, (28) and the other probabilities P (ν µ → ν e ), P (ν τ → ν e ) and P (ν τ → ν µ ) are obtained by the replacements S ′ αβ → S ′ βα and δ → −δ in P (ν e → ν µ ), P (ν e → ν τ ) and P (ν µ → ν τ ), respectively. Here all coefficients A eµ , · · · , E µτ are constructed from the mixing angle θ 23 and S ′ including matter effects. See appendix A for detail. The oscillation probabilities for "anti-neutrino" are also obtained by the replacements δ → −δ and a(t) → −a(t).
We also comment the CP trajectory introduced by Minakata and Nunokawa. This is an orbit in the bi-probability space when δ changes from 0 to 2π [33,34]. Eq. (23) shows that CP trajectory is exactly elliptic even in arbitrary matter profile. In addition, we point out that the dependence of θ 23 for the oscillation probabilities is completely understood from eqs.(23)- (27). See appendix A for detail.
It is also noted that there are two features in asymmetric matter profile. First, the terms proportional to sin δ and sin 2δ are appeared in P (ν µ → ν µ ) and P (ν τ → ν τ ). The term proportional to sin 2δ are appeared in P (ν µ → ν τ ) and P (ν τ → ν µ ). These terms do not exist in constant matter [31,32]. Second, ∆P T αβ is not proportional to sin δ in asymmetric matter as in constant matter [12]. In the next section, we describe these features in more detail.

CP Dependence in Symmetric Matter Profile
In this section, we study the CP dependence of P (ν α → ν β ) in symmetric matter profile as special case of the previous section. In the case of symmetric matter along neutrino path, the time evolution operator S ′ becomes symmetric matrix for flavour indices [28,35]. As results, the relations between the coefficients of P (ν α → ν e ) eq.(23)- (25) and P (ν e → ν α ) are given by A τ e = A eτ , B τ e = −B eτ , C τ e = C eτ .
See appendix B for detail calculation. The probabirity P (ν e → ν α ) have the same form with respect to δ as eqs.(23)- (25) in arbitrary matter profile.
On the other hand, applying the condition (29) to the probability (26)- (28), we obtain the remarkable relations where the detailed calculation is given in the appendix B. Using these relations, the oscillation probabilities (26)- (28) have more simple form such as and P (ν τ → ν µ ) is simply obtained by replacements A αβ , B αβ , C αβ and D αβ . Then, the coefficients of P (ν τ → ν µ ) are given by where we use the condition (29) for eqs.(36) or the unitarity for last equation. Here, the point is that the term proportinal to sin 2δ is dropped in eq.(35) comparing with eq. (28). The other point is that the terms proportional to sin δ and sin 2δ do not exist in P (ν µ → ν µ ) and P (ν τ → ν τ ), As the results, the CP dependence of the oscillation probability for each channel in symmetric matter reduces to the same form as in constant matter [32]. This is the generalization of the result in our previous paper.
Finally, we study the T-odd asymmetry ∆P T αβ = P (ν α → ν β ) − P (ν β → ν α ). From the unitarity relation, we easily obtain and in symmetric matter profile we obtain where we use the relations (30) in eq.(39). In constant matter, Krastev and Petcov [12] have pointed out that ∆P T αβ is proportional to sin δ. Our result is applicable to the symmetric matter profile, which corresponds to the generalization of their result, even if the oscillation is nonadiabatic.
Let us turn the case of arbitrary matter profile. ∆P T αβ in asymmetric matter is not proportional to sin δ because the time evolution operator S ′ is not symmetric, namely S ′ αβ = S ′ βα . More concretely speaking, the coefficients are not symmetric for flavor indices A αβ = A βα , B αβ = −B βα , C αβ = C βα . We clarify the exact CP dependence of ∆P T αβ in asymmetric matter although this fact is suggested using approximation [27,28].

Summary
We summarize the results obtained in this paper. We have studied the CP dependence of the oscillation probability P (ν α → ν β ) both in arbitrary and in symmetric matter profile.
It is remarkable that the oscillation probability for each channel in symmetric matter reduces to the same form as in constant matter.