On Neutrino Mixing in Matter and CP and T Violation Effects in Neutrino Oscillations

Aspects of 3-neutrino mixing and oscillations in vacuum and in matter with constant density are investigated working with a real form of the neutrino Hamiltonian. We find the (approximate) equalities $\theta^m_{23} = \theta_{23}$ and $\delta^m = \delta$, $\theta_{23}$ ($\theta^m_{23}$) and $\delta$ ($\delta^m$) being respectively the atmospheric neutrino mixing angle and the Dirac CP violation phase in vacuum (in matter) of the neutrino mixing matrix, which are shown to represent excellent approximations for the conditions of the T2K (T2HK), T2HKK, NO$\nu$A and DUNE neutrino oscillation experiments. A new derivation of the known relation $\sin2\theta^m_{23} \sin\delta^m = \sin2\theta_{23} \sin\delta$ is presented and it is used to obtain a correlation between the shifts of $\theta_{23}$ and $\delta$ due to the matter effect. A derivation of the relation between the rephasing invariants which determine the magnitude of CP and T violating effects in 3-flavour neutrino oscillations in vacuum, $J_{\rm CP}$, and of the T violating effects in matter with constant density, $J^{m}_{\rm T} \equiv J^{m}$, reported in \cite{Krastev:1988yu} without a proof, is presented. It is shown that the function $F$ which appears in this relation, $J^{m} = J_{\rm CP}\,F$, and whose explicit form was given in \cite{Krastev:1988yu}, coincides with the function $\tilde{F}$ in the similar relation $J^{m} = J_{\rm CP}\tilde{F}$ derived in \cite{Naumov:1991ju}, although $F$ and $\tilde{F}$ are expressed in terms of different sets of neutrino mass and mixing parameters and have completely different forms.


Introduction and Preliminary Remarks
It was shown in 1988 in ref. [1] that in the case of what is currently referred to as the reference 3-neutrino mixing (see, e.g., [3]), the magnitude of the CP and T violating (T violating) effects in neutrino oscillations in vacuum (in matter with constant density) are controlled by the rephasing invariant J CP (J m T ≡ J m ) associated with the Dirac CP violation phase present in the Pontecorvo, Maki, Nakagawa and Sakata (PMNS) [4,5] neutrino mixing matrix: where U (m) li , l = e, µ, τ , i = 1, 2, 3, are the elements of the PMNS matrix in vacuum (in matter) U (m) . The CP violating asymmtries in the case of neutrino oscillations in vacuum, for example, A (l,l ) CPvac = P vac (ν l → ν l ) − P vac (ν l →ν l ) , l = l and l, l = e, µ, τ , P vac (ν l → ν l ) and P vac (ν l →ν l ) being the probabilities of respectively ν l → ν l and ν l →ν l oscillations, were shown to be given by [1]: A (e,µ) with Φ vac osc = sin ∆m 2 21 L 2E + sin ∆m 2 32 L 2E + sin ∆m 2 13 L 2E .
where ∆m 2 ij = m 2 i − m 2 j , i = j, m i , i = 1, 2, 3, is the mass of the neutrino ν i with definite mass in vacuum, E is the neutrino energy and L is the distance travelled by the neutrinos. In [1] similar results were shown to be valid for the T-violating asymmetries in oscillations in vacuum (in matter), A where Φ m osc has the same form as Φ vac osc in eq. (4) with ∆m 2 ij replaced by ∆M 2 ij = M 2 i −M 2 j , M i , j = 1, 2, 3, being the masses of the three neutrino mass-eigenstates in matter. In vacuum the T violating asymmetries in antineutrino oscillations,Ā (l ,l) Tvac = P vac (ν l →ν l )− P vac (ν l →ν l ), are related to those in neutrino oscillations owing to the CPT invariance: A (l ,l) Tvac . In ordinary matter (Earth, Sun) 2 the presence of matter causes CP and CPT violating effects in neutrino oscillations [6] and |Ā Tm |. However, in ordinary matter with constant density or with density profile which is spherically symmetric, like the matter of the Earth, the matter effects preserve the T symmetry and do not generate T violating effects in neutrino oscillations [1]. Thus, T violating effects in the flavour neutrino oscilations taking place when the neutrinos traverse, e.g., the Earth mantle or the Earth core can be caused in the case of 3-neutrino mixing only by the Dirac phase in the PMNS matrix.
The J CP -factor in the expressions for A (l,l ) CP(T) vac , l = l , is analogous to the rephasing invariant associated with the Dirac phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix, introduced in [7]. In the standard parametrization of the PMNS mixing matrix (see, e.g., [3]) it has the form: cos θ 13 sin 2θ 12 sin 2θ 23 sin 2θ 13 sin δ , where θ 12 , θ 23 and θ 13 are the solar, atmospheric and reactor neutrino mixing angles and δ is the Dirac CP violation phase. The expression for the J CP -factor is the same in the parametrisation of the PMNS matrix U PMNS ≡ U employed in [1]: where 2 By "ordinary" we mean matter which does not contain antiprotons, antineutrons and positrons. The expression of the PMNS matrix in the standard parametrisation U sp , is related to the expression in the parametrisation in eq. (7) as follows: U sp = U P * 33 (δ). In eq. (7) the two CP violation (CPV) Majorana phases present in U PMNS in the case of massive Majorana neutrinos [8] were omitted since, as was shown in [8,6], the probabilities of flavour neutrino oscillations of interest for the study performed in [1] and for the present study, do not depend on the Majorana phases. Thus, the results presented in [1] and the new results derived in the present article are valid for both Dirac and Majorana neutrinos with definite masses in vacuum.
In ref. [1] the following relation between the rephasing invariants in vacuum and in matter with constant density, J CP and J m T ≡ J m , has been reported: where A = 2E √ 2 G F N e is the matter term [9,10,11], G F and N e being respectively the Fermi constant and the electron number density of matter. The function F in eq. (10) was given in the following explicit form in [1]: where As was noticed in [1], the function F 3 can formally be obtained from the function F 2 by interchanging m 2 2 and m 2 3 , M 2 2 and M 2 3 , and |U e2 | 2 and |U e3 | 2 . In the parametrisation (7) used in [1] and, thus in eqs. (13) - (15), U e1 , U e2 and U e3 are real quantities: U e1 = c 12 c 13 , U e2 = s 12 c 13 and U e3 = s 13 , where c ij ≡ cos θ ij and s ij ≡ sin θ ij . Thus, |U ei | 2 = U 2 ei , i = 1, 2, 3. The function F (θ 12 , θ 13 , ∆m 2 21 , ∆m 2 31 , A) as defined by eqs. (11) -(15), depends, in particular, on the differences between the squares of the neutrino masses in vacuum and in matter, However, as it follows from the form of the Hamiltonian of the neutrino system in matter with constant density, whose eigenvalues are M 2 j /(2E) (see further), as well as from the explicit analytic expressions for M 2 j derived in [10], the mass squared differences D ij of interest are functions of θ 12 , θ 13 , ∆m 2 21 , ∆m 2 31 and A and do not depend on θ 23 and δ. As a consequence, the function F in eq. (10) is independent on θ 23 and δ [1]: F = F (θ 12 , θ 13 , ∆m 2 21 , ∆m 2 31 , A). In deriving the relation (10), the following parametrisation of the neutrino mixing matrix in matter U m was used: where θ m 23 , θ m 13 , θ m 12 , δ m are the neutrino mixing angles and the Dirac CPV phase in matter and the Majorana CPV phases were omitted. The phases β 2 and β 3 in the matrix Q are unphysical and do not play any role in the derivation of relation (10). They ensure that the matrix U m can be cast in the form given in eq. (16) [12] (see also [13]). Obviously, the parametrisation of U m in eq. (16) is analogous to the parametrisation (7) of the neutrino mixing matrix in vacuum.
It follows from eqs. (11) -(15) that [1] in the case of oscillations in vacuum, i.e., for N e = 0 (A = 0), one has and that F is symmetric with respect to the interchange of m 2 2 and m 2 3 , M 2 2 and M 2 3 and of |U e2 | 2 and |U e3 | 2 .
The relation (10) between J m and J CP implies, in particular, that we can have J m = 0 only if J CP = 0, i.e., T violation effects can be present in neutrino oscillations taking place in matter with constant density or spherically symmetric density distribution (like in the Earth) only if CP and T violation effects are present in neutrino oscillations taking place in vacuum. It was shown also in [1] that the presence of matter can enhance somewhat |J m | with respect to its vacuum value |J CP |: in the example considered in [1] the enhancement was by a factor of 3. Taking the best fit values of neutrino oscillation parameters for neutrino mass spectrum with normal ordering (inverted ordering) 3 obtained in the global analysis in [14], one always has for the ratio |J m /J CP | < 1.2 [15]. This result persists even if we fix δ to its best fit value and vary the other neutrino oscillation parameters in their 3σ allowed ranges determined in [14]. Relaxing arbitrarily the 3σ experimental constraints on the allowed ranges of ∆m 2 21 and ∆m 2 31 , we find that indeed the maximal enhancement factor |J m /J CP | is 3.6 for neutrino mass spectrum with normal ordering (NO) and 2.9 for spectrum with inverted ordering (IO). In both cases, the maximal enhancement corresponds to J m reaching its theoretical maximal value max(|J m |) = 1/(6 √ 3). In 1991 in [2] a relation similar to that given in eq. (10) was obtained: The functionF was given in the following form: where The relation (10) between the rephasing invariants J m and J CP was presented in [1] without a proof. In the present article, after discussing certain aspects of neutrino mixing in matter, we provide a derivation of the relation (10). Further, we show that the function F in eq. (10), as defined in eqs. (13) - (15), coincides with the functionF in the relation (19) obtained in [2], i.e., that the function F is just another representation of the functionF .

On the 3-Neutrino Mixing in Matter
In [1] the analysis was performed starting with the following Hamiltonian of the neutrino system in matter diagonalised with the help of the neutrino mixing matrix in matter U m [6]: (22) It follows from the preceding equation 4 that the Hamiltonian of the neutrino system, is diagonalised by the matrix U † U m and its eigenvalues are In the parametrisation (7) of the PMNS matrix U e1 , U e2 and U e3 are real quantities: U e1 = c 12 c 13 , U e2 = s 12 c 13 and U e3 = s 13 , where c ij ≡ cos θ ij and s ij ≡ sin θ ij . As a consequence, the Hamiltonian H is a real symmetric matrix 5 . This implies that the matrix U † U m , which diagonalises H, is a real orthogonal matrix: Since H does not depend on θ 23 and δ, O = U † U m should not depend on θ 23 and δ either. The fact that the matrix O in eq. (25) is a real orthogonal matrix implies that in the parametrisations (7) and (16) of the PMNS matrix in vacuum and in matter, the matrix is a real orthogonal matrix. The requirement of reality of the nondiagonal elements ofÕ leads to the conditions: which imply, in particular: The last condition has two solutions: The requirement of reality of the diagonal elements ofÕ leads to: These conditions also lead, in particular, to the constraint given in eq. (29) and to the solutions (30) and (31). It should be clear that satisfying the constraint (30) or (31) is not enough to ensure the reality of the matrixÕ. Consider first the consequences of the constraint in eq. (30). Requiring in addition that the determinant ofÕ is a real quantity implies: The constraint in eq. (30) for k = 0, for example, and the conditions of reality of the elements (Õ) 23(32) and (Õ) 22(33) ofÕ lead to: Numerical results on the dependence of θ m 23 , δ m , θ m 23 and δ m on the matter potential A, θ m 23 and δ m being the corresponding antineutrino mixing angle and CP violation phase, show that the relations (36) and (37) cannot be exact. However, they are fulfilled with extremely high precision for the mixing of neutrinos (antineutrinos) in matter in the case of IO (NO) neutrino mass spectrum. We find that in this case for any A/∆m 2 21 and the best fit values of the neutrino oscillation parameters quoted in eq. (18) we have: 6 An alternative solution to the discussed constraints is θ m 23 = θ 23 , β 2 = qπ, q = 0, 1, 2, ..., δ = k π, k = 0, 1, 2, δ m =k π,k = 0, 1, 2. It corresponds to CP (T) conserving values of δ (δ m ).
For the mixing of neutrinos (antineutrinos) in matter and spectrum with normal (inverted) ordering, eqs. (36) and (37) are fulfilled also with extremely high precision for A/∆m 2 21 < 30: For A/∆m 2 21 ∼ > 30 and mixing of neutrinos (antineutrinos) in matter and NO (IO) neutrino mass spectrum we have: Setting δ to its best fit value given in eq. (18) and varying the other neutrino oscillation parameters in their 3σ allowed ranges determied in [14] does not change significantly the results quoted in eqs. (40) -(44). Indeed, for mixing of neutrinos (antineutrinos) in matter in the case of IO (NO) neutrino mass spectrum and any A/∆m 2 21 we find that | These results are illustrated in Figs. 1 and 2 (3 and 4) where the ratios θ m 23 /θ 23 and δ m /δ (the ratios θ m 23 /θ 23 and δ m /δ) are shown as functions of A/∆m 2 21 in the case of mixing of neutrinos and NO (left panel) and IO (right panel) neutrino mass spectrum. We used the best fit values of neutrino oscillation parameters ∆m 2 31 , ∆m 2 21 , θ 12 and θ 13 from [14] and the analytic expressions for M 2 i , i = 1, 2, 3, from [12]. The approximate ranges of values of A/∆m 2 21 relevant for the T2K (T2HK) [17,18], NOνA [19] and DUNE [20]   Consider next the implications of the second condition (31) related to the requirement of reality of the matrixÕ. As can be easily shown, this condition alone i) ensures the reality of det(Õ), and ii) makes identical the two conditions in eq. (28) and the two conditions in eq. (32). Thus, after using condition (31) there are still two independent conditions to be satisfied to ensure the reality of the matrixÕ. We well derive next a condition that can substitute one of the required two conditions. The second condition then can be either the condition in eq. (28) or the condition in eq. (32) (after eq. (31) has been used).
The condition of orthogonality ofÕ,Õ(Õ) T = diag(1, 1, 1), as can be shown, leads to the following additional constraints: This result was derived in [21] (see also [13]) using the parametisations (7) and (16) introduced in [1] but employing a different method 7 . The equality (48) implies that the product sin 2θ 23 sin δ does not depend on the matter potential, i.e., is the same for neutrino oscillations taking place in vacuum and in matter with constant density. It is valid for neutrino and antineutrino mixing in matter independently of the type of spectrum neutrino masses obey -with NO or IO. From eq. (10) using the parametrisations defined in eqs. (7) and (16) we find: sin 2θ m 23 sin δ m sin 2θ 23 sin δ = F cos θ 13 sin 2θ 12 sin 2θ 13 cos θ m 13 sin 2θ m 12 sin 2θ m From this result and eq. (48) we obtain yet another equivalent representation of the function F (θ 12 , θ 13 , ∆m 2 21 , ∆m 2 31 , A):   [23].

The Relation between J m and J CP
Equation (22) can be cast in the form: One possible relatively simple way to derive the relation between J m and J CP given in eq. (19) and reported in [1] is to exploit the fact that the column matrices ) T are eigenvectors of the Hamiltonian H defined in eq. (24), corresponding to the eigenvalues M 2 i /(2E), i = 1, 2, 3. Using this observation it is possible to derive from eq. (52) explicit expressions for the elements of the neutrino mixing matrix in matter U m . They read: where and The function D 2 2 , which, as it follows from eqs. (1) and (56) -(59), enters into the expression for J m , is given by: It is easy to check that expression (60) for the function D 2 2 coincides with expression (14) for the function F 2 , i.e., that we have One can show in a similar way that the function D 2 3 coincides with the function F 3 given in eq. (15), i.e., that Using the fact that the function R in eq. (64), after some algebra, can be brought to the form: Equations (11), (10), (63) and the equalities F 2 = D 2 2 and F 3 = D 2 3 proven above, together with the equalities D 32 − D 12 = ∆m 2 31 and D 23 − D 13 = ∆m 2 21 , imply that R = F 1 . This completes the proof of the result reported in [1] and given in eqs. (10) and (11).
Two comments are in order. First, the function F (θ 12 , θ 13 , ∆m 2 21 , ∆m 2 31 , A), as determined in eqs. (11) is positive. Indeed, it follows from eqs. (61) and (62) that the functions F 2 and F 3 are positive. For A = 0, we have also F 1 D 12 D 13 D 23 D 32 > 0. One can show that this inequality holds also for A = 0, which leads to F > 0. This implies that the J CP -factors in vacuum and in matter have the same sign: This result is valid both for neutrino mass spetra with normal ordering (∆m 2 31 > 0) and with inverted ordering (∆m 2 31 < 0).
Second, the function F (θ 12 , θ 13 , ∆m 2 21 , ∆m 2 31 , A) in eq. (10) has different equivalent representations. This should be clear from the fact that and the derivation presented above. Indeed, we can use the second or the third form of J CP (J m ) in eq. (68) to obtain the relation given in eq. (10). The function F thus derived will differ in form from, but will be equal to, the function F defined in eqs. (11) - (15). It follows from eqs. (10) and (19) i.e., that the function F (θ 12 , θ 13 , ∆m 2 21 , ∆m 2 31 , A) found in [1] is another representation of the functionF found in [2]. The functions F andF have very different forms. Nevertheless, as we have verified, they coincide numerically. This is illustrated in Fig. 6 where we show the functions F (eq. (11)),F (eq. (20)) and the difference (F −F ) versus A/∆m 2 21 . We used the analytic expressions for M 2 i , i = 1, 2, 3, in terms of m 2 1 , A and the neutrino oscillation parameters ∆m 2 31 , ∆m 2 21 , θ 12 and θ 13 derived in [12]. It should be clear from eq. (22) that, as we have already discussed, in the parametrisation (7) employed in [1] the mass parameters M 2 i , i = 1, 2, 3, do not depend on θ 23 and δ. In Fig. 6, the neutrino oscillation parameters on which the functions F andF depend were set to their best fit values found in the global analysis of the neutrino oscillation data in [14] in the cases of NO and IO neutrino mass spectra.
As is suggested by Fig. 6 and we have commented earlier, our numerical results show that the function F is positive.

The Case of Antineutrino Mixing in Matter
In the preceding Sections we have focused primarily on the mixing and oscillations in matter of flavour neutrinos. In this Section we will discuss briefly the case of mixing and oscillations in matter of flavour antineutrinos.
In ordinary matter (of, e.g., the Earth, the Sun) the mixing of antineutrinos in matter differs from the mixing of neutrinos in matter as a consequence of the fact that ordinary matter is not charge conjugation invariant: it contains protons, neutrons and electrons, but does not contain their antiparticles. This causes CP and CPT violating effects in the mixing and oscillations of neutrinos in matter [6]. As a consequence the neutrino and antineutrino mixing angles, as well the masses of the respective neutrino mass-eigenstates, in matter differ. The expressions for the antineutrino mixing angles in matter, θ ij , the neutrino masses in this case, M k , and the corresponding J-factor, J m , can be obtained from those corresponding to neutrino mixing in matter, as is well known, by replacing the potential A with (−A).
Since the derivations of the results given in eqs. (69) -(71) do not depend on the sign of the matter term A, these results are valid also for mixing of antineutrinos in matter and for oscillations of antineutrinos ν l in matter with constant density. Thus, we have: where U m lj are the elements of the antineutrino mixing matrix in matter U m , δ m is the Dirac phase present in U m , and θ m ij are the antineutrino mixing angles in matter. We also have: sin 2θ m 23 sin δ m = sin 2θ 23 sin δ , Finally, as in the neutrino mixing in matter case, the equalities although not exact, represent an excellent approximations for the ranges of values of A/∆m 2 21 relevent for the T2K (T2HK), NOνA and DUNE neutrino oscillation experiments.

Summary
In the present article we have analysed aspects of 3-neutrino mixing in matter and of CP and T violation in 3-flavour neutrino oscillations in vacuum and in matter with constant density. The analyses have been performed in the parametrisation of the PMNS neutrino mixing matrix U PMNS ≡ U specified in eq. (7) and introduced in [1]. However, as we have shown, the results obtained in our study are valid (in some cases with trivial modifications) also in the standard parametrisation of the PMNS matrix (see, e.g., [3]).
Investigating the case of 3-neutrino mixing in matter with constant density we have derived first the relations θ m 23 = θ 23 and δ m = δ, θ 23 (θ m 23 ) and δ (δ m ) being respectively the atmospheric neutrino mixing angle and the Dirac CP violation phase in vacuum (in matter) present in the PMNS neutrino mixing matrix. Performing a detailed numerical analysis we have shown that although these equalities are not exact, they represent excellent approximations for the ranges of values of A/∆m 2 21 < 30 relevent for the T2K (T2HK), NOνA and DUNE neutrino oscillation experiments, the deviations from each of the two relations not exceeding respectively 1.3 × 10 −2 and 1.3 × 10 −3 (Figs. 1 and  2)). Similar conclusion is valid for the corresponding parameters θ m 23 and δ m in the case of mixing of antineutrinos (Figs. 3 and 4)).
We have derived next the relation sin 2θ m 23 sin δ m = sin 2θ 23 sin δ, and have shown numerically that it is exact (Fig. 5). The relation is well known in the literature (see [21,13]). We have presented a new derivation of this result. Using the indicated relation and the fact that the deviations of θ m 23 from θ 23 , 23 (A/∆m 2 21 ), and of δ m from δ, δ (A/∆m 2 21 ), are small, | 23 |, | δ | 1, we have derived a relation between 23 and δ working in leading order in these two parameters (eq. (51)). It follows from this relation, in particular, that for θ 23 = π/4, the leading order matter correction to δ vanishes, while for δ = 3π/2 (π/2) and θ 23 = π/4, the leading order matter correction to θ 23 vanishes.
We have discussed further the relation between the rephasing invariants, associated with the Dirac phase in the neutrino mixing matrix, which determine the magnitude of CP and T violating effects in 3-flavour neutrino oscillations in vacuum, J CP , and of the T violating effects in matter with constant density, J m T ≡ J m , obtained in [1]: J m = J CP F . F is a function whose explicit form in terms of the squared masses in vacuum and in matter of the mass-eigenstate neutrinos, of the solar and reactor neutrino mixing angles and of the neutrino matter potential (eq. (11)) was given in [1]. The quoted relation between J m and J CP was reported in [1] without a proof. We have presented a derivation of this relation. We have shown also that the function F = F (θ 12 , θ 13 , ∆m 2 21 , ∆m 2 31 , A) i) is positive, F > 0, which implies that J m and J CP have the same sign, sgn(J m ) = sgn(J CP ), and that ii) it can have different forms. We have proven also that the function F as given in [1] is another representation of the so-called called "Naumov factor" (Fig. 6): F = ∆m 2 12 ∆m 2 23 ∆m 2 31 (∆M 2 12 ∆M 2 23 ∆M 2 31 ) −1 , where ∆m 2 ij = m 2 i − m 2 j , ∆M 2 ij = M 2 i − M 2 j , m i and M i , i = 1, 2, 3, being the masses of the three mass eigenstate neutrinos in vacuum and in matter.
Finally, we have considered briefly the case of antineutrino mixing in matter and have shown that results similar to those derived for the mixing of neutrinos in matter are valid also in this case.
The results of the present study contribute to the understanding of the neutrino mixing in matter and flavour neutrino oscillations in matter with constant density, widely explored in the literature on the subject. They could be useful for the studies of neutrino oscillations in long baseline neutrino oscillation experiments (T2K, NOνA, T2HK, DUNE).