Features of Neutrino Mixing

The elements (squared) of the neutrino mixing matrix are found to satisfy, as functions of the induced mass, a set of differential equations. They show clearly the dominance of pole terms when the neutrino masses"cross". Using the known vacuum mixing parameters as initial conditions, it is found that these equations have very good approximate solutions, for all values of the induced mass. The results are applicable to Long Baseline Experiments (LBL).


I. INTRODUCTION
Advances in neutrino oscillation experiments have yielded a wealth of information on the intrinsic neutrino properties, their masses and mixings. Two mass differences are well measured. Neutrino mixing is described by the 3 × 3 unitary PMNS matrix, V P M N S , which, because of rephasing invariance, contains only four physical variables. Thus, instead of the matrix elements V αi (α = e, µ, τ ; i = 1, 2, 3), only rephasing invariant combinations thereof, such as |V αi | or |V αi | 2 = W αi , are physically measurable, and they can be expressed in terms of four physical parameters. In an ideal situation, where all the parameters are precisely known, one can choose to use any set and arrive at the same result for the exact |V αi | values. In reality, however, our knowledge about |V αi | is far from uniform. While the elements |V ei |, (i = 1, 2, 3), and |V α3 |, (α = µ, τ ) (group I), are experimentally accessible and well determined, the remaining four elements (|V µ1 |, |V µ2 |, |V τ 1 |, |V τ 2 |) (group II) suffer from large uncertainties (see, e.g. [1][2][3]). In the widely used standard parametrization (SP), the elements in group I are simple functions of SP, so that the angles (θ 12 , θ 23 , θ 13 ) are all well determined, while elements in group II are complicated functions of SP, making it very hard to estimate the remaining phase (the Dirac δ). Note that the unitarity conditions on W αi = |V αi | 2 , which may help to constraint group II elements, are hard to implement in terms of SP. The above comments are given assuming neutrinos are Dirac particles. However, there are likely scenarios in which neutrinos are Majorana particles, and there may be extra neutrinos in addition to those in the standard model [4]. In the first case, phases of Majorana neutrinos are physical, and rephasing invariance is lost. In the second case, unitarity of W ν is broken. Thus, the results in this paper are valid only if extra neutrinos do not exist, and, for Majorana neutrinos, only when we consider neutrino oscillations which conserve the lepton number.
In this paper, we propose to parametrize W directly and simply. The linear dependence of these parameters facilitates the implementation of the unitarity condition so that the group II elements are woven into the structure of W . It is found that, given the known values of group I, these elements are already significantly constrained. They are also tightly correlated. For neutrino propagation in matter, we establish a set of differential equations for the evolution of the elements W αi , as functions of the induced neutrino mass. These equations are simple and compact in form, so that one can visualize the properties of their solutions with ease. It is found that the result corresponds to two well-separated levelcrossing solutions. The mixing parameters change rapidly only in the neighborhood of two resonances, while in the regions outside of those they are mostly flat. Another interesting consequence of level crossing is the decoupling effects, which tend to suppress the influence of initial conditions. Thus, it will be shown that the mixing matrix in matter is actually simpler than that in vacuum. This paper is organized as follows. In Section II, the general properties of a set of rephasing-invariant parameters are briefly introduced. In Section III, the physical variables |V αi | 2 = W αi are parametrized by imposing the unitary conditions. In Section IV, a set of differential equations for matter effects are derived, and the approximate solutions are obtained. The numerical solutions for the differential equations are shown in Section V. We then outline the possible applications of our formulations to experiments in Section VI and summarize this work in Section VII.

II. NOTATIONS
It is well known that physical observables are independent of rephasing transformations on the mixing matrices of quantum-mechanical states. Whereas there is nothing wrong with using these matrix elements in intermediate steps of a calculation, at the end of the day, they must form rephasing-invariant combinations in physical quantities. This situation is similar to that in gauge theory, where one often resorts to a particular gauge choice for certain problems. The final results, however, must be gauge invariant. In this paper, we propose to use, from the outset, parameters that are rephasing invariant. The use of only physical variables has another interesting consequence. As we will show in Sec. IV, as functions of the induced neutrino mass, the physical variables obey a simple set of differential equations, while one expects that they would be rather complicated when written in terms of the SP variables. Note that there is a similar simplification for the RGE of neutrinos and of quarks [5,6].
We turn now to Ref. [7], where it was pointed out that six rephasing invariant combination can be constructed from a 3 × 3 unitary mixing matrix V , which, for ν mixing, are given by where (α, β, γ) = (e, µ, τ ), (i, j, k) are cyclic permutations of (1, 2, 3), and detV = +1 is imposed. The common imaginary part is identified with the Jarlskog invariant [8], and the real parts are defined as The (x i , y j ) parameters are bounded, −1 ≤ (x i , y j ) ≤ 1, with x i ≥ y j for any pair of (i, j). It is also found that the six parameters satisfy two conditions, leaving four independent parameters for the mixing matrix. They are related to the Jarlskog invariant, and the squared elements of V , The matrix of the cofactors of W , denoted as w with w T W = (detW )I, is given by The elements of w are also bounded, −1 ≤ w αi ≤ +1, and The relations between (x i , y j ) and the standard parametrization can be found in Ref. [9]. There are other rephasing-invariant combinations that are useful. One first considers the product of four mixing elements [8] which can be reduced to In addition, for α = β = γ and i = j = k, we define Since Re(Π αβ ij ) takes the forms, we have In terms of the (x, y) variables, where (x a , y l ) comes from |V γk | 2 = x a − y l , and a = b = c, l = m = n. Another interesting combination is given by Here if |V αi | 2 = x a − y l , then b = c = a, m = n = l. This means that if one takes the αth row and the ith column, complex conjugates the vertex (V * αi ), then the product is rephasing invariant and has a well-defined imaginary part. Of particular interest is Ξ e3 . If we write Ξ e3 = |Ξ e3 |e iδ , then Im[Ξ e3 ] = J(1 − W e3 ) = sin δ|Ξ e3 |. Thus, the (rephasing invariant) phase of Ξ e3 is identified with the Dirac phase in the SP. Also, using vacuum values, |Ξ 0 e3 | 2 ∼ = 1.1 × 10 −3 and Re[Ξ 0 e3 ] ∼ = w 0 e3 /2, it was found [10] that As a result, the leptonic CP violation depends crucially on the determination of w 0 e3 .

III. PARAMETRIZATION OF THE NEUTRINO MIXING MATRIX
Neutrino mixing is described by the 3 × 3 unitary PMNS matrix, V P M N S , or V ν . Because of the rephasing invariance, only four parameters contained therein are physical. If these parameters are all precisely known, using different sets will not make much difference. In reality, the choice of them depends on how best they can be used to incorporate our partial knowledge gleaned from available experimental data. The widely used standard parametrization (SP) emphasizes the matrix elements V ei and V α3 (i = 1, 2, 3; α = e, µ, τ ), because (the absolute squares of) these elements have been well measured. The remaining elements (V µ1 , V µ2 , V τ 1 , V τ 2 ), however, are complicated functions of the SP and are hard to pin down (see, e.g., [1][2][3]), given that their possible errors are not related to those of the SP in any simple way.
In this paper we propose a parametrization by concentrating directly on the physical variables |V αi | or |V αi | 2 = W αi . By imposing the unitarity conditions uniformly, it is seen that the errors of all W αi are strictly and simply correlated. To include the matter effects, these parameters are considered to be functions of the induced neutrino mass A = 2 √ 2G F n e E. In the next section, we show that they obey simple differential equations which, with the known vacuum neutrino parameters as inputs, have good approximate solutions.
A general parametrization of W ν can be written in the following form where we choose and Here The constant matrices are chosen to take into account the known features of vacuum neutrino mixing. In particular, the matrix [ W 0 ] is a well-known approximation to the vacuum mixing matrix [W ν (0)]. Thus, all the vacuum values (b 0 , c 0 , d 0 , e 0 ) are small. Indeed, the estimated values [11] of (b 0 , c 0 ) are b 0 ∼ = 0.01 and c 0 ∼ = 0.02. While the sign of d 0 is ambiguous, its absolute value is favored to be d 0 ∼ = 0.05. Also, there is a bound on w 0 [B], and [C]. Also, (d, e) are µ − τ symmetry-breaking parameters.
As we shall see in the next section, as A varies, (d, e) remain small, while (b, c) will undergo substantial changes.
Putting it all together, we have Using the matrices [W ν ] and [w ν ], we can readily express the variables (x i , y j ) in terms of the set (b, c, d, e), which we will not write down explicitly here.

IV. DIFFERENTIAL EQUATIONS FOR MATTER EFFECTS
When neutrinos propagate in a medium of constant density, their interactions induce a term in the effective Hamiltonian, H = 1 2E M ν M † ν , given by [12,13] Thus, the neutrino mass eigenvalues squared (D i = m 2 i ) and mixing matrix are functions of A. It was shown [14] that they satisfy a set of differential equations, given by Here, Eq. (24) is not rephasing invariant and should be used by making rephasing invariant combinations constructed from V αi . This was done for the (x i , y j ) variables in Ref. [10], as well as for W ei and w ei . In this paper, we study the corresponding equations for W αi = |V αi | 2 . We find, from Eq. (24), where we have used the definitions Π αe ik = V αi V * αk V ek V * ei and the relation (Π αe ik ) = (Π αe ki ) * . We may further simplify the results by using Also, from the identity (which follows from α V * We may now collect these results in a very compact form, The equation for J was also computed [9]. It reads Eq. (29) has a simple structure-it consists of pole terms with numerators being quadratic functions of W αi . In addition, with e singled out by (δH) ee = 0, it exhibits permutation symmetry under the exchanges µ ↔ τ and i ↔ j ↔ k. This can be made explicit by rewriting Eq. (29) in the form,  (30), we infer the "matter invariant" [14][15][16][17], Another "matter invariant" follows immediately from Eqs. (29) and (30), We turn now to a more detailed analysis of these differential equations. We note first that the group (W ei , D i ), (i = 1, 2, 3), according to Eqs. (23) and (29), forms a closed set (see also Eq. (30) for J. With the known vacuum values of W ei and (D i − D j ), we can thus solve for these parameters as functions of A. Armed with these results, the remaining elements W αi (α = µ, τ , and i = 1, 2, 3) can be analyzed.
Consider explicitly the equations for (W e1 , W e2 , For small A values, the term ∝ 1/(D 2 − D 1 ) dominates, since here D 3 ≫ (D 2 , D 1 ). In addition, W e3 ≪ 1, so that there is a "double suppression" for the second term in Eqs. (35,36), which can be well-approximated by Eqs. (34) and (37) are exactly those for a level-crossing problem for two flavors. The solutions, as given in Eq. (22) of Ref. [9], with the approximate initial conditions W e1 (0) = 2/3, W e2 (0) = 1/3, δm 2 21 = δ 0 = 7.53 × 10 −5 eV 2 , plus the definition ∆ 21 = D 2 − D 1 , are Note also that (d/dA)(W e1 W e2 ∆ 2 21 ) = 0. Thus, we find a typical resonance behavior near the (lower) resonance point, A = A l ∼ = δ 0 /3. Away from A l , ∆ 21 ≃ D 2 → A, W e2 → 1, and W e1 → W 0 e1 W 0 e2 /∆ 2 21 ∼ 2/(9A 2 ). Notice the effects of decoupling. As A pulls away from A l , the state |2 approaches a pure |e state. All the parameters tend to their limiting values of no mixing, independent of their initial configurations. Similarly (for normal ordering), as A increases further, when D 32 reaches a minimum near D 2 ≃ A ∼ m 2 3 , we have the equations, The initial conditions for these equations are the values of W ei and D i obtained for A ≫ A l . Again, the dropped terms are doubly suppressed from (D 2 , D 3 ) ≫ D 1 and W e1 → 0. So now we have another (higher) resonance near A ≃ ∆ 0 = 2.45 × 10 −3 eV 2 ∼ = 31δ 0 , where q h ∼ = 1 − 2W e3 (0). Note that the contributions from the pole term ∝ 1/(D 3 − D 1 ), according to Eq. (29), are always doubly suppressed. First, from the denominator, and second, from W e1 W e3 ≪ 1, for all A values. In summary, the set (W ei , ∆ ij ), according to Eqs. (38-40) and (44-46), can be described in terms of two well separated level-crossing problems. While the mass eigenvalues D i take turns to rise proportionally to A, the W ei change rapidly only near A l ( ∼ = δ 0 /3, the "lower resonance") and A h ( ∼ = ∆ 0 ∼ = 31δ 0 , the "higher resonance"). There are two regions 1) A i , with A l < A i < A h ; and 2) A d , with A d ≫ A h , in which W ei are stationary. The span of A i and A d can be obtained from the positions and widths of the two resonances. A conservative estimate yields: 5δ 0 < ∼ A i < ∼ 15δ 0 , A d > ∼ 50δ 0 . In terms of the parameters b and c, we see that b → − 2 3 + c 0 2 and c = c 0 as A → A i , so that W e1 → 0 and W e2 → 1 − c 0 . After the higher resonance, We now turn to the matrix elements W µi and W τ i . Up to the region A ∼ A i , from Eq. (29), for W µ3 and W τ 3 , there is no contribution from the dominant pole term (∝ 1/(D 2 − D 1 )), contributions from the other pole terms (∝ 1/(D 3 − D 1 ) and 1/(D 3 − D 2 )) are also doubly suppressed (see Eq. (52) below). Thus, to a very good approximation, (47) Similarly, And, positivity demands i.e., e(A i ) → −(d 0 /2) + ε 0 . This also fixes the elements W µ1 and W µ2 . To within an accuracy of 0.01, we may ignore ε 0 and obtain This shows that, after the lower resonance, the matrix W assumes a very simple form, depending only on the small vacuum parameters (c 0 , d 0 ). As a check on the stability of W over the range That all elements Λ µi and Λ τ i are small is consistent with the constancy of W µi and W τ i . Finally, after the higher resonance (for NO), A ∼ A d , W α1 is unchanged, while W e3 → 1 and W µ3 and W τ 3 → 0, but (W µ2 + W µ3 ) and (W τ 2 + W τ 3 ) are stationary. This means that To summarize, Eq. (23) and Eq. (29), with the known (approximate) vacuum values (δ 0 , ∆ 0 , W 0 e1 , W 0 e3 ) as initial conditions, turn out to have very good approximate solutions for all A values. The mass eigenvalues squared, D i , rise proportionally to A, successively. The mixing matrix, [W (A)], has two well-separated regions (around A l and A h ) in which some matrix elements evolve rapidly. There are two regions, A ∼ A i and A > ∼ A d , wherein all the matrix elements are nearly stationary. These matrices are given in Eqs.

V. NUMERICAL SOLUTIONS
The general features of D i and W αi for the ν sector in matter are plotted as functions of A/δ 0 in Fig. 1 and Fig. 2, respectively, under both the normal (NO) and the inverted (IO) orderings. It is seen that D i goes through both lower and higher resonances under NO, while there is no resonance under IO. The elements W αi may go through the resonance at A < ∼ δ 0 or A ∼ ∆ 0 , or both, depending on the neutrino types (ν orν) and the mass orderings (NO or IO). We will not show the plots for theν sector, in which there is a higher resonance for D i under IO. The behavior of W αi for theν sector can be summarized as follows: (i) For NO, there is no resonance. (ii) For IO, only W α1 and W α3 go though the higher resonance. It should be emphasized that at the present accuracy, it is unlikely to reach the vacuum values of all the [W ν ] elements to the same satisfactory level. Thus, for illustration purpose only, we roughly estimate the values of W αi in vacuum based on the available analyses (see, As will be seen in the next section, the quantities Λ γk defined in Eq. (26), , play important roles in the transition probability for neutrino oscillation. They represent the relative weight of each sin 2 Φ ij component in the probability function. Explicitly, We plot Λ γk as functions of A/δ 0 in Fig. 3.

VI. APPLICATIONS TO THE EXPERIMENTS
As the neutrinos travel through a baseline, the flavor transition probability is given by the well-known expression, where the explicit form of , L is the baseline length, and E is the neutrino energy. More explicitly, for the disappearance channel, and for the appearance channel, α = β = γ, For neutrinos in vacuum, the [W ν ] matrix is parametrized by Eq. (21), and the values of Λ 0 = Λ(0) are given by Using Eq. (52), it can be verified that with A ∼ A i , the appearance channels, ν e → ν µ and ν e → ν τ , are insignificant and their probabilities are only of order ∼ c 0 or less. In addition, for P (ν µ → ν τ ), the contribution from the dominant term (sin 2 Φ 31 ) is of order ∼ 1, while that from sin 2 Φ 21 and sin 2 Φ 32 terms are only of order ∼ c 2 0 . Thus, the appearance channel ν µ → ν τ could be significant at A ∼ A i if the experimental setup is properly chosen so that sin 2 Φ 31 is large. On the other hand in P (ν µ → ν e ) the contribution from sin 2 Φ 31 and sin 2 Φ 21 terms become of order ∼ c 0 d 0 , while that from sin 2 Φ 32 terms are of order ∼ c 0 .
In addition to the experiments involving terrestrial neutrino sources, intensive effort has also been devoted to the study of extraterrestrial neutrino sources such as the neutrinos from a core-collapsed supernova [20,21]. One of the characteristics of these neutrino fluxes is that they travel through a very dense media before they exit. Thus, the induced mass corresponds to A > ∼ A d . Our results are therefore relevant to such processes, especially in regard to the question of NO vs IO.
By using the W -centric parametrization, a proper estimation of W αi in matter leads to simple expressions of ν oscillation probabilities to within the accuracy of ∼ 0.01. The expressions reveal explicitly the relative order of magnitude of the contributions from sin 2 Φ ij . Thus, by choosing a proper experimental setup which leads to a significant magnitude of sin 2 Φ ij , a careful analysis of data may shed some light on the parameter d 0 , which is closely related to the µ − τ asymmetry, J, and θ 23 . We shall leave the detailed analysis to a future work.

VII. CONCLUSION
The central issue in flavor physics is the determination of the mixing matrices of quarks and neutrinos; or rather because of the rephasing invariance, the measurement of the absolute values of their matrix elements. In the quark sector, this effort has culminated in extremely accurate results for the squared CKM matrix elements (to order 10 −5 ), which will be summarized in Appendix A. For the neutrino sector, despite its "new comer" status, our knowledge on |V αi | 2 is nevertheless quite substantial. Of the elements of [W ν ] (W αi = |V αi | 2 ), five are rather accurately known. In this paper we introduce a parametrization of [W ν ] in which the unitarity conditions are strictly imposed. This brings out explicitly the strong correlations between the elements of [W ν ]. A precision measurement on one of the lessor known elements would go a long way toward fixing the whole matrix.
Another interesting subject is the study of neutrino propagation in matter, for which its parameters become functions of A, the induced neutrino mass. In this paper, we derive a set of differential equations obeyed by the elements W αi . The distinctive feature of these equations is their dependence on the variables Λ γk , which also play central roles in the formulas of neutrino oscillation probabilities, in addition to the renormalization group equations of quarks and neutrinos. Note also that Λ γk are simple functions of rephasing invariant variables (W αi or (x i , y j )), instead of their complicated forms in the SP. Thus, it would be worthwhile to reanalyze the experiments directly using rephasing invariant variables so as to avoid losing information in translation.
As for solving these differential equations, it is found, somewhat fortuitously, that they have very good approximate solutions for all values of A, when the initial conditions are taken to be the currently available (albeit incomplete) values for vacuum neutrino parameters. The results (for NO) are dominated by two well-separated level-crossing (resonance) solutions. Outside of these resonance regions, all the mixing parameters are nearly stationary. It is noteworthy that several LBL experiments operate in the A range which coincides with the stability region. This situation should be helpful in deciphering the implications of the experimental results.
It is tempting to follow the same methodology in order to convert the known [W ν ] elements into values of (x i , y j ). However, at the present level of accuracy, a consistent solution is not available.
While the (x i , y j ) parametrization is applicable in general, for quark mixing, we may also use another parametrization that incorporates the feature of |V CKM |, similar to Eq. (18)