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More Straightforward Extraction of the Fundamental Lepton Mixing Parameters from Long-Baseline Neutrino Oscillations

We point out the simple reversibility between the fundamental neutrino mixing parameters in vacuum and their effective counterparts in matter. The former can therefore be expressed in terms of the latter, allowing more straightforward extraction of the genuine lepton mixing quantities from a variety of long-baseline neutrino oscillation experiments. In addition to the parametrization-independent results, we present the formulas based on the standard parametrization of the lepton flavor mixing matrix and give a typical numerical illustration.


I. INTRODUCTION
The observation of solar and atmospheric neutrino oscillations in the Super-Kamiokande experiment has provided the most convincing evidence that neutrinos are massive and lepton flavors are mixed [1] 1 . It opens an important window for deep insight into the dynamics of fermion mass generation and flavor mixing in particle physics, and has important implications in astrophysics and cosmology.
Although some useful constraints on the lepton flavor mixing matrix can be obtained from the Super-Kamiokande measurements and other non-accelerator neutrino experiments, a precise determination of its parameters has to rely on the new generation of accelerator neutrino experiments with very long baselines [3], including possible neutrino factories [4]. The terrestrial matter effects in all long-baseline neutrino experiments must be taken into acount, since they unavoidably modify the genuine behaviors of neutrino oscillations in vacuum.
To express the pattern of neutrino oscillations in matter in the same form as that in vacuum, one may define the effective neutrino massesm i and the effective lepton flavor mixing matrixṼ in which the terrestrial matter effects are already included. In this commonlyaccepted approach, it is necessary to find out the concrete relations between the fundamental quantities of lepton mixing in vacuum (m i and V ) and their effective counterparts in matter (m i andṼ ). The analytically exact formulas ofm i andṼ as functions of m i and V have been achieved in the three-neutrino mixing scheme [5][6][7], but no effort has been made to derive the expressions of m i and V in terms ofm i andṼ . The latter case is equivalently interesting in phenomenology, because our physical purpose is to determine the fundamental parameters of lepton flavor mixing from the effective ones, which can directly be observed from a variety of long-baseline neutrino oscillation experiments.
This paper aims to present the analytically exact results of the fundamental quantities m i and V in terms of the effective quantitiesm i andṼ . In Section 2, we first point out the interesting reversibility between (m i , V ) and (m i ,Ṽ ), and then derive their explicit relations in a way independent of any specific parametrizations of the lepton flavor mixing matrix. In Section 3, the standard parametrization is introduced to describe V andṼ in a parallel form. We calculate the mixing angles and the CP-violating phase(s) of V as functions of the relevant parameters ofṼ , and give a typical numerical illustration. Section 4 is devoted to a brief summary.

II. REVERSIBILITY AND PARAMETRIZATION-INDEPENDENT FORMULAS
Let us concentrate on the flavor mixing between three active neutrinos and three charged leptons. The 3 × 3 lepton flavor mixing matrix V in vacuum is defined to link the neutrino mass eigenstates (ν 1 , ν 2 , ν 3 ) to the neutrino flavor eigenstates (ν e , ν µ , ν τ ): A similar definition can be made forṼ , the effective counterpart of V in matter. If neutrinos are massive Dirac fermions, V orṼ can be parametrized in terms of three rotation angles and one CP-violating phase. If neutrinos are Majorana fermions, however, two additional CP-violating phases are in general needed to fully parametrize V orṼ . The strength of CP or T violation in neutrino oscillations, no matter whether neutrinos are Dirac or Majorana particles, depends only upon a universal parameter J (in vacuum) orJ (in matter) [8]: where the Greek subscripts (α, β, γ) and the Latin subscripts (i, j, k) run over (e, µ, τ ) and (1, 2, 3), respectively. Of course, J andJ are two rephasing-invariant quantities. The effective Hamiltonian responsible for the propagation of neutrinos in vacuum or in matter can be written as where E is the neutrino beam energy; m i andm i (for i = 1, 2, 3) denote the neutrino masses in vacuum and those in matter, respectively. The deviation ofH ν from H ν results non-trivially from the charged-current contribution to the ν e e − forward scattering [9], when neutrinos travel through a normal material medium like the earth: where a = √ 2G F N e with N e being the background density of electrons. Subsequently we assume a constant earth density profile (i.e., N e = constant), which is a very good approximation for all of the presently-proposed long-baseline neutrino experiments 2 .
With the help of Eqs. (3) and (4), we obtain the following relations: where A = 2aE depending linearly on the neutrino beam energy E. This result implies that the expression ofm i as a function of A, m j and V ej is reversible with that of m i , which must behave via the same function of variables A,m j andṼ ej ; namely, As m i is real and positive, it depends only upon the absolute values ofṼ αj . The specific texture of ∆H ν assures that only the matrix elements |Ṽ ej |, which are always associated with the matter parameter A, can affect m i . The similar argument is valid form i . Analogously, there exists the simple reversibility between the expressions of V αi andṼ αi : We observe that the reversible relations in Eqs. (6) and (7) coincide formally with the sum rules of neutrino mass and mixing parameters obtained in Ref. [11]: where α = β running over (e, µ, τ ). In the limit A = 0,m i = m i andṼ αi = V αi must hold. Hence the functions f i and F αi might have transparent dependence on A. To find out the explicit forms of f i and F αi is simply an algebraic exercise in the three-neutrino mixing scheme [5][6][7]. Using Eqs. (6) and (7), one may obtain the formulas of m i and V αi straightforwardly from those ofm i andṼ αi , which have been presented in Ref. [7] in a parametrization-independent way. We then arrive at wherex,ỹ andz are given by with ∆m 2 21 ≡m 2 2 −m 2 1 and ∆m 2 31 ≡m 2 3 −m 2 1 . Furthermore, V reads as follows: where α runs over (e, µ, τ ); i, j or k runs over (1,2,3) with i = j = k; and 3 Obviously A = 0 leads to V =Ṽ . The exact and compact formulas obtained above show clearly how the flavor mixing matrix in vacuum is connected to that in matter. Some instructive analytical approximations can be made for Eq. (11), once the hierarchy of effective neutrino masses and that of effective flavor mixing matrix elements are experimentally known.
With the help of Eq. (11), one may calculate the universal CP-violating parameter J in vacuum. Indeed it is easier to derive the relationship between J andJ by use of either the sum rules in Eq. (8) [11] or the equality between the determinants of lepton mass matrices in matter and in vacuum [12]. The result is where the neutrino mass-squared differences in vacuum can be read off from Eq. (9). Note that the afore-obtained results are valid only for neutrinos propagating in vacuum and interacting with matter. As for antineutrinos, the respective formulas for m 2 i , V and J can straightforwardly be obtained from Eqs. (9) -(13) through the replacementsṼ =⇒Ṽ * and A =⇒ −A.

III. STANDARD PARAMETRIZATION AND NUMERICAL ILLUSTRATION
It is well known that the 3×3 lepton flavor mixing matrix can be parametrized in terms of three rotation angles (θ 1 , θ 2 , θ 3 ) and three phase angles (δ, ρ, σ), if neutrinos are Majorana particles. There are nine distinct parametrizations of this nature [13], but only one of them is particularly convenient in the analyses of experimental data on the neutrinoless double beta decay and neutrino oscillations. This "standard" parametrization reads in vacuum as with s i ≡ sin θ i and c i ≡ cos θ i (for i = 1, 2, 3). Without loss of generality, the three mixing angles (θ 1 , θ 2 , θ 3 ) can all be arranged to lie in the first quadrant. Arbitrary values between −π and +π are allowed for the CP-violating phases δ, ρ and σ. Note that the location of the Dirac-type phase δ in V is different from that advocated by the Particle Data Group in Ref. [14]. The advantage of our present phase assignment is that δ itself does not appear in the effective Majorana mass term of the neutrinoless double beta decay [13]. Note also that normal neutrino oscillations are completely insensitive to the Majorana-type phases ρ and σ, therefore the magnitudes of these two parameters keep unchanged even when neutrinos propagate in matter [15]. On the other hand, the other four parameters θ 1 , θ 2 , θ 3 and δ may be contaminated by the terrestrial matter effects in realistic long-baseline neutrino oscillation experiments. In analogy to Eq. (14), the effective (matter-corrected) lepton flavor mixing matrix can be parametrized as follows: The CP-and T-violating asymmetries of neutrino oscillations are proportional to J in vacuum andJ in matter. Now let us proceed to figure out the analytical expressions of the fundamental parameters (θ 1 , θ 2 , θ 3 , δ) in terms of the effective parameters (θ 1 ,θ 2 ,θ 3 ,δ), which can directly be measured from a variety of long-baseline neutrino oscillation experiments. To do so, we simply apply the standard parametrization ofṼ to the parametrization-independent formula in Eq. (11). Then the matrix elements of V read explicitly as whereÑ i andD i have been given in Eq. (12), P ≡ Diag{1, e iρ , e iσ } denotes the diagonal Majorana phase matrix, and the matter-associated quantitiesT αk are given as It is worthwhile at this point to remark two advantages of the results presented above over those obtained by Zaglauer and Schwarzer in Ref. [6]. First, our relations clearly show the proportionality between the sine or tangent functions of (θ 1 , θ 2 , θ 3 , δ) and (θ 1 ,θ 2 ,θ 3 ,δ). Second, our relations make the dependence ofθ 1 ,θ 2 ,θ 3 andδ on the matter parameter A more transparent. Therefore our results are expected to be more useful for the analytical study of terrestrial matter effects on lepton flavor mixing and CP violation.
For numerical illustration, we assume the matter density of the earth's crust to be constant. Then A ≈ 2.28 · 10 −4 eV 2 E/[GeV] is a good approximation [16], where E is the neutrino beam energy. We typically take E = 5 GeV. The input values of the effective neutrino mixing parameters in matter are listed on the left side of Table 1. With the help of Eqs. (9) and (10) as well as Eqs. (19) and (20), one may calculate the fundamental neutrino mixing parameters in vacuum. We list our numerical results on the right side of Table 1 for both the case of neutrinos (+A) and that of antineutrinos (−A). One can see that the input values of (∆m 2 21 , ∆m 2 31 ) and (θ 1 ,θ 2 ,θ 3 ,δ) are rather ad hoc. The reason is simply that we contrive to get the appropriate sizes of (∆m 2 21 , ∆m 2 31 ) and (θ 1 , θ 2 , θ 3 , δ), which are compatible with the present Super-Kamiokande data on solar and atmospheric neutrino oscillations [1]. Indeed we observe from Table 1 that ∆m 2 sun ≈ ∆m 2 21 ≈ 4.62 · 10 −5 eV 2 (or 5.05 · 10 −5 eV 2 ) and sin 2 2θ sun ≈ sin 2 2θ 1 ≈ 0.75 (or 0.90) do agree with the large-angle MSW solution to the solar neutrino problem, and ∆m 2 atm ≈ ∆m 2 31 ≈ 3.00 · 10 −3 eV 2 (or 2.89 · 10 −3 eV 2 ) together with sin 2 2θ atm ≈ sin 2 2θ 2 ≈ 0.99 (or 0.97) is consistent with the atmospheric neutrino oscillation data. Moreover, sin 2 2θ CHOOZ ≈ sin 2 2θ 3 ≈ 0.03 < 0.1 coincides with the CHOOZ and Palo Verde reactor experiments [17]. Note that ∆m 2 21 ∼ A ≫ ∆m 2 21 holds for the given value of E, hence there appears a substantial difference between θ 1 andθ 1 . On the other hand, θ 2 ≈θ 2 and δ ≈δ hold to an excellent degree of accuracy. The tiny difference between θ 2 andθ 2 in our numerical illustration can easily be understood: the special input δ = ±90 • assures the vanishing of the term proportional to A cosδ on the right-hand side of the expression for tan θ 2 / tanθ 2 in Eq. (19), leading straightforwardly to θ 2 =θ 2 in the next-to-leading order approximation. In general, θ 2 ≈θ 2 remains a good approximation, if the condition |∆m 2 21 | ≪ |∆m 2 31 | is satisfied [18]. One may numerically check that δ ≈δ is generally valid as well, provided that |∆m 2 21 | ≪ |∆m 2 31 | holds. Analytically, however, we have not found a transparent way to show the approximate equality between δ andδ. Some more effort is obviously desirable, in order to understand why the CP-violating phase δ is in most cases insensitive to the terrestrial matter effects [19].
Our numerical results illustrate that the fundamental neutrino mixing parameters can straightforwardly be extracted from their matter-corrected counterparts, once the latter are measured from the future long-baseline neutrino oscillation experiments. In particular, |∆m 2 21 | ≪ |∆m 2 31 | assures thatθ 2 ≈ θ 2 andδ ≈ δ hold in the standard parametrization of lepton flavor mixing.

IV. SUMMARY
We have pointed out that there exists the reversibility between the fundamental neutrino mixing parameters in vacuum and their effective counterparts in matter. The former can therefore be expressed in terms of the latter, allowing more straightforward extraction of the genuine lepton mixing quantities from a variety of long-baseline neutrino oscillation experiments. Besides the parametrization-independent formalism, we have presented the formulas based on the standard parametrization of the neutrino mixing matrix and given a typical numerical illustration. Our results are expected to be a useful addition to the phenomenology of lepton flavor mixing and neutrino oscillations. TABLES TABLE I. Numerical illustration of extracting the fundamental neutrino mixing parameters in vacuum from their effective counterparts in matter, where A = 2.28 · 10 −4 eV 2 E/[GeV] with E = 5 GeV has typically been used.