A new Wolfenstein-like expansion of lepton flavor mixing towards understanding its fine structure

Taking the tri-bimaximal flavor mixing pattern as a particular basis, we propose a new way to expand the $3\times 3$ unitary Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton flavor mixing matrix $U$ in powers of the magnitude of its smallest element $\xi \equiv \left|U^{}_{e 3}\right| \simeq 0.149$. Such a Wolfenstein-like parametrization of $U$ allows us to easily describe the salient features and fine structures of flavor mixing and CP violation, both in vacuum and in matter.


Motivation
Among the three Euler-like rotation angles of the Cabibbo-Kobayashi-Maskawa (CKM) quark flavor mixing matrix V [1,2], it is the largest one -the Cabibbo angle θ C ≃ 13 • that was most accurately measured from the very beginning [3].That is why Wolfenstein proposed a remarkable parametrization of the CKM matrix V in 1983 by expanding its nine matrix elements in powers of a small parameter λ ≡ sin θ C ≃ 0.225 [4], from which the hierarchical structure of quark flavor mixing can be well understood.For example, one may easily arrive at the four-layered ordering of respective O(1), O(λ), O(λ2 ) and O(λ 3 ) as a natural consequence of the unitarity of V [5].
In particular, the CKM matrix V approaches the unique identity matrix I in the λ → 0 limit, implying that the up-and down-type quark sectors should have an underlying parallelism between their flavor textures.Such a conceptually interesting limit is quite suggestive, and it has widely been considered for explicit model building [6].
In comparison, two of the three Euler-like rotation angles of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton flavor mixing matrix U (i.e., θ 12 ≃ 33.4 • and θ 23 ∼ 45 • [3]) are so large that a naive Wolfenstein-like parametrization of U seems quite unlikely 1 .A very real possibility is that the leading-order term of the PMNS matrix U is a constant flavor mixing pattern U 0 consisting of two large angles and originating from a kind of discrete flavor symmetry, while the smallest flavor mixing angle θ 13 ≃ 8.6 • and CP-violating effects arise after small perturbations or quantum corrections to U 0 are taken into account [10,11].From the point of view of model building [6,12,13,14], the most popular choice of U 0 has been the tri-bimaximal flavor mixing pattern U TBM [15,16,17] which predicts θ 13 = 0 • and θ (0) 23 = 45 • .Given the smallness of θ 13 , several attempts have been made along the above line of thought to expand the U TBM -based PMNS matrix U in powers of the small Wolfenstein parameter λ that is borrowed from quark flavor mixing (see, e.g., Refs.[18,19,20,21,22]).
Guided by the best-fit values and 1σ intervals of three lepton flavor mixing angles extracted from a global analysis of the currently available experimental data on neutrino oscillations [23,24], NMO : θ 12 = 0.583 +0.013 −0.013 , θ 13 = 0.150 +0.002 −0.002 , θ 23 = 0.736 +0.019 for the normal mass ordering (NMO) of three active neutrinos, or IMO : θ 12 = 0.583 +0.013 −0.013 , θ 13 = 0.150 +0.002 −0.002 , θ 23 = 0.855 +0.018 in the inverted mass ordering (IMO) case, we find that the smallest angles θ 13 is most accurately determined and thus suitable for serving as an optimal expansion parameter 2 .So we are going to study an expansion of the U TBM -based PMNS matrix U in powers of the small parameter ξ ≡ |U e3 | = sin θ 13 ≃ 0.149 ≃ 0.662λ.Moreover, the best-fit values of θ 12 and θ 23 lead us to NMO : (4) as well as (5) Eqs. ( 4) and ( 5 It is obvious that B ̸ = 0 characterizes the effect of µ-τ permutation symmetry breaking of U , and the sign of B determines the octant of θ 23 [27,28].In this case the PMNS matrix U will be expressed in terms of the tri-bimaximal flavor mixing pattern U TBM , the three real parameters ξ, A and B, and the poorly known CP-violating phase δ.Here we leave aside the two possible extra CP phases associated with the Majorana nature of massive neutrinos, as they are completely unknown and have little effect on the fine structure of U .

The expansion of U
As advocated by the Particle Data Group (PDG), the standard Euler-like parametrization of the Substituting Eq. ( 7) into Eq.( 6), we immediately arrive at the nine elements of U in the PDGadvocated phase convention as follows: up to O(ξ 6 ) or equivalently O(10 −5 ).This degree of precision and accuracy for the elements of U should be good enough to confront the present and future precision measurements of various neutrino oscillation channels.Some comments and discussions are in order.
• The two off-diagonal asymmetries of the unitary PMNS matrix U , which largely characterize its geometrical structure about the U e1 -U µ2 -U τ 3 and U e3 -U µ2 -U τ 1 axes, are given by In comparison, the corresponding off-diagonal asymmetries of the CKM quark flavor mixing matrix V are respectively of O(λ 6 ) and O(λ 2 ) [5].So the PMNS matrix U is geometrically not so symmetrical as the CKM matrix V .Given the fact that either A 1 = 0 or A 2 = 0 would imply the congruence of three pairs of the PMNS unitarity triangles in the complex plane [29], we find that the relatively large off-diagonal asymmetries of U means that its six unitarity triangles are not very similar to one another in shape.
• The well-known Jarlskog invariant of leptonic CP violation [32,33], which measures the universal strength of CP-violating effects in neutrino oscillations, is given as So J ν ≃ 3.5 × 10 −2 sin δ holds in the leading-order approximation.Given δ ∼ −π/2, for instance, the size of the leptonic Jarlskog invariant will be about a thousand times larger than that of its counterpart in the quark sector [3].
Note that one only needs to keep the leading-order terms of A 1 , A 2 , ∆ 1 , ∆ 2 , ∆ 3 and J ν in most cases, as they are analytically simple enough and numerically accurate enough.
3 The ordering of |U αi | Taking account of the U TBM -based expansion of the PMNS matrix U in powers of ξ ≃ 0.149 in Eq. ( 8), we find that the analytical approximation • For the matrix elements in the third column of U , the sign of B is crucial as it determines whether |U µ3 | is larger or smaller than its counterpart |U τ 3 |.This point is obviously (10).We are therefore left with • For the matrix elements in the second and third rows of U , the smallness of ξ assures that This observation is independent of the values of A, B and δ in the Wolfenstein-like expansion of U proposed above.
• To compare between the magnitudes of U τ i and U µi (for i = 1, 2), we may simplify the expressions of ∆ 1 and ∆ 2 in Eq. ( 10) as follows: holds to the accuracy of O(ξ 2 ).In case of cos δ = 0, however, the sign of B will play a part.
A summary of the above discussions leads us to the following most likely ordering of the nine matrix elements of U in magnitude: where the ordering of the bracketed moduli remains unidentifiable from the present neutrino oscillation data.That is why the next-generation long-baseline neutrino oscillation experiments aim to pin down the octant of θ 23 (or equivalently, the sign of B) and the quadrant of δ.
To be more specific, let us simply take the best-fit value of δ to illustrate the ordering of the nine PMNS moduli |U αi | (for α = e, µ, τ and i = 1, 2, 3).Given [23,24] NMO : where the values of A and B are directly extracted from Eq. ( 4), we obtain 14) and |U τ 3 | > |U µ3 | from Eq. ( 10).As a result, NMO : In the IMO case, we input [23,24] IMO : where the values of A and B are directly extracted from Eq. ( 5), and then find |U We see that the nine PMNS matrix elements do not have a clearly layered hierarchy in magnitude, as compared with the four-layered ordering of the nine CKM moduli shown in Eq. ( 1).The reason behind this difference should be closely related to the underlying mechanism responsible for the origin of tiny neutrino masses, although it remains vague and unclear at present.

The unitarity triangle
The "appearance" neutrino oscillation ν µ → ν e and its CP-conjugated process ν µ → ν e are the only realistic channels to measure leptonic CP violation in a long-baseline oscillation experiment like T2K [31].It is the so-called unitarity triangle △ τ [35] defined by the orthogonality relation the complex plane that is directly related to ν µ → ν e and ν µ → ν e oscillations.In view of the PDG-advocated phase convention of U taken in Eq. ( 6) or (8), we find that it is more convenient to use the side U e3 U * µ3 to rescale the three sides of △ τ [36].In this case, we simply arrive at Figure 1: An illustration of the geometrical shapes of the rescaled unitarity triangle △ ′ τ (solid black) and its effective matter-corrected counterpart △ ′ τ (dashed red for the NMO case and dashed blue for the IMO case) in the complex plane, where δ ≃ −π/2 has typically been input.
where the two sloping sides are given by to a good degree of accuracy.Just taking δ ≃ −π/2 and ξ ≃ 0.149 for example, we obtain the Namely, two of the three sides of △ ′ τ are about three times longer than the shortest one in magnitude, as illustrated by the solid black triangle in Fig. 1.Note that the height of △ ′ τ , denoted as J ′ ν , is correlated with the Jarlskog invariant J ν as follows: where the expression of J ν obtained in Eq. ( 11) has been used.We are then left with the result √ 2 sin δ/ (3ξ) in the leading-order approximation, as clearly indicated by the imaginary parts of the two sloping sides of △ ′ τ in Eq. ( 21).It is well known that the terrestrial matter effects on ν µ → ν e and ν µ → ν e oscillations are not very significant in the T2K and Hyper-Kamiokande experiments with the baseline length L ≃ 295 km and the typical beam energy E ≃ 0.6 GeV [3], but such effects can modify the shape of the above unitarity triangle and thus modify the Jarlskog invariant of CP violation to some extent.To see this point more clearly, let us consider the effective rescaled unitarity triangle where U ei and U µi (for i = 1, 2, 3) denote the effective PMNS matrix elements in matter.Following the analytical approximations made in Ref. [37], we obtain to a good degree of accuracy, where with ∆m 2 ij (for i, j = 1, 2, 3) being the neutrino mass-squared differences, G F being the Fermi constant, N e being the terrestrial background density of electrons, and E being the neutrino beam energy [38,39].Substituting Eq. ( 21) into Eq.( 24), we arrive at This result shows that the terrestrial matter effects can obviously modify the rescaled unitarity triangle △ ′ τ in vacuum.Taking account of J ν /J ν ≃ |α|/ϵ obtained in Ref. [37], where J ν denotes the effective Jarlskog invariant in matter, we may similarly achieve the height of △ ′ τ : whose leading term is certainly consistent with the imaginary parts of the two sloping sides of △ ′ τ that can directly be seen from Eq. (26).
To illustrate, one may typically take A ≃ 2.28 × 10 −4 eV 2 (E/GeV) for a neutrino trajectory through the Earth's crust [40], which is suitable for the realistic ongoing and upcoming long-  2) and (3).Fig. 1 illustrates how the geometrical shape of △ ′ τ changes as a consequence of the terrestrial matter effects on neutrino oscillations in the NMO and IMO cases.It becomes clear that the area of △ ′ τ is remarkably smaller than that of △ ′ τ , and their ratio is simply governed by J ν /J ν ≃ |α|/ϵ as discussed above.It is worth pointing out that θ 13 , θ 23 and δ are essentially insensitive to terrestrial matter effects in the E ≲ 1 GeV region [41,42,43], and sin 2 θ 12 / sin 2θ 12 ≃ J ν /J ν ≃ |α|/ϵ holds as a good approximation [37].These observations imply that ξ, B and δ in our Wolfenstein-like expansion of the PMNS matrix U are also expected to be insensitive to terrestrial matter effects in an accelerator-based neutrino oscillation experiment with E ≲ 1 GeV, and only A is an exception.To be specific, we take θ 12 ≃ θ (0) 12 − Aξ 2 and then obtain The second term on the right-hand side of Eq. ( 28) is in general unsuppressed as it arises from θ (0) 12 ≃ 35.26 • .Although A itself may be far above O(1), it does not really point to a significant matter effect simply because it is not directly related to any observable of neutrino oscillations.
Note that the terrestrial matter effects are expected to be far more significant in the upcoming DUNE neutrino oscillation experiment with the baseline length L ≃ 1300 km and the beam energy E ∈ [2, 3] GeV [44].In this case one may instead use Freund's analytical approximations for the neutrino oscillation parameters in matter [45] to discuss the matter-modified PMNS lepton flavor mixing matrix and the corresponding unitarity triangles.

Further discussions
Motivated by the fact that the fine structure of quark flavor mixing can well be understood in the Wolfenstein expansion of the CKM matrix V , we have proposed a similar expansion of the PMNS matrix U in the basis of the tri-bimaximal mixing pattern U TMB towards understanding the fine structure of lepton flavor mixing.The corresponding expansion parameters are λ ≃ 0.225 for V and ξ ≃ 0.149 for U , and thus we can directly arrive at in the limits of vanishing or vanishingly small λ and ξ.While I is unique in the quark sector, U TBM is just our choice in the lepton sector.However, we have argued that it is rather reasonable to choose U TBM as an expansion basis of U since this constant pattern is particularly favored from the point of view of model building with the help of an underlying discrete flavor symmetry.
It makes sense to ask what new information can be achieved from the proposed Wolfensteinlike expansion of the U TBM -based PMNS matrix U as compared with the exact PDG-advocated parametrization of U in terms of the Euler-like angles and phases.We find that our Wolfensteinlike parametrization does have some phenomenological merits.11) and (12).In comparison, it is difficult to obtain such instructive relations from the exact Euler-like parametrization in Eq. ( 6).
• It can largely simplify all the leading-order calculations and some of the next-to-leadingorder calculations of the observable effects regarding lepton flavor mixing and CP violation.
A typical example of this kind -the leptonic unitarity triangle △ τ has been presented in section 4, from which one can see that even the analysis of terrestrial matter effects is simplified to some extent in our Wolfenstein-like parametrization of U .The reason for this simplification is of course that U has been expanded in the basis of the constant flavor mixing pattern U TBM and in powers of the well-measured small parameter ξ ≡ |U e3 |.
So our approach proves to be useful, and more of its applications in neutrino phenomenology (e.g., on the flavor distribution of ultrahigh-energy cosmic neutrinos) will be explored elsewhere.
Finally, it is worth pointing out that the small expansion parameters like λ and ξ usually play a role of characterizing possible flavor symmetry breaking effects or quantum corrections in building an explicit top-down model of fermion mass generation and flavor mixing at a proper energy scale.In this case the CKM matrix V and the PMNS matrix U are very likely to satisfy the intriguing limits like Eq. ( 29).Although U TBM itself is not a unique choice along this line of thought, the Wolfenstein-like expansions of V and U should make sense in any case.

√ 2 ≃) 23 −
) imply that the observed values of θ 12 and θ 23 are most likely to deviate from their respective tri-bimaximal flavor mixing limits at the level of O(ξ 2 ).This observation provides us with a new angle of view, which is quite different from those in the previous attempts, to expand U in the basis of U TBM .In what follows we shall propose a new expansion of the PMNS matrix U by starting from the standard Euler-like parametrization of U and taking account of • sin θ 13 ≡ ξ ≃ 0.149 as the lepton flavor mixing expansion parameter; • θ 12 = θ 35.26 • and A ∼ O(1); • θ 23 = θ (0Bξ 2 with θ (0) 23 = 45 • and |B| ∼ O(1).

•
is actually good enough to fit current neutrino oscillation data.Let us examine to what extent one may identify the ordering of |U αi | (for α = e, µ, τ and i = 1, 2, 3) 3 .For the matrix elements in the first row of U , it is easy to identify |U e1 | > |U e2 | > |U e3 |.In fact, |U e1 | and |U e3 | are the largest and smallest moduli among the nine elements of U .

•
It can easily reflect the relative magnitudes of the nine matrix elements of U to a reasonable degree of accuracy.For example, the ratios |U e1 | : |U µ1 | : |U τ 1 | ≃ 2 : 1 : 1 and |U e3 | : |U µ3 | : |U τ 3 | ≃ √ 2 ξ : 1 : 1 are the straightforward consequences of Eq. (12) in no need of doing any algebraic calculations, and they are essentially consistent with the numerical results obtained from a global fit of current neutrino oscillation data [23, 24].Such an advantage is analogous to the fact that |V ub | : |V cb | : |V us | : |V ud | ≃ λ 3 : λ 2 : λ : 1 can be directly read off from the Wolfenstein parametrization of the CKM matrix V .• It can help establish some simple and testable relations between different PMNS matrix elements which are associated with different neutrino oscillation experiments.For instance, the relation |U µ2 | 2 + |U τ 2 | 2 ≃ 2|U e2 | 2 holds up to a small correction of O(ξ 2 ), as can be seen from Eq. (14); and J ν ≃ |U e1 ||U e2 ||U e3 | sin δ holds to the same degree of accuracy, as can be easily derived from Eqs. (