Rephasing invariance and permutation symmetry in flavor physics

With some modifications, the arguments for rephasing invariance can be used to establish permutation symmetry for the standard model. The laws of evolution of physical variables, which transform as tensors under permutation, are found to obey the symmetry, explicitly. We also propose to use a set of four mixing parameters, with unique properties, which may serve to characterize flavor mixing.


II. NOTATION AND MATHEMATICAL PRELIMINARIES
In order to facilitate the study of rephasing invariance and permutation, it is important to choose variables which have simple properties under their operations. Let us start with V αi (To avoid repetition, we will use V αi to denote elements of the PMNS matrix. The case for CKM is the same.) We will impose, without loss of generality, the condition [2] detV = +1, (1) so that and rephasing is given by with φ α = φ i = 0. From Eq.
(2), we may eliminate V * βj from any expression in favor of only products of the V αi 's. One can then construct basic rephasing invariant combinations (RIC) in the form (no summation over capital indices) where E αβγ is the symmetric Levi-Civita symbol [1], which is symmetric under exchange, and E αβγ = 1, α = β = γ; = 0, otherwise Thus, Γ contains the index of each row (and each column) once, and only once. It is also seen that the difference of two Γ's is equal to some |V αi | 2 = W αi , so that all the Γ's must have the same imaginary part, which can be identified with J, the Jarlskog invariant [3]. We write Γ eµτ IJK = R IJK − iJ, with the definition (R 123 , R 231 , R 312 ; R 132 , R 213 , R 321 ) = (x 1 , x 2 , x 3 ; y 1 , y 2 , y 3 ), which satisfy the consistency conditions We now turn to the properties of V αi under permutation. Under an exchange, (ij), we have V αi ↔ (phase)V αj , where an arbitrary phase is associated with the exchange operator. In order to maintain detV = +1, we choose where further possible rephasing is contained in (phase) ′ . However, if we include V αi in an RIC, these phases cancel out and we have (V αi , V αk ) ←→ −(V αj , V αk )(in rephasing invariant combinations).
In analogy to V * αi (Eq. (2)), we can define which transforms as 3 × 3. Now we have These are identified with the variables (W AI , w AI ) introduced earlier [2], and we can verify by using properties of the (x i , y j ) variables. Also, The relation, shows that D ∼ 1 × 1 and J ∼ 1 × 1. While the transformation property of D is expected, that J behaves like a pseudo-P-scalar is very interesting. We will return to this point in our analysis later. It is useful to introduce, explicitly, a set of 3 × 3 matrices which are representations of S 3 (with elements (e, (1,2,3), (1,3,2); (1, 2), (2, 3), (3, 1)): is for 3. Thus, if we write V αi as a matrix, [V ], a permutation of the index i is given by while that on the α index is given by left multiplication The matrices for W αi and w αi can be written as Permutations on the index i are given by while for index α one would have [X] and [ X] multiplying from the left. It follows that under either (ij) or (αβ), x l ↔ −y m for appropriate l and m. Also, x i ↔ − y i , from (ij) or (αβ). Thus, D = (x i + y i ) is a pseudo-P-scalar, as before. We also find Q 2 = x i x j + y i y j → Q 2 , and J 2 = Πx i − Πy i → J 2 , so that both are P-scalars.
Another familiar RIC [3,5] is, for (A = B = Γ, I = J = K), Under permutation, it transforms ∼ 3 × 3. It turns out that we can express Λ AI in terms of W , A similar construction using w is Some other tensors will also appear in RGE calculations. There is a tensor (∼ 3× 3+ 1× 1) which was discussed earlier [5]: Another useful tensor (∼ 3 × 3 + 1 × 1) was also used [5], Finally, one also needs the tensor (3 × 3), in addition to the identity These relations and other identities will be further discussed in Appendix A.

III. REPHASING INVARIANCE AND PERMUTATION SYMMETRY
We now turn to a systematic study of rephasing and permutation. We concentrate on the SM in a minimally extended version, with massive Dirac neutrinos. For our purposes only the EW interactions need to be considered, so that we will only study the lepton sector explicitly, bringing in the parallel quark sector when appropriate. In this case, of the many parts of the SM Lagrangian, L SM , we can focus on the leptonic EW interactions in the mass eigenstate basis. Schematically, we write Here, to simplify the notation, we omit the gauge fields (W µ , in J µ W † µ + h.c.) and proper Dirac matrices. Also, α = (e, µ, τ ), ψ i refers to ν i , and m α (m i ) are their masses. h denotes the Higgs field in the physical gauge, v is the VEV, and V αi is an element of the PMNS matrix. In an obvious matrix notation, Eq. (34) is the result of diagonalizing the Higgs-Fermion coupling by U(3) transformations on Ψ α and ψ i . However, this procedure is not unique. A familiar example is the rephasing transformation: where (P, P ′ ) are diagonal phase matrices which, to maintain detV = +1, satisfy detP = detP ′ = +1. The Lagrangian is invariant in form provided that V also changes according to Similarly, there is another transformation on [Ψ] and [ψ] which leaves L SM invariant in form: It is noteworthy that rephasing invariance and permutation symmetry are so closely related. While rephasing can be balanced out by a corresponding operation on V , a permutation of the wave functions can be countered with similar actions on V and on the masses. Among the possible U(3) operation on the wave functions, rephasing and permutation are unique since any unitary transformation on a matrix cannot change its eigenvalues, except possibly their ordering.
As noted before, the basic RIC, given in Eq. (4), contains each family index once, and only once. A permutation of indices on any RIC yields another RIC. Since physical variables are composed of products of RIC's, starting from any variable, repeated permutations generates a tensor. That is, physical variables are naturally grouped into tensors under S 3 × S 3 . Coupled with the invariance of L SM , this means that the evolution of these tensors is regulated by the permutation symmetry. As we will demonstrate explicitly in the next section, there are many known examples which exhibit the symmetry. Considerations on permutation offers interesting insights into these equations, they can also serve as useful tools to check the validity of future calculations in flavor physics.

IV. EXAMPLES
In this section, we investigate the implications of the permutation symmetry as applied to some known results in flavor physics. At first sight, a permutation, such as (ψ i , V αi , m i ) → (ψ j , V αj , m j ), seems rather innocuous and inconsequential. Indeed, if one considers, e.g., the decays b → u + ℓν and b → c + ℓν, at the tree level, it is obvious that the two procedures are related by the permutation (u, V ub ) → (c, V cb ). However, a more interesting situation arises when a calculation involves internal fermions. In this case, the vestige of their participation is contained in a function f (V αi , m i ), corresponding to using ψ i , in a certain order, as the basis of calculation. Had one chosen the basis [ X]ψ i , one would have obtained the function f with permuted indices. With permutation symmetry, this implies that f must be a function of permutation invariants, such as f ( m i W αi ). In the following we will revisit some examples that were studied before [1], with further comments. Additional examples will also be given.

A. Neutrino oscillation in vacuum
The probability function for neutrino oscillation is well-known. In tensor notation, it can be written [1] as (for α = β) ). This result is obtained by considering the propagation (internally) of the mass eigenstate, |ν i , from t = 0 to t = L. Had we used an equivalent but permuted basis, X|ν i , we would have obtained the same probability function, only with the indices permuted. The S 3 (ν) symmetry then dictates that the function must be a P-scalar, which is indeed the case. Also as we have emphasized before, the permutation property of J, that J → −J under any exchange α ↔ β (and i ↔ j), implies that the second term in P (ν α → ν β ) is T (and CP) violating, from symmetry arguments without the need to do any calculation.

B. Neutrino oscillation in matter
When a neutrino passes through a medium, an effective mass for ν e is generated. This produces changes in the neutrino parameters. For an infinitesimal changes, dA, A = 2 √ 2G F N e E, with (δH D ) ξ = (dA, 0, 0) = (δH D ee , 0, 0), the induced changes in neutrino parameters are given by ( [1,6], see also [7], for an analysis in the PDG variables) . These equations are invariant in form under S 3 (ℓ) × S 3 (ν). To arrive at these results, one chooses a certain basis |ν i . However, one could have chosen to use X|ν i as basis, leading to the same physics. This freedom of choice is reflected in the symmetric tensor forms of these equations, It is also interesting to note that there are two "matter invariants" obtained a long time ago [8][9][10][11][12][13]. They can be written in the tensor notation (see Eqs. (32-33) in Ref. [6]) It is seen that both expressions are invariant under S 3 (ν). Moreover, note the pairing of J(∼ 1) with E ijk D i D j D k , which is another pseudo-P-scalar, and that of J 2 (∼ 1) with E ijk W ei W ej W ek .

C. One-loop RGE for quarks
The evolution of the physical parameters of fermions has been studied for a long time. We will discuss here only the RGE for quarks, since with Dirac neutrinos, the leptons behave just like the quarks. Traditionally, these equations were given using the PDG parameters. The results are rather complicated (e.g., Ref. [14]). In terms of the tensor notation, the equations for the mixing parameters can be written in the form [1] Here, ∆ m 2 , etc. It is clear that these equations obey the permutation symmetry S 3 (u) × S 3 (d). In this connection we recall the well-known result [15] about the evolution of J, which can be written in the form [16] Note how J(∼ 1 × 1) combine with Π(∆m 2 αβ )Π(∆m 2 ij )(∼ 1 × 1) to form a scalar under S 3 (u) × S 3 (d), while the other mass combinations in Eq. (51) are all scalars.

D. A two-loop RGE
Although most RGE results come from one-loop calculations, there is one two-loop calculation [17] in the literature. The results, written in a way suitable for analyzing their properties under permutation, were given in [16]. Using the same manipulations as in [4], we can write the RGE for W αi in the form Here, DW αi denotes the one-loop contribution as in Eq. (40). Without going into details (see Ref. [4]), the contribution from two-loop is similar in form, except for the introduction of the primed functions, where ∆ f ′ β (∆ g ′ k ) and G ′ k ( H ′ β ) are modified forms of ∆ m 2 β (∆ m 2 k ) and G k (H k ), but which transform the same way. This example suggests that, for any multi-loop calculations, while the details may differ, one would expect that the result will obey the permutation symmetry.

V. PARAMETRIZATION OF FLAVOR MIXING
Although the SM Lagrangian is given in terms of the mixing matrix V , it contains only four physical variables, and a general problem is the lack of a criterion to pick an appropriate subset of four parameters amongst those in V . A natural starting point seems to be the use of W αi , which are rephasing invariant and have clear physical meanings, and try to further pare down the set. Such a reduction was proposed earlier [2], giving rise to a six-parameter set, (x i , y j ), with two consistency conditions. Permutation symmetry suggests a further reduction, the use of singlets as parameters. It turns out that, out of the set (x i , y j ), one can construct six singlets, x i ± y j , x i x j ± y i y j , and Πx i ± Πy j , two of which are fixed by the consistency conditions. (Note that the condition x i x j − y i y j = const. is consistent with ( x i x j − y i y j ) being a pseudo-P-scalar only if const. = 0.) We now propose the following parameter set for flavor mixing, Here, D and K transform as pseudo-P-scalars (∼ 1 × 1), while Q 2 and J 2 are P-scalars (∼ 1×1). It can be shown that (see Ref. [2] and Appendix B), |K| assumes the value K M . The value of (D, Q 2 , J 2 .K) for various special matrices [W ] can be summarized in Table  I. Note that W AI = 0 implies V AI = 0, so that there is a pair of vanishing (x a , y b ), and We now turn to a discussion of the RGE evolution of the variables {D, Q 2 , J 2 , K}. To do that we can use the results of [4], where the RGE for (x i , y j ) were obtained ( see Eqs. (26)-(27)): Here, the notation is as in Eqs.  Table II, which is adopted from a similar table in Ref. [4], written in a notation which is consistent with this paper.
Here, [w 2 ] does not mean matrix multiplication, but indicates products of individual matrix elements. E.g.,

VI. CONCLUSION
In the SM, the physics of flavor is derived from the mixing matrices and the mass terms in the Lagrangian. Rephasing of the wave functions, which leave the mass terms intact, can be cancelled by corresponding phases applied to V αi . This results in rephasing invariance, whereby any V αi 's which differ by rephasing are physically equivalent. Similarly, if we subject the wave function to a permutation, it can be neutralized by a permutation both in V αi and in the masses, and we have permutation symmetry. Physically, the validity of this argument is owing to the absence of a mechanism in L SM which can pre-assign the order of the families. This freedom to rearrange the family order gives rise to the permutation symmetry.
In this paper we emphasize the close analogy between rephasing invariance and permutation symmetry. While physical variables are invariants under rephasing (abelian), they are found to be tensors under permutation (nonabelian). A prime example is J, the Jarlskog invariant, which transforms as a pseudo-P-scalar. This assignment is verified in many examples, and offers insights about these relations. More generally, the laws of evolution of these physical tensors are shown to obey the permutation symmetry. As tensor equations, they are also much simpler in form than the evolution equations written in other variables.
As for the physical applications, we note that the implications of permutation symmetry are most noticeable when we consider processes which include internal fermions. In this case, the final results do not contain the wave functions, but are functions of masses and mixing parameters only. Permutation symmetry then implies that these functions must contain only invariants, such as f (m α Λ αi m i ). This is borne out by the examples given in the text.
Lastly, permutation properties can be used to identify a unique set of four mixing parameters, {D, Q 2 , J 2 , K}. Properties of the set were given. rephasing invariant and exhibits permutation symmetry. In this Appendix we discuss some detailed properties of this set.
We start by recalling the properties of the variables (x i , y j ), which are rephasing invariant but behave as mixed tensors under permutation. Under simple exchanges ((αβ) or (ij)), x a ↔ −y b , while for cyclic permutations, x a ↔ x c and y b ↔ y d , for some indices (a, b, c, d). Using these values, this conjecture can be verified as follows, Starting from any given (x i , y j ), we consider the variation (x i + δx i , y j + δy j ). To satisfy the consistency conditions (Eqs.(6)-(7)), we have i =j With Q 2 K = (2/3) 5 , this shows that [Z 0 (Λ − Q 2 2 )] K = − 5·(2) 6 (3) 10 · W M , whose contribution to Eq.(66) indeed vanishes.