Flavor Mixing and the Permutation Symmetry among Generations

In the standard model, the permutation symmetry among the three generations of fundamental fermions is broken by the Higgs couplings. It is found that the symmetry is restored if we include the mass matrix parameters as dynamical variables which transform appropriately under the symmetry operation. Known relations between these variables, such as the renormalization group equations, as well as formulas for neutrino oscillations (in vacuum and in matter), are shown to be covariant tensor equations under the permutation symmetry group.


Introduction
One of the long-standing puzzles in the Standard Model (SM) is the existence of three generations of fermions which behave identically under the gauge interactions. The resulting exchange symmetry will be dubbed as the g-permutation symmetry in this paper. This symmetry is spontaneously broken by the Higgs coupling through its vacuum expectation value (VEV), resulting in four mass matrices and a plethora of physical parameters, viz., the fermion masses and the mixing matrices for quarks (V CKM ) and for neutrinos (V P MN S ). If these parameters are considered as fixed entities, then they would seem to be a collection of arbitrary numbers, and the g-permutation would just be broken. However, there are at least three classes of physical phenomena which suggest an alternative interpretation. 1) Neutrino oscillation in vacuum. Here, as a neutrino beam travels, the mixing parameters evolve along and are not static. 2) Neutrino oscillation in matter (see, e.g., [1,2]). When neutrinos propagate in a medium, an induced mass is obtained which changes the mixing pattern.
3) The renormalization group equations (RGE) for quarks (see, e.g., [3][4][5][6][7]) and for neutrinos (see, e.g., [8,9]). A change in energy scales entails a new set of parameters and are governed by the RGE. For cases 2) and 3), one could say that the physical "vacuum" itself is evolving. In all of these examples, conceptually, it is more natural to regard the mass matrix parameters as dynamical variables. And, when one considers g-permutation, these variables should also transform under the symmetry operations. Once we do that, it becomes clear that they have natural assignments as tensors under S 3 , the permutation group of three objects. With this interpretation one can show that the g-permutation symmetry, now operating on both the fundamental fermions and the mass matrix parameters, is restored.
In the literature, there are numerous relations amongst the mass matrix parameters associated with neutrino os- * tkkuo@purdue.edu † schiu@mail.cgu.edu.tw cillations and RGE of quarks and neutrinos (see, e.g., [7,9,10], and the references therein). These are obtained by direct and explicit calculations. When written in appropriate variables, hints of a permutation symmetry seem ubiquitous. In this paper we present a general analysis of these equations. It is found that the SM has a gpermutation symmetry group , where the factors denote permutations in sectors of the u− and d−type quarks, the charged leptons and the neutrinos, respectively. Also, the relations mentioned above are all covariant tensor equations under [S 3 ] 4 , just like the tensor equations in theories with rotational symmetry. Our considerations are similar to those of another example of symmetry restoration in a familiar setting. Consider the case of an atom in an external B-field. The interaction term is proportional to B · σ. If we treat B as a fixed external field, then this term breaks the rotational symmetry. On the other hand, we can include B as a dynamical variable in the atomic system, transforming as a vector, then rotational symmetry is restored. The transformation of the mass matrix parameters under gpermutation is analogous to the rotation of the B-field.
We add that the Particle Data Group (PDG) parametrization [11] is ill-equipped to exploit the gpermutation symmetry. Besides being rephasing dependent, the PDG variables θ ij 's, despite their appearances, have very complicated behaviour under g-permutation, making it difficult to uncover possible symmetries in an equation.
2. Tensor Analysis of S 3 We turn now to an analysis of the representation of the g-permutation group, which is based on S 3 . The elements of S 3 operate on three objects, say B i , i = 1, 2, 3. For our purposes, it suffices to concentrate on the exchange operators: To borrow the terminology of O(3), we will call B i a Pvector or B i ∼ 3. The three-dimensional representation of S 3 , however, is reducible ( B i = invariant). Nevertheless, it is convenient to use the reducible 3 and develop a tensor analysis for S 3 , similar to that for O(3). This is useful because, as it turns out, the physical variables behave like P-tensors under permutations, and relations between them are covariant P-tensor equations.
To begin, we note that, different from the linear algebra of O(3), simple functions of B i behave like B i under permutations, and are also P-vectors. E.g., Next, out of two P-vectors, B i and C i , we can construct rank-two P-tensors such as where f is some regular function. The simplest of these tensors are B i ± C i or B i C j . Thus, the product B i C j can be decomposed into three P-vectors: 1) Diagonal: Their transformation properties may be further elucidated by the use of invariant tensors in S 3 . In addition to the familiar O(3) tensors δ ij and e ijk , in S 3 there is a third, E ijk which is defined as: This may be dubbed as the "symmetrical Levi-Civita symbol". Using these, we have the following: A i is a pseudo-P-vector since under the exchange operator, Thus, the rank-two P-tensor B i C j is decomposed into three 3's under S 3 , two of them are P-vectors, while the third is a pseudo-P-vector. Other useful constructions are In addition, for odd or even functions of A ij , e.g., In summary, the tensor analysis of S 3 has a lot in common with that of O(3), though with two important differences: 1) The existence of the symmetric Levi-Civita symbol E ijk ; 2) The linear tensor algebra is generalized to include functions of tensors for S 3 . Once these two differences are properly managed, the implementation of the S 3 symmetry amounts to demanding that all relations are covariant tensor equations, just like the familiar equations which are covariant under rotation.
3. The Spontaneously Broken g-Permutation Symmetry and Its Restoration We will now turn to the gpermutation symmetry in the SM. To accommodate neutrino oscillations which are central to the consideration in this paper, the SM will be augmented by the inclusion of Dirac neutrino mass term. This minimal extension of the SM brings the leptonic sector on a par with the quark sector, and will facilitate the ensuing discussions. The interesting possibility of neutrinos being Majorana particles will not be dealt with here, but could hopefully be the topic of a future investigation.
In order not to clutter our notations, we will first concentrate our discussion to the leptonic sector of the SM. The parallel case of the quark sector will be brought in when appropriate. Also, to study the effect of gpermutation, one need only to focus on the part of the SM Lagrangian which contains the fermion-Higgs interaction, after it has acquired its VEV. The result, after diagonalization of the mass matrix, can be represented by the following terms in the Lagrangian (see, e.g., Ref. [12]), schematically, Here, to highlight the part of the lepton-Higgs L which is relevant for our discussion, we omit the gauge fields and proper Dirac matrices in L(l, H). Also, α = (e, µ, τ ), ψ i refers to ν i , m α and m i are their masses, h denotes the Higgs field and v is its VEV, and V αi is an element of the PMNS matrix.
We may now study the action of g-permutation on L(l, H). If the permutation only acts on the fermions then clearly L(l, H) is not invariant and the gpermutation symmetry is (spontaneously) broken. However, we may include (m α , m i , V αi ) as dynamical variables which also transform under the action of X ij and X αβ , The structure of L(l, H) suggests that they transform like P-tensor. So now we have With these assignments and referring to the tensor analysis in Sec. 2, it is evident that L(l, H) is invariant: The symmetry group here is S 3 (l) × S 3 (ν). Exactly the same argument can be given for the quark sector, with the replacement of (e, µ, τ ) by (u, c, t) and (ν 1 , ν 2 , ν 3 ) by (d, s, b). We conclude that the SM has a g-permutation symmetry group, given by , that operate not only on the fundamental fermions, but also on the masses and mixing parameters, which transform like P-vectors, as indicated by the indices they carry.
We now pause to consider the effect of rephasing invariance, which was glossed over earlier. With rephasing, the transformation of V αi can acquire a phase: This means that only rephasing invariant combinations of a set of V αi 's can have definite transformation laws under exchange. Two well-known combinations are W αi = |V αi | 2 and the Jarlskog invariant [13], defined by Thus, for physical variables, the transformation laws under exchange are (for the lepton sector): Note that, in the terminology of Sec. 2, J is a pseudo-P-scalar. Also, for the quark sector, the corresponding invariant, J (q) , is also a pseudo-P-scalar: While the transformation of W αi is not surprising, J → −J under any exchange is a remarkable property. Another interesting aspect of rephasing invariance is that one can take out an overall phase from V and demand, without loss of generality, that detV = +1, while restricting further rephasing by detP = +1, where P is a diagonal phase matrix [14]. Under this condition there are the following rephasing invariants: where α = β = γ, i = j = k. This yields an alternative definition for J: Since detV changes sign under both V → −V and the exchange of rows or columns, to keep detV = +1 under the exchange operation, we have now Note that for V → −V , the invariants J and W αi are unaffected. The variables (x i , y j ) [14], which was defined in terms of Re[Γ αβγ ijk ], now have the transformation laws: where (a, b, c) is a permutation of (1, 2, 3). This implies, in particular, that In summary, the SM has the symmetry group [S 3 ] 4 when the physical mass matrix parameters behave as tensors. This set includes the fermion masses, the mixing matrices [W (CKM) ] and [W (P MN S) ], and two signs for the Jarlskog invariants, J = ± √ J 2 (J 2 = function of [W ]), in the quark and lepton sectors, respectively. The transformation laws are given in Eqs. (11), (15), and (19).

Composite Tensors
To apply the g-permutation symmetry to physical processes, it turns out that, besides the basic tensors, certain of their combinations make frequent appearances. We now present a brief discussion of their properties.
A) For the masses m α (and similarly for m i ), we have a scalar, m α , and an anti-symmetric tensor ∆m βγ = m β − m γ , which becomes a pseudo-P-vector: This combination will appear repeatedly in applications. B) Out of two W αi 's, we can have: • W αi −W αj , or 1 2 e ijk (W αj −W αk ), which transforms as a 3 in S 3 (l) and a 3 in S 3 (ν).
• 1 2 e αβγ e ijk W βj W γk = w αi . Here, w αi is an element of the cofactor matrix of W , as defined before [14]. Sums of its rows and columns are given by α w αi = i w αi = detW . It transforms as the product of pseudo-P-vectors 3(l) × 3(ν), These properties will be useful later.
This combination was also used before [7,9,10]. It played an essential role in many of the formulas in neutrino oscillation and in RGE. The transformation properties of Λ αi are exactly like W αi . What sets them apart is its structure for specific forms of the . To prove i), note that W αi = 0 implies V αi = 0. One can then use the alternative definition Λ βj = Re[V γk V δl V * γl V * δk ] to deduce Λ βj = 0. As for ii), if W αi = 1, then W αj = W βi = 0, α = β, i = j. Using i), ii) follows.
• W αj W αk , or E ijk W αj W αk . This is yet another composite which transforms like W αi . It is, however, not independent because of the relation: Nevertheless, it is sometimes used for simplicity.
C) We can also construct Under X αβ or X ij , detW ↔ −detW , so that detW is a pseudo-P-scalar. So far, no practical use has been found for detW , nor for other higher rank tensors out of the basic ones.

Applications
We may now turn to detailed analyses of neutrino oscillations and RGE, in which one can arrange to vary certain parameters to induce changes to all the parameters as a set. The resulting equations will now be written in the tensor notation. This new look offers fresh insights into their structure, making them more understandable. In the following we will study these issues case by case.
a. Neutrino Oscillation in Vacuum When a neutrino beam travels down a path, the neutrino mass eigenstates pick up a phase, exp(2iφ i ), φ i = m 2 i L/4E, which then causes change of the mixing matrix. This effect depends only on the phase difference, ∆φ ij = φ i − φ j . In the tensor terminology, neutrino oscillation is driven by the pseudo-P-vector The probability P (ν α → ν β ) is well known, and, in a notation that is adoptable for our use, given by Eqs. (58) and (59) of Ref. [10], To transcribe these equations, we start with P (ν α → ν β ), α = β. In tensor notation it reads Thus, first of all, P (ν α → ν β ) is a P-scalar under S 3 (ν), which is reasonable. This is achieved by combining Λ αi with sin 2 Φ i (∼ 3) and J(∼ 1) with i sin 2 Φ i (also ∼ 1). Note also that P (ν α → ν β ) = 0

if [W ] = [I], and [Λ] is the unique matrix (not [W ] or [w]) which also vanishes if [W ] = [I].
The formula for P (ν α → ν β ) shows that it consists of a symmetric part, ∼ S αβ , and an antisymmetric part, ∼ J ∼ 1. The antisymmetric part is CP and T violating. With J being a pseudo-P-scalar, we can apply X αα ′ repeatedly and obtain which is a well-known result. It should also be noted that, in the quark sector, the CP-measure, J ·Π∆m 2 αβ ·Π∆m 2 ij , is a P-scalar under S 3 (u) × S 3 (d), again a reasonable requirement for general CP violations.
b. Neutrino Oscillation in Matter When neutrinos propagate in a medium rich in electrons, the effective Hamiltonian acquires an induced mass (δH) ee = A. We may regard this as the first component of a P-vector, (δH D ee , δH D µµ , δH D τ τ ) = (δH D ) ξ ∼ 3, which also covers the possibility of "gedanken media" that are rich in µ and/or in τ . The addition of (δH D ) ξ generates changes in the physical variables. These were expressed [10] as differential equations given by, with dA = (δH D ) ee and In tensor notation, these equation read Here, ∆ D k = e klm (D l −D m ), ∆ W ξk = e klm (W ξl −W ξm ). For normal medium we will take (δH D ) ξ = (dA, 0, 0). In this case, Eq. (33) does not cover δW ei , which can be obtained by We can now discuss the salient features of these equations. To begin with, it is clear that they are covariant tensor (including P-parity) equations under S 3 (l)×S 3 (ν). It is noteworthy that these equations utilize the tensors W αi , Λ αi , and ∆ W αi (but not w αi or E ijk W αj W αk ), all of which have similar transformation properties. It turns out that there are consistency conditions which dictate where they belong. For Eq. (33), we note that since 0 ≤ W αi ≤ 1, it is necessary that δW αi = 0 at the boundary W αi = 0 or 1. From the properties listed in Sec. II, we find that Λ ηj = 0 if W αi = 0 or 1, for η = α, j = i. These conditions are exactly met owing to the factor E αξη e ijk , so that from Eq. (33), δW αi = 0 if W αi = 0 or 1. Finally, for Eq. (34), there are also two consistency checks. First, J 2 is known to have a maximum at [W ] = [D 0 ]/3, with J 2 max = 1/108. Also, J < 0 and J > 0 belong to two separate regimes reachable by discrete transformations, but not by infinitesimal increments. It follows that we must demand that δJ = 0 for |J| = J max or J = 0. This also means that δJ = 0 if any W αi = 0 or 1, since these last conditions imply J = 0. To satisfy the requirement at J max , ∆ W ξk is a possible (with wrong parity?) candidate in Eq. (34). But the second condition, that δJ = 0 if any W αi = 0 or 1, can not be fulfilled by any tensor. Eq. (34) solves this problem (and the P-parity problem) by using ln J so that δJ ∼ J[......], and δJ = 0 if J = 0. It is remarkable how the symmetry argument and the direct calculations reinforce each other in confirming these equations.
c. RGE for Quarks and Neutrinos In this section, we will deal exclusively with the RGE for quarks. The RGE for Dirac neutrinos are almost identical (see Eqs. (26) and (33) in Ref. [9]), but are simpler since terms proportional to neutrino masses can be dropped. It is thus sufficient to consider quarks only.
One loop RGE for quark mass matrices have been studied for a long time. When written in the matrix form, they are given by [3][4][5] constants (a u , a d , b, c).
Although the RGE for the mass matrices are simple, they are not directly useful since the matrices contain a large number of unphysical degrees of freedom. One needs to extract the RGE for the physical parameters. This was carried out, but usually in variables which mask the underlying symmetry. The easiest for our adaptation are the equations obtained in Ref. [6,7]. These equations describe the variations of the mass ratios, the mixing parameters, and J.
We now write down the tensor form of these equations, and then justify them by comparing with the established ones which were obtained by direct and explicit calculations. In the following, as before, the indices (α, β, γ) and (i, j, k) refer to (u, c, t) and (d, s, b), respectively. We define the mass ratios, Also, the mass differences, Finally, the combinations, Note that they transform as tensors according to the indices they carry, including pseudo-P-tensors which are identified with a "∼" symbol. With these we can write down the following RGE in tensor form, These equations are just the equations (3.17) and (3.18) in Ref. [6] and equations (28) and (29) in Ref. [7], although the indices α and i were not distinguished, nor were the tensors clearly identified. Also, b ′ = b/v 2 , c ′ = c/v 2 , since masses are used directly here. The first thing that catches the eye in these equations is that they are manifestly covariant tensor equations under S 3 (u) × S 3 (d). Let us now concentrate on the cdependent part of them. The resemblance to Eqs.  46)], while keeping Λ γj intact so that δW αi = 0 if W αi = 0 or 1. As for D(ln J), P-parity calls for switching ∆ W ξk to w αi , and all consistency requirements are met. Now a comment on terms that depend on a or b in Eqs. (36) and (37). These terms do not contribute to changes in mixing, since the diagonalization of M is the same as that of a polynomial in M . This is also why only mass differences, ∆ m 2 α and ∆ m 2 i , appear in these equations. A common mass in m 2 α or m 2 i , according to Eqs. (36) and (37), can always be absorbed in a d and a u , respectively.
Just as for neutrino oscillations, it is impressive to see how the results of direct calculations fit into the framework of permutation symmetry. Conversely, except for some overall constants, and barring the use of higherrank tensors [e.g., (detW ) 2 Λ αi ], one could almost write down these equations without any detailed computations.
6. Conclusion The SM is notorious for having a multitude of parameters. They originate from the spontaneous breaking of the g-permutation symmetry by the Higgs interaction. In this paper we suggest that, instead of regarding them as fixed numbers, these parameters can be included as dynamical variables which also transform under the actions of the g-permutation operation. By assigning them as appropriate tensors, the symmetry is shown to be restored. Indeed, using this procedure, the SM is found to have the discrete symmetry, We apply the symmetry to physical processes, including neutrino oscillations and RGE, for which there are established relations between the parameters due to a change in certain variables. These relations are the results of direct calculations. When we rewrite them in terms of tensors, it is revealed that they are indeed covariant tensor equations under [S 4 3 ]. The tensor notation helps to make them very compact and simple in form. In addition, one gains insights that could otherwise be obscured by a different notation. It is hoped that there will be further developments and applications of the symmetry principle in other areas of flavor physics.