# Flavor Mixing and the Permutation Symmetry among Generations

Authors: T. K. Kuo; S. H. Chiu
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

In the standard model, the permutation symmetry among the three generations of fundamental fermions is usually regarded to be broken by the Higgs couplings. It is found that the symmetry is restored if we include the mass matrix parameters as physical 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.


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## Body

## 1. 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 commonly regarded as 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 (VCKM) and for neutrinos (VPMNS). If these parameters are considered as fixed entities, then they would seem to be a collection of arbitrary numbers, which do not transform under the symmetry operation, 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., [1–3]): 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., [4–8]) and for neutrinos (see, e.g., [9, 10]): 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 changeable physical 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 S3, 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 this connection, it should be noted that, in SM, the mass parameters are, apart from a common Higgs VEV, identified with the “coupling constants” of the Higgs to the fermions. The permutation operation, ψi,mi⟶ψj,mj, is not unlike the charge conjugation transformation, ψ,e⟶ψc,−e. Thus, it is reasonable to include masses in a permutation operation.In the literature, there are numerous relations amongst the mass matrix parameters associated with neutrino oscillations and RGE of quarks and neutrinos (see, e.g., [8, 10, 11], 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 g-permutation symmetry group S3u×S3d×S3l×S3ν (or S34), 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 S34, just like the tensor equations in theories with rotational symmetry. It should be emphasized that, to establish a symmetry for a given Lagrangian, it is necessary to assign appropriate transformation properties to the variables contain therein. The symmetry S34 would be broken if one assigns the mass matrix variables as singlets under its operation.Our considerations are similar to those of another example of symmetry restoration in a familiar setting. Consider the case of an atom in an externalB→-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 g-permutation is analogous to the rotation of the B→-field.We add that the Particle Data Group (PDG) parametrization [12] is ill-equipped to exploit the g-permutation symmetry. Besides being rephasing dependent, the PDG variables θij’s, despite their appearances, have very complicated behaviour under g-permutation, making it difficult to uncover possible symmetries in an equation.


## 2. Tensor Analysis ofS3

We turn now to an analysis of the representation of the g-permutation group, which is based onS3. The elements of S3 operate on three objects, say Bi, i=1,2,3. For our purposes, it suffices to concentrate on the exchange operators:
(1)Xij:Bi⟷Bj,Bk⟷Bk,i≠j≠k.To borrow the terminology ofO3, we will call Bi a P-vector or Bi~3. The three-dimensional representation of S3, however, is reducible (∑Bi=invariant). Nevertheless, it is convenient to use the reducible 3 and develop a tensor analysis for S3, similar to that for O3. 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 ofO3, simple functions of Bi behave like Bi under permutations and are also P-vectors, e.g.,
(2)1Bi=1B1,1B2,1B3~3,sinBi=sinB1,sinB2,sinB3~3,sin2Bi=sin2B1,sin2B2,sin2B3~3.Next, out of two P-vectors,Bi and Ci, we can construct rank-two P-tensors such as fBi±fCj or fBifCj, where f is some regular function. The simplest of these tensors are Bi±Ci or BiCj. Thus, the product BiCj can be decomposed into three P-vectors: (1) diagonal: B1C1,B2C2,B3C3=Dii (no sum), (2) symmetrical: Sij=BiCj+BjCi=Sji,i≠j, and (3) antisymmetrical: Aij=BiCj−BjCi=−Aji,i≠j. Their transformation properties may be further elucidated by the use of invariant tensors in S3. In addition to the familiar O3 tensors δij and eijk, in S3 there is a third, Eijk which is defined as
(3)Eijk=1,i≠j≠k,evenunderexchangeofindices,0,anyrepeatedindex.This may be dubbed as the “symmetrical Levi-Civita symbol.” Using these, we have the following:(4)δijBiCj=∑iDii~1,P−scalar;EijkSjk~S¯i~3,P‐vector;eijkAjk~A~i~3~,pseudo‐P‐vector.A~i is a pseudo-P-vector since under the exchange operator,
(5)Xij:A~i⟷−A~j,A~k⟷−A~k.Thus, the rank-two P-tensorBiCj is decomposed into three 3’s under S3, two of them are P-vectors, while the third is a pseudo-P-vector.Other useful constructions are(6)F=EijkBiCjDk~1,P‐scalarXij:F⟶+F,(7)G=eijkBiCjDk~1~,pseudo‐P‐scalarXij:G⟶−G.In addition, for odd or even functions ofAij, e.g.,
(8)sinAij~3~,sin2Aij~3.In summary, the tensor analysis ofS3 has a lot in common with that of O3, though with two important differences: (1) the existence of the symmetric Levi-Civita symbol Eijk; (2) the linear tensor algebra is generalized to include functions of tensors for S3. Once these two differences are properly managed, the implementation of the S3 symmetry amounts to demanding that all relations are covariant tensor equations, just like the familiar equations which are covariant under rotation.


## 3. The Broken g-Permutation Symmetry and Its Restoration

We will now turn to the g-permutation 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 g-permutation, 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. [13]), schematically,
(9)Ll,H~JμWμ†+h.c.−1+hv∑αmαψ¯αψα+∑imiψ¯iψi,Jμ~∑α,iψ¯αVαiψi.Here, to highlight the part of the lepton-HiggsL which is relevant for our discussion, we omit the gauge coupling constants and proper Dirac matrices in Ll,H. Also, α=e,μ,τ, ψi refers to νi, mα and mi are their masses, Wμ refers to the W-boson, h denotes the Higgs field and v is its VEV, and Vαi is an element of the PMNS matrix.We may now study the action of g-permutation onLl,H. If the permutation only acts on the fermions (with Xij=Xij0, Xαβ=Xαβ0),
(10)Xij0:ψi⟷ψj,ψk⟷ψk,Xαβ0:ψα⟷ψβ,ψγ⟷ψγ,then clearly Ll,H is not invariant and the g-permutation symmetry is broken. However, we may include mα,mi,Vαi as dynamical variables which also transform under the action of Xij and Xαβ. The structure of Ll,H suggests that they transform like P-tensor. So now we have
(11)Xij:ψi⟷ψj;mi⟷mj;Vαi⟷Vαj,Xαβ:ψα⟷ψβ;mα⟷mβ;Vαi⟷Vβi.With these assignments and referring to the tensor analysis in Section2, it is evident that Ll,H is invariant:
(12)Xij,Xαβ:Ll,H⟷Ll,H.The symmetry group here isS3l×S3ν. Exactly the same argument can be given for the quark sector, with the replacement of e,μ,τ by u,c,t and ν1,ν2,ν3 by d,s,b. We conclude that the SM has a g-permutation symmetry group, given by S3u×S3d×S3l×S3ν=S34 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. A more concrete interpretation of Equation (12) is to regard Ll,H as an effective Lagrangian. It is used as a starting point for calculations in flavor physics. The results thus obtained are expressed in terms of the physical parameters contained in Ll,H. Equation (12) then implies that these results must exhibit the S34 symmetry. Some examples are cited in Section 5.The existence of the symmetry groupS34 can also be established from another approach. The diagonalization of the Yukawa coupling ~ψ¯LYψRH,
(13)Y=uL†YDuR,and the absorption of uL,uR into the wave functions yield the Lagrangian in the mass eigenstate basis. This procedure also constrains additional U3 transformations on ψ, so that the global symmetry U3×U3×.⋯ of the gauge interaction Lagrangian is broken down to UB1×UL1, as stated in the literature. However, the solution to the diagonalization of a 3×3 matrix has a six 3!-fold symmetry,
(14)YD=X†Y′DX,where X is a 3×3 permutation matrix, e.g.,
(15)X12=010100001.The replacement ofuL,uR by XuL,XuR corresponds to an operation of S3, which survives the diagonalization process, and is a symmetry of the Lagrangian. Thus, the SM (with Dirac neutrinos) is found to have the global symmetry group S34×UB1×UL1. The action of S34 is given by Equation (11).We now pause to consider the effect of rephasing invariance, which was glossed over earlier. With rephasing, the transformation ofVαi can acquire a phase:
(16)Xij:Vαi⟶phase·Vαj.This means that only rephasing invariant combinations of a set ofVαi’s can have definite transformation laws under exchange. Two well-known combinations are Wαi=Vαi2 and the Jarlskog invariant [14], defined by
(17)ImVαiVβjVαj∗Vβi∗=J∑γkeαβγeijk.Thus, for physical variables, the transformation laws under exchange are (for the lepton sector)(18)Xij:Wαi⟷Wαj;J⟷−J,Xαβ:Wαi⟷Wβi;J⟷−J.Note that, in the terminology of Section2, J is a pseudo-P-scalar. Also, for the quark sector, the corresponding invariant, Jq, is also a pseudo-P-scalar:
(19)Xijq,Xαβq:Jq⟷−Jq.While the transformation ofWαi 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 fromV and demand, without loss of generality, that detV=+1, while restricting further rephasing by detP=+1, where P is a diagonal phase matrix [15]. Under this condition, there are rephasing invariants:
(20)Γijkαβγ=VαiVβjVγk=ReΓijkαβγ−iJ,where α≠β≠γ, i≠j≠k. This yields an alternative definition for J:
(21)J=−ImVαiVβjVγk,detV=+1.SincedetV changes sign under both V⟶−V and the exchange of rows or columns, to keep detV=+1 under the exchange operation, we have now
(22)Xij:Vαi⟷VαjandV⟷−V,Xαβ:Vαi⟷VβiandV⟷−V.Note that forV⟶−V, the invariants J and Wαi are unaffected. The variables xi,yj [15], which was defined in terms of ReΓijkαβγ, now have the transformation laws:
(23)XijorXαβ:x1,x2,x3⟷−ya,yb,yc,where a,b,c is a permutation of 1,2,3. This implies, in particular, that ∑xi⟷−∑yj and x1x2x3⟷−y1y2y3. With J2=x1x2x3−y1y2y3, we have J2⟶J2, consistent with J⟷−J.In summary, the SM has the symmetry groupS34 when the physical mass matrix parameters behave as tensors. This set includes the fermion masses, the mixing matrices WCKM and WPMNS, and two signs for the Jarlskog invariants, J=±J2J2=functionofW, in the quark and lepton sectors, respectively. The transformation laws are given in Equations (11), (18), and (22).


## 4. Composite Tensors

To apply the g-permutation symmetry to physical processes, it turns out that, besides the basic tensors, certain combinations make frequent appearances. We now present a brief discussion of their properties.(a)
For the massesmα (and similarly for mi), we have a scalar, ∑mα, and an antisymmetric tensor Δmβγ=mβ−mγ, which becomes a pseudo-P-vector:
(24)Δm~α=12eαβγΔmβγ~3~.
This combination will appear repeatedly in applications
(b)
Out of twoWαi’s, we have
(i)
Wαi−Wαj, or 1/2eijkWαj−Wαk, which transforms as a 3 in S3l and a 3~ in S3ν
(ii)
1/2eαβγeijkWβjWγk=wαi. Here, wαi is an element of the cofactor matrix of W, as defined before [15]. Sums of its rows and columns are given by ∑αwαi=∑iwαi=detW. It transforms as the product of pseudo-P-vectors 3~l×3~ν, or 3~u×3~d in the quark sector. For specific forms of the W matrix, with (a) W=I and (b)W=D0/3 (maximal mixing, D0αi=1, for all α and i), we have (a) w=I and (b) w=0. These properties will be useful later
(iii)
1/21/2EαβγEijkWβjWγk−Wαi=Λαi. This combination was also used before [8, 10, 11]. It played an essential role in many of the formulas in neutrino oscillation and in RGE. The transformation properties of Λαi are exactly like Wαi. What sets them apart is its structure for specific forms of the W matrix. Of particular interests are: (i) if Wαi=0, then Λβj=0, α≠β, i≠j, and (ii) if Wαi=1, then except possibly for Λαi, all other Λβj=0. (iii) If W=D0/3, for maximal mixing, then Λ=−D0/18. If W=I, then Λ=0. To prove (i), note that Wαi=0 implies Vαi=0. One can then use the alternative definition Λβj=ReVγkVδlVγl∗Vδk∗ to deduce Λβj=0. As for (ii), if Wαi=1, then Wαj=Wβi=0, α≠β, i≠j. Using (i), (ii) follows
(iv)
WαjWαk, or EijkWαjWαk. This is yet another composite which transforms like Wαi. It is, however, not independent because of the relation:
(25)Λβi+Λγi=−WαjWαk.
Nevertheless, it is sometimes used for simplicity
(c)
We can also construct(26)13!eαβγeijkWαiWβjWγk=detW.
UnderXαβ or Xij, 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


## 5. 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.


### 5.1. Neutrino Oscillation in Vacuum

When a neutrino beam travels down a path, the neutrino mass eigenstates pick up a phase,exp2iϕi, ϕi=mi2L/4E, which then causes change of the mixing matrix. This effect depends only on the phase difference, Δϕij=ϕi−ϕj. In the tensor terminology, neutrino oscillation is driven by the pseudo-P-vector
(27)Φ~i=12!eijkΔϕjk=Δm~i2L4E~3~.The probabilityPνα⟶νβ is well known, and, in a notation that is adoptable for our use, given by Equations (58) and (59) of Ref. [11],
(28)Pνα⟶να=1−4Wα1Wα2sin2Φ21+Wα1Wα3sin2Φ31+Wα2Wα3sin2Φ32,Pνα⟶νβ=−4Λγ3sin2Φ21+Λγ2sin2Φ31+Λγ1sin2Φ32+2Jsin2Φ21+sin2Φ13+sin2Φ32,α≠β≠γ.To transcribe these equations, we start withPνα⟶νβ, α≠β. In tensor notation it reads
(29)Pνα⟶νβ=−4EαβγδijΛγisin2Φ~j+2J∑isin2Φ~i.Thus, first of all,Pνα⟶νβ is a P-scalar under S3ν, which is reasonable. This is achieved by combining Λαi with sin2Φ~i~3 and J~1~ with ∑isin2Φ~i (also ~1~). Note also that Pνα⟶νβ=0 if W=I, and Λ is the unique matrix (not W or w) which also vanishes if W=I. The formula for Pνα⟶νβ shows that it consists of a symmetric part, ~Sαβ, and an antisymmetric part, ~J~1~. The antisymmetric part is CP and T violating. With J being a pseudo-P-scalar, we can apply Xαα′ repeatedly and obtain
(30)Pνα⟶νβ=Pνβ⟶νγ=Pνγ⟶να=−Pνβ⟶να=⋯,which is a well-known result. It should also be noted that, in the quark sector, the CP-measure, J·ΠΔmαβ2·ΠΔmij2, is a P-scalar under S3u×S3d, again a reasonable requirement for general CP violations.Finally, the probabilityPνα⟶να can be obtained by unitarity, ∑βPνα⟶νβ=1, with use of the relation in Equation (25).


### 5.2. Neutrino Oscillation in Matter

When neutrinos propagate in a medium rich in electrons, the effective Hamiltonian acquires an induced massδHee=A. We may regard this as the first component of a P-vector, δHeeD,δHμμD,δHττD=δHDξ~3, which also covers the possibility of “gedanken media” that are rich in μ and/or in τ. The addition of δHDξ generates changes in the physical variables. These were expressed [11] as differential equations given by, with dA=δHDee and Di=mi2,
(31)dDidA=Wei,(32)12ddAWe1We2We3Wμ1Wμ2Wμ3Wτ1Wτ2Wτ3=1D1−D2We1We2,−We1We2,0Λτ3,−Λτ3,0Λμ3,−Λμ3,0+1D2−D30,We2We3,−We2We30,Λτ1,−Λτ10,Λμ1,−Λμ1+1D3−D1−We1We3,0,We1We3−Λτ2,0,Λτ2−Λμ2,0,Λμ2,(33)ddAlnJ=−∑i≠jWei−WejDi−Dj.In tensor notation, these equations read(34)δmi2=δHDξWξi,(35)δWαi=2EαξηeijkδHDξΛηjΔD~k,(36)δlnJ=−δHDξΔW~ξkΔD~k.Here,ΔD~k=1/2eklmDl−Dm, ΔW~ξk=1/2eklmWξl−Wξm. For normal medium, we will take δHDξ=dA,0,0. In this case, Equation (35) does not cover δWei, which can be obtained by
(37)δWei=−δWμi+Wτi.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 underS3l×S3ν. It is noteworthy that these equations utilize the tensors Wαi, Λαi, and ΔW~αi (but not wαi or EijkWαjWαk), all of which have similar transformation properties. It turns out that there are consistency conditions which dictate where they belong. For Equation (34), in the special case W=I, it is known that δm12=δHDee, thus ruling out the use of Λξi, since Λ=0 for W=I. For Equation (35), we note that since 0≤Wαi≤1, it is necessary that δWαi=0 at the boundary Wαi=0 or 1. From the properties listed in Section 2, we find that Ληj=0 if Wαi=0 or 1, for η≠α, j≠i. These conditions are exactly met owing to the factor Eαξηeijk, so that from Equation (35), δWαi=0 if Wαi=0 or 1. Finally, for Equation (36), there are also two consistency checks. First, J2 is known to have a maximum at W=D0/3, with Jmax2=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∣=Jmax or J=0. This also means that δJ=0 if any Wαi=0 or 1, since these last conditions imply J=0. To satisfy the requirement at Jmax, ΔW~ξk is a possible (with wrong parity?) candidate in Equation (36). But the second condition, that δJ=0 if any Wαi=0 or 1, cannot be fulfilled by any tensor. Equation (36) solves this problem (and the P-parity problem) by using lnJ 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.


### 5.3. 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 Equations (26) and (32) in Ref. [10]) 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 has been studied for a long time. When written in the matrix form, they are given by [4–6]
(38)DMu=auMu+bMu2+cMu,Md,(39)DMd=adMd+bMd2+cMu,Md.Here,Mu=YuYu† (Md=YdYd†), Yu (Yd) is the Yukawa coupling matrix for the u-type (d-type) quarks; D=1/16π2d/dt, t=lnμ/MW, μ is the energy scale, and MW is the W boson mass. The model dependence of the RGE is contained in the constants (au,ad,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. [7, 8]. 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 indicesα,β,γ and i,j,k refer to u,c,t and d,s,b, respectively. We define the mass ratios,
(40)Rαβ=mα2mβ2,lnR~α=12eαβγlnRβγ,rij=mi2mj2,lnr~i=12eijklnrjk.Also, the mass differences,(41)Δm~α2=12eαβγmβ2−mγ2,Δm~i2=12eijkmj2−mk2.Finally, the combinations,(42)H~α=eαβγHβγ,Hβγ=mβ2+mγ2mβ2−mγ2,G~i=eijkGjk,Gjk=mj2+mk2mj2−mk2.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(43)DlnR~α=b′Δm~α2+2c′·wαiΔm~i2,(44)Dlnr~i=b′Δm~i2+2c′·wiαTΔm~α2,(45)DWαi=−2c′·eαβγeijkΔm~β2ΛγjG~k+Δm~kΛjγTH~β,(46)DlnJ=−c′·Δm~α2wαiG~i+Δm~i2wiαTH~α.These equations are just Equations (3.17) and (3.18) in Ref. [7] and Equations (28) and (29) in Ref. [8], although the indices α and i were not distinguished nor were the tensors clearly identified. Also, b′=b/v2, c′=c/v2, since masses are used directly here.The first thing that catches the eye in these equations is that they are manifestly covariant tensor equations underS3u×S3d. Let us now concentrate on the c-dependent part of them. The resemblance to Equations (34), (35), and (36) is striking, although there are also differences. For Equations (43), (44), (45), and (46), there are two “source” terms, Δm~i2 and Δm~α2, which generate the changes. Their effects on Vu† and Vd are combined in VCKM, and thus in DWαi and DlnJ. Also, the simple pole terms 1/Di−Dj in Equations (35) and (36) are replaced by Gij and Hij, reflecting the nature of the new situation, while keeping the singular behaviour if Di−Dj⟶0. In addition, δHDξ~3 for neutrino oscillations, while Δm~α2 and Δm~i2~3~ for RGE. This results in replacing Eαβγ (in Equation (35)) by eαβγ (in Equation (45)), while keeping Λγj intact so that δWαi=0 if Wαi=0 or 1. As for DlnJ, P-parity calls for switching ΔW~ξk to wαi, and all consistency requirements are met. Now a comment on terms that depend on a or b in Equations (38) and (39). 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, Δm~α2 and Δm~i2, appear in these equations. A common mass in mα2 or mi2, according to Equations (38) and (39), can always be absorbed in ad and au, 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 higher-rank tensors (e.g.,detW2Λαi), one could almost write down these equations without any detailed computations.Another consequence of permutation symmetry is that tensors are the entities being measured, e.g., neutrino oscillations determine the tensorsΛγk. They, in turn, are simple functions of Wαi. It is therefore useful to analyze data directly in terms of Wαi, thereby avoiding the possible loss of information in translation. Some of the issues were also discussed elsewhere [11].


## 5.1. Neutrino Oscillation in Vacuum

When a neutrino beam travels down a path, the neutrino mass eigenstates pick up a phase,exp2iϕi, ϕi=mi2L/4E, which then causes change of the mixing matrix. This effect depends only on the phase difference, Δϕij=ϕi−ϕj. In the tensor terminology, neutrino oscillation is driven by the pseudo-P-vector
(27)Φ~i=12!eijkΔϕjk=Δm~i2L4E~3~.The probabilityPνα⟶νβ is well known, and, in a notation that is adoptable for our use, given by Equations (58) and (59) of Ref. [11],
(28)Pνα⟶να=1−4Wα1Wα2sin2Φ21+Wα1Wα3sin2Φ31+Wα2Wα3sin2Φ32,Pνα⟶νβ=−4Λγ3sin2Φ21+Λγ2sin2Φ31+Λγ1sin2Φ32+2Jsin2Φ21+sin2Φ13+sin2Φ32,α≠β≠γ.To transcribe these equations, we start withPνα⟶νβ, α≠β. In tensor notation it reads
(29)Pνα⟶νβ=−4EαβγδijΛγisin2Φ~j+2J∑isin2Φ~i.Thus, first of all,Pνα⟶νβ is a P-scalar under S3ν, which is reasonable. This is achieved by combining Λαi with sin2Φ~i~3 and J~1~ with ∑isin2Φ~i (also ~1~). Note also that Pνα⟶νβ=0 if W=I, and Λ is the unique matrix (not W or w) which also vanishes if W=I. The formula for Pνα⟶νβ shows that it consists of a symmetric part, ~Sαβ, and an antisymmetric part, ~J~1~. The antisymmetric part is CP and T violating. With J being a pseudo-P-scalar, we can apply Xαα′ repeatedly and obtain
(30)Pνα⟶νβ=Pνβ⟶νγ=Pνγ⟶να=−Pνβ⟶να=⋯,which is a well-known result. It should also be noted that, in the quark sector, the CP-measure, J·ΠΔmαβ2·ΠΔmij2, is a P-scalar under S3u×S3d, again a reasonable requirement for general CP violations.Finally, the probabilityPνα⟶να can be obtained by unitarity, ∑βPνα⟶νβ=1, with use of the relation in Equation (25).


## 5.2. Neutrino Oscillation in Matter

When neutrinos propagate in a medium rich in electrons, the effective Hamiltonian acquires an induced massδHee=A. We may regard this as the first component of a P-vector, δHeeD,δHμμD,δHττD=δHDξ~3, which also covers the possibility of “gedanken media” that are rich in μ and/or in τ. The addition of δHDξ generates changes in the physical variables. These were expressed [11] as differential equations given by, with dA=δHDee and Di=mi2,
(31)dDidA=Wei,(32)12ddAWe1We2We3Wμ1Wμ2Wμ3Wτ1Wτ2Wτ3=1D1−D2We1We2,−We1We2,0Λτ3,−Λτ3,0Λμ3,−Λμ3,0+1D2−D30,We2We3,−We2We30,Λτ1,−Λτ10,Λμ1,−Λμ1+1D3−D1−We1We3,0,We1We3−Λτ2,0,Λτ2−Λμ2,0,Λμ2,(33)ddAlnJ=−∑i≠jWei−WejDi−Dj.In tensor notation, these equations read(34)δmi2=δHDξWξi,(35)δWαi=2EαξηeijkδHDξΛηjΔD~k,(36)δlnJ=−δHDξΔW~ξkΔD~k.Here,ΔD~k=1/2eklmDl−Dm, ΔW~ξk=1/2eklmWξl−Wξm. For normal medium, we will take δHDξ=dA,0,0. In this case, Equation (35) does not cover δWei, which can be obtained by
(37)δWei=−δWμi+Wτi.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 underS3l×S3ν. It is noteworthy that these equations utilize the tensors Wαi, Λαi, and ΔW~αi (but not wαi or EijkWαjWαk), all of which have similar transformation properties. It turns out that there are consistency conditions which dictate where they belong. For Equation (34), in the special case W=I, it is known that δm12=δHDee, thus ruling out the use of Λξi, since Λ=0 for W=I. For Equation (35), we note that since 0≤Wαi≤1, it is necessary that δWαi=0 at the boundary Wαi=0 or 1. From the properties listed in Section 2, we find that Ληj=0 if Wαi=0 or 1, for η≠α, j≠i. These conditions are exactly met owing to the factor Eαξηeijk, so that from Equation (35), δWαi=0 if Wαi=0 or 1. Finally, for Equation (36), there are also two consistency checks. First, J2 is known to have a maximum at W=D0/3, with Jmax2=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∣=Jmax or J=0. This also means that δJ=0 if any Wαi=0 or 1, since these last conditions imply J=0. To satisfy the requirement at Jmax, ΔW~ξk is a possible (with wrong parity?) candidate in Equation (36). But the second condition, that δJ=0 if any Wαi=0 or 1, cannot be fulfilled by any tensor. Equation (36) solves this problem (and the P-parity problem) by using lnJ 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.


## 5.3. 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 Equations (26) and (32) in Ref. [10]) 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 has been studied for a long time. When written in the matrix form, they are given by [4–6]
(38)DMu=auMu+bMu2+cMu,Md,(39)DMd=adMd+bMd2+cMu,Md.Here,Mu=YuYu† (Md=YdYd†), Yu (Yd) is the Yukawa coupling matrix for the u-type (d-type) quarks; D=1/16π2d/dt, t=lnμ/MW, μ is the energy scale, and MW is the W boson mass. The model dependence of the RGE is contained in the constants (au,ad,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. [7, 8]. 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 indicesα,β,γ and i,j,k refer to u,c,t and d,s,b, respectively. We define the mass ratios,
(40)Rαβ=mα2mβ2,lnR~α=12eαβγlnRβγ,rij=mi2mj2,lnr~i=12eijklnrjk.Also, the mass differences,(41)Δm~α2=12eαβγmβ2−mγ2,Δm~i2=12eijkmj2−mk2.Finally, the combinations,(42)H~α=eαβγHβγ,Hβγ=mβ2+mγ2mβ2−mγ2,G~i=eijkGjk,Gjk=mj2+mk2mj2−mk2.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(43)DlnR~α=b′Δm~α2+2c′·wαiΔm~i2,(44)Dlnr~i=b′Δm~i2+2c′·wiαTΔm~α2,(45)DWαi=−2c′·eαβγeijkΔm~β2ΛγjG~k+Δm~kΛjγTH~β,(46)DlnJ=−c′·Δm~α2wαiG~i+Δm~i2wiαTH~α.These equations are just Equations (3.17) and (3.18) in Ref. [7] and Equations (28) and (29) in Ref. [8], although the indices α and i were not distinguished nor were the tensors clearly identified. Also, b′=b/v2, c′=c/v2, since masses are used directly here.The first thing that catches the eye in these equations is that they are manifestly covariant tensor equations underS3u×S3d. Let us now concentrate on the c-dependent part of them. The resemblance to Equations (34), (35), and (36) is striking, although there are also differences. For Equations (43), (44), (45), and (46), there are two “source” terms, Δm~i2 and Δm~α2, which generate the changes. Their effects on Vu† and Vd are combined in VCKM, and thus in DWαi and DlnJ. Also, the simple pole terms 1/Di−Dj in Equations (35) and (36) are replaced by Gij and Hij, reflecting the nature of the new situation, while keeping the singular behaviour if Di−Dj⟶0. In addition, δHDξ~3 for neutrino oscillations, while Δm~α2 and Δm~i2~3~ for RGE. This results in replacing Eαβγ (in Equation (35)) by eαβγ (in Equation (45)), while keeping Λγj intact so that δWαi=0 if Wαi=0 or 1. As for DlnJ, P-parity calls for switching ΔW~ξk to wαi, and all consistency requirements are met. Now a comment on terms that depend on a or b in Equations (38) and (39). 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, Δm~α2 and Δm~i2, appear in these equations. A common mass in mα2 or mi2, according to Equations (38) and (39), can always be absorbed in ad and au, 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 higher-rank tensors (e.g.,detW2Λαi), one could almost write down these equations without any detailed computations.Another consequence of permutation symmetry is that tensors are the entities being measured, e.g., neutrino oscillations determine the tensorsΛγk. They, in turn, are simple functions of Wαi. It is therefore useful to analyze data directly in terms of Wαi, thereby avoiding the possible loss of information in translation. Some of the issues were also discussed elsewhere [11].


## 6. Conclusion

The SM is notorious for having a multitude of parameters. They originate from the 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 physical 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 (with inclusion of Dirac neutrino mass terms) is found to have the discrete symmetry,S3u×S3d×S3l×S3ν=S34.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 underS34. 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.



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