Renormalization of the Neutrino Mass Matrix

In terms of a rephasing invariant parametrization, the set of renormalization group equations (RGE) for Dirac neutrino parameters can be cast in a compact and simple form. These equations exhibit manifest symmetry under flavor permutations. We obtain both exact and approximate RGE invariants, in addition to some approximate solutions and examples of numerical solutions.


I. INTRODUCTION
Despite the tremendous progress made recently regarding the intrinsic properties of the neutrinos, much remains to be learned. For instance, it is not known whether the neutrinos are Dirac or Majorana particles. The pattern of masses and mixing parameters is also a mystery. Within the framework of the standard model (SM), it is well-established that these parameters are functions of the energy scale, and are governed by the renormalization group equations (RGE). Indeed, the evolution of the gauge couplings (e.g., asymptotic freedom and gauge coupling unification) has been very successfully applied to particle physics, forming an essential part of the foundation of the standard model. One would expect similar considerations to apply to the Higgs couplings (i.e., the mass matrices). In addition, these effects are indispensable toward a theoretical understanding of the mass matrices. Unfortunately, to this date existing results are usually numerical in nature and not very general. The main difficulty lies in the complexity of the RGE for these parameters. While the RGE in terms of the mass matrices are reasonably simple, these matrices contain a large number of unphysical degrees of freedom, and the RGE for physical variables are very complicated indeed. For instance, the mixing matrix is rephasing invariant, which is characteristic of the mixing of quantum states, for which the phase of an individual state vector is unobservable. To extract the physical parameters, one may proceed by fixing some phases. This is similar to "fixing a gauge" in gauge theories. Thus, in the standard parametrization, one sets four phases in the mixing matrix (those of V 11 , V 12 , V 23 , V 33 ) to vanish, arriving at three angles and a phase. When applied to the analysis of the RGE, one is enforcing this "gauge" at every energy scale. The lack of a theoretical rationale for this choice, we believe, contributes to the complexity of the RGE, when expressed in terms of the standard parameters.
In this work we investigate the one-loop RGE evolution of Dirac neutrinos, using a rephasing invariant parametrization introduced earlier. What characterizes this parametrization is its symmetry structure under flavor permutations. Indeed, we establish a set of RGE which exhibits a simple structure with built-in permutation symmetry. This set of equations is not as formidable as the one written in terms of the standard parametrization. Its solutions are studied both analytically and numerically. We found a RGE invariant, in addition to obtaining some approximate solutions. Numerical examples are also presented.
This work is organized as follows. In Section II, we briefly introduce the rephasing invariant parametrization that will be used in this work. Their properties are also discussed. In Section III, we obtain the RGE in a compact and simple form. From these we derive both exact and approximate RGE invariants in Section IV. In Section V, we further obtain approximate solutions of the RGE and provide some numerical examples. We then summarize the work in Section VI.
The (x i , y j ) variables are constrained by two conditions: and they are related to the Jarlskog invariant, In addition, the (x i , y j ) variables are bounded by ±1: −1 ≤ (x i , y j ) ≤ +1, with x i ≥ y j for any pair of (i, j).
It is convenient to write |V ij | 2 in a matrix form with elements x i − y j : The matrix of the cofactors of W , denoted as w with w T W = (detW )I, is given by The elements of w are also bounded, −1 ≤ w αi ≤ +1, and Note that the constraint equations, Eq. (3) and Eq. (4), have been used here. The relations between this (x i , y j ) parametrization and the standard ones, θ 12 , θ 23 , θ 13 , and the Dirac CP phase δ are shown in Appendix A.
One may further obtain useful expressions of the rephasing invariant combination formed by products of four mixing elements [6], which can be reduced to where the second term in either expression is one of the Γ's (Γ * 's) defined in Eq. (1). In addition, the combination of five elements can be written in the following form: Here if |V αi | 2 = x a − y l , then b = c = a, m = n = l. This means that if one takes the αth row and the ith column, complex conjugates the vertex (V * αi ), then the product is rephasing invariant and has a well-defined imaginary part. Certain intriguing properties of Ξ are shown also in Appendix A.

III. RGES FOR THE NEUTRINO PARAMETERS
The RGE for the Hermitian matrix M ≡ Y † ν Y ν , where Y ν is the neutrino Yukawa coupling matrix, is given by (see, e.g., [7][8][9][10][11][12][13]) at the one-loop level. Here, α is real and model-independent, P = CY † l Y l + C ′ Y † ν Y ν , with model-dependent coefficients C and C ′ , and Y l is the charged lepton Yukawa matrix. Following [7][8][9][10][11][12][13], we will ignore the term C ′ Y † ν Y ν so that Equation ( 13) is very simple in form. However, since M contains a number of unphysical parameters, it is necessary to extract its physical parts. To this end we may choose the basis where Y l is diagonal, for all energy scales. One is left then to diagonalize M by the mixing matrix V : where h 2 i are the eigenvalues of M. Further, one needs to separate the physical, rephasing invariant, parts of V . To do this we will follows the procedure of [7][8][9][10][11][12] but regroup the equations for rephasing invariant variables at the end.
The evolution of the mixing matrix V satisfies the relation, here the matrix T is anti-Hermitian. One may define D = 16π 2 d dt with t = ln(µ/M W ), and compares the diagonal elements of D(V [diag(h 2 1 , h 2 2 , h 2 3 )]V † ) with that of DM to obtain where f 2 i are the eigenvalues of the matrix Y † l Y l , and the C ′ terms have been ignored. From the off-diagonal elements, we obtain the expression of T ij : with P ′ = V † P V and To derive the RGE for neutrino mixing parameters, we may start with By using Eq. (16), Eq. (18), and Eq. (57), one reaches the following form, The real part of DΓ ijk gives rise to Dx i and Dy i in the following matrix forms: where we define Here We have taken over the results of Ref. [19], which are also valid for the Dirac neutrino problem. Thus, the matrices matrices [A i ] and [A ′ i ] are given in Table II of Ref. [19], and are reproduced here as Table III in Appendix B. Note that, to compensate for the usual convention of the neutrino mixing matrix, whereby it corresponds to the conjugate of that for the quarks, we have adapted the results of Ref. [19] by making the correspondences charged leptons ↔ u-type quarks and neutrinos ↔ d-type quarks. In addition, the imaginary part of DΓ ijk leads to It should be noted that, since ∆f 2 ij = 0, the evolution equations are invariant when a constant is added to any column of It turns out that we may recast Eqs. (22) and (23) into a more symmetrical and suggestive form. To do that we start with Eq. (10) and separate its real and imaginary parts, for Since Re(π αβ ij ) takes the following forms, we have In terms of the (x, y) variables, we find where (x a , y j ) comes from |V γk | 2 = x a −y j , and a = b = c, j = k = l. Another useful identity is Using Eqs. (27) and (31) By expressing (x, y) in terms of W ij , we further obtain Here, [S ij ] are given in Table I To exhibit the structure of these equations, let's write down explicitly the evolution equation of x 1 , e.g., Also, for |V 11 | 2 , The simple forms of these evolution equations mirror the permutation patterns contained in the definitions of (x i , y j ), such as x 1 = Re(Γ 123 ) = Re(V 11 V 22 V 33 ). It is also noticeable that Λ αi , which are the real parts of the Jarlskog variables, π βγ jk (Eq. (28)), and are directly measurable in neutrino oscillation experiments, play such a prominent role in these evolution equations. An instructive comparison can be made to the two flavor problem. Here, the familiar 2 × 2 (real) mixing matrix can be parametrized as Let us adapt the results (for Majorana neutrinos) of Ref. [14]. With real mass matrices (and m → m 2 ), Eq.(29) in Ref. [14] becomes Thus, there is a clear lineage between Eqs.( 34)-( 35) and the two flavor RGE. It is interesting, and somewhat surprising, that, allowing for the effects of permutation, the two flavor RGE act simply like building blocks for the three flavor RGE.
With the above alternative expressions, we may verify that and For completeness, we also present the evolution equations for the elements of the cofactor matrix [w]: where we have used w γk = |V αi | 2 |V βj | 2 − |V αj | 2 |V βi | 2 to obtain the matrix [G ij ]. They are listed explicitly in Table IV of Appendix B.

IV. THE RGE INVARIANTS
RGE describe the evolution of a multitude of variables as functions of a single parameter, t. As such one might find combinations of physical variables which become independent of t. These RGE invariant prescribe correlations among physical variables and can serve as powerful constraints on possible theories at high energy. They have also drawn certain attention in recent literature [14][15][16][17][18]). To search for neutrino RGE invariants using our parametrization, and to further pave the way for the analytic, approximate solutions for W ij , we first define the neutrino mass ratio i.e., J 2 (Π ij sinh 2 ln r ij ) is a RGE invariant. Furthermore, one notes that any quantity that is identical to J 2 also gives rise to an invariant when multiplied by Π ij sinh 2 ln r ij in Eq. (47). With the expression of Λ γk in Eq. (30), it is straightforward to write down nine different forms of J 2 = π 2 γk − Λ 2 γk , which correspond to nine different combinations of (γ, k), This leads to nine RGE invariants which consist of |V ij | 2 and the mass ratios ln r 2 ij : Under the same approximation, we find, e.g., where we used the relations Λ 12 +Λ 22 = −|V 31 | 2 |V 33 | 2 and Λ 13 +Λ 23 = −|V 31 | 2 |V 32 | 2 . Together with similar results for D ln |V 22 | 2 and D ln |V 33 | 2 , we find the following approximate RGE invariant: where C ′ = 1/(2Cf 2 3 ). One observes that the six quantities, D ln W 11 , D ln W 13 , D ln W 21 , D ln W 23 , D ln W 31 , and D ln W 33 depend only on W 31 and W 33 . It is also seen that the following two relations hold: One notes that Eq. (64) has an approximate, analytic solution for a range of t-values inside which f 2 3 ≈constant is a good approximation: where C ′′ = 16π 2 C ′ , and a ij is the initial value of W ij at t = t 0 . Similarly, we can solve for W 33 from Eq. (66): With the explicit expressions for W 31 in Eq. (69), we may rewrite Eq. (58) as: which can be solved for W 11 : Similarly, W 13 can be solved by using the explicit solution of W 33 in Eq. (70), The rest of W ij can be obtained directly by using i W ij = 1 and j W ij = 1. A confidence builder for these approximate solutions is the numerical solutions of RGE with initial conditions incorporating the approximations in Eqs. (56) and (57). Note that to assess the general nature of the RGE, it seems more appropriate to start from a point with fast evolution, so that most changes may be accomplished in its neighborhood, with minor corrections afterwards. The low energy physics values are close to a fixed point of the RGE, and it is crucial to study how they are approached from the high energy values. However, without detailed knowledge of the initial values at high energy in the scope of theoretical framework, it is difficult to assign a suitable region of initial parameter space that yields all the measured values as the RGE evolve down to the low energy.
At high energy, either hierarchical or near degenerate masses and mixing parameters are both motivated by various models [20]. For the purpose of illustration, we simply assume C = −3/2 in analogy to the quark RGE and adopt the following initial input parameters at high energy: • f 2 3 = 10 −4 , f 2 3 = 10 −2 , and f 2 3 = 1 are adopted for calculating the approximate solutions, which will be compared with the full solution using the same values of f 2 3 as the initial input at t 0 = 30.
With the above inputs, we present examples of approximate and full numerical solutions for |V 11 | 2 and |V 13 | 2 in Fig.1, and summarize the results in the following.
• The approximate solutions agree well with the full solutions for small f 2 3 . However, deviation begins to enlarge for larger f 2 3 , e.g., when f 2 3 ∼ 1. • Although the quark mixing elements can evolve quite significantly from high energy [19], the neutrino mixing parameters (x i and y i ) only evolve slightly and thus the elements W ij = |V ij | 2 do not evolve much from the initial values. This general behavior is in agreement with the expectation of recent study (see, e.g., Ref [21]).
• Note that with the chosen input values, some of the parameters do not evolve to the observed values at low energies. It appears that the RGE evolution may be sensitive to the initial parameters and that fine-tuning the may be required.
In this case, we obtain the following simple forms for DW ij : The RGE (Eq. (34)-(35)) are valid for a class of theories that are similar to the SM. The examples that we presented also show that these equations have simple approximate solutions under various sets of input parameters. In the search of theories beyond the standard model, they should be helpful to identify those that can yield viable sets of physical parameters at low energies.

VI. CONCLUSION
One of the most important questions in the study of neutrinos is an assessment of the RGE evolution of neutrino masses and mixing. In contrast to the corresponding problem for gauge couplings, owing to the complexity of the RGE, the evolution of the neutrino parameters is rather poorly understood. It is thus not easy to analyze models at high energies which may start with a variety of mass and/or mixing patterns. We studied the one-loop RGE of Dirac neutrinos, using a rephasing invariant parametrization introduced earlier. Because of the symmetry structure (under flavor permutation) of this parametrizations, it is found that the resulting equations (Eqs. (26), (33), (34), (35)) can be arranged into a highly symmetric matrix form, which can facilitate both theoretical and numerical studies of the solutions. In particular, they should provide a useful guide in the search of extensions of the SM. We obtained a RGE invariant, as well as some approximate solutions. In addition, some numerical results are presented.
Work is in progress to generalize our results to the case of Majorana neutrinos, as well as the evolution of the quark mass matrices. We hope to present these studies in a future publication.