Neutrino Mass Operator Renormalization Revisited

We re-derive the renormalization group equation for the effective coupling of the dimension five operator which corresponds to a Majorana mass matrix for the Standard Model neutrinos. We find a result which differs somewhat from earlier calculations, leading to modifications in the evolution of leptonic mixing angles and CP phases. We also present a general method for calculating beta-functions from counterterms in MS-like renormalization schemes, which works for tensorial quantities.


Introduction
The Standard Model (SM) is most likely an effective theory up to some scale Λ, above which new physics has to be taken into account. The discovery of neutrino masses requires an extension of the SM, which may involve right-handed neutrinos or other new fields. Introducing right-handed neutrinos allows Dirac masses m D via Yukawa couplings analogous to the quark sector. In general, lepton number need not be conserved, so that Majorana masses are possible. Another, less model dependent approach is to study the effective field theory with higher dimensional operators of SM fields. If lepton number is not conserved, some of these generate Majorana neutrino masses. The lowest dimensional operator of this kind has dimension 5 and couples two lepton and two Higgs doublets. It appears e.g. in the see-saw mechanism by integrating out the heavy right-handed neutrinos.
As quarks have only small mixings, it is somewhat surprising that neutrinos most likely have two large mixing angles [2][3][4]. It is interesting to investigate mechanisms which can produce such large or maximal mixings. These mechanisms operate, however, typically at the embedding scale Λ. For a comparison of experimental results with high energy predictions from unified theories, it is thus essential to evolve the predictions to low energies with the relevant renormalization group equations (RGE's). This evolution is related to the running of the leading dimension 5 operator. Therefore, we calculate in this letter the RGE that governs this running above the electroweak scale at one-loop order in the SM.

Lagrangian and Counterterms
Let ℓ f L , f ∈ {1, 2, 3}, be the SU(2) L -doublets of SM leptons, e f R the SU(2) Lsinglet (right-handed) charged leptons, and φ the Higgs doublet. The dimension 5 operator that gives Majorana masses to the SM neutrinos is given by where κ is symmetric under interchange of the generation indices f and g, ε is the totally antisymmetric tensor in 2 dimensions, and ℓ C L := (ℓ L ) C is the charge conjugate of the lepton doublet. a, b, c, d ∈ {1, 2} are SU(2) indices. They will only be written explicitly in terms with a non-trivial SU(2) structure. Summation over repeated indices is implied throughout this letter.
L κ gives rise to the vertex shown in Fig. 1, and an analogous one for the Hermitian conjugate term. The complete Lagrangian consists of L κ , the SM Lagrangian L SM and proper counterterms C , In the following, we omit most of those parts that yield only flavour diagonal contributions to the β-function and therefore do not contribute to the running of mixing angles, in particular terms involving quarks and gauge bosons. The remaining ones are δZ i (i ∈ {ℓ L , φ}) determine the wavefunction renormalization constants Z i = ½ + δZ i , defined in the usual way. Note that Z ℓ L is a matrix in flavour space.
δκ satisfies the relation where the factor µ ǫ is due to dimensional regularization, with µ denoting the renormalization scale and ǫ := 4 − d. The subscript B denotes a bare quantity. Note that the usual ansatz κ B ∼ Z κ κ is not possible in this case, as it would obviously spoil the symmetry of κ B or κ with respect to interchange of the flavour indices.

Calculation of the Counterterms
In the MS scheme, the quantity δκ can be computed at one-loop order from the requirement that the sum of diagrams in Fig. 2 be ultraviolet finite. Using FeynCalc [6] we obtain where C κ denotes the contribution from gauge interactions. The usual calculation of the wavefunction renormalization constants yields Again, C φ and C ℓ L represent terms from quarks and gauge interactions, which are diagonal in flavour space.

Calculating RGE's from Counterterms with Tensorial Structure
The calculation of the β-function involves some subtle points, which are related to the matrix structure of the counterterm Lagrangian. Before presenting our result in Sec. 5, we provide now some details of the calculation, which should be of general interest and which are essential for verifying our result. In particular, we generalize the usual formalism for calculating β-functions to include tensorial quantities as well as non-multiplicative renormalization.
We are interested in the β-function for a quantity Q, β Q := µ dQ dµ . In general, the bare and the renormalized quantity are related by where I = {1, . . . , M}, J = {M + 1, . . . , N} and D Q is related to the mass dimension of Q. δQ and the wavefunction renormalization constants depend on Q and some additional variables {V A }, Note that Q = Q(µ) and V A = V A (µ) are functions of the renormalization scale µ, but δQ and Z φ i do not depend explicitly on µ in an MS-like renormalization scheme. Taking the derivative of equation (10) yields Here we have introduced the notation We will solve equation (12) and the corresponding expression for V A by expanding all quantities in powers of ǫ. In the MS-scheme the quantities δQ and Z φ i can be expanded as with higher powers of 1 ǫ corresponding to higher powers in perturbation theory. On the other hand, β-functions are finite as ǫ → 0. We can therefore make the ansatz where n is an arbitrary integer. Note that in this case the power of ǫ is not related to the order of perturbation theory. From (14) and (15) we find that where the lowest possible power of 1 ǫ appearing on the right side of (16) is 1. An analogous relation holds for Q ↔ V A . Our analysis of equation (12), starting with the inspection of the ǫ n term, then shows that β (n) Q vanishes. The analog of equation (12) for β V A implies that β Note that these terms do not contribute to the β-function in 4 dimensions, i.e. for ǫ → 0, but they are necessary to read off β (0) Q from equation (12), leading to the result Note that for complex quantities Q and V A we have to treat the complex conjugates Q * and V * A as additional independent variables.

Renormalization Group Equation
The RGE for the effective coupling κ is Using equations (18) and (7) - (9), we obtain for the contributions from vertex and wavefunction renormalization (omitting terms from C κ , C φ and C ℓ L ): Adding the terms involving quarks and gauge bosons [7,8], we obtain the final result where g 2 is the SU(2) gauge coupling constant and where Y u , Y d are the Yukawa matrices for the up and the down quarks. Thus, compared to earlier results [7], we find a coefficient − 3 2 instead of − 1 2 in front of the non-diagonal term κ(Y † e Y e )+(Y † e Y e ) T κ. Note that the difference in the λκ-term is due to a different convention for the Higgs self-interaction used in this work.
We have checked our results by calculating the essential parts of the same βfunctions from the finite parts of the relevant diagrams in the framework of an underlying renormalizable theory. This calculation as well as the application to the MSSM and the two Higgs SM will be presented in a future paper [9].

Discussion and Conclusions
We have calculated in the SM the β-function for the effective coupling κ of the dimension 5 operator which corresponds to a Majorana mass matrix for neutrinos. We have explicitly presented our calculations for the non-diagonal part of the β-function, where our result disagrees with the previous one in [7] by a factor of 3. This part is responsible for the evolution of neutrino mixing angles and CP phases. Therefore, our result modifies the renormalization group running of these quantities between predictions of models at high energies and experimental data at low energies. Consequently, our work affects the SM results of previous studies based on the existing RGE's, e.g. [10][11][12][13][14][15].