Neutrino Mass Operator Renormalization in Two Higgs Doublet Models and the MSSM

In a recent re-analysis of the Standard Model (SM) beta-function for the effective neutrino mass operator, we found that the previous results were not entirely correct. Therefore, we consider the analogous dimension five operators in a class of Two Higgs Doublet Models (2HDM's) and the Minimal Supersymmetric Standard Model (MSSM). Deriving the renormalization group equations for these effective operators, we confirm the existing result in the case of the MSSM. Some of our 2HDM results are new, while others differ from earlier calculations. This leads to modifications in the renormalization group evolution of leptonic mixing angles and CP phases in the 2HDM's.


Introduction
The discovery of neutrino masses requires an extension of the Standard Model (SM). The most promising scenario for giving masses to neutrinos is the seesaw mechanism [1], which provides a convincing explanation for their smallness. It typically introduces heavy, gauge-singlet neutrinos and thereby gives small Majorana masses to the SM neutrinos. When the SM is viewed as an effective field theory, Majorana masses for the neutrinos can be introduced via higher dimensional operators of SM fields. 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 singlets.
The experimental results in the neutrino sector provide an interesting new way for testing theories beyond the SM. In order to compare the experimental results with predictions from models beyond the SM, like unified theories, it is essential to evolve the masses, mixing angles and CP phases from high to low energies. This is accomplished with the renormalization group equations (RGE's) for the neutrino mass operators in the theory valid at intermediate energy scales. This theory may be the SM, but can also be an extension like a Two Higgs Doublet Model (2HDM) or the Minimal Supersymmetric Standard Model (MSSM).
In a recent letter [2] we discussed the derivation of the RGE for the dimension 5 neutrino mass operator in the SM. In this letter we derive the RGE's for the corresponding operators in a class of 2HDM's and the MSSM.

Effective Neutrino Mass Operators in 2HDM's
In many extensions of the SM, the Higgs sector is enlarged by introducing additional SU(2) L doublet scalar fields φ (i) (1 ≤ i ≤ N H ). These can couple to the SM fermions via the Yukawa couplings ℓ f L and Q f L , f ∈ {1, 2, 3} are the SU(2) L doublets of SM leptons and quarks, respectively. e f R , d f R and u f R denote the SU(2) L -singlet (right-handed) charged leptons, down-type quarks and up-type quarks. ε is the totally antisymmetric tensor in 2 dimensions and a, b, c, d ∈ {1, 2} are SU(2) indices. Summation over repeated indices is implied throughout this letter. We have chosen the notation in equation (1) in such a way that all φ (i) transform as (2, 1 2 ) under SU(2) L ⊗ U(1) Y . In particular, for N H = 1 we obtain the SM.
Note that there are tight phenomenological constraints on Yukawa couplings. As pointed out in [3][4][5], it is very hard to construct viable models in which one type of SM fermions e, d and u couples to two or more Higgs bosons, since this in general leads to tree-level flavor-changing neutral currents (FCNC's). Therefore, we will only consider models in which the fermions couple to at most one Higgs. As a consequence, the suffix "(i)" on the Yukawa couplings in equation (1) is redundant and will be omitted in the following.

Classification of 2HDM's
We concentrate on models with two Higgs doublets for simplicity, i.e. N H = 2, and consider only schemes in which each of the right-handed SM fermions couples to exactly one Higgs boson. All inequivalent possibilities are classified in table 1. By convention, the scalar which couples to e is defined to be φ (1) . In order to avoid FCNC's, we impose the 2 symmetry and corresponding transformations in the fermion sector. For example, in scheme (ii) all fields transform trivially except for The most general Higgs self-interaction Lagrangian is then

Effective Neutrino Mass Operators
The lowest dimensional effective neutrino mass operators compatible with the symmetry (2) are given by where ℓ C L is the charge conjugate of the lepton doublet, and κ (ii) gf are symmetric matrices with respect to the generation indices g and f . Note that it is possible that only one of these operators, e.g. L (22) κ , arises from integrating out heavy degrees of freedom in a specific model. However, as we shall see, both mix due to the renormalization group evolution and therefore have to be taken into account simultaneously.
As long as the symmetry (2) is valid, L (11) κ and L (22) κ represent the only possible dimension 5 operators containing two ℓ L fields. If this symmetry was broken, further couplings would appear in the Higgs interaction Lagrangian (4).

Calculation of the RGE
We work in the MS renormalization scheme at the one-loop level. The wavefunction renormalization constants Z i = 1 + δZ i are defined in the usual way. For the Higgs fields φ (i) we obtain where ǫ := 4 − d is the deviation from 4 dimensions in dimensional regularization, and ξ B and ξ W are the gauge fixing parameters used in R ξ gauge. g 1 and g 2 are the U(1) Y and SU(2) L gauge coupling constants, respectively. The coefficients z (i) f are defined to be 1 if the fermion f couples to the Higgs boson φ (i) and 0 otherwise. For the models classified in table 1 they are given by  table 2. The wavefunction renormalization for ℓ L is given by With the counterterms for the effective vertices defined by we find for the vertex correction The relevant Feynman diagrams are shown in figure 1. Using the technique described in [2], the β-functions β κ (ii) = µ d dµ κ (ii) with µ denoting the renormalization scale, can be obtained from the counterterms, The terms proportional to λ 5 are responsible for the mixing of the effective operators mentioned before. Our result for β κ (11) differs from the one in [7] by a factor of 3 because of the term 1 2 − 2δ i1 in the first line. We had earlier found [2] an analogous discrepancy in β κ for the SM, which has recently been confirmed [8].
Note that in 2HDM's running effects are in general larger than in the SM due to the fact that the Yukawa couplings are enhanced, e.g. (Y e ) 2HDM = (Y e ) SM / cos β, where tan β = v (1) /v (2) with v (i) being the vacuum expectation value of the Higgs field φ (i) .  . The gray arrow indicates the fermion flow as defined in [6].
In the MSSM, the effective dimension 5 operator that gives Majorana masses to the SM neutrinos is contained in the F -term of the superpotential L and H (2) are the chiral superfields that contain the SU(2) L doublets, the Higgs doublet with weak hypercharge + 1 2 and the corresponding superpartners. The part of the superpotential describing the Yukawa interactions is given by The superfields E C , D C and U C contain the SU(2) L -singlet charged leptons, down-type quarks and up-type quarks, respectively, and Q contains the SU(2) L quark doublets. The Higgs superfield H (1) has weak hypercharge − 1 2 . Calculating the RGE in the MSSM yields confirming the existing MSSM result [9,7].

Conclusion
A recent check of the SM β-function [8] showed that previously published results were not quite correct. Therefore, in this letter we have derived the RGE's for the effective dimension 5 operators which yield a Majorana mass for neutrinos after the electroweak symmetry breaking in four types of 2HDM's and in the MSSM. For the MSSM, we confirmed the earlier result [9,7]. However, when we applied our general result (11) to the 2HDM discussed in [7], we found that the non-diagonal part of one of the β-functions, relevant for the evolution of the mixing angles, is enhanced by a factor of 3.