Renormalization group evolution of dimension-six baryon number violating operators

We calculate the one-loop anomalous dimension matrix for the dimension-six baryon number violating operators of the Standard Model effective field theory, including right-handed neutrino fields. We discuss the flavor structure of the renormalization group evolution in the contexts of minimal flavor violation and unification.


I. INTRODUCTION
The baryon asymmetry of the universe hints at baryon number violating (BNV) interactions beyond the Standard Model (SM) of particle physics. Baryon number is an accidental symmetry of the SM violated by quantum effects [1], and there is no fundamental reason why it cannot be violated in extensions of the SM. Indeed, well-motivated theories like grand unified theories [2][3][4] violate baryon number at tree level through the exchange of very massive gauge bosons.
There has been no direct experimental observation of baryon number violation to date. The large lower bound for the lifetime of the proton [5,6] requires that the scale of baryon number violation M / B be much greater than accessible energy scales, and, in particular, much greater than the SM electroweak scale M Z . The decay of baryons (such as the proton) can then be computed using an Effective Field Theory (EFT) formalism. In the modelindependent treatment of EFT, the SM Lagrangian is extended by higher dimensional non-renormalizable operators (d ≥ 5) suppressed by inverse powers of the new physics scale.
The leading order BNV operators arise at dimension d = 6. The most general dimension-six Lagrangian can be cast in 63 independent operators [7][8][9][10][11]. Out of these 63 operators, 59 operators preserve baryon number, and the complete set of one-loop renormalization group equations for these 59 operators was recently computed in Refs. [12][13][14][15]. In the present work, we focus on the four BNV operators [9][10][11], and we extend the one-loop renormalization group evolution (RGE) analysis to these remaining dimension-six operators.
The four BNV operators can be written 1 as [11] Q duqℓ prst = ǫ αβγ ǫ ij (d α p Cu β r )(q iγ s Cℓ j t ) , where C is the Dirac matrix of charge conjugation, q and ℓ are the quark and lepton left-handed doublets, and we 1 The connection with the basis of Ref. [9] is given in Appendix A.
use u, d and e for up-type, down-type, and charged lepton right-handed fermions. Greek letters denote SU (3) c color indices and Roman letters from i to l refer to SU (2) L indices. Roman letters towards the end of the alphabet p-w refer to flavor (generation) indices and take on values from 1, . . . , n g = 3.
In this work, we also will accommodate neutrino masses for the light neutrinos by including singlet fermions N (right-handed neutrinos) under the SM gauge group. Including singlet N fields, two additional dimension-six BNV operators can be constructed: The singlet neutrinos N , in contrast to the SM fermions, are allowed a Majorana mass M N by the SM gauge symmetry. M N can range from a very high scale as in the standard type-I seesaw model [16][17][18][19] to the Dirac neutrino limit for which it vanishes -see Ref. [20] for a general parametrization in terms of light masses and mixing angles. Even in the case of a very high Majorana mass scale M N , naïve estimates of proton decay and light neutrino masses imply that M N < M / B . This hierarchy of scales implies that an EFT with the operators in Eq. (2) holds in the energy regime M N < µ < M / B . Below the scale M N , one integrates out the N fields, matching onto the EFT containing only the four operators of Eq. (1), and drops the terms of Eq. (2) in the renormalization group equations.
We will use the conventions of Ref. [12], generalized to include singlet fermions N at energies above M N . Specifically, for µ > M N , the L d≤4 SM Lagrangian includes a Majorana mass term M N for the N fermions as well as Yukawa couplings Y N for the N and ℓ fermions to the electroweak Higgs doublet H. For µ < M N , the N fields are integrated out of the EFT, and L d≤4 reduces to the conventional SM Lagrangian.
Baryon number is an (anomalous) symmetry that is preserved by the one-loop renormalization group equations, so the dimension-six BNV operators only mix among themselves. The gauge contribution to the anomalous dimensions of Eq. (1) was computed in Ref. [11], and we agree with those results. In addition, we compute the anomalous dimensions of Eq. (2), and the Yukawa terms. We also classify the operators in terms of representations of the permutation group, which diago- nalizes the gauge contributions to the anomalous dimension matrix.

II. RESULTS
The one-loop anomalous dimension matrix of the BNV operators decomposes into a sum of gauge and Yukawa terms. The gauge anomalous dimension matrix of the operators in Eq. (1) was computed in Ref. [11]. The gauge terms for Eq. (2) have not been computed previously. The Yukawa terms are generated by the diagram in Fig. 1, where all the fermion lines are incoming, because of the chiral structure of the BNV operators. The gauge coupling dependence is obtained from an analogous diagram with the scalar replaced by a gauge boson.
The calculation is done using dimensional regularization in d = 4 − 2ǫ dimensions in a general ξ gauge. Cancellation of the gauge parameter ξ provides a check on the calculation. The sum of the hypercharges y i of the four fermions for each operator is constrained to be equal to zero for the ξ-dependence to cancel. Furthermore, the number of colors N c = 3 for the operator to be SU (3) gauge invariant. The RGE for the operator coefficients C duue prst = − C duue prst 4g 2 3 − 2 2y d y u + 2y e y u + y 2 u + y e y d g 2 1 + 4C duue psrt (y d + y e )y u − y 2 u − y e y d g 2 A non-trivial check on these equations is provided by the custodial symmetry limit (Y u(N ) → Y d(e) , g 1 → 0). In order to respect the custodial symmetry, the BNV operator coefficients have to satisfy certain relations given in appendix A, and the RGE flow should preserve these relations. Remarkably, the construction of custodial invariant operators is compatible with U (1) Y invariance.
The structure of the anomalous dimensions can be clarified by studying the symmetry properties of the BNV operators. The operators Q qque and Q qqdN are symmetric in the two q indices [11], The operator Q qqqℓ satisfies the relation [11], Q qqqℓ has three q indices, and so transforms like ⊗ ⊗ , which gives one completely symmetric, one completely antisymmetric, and two mixed symmetry tensors. Eq. (10) implies that one of the mixed symmetry tensors vanishes. The allowed representations of the BNV operators are shown in Table I.
The coefficients C duue prst and C uddN prst can be decomposed into the symmetric and antisymmetric combinations, The coefficient C qqqℓ prst can be decomposed into terms with definite symmetry under permutations, where S qqqℓ prst is totally symmetric in (p, r, s), A qqqℓ prst is totally antisymmetric in (p, r, s), and M qqqℓ prst and N qqqℓ prst have mixed symmetry.
A convenient choice of basis is The coefficient M qqqℓ prst is obtained by first antisymmetrizing C qqqℓ prst in (p, r), and then symmetrizing in (p, s). Likewise, N qqqℓ prst is obtained by first antisymmetrizing in (p, s), and then symmetrizing in (p, r). Eq. (10) implies that N qqqℓ prst vanishes. The gauge contributions to the anomalous dimensions respect the flavor symmetry of the operators. With the decomposition Eq. (13), the gauge contribution to the anomalous dimension matrix diagonalizes, The " · · · " refers to the Yukawa contributions, which can mix different permutation representations.

III. DISCUSSION
The renormalization group equations presented here have an involved flavor structure; to better understand the generic features, we turn now to certain simplifying hypotheses and models that produce a simple subclass of BNV operators.   (2), for a total of 408 ∆B = 1 operators with complex coefficients. One coefficient can be made real by a phase rotation of fields proportional to baryon number.

A. Minimal Flavor Violation
The SM has an SU (3) 5 flavor symmetry for the q, u, d, l, and e fields, broken only by the Higgs Yukawa interactions. The symmetry is preserved if we promote the Yukawa coupling matrices to spurions that transform appropriately under the flavor group. Minimal flavor violation (MFV) [21,22] is the hypothesis that any new physics beyond the SM preserves this symmetry, so the Yukawa coupling matrices are the only spurions.
Dimension-six BNV operators do not satisfy naïve minimal flavor violation because of triality. The argument proceeds as follows: under every SU (3) i flavor transformation, each BNV operator transforms as a representation of SU (3) i with n i upper indices and m i lower indices. All BNV operators satisfy No combination of Yukawa matrices (or other invariant tensors) can change this into a singlet, as they all have (n − m) ≡ 0 (mod 3).
In extensions of the MFV hypothesis to account for massive neutrinos [23][24][25], a Majorana mass term introduces a spurion with (n − m) ≡ 2 (mod 3). This in turn allows for the implementation of MFV, as pointed out in Ref. [26]. Note also that if the Yukawa spurions are built out of objects with simpler flavor-transformation properties [27], a variant of minimal flavor violation is possible without Lepton number violation.
Finally, there is the possibility that the fermion fields do not each separately have an SU (3) flavor symmetry, but that some transform simultaneously [28]. The latter is an attractive option that is realized in Grand Unified Theories (GUTs), and we explore this possibility in the next subsection.

B. Grand Unified Theories
The Georgi-Glashow SU (5) theory [2] places u c , q, and e c in a 10 representation of SU (5), and d c and l in a 5. In the context of the type-I seesaw, N is a 1. The flavor group in this case cannot be that of putative MFV since the fields in each SU (5) representation must transform simultaneously. The flavor symmetry is instead SU (3) 3 = SU (3) 10 ⊗ SU (3)5 ⊗ SU (3) 1 , where each SU (3) stands for transformations in flavor space of the corresponding SU (5) representation [28]. The fermions and spurions then fall into the representations where the right-handed neutrino Majorana mass M N also needs to be promoted to a spurion. Note that the triality argument given previously does not apply to the Yukawa matrices in this scenario. With the SU (5) GUT in mind, we will relabel the Yukawas Y u → Y 10 , (Y d , Y T e ) → Y 5 , and Y N → Y 1 .

The operators transform as
which now can be combined with Yukawa couplings to build up invariant terms in the Lagrangian. Explicitly, the coefficients of the operators in terms of Yukawa matrices up to second order are Notice that only C duqℓ and C qque can be constructed out of flavor singlets. These are the only two operators that can be generated by integrating out heavy gauge bosons in the context of SU (5) or, in general, by flavorblind SU (5) invariant dynamics. In addition, these are the only two coefficients that remain in the limit To close this section, let us comment on the implications for supersymmetric GUTs in our framework. BNV dimension-five operators are produced by integrating out GUT particles in supersymmetric theories in the absence of selection rules like R-parity [29][30][31]. Below the supersymmetry breaking scale, these will translate into the operators Q qqqℓ , Q duue and Q uddN in terms of the SM EFT Lagrangian, being only suppressed by one power of the BNV scale: 1/(M / B M SUSY ). A feature of this scenario is that, as a result of the supersymmetric origin of the operators, all diagonal entries in flavor vanish [30], so that proton decay would require a strange particle. The renormalization group equations presented here only apply in the regime µ < M SUSY since they depend on the spectrum of the theory, and we have assumed only dynamical SM particles. See Ref. [32] for a RGE study of BNV effects in the context of supersymmetry.

C. Magnitude of Effects
In this subsection, we simplify the RGE to estimate the magnitude of running a BNV operator coefficient from the GUT scale to the electroweak scale. Working in the context of a MFV GUT discussed in Sec. III B, we set Y d = Y e = Y N = 0, assuming top-Yukawa dominance.
In that limit, the only two non-vanishing operators are Q duqℓ prst and Q qque prst , whose RGE equations decouple. The coefficients of these two operators are given by appropriate combinations of Y 10 which transforms as the symmetric representation,6.
As an example, we focus on Q duqℓ prst , whose coefficient takes on a simple form: and f (0) rs ∝ δ rs . The RGE of this coefficient becomeṡ We can now choose the basis Y 10 = Y u = diag(0, 0, y t ), where y t is the top-quark Yukawa coupling and lighter up-type quark masses are neglected. With this simplification, C duqℓ rs is a diagonal matrix. Setting M GUT ≈ 10 15 GeV, the C duqℓ coefficients at the electroweak and GUT scales are related by The first factor in parentheses comes from the gauge contribution alone, is dominated by the QCD coupling, and is common to all flavor coefficients. The second factor is the extra correction from including the Yukawa contribution, with only the top entry sizeable. Whereas the gauge contribution to the RGE enhances the C duqℓ rs coefficient at lower energy scales, the Yukawa contribution gives a small suppression.
The Yukawa-induced running will in general be negligible for the lightest generation coefficients and processes like proton or neutron decay are unaffected. The Yukawa running gives a small correction for heavier generations. Note that the relatively small correction from Yukawa running compared to gauge-induced running stems from the different numerical coefficients of the anomalous dimension, since g 3 ∼ y t . For example, in Eq. (19), the color and SU (2) L gauge contributions have each a prefactor ∼ 8 times that of the Yukawas. These numerical factors cannot be estimated and require the explicit computation presented here.
The Yukawa running studied in this section have the most impact in heavy flavor BNV transitions, which are searched for experimentally [33,34]. In this regard, the fact that W boson exchange below the electroweak symmetry-breaking scale produces flavor mixing is relevant. In particular, at two-loop order, proton or neutron decay is sensitive to BNV operators with arbitrary flavor. Even though a two-loop effect, this places a strong bound on heavy flavor BNV. Discussions of heavy BNV transitions taking into account these effects can be found in Refs. [35][36][37].

IV. CONCLUSIONS
In this letter, we have included the Yukawa contribution to the anomalous dimension matrix of baryon number violating operators and have thus completed the oneloop renormalization group evolution. Together with the computation of Refs. [12][13][14], this completes the anomalous dimension matrix for the totality of dimension-six operators of the SM. We included right-handed neutrinos and therefore two new BNV operators, and classified all the operators under flavor symmetry. None of the operators satisfies SU (3) 5 minimal flavor violation, but it is possible to impose a weaker grand unified theory variant of MFV. The Yukawa coupling corrections only give small corrections to the operator evolution.