Baryon Number, Lepton Number, and Operator Dimension in the Standard Model

We prove that for a given operator in the Standard Model (SM) with baryon number B and lepton number L, that the operator's dimension is even (odd) if (B-L)/2 is even (odd). Consequently, this establishes the veracity of statements that were long observed or expected to be true, but not proven, e.g., operators with B-L=0 are of even dimension, B-L must be an even number, etc. These results remain true even if the SM is augmented by any number of right-handed neutrinos with L=1.

though it is not absolutely necessary, to explain the observation that neutrinos have nonzero mass. When these building blocks are put together to form an operator of any mass dimension in the SM, one must conserve Lorentz invariance and ensure that the operator transforms trivially under the SU (3) c ⊗ SU (2) L ⊗ U (1) Y group symmetries.
These requirements for an SM operator can be easily imposed by noting that they can be interpreted as global constraints on the number N of fermion fields, Higgs fields, covariant derivatives, and field-strength tensors in the operator. For example, the requirement that the operator is invariant under the U (1) Y weak hypercharge symmetry (hypercharge henceforth) can be stated as 2 Interestingly, the constraints placed on an SM operator by requiring SU (3) c ⊗ SU (2) L invariance are, in fact, contained in Eq. (1). To see how, note that because all N 's must be a positive integer, and the last three terms in Eq. (1) sum to an integer, it immediately follows that the sum of the first three terms in Eq. (1) must also be an integer, which can be rewritten as and it immediately follows that Eq. (3) is exactly the requirement that the effective operator is invariant under SU (3) c transformations. Likewise, we can begin again with Eq. (1), and rewrite it as Since the last three terms in Eq. (4) must sum to an even integer, the sum of the first three terms must also sum to an even integer. To show this will always hold, here we can note that the requirement of invariance under the SU (2) L transformations is no more than the requirement that there are an even number of SU (2) L doublets: Since an even number of fields can be added to both sides of Eq. (5), it implies that should also hold. Therefore, Eq. (1) also contains the SU (2) L constraint. This shows that the hypercharge assignments for the fields in the SM contain all the necessary information needed when determining whether a given operator is a SM singlet under SU (3) c ⊗ SU (2) L ⊗ U (1) Y group transformations. Thus, we only make use of only the U (1) Y hypercharge constraint to ensure gauge invariance. We now turn to the discussion of Lorentz invariance. If there are no σ µ matrices in the operator, then the number of right-and left-handed fields must each be even in order for the operator to be Lorentz invariant. Because there must be at least one right-and left-handed field each for every σ µ matrix in the operator to form a Lorentz singlet, it implies that if there are an even (odd) number of σ µ matrices in the operator, then there are an even (odd) number of right-handed fields and an even (odd) number of left-handed fields. Additionally, we can note that the only way to contract a Lorentz index associated with a derivative is to contract it with either a σ µ matrix or a field strength tensor. Because a σ µ matrix has one Lorentz index and a field strength tensor has two, an even (odd) number of Lorentz indices associated with derivatives must correspond to an even (odd) number of σ µ matrices in the operator. If the operator has an odd number of derivatives, then the operator must have an odd number of σ µ matrices. Therefore, from these observations, Lorentz invariance can be stated as and if N D is even (odd), then ( Here, N D stands for the number of covariant derivatives, and N ν (N ν † ) counts the total number of left-handed antineutrinos (right-handed neutrinos). Together, the constraint of hypercharge invariance, Eq. (1), and Lorentz invariance, Eqs. (7) and (8), constitute the necessary and sufficient conditions for operators in the SM. In order to discuss any relationships between ∆B, ∆L, and the operator mass dimension d, we begin by defining d as Here, N X indicates the number of field-strength tensors, i.e., B µν , W µν , or G µν . Given the definitions of baryon number ∆B and lepton number ∆L, we know from its definition that ∆L is an integer (∆L ∈ Z), and from Eq. (3) that hypercharge invariance implies that ∆B must be an integer (∆B ∈ Z). From the definition of d in Eq. (9) and the fact that ∆B ∈ Z, we can note that for an operator with a given value of ∆B and ∆L, the minimum value of its mass dimension d min is Eq. (12) is a weak lower bound, but is an exact equality when the operator contains only fermionic matter fields and when ∆B or ∆L are nonzero. While it is commonly noted that there is a value of d min for a given value of ∆B and ∆L, e.g., when ∆B = 0 and |∆L| = 2, d min = 3 (5) if there are (no) right-handed neutrinos [1], it is not trivial to determine whether or not there is an analytical function for d min as a function of only ∆B and ∆L. Finding such a solution would require solving a discrete optimization problem, but with 16 free parameters subject to a few constraints, i.e., hypercharge invariance, Lorentz invariance, and definitions of ∆B and ∆L, it remains an open question whether there exists an analytical solution. Since one may not be able to derive a closed analytical expression for d min , the next best thing may be to determine whether d is even or odd, given only values of ∆B and ∆L. Inserting the constraint of hypercharge invariance in Eq. (1) and the definitions of ∆B and ∆L in Eqs. (11) and (10) into the definition of d in Eq. (9) gives: At this point, determining whether d is even or odd requires determining whether the number is even or odd. The requirement of Lorentz invariance in Eq. (8) proves, in fact, that Eq. (14) is always even. Because d is a positive integer (d ∈ N), one therefore can conclude: Or, equivalently, These statements can be further summarized succinctly by the following statements: There are some useful takeaways, that are direct consequences from Eq. (12) and Eqs. (15) -(18): • |L| = 2 operators responsible for Majorana neutrino masses occur at odd mass dimension. This was observed to be true in Refs. [8,9,12] for those operators up to and including d = 11, but not including those that contain derivatives, gauge bosons, nor non-trivial Lorentz structure.
• Operators in the SM with odd mass dimension have either nonzero ∆B or ∆L. This was proven in Ref. [11] and observed to be true when d = 7 in Refs. [15,16] for the SM without right-handed neutrinos.
• If an operator comprised of only N f number of fermion fields, where N f /2 is odd, then it has nonzero (∆B − ∆L). This was proven in Ref. [6].
• Nucleon decays where ∆B = −∆L can be described by operators with odd dimension. This was verified for d = 7 [3,4].
• An operator with even (odd) ∆B must also have even (odd) ∆L.
• Neither the value of (∆B − ∆L) nor (∆B + ∆L) can be an odd number for any operator in the SM.
To reiterate, these results are a direct consequence of only two assumptions: hypercharge invariance and Lorentz invariance, and remain true if any number of right-handed neutrinos augments the SM. Similar results were found for the small subset of SM operators that contain no covariant derivatives, no field strength tensors, and have trivial Lorentz structure, i.e., no σ µ matrices [14]. These results only apply to extensions of the SM that utilize the SM particle content within operators invariant under the SM gauge symmetries, i.e., effective operators. Of course, there many instances of model building beyond the SM that do not fall into this category, e.g., if the SM is expanded in a way that (1) introduces new particles that are charged under the SM gauge symmetries, (2) the SM gauge symmetries are embedded within a larger group, (3) contain particles with non-trivial ∆B or ∆L assignments, and the list goes on. These types of possibilities are necessary to consider if one wishes to build a renormalizable model that can give rise to ∆B-or ∆L-violating processes.