Weyl Consistency Conditions and $\gamma_{5}$

The treatment of $\gamma_{5}$ in Dimensional Regularization leads to ambiguities in field-theoretic calculations, of which one example is the coefficient of a particular term in the four-loop gauge $\beta$-functions of the Standard Model. Using Weyl Consistency Conditions, we present a scheme-independent relation between the coefficient of this term and a corresponding term in the three-loop Yukawa $\beta$-functions, where a semi-na\"ive treatment of $\gamma_{5}$ is sufficient, thereby fixing this ambiguity. We briefly outline an argument by which the same method fixes similar ambiguities at higher orders.


Introduction
The treatment of γ 5 in Dimensional Regularization is a well-known theoretical issue [1], and can be summarized in the following statement: given a four-dimensional, Poincaréinvariant quantum field theory, there is no gauge-invariant regularization method that preserves chiral symmetry [2]. The precise connection between the two is most easily demonstrated using the ABJ anomaly, the derivation of which requires tr [γ µ γ ν γ ρ γ σ γ 5 ] = 4iǫ µνρσ , ǫ 0123 = −ǫ 0123 = 1 (1.1) in four dimensions, whereas the d-dimensional γ-matrix algebra {γ µ , γ ν } = 2g µν 1, g µν g µν = d, {γ µ , γ 5 } = 0 (1.2) plus trace-cyclicity directly implies tr [γ µ γ ν γ ρ γ σ γ 5 ] = 0 (1.3) even when d → 4. Thus, if one wishes to renormalize a gauge theory with chiral fermions, one must sacrifice either cyclicity of the trace over Dirac matrices involving γ 5 , or break gauge invariance at intermediate stages of a calculation in perturbation theory. The former option is preferable for the purpose of calculating higher-order perturbative corrections, but will inevitably give rise to ambiguities in loop integrals stemming from the precise location of γ 5 in the Dirac traces. Such ambiguities may appear for the first time at three loops, however the β-functions of the gauge [3] and scalar [4] couplings in the Standard Model are spared, due to the cancellation of the ABJ anomaly. Furthermore, for the Yukawa couplings, one can use a "semi-naïve" treatment of γ 5 , (1.4) in order to show that the resulting ambiguity in the relevant Feynman integral is O(ǫ), and hence cannot affect the Yukawa β-function [5]. Unfortunately, such minor miracles no longer hold at four loops; by parametrizing the integrals according to the position of γ 5 , the resulting ambiguity in the four-loop strong-coupling β-function, β α S , has been explicitly calculated as [7,8] While the pursuit of higher-order loop calculations has motivated many significant computational developments, there have also been notable advances in our understanding of renormalization itself, which are not yet as well-known in the phenomenological community. One such development is the notion of Weyl Consistency Conditions [9]: if one extend a theory to curved spacetime and local couplings, then the Wess-Zumino consistency conditions for the trace anomaly imply a plethora of relations between various RG quantities, amongst them Osborn's equation 1 where g I labels the marginal couplings of the theory. For the purpose of calculation, it is easier to work with an equivalent equation, obtained by multiplying (1.6) by dg I : This equation therefore demonstrates the existence of a function,Ã, of the couplings in a general renormalizable theory, which places constraints on the corresponding β-functions.
Central to these constraints is the "3-2-1" phenomenon, where the gauge β-function is related to the Yukawa β-function one loop below, and the scalar β-function two loops below. The reason for this ordering is topological, and is thus manifestly preserved to all orders; consequently, given enough information at lower orders, one can use (1.6) to predict coefficients of terms at higher orders. Most importantly, the β-functions in (1.6) are precisely the four-dimensional functions that one should obtain after taking the ǫ → 0 limit of Dimensional Regularization. This is the crux of our approach: if there exists a consistency condition relating the ambiguous term in β (4) α S to lower-order β-function coefficients, and if the consistency condition is simple enough, then it may be possible to fix the ambiguity inherent in the treatment of γ 5 .

Constraints from Weyl Consistency Conditions
In order to derive constraints on the four-loop gauge β-function, one must constructÃ at five loops. This is already a somewhat awkward task, but there is a further complication: in order to isolate particular contributions to the β-function, such as those stemming from the integrals involving γ 5 , one must work with a completely general theory, described in terms of tensor couplings between arbitrary multiplets of matter fields 2 . Expressing the matter content as n φ real scalars φ a and n ψ Weyl fermions ψ j , the Lagrangian density of a general theory with a semi-simple gauge symmetry group G = G 1 × . . . × G n , containing at most one U(1) factor 3 , is given by where we have assumed that G 1 = U(1). The fermions transform under a representation R α of the corresponding gauge group G α , with Hermitian generators (R α ) aα † ij = (R α ) aα ij , and the scalars likewise transform under a representation S α with antisymmetric, Hermitian generators (S α ) aαT ab = −(S α ) aα ab . When constructingÃ, it proves convenient to assemble the Yukawa couplings and fermion generators into matrices, so that there is a single Yukawa interaction involving Weyl fermions assembled into Majorana spinors Ψ T = ψ T i , (ψ i ) T [10]. Yet another complication is the identities that the gauge generators must satisfy, in order for a theory with scalar and Yukawa interactions to be gauge-invariant: These identities relate various gauge-dependent tensor structures, leading to redundancies; one must therefore reduce the set of tensors in each β-function to a basis. Taking all this into account, the construction ofÃ for a general four-dimensional theory with a single gauge group has been done at four loops, using a diagrammatic representation of the tensor couplings to express the β-functions as a sum over tensor structures, each multiplied by a coefficient 4 [10]. The authors extracted scheme-independent relations between the coefficients of β aij , and β g , and used MS results to show how one could deduce many of the coefficients in β (3) g without explicit calculation. By expressing the gauge couplings as entries in a diagonal matrix G k αβ ≡ diag g k 1 , . . . , g k n , repesenting β G as a two-point tensor, and adopting the convention that contracted gauge lines sum over all gauge couplings and associated generators, we have extended the notation of [10] to a general semi-simple gauge group, whereby a dot on a gauge line with label k represents the new coupling matrix G k αβ , and have written a bespoke Mathematica procedure to automate the generation of consistency conditions in the same manner [15].
The diagram of interest in β (4) G is given in Fig. 1a, and indeed such a diagram appears in the basis of terms generated by our program. In line with our extended notation, it is easy to see thatÃ will receive a contribution to the diagram in Fig. 1b by contracting β   (4) G with the leading-order tensor T (1) αβ δ γδ , at which point certain special features become obvious. Fig. 1b is in fact topologically equivalent to a cube, hence has no subdiagrams in the form of a subdivergence, and therefore has no other contributions from higher-order terms in T GJ . Similarly, the basis of terms in the general three-loop Yukawa β-function β (3) Y contains the tensor shown in Fig. 1c, soÃ receives the same contribution by contracting β (3) Y with the leading-order tensor T Y δ Y Y , and there are again no possible higherorder contributions from T Y J 5 . Consequently, (1.6) and (1.7) imply that, if b (4) is the coefficient of Fig. 1a and y (3) the coefficient of Fig. 1c, then the coefficient a (5) Using the leading-order calculations of T IJ in [11], we obtain the desired scheme-independent consistency condition y (3) = 12b (4) (2.6) 4 Scheme-dependence of the β-functions then simply corresponds to changes in these coefficients. 5 In [12], the topologically-equivalent case of constructingÃ for six-dimensional φ 3 theory demonstrated the exact same behaviour, whereby the only contribution to A We have, of course, used our program to generate the full set of consistency conditions for a completely general theory, and found precisely this condition [15].
3 Standard Model β-functions and γ 5 The consistency condition (2.6) relates two tensor structures that may receive non-trivial contributions from integrals involving γ 5 , and holds for a completely general renormalizable theory with a semi-simple gauge group. The Standard Model is, of course, precisely such a theory, and so by inserting the SM matter content we may extract relations between various terms in the SM β-functions. As indicated in [3][4][5][6][7][8], the SM matter content is such that integrals involving γ 5 only contribute to these tensors, and so (2.6) directly relates the ambiguous treatment in β (4) α S to the semi-naïve treatment in β aij that contain four generators, a trace over two Yukawa tensors, and an additional untraced Yukawa tensor, we can extract the MS coefficient y (3) by using the results in appendix D of [3] and matching with the SM calculations in [6]: so expanding out the tensor structure in Fig. 1a and multiplying by (3.2) gives By comparing with (1.5), we are therefore forced to take R = 3 (3.4) in the β (4) α S calculation of [7,8], corresponding to a reading of the traces whereby one insert γ 5 at any of the internal vertices. While [7] gave some theoretical justifications for preferring this value of R, we believe this constitutes the first proof that it must be so. We stress that there is no wiggle-room in the conclusion: (2.6) relates the final β-function coefficients after removal of the regulator, and holds for all perturbative renormalization schemes, thus the four-loop integral involving γ 5 must be treated in this manner.
The topological argument guaranteeing that no higher-order T IJ contributions influence the consistency condition can easily be extended to higher loops: if the tensor structure inÃ (n) is topologically equivalent to a connected symmetric graph 6 , and the associated 6 A symmetric graph generally refers to a graph with a set number of edges connected to each vertex, such that the automorphism group acts transitively on both the associated vertex-and edge-graph; a connected symmetric graph is then a symmetric graph with no disconnected vertices or subgraphs. Due to the multiple interaction types, the graph topologies that contribute to the A-function and lead to a simple consistency condition like (2.6) are more general -we are unaware of a classification scheme for all such topologies, but the connected symmetric graphs form a well-defined subset.
primitive tensors in β (n−1) G , β (n−2) aij and/or β (n−3) abcd contain non-trivial contributions from γ 5 , then one can quickly derive an analogous consistency condition to fix the potential ambiguity, as parametrized by the same trace-cutting procedure used at four loops. It may of course be possible that, for a particular theory, γ 5 does not contribute to the terms in these simple conditions. If this is so, it is still possible to use the full set of consistency conditions to infer a consistent treatment, although the amount of work required will be dramatically increased.