CCP -Origins-2019-1 DNRF90 Weyl Consistency Conditions and γ C. Poole ∗ and A. E. Thomsen † CP -Origins, University of Southern Denmark,Campusvej 55, DK-5230 Odense M, Denmark Abstract
The treatment of γ in Dimensional Regularization leads to ambiguities in field-theoretic calculations, of which one example is the coefficient of a particular termin the four-loop gauge β -functions of the Standard Model. Using Weyl ConsistencyConditions, we present a scheme-independent relation between the coefficient of thisterm and a corresponding term in the three-loop Yukawa β -functions, where a semi-na¨ıve treatment of γ is sufficient, thereby fixing this ambiguity. We briefly outlinean argument by which the same method fixes similar ambiguities at higher orders. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] J u l Introduction
The treatment of γ in Dimensional Regularization is a well-known theoretical issue [1],and can be summarized in the following statement: given a four-dimensional, Poincar´e-invariant quantum field theory, there is no gauge-invariant regularization method thatpreserves chiral symmetry [2]. The precise connection is most easily demonstrated usingthe ABJ anomaly, the derivation of which requirestr [ γ µ γ ν γ ρ γ σ γ ] = 4 i(cid:15) µνρσ , (cid:15) = − (cid:15) = 1 (1.1)in four dimensions, whereas the d -dimensional γ -matrix algebra { γ µ , γ ν } = 2 g µν , g µν g µν = d, { γ µ , γ } = 0 (1.2)combined with trace-cyclicity directly impliestr [ γ µ γ ν γ ρ γ σ γ ] = 0 (1.3)even when d →
4. The correct trace-relation may be obtained by using the ’t Hooft-Veltman algebra, which decomposes the d -dimensional space into a four-dimensional sub-space containing γ µ = ˆ γ µ , µ ∈ { , , , } and an orthogonal subspace containing γ µ =¯ γ µ , µ ∈ { , . . . , d − } . In this case, if we define the anticommutation relation as { γ µ , γ } = (cid:40) , µ ∈ , , , γ µ γ , otherwise , γ ≡ i (cid:15) µνρσ ˆ γ µ ˆ γ ν ˆ γ ρ ˆ γ σ (1.4)then trace-cyclicity is consistent with (1.1). However, this method forces the fermionpropagator to take its four-dimensional form for all d , hence fermion loops cannot be regu-larized; attempting to use the d -dimensional propagator in loop integrals is then equivalentto adding an additional term δ L = ¯ ψi ¯ γ µ ∂ µ ψ to the Lagrangian density, explicitly breakinggauge invariance [2].Thus, if one wishes to renormalize a gauge theory with chiral fermions, one must sacrificeeither cyclicity of the trace over Dirac matrices involving γ , or break gauge invariance atintermediate stages of a calculation in perturbation theory. The former option is preferablefor the purpose of calculating higher-order perturbative corrections, but will inevitably giverise to ambiguities in loop integrals stemming from the precise location of γ in the Diractraces. 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, dueto the cancellation of the ABJ anomaly. Furthermore, for the Yukawa couplings, one canuse a “semi-na¨ıve” treatment of γ ,tr [ γ µ γ ν γ ρ γ σ γ ] = 4 i ˜ (cid:15) µνρσ + O ( (cid:15) ) , ˜ (cid:15) µνρσ ˜ (cid:15) αβγδ = g [ µ [ α g νβ g ργ g σ ] δ ] , ˜ (cid:15) µνρσ d → −−→ (cid:15) µνρσ (1.5)in order to show that the resulting ambiguity in the relevant Feynman integral is O ( (cid:15) ),and hence cannot affect the Yukawa β -function [5]. Unfortunately, such minor miracles1o longer hold at four loops; by parametrizing the integrals according to the “readingpoint” of the traces, the resulting ambiguity in the four-loop strong-coupling β -function, β (4) a S , has been explicitly calculated [7, 8]. In the conventions of [7], given rescaled couplings(16 π ) a = { g S , y t } and β -functions defined by d a S d ln µ = β a S a S , one finds β (4) a S ⊃ R (cid:18)
163 + 32 ζ (cid:19) T F a S a t , R ? = 1 , , or 3 . (1.6)While the pursuit of higher-order loop calculations has motivated many significant com-putational developments, there have also been notable advances in our understanding ofrenormalization 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 extends atheory to curved spacetime and local couplings, then the Wess-Zumino consistency condi-tions for the trace anomaly imply a plethora of relations between various RG quantities,amongst them Osborn’s equation ∂ I ˜ A ≡ ∂ ˜ A∂g I = T IJ β J , (1.7)where g I labels the marginal couplings of the theory. This equation therefore demonstratesthe existence of a scalar function, ˜ A , of the couplings in a general renormalizable theory,which places constraints on the corresponding β -functions – T IJ is simply a function ofthe couplings, and the form of its perturbative expansion is fixed by (1.7). Central tothese constraints is the “3–2–1” phenomenon, where the gauge β -function is related tothe Yukawa β -function one loop below, and the scalar β -function two loops below. Thereason for this ordering is topological, and is thus manifestly preserved to all orders; conse-quently, given enough information at lower orders, one can use (1.7) to predict coefficientsof terms at higher orders. Most importantly, the β -functions in (1.7) are precisely the four-dimensional functions that one should obtain after taking the (cid:15) → β (4) a S to lower-order β -function coefficients, and if the con-sistency condition is simple enough, then it may be possible to fix the ambiguity inherentin the treatment of γ .This paper is essentially a companion piece to [16], which contains a full and detailedanalysis of the β -function constraints imposed by Osborn’s equation, with all non-trivialmodifications, up to order ˜ A (5) . We begin with a summary of the Lagrangian densityfor a general, renormalizable, four-dimensional theory, then introduce our diagrammaticnotation for the associated tensor structures appearing in (1.7). We then quickly re-derivethe constraints on γ contributions, using a topological shortcut. Finally, we use [6] to This equation is not generally known by any set name, but one of us (CP) is fed up of using phrasessuch as the technically incorrect “gradient-flow equation”, the correct-but-cumbersome “gradient-flow-likeequation”, and the frankly horrific “equation defining the four-dimensional perturbative a -function”. Asthe power of this equation is only now being realized, we feel it appropriate that its author be suitablyrecognised. MS coefficients for all terms in the general 3-loop Yukawa β -function thatinvolve γ ; the constraints then uniquely determine all γ contributions to the general4-loop gauge β -function, from which we extract the unique, consistent value for R in (1.6). In order to derive constraints on the four-loop gauge β -function, one must construct ˜ A atfive 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 fromthe integrals involving γ , one must work with a completely general theory, described interms of tensor couplings between arbitrary multiplets of matter fields. The most general,renormalizable, four-dimensional Lagrangian density, based on a compact gauge group G = × u G u with any number of Abelian and non-Abelian factors, can be written as L = − (cid:88) u F A u u,µν F A u µνu + ( D µ φ ) a ( D µ φ ) a + i ¯ ψ i ¯ σ µ ( D µ ψ ) i − (cid:0) Y aij φ a ψ i ψ j + ¯ Y aij φ a ¯ ψ i ¯ ψ j (cid:1) − λ abcd φ a φ b φ c φ d + mass terms + relevant operators + gauge-fixing + ghost terms (2.1)with covariant derivatives D µ φ a = ∂ µ φ a − i (cid:88) u g u V A u u,µ ( T A u φ,u ) ab φ b , D µ ψ i = ∂ µ ψ i − i (cid:88) u g u V A u u,µ ( T A u ψ,u ) ij ψ j . (2.2)The fermions transform under a representation R u of the corresponding gauge group G u ,with Hermitian generators ( T A u ψ,u ) † = ( T A u ψ,u ); likewise, the scalars transform under a realrepresentation S u with antisymmetric, Hermitian generators ( T A u φ,u ) T = − ( T A u φ,u ).For our purposes, it proves convenient to assemble the Yukawa couplings and fermiongenerators into larger matrices, y a = (cid:18) Y a Y ∗ a (cid:19) , T A u u = (cid:18) T A u ψ,u − ( T A u ψ,u ) ∗ (cid:19) , ˜ y a = σ y a σ = (cid:18) Y ∗ a Y a (cid:19) , ˜ T A = σ T A σ = − ( T A ) T , (2.3)so that there is a single Yukawa interaction between Majorana-like spinors Ψ i = (cid:0) ψ ¯ ψ (cid:1) .Similarly, by arranging G such that the first n factors are Abelian, we may define gaugefield multiplets A Aµ with a generalized adjoint index A : A ∈ { ( u, A u ) : A u ≤ d ( G u ) } with summation convention (cid:88) A = (cid:88) u d ( G u ) (cid:88) A u =1 . (2.4)3he gauge couplings may then be assembled into a block-diagonal matrix, G AB = (cid:40) h uv for A, B ≤ ng u δ uv δ A u B v for A > n , (2.5)where h uv is a symmetric n × n matrix of U (1) couplings, allowing us to incorporatethe effects of kinetic mixing ; generalized group Casimirs [ C ( G )] AB and Dynkin indices[ S ( R )] AB , R ∈ { F, S } may defined analogously. The general Lagrangian density (2.1) maynow be re-written in the form L = − G − AB F Aµν F Bµν + ( D µ φ ) a ( D µ φ ) a + i Ψ T (cid:18) σ µ ¯ σ µ (cid:19) D µ Ψ − φ a Ψ T y a Ψ − λ abcd φ a φ b φ c φ d + . . . (2.6)with general tensor couplings g I = { G AB , y aij , λ abcd } , and the associated β -functions definedaccording to β I = d g I d ln µ .When evaluating loop integrals, the Feynman rules for each diagram produce contrac-tions between the various tensor couplings and group factors, which we refer to as TensorStructures (TSs); the β -functions may therefore be expressed as a sum over particular TSs,each weighted by a coefficient . To maintain gauge invariance of a theory with multipleinteractions, the generators and tensor couplings must satisfy0 = − ˜ T A y a + y a T A + y b ( T Aφ ) ba , T Aφ ) ae λ ebcd + ( T Aφ ) be λ aecd + ( T Aφ ) ce λ abed + ( T Aφ ) ad λ abce . (2.7)Thus, there will be non-trivial relations between TSs, and one must reduce the full setof TSs in each β -function to a basis. Once done, the TSs may be represented using aconvenient diagrammatic notation, based on the following identifications: A B = G AB , i j = δ ij , a b = δ ab , i ja = y aij , i ja = ( σ y a ) ij , ab cd = λ abcd , i jA = ( T A ) ij , a bA = ( T Aφ ) ab , AB C = G − AD f DBC . (2.8)Contracted indices in TSs (with summation implied) are then represented by lines con-necting different vertices. Since the Feynman rules for (2.6) introduce a σ between twoYukawa couplings, TSs implicitly alternate between y a and ˜ y a along a fermion line. Note The effect of kinetic mixing on the β -functions was first derived in [13], and was reinterpreted in termsof a unified matrix coupling in [14]. Scheme-dependence of the β -functions then simply corresponds to changes in these coefficients. γ contributions givethe opposite sign. To accommodate this, we simply insert a σ (˜ σ ) factor along with y a (˜ y a ),representing such insertions with a blob on the Yukawa vertex.The two main advantage of this representation are that it becomes substantially easierto represent and visualize β -functions at higher loop-orders, and that the notation may beextended to all other quantities appearing in (1.7). A simplified version of this notationwas used in [10] to construct ˜ A at four loops for a theory with a simple gauge group, and toderive the associated consistency conditions. Taking all this into account, the β -functionsof the general theory described by (2.6) may be expanded in the form β AB = d G AB d ln µ = 12 (cid:88) perm (cid:88) (cid:96) G AC β ( (cid:96) ) CD (4 π ) (cid:96) G DB , β aij = d y aij d ln µ = 12 (cid:88) perm (cid:88) (cid:96) β ( (cid:96) ) aij (4 π ) (cid:96) , (2.9)with associated (cid:96) -loop diagrammatic representations given by β ( (cid:96) ) AB = (cid:88) n g ( (cid:96) ) n A B ( ‘, n ) , β ( (cid:96) ) aij = (cid:88) n y ( (cid:96) ) n i ja ( ‘, n ) . (2.10)The blobs represent a suitable basis of TSs with the correct external legs, constructed usingthe identifications in (2.8), each multiplied by a coefficient. To construct ˜ A in Osborn’s equation, it is more convenient to work with an equivalentstatement, obtained by multiplying both sides of (1.7) by d g I :d ˜ A ≡ d g I ∂ I ˜ A = d g I T IJ β J . (3.1)Since ˜ A is a scalar function of the couplings, its contributions may be represented bytotally contracted TSs, and its total derivative should consist of contractions between thedifferentials d g I and the β -functions β J . The (cid:96) -loop contributions to ˜ A are therefore givenby all possible (cid:96) -loop contractions between d g I and the (cid:96) -loop β J , such that (cid:96) = (cid:96) + (cid:96) ;each possible contraction represents a term in the tensor T IJ . At leading order in eachtensor coupling, and using the same conventions as [16], we have T (1) GG = T (1) gg G − AC G − BD , T (2) yy = T (2) yy δ ab δ ik δ jl . (3.2)To identify β -function TSs from γ contributions, one need only consider the possibleLorentz indices that could lead to two fully contracted (cid:15) -tensors. Of all possible diagramsat this order, only those with two fermion lines containing at least two gauge generators andone Yukawa tensor will contribute: all other possibilities vanish by anomaly cancellation.5 (4)199 g (4)200 g (4)201 g (4)202 y (3)304 y (3)305 y (3)306 y (3)307 y (3)308 A (5)258 A (5)259 A (5)260 A (5)261 Figure 1: Complete set of Tensor Structures (TSs) related to the first non-trivial contri-butions involving γ , taken from [16]. The first two rows are contributions to β (4) AB and β (3) aij respectively, while the last row gives the 4 corresponding TSs in ˜ A (5) . The blobs onYukawa vertices symbolize σ insertions.A convenient basis for the relevant diagrams occurring in β (3) aij and β (4) AB , plus the associatedterms in ˜ A (5) , is given in Fig. 1.It is at this point that certain special features of these terms become obvious. Inprinciple, ˜ A (5) receives additional contributions from inserting lower-loop β -functions intohigher-loop T IJ contractions. However, the contributions ˜ A (5)258 − are topologically equiv-alent to cubes and M¨obius-ladders: both graphs are simple and vertex-transitive, thusthe only way each tensor could receive additional contributions would be if the β -functiongraphs contained subgraphs that also appeared at lower loops. As each β -function graphis primitive, this is clearly not the case, hence there are no other possible contributions tothese tensors. Substituting the terms from Fig. 1 into (1.7) therefore gives2 A (5)258 = T (1) gg, g (4)199 , A (5)259 = T (1) gg, g (4)200 , A (5)260 = T (1) gg, g (4)201 , A (5)261 = T (1) gg, g (4)202 , A (5)258 = T (1) yy, y (3)304 , A (5)259 = T (1) yy, y (3)305 , A (5)260 = T (1) yy, y (3)306 , A (5)261 = T (1) yy, y (3)307 , (3.3)2 A (5)261 = T (1) yy, y (3)308 . A (5) coefficients and substituting in the (scheme-independent) T IJ coeffi-cients [11] then leaves five consistency conditions, completely determining the γ contribu-tions to β (4) AB : g (4)199 = 16 y (3)304 , g (4)200 = 16 y (3)305 , g (4)201 = 16 y (3)306 , g (4)202 = 13 y (3)307 , y (3)307 = y (3)308 . (3.4) β -functions and γ γ , and hold for a completely general renormalizable theory witha compact gauge group. The Standard Model is, of course, precisely such a theory, so wemay deduce y (3)304 − by substituting the SM couplings into a basis of relevant β (3) aij TSs ,and comparing with the known SM results of [6]. It is sufficient to focus on TSs containingfour powers of the gauge couplings and a trace over two Yukawa matrices; the comparisonmay be done using either Appendix D of [3], or the general methods of [15]. Our finalresult is y (3)304 = − , y (3)305 = − , y (3)306 = y (3)307 = y (3)308 = 8 − ζ , (4.1)hence equation (3.4) requires that g (4)199 = − , g (4)200 = − , g (4)201 = − ζ , g (4)202 = − ζ , (4.2)as well as confirming that the final condition, y (3)307 = y (3)308 , is indeed satisfied. Substitutingthe SM couplings into the β (4) AB TSs, multiplying by the coefficients in (4.2), and convertingto the conventions of [7] then gives β (4) a S ⊃ (16 + 96 ζ ) T F a S a t . (4.3)Thus, comparison with (1.6) forces the choice R = 3 (4.4)in the β (4) a S calculation of [7, 8], corresponding to a reading whereby one cuts the traces atany of the internal vertices. While [7] gave some theoretical justifications for preferring thisvalue of R , we believe this constitutes the first proof that it must be so; furthermore, ourresults determine all γ contributions to any gauge β -function at four loops, including allthree SM gauge couplings with full matter content. We stress that there is no wiggle-roomin the conclusion: (3.4) relates the final β -function coefficients after removal of the regu-lator, and holds for all perturbative renormalization schemes, thus the four-loop integralinvolving γ must be treated in this manner. The γ contributions to β (3) aij are listed in Fig. 1, and the full basis of β (3) aij TSs is given in the ancillaryfile of [16]. T IJ contributions influencethe consistency conditions can easily be extended to higher loops: if the tensor structurein ˜ A ( n ) is topologically equivalent to a connected symmetric graph , and the associatedprimitive tensors in β ( n − AB , β ( n − aij and/or β ( n − abcd contain non-trivial contributions from γ , then one can quickly derive an analogous consistency condition to fix the potentialambiguity, as parametrized by the same trace-cutting procedure used at four loops. It mayof course be possible that, at higher orders, γ contributions also appear in more complexconditions than those similar to (3.4). If this is so, it is still possible to use the full setof consistency conditions to infer a consistent treatment, although the amount of workrequired will be dramatically increased. Acknowledgements
We are very grateful to Florian Herren for useful discussions, helping clarify the treatmentof γ in general theories. CP would like to thank Joshua Davies for his off-the-cuff questionthat eventually led to the result in this paper, and Ian Jack for his careful reading ofthe manuscript. AET would like to thank Fermi National Accelerator Lab for hostinghim during the completion of this paper, and gratefully acknowledges financial supportfrom the Danish Ministry of Higher Education and Science through an EliteForsk TravelGrant. This work is partially supported by the Danish National Research Foundation grantDNRF90. References [1] G. ’t Hooft and M. Veltman,
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