Gravity-gauge Anomaly Constraints on the Energy-momentum Tensor

We derive constraints on the four dimensional energy-momentum tensor from gravitational and gauge anomalies. Our work can be considered an extension of Duff's analysis [1] to include parity-odd terms and explicit symmetry breaking. The constraints imply the absence of the parity-odd $R\tilde R$ and $F\tilde F$ terms, for theories whose symmetries are compatible with dimensional regularisation, in a model-independent way. Remarkably, even in the case of explicit symmetry breaking the $\Box R$-anomaly is found to be finite and unambiguous after applying the symmetry constraints. We compute mixed gravity-gauge anomalies at leading order and deduce phenomenological consequences for vector bosons associated with global chiral symmetries.


Introduction
Weyl anomalies in four dimensions have been discovered fifty years ago [2], they are connected to many fields of physics [3] and are still an active field of research.The anomaly appears in the divergence of the dilatation current which is given by the trace of the energy momentum tensor (TEMT) where E, R 2 , W 2 , R R and □R are invariants of the Riemann tensor and F 2 and F F are the standard field strength tensor invariants (all defined in the main text).The coefficients are model-dependent and a non-zero value signals anomalous breaking of Weyl invariance in the case where the theory is classically Weyl invariant.
Constraints on the TEMT were worked out at leading order (LO) in a classic paper by Duff within dimensional regularisation [1].The main goal of our work is to extend it to include: i) the case of parity-odd terms and ii) explicit symmetry breaking.The parity-odd terms consist of the topological Pontryagin densities R R and F F which each contain a single Levi-Civita tensor.If their respective coefficients are real (imaginary) then the terms violate CP (CPT) in addition.There has been a recent controversy as to whether Weyl fermions can give rise to an R R-term.The group of Bonora has found that there ought to be [4][5][6][7][8][9] whereas all other investigations have concluded to the contrary [10][11][12][13][14][15].In our previous work [15] we have analysed the issue via the path integral paying attention to the definition of the Weyl determinant.In this work we derive the vanishing e = 0 from the decomposition of the EMT-counterterms and our results hold for all theories whose symmetries are compatible with dimensional regularisation, which includes the spin-3/2 case (and also spin-2) [16][17][18][19][20][21][22][23][24][25].In addition we consider the parity-even sector with explicit breaking of Weyl symmetry.Constraints are implied by the finiteness of the diffeomorphism (diffeo) and Lorentz anomalies.A priori this is not obvious as one might think that their presence invalidates the theory.However, this is not the case since they can, for example, be seen as low energy effective theories of decoupled chiral fermions in the context of the Higgs mechanism [26,27].On a practical level finiteness of the diffeo, Lorentz and gauge anomalies is due to their topological nature [28,29].In the second part of the paper we include gauge fields and work out diffeo, Lorentz and gauge anomaly constraints which automatically includes mixed anomalies.Where possible we find agreement with the literature but we believe that our diffeo-anomaly constraints are more complete.Phenomenological implications for vector bosons associated with global chiral symmetries are briefly outlined.
The paper is organised as follows.Sec. 2 contains definitions and the derivation of a formula, widely used in the literature, for the Weyl anomaly in the presence of explicit symmetry breaking.Sec. 3 is the central part of this work where all possible terms of the EMT-counterterms are considered and constraints due to the diffeo and the Lorentz anomalies are worked out.In Sec. 4 the analysis is extended to include a gauge sector.Finally, in Sec. 5 we explicitly compute the mixed gravity-gauge anomalies for a Weyl fermion, compare to the literature and deduce phenomenological consequences.The paper ends with summary and conclusions in Sec. 6.Further derivations and constraints of the EMT are deferred to Apps.A and B respectively.

The Weyl Anomaly and (no-)explicit Symmetry Breaking
In Sec.2.1 some basic definitions are given before discussing the Weyl anomaly with and without explicit symmetry breaking in Secs.2.2 and 2.3 respectively.

Preliminary definitions
Let us consider a theory where quantum fields are collectively denoted by ϕ.The metric g µν and the gauge field A µ are considered as external fields, with Weyl transformations In a space with Euclidean signature1 the quantum effective action W is given by The renormalised effective action is defined as the sum of the bare and the counterterm effective action which is finite when the regulator is removed (d → 4 in dimensional regularisation with d = 4 − 2ϵ adopted throughout).The separate pieces W(d) and and the integrand is a local polynomial in the background fields.Note that no □R-term is included at the level of the action since it is a total derivative.The possibility of adding a P and CP -violating term R R will be discussed in Sec. 3.For a classically Weyl invariant theory, the Weyl consistency conditions imply b = 0 [1].The remaining two terms E and C are given by where E(4) reduces to the topological Euler density and C(4) to the Weyl tensor squared.The curvature invariants in non-integer dimension are understood as power series in the metric [33].The (unrenormalised) EMT and its vacuum expectation value (VEV) are given by the metric variation of the (effective) action Above g denotes the metric determinant.The Weyl variation of the action yields the (classical) TEMT when ϕ is on-shell In the literature the Weyl anomaly is often defined as [3,[34][35][36] within dimensional regularisation.The objective of this section is to formally derive this formula accounting for explicit symmetry breaking -an aspect seemingly absent in the literature. 5

Weyl anomaly with classical Weyl invariance
Let us first consider the case of a theory which is classically Weyl invariant: δ W σ S = 0. To define the anomaly from the unrenormalised EMT bears an ambiguity in that the Weyl variation can be performed in four or d dimension: This ambiguity is absent if one directly defines the Weyl anomaly from the renormalised effective action since the finiteness of W ren guarantees that the Weyl variation and the d → 4 limit commute.Applying the first definition results in 6 since classical Weyl invariance implies [1] g (4)   µν δW ct δg µν d=4 = 0 . (2.11) If we apply the second definition we obtain, the expression used in [1], 3 The superscript denotes the dimension in which the metric lives and we have g µν + O(d − 4).When the superscript is omitted d-dimensional tensors are assumed. 4Note that whereas for d = 4 − 2ϵ one has g µν T µν ⟩ = ⟨T ρ ρ ⟩, this no longer holds for g µν . 5A new definition of the anomaly beyond dimensional regularisation has very recently been proposed in [37]. 6This expression of the anomaly was found in [33] in d = 2 − ϵ using a Taylor expansion in ϵ.This assumes continuity of the tensors which may not always hold, as we will discuss in Sec. 3.
by classical Weyl invariance, with more details in App. A. This demonstrates formula (2.8) for a classically Weyl invariant theory and clarifies why the Weyl anomaly can be obtained in different ways.Similar reasonings have been applied in [36] (cf.Sec.1.2.3.) to show that the anomaly can be obtained in both ways.Note that the form of the anomaly is constrained by the Wess-Zumino consistency conditions [38] in a classically Weyl invariant theory, since it is given by the variation of a functional (2.9). 7This aspect will be exploited in Sec.3.2.Anomalies are obtained by the variation of the bare effective action W and since they are finite, the variation of W ct does not enter in general.We will use this invariance of W ct under the diffeo, Lorentz and gauge transformations later on.For the Weyl anomaly, it manifests itself in (2.11).On the other hand, the Weyl anomaly is known to be related to the renormalisation of the theory (e.g [39,40]).This is apparent in the second expression of the Weyl anomaly (2.12), which is finite since the d-dimensional variation produces a term proportional to d − 4.

Weyl anomaly without classical Weyl invariance
We turn to the case of a theory with explicit Weyl symmetry breaking: δ W σ S ̸ = 0.In that case, formula (2.9) is amended 8 to include a term that parametrises the explicit breaking.It is worthwhile to emphasise that both A Weyl and E Weyl are finite quantities.The results in the previous section, that is Eqs.(2.11) and (2.13), will not apply as W ct does not need to consist of Weyl invariant terms.Applying the two definitions one gets (4)  µν ⟨T µν ⟩ + g (4)   µν upon using Eq.(2.3).In each line, both terms are separately divergent but finite in their sum.The main idea is to define E Weyl from the finite part of ⟨T µ µ ⟩ and, using the second line of (2.15), this further implies 9 where the subscript fin (div) indicates that only the finite (divergent) part is taken.Combining with the first line of (2.15) one gets (2.17) 7 The Wess-Zumino consistency conditions for the Weyl transformation have been investigated for a Weyl invariant theory in full generality some time ago [39]. 8Note that (2.14) equals to g If we further use g (4)   µν which follows from the fact that the left-hand side vanishes in a classically Weyl invariant theory, we then get the formula (2.8) which we promised to obtain by derivation.More details are provided in App. A. This formula is valid to all order in perturbation theory in dimensional regularisation and can be expected to be valid non-perturbatively.Note that now the Weyl anomaly is not the variation of a functional anymore, due to the explicit breaking in (2.14).
In that case, the Weyl anomaly does not generally respect the Wess-Zumino consistency conditions.
3 Curved-space Anomaly Constraints on the Weyl Anomaly The goal of this section is to extend the results in [1] to theories without classical Weyl invariance and to consider the possibility of adding the P -and CP -odd term to the counterterm action given in (2.4). 10 Unlike the other operators the extension of R R to d-dimension is ambiguous since the Levi-Civita tensor is intrinsically tied to d = 4.We may however proceed formally, without committing to a specific extension, as follows with L odd (4) ∝ R R as this is the only possible parity-odd term in d = 4 (cf.Ref. [15]).
Note that (3.3) does not imply that L odd (d) is of O(d−4).For example, the extension of the P -even topological Euler density to d dimensions of is discontinuous around d = 4 [42,43].However, we may parametrise the metric variation of W ct,odd in terms of a continuous (d − 4) × V αβ and a discontinuous piece The 2-tensors V αβ (4) and U αβ (d) are parity-odd, of mass dimension four, and one has owing to (3.3).Using Bianchi identities, algebra, and intrinsically 4-dimensional identities [44][45][46] one can show that the only possible operator (cf.App.B.1.1)at d = 4 is where e is a constant to be determined.For our purposes it is not necessary to specify the extension of the ϵ-tensor to d dimensions in parametrising U αβ (this is similar to [47,48] whereby one uses free parameters in order to remain independent of a specific γ 5 -scheme) Using Bianchi identities, which are d-independent, we may write where round brackets denote symmetrisation t (µν) = 1 2 (t µν + t νµ ) and and any other P -odd symmetric 2-tensor is related to these by algebra and Bianchi identities (see App. B).At d = 4, the Schouten identity reduces all these tensors to and thus U αβ (4) = 0 implies the constraint 2 e 1 + e 2 + e 3 = 0 . (3.9) As expected, this defines a tensor that vanishes at d = 4, but does not scale as O(d − 4).Similarly, we can choose a basis of independent P -even operators, and then write a general ansatz for the metric variation of W ct as follows We would like to add the following remarks: • The coefficients {a i , b i , c i } are function of the parameters in W ct and can depend on d, such that the parity-even part T αβ ct can contain finite pieces despite the global 1/(d − 4)-factor.
• The coefficients {e, e i } depend on the parameters in W ct and the scheme chosen for the ϵ-tensor.For convenience, the e i are taken to be independent of d, since this would only amount to relabel e.
• It is possible to write δW ct /δg αβ under this form because the counterterms W ct allowed by renormalisation are local polynomials.This is not the case of the effective action W which is non-local, e.g.[33].
• The ansatz is valid for both theories with and without classical Weyl invariance.
• We only included tensors in T ct , i.e covariant quantities.This does not amount to the absence of diffeo and Lorentz anomalies, but restricts them to their covariant form [49].
We may trace T ct in (3.10), using g where we used upon using the Bianchi identities only.Incidentally, we see that there remains no P -odd term in the divergent part upon using (3.9).In other words, the topological nature of the P -odd term in W ct (Eq.(3.3)), forbids the presence of P -odd divergent terms in the trace of T ct .From (3.11) we can compute the Weyl anomaly.That is which follow from Eqs. (2.12) and (2.16) respectively.In Sec.3.1 and Sec.3.2 the parameters {e, e i } and {a i , b i , c i } will be subjected to diffeo constraints and the Wess-Zumino consistency conditions.

Constraints from the diffeomorphism anomaly
The diffeo variation of the effective action is given by where δ d ξ is the (active) diffeo transformation, and we used δ d ξ g µν = D µ ξ ν + D ν ξ µ .Note that, Eq. (3.14) may be non-zero but remains finite and one-loop exact (it is non-zero in theories in d = 2 + 4k with Weyl fermions for example [29]).Since it is finite, it should not be altered by W ct as there is no divergence to remove, and therefore must hold.We thus compute the divergence of T ct and then enforce The P -odd and -even pieces are independent of each other.Enforcing (3.16) on the P -even sector yields seven constraints Remarkably, the same procedure applied to the P -odd sector only admits the trivial solution which implies the absence of P -odd operators in both the counterterms to the EMT, and the Weyl anomaly, for both classically Weyl invariant and non-invariant theories.We stress that this result is independent of both, the underlying theory and the ϵ-tensor scheme.In our previous work [15] we have obtained the same result specifically for a theory with a Weyl fermion only.Inserting these constraints in (3.11) we obtain an expression in terms of the three coefficients a 1,2,3 .The 1/(d − 4)-term is addressed just below.

Constraints from classical (non-)invariance
As noted earlier anomalies are finite and thus the 1/(d − 4)-term in (3.19) ought to vanish.

Classical Weyl invariance (2 anomaly coefficients)
The Weyl anomaly for a classically Weyl invariant theory is given by Eq. (3.13) and must respect the Wess-Zumino consistency condition.In Eq. (3.19) it is only the R 2 -term which does not respect the consistency conditions [39,[50][51][52][53] and therefore must hold which indeed removes the divergent □R-term.The result assumes the form which depends on the two parameters a 1 and a 2 which are model-dependent.This is the result obtained by Duff in [1], supplemented by the additional constraint that R R is absent.
The question that poses itself is whether the results apply beyond LO.We would think that the answer is affirmative to all orders in perturbation theory for theories whose nonanomalous symmetries are compatible with dimensional regularisation. 11Beyond LO, W ct and the ansatz (3.10) contains higher order poles in d−4.However, since the Weyl anomaly is finite, their contributions have to cancel in analogy to the computation of an anomalous dimension of a parameter or an operator (e.g .[56]).

Broken classical Weyl symmetry (3 anomaly coefficients)
In a theory that explicitly breaks Weyl invariance, the Wess-Zumino conditions do generally not apply.However, since the Weyl anomaly can be obtained as in Eq. (3.13), it is automatically given by the finite part and thus the Weyl anomaly depends on the three parameters a 1,2,3 which are again model-dependent.Note that the □R-term is fixed in terms of others which we believe to be a new observation in the presence of explicit breaking.With regards to the validity beyond LO the same remarks apply as in the previous section.

Constraints from the Lorentz anomaly
In the ansatz (3.10), the assumption is made that the EMT is symmetric in its indices.To remain the most generic, we should assume that the theory may exhibit a Lorentz anomaly, which is the breaking of rotational symmetry at the quantum level and manifests itself in the antisymmetry of ⟨T µν ⟩.In practice, the antisymmetry can arise in the presence of fermions since then the vierbein e a ν replace the metric, and the EMT is not automatically symmetric by metric-variation but follows from where e aα = g αβ e a β with latin indices referring to the tangent frame.The Lorentz anomaly is then defined by where α ab (x) = −α ba (x) is the Lorentz transformation parameter, and δ L α e a µ = −α a b e b µ [49] was made use of.Note that since the vierbein is not diffeo invariant, the diffeo anomaly becomes where ω µab = −ω µba is the spin-connection.
The same argument as for the diffeo anomaly applies: since the Lorentz anomaly is finite, the counterterms must not contribute (i.e δ L α W ct = 0) yielding the Lorentz constraint Enforcing the Lorentz constraint is equivalent to considering directly the metric variation as was done in this Section and Sec. 2, since the vierbein variation can be split into its symmetric and antisymmetric parts δe a ν = 1 2 δg µν e aµ − α a b e b ν [57].Besides, the diffeo constraint (3.16) is unchanged when the Lorentz constraint is verified.

The non-removable □R-term
The constraints on the Weyl anomaly in the case of a classically Weyl invariant theory imply that the coefficient of the □R is fixed with respect to the coefficients of R 2 , R µν R µν and R µνρσ R µνρσ (3.21) which has been known since a long time [1].We believe that it is a new result that the same holds true for a theory which is not classically Weyl invariant (3.22).Further notice that the usual ambiguity in □R, due to the possibility of adding a local term in the action, which shifts □R, is not present in the case at hand since this term cancels in the formula (2.19).This is the case since the term in (3.27) enters the explicit breaking E Weyl and not the anomalous breaking in (2.6).That is (3.27) is intrinsically 4-dimensional and independent of the quantum fields, hence only affects the EMT at tree level.
The finding that the term (3.27) does not alter the anomalous part is related to the possibility that a □R flow theorem may exist [58].However, for the latter there is a problem when one considers the flow of QCD.Pions cannot be coupled conformally, that is a quadratic term L ⊃ 1 2 (D α πD α π + ξRπ 2 ) with ξ = 1 6 (or any ξ ̸ = 0) is not allowed by the shift symmetry for Goldstone bosons.However ξ ̸ = 1  6 renders the integral for the flow term infrared divergent.In the purely anomalous part the non-conformal terms would drop out and suggest that it might be worthwhile to consider whether one can formulate flow theorems in terms of the anomalous part only. 12Effectively the infrared divergence is then shifted into the explicit breaking part which has no relevance for the definition of the (pure) Weyl anomaly.
Since we believe that the □R-anomaly is calculable, the constraints on □R in (3.21) and (3.22) have to hold when computed with other regularisations provided they respect the non-anomalous symmetries of the theory.For example, the ζ-function regularisation employed in [40,60] satisfies the constraint for a spin-1/2 fermion and a spin-1 vector in the ( 12 , 0) ⊕ (0, 1 2 ) and ( 1 2 , 1 2 ) Lorentz representations respectively.This is sometimes difficult to see when it is the total breaking A Weyl + E Weyl that is given.That is the case for example for the spin 0 scalar given in [18] (Tab. 1 in chapter 5).However, since ξ = 1 6 removes the explicit breaking it is readily verified that in this case the constraint is satisfied.In [34], uniquely A Weyl is determined, and the constraint is satisfied for any ξ.Similarly, it was pointed in [36] that the heat kernel leads to a different definition to (2.8) in the presence of explicit breaking (the second term is replaced by ⟨g (4) µν T µν ⟩), and a different value for the □R.According to our derivation in Sec.2.3, this implies that some explicit breaking is included in the heat kernel anomaly definition.Furthermore, since ghosts and gauge fixing may affect explicit breaking they require careful assessment as well.Hence, care must be taken when comparing the literature.Note also that the sign of □R is dependent on the sign convention of the metric.

Curved-space Anomaly Including a Gauge Sector
In this section, we apply the same method in the presence of a gauge sector.This involves adding gauge field terms in W ct , or the decomposition in (3.10).

Diffeomorphism anomaly with a gauge sector
In the presence of a background (abelian or non-abelian) gauge field A µ , the Lorentz and Weyl anomalies keep the same expression as in pure gravity since the gauge field is not affected by their transformation: The gauge index i is to be omitted for an abelian gauge group.Upon using δ d ξ e a µ = ξ ν ∂ ν e a µ + e a ν ∂ µ ξ ν [40,49], one finds where is the gauge field strength.The third term can be written precisely as the gauge anomaly with using √ gH i,µ ≡ δW/δA i µ as a shorthand.Similar expressions arise for the counterterms W ct which have to be diffeo, Lorentz and gauge invariant as a consequence of the finiteness of these potential anomalies.Therefore, we may write the analogue of (4.2) in terms of where √ gH i,µ ct ≡ δW ct /δA i µ .The Lorentz and gauge constraints read respectively such that when combined one finally gets the main equation of this section, representing the diffeo, the Lorentz and the gauge anomaly constraints on the form of the Weyl anomaly.

Constraints from the diffeomorphism anomaly on the gauge sector
In this section we apply the constraint (4.6).In the P -even case there exists a single counterterm from which we deduce the metric variation and gauge field variation of the pure gauge part.The diffeo constraint (4.6) can be seen to be satisfied (after using Bianchi identities) since (4.7) is indeed diffeo invariant.The gauge field dependent part of the Weyl anomaly is directly finite and reads As is well-known, in the full theory f is proportional to the β function of the running gauge coupling.
Let us now consider the more subtle case of P -odd operators.We proceed as for pure gravity with the addition of gauge field dependent terms Since in d = 4, the only P -odd gauge dependent operator of mass dimension 4 is the Pontryagin density F F ≡ 1 2 ϵ µνρσ F µν F ρσ , we must have L odd (4) ∝ tr F F .Its topological nature implies With the same reasoning as for pure gravity, we can express metric and gauge variations of d d x L odd (d) in terms of a continuous and a discontinuous term with and likewise for N .Using the intrinsically 4-dimensional Schouten identity, one can show that the only possible P -odd gauge covariant 2-tensor of mass dimension four is which is the analogue of (3.6).Without assuming a specific scheme for the ϵ-tensor, we may write without loss of generality using d-independent identities.Note that the trace has to be ignored if the gauge group is abelian.We only consider symmetric tensors such that the Lorentz constraint (cf.Sec.3.3) is directly enforced.At d = 4, using the Schouten identity we obtain (U αβ (4) = 0) In principle, one should include mixed gravity-gauge 2-tensors in V αβ (4) and U αβ (d).They are however independent from the pure gauge operators, and vanish under the metric contraction, hence cannot contribute to the Weyl anomaly (see App. B.2.1).
Turning to H ct , the second term in our constraint equation (4.6), we first notice that these mixed gravity-gauge terms can be ignored for the same reason.Secondly, the only P -odd 1-tensor of mass dimension 3 we could be tempted to write is which vanishes by virtue of the Bianchi identity.As a result we also have M i,α = 0.The gauge dependent piece of the Weyl anomaly using this parametrisation is where like in the pure gravity case, the divergent part is zero by 4 h 1 + h 2 = 0 (4.17).Finally, the diffeo constraint (4.6) on the pure gauge terms reads and leads to a system that only admits the solution We thus conclude on the absence of P -odd gauge dependent terms in the counterterms and the Weyl anomaly, for both non-abelian and abelian gauge groups.

Explicit Gravity-gauge Anomalies for a Weyl fermion
In this section, we compute Weyl, diffeo and Lorentz anomalies for a ( 1 2 , 0) Weyl fermion including a gauge sector.Note that a Weyl fermion is not the same as a Dirac fermion coupled chirally to gauge or gravitational fields [61] (where parity odd terms are absent) as emphasised in [15].Since these anomalies involve gauge and gravitational fields, they are sometimes referred to as mixed gravity-gauge anomalies in the literature.We follow the notation in our previous work [15].The action of the Weyl fermion is given by where ψL = √ eψ L is the two-component Weyl fermion appropriate for avoiding spurious diffeo anomalies in the path integral [40,60,62].Here, the variable e = det e a µ = √ g is the vierbein determinant.The Dirac-Weyl operator is defined by For Dirac fermions the functional determinant of an operator is defined by expressing it as the product of its eigenvalues.In our case, D maps left-handed onto right-handed Weyl fermions and hence the eigenvalue equation Dϕ = λϕ is meaningless [29].Even though W is ill-defined, its variation δW = −Tr δDD −1 , (5.4) is well-defined, since δDD −1 maps fermions of the same chirality into each other [57,63].This quantity can be regularised by introducing a smooth function f , with f (0) = 1 and ∀ n ≥ 0, lim x→∞ x n f (n) (x) = 0, and making use of the formal expression [57] This is well-defined since DD † is a positive operator provided that the zero modes are excluded (i.e perturbative set-up).For the practical computation we will use f (x) = 1/(1 + x), suitable for the Covariant Derivative Expansion (CDE) technique for curved space (e.g.[64]).Finally, from (5.4) and (5.5) one infers the regularised effective action variation which will be the basis for the computations in the remaining part of this section.

Weyl, diffeomorphism, Lorentz and gauge anomalies
From the Weyl-, diffeo-and Lorentz-variations of the background fields e a µ and A µ , we obtain the Dirac-Weyl operator variations [15] δ where λ ab ≡ 1 4 (σ a σ b − σb σ a ), µ ab ≡ 1 4 (σ a σb − σ b σa ) and ∇ µ is the diffeo-covariant derivative. 13The gauge variation, not treated in [15], reads When inserting these expressions in (5.6) it turns out that one can recast the equations in terms of the Dirac operator [15] where Note that by writing ∇ µ = D µ − ω µ − iA µ in the second line of (5.10), the diffeo anomaly can be recast in terms of the covariant derivative and gauge and Lorentz anomalies (with parameters θ = ξ µ iA µ and α ab = ξ µ ω µab respectively).The appearance of the gauge anomaly, as noticed earlier, is due the gauge field being diffeo-variant.We Wick rotate back to Minkowski signature for the following results.For the Weyl anomaly we obtain using the CDE where the sign in F 2 differs from some of the literature since F is defined in Sec.4.1 with an extra factor of i with respect to the (standard) convention.This extends the results from [15] to include a gauge sector.In analogy to the absence of the R R-term we find by explicit computation that the F F -term is absent, in agreement with Sec.4.2.The other anomalies in (5.10) evaluate to where α µν = α ab e a µ e b ν .Traces run over gauge indices (and is to be ignored if the gauge group is abelian), and the same remark with respect to the sign of F F applies as above. 14ote that a chiral Dirac fermion coupling to A as S D = ψi / ∇ + / ω + i / AP L ψ has the same mixed anomalies (and only the mixed ones) as the Weyl fermion (5.1).Indeed S D is the sum of (5.1), and of a right-handed Weyl fermion not coupling to A which produces no additional mixed anomalies.The Lorentz anomaly in (5.12) agrees with [65][66][67].The gauge anomaly is standard, the R R-term corresponds to the mixed-axial gravitational anomaly [40,49] and vanishes if the gauge group is non-abelian.To the best of our knowledge, this is the first time that the diffeo anomaly of a Weyl fermion is explicitly computed and shown to bear mixed terms (the same applies to a chiral Dirac fermion as mentioned earlier).In [66] the divergence of the EMT was considered which is only a part of the diffeo anomaly (4.2). 15

Phenomenological considerations
Note that whereas the results for the Weyl anomaly are physical, the other anomalies (5.12) must cancel in a ultraviolet complete theory.Another difference is that the anomalies in (5.12) are one-loop (or LO) exact where the Weyl anomaly has in general contributions to all orders and beyond.The main points for abelian and non-abelian cases are: a) Abelian case: the Lorentz and diffeo anomaly cancellation requires i Q i = 0 where Q i are the U (1)-charges of the Weyl fermions (or the Dirac fermions coupled to the U (1) via a chiral projector).If we had right-handed fermions in addition then the condition would read: The diffeo and the Lorentz anomaly cancellation thus gives the same contraints as the gauge anomaly cancellation.b) Non-abelian case: the Lorentz anomaly vanishes and the gauge anomaly is equivalent to the standard one.However, the diffeo anomaly does not vanish due to the δ g ξ µ A i µ Wterm.For a gauge group SU (n) this leads to the condition tr[T a {T b , T c }] ∝ d abc = 0, often referred to as SU (n) ⊗3 -anomaly condition.Hence the diffeo anomaly cancellation condition is the same as the gauge anomaly one.
As emphasised above there are effectively no new gauge constraints.However, in the case where one has a vector field associated to a global chiral symmetry there are interesting phenomenological consequences.Let us consider a global chiral vector field.Whereas the gauge anomaly δ g θ W (5.12) is not relevant, the Lorentz and diffeo anomalies are and thus the constraints found in a) and b) do apply.Should the chiral field theory not satisfy these constraints this implies that the theory cannot be ultraviolet complete.Examples of where such models have been considered include [71][72][73][74][75].However, these models are still meaningful as effective theories since they can arise as low energy theories of a Higgs mechanism where the anomaly-cancelling fermions have been integrated out [26,27].The only deficiency of such theories are that they are not renormalisable and have a maximum energy up to which they can be used.In these references only gauge anomalies were considered but it would seem that the very same reasoning does equally apply to the case of Lorentz and diffeo anomalies.

Summary and Conclusion
In this work we have revisited the constraints on the EMT arising from the finiteness of the diffeomorphism, the Lorentz and the gauge anomaly.Our work extends Ref. [1] to include parity-odd terms and explicit symmetry breaking.We derive the formula for the pure Weyl anomaly (2.19), widely used in the literature, within dimensional regularisation.The most general decomposition of the EMT counterterms (3.10) consists of ten parityeven and one parity-odd term.After applying the diffeomorphism anomaly and Wess-Zumino consistency constraints it is found that the parity-odd term vanishes and that the parity-even terms reduce to three (two) unknowns in the case of (no-)explicit symmetry breaking.In particular, this implies the absence of the R R-term from the Weyl anomaly in a model-independent way.The □R-anomaly is found to be finite, unambiguous and determined through the Weyl and Euler coefficients.That this still holds in the case of explicit Weyl breaking is a new result (Sec.3.4).It was argued that the constraints should extend to all orders in theories whose non-anomalous symmetries are respected by dimensional regularisation.In Sec. 4 the method was extended to include a gauge sector, ruling out the F F -term.Mixed gravity-gauge anomalies for Weyl fermions were computed to leading order (Sec.5).We believe that our results for the diffeomorphism anomaly are more complete than in the literature.We deduce phenomenological constraints on UV-completeness of theories that include vector bosons associated with global chiral symmetries.
The authors are grateful to Latham Boyle, Franz Herzog and Neil Turok for helpful discussions, and we acknowledge valuable comments by the referee.The work of RL and JQ is supported by the project AFFIRM of the Programme National GRAM of CNRS/INSU with INP and IN2P3 co-funded by CNES and by the project EFFORT supported by the programme IRGA from UGA. JQ and RZ acknowledge the support of CERN associateships.The work of RZ is supported by the STFC Consolidated Grant, ST/P0000630/1.Many manipulations were carried out with the help of Mathematica and the package xAct [76].

A Anomaly Definition
The Weyl variation of W ren (d) is defined by (2.14) with explicit symmetry breaking, and (2.9) without.Either way, if we take the limit before the Weyl variation we obtain where we used the fact that W ren (4) is finite to commute the metric variation and the metric contraction inside the limit.Alternatively, the Weyl anomaly can be written by taking the limit after the Weyl variation In the following we prove some identities on the counterterms that are used in the derivation of Eqs.(2.10), (2.12), (2.16) and (2.19).

A.1 Classical Weyl invariance
The most generic form of the counterterms in a classically Weyl invariant theory reads [1] W where we included only P -even terms.We omit P -odd operator since we have showed that they must be absent at the level of the metric variation of where X(a, c, f ) denotes the contributions from the E, C and F 2 terms which are finite (A.2).As expected, we confirm (A.22) This is the contribution from W ct that cancels the divergence of the explicit breaking ⟨T ρ ρ ⟩.Once again, this relation still holds when P -odd operators are included in W ct since they vanish at the level of δW ct /δg µν .

B Constraints on the Energy-momentum Tensor
In this appendix, we discuss the basis of P -odd operators, and we detail how to enforce the diffeo-Lorentz-gauge constraints.

B.1 Pure gravity case
Let us recall the generic ansatz (3.10) with the symmetrisation of indices t (µν) = 1 2 (t µν + t νµ ) .The P -odd coefficients are constrained by (3.Let us try to write all possible parity-odd 2-tensors of mass dimension four depending only on curvature invariants.Firstly, by enumeration it is possible to show that there exists no P -even antisymmetric 2-tensor, therefore we have ϵ αβµν O µν = 0 where O is a P -even 2-tensor.Secondly, due to the Bianchi identities one has ϵ ανρσ R βνρσ = 0 and ϵ µνρσ D ν R αβρσ = 0. Using this result, any operator can be related to these by Bianchi identities and algebra 3) However, upon using the intrinsically four-dimensional Schouten identity [44], we obtain showing that these operators are not independent in d = 4.These relations can also directly be obtained from Ref. [46], or from useful applications of the Schouten identities (e.g.[45]).

B.1.2 Diffeomorphism anomaly constraint
The finiteness of the diffeo anomaly is ensured by Using Bianchi identities and algebra, it is possible to write the P -even part of the divergence of T ct in terms of independent operators as Note that it is not allowed to use integrations by parts at this level (besides D α T αβ ct is itself a boundary term).Imposing D α T αβ ct P −even = 0 yields seven constraints on the ten parameters associated to the P -even operators.They can be recast to obtain (3.17).
Concerning the P -odd part, it is possible to write the divergence as where Some of these operators may be related by the Schouten identity in d = 4 only.They are not related by Bianchi identities either and are thus independent.Additionally, the pole and the finite parts in (B.7) are independent.Contrary to the P -even part, the system (B.

B.2 Gauge sector
Let us now consider the gauge sector.

B.2.1 P -odd operator basis
Keeping in mind, the Bianchi identity ϵ µνρσ D ν F ρσ = 0 and that [F, F ]-commutator terms vanish for both abelian and non-abelian gauge groups (by trace-cyclicity), it is possible to show by enumeration that the operators g µν tr F F , tr F Mixed gravity-gauge symmetric 2-tensors can be ignored since they all vanish under the trace, as mentionned in Sec.4.2.This is the case since there exists no mixed gravitygauge scalar of mass dimension four.For example, for an abelian gauge group we have the symmetric 2-tensors ϵ ανρσ R β λρσ F λ ν + (α ↔ β) and ϵ ανρσ R β ν F ρσ + (α ↔ β).They however vanish (by Bianchi for the first one) when contracted with g αβ .