On the Trace Anomaly of a Weyl Fermion in a Gauge Background

Eur. Phys. J. C (2019) 79:292 https://doi.org/10.1140/epjc/s10052-019-6799-z

Regular Article - Theoretical Physics

On the trace anomaly of a Weyl fermion in a gauge background

Fiorenzo Bastianelli1,2,3,a, Matteo Broccoli1,3,b 1 Dipartimento di Fisica e Astronomia, Università di Bologna, via Irnerio 46, 40126 Bologna, Italy 2 INFN, Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy 3 Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Am Mühlenberg 1, Golm, 14476 Potsdam, Germany

Received: 30 October 2018 / Accepted: 19 March 2019 © The Author(s) 2019

Abstract We study the trace anomaly of a Weyl fermion antiparticle, expected to contribute oppositely to any chiral in an abelian gauge background. Although the presence of imbalance in the coupling to gravity. To see that, one may cast the chiral anomaly implies a breakdown of gauge invari- the quantum field theory of a Weyl fermion as the quantum ance, we find that the trace anomaly can be cast in a gauge theory of a Majorana fermion. The latter shows no sign of an invariant form. In particular, we find that it does not con- odd-parity trace anomaly. Indeed, the functional determinant tain any odd-parity contribution proportional to the Chern– that arises in a path integral quantization can be regulated Pontryagin density, which would be allowed by the consis- using Pauli–Villars Majorana fermions with Majorana mass, tency conditions. We perform our calculations using Pauli– so to keep the determinant manifestly real, thereby exclud- Villars regularization and heat kernel methods. The issue is ing the appearance of a phase that might produce an anomaly analogous to the one recently discussed in the literature about (the odd-parity term would carry an imaginary coefficient) the trace anomaly of a Weyl fermion in curved backgrounds. [7]. Recently, this has been verified again using Feynman diagrams [8], which confirms the results of [6]. An additional piece of evidence comes from studies of the 3-point 1 Introduction correlation functions of conserved currents in four dimensional CFTs, which exclude odd-parity terms in the correla- In this paper we study the trace anomaly of a chiral fermion tion function of three stress tensors at non-coinciding points coupled to an abelian gauge field in four dimensions. It is [9,10], seemingly excluding its presence also in the trace well-known that the model contains an anomaly in the axial anomaly (see however [11]). gauge symmetry, thus preventing the quantization of the Here we analyze the analogous situation of a Weyl fermion gauge field in a consistent manner. Nevertheless, it is useful coupled to an abelian U(1) gauge background. The theory to study the explicit structure of the trace anomaly emerging exhibits a chiral anomaly that implies a breakdown of gauge in the axial U(1) background. invariance. It is nevertheless interesting to compute its trace One reason to study the problem is that an analogous sit- anomaly. Apart from the standard gauge invariant contribu- uation has recently been addressed for a Weyl fermion cou- tion (∼ F2) and possible gauge noninvariant terms, which as pled to gravity. In particular, the presence of an odd-parity we shall show can be canceled by counterterms, one might term (the Pontryagin density of the curved background) in expect a contribution from the odd-parity Chern–Pontryagin the trace anomaly has been reported in [1], and further elab- density F F˜. Indeed the latter satisfies the consistency con- orated upon in [2,3]. This anomaly was envisaged also in ditions for trace anomalies. In addition, the fermionic func- [4], and discussed more recently in [5]. However, there are tional determinant is now complex in euclidean space, and many indications that such an anomaly cannot be present in thus carries a phase (which is responsible for the known U(1) the theory of a Weyl fermion. The explicit calculation carried axial anomaly). On the other hand, the structure of the 3- out in [6] confirms this last point of view. point correlation function of the stress tensor with two U(1) One of the reasons why one does not expect the odd- currents in generic CFTs does not allow for odd-parity terms parity contribution to the trace anomaly is that by CPT in [9,10] that could signal a corresponding anomaly in the trace four dimensions a left handed fermion has a right handed of the stress tensor in a U(1) background. Apart from a few differences, the case seems analogous to that of the chiral a e-mail: [email protected] fermion in curved space, and thus it is worth addressing. b e-mail: [email protected]

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To ascertain the situation we compute explicitly the trace where the chirality of the spinor is defined by the constraint anomaly of a Weyl fermion coupled to a U(1) gauge field. γ 5λ = λ λ = 1+γ 5 λ , or equivalently by 2 . It is classically Using a Pauli–Villars regularization we find that no odd- gauge invariant and conformally invariant. Both symmetries parity term emerges in the quantum trace of the stress tensor. become anomalous at the quantum level. We use a Majorana mass for computing the trace anomaly, as In the following we find it convenient to use the charge this mass term can be covariantized (to curved space) without conjugated spinor λc, which has the opposite chirality of λ the need of introducing additional fields of opposite chirality, −1 T 5 as required by a Dirac mass. The coupling to gravity (needed λc = C λ ,γλc =−λc. (2) only at linear order) is used to treat the metric (or vierbein) as λ an external source for the stress tensor, and to relate the trace The lagrangian can be cast in equivalent forms using c rather λ of the latter to a Weyl rescaling of the metric (or vierbein). then The manifest covariance of the Majorana mass guarantees L = λT /( )λ = λT /(− )λ that the stress tensor can be kept conserved and symmet- W c CD A CD A c ric also at the quantum level, i.e. without general coordinate 1 T T = λ CD/(A)λ + λ CD/(−A)λc (3) (Einstein) and local Lorentz anomalies. We repeat part of our 2 c calculations with a Dirac mass as well. In addition, we calculate also the anomalies of a massless Dirac fermion which, where the last two forms are valid up to boundary terms (we while well-known, serve for comparison and as a test on the perform partial integrations in the action and drop boundary scheme adopted. We verify the consistency of the different terms). We use the last form in our calculations. regularizations, and report the local counterterms that relate The gauge transformations can be written as them. ⎧ α( ) ⎪ λ(x) → λ (x) = ei x λ(x) We organize the paper as follows. In Sect. 2 we set up ⎪ ⎨⎪ − α( ) the stage and review the lagrangians of the Weyl and Dirac λ(x) → λ (x) = e i x λ(x) (4) fermions, respectively, and identify the relevant differential ⎪ λ ( ) → λ ( ) = −iα(x)λ ( ) ⎪ c x c x e c x operators that enter our regularization schemes. In Sect. 3 we ⎩ A (x) → A (x) = A (x) + ∂ α(x) review the method that we choose for computing the chiral a a a a and trace anomalies. In Sect. 4 we present our results. We 4 and the action SW = d x LW is gauge invariant. Recall conclude in Sect. 5, confining to the appendices notational also that Aa can be used as an external source for the current conventions, heat kernels formulas, and sample calculations. J a = iλγ aλ. (5)

Varying in the action only Aa with a gauge transformation 2 Actions and symmetries with inﬁnitesimal parameter α(x) produces

(A) 4 a We first present the classical models that we wish to con- δα SW =− d x α(x)∂a J (x) (6) sider, and review their main properties to set up the stage for our calculations. The model of main interest is a mass- and the full gauge symmetry (δα SW = 0) guarantees that the less Weyl fermion coupled to an abelian gauge field. We first U(1) current is conserved on-shell (i.e. using the fermion describe its symmetries, and then the mass terms to be used equations of motion) in a Pauli–Villars regularization. For comparison, we con- a sider also a massless Dirac fermion coupled to vector and ∂a J (x) = 0. (7) axial abelian gauge fields, a set-up used by Bardeen to com- Similarly, one may check that the action is classically con- pute systematically the anomalies in vector and axial currents formal invariant and that the stress tensor has a vanishing [12]. Our notation is commented upon and recapitulated in trace. To see this one couples the model to gravity by intro- “Appendix A”. ducing the vierbein eμa (and related spin connection ωμab), and realizes that the action is invariant under general coor- 2.1 The Weyl fermion dinate, local Lorentz, and Weyl transformations. The stress tensor, or energy momentum tensor, is defined as usual by The lagrangian of a left handed Weyl spinor λ coupled to a μ 1 δSW U(1) gauge field is T a(x) = (8) e δeμa(x) L =−λγ a(∂ − )λ =−λγ a ( )λ =−λ /( )λ W a iAa Da A D A where e is the determinant of the vierbein, and is covari- (1) antly conserved, symmetric, and traceless on-shell, as con- 123 Eur. Phys. J. C (2019) 79:292 Page 3 of 12 292 sequence of diffeomorphisms, local Lorentz invariance, and can take as PV field a Weyl fermion of the same chirality, with Weyl symmetry, respectively a Majorana mass added. The mass term is Lorentz invari- μa a ant, but breaks the gauge and conformal symmetries. It takes ∇μT = 0, T = T , T = 0(9) ab ba a many equivalent forms (indices are made “curved” or “flat” by using the vierbein and M T its inverse). The vierbein can be used as an external source for ΔM LW = λ Cλ + h.c. 2 the stress tensor, and an infinitesimal Weyl transformation on M − T the vierbein acts as a source for the trace T a . In the following = λT Cλ − λC 1λ (16) a 2 we only need a linearized coupling to gravity to produce a M T T single insertion of the stress tensor in correlation functions. = λ Cλ + λ Cλc 2 c Otherwise, we are interested in flat space results only. In any case, the full coupling to gravity reads where h.c. denotes the hermitian conjugate and M is a real mass parameter. Since the charge conjugation matrix C is L =− λγ μ∇ λ W e μ (10) antisymmetric this term is nonvanishing for an anticommut- 1 μ μ a ing spinor. where γ = e aγ are the gamma matrices with curved μ Casting the full massive PV action L = L + Δ L indices, e a is the inverse vierbein, and ∇μ is the covariant PV W M W derivative containing both the U(1) gauge field Aμ and spin in the compact form connection ωμab 1 T 1 T LPV = φ T Oφ + Mφ T φ, (18) 1 a b ∇μ = ∂μ − iAμ + ωμabγ γ . (11) 2 2 4 φ λ λ φ The local Weyl symmetry is given by where is a column vector containing both and c (thus ⎧ is a 8 dimensional vector) ⎪ − 3 σ(x) ⎪ λ(x) → λ (x) = e 2 λ(x) ⎪ λ ⎨ − 3 σ(x) φ = , λ(x) → λ (x) = e 2 λ(x) λ (19) (12) c ⎪ ⎪ A (x) → A (x) = A (x) ⎪ a a a permits the identification of the operators ⎩ a a σ(x) a eμ (x) → e μ (x) = e eμ (x) 0 CD/(−A)PR CPL 0 where σ(x) is an arbitrary function. Varying in the action T O = , T = CD/(A)PL 0 0 CPR only the vierbein with an infinitesimal Weyl transformation (20) produces the trace of the stress tensor

( ) and δ e S =− d4xeσ(x)T a (x) (13) σ W a /(− ) O = 0 D A PR , and the full Weyl symmetry of the action (δσ S = 0) guar- D/(A)PL 0 W (21) antees that it is traceless on-shell /(− ) /( ) O2 = D A D A PL 0 . a 0 D/(A)D/(−A)PR T a(x) = 0. (14) For completeness, we record the form of the stress tensor in The latter will be used in our anomaly calculations. The chiral flat space emerging from the previous considerations, sim- projectors PL and PR plified by using the equations of motion, 1 + γ 5 1 − γ 5 PL = , PR = (22) 1 ↔ ↔ 2 2 Tab = λ γa Db + γb Da λ (15) 4 have been introduced to stress that the matrix T is not invert- ↔ ← ible in the full 8 dimensional space on which φ lives. An where Da = Da − Da (in terms of the gauge covariant derivative). It is traceless on-shell. 1 In terms of the 2-component left handed Weyl spinor lα this mass 2.1.1 Mass terms term reads as M 2 αβ ∗ 2 α˙ β˙ ∗ ΔM LW = lα(−iσ ) lβ + lα˙ (iσ ) l ˙ (17) To compute the anomalies in the quantum theory we regular- 2 β ize the latter introducing massive Pauli–Villars (PV) fields, and it does not contain any other spinor apart from lα and its complex ∗ with the anomalies eventually coming from the noninvari- conjugate lα˙ . In the chiral representation of the gamma matrices the ance of the mass term. For the massless Weyl fermion, one 2-component spinor lα sits inside λ,asinEq.(A.12). 123 292 Page 4 of 12 Eur. Phys. J. C (2019) 79:292 advantage of the Majorana mass term is that it can be con- ator were studied with the purpose of addressing the chiral structed without the need of introducing extra degrees of free- anomalies. dom (as required by the Dirac mass term discussed below). A drawback of the Dirac mass term, as regulator of the Moreover, it can be covariantized under Einstein (general Weyl theory, is that one cannot covariantize it while keeping coordinate) and local Lorentz symmetries. The covariantiza- the auxiliary right handed spinor ρ free in the kinetic term tion is achieved by multiplying it with the determinant of the (it cannot be coupled to gravity, otherwise it would not reg- vierbein e ulate properly the original chiral theory). One can still use the regularization keeping ρ free in the kinetic term but, as eM T T ΔM LW = λ Cλ + λ Cλc . (23) 2 c the mass term breaks the Einstein and local Lorentz symmetries explicitly, one would get anomalies in the conservation An alternative mass term is the Dirac mass. To use it one (∂ T ab) and antisymmetric part (T [ab]) of the stress tensor. must introduce in addition also an uncoupled right handed a Then, one is forced to study the counterterms to reinstate con- PV fermion ρ (satisfying ρ = PRρ), so that the full massive servation and symmetry of the stress tensor (it can always be PV lagrangian reads done in 4 dimensions [7,14]), and eventually check which ˜ LPV =−λD/(A)λ − ρ∂/ρ − M(λρ + ρλ) (24) trace anomaly one is left with. As this is rather laborious, we do not use this mass term to calculate the trace anomaly in or, equivalently, 2 the Weyl theory. L˜ = 1 λT /( )λ + λT /(− )λ PV c CD A CD A c 2 2.2 The Dirac fermion + 1 ρT ∂/ρ + ρT ∂/ρ c C C c (25) 2 We consider also the more general model of a massless Dirac M + (λT ρ + ρT λ + ρT λ + λT ρ ). fermion coupled to vector and axial U(1) gauge fields Aa and c C C c c C C c 2 Ba. The lagrangian is The latter expression allows to cast the PV lagrangian in the L =−ψγa(∂ − − γ 5)ψ =−ψ /( , )ψ general form (18) with D a iAa iBa D A B ⎛ ⎞ 1 T 1 T λ = ψ CD/(A, B)ψ + ψ CD/(−A, B)ψc (31) 2 c 2 ⎜ λ ⎟ φ = ⎜ c ⎟ ⎝ ρ ⎠ (26) where the last form is valid up to boundary terms. A chiral =± ρ projector emerges when Aa Ba, and we use this model c to address again the issue of the chiral fermion in flat space = → Aa where each entry is a 4 dimensional Dirac spinor (with chiral (the limit Aa Ba 2 reproduces the massless part of projectors attached), and one finds (24)). ⎛ ⎞ The lagrangian is invariant under the local U(1)V vector 0 CD/(−A)PR 0 0 ⎜ /( ) ⎟ transformations O = ⎜CD A PL 0 0 0 ⎟ ⎧ T ⎝ ∂/ ⎠ (27) ⎪ ψ( ) → ψ( ) = iα(x)ψ( ) 0 0 0 C PL ⎪ x x e x ⎪ 0 0 C∂/PR 0 ⎪ ψ( ) → ψ ( ) = −iα(x)ψ( ) ⎛ ⎞ ⎨⎪ x x e x 000CPL ψ (x) → ψ (x) = e−iα(x)ψ (x) (32) ⎜ ⎟ ⎪ c c c = ⎜ 00CPR 0 ⎟ ⎪ T ⎝ ⎠ (28) ⎪ Aa(x) → A (x) = Aa(x) + ∂aα(x) 0 CPR 00 ⎩⎪ a ( ) → ( ) = ( ) CP 000 Ba x Ba x Ba x ⎛ L ⎞ 0 0 ∂/PR 0 and local U(1)A axial transformations ⎜ ∂/ ⎟ ⎧ ⎜ 0 0 0 PL ⎟ 5 O = ⎝ ⎠ (29) ⎪ ψ( ) → ψ( ) = iβ(x)γ ψ( ) D/(A)P 0 0 0 ⎪ x x e x L ⎪ 0 D/(−A)P 0 0 ⎪ ψ( ) → ψ( ) = ψ( ) iβ(x)γ 5 ⎛ R ⎞ ⎨ x x x e ∂/ /( ) β( )γ 5 D A PL 0 0 0 ⎪ ψ (x) → ψ (x) = ei x ψ (x) (33) ⎜ ∂/ /(− ) ⎟ ⎪ c c c O2 = ⎜ 0 D A PR 0 0 ⎟ . ⎪ ⎝ ⎠ ⎪ Aa(x) → A (x) = Aa(x) 0 0 D/(A)∂/PR 0 ⎪ a ⎩ ( ) → ( ) = ( ) + ∂ β( ). 0 0 0 D/(−A)∂/PL Ba x Ba x Ba x a x (30) 2 A possibility to simplify the calculation would be to use the axial met- O2 The differential operators in have appeared also in [13], ric background introduced in [2,3], but we will not follow this direction where definitions for the determinant of a chiral Dirac oper- either. 123 Eur. 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a a Again one can use Aa and Ba as sources for J = iψγ ψ Thus, it is useful to consider a Majorana mass as well. It a = ψγaγ 5ψ and J5 i , respectively. Under inﬁnitesimal vari- breaks both vector and axial symmetries ation of these external sources one ﬁnds

Δ˜ L = M (ψ T ψ + . .) = M (ψ T ψ + ψ T ψ ) δ(A) =− 4 α( )∂ a( ) M D C h c C c C c α SD d x x a J x 2 2 (34) (41) δ(B) =− 4 β( )∂ a( ) β SD d x x a J5 x ˜ and one ﬁnds from the alternative PV lagrangian LPV = a a L + Δ˜ L and the classical gauge symmetries imply that J and J5 are D M D the operators conserved on-shell a 0 CD/(−A, B) C 0 ∂a J (x) = 0 T O = , T = (42) (35) CD/(A, B) 0 0 C ∂ a( ) = . a J5 x 0 A coupling to gravity shows that the stress tensor is trace- and less because of the Weyl symmetry. The Weyl transforma- 0 D/(−A, B) tions rules have the same form as in (12), with Ba left invari- O = , D/(A, B) 0 ant. An inﬁnitesimal Weyl variation on the vierbein produces (43) the trace of the stress tensor D/(−A, B)D/(A, B) 0 O2 = . 0 D/(A, B)D/(−A, B) (e) 4 a δσ SD =− d xeσ(x)T a(x). (36) Covariantization to gravity does not mix the chiral parts of and the Weyl symmetry implies that it vanishes on-shell the Dirac fermion, and a decoupling limit to the chiral theory λ a of a Weyl fermion is now attainable. T a(x) = 0. (37)

2.2.1 Mass terms 3 Regulators and consistent anomalies To regulate the one-loop graphs we introduces massive PV fields. The standard Dirac mass term To compute the anomalies we employ a Pauli–Villars regularization [15]. Following the scheme of Refs. [16,17]we M T T ΔM LD =−Mψψ = (ψ Cψ + ψ Cψc) (38) cast the calculation in the same form as the one obtained 2 c by Fujikawa in analyzing the measure of the path integral preserves vector gauge invariance, and casting the PV [18,19]. This makes it easier to use heat kernel formulas L = L + Δ L lagrangian PV D M D in the form (18), now with [20,21] to evaluate the anomalies. At the same time, the ψ φ = , allows to recognize the operators method guarantees one to obtain consistent anomalies, i.e. ψc anomalies that satisfy the consistency conditions [22,23]. Let us review the scheme of Ref. [16]. One considers a 0 CD/(−A, B) 0 C T O = , T = (39) lagrangian for a field ϕ CD/(A, B) 0 C 0 1 and L = ϕT T Oϕ (44) 2 /( , ) O = D A B 0 , 0 D/(−A, B) which is invariant under a linear symmetry (40) /( , )2 O2 = D A B 0 . 0 D/(−A, B)2 δϕ = K ϕ (45) λ ρ This mass term mixes the two chiral parts and of the Dirac that generically acts also on the operator T O,asitmay ψ = λ+ρ fermion ,seeEqs.(24)or(25) that makes it visible. depend on background fields. The one-loop effective action After covariantization to gravity the decoupling of the two can be regulated by subtracting a loop of a massive PV field chiralities is not easily achievable, and relations between the φ with action trace anomaly of a Dirac fermion and the trace anomaly of a Weyl fermion cannot be studied directly by using a Dirac 1 1 L = φT T Oφ + MφT T φ (46) mass in the PV regularization. PV 2 2 123 292 Page 6 of 12 Eur. Phys. J. C (2019) 79:292

3 = ( − O )( − O )−1 where M is a real parameter. The mass term identifies the obtained by inserting the identity 1 1 M 1 M operator T , that in turn allows to find the operator O.Aswe and using the invariance of the massless action shall see, in fermionic theories with a first order differential O O2 1 1 operator in the kinetic term, the operator acts as a δL = ϕT T O K + δT O + T δO ϕ = 0. (52) regulator in the final formula for the anomaly. The invariance 2 2 of the original action extends to an invariance of the massless In deriving these expressions, we have considered a fermionic part of the PV action by defining theory, used the PV propagator δφ = K φ (47) i φφT = , (53) so that only the mass term may break the symmetry T O + TM 1 taken into account the opposite sign for the PV field in the δL = MφT (TK + K T T + δT )φ PV 2 loop, and considered an invertible mass matrix T . (48) O2 − 1 In the limit M →∞the regulating term (1 − ) 1 = MφT TK + δT φ. M2 2 O 2 inside (51) can be replaced by e M2 . This is allowed as, for The path integral Z and the one-loop effective action Γ the purpose of extracting the limit, these regulators cut off are regulated by the PV field the ultraviolet frequencies in an equivalent way (we assume that O2 is negative definite after a Wick rotation to euclidean Z = eiΓ = Dϕ eiS ⇒ Z = eiΓ space) [16]. Clearly, if one finds a symmetrical mass term, then the symmetry would remain automatically anomaly free. ( + ) = DϕDφ ei S SPV (49) Heat kernel formulas may now be directly applied. Denot- ing where it is understood that one should take the M →∞ 1 − 1 δO limit, with all divergences canceled as explained in the foot- J = K + T 1δT + , R =−O2 (54) 2 2 M note. The anomalous response of the path integral under a symmetry is due to the PV mass term only, as one can define the anomaly is related to the trace of the heat kernel of the the measure of the PV field so to make the whole path inte- regulator R with an insertion of J gral measure invariant [16]. In a hypercondensed notation, T − R where a term like φ φ includes in the sum of the indices a iδΓ = i δS =− lim Tr[Je M2 ]. (55) →∞ spacetime integration as well, a lagrangian like the one in M (46) is equivalent to the action, and one may compute the This has the same form that appears in Fujikawa’s method symmetry variation of the regulated path integral to obtain for computing anomalies [18,19], where J is the infinitesi- 1 mal part of the fermionic jacobian arising from a change of iδΓ = i δS = lim iM φT (TK + δT )φ the path integral variables under a symmetry transformation, M→∞ 2 −1 (50) and R is the regulator. The limit extracts the mass indepen- 1 − O =− lim Tr K + T 1δT 1 + dent term (negative powers of the mass vanish in the limit, M→∞ 2 M while positive (diverging) powers are made to cancel by using where brackets ... denote normalized correlation func- additional PV fields). The PV method guarantees that the tions. For our purposes, it is convenient to cast it in an equiv- regulator R together with J produces consistent anomalies, alent form [17] which follows from the fact that one is computing directly the variation of the effective action. iδΓ = i δS The heat kernel formulas that we need in the anomaly cal- 2 −1 culation are well-known, and we report them in “Appendix B” 1 − 1 δO O =− lim Tr K + T 1δT + 1 − using a minkowskian time. In particular, in four dimensions M→∞ 2 2 M M2 we need the Seeley–DeWitt coefficients a2(R), correspond- (51) ing to the various regulators R associated to the fields assem- bled into φ. These are the only coefficients that survive in the 3 To be precise, one should employ a set of PV fields with mass Mi limit M →∞(as said, diverging pieces are removed by the and relative weight ci in the loop to be able to regulate and cancel all PV renomalization). Running through the various cases pre- possible one-loop divergences [15]. For simplicity, we consider only sented in the previous section, we extract the “jacobians” J one PV field with relative weight c =−1, as this is enough for our purposes. The weight c =−1 means that we are subtracting a massive and regulators R to find the structure of the anomalies. For PV loop from the original one. the Weyl theory we find 123 Eur. Phys. J. C (2019) 79:292 Page 7 of 12 292 a i 4.1 Chiral and trace anomalies of a Weyl fermion ∂a J = tr [PL a (Rλ)]−tr [PRa (Rλ )] (4π)2 2 2 c (56) a 1 We consider first the case of a Weyl fermion. T a =− tr [PL a (Rλ)]+tr [PRa (Rλ )] . 2(4π)2 2 2 c 4.1.1 PV regularization with Majorana mass These formulas are obtained by considering that for the U(1) symmetry the jacobian J in (54) is found from the symmetry The regularization of the Weyl fermion coupled to an abelian transformations of λ and λc in (4) gauge field is achieved in the most minimal way by using a iαP 0 PV fermion of the same chirality with the Majorana mass J = L . (57) 0 −iαPR term in Eq. (16) added. This set-up was already used in [6] to address the case of a Weyl fermion in a gravitational back- Only K contributes, as δT vanishes as well as the contribution ground, but without the abelian gauge coupling. The mass from δO (it vanishes after taking the traces in (56), as will term is Lorentz invariant and does not introduce additional be checked in the next section). The infinitesimal parameter chiralities, but breaks the gauge and conformal (Weyl) sym- α is eventually factorized away from (55) to obtain the local metries. Therefore, one expects chiral and trace anomalies. form in (56). In computing J from (54), it is enough to check To obtain the anomalies we have to compute the expres- that the mass matrix T is invertible on the relevant chiral sions in (56) with the regulators contained inside O2 of Eq. spaces (extracted by the projectors PL and PR). For the Weyl (21). They read symmetry one uses instead the transformation laws in (12) Rλ =−D/(−A)D/(A)PL to find (60) Rλ =−D/(A)D/(−A)PR. 1 σ P 0 c J = 2 L , (58) 1 σ Using the Seeley-DeWitt coefficients a of these regulators 0 2 PR 2 we find the chiral anomaly where it is now crucial to consider that the covariant (under gravity) extension of the mass terms contains a factor of e, 1 1 8 2 − ∂ J a = εabcd F F − ∂ (Aa A2) + (∂ A) 1 1δ a 2 ab cd a see Eq. (23), which brings in a contribution from 2 T T (4π) 6 3 3 to J. This contribution is necessary to guarantee that general (61) coordinate invariance is kept anomaly free in the regulariza- δO tion ( is neglected again for the same reason as before). where Fab = ∂a Ab − ∂b Aa (see “Appendix C” for an outline The infinitesimal Weyl parameter σ is then factorized away of the calculation). It contains normal-parity terms that can from (55) to obtain the second equation in (56). be canceled by the gauge variation of the local counterterm Proceeding in a similar way, we find for the Dirac model 4 d x 2 4 1 a Γ1 = A − A Aa , (62) a i (4π)2 3 3 ∂a J = [tr a (Rψ ) − tr a (Rψ )] (4π)2 2 2 c so that the chiral gauge anomaly takes the form a i 5 5 ∂ J = tr [γ a (Rψ )]+tr γ [a (Rψ )] (59) a 5 ( π)2 2 2 c 1 4 ∂ J a = εabcd F F . (63) a π 2 ab cd a 1 96 T a =− [tr a2(Rψ ) + tr a2(Rψ )]. 2(4π)2 c This is the standard result. Similarly, we compute the trace anomaly, given by All remaining traces are traces on the gamma matrices taken in the standard four dimensional Dirac spinor space. a 1 2 a b 2 2 2 2 T a =− (∂a Ab)(∂ A ) − (∂ A) − A . (4π)2 3 3 3 (64) 4 Anomalies It does not contain any odd-parity contribution. Gauge invariance is broken by the chiral anomaly, but we find that the trace In this section we compute systematically the chiral and trace anomaly can be cast in a gauge invariant form by varying anomalies for the Weyl and Dirac theories described ear- a local counterterm with a Weyl transformation (and then lier. We use, when applicable, two different versions of the restricting to flat space). The (gravity covariant but gauge Pauli–Villars regularization with different mass terms. We noninvariant) counterterm is given by verify that the final results are consistent with each other, √ 4 and coincide after taking into account the variation of local d x g 1 μ ν 1 2 Γ = (∇ A )(∇μ Aν) + RA (65) counterterms. 2 (4π)2 3 6 123 292 Page 8 of 12 Eur. Phys. J. C (2019) 79:292 and the trace anomaly takes the form 4.2 Chiral and trace anomalies of a Dirac fermion 1 T a =− F Fab. (66) For completeness, we now consider the case of the massless a π 2 ab 48 Dirac spinor coupled to vector and axial gauge fields with The counterterms Γ1 and Γ2 are consistent with each other, lagrangian given in Eq. (31). The results are well-known, but and merge into the unique counterterm (needed only at linear we wish to present them for comparison and as a check on order in the metric) our method. The most natural regularization is obtained by √ 4 employing a Dirac mass for the PV fields, but we employ d x g 2 4 1 μ ν 1 2 Γ = A + (∇ A )(∇μ Aν) + RA also a Majorana mass. The latter allows to take a chiral limit 3 ( π)2 4 3 3 6 in a simple way, which we use to rederive the previous results (67) on the Weyl fermion. μν where, of course, A2 = g Aμ Aν and A4 = (A2)2. Thus, we have seen that the trace anomaly of a Weyl 4.2.1 PV regularization with Dirac mass fermion does not contain any contribution from the topological density F F˜ (which on the other hand enters the chiral The relevant regulators are obtained from (40) and read 2 anomaly in (63), as well-known). It can be presented in a Rψ =−D/(A, B) (72) gauge invariant form by the variation of a local countert- 2 Rψ =−D/(−A, B) . erm, and equals half the standard trace anomaly of a Dirac c fermion. These are the main results of our paper. The vector symmetry is guaranteed to remain anomaly free by the invariance of the mass term, while the chiral anomaly 4.1.2 PV regularization with Dirac mass from (59) becomes a 1 abcd For using a Dirac mass we have to include a right handed ∂a J = ε Fab(A)Fcd(A) 5 (4π)2 free fermion in the PV lagrangian as well. The lagrangian is 1 abcd given in (24), and from Eq. (27) one finds the regulators + ε Fab(B)Fcd(B) 3 =−∂/ /( ) Rλ D A PL −16∂ ( a 2) + 4(∂ ) . (68) a B B B (73) =−∂/ /(− ) . 3 3 Rλc D A PR It contains normal-parity terms in the B field. They are can- Then, from the corresponding heat kernel coefficients a2 we celed by the variation of a local counterterm find the chiral anomaly

4 d x 4 4 2 a a 1 1 abcd 1 a 2 1 Γ5 = B − B Ba (74) ∂a J = ε Fab Fcd − ∂a(A A ) + (∂ A) . (4π)2 3 3 (4π)2 6 3 3 (69) so that one ends up with

a It contains noncovariant normal-parity terms, that are can- ∂a J =0 (75) celed by the variation of the local counterterm a 1 abcd ∂a J = ε Fab(A)Fcd(A) d4x 1 1 5 (4π)2 Γ = A4 − AaA 4 2 a (70) (4π) 12 6 1 abcd + ε Fab(B)Fcd(B) . (76) 3 so that the anomaly takes the standard form

a 1 abcd As for the trace anomaly, we ﬁnd from (59) ∂a J = ε FabFcd (71) 96π 2 a 1 2 ab as in the previous section. T a =− Fab(A)F (A) (4π)2 3 Unfortunately, we cannot proceed to compute in a sim- 4 a b 4 2 4 2 ple way the trace anomaly using this regularization, as the + (∂a Bb)(∂ B ) − (∂ B) − B (77) mass term breaks the Einstein and local Lorentz symmetries 3 3 3 as well. The ensuing anomalies should then be computed with the counterterm and canceled by local counterterms, to ﬁnd eventually the √ 4 expected agreement of the remaining trace anomaly with the d x g 2 μ ν 1 2 Γ = (∇ B )(∇μ Bν) + RB (78) one found in the previous section. 6 (4π)2 3 3 123 Eur. Phys. J. C (2019) 79:292 Page 9 of 12 292 that brings it into the gauge invariant form a 1 abcd ∂a J = ε Fab(A)Fcd(A) 5 (4π)2 1 a =− ( ) ab( ) + ( ) ab( ) . 1 abcd T a Fab A F A Fab B F B (79) + ε Fab(B)Fcd(B) . (86) 24π 2 3

All these counterterms merge naturally into the complete As for the trace anomaly, we find counterterm √ 4 1 4 4 4 d x g 4 4 2 μ ν 1 2 T a =− (∂ A )(∂a Ab) − (∂ A)2 − A2 Γ7 = B + (∇ B )(∇μ Bν) + RB . a ( π)2 a b (4π)2 3 3 3 4 3 3 3 (80) 4 a b 4 2 4 2 + (∂a Bb)(∂ B ) − (∂ B) − B (87) 3 3 3 4.2.2 PV regularization with Majorana mass and using the counterterm √ 4 At last, we consider the regularization with a Majorana mass. d x g 2 μ ν 2 μ ν Γ = (∇ A )(∇μ Aν) + (∇ B )(∇μ Bν) As both vector and chiral symmetries are broken by the mass 10 (4π)2 3 3 term, we expect anomalies in both U(1) currents. From Eq. 1 (43) we find the regulators + R(A2 + B2) (88) 3 Rψ =−D/(−A, B)D/(A, B) (81) we get the final gauge invariant form Rψ =−D/(A, B)D/(−A, B). c 1 Then, we compute from (59) T a =− F (A)Fab(A) + F (B)Fab(B) . (89) a π 2 ab ab 24 a 1 2 abcd 4 ∂a J = ε Fab(A)Fcd(B) + (∂ A) Also the counterterms employed in this section are con- (4π)2 3 3 sistent with each other, and combine into a unique final coun- 16 a 2 2 32 a b − ∂a[A (A + B )]− ∂a(B Ab B ) terterm, which we report for completeness 3 3 √ (82) 4 d x g 2 μ ν 2 μ ν Γ11 = (∇ A )(∇μ Aν) + (∇ B )(∇μ Bν) and (4π)2 3 3 1 2 2 4 2 2 2 16 μ 2 + R(A + B ) + (A + B ) + (A Bμ) a 1 1 abcd 3 3 3 ∂a J = ε Fab(A)Fcd(A) 5 (4π)2 3 4 εμνρσ + √ Bμ Aν Fρσ (A) . (90) 1 abcd 4 3 g + ε Fab(B)Fcd(B) + (∂ B) 3 3 16 a 2 2 32 a b Evidently, the anomalies computed with the Majorana − ∂a[B (A + B )]− ∂a(A Ab B ) . 3 3 mass coincide with those obtained with the Dirac mass, after (83) using local counterterms. The results of this section can be projected consistently

The counterterm Γ8 + Γ9 to recover the chiral and trace anomalies of a Weyl fermion. = → 1 Indeed, one can consider the limit Aa Ba 2 Aa.In +γ 5 d4x 4 16 2 this limit, a chiral projector P = 1 emerges inside the Γ = (A2 + B2)2 + (Aa B )2 − AaA L 2 8 ( π)2 a a Dirac lagrangian (31) to reproduce the Weyl lagrangian (1). 4 3 3 3 In addition, in the coupling to gravity, the right handed com- 2 a − B Ba 3 ponent of the Dirac ﬁeld can be kept free, both in the kinetic and PV mass term, preserving at the same time covariance of d4x 8 Γ = εabcd B A (∂ A ) (84) the mass term of the left handed part of the PV Dirac fermion. 9 ( π)2 a b c d 4 3 Then, the right handed part decouples completely and can be ignored altogether. Thus, one may verify that the anomalies allows to recover vector gauge invariance, and the anomalies in Sect. 4.1.1 are reproduced by those computed here, includ- take the form = → 1 ing the counterterms, by setting Aa Ba 2 Aa (note that the current J a in Sect. 4.1.1 corresponds to half the sum of a ∂a J =0 (85) a a J and J5 of this section). 123 292 Page 10 of 12 Eur. Phys. J. C (2019) 79:292

As ﬁnal remark, we have checked that terms proportional The hermitian chiral matrix γ 5 is given by to δO in (54) never contribute to the anomalies computed γ 5 =−γ 0γ 1γ 2γ 3 thus far, as the extra terms vanish under the Dirac trace. i (A.3) and used to deﬁne the chiral projectors

5 Conclusions 1 + γ5 1 − γ5 PL = , PR = (A.4) 2 2 We have calculated the trace anomaly of a Weyl fermion that split a Dirac spinor ψ into its left and right Weyl com- coupled to an abelian gauge field. We have found that the ponents anomaly does not contain any odd-parity contribution. In particular, we have shown that the Chern–Pontryagin term ψ = λ + ρ, λ = PL ψ, ρ = PRψ. (A.5) F F˜ is absent, notwithstanding the fact that it satisfies the consistency conditions for Weyl anomalies. Of course, the The charge conjugation matrix C satisfies chiral anomaly implies that gauge invariance is broken. Nev- − Cγ aC 1 =−γ aT , (A.6) ertheless the trace anomaly can be cast in a gauge invariant form, equal to half the standard contribution of a nonchiral it is antisymmetric and used to define the charge conjugation Dirac fermion. of the spinor ψ by While this result seems to have no direct implications for −1 T the analogous case in curved background, it strengthens the ψc = C ψ (A.7) findings of ref. [6]. where the roles of particle and antiparticle are interchanged. Recently, a generalized axial metric background has been Note that a chiral spinor λ has its charge conjugated field λ developed in [2,3] to motivate and explain the appearance of c of opposite chirality. A Majorana spinor μ is a spinor that the Pontryagin term in the trace anomaly of a Weyl fermion, equals its charged conjugated spinor which however is in contradiction with the explicit calculation of [6]. Perhaps it would be useful to apply the methods μ = μc. (A.8) used here in the axial metric background to clarify the situation in that context, and spot the source of disagreement. This constraint is incompatible with the chiral constraint, and Majorana–Weyl spinors do not exist in 4 dimensions. Acknowledgements We thank Stefan Theisen for extensive discus- We find it convenient, as a check on our formulas, to use sions, suggestions, critical reading of the manuscript, and hospitality at the chiral representation of the gamma matrices. In terms of AEI. In addition, FB would like to thank Loriano Bonora and Claudio × Corianò for useful discussions, and MB acknowledges Lorenzo Casarin 2 2 blocks they are given by for interesting discussions and insights into anomalies and heat kernel. 1 σ i γ 0 =− 0 ,γi =− 0 i i i (A.9) Data Availability Statement This manuscript has associated data in a 1 0 −σ 0 data repository. [Authors’ comment: The datasets generated and anal- ysed during the current study are available from the authors on reason- where σ i are the Pauli matrices, so that able request.] 1 0 0 1 γ 5 = ,β= iγ 0 = . (A.10) Open Access This article is distributed under the terms of the Creative 0 −1 1 0 Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, The chiral representation makes evident that the Lorentz gen- and reproduction in any medium, provided you give appropriate credit erators in the spinor space Mab = 1 [γ a,γb]= 1 γ ab take a to the original author(s) and the source, provide a link to the Creative 4 2 Commons license, and indicate if changes were made. block diagonal form Funded by SCOAP3. 1 σ i 0 i σ k 0 M0i = , Mij = εijk (A.11) 2 0 −σ i 2 0 σ k Appendix A: Conventions and do not mix the chiral components of a Dirac spinor (as 5 We use a mostly plus Minkowski metric ηab. The Dirac matri- γ is also block diagonal). The usual two-dimensional Weyl ces γ a satisfy spinors appear inside a four-dimensional Dirac spinor as fol- a b ab lows {γ ,γ }=2η (A.1) l l 0 and the conjugate Dirac spinor ψ is defined using β = iγ 0 ψ = ,λ= ,ρ= (A.12) r 0 r by ψ = ψ†β. (A.2) where l and r indicate two-dimensional independent spinors of opposite chirality. In the chiral representation one may 123 Eur. Phys. J. C (2019) 79:292 Page 11 of 12 292

2 a take the charge conjugation matrix C to be given by with V a matrix potential and ∇ =∇ ∇a constructed with ∇ = ∂ + σ 2 a gauge covariant derivative a a Wa that satisfies = γ 2β =− 0 C i −σ 2 (A.13) 0 [∇a, ∇b]=∂a Wb − ∂bWa +[Wa, Wb]=Fab. (B.23) and satisfies The trace of the corresponding heat kernel is perturba- − ∗ C =−C T =−C 1 =−C† = C (A.14) tively given by (some of these relations are representation dependent). In the − − Tr Je isH = d D x tr J(x) x|e isH|x chiral representation the Majorana constraint (A.8) takes the form ∞ D i n ∗ = d x tr [J(x)an(x, H)](is) l iσ 2r ( π ) D μ = μ → = (A.15) 4 is 2 n=0 c r −iσ 2l∗ d D x = i tr [J(x)(a (x, H) which shows that the two-dimensional spinors l and r cannot D 0 (4πis) 2 be independent. The Majorana condition can be solved in + ( , ) + ( , )( )2 terms of the single two-dimensional left-handed spinor l by a1 x H is a2 x H is + a (x, H)(is)3 +···)] (B.24) l 3 μ = (A.16) −iσ 2l∗ where the symbol “tr” is the trace on the remaining dis- which, evidently, contains the four-dimensional chiral spinors crete matrix indices, J(x) is an arbitrary matrix function, λ and λc defined by and an(x, H) are the so-called Seeley-DeWitt, or heat kernel, coefficients. They are matrix valued, and the first few l 0 λ = ,λ= ∗ . (A.17) ones are given by 0 c −iσ 2l ( , ) = 1 In a four-dimensional spinors notation one can write a0 x H a1(x, H) =−V μ = λ + λ . (A.18) (B.25) c 1 1 1 a (x, H) = V 2 − ∇2V + F 2 . Alternatively, the Majorana condition can be solved in terms 2 2 6 12 ab of the two-dimensional right-handed spinor r by ∇ = ∂ +[ , ] As V is allowed to be a matrix, then a V a V Wa V , iσ 2r ∗ etc.. μ = (A.19) r In the main text, the role of the hamiltonian H is played by the various regulators R, and is ∼ 1 ,seeEq.(55). ρ ρ M2 which contains the four-dimensional chiral spinors and c = In D 4thes-independent term is precisely the one with 2 ∗ 0 iσ r a2(x, H), which is the coefficient producing the anomalies ρ = ,ρ= (A.20) r c 0 in 4 dimensions (we use a minkowskian set-up, but justify the heat kernel formulas by Wick rotating to euclidean time and μ = ρ + ρ . This solution is of course the same as the c and back, when necessary). previous one, as one identifies λ = ρ . c More details on the heat kernel expansion are found in The explicit dictionary between Weyl and Majorana [20,21], where the coefficients appear with the additional spinors shows clearly that the field theory of a Weyl spinor is coupling to a background metric. They have been recom- equivalent to that of a Majorana spinor, as Lorentz symmetry puted with quantum mechanical path integrals in [24], a use- fixes uniquely their actions, which are bound to be identical. ful report is [25], while in [26] one may find the explicit Finally, we normalize our ε symbols by ε =−1 and 0123 expression for a (x, H), originally calculated by Gilkey [27], ε0123 = 1, so that 3 which is relevant for calculations of anomalies in 6 dimen- 1 tr (γ 5γ aγ bγ cγ d ) = iεabcd. (A.21) sions. 4

Appendix C: Sample calculations Appendix B: The heat kernel As an example of the calculations leading to the results of We consider an operator in ﬂat D dimensional spacetime of Sect. 4, we consider the case of the PV regularization with the form Majorana mass used for the Weyl model in Sect. 4.1.1.One H =−∇2 + V (B.22) regulator needed there is 123 292 Page 12 of 12 Eur. Phys. J. C (2019) 79:292

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