Emergent Adler-Bardeen theorem

We consider a QEDd+1, d = 1, 3 lattice model with emergent Lorentz or chiral symmetry, both when the interaction is irrelevant or marginal. While the correlations present symmetry breaking corrections, we prove that the Adler-Bardeen (AB) non-renormalization property holds at a non-perturbative level even at finite lattice: all radiative corrections to the anomaly are vanishing. The analysis uses a new technique based on the combination of non-perturbative regularity properties obtained by exact renormalization Group methods and Ward Identities. The AB property, essential for the renormalizability of the standard model, is therefore a robust feature imposing no constraints on possible symmetry breaking terms, at least in the class of lattice models considered.


JHEP03(2020)095
Our results are based on a novel technique based on the combination of constructive regularity properties obtained by exact Renormalization Group (RG) methods and Ward Identities. The contribution of irrelevant terms at each step is essential and fully taken into account. The results are fully non-perturbative, as physical quantities are expressed in terms of series whose convergence is established in presence of a finite photon mass, as consequence of cancellations due to Pauli principle (see [12] for an introduction to such methods). This is a major difference with respect to other approaches to the anomaly which give results valid only order by order, see e.g. [13,14]. The strategy of proof was used in [15] for irrelevant interactions and is here extended to the marginal case. Even if the validity of the AB property is proved for the U(1) gauge group and with a photon mass regularization, the result indicates that the anomaly cancellation condition is common between the continuum theory and the lattice theory and suggests that anomaly-free chiral gauge theory, as the Standard model, can be formulated non-perturbatively by lattice formulation.

Lattice models and anomaly non-renormalization
The massless lattice QED d+1 model we consider is the interacting extension of the Nielsen-Ninomiya anomaly simulation [11], where the interaction with a quantum photon field is included. The detailed form of the lattice has no importance and we do a specific choice just for definiteness.

JHEP03(2020)095
In addition to Lorentz invariance, Dirac particles verify gauge and chiral symmetry, implying the conservations of the d + 1 current J µ =Ψγ µ Ψ and axial current J µ =Ψγ µ γ 5 Ψ. It is also convenient to write Ψ = (Ψ + , Ψ − ) andΨ = Ψ + γ 0 , so that the Dirac propagator can be written as, in d = 1 and d = 3 respectively. Let us look at the lattice propagator (2.9) restricting to momenta close to ±ζ/a. We introduce a smooth compact support function χ ω (k), with ω = ± non vanishing only for |k − ωζ/a| ≤ 1/(10a) withζ = (0, ζ) in d = 1 orζ = (0, 0, 0, ζ) in d = 3. In d = 1 we define g ω (k − ωζ/a) = χ ω (k) g(k) and in d = 3 we define with δ i,j,+ = 1, δ i,j,− = (−1) i+j . The function g ω is the propagator restricted to momenta around ±ζ/a and, setting v = v 1 = 1 we get, calling k − ωζ/a = q The lattice models (2.1) and (2.4) admit therefore an emerging description in terms of massless Dirac partcles; the propagator for momenta far from the inverse spacing has a Lorentz invariant part up to corrections which are small but non vanishing. Let us see what happens to the conservation of the currents. The current in a lattice theory can be introduced using the Peierls substitution. In d = 1 one introduces an interaction with an external gauge field by writing with a similar expression holding in d = 3; the current is defined as j x = ∂H 0 (A) ∂A(x) | A(x)=0 , and the lattice density is ρ x = ψ + x ψ − x , and they can be combined in j µ = (ρ, j 1 , . . . , j d ). The lattice density and current vertex are close, in the sense of correlations and up to corrections as in (2.12), to the Dirac onesΨγ µ Ψ. Such corrections however do not prevent the conservation of the lattice current in the sense of Ward Identities (see (2.26) below), as the Peierls substitution ensures gauge invariance at a lattice level.
A different situation is encountered in the case of chiral currents. Following [11] one can indeed define an analogue of the chiral density and current in the lattice model, by the requirement that it is close to the Dirac chiral currentΨγ µ γ 5 Ψ in the sense of correlations, up to corrections. The lattice chiral density can be defined as the difference of densities of fermions around ±ζ/a, that is in d = 1 . The definition of the axial current is given in a similar

JHEP03(2020)095
way inserting a factor sin ka or sin k 3 a in the Fourier transform of the current. The axial symmetry is however broken and there is no conservation of axial current. Let us introduce now a dynamical photon fieldĀ µ (x) (not to be confused with the external field A µ ) with integration P (dĀ) and propagator is a cut-off function vanishing for momenta larger than O(1/a) and M a = M in d = 1 and M a = a −1 M in d = 3 is a regularizing mass (such a regularization is the one adopted in [5]). For a non-perturbative analysis we find convenient to integrate out the boson field getting a purely fermionic theory, that is The lattice model we consider is therefore defined by the following generating function x,i , ψ ε y,j } = 0, ε, ε = ± (with abuse of notation we denote the Grassmann variables with the same symbol as fields), P (dψ) is the fermionic integration with propagator (2.9) and where the first term is the interaction, j µ = (ρ, j 1 , . . . , j d ) are the lattice density and current expressed in terms of Grassmann variables, j µ (A) is obtained by j µ by the Perierls substitution, dx is a notation for dx 0 x , λ = e 2 is the coupling and the second term is a counterterm to fix the singularity of the propagator, Finally A µ , A 5 µ , φ are external fields (φ is a Grassman variable) and derivatives of W with respect to A µ , A 5 µ , φ give the correlations of the current, chiral current or fermionic field respectively. In order to ensure gauge invariance for the external field A µ (see (2.25) below) we define with H 0 (A) given by (2.13) with Grassmann variables replacing fields and j 5 µ,x (A) is obtained by j 5 µ,x by the Peierls substitution; in particular the gauge invariant chiral density is with A 3 (s) = A 3 (x 0 , x 1 , x 2 , s) and Z 5 0 is a renormalization to be properly fixed, see below; a similar expression holds for the axial current j 5 i,x (A), and Z 5 i are the corresponding renormalizations.

JHEP03(2020)095
The correlations are obtained by differentiating the generating function with respect to the external fields; in particular Similarly we introduce the current correlations By Feynman graph expansion one can see that the correlations of (2.17) coincide in the formal limit in which regularizations are removed a → 0, M → 0 with massless QED in the Feynman gauge. The lattice breaks the Lorentz symmetry, so that the parameters t, t have to be chosen as function of the coupling λ to fix the light velocity equal to c = 1; ν is a counterterm to fix the position of the singularity. The chiral symmetry is also broken and one has to fix the constants Z 5 µ in order to ensure the following condition, if k = q + ωζ/a, q, p small, ω = ± The AB non-renormalization means that the anomaly acquires no corrections provided that the normalizations are fixed so that (2.24) holds, see e.g. [6]. While the lattice breaks chiral and Lorentz symmetry (which are only emergent), our model respects exactly gauge symmetry, as by construction and from this we get the following Ward Identity expressing the conservation of the current p µ Γ µ,µ 1 ,...,µn = 0 (2.26) and the relation The chiral symmetry is broken by the lattice so that the analogue of (2.26) for the chiral current is not true. In the emergent continuum theory the chiral symmetry holds exactly but nevertheless p µ Γ 5 µ,µ 1 ,...,µn is non vanishing, what is precisely the quantum anomaly [4]. In [11] it was shown that, in the non-interacting case, one has in the lattice theory p µ Γ 5 µ,ν (p) = 1 π ε µ,ν p µ in d = 1 and p µ Γ 5 µ,ν,σ (p 1 , p 2 ) = 1 2π 2 p 1,α p 2,β ε α,β,ν,σ in d = 3, that is one gets the same result as the continuum theory. We investigate what happens to the anomaly in presence of interaction with a finite lattice.

JHEP03(2020)095
Theorem. For small λ and suitable ν, t, t and Z 5 µ chosen so that (2.24) holds, the correlations of (2.17) are, respectively for d = 1 and d = 3 and R(q) non vanishing and |R(q)| ≤ Ca|q|; moreover, up to higher order terms in p The above result is an emergent Adler-Bardeen theorem, as (2.29) says that there are no interaction corrections to the anomaly, even in presence of a finite lattice; its value coincides with the one of non interacting Dirac fermions. In contrast symmetry breaking terms produce non vanishing corrections to the correlations, see (2.28). The above result is rigorous, as the presence of the lattice allows to get a full non-perturbative control on the functional integrals.
In the rest of the paper a proof of the above result is provided. In section 3 we describe the Renormalization Group analysis for the lattice model (2.17), and we get the main regularity properties for the kernels of the effective potential. In section 4 we get the anomaly non-renormalization in the d = 3 case, and in section 5 in the d = 1 case; finally section 6 is devoted to conclusions.

Renormalization Group
As we are interested in the possible breaking of the AB property due the irrelevant terms, one needs an exact RG analysis in order to take them fully into account [17,18]. The starting point is the decomposition of the propagator in higher and lower energy degrees of freedom, that is where g (N ) (k) and g (≤N −1) (k) are equal to g(k) times f N (k) and χ N −1 (k) respectively, where χ N −1 (k) is a compact support function selecting momenta such that |k−ωζ/a| ≤ γ N with γ > 1, γ N = 1/(10a) and f N = 1 − χ N −1 . We can use the decomposition property P (dψ) = P (dψ (≤N −1) )P (dψ (N ) ), where P (dψ (≤N −1) ) and P (dψ (N ) ) have propagator g (≤N −1) (x) and g (N ) (x). The field ψ (N ) represents the highest energy degree of freedom; its propagator g (N ) (x) decays at large distances faster than any power with rate γ N and is bounded by γ dN , and it can be integrated out safely. Note that χ N −1 (k) as a function of k has support in two disconnected regions around ±ζ/a; we can therefore, after shifting the momenta, write

JHEP03(2020)095
In conclusion we get n! E T N (V ; n) and E T N is the truncated expectation, that is the sum of connected Feynman graphs. The effective potential V (N −1) is given by where x = x 1 , . . . , x l , y = y 1 , . . . , y m , j = 1, 2 in d = 3 or j = 1 in d = 1, ε i = ±, µ = 0, 1 in d = 1 and µ = 0, 1, 2, 3 in d = 3, ω = ± and σ = 0, 5 (A 0 µ,y ≡ A µ,y ). Note that the RG integration step has two effects; the first is that the potential is now expressed as sum over monomials of fields of every order and the second that the field is splitted in two components labeled by ω = ±. The kernels W (N −1) l,m are expressed by convergent series in λ; this follows from the representation g (N ) (x − y) = (f x , g y ) where (, ) is a suitable scalar product and the fact that fermionic expectation can be written as the determinant of a Gram matrix M with elements (f [19] or [12].
The effective potential V h can be decomposed in an irrelevant part, containing all the monomials with negative scaling dimension D = (d + 1) − dn/2 − m, and a relevant and marginal part D ≥ 0. The marginal term linear in A have the form

JHEP03(2020)095
The factors Z µ,h or Z 5 µ,h are the renormalizations of the current and axial current respectively. The relevant term is γ h ν h ω dxψ + x,ωᾱ ψ − x,ω withᾱ = 1 in d = 1 and σ 3 in d = 3 and ν has to fixed so that so that l,m are obtained, see e.g. [12], by contracting the effective potentials at previous scales, and one can distinguish the contributions W b,l,m obtained contracting at least an irrelevant or relevant ν term; the series expansion are convergent and the following bound holds [12] d(x/x 1 )|W Note that there is an essential difference between the d = 3 and d = 1 case; in the first case to W (h) a,n,m no vertices with more than two fermionic lines contribute, while in the second also the local vertices quartic in ψ contribute.
The flow of the running coupling constants and renormalizations is quite different. In the d = 3 case [20,21] the terms with more than 2 fields have negative dimension so that by (3.9). We choose the parameters so that and similar expressions for Z h , Z µ,h . In the d = 1 case [22][23][24] in contrast the interaction is marginal and the beta function of the renormalizations is given by and similar expressions holds for Z µ,h and Z 5 µ,h . It turns out that, as a consequence of the emerging chiral symmetry, the beta function for λ h is asymptotically vanishing λ h−1 = λ h +O(λ 2 γ h−N ) and the same is true for the velocity. Note that, as λ h → λ −∞ = λ+O(λ 2 ), then the renormalization can be singular as h → −∞; in particular The conclusion of the above analysis is that, if we suitable fix the velocities v 0 and the counterterms ν one gets (2.28), that is Lorentz invariance emerges up to corrections which are small if q is far from the lattice scale.

JHEP03(2020)095
4 Anomaly non-renormalization; the irrelevant case In d = 3 the interaction is irrelevant and, by (3.9), for k ∼ ωζ/a, p ∼ 0, ω = ± G 2 (k) = 1 Z g(k)(1 + O(aq)) and with |R| ≤ Ca(|q|, |q + p|). Note the perfect proportionality of the vertex function to Z µ , Z 5 µ which is not true in the marginal case (the R term is not subdominant). We know from the previous section that Z, Z µ , Z 5 µ are expressed by convergent series depending on all details at the lattice scale; the Ward Identity (2.27) implies the exact relation A similar identity is not true for Z 5 µ and generically Z 5 µ /Z µ is a non trivial function of λ. Therefore in order to ensure the validity of (2.24) we choose The anomaly coefficient is expressed in terms of By (3.9) it is bounded by so that it is continuous as a function of p 1 , p 2 ; it is however not differentiable as each derivative produces an extra γ −h . The continuity combined with Ward Identites (2.25) are sufficient to prove that Γ 5 µ,µ 1 ,µ 2 (0, 0) = 0 without any explicit computation: it is sufficient to write from (2.25) p 1,µ 1 Γ 5 µ,µ 1 ,µ 2 (p 1 , p 2 ) = 0 at p 1,1 =p 1 and zero otherwise and use continuity. One would be tempted to iterate this argument for the derivative of Γ 5 µ,µ 1 ,µ 2 , but that is impossibile for the lack of differentiability, and indeed Γ 5 µ,µ 1 ,µ 2 has non vanishing derivatives.
Regularity properties are a very efficient tool to get information on the property of the anomalies, once that Γ 5 µ,µ 1 ,µ 2 (p 1 , p 2 ) is suitable decomposed in order to get advantage from the dimensional gain in (3.9). We write, p = p 1 + p 2 where ∆ is the Schwinger term and j the interacting current (obtained by the derivative in A). ∆ has the same bound as the terms with m = 2, 1 hence they are differentiable.
JHEP03(2020)095 In absence of interaction λ = 0  5 µ,p ;  ν,p 1 ;  σ,p 2 is expressed by the triangle graph. In presence of interaction, the RG analysis of the previous section says that where the first term, containing only marginal source terms, is the triangle graph with propagators g (h) /Z h and vertices associated to Z µ,h , Z 5 µ,h , while the second is a series of terms with an arbitrary number of quartic interactions, see figure 1. According to the bound (3.9) we have a,0,3 is not. We can replace in the renormalized triangle graph the values of Z µ,h , Z 5 µ,h , v h with their limiting value; the difference has again an extra O(γ h−N ) so gives a differentiable contribution. Summing over the scale h has the effect that the cut-off f h of single scale propagators add up to χ = N h=−∞ f h so that we get at the end where the second term is differentiable while I µ,ν,σ (p 1 , p 2 ) is the relativistic triangle graph with propagators χ(k) −i k , that is with a momentum cut-off. In conclusion where H 5 µ,ν,σ is continuously differentiable. By (4.2), (4.3) we get In addition the contribution from the first term in (4.10) can be explicitly computed, see [15], and one gets p µ I µ,ν,σ (p 1 , p 2 ) = 1 6π 2 p 1,α p 2,β ε αβνσ p 1,ν I µ,ν,σ (p 1 , p 2 ) = 1 6π 2 p 1,α p 2,β ε αβµσ (4.12)
JHEP03(2020)095 We can take advantage from the fact that the model (5.3) verifies global and axial symmetries; however local symmetries are broken by the presence of the momentum cutoff and this produces extra anomalous terms in the WI for the global and axial current. Note indeed that, if D ω (k) = −ik 0 + ωk ω (k + p) (5.6) with C(k, p) = D ω (k)(χ −1 K (k) − 1) − D ω (k + p)(χ −1 K (k + p) − 1) (the r.h.s.would be zero in absence of cut-off). The presence of this extra term produce an additional factor in the WI, see figure 2; as proven in [22][23][24] in the K → ∞ limit the following WI for the vertex and chiral vertex are obtained and τ = λ ∞ /4π. The extra term in the WI produced by the C-term reduces, in the limit K → ∞, to the vertex function times the constant τ (which is the graph for the anomaly in d = 1 with momentum cut-off). The fact that the vertex and 2-point function of (5.7) and lattice model (computed at q + ωζ/a with q small) are close up to O(aq) terms says that the first of the WI (5.7) coincides with (2.27); this imposes constraints for the parameters of effective QFT (5.3), that is We have now to choose Z 5 µ by (2.24); from (5.7) in the limit p 0 → 0, p → 0