Vanishing of the anomaly in lattice chiral gauge theory

The anomaly cancellation is a basic property of the Standard Model, crucial for its consistence. We consider a lattice chiral gauge theory of massless Wilson fermions interacting with a non-compact massive U(1) field coupled with left and right handed fermions in four dimensions. We prove in the infinite volume limit, for weak coupling and inverse lattice step of the order of boson mass, that the anomaly vanishes up to subleading corrections and under the same condition as in the continuum. The proof is based on a combination of exact Renormalization Group, non perturbative decay bounds of correlations and lattice symmetries.


Introduction and Main results
1.1.Chiral gauge theory.The perturbative consistence (renormalizability) of the Standard Model relies on the vanishing of the anomalies, achieved under certain algebraic conditions [1] severely constraining the elementary particles charges and providing a partial explanation of the charge quantization.In order to go beyond a purely perturbative framework in terms of diverging series [2], one needs a lattice formulation with functional integrals with cut-off much higher than the experiments scale; due to triviality [3], [4], the cut-off cannot be completely removed, at least in the Electroweak sector, hence the theory can be seen as an effective one.
One expects a relation between the perturbative renormalizability properties and the size of the cut-off.The electroweak theory is renormalizable [5], [6] so that a construction up to exponentially large cut-off could be in principle possible, and such cut-off is much higher than the scales of experiments.However, this requires as a crucial prerequisite that the anomalies cancel, at least to a certain extent.This rises the natural question: does the anomaly cancel at a non-perturbative level with finite lattice, under the same condition as in the continuum?
In the continuum, the cancellation is based on compensations at every order [7] based on dimensional regularizations and symmetries, but finite lattice cut-off produce corrections and the question is if they cancel or not.Jacobian arguments are used to support vanishing of higher orders contributions to anomalies but are essentially one loop results, as shown in [8].Topological arguments explain the anomaly cancellation on a lattice [9] with classical gauge fields, but in the quantum case they work only at lowest order (one loop).The cancellation would be obtained if a non-perturbative regulator for lattice chiral gauge theories could be found, but this is a long standing unsolved problem and only order by order results are known [10], [11].
We consider a lattice chiral gauge theory, given by 2N massless fermions in four dimensions, labeled by an index i = 1, ..., 2N ; we also define the indices i 1 = 1, ..., N and i 2 = N +1, ..., 2N .If the gamma matrices are and σ L µ = (σ 0 , iσ), σ R µ = (σ 0 , −iσ), 1 the formal continuum action is given by the following expression: with µ = (0, 1, 2, 3) and F µν = ∂ µ A ν − ∂ ν A µ .Note that the R fermions of kind i 1 and the L fermions of kind i 2 decouple and are fictitious, non interacting degrees of freedom, which are convenient to introduce in view of the lattice regularization, see eg [12], [13].The total current coupled to A µ is and the axial and vector part of the current is with ε i 1 = − ε i 2 = 1, j µ,i,x = ψi,x γ µ ψ i,x , j 5 µ,i,x = ψi,x γ 5 γ µ ψ i,x and ψ i,x = (ψ − i,L,x , ψ − i,R,x ), ψi,x = (ψ + i,L,x , ψ + i,R,x )γ 0 .Note the chiral nature of the theory, as in the current the fermion with different chirality have different charges.An example of chiral theory is obtained setting Q i 2 = 0; in such a case one is describing N fermions with the same chirality interacting with a gauge field.A physically more important example is given by the U (1) sector of the Standard Model with no Higgs and massless fermions; in this case N = 4, i 1 = (ν 1 , e 1 , u 1 , d 1 ) are the left handed components and i 2 = (ν 2 , e 2 , u 2 , d 2 ) the right handed of the leptons and quarks.A formal application of Noether theorem with classical fermions and bosons says that the invariance under phase and chiral symmetry,implying the current conservation ∂ µ j T µ = 0.If the fermions are quantum (and the bosons classical) the conservation of current is reflected in Ward Identities, and it turns out that anomalies generically break the conservation of j T µ,i,x In the elecroweak sector the physical values (6), if Q are the hypercharges and an index for the three colours of quarks is added.Remarkably the hyperchrges (and therefore the charges) are constrained to physical values by purely quantum effects.The question is therefore if in a lattice regularization of (3) and considering A µ a quantum field, the chiral current is conserved under the same condition (6) at a non-perturbative level.
1.2.The lattice chiral gauge theory.The lattice chiral gauge theory is defined by its generating function e W(J,J 5 ,φ) = P (dA) P (dψ)e V (ψ,A,J)+Vc(ψ)+B(J 5 ,ψ)+(ψ,φ) where periodic boundary conditions) and the bosonic integration is with is the action of a non-compact lattice U (1) gauge field with a gauge fixing and a mass term, is expressed by the Wick rule with covariance 4 .The bosonic truncated expectation is expressed by the Wick rule restricted to the connected terms.We denote by ψ ± i,s,x the Grassmann variables, with i = 1, .., 2N the particle index; s = L, R the chiral index; anti-periodic boundary conditions are imposed and We define 4 .The fermionic gaussian measure is defined as, where N ψ a normalization and, if We can write therefore The fermionic simple expectation is expressed by the anticommutative Wick rule with covariance with The interaction is with with, if i 1 = 1, , N and and : e ±iaλQ i Aµ(x) := e ±iλQ i aAµ(x) e Finally the source term is . ν i and Z 5 i,s are parameters to be fixed by the renormlization conditions, see below.
Remark.The term proportional to r in S F (16) is called Wilson term.If r = 0 the fermionic propagator g i,k has, in the L → ∞ limit, several poles; this has the effect that the low energy behaviour of the lattice theory would not correspond to the continuum target theory (3); the presence of the Wilson term r = 0 has the effect that only the physical pole k = 0 is present but the chiral symmetry is broken [14].
1.3.Physical observables.The fermionic 2-point function is and the Fourier transform is The vertex functions are The Fourier transform is and similarly is defined Γ 5,Λ µ,i ′ s (k, p).The three current vector V V V and axial AV V correlations are and 1.4.Ward Identities.The correlations are connected by relations known as Ward Identities.
They can be obtained by performing the change of variables with α x is a function on aZ 4 , with the periodicity of Λ.Let Q(ψ + , ψ − ) be a monomial in the Grassmann variables and Q α (ψ + , ψ − ) be the monomial obtained performing the replacement (31) in as both the left-hand side and the right-hand side of (32) are zero unless the same Grassmann field appears once in the monomial, hence the fields ψ + i,s,x ,ψ − i,s,x come in pairs and the α dependence cancels.By linearity of the Grassmann integration, the property (32) implies fhe following identity, valid for any function f on the finite Grassmann algebra: with f α (ψ) the function obtained from f (ψ), after the transformation (31).We apply now (33) to (7); the phase in the non-local terms can be exactly compensated by modifying J, that is we get W (J, J 5 , φ) = W (J + d µ α, J 5 , e iQα φ) (34) where J + d µ α is a shorthand for J µ,x + d µ α x and e iQα φ is a shorthand for e ±iQ i αx φ ± i,s,x ; by differentiating we get the Ward Identities (WI) Remark The above Ward Identities represent the conservation of the vector part of the current coupled to the gauge field A µ ; in particular the first is the lattice counterpart of ∂ µ < j T,V µ ; j T,V ν 1 ; ...j T,V νn > T = 0, see (5).
1.5.Main result.Our main result is the following, denoting by lim L→∞ S Λ i,s,s ′ (k) and similarly the other correlations.
Theorem 1.1.Let us fix r = 1 and M a ≥ 1.There exists λ 0 , C independent on L, a, M such that, for |λ| ≤ λ 0 (M a), it is possible to find ν i , Z 5 i,s continuous functions in λ such that 2) The AVV correlation verifies

Remarks
(1) The correlations are written in the form of expansions which are convergent in the limit of infinite volume provided that the lattice cut-off is smaller than the boson mass.(2) The counterterms ν i are chosen so that the fermions remain massless in presence of interactions; the parameters Z 5 i,s are fixed so that the charge associated to the vector and axial current are the same, a condition present also at a perturbative level [7].
(3) Under the condition [ ) expressing the conservation of the chiral current in the sense of correlations and up to subdominant terms for momenta far from the cut-off.The vanishing of the anomaly, obtained up to now only at a purely perturbative level, is proved with a finite lattice cut-off, even if the cut-off breaks important symmetries [14] on which the perturbative cancellation were based, like the Lorentz or the chiral one, and excluding non perturbative effects.The anomaly cancellation condition is the same as in the continuum case.The lattice regularization plays an essential role; with momentum one a much weaker result holds [15].(4) Anomalies are strongly connected with transport properties in condensed matter [16]- [18] and we use indeed techniques recently developed for the proof of universality properties in metals to the anomaly cancellation on a lattice [19]- [27].Such methods have their roots in the Gallavotti tree expansion [28], the Battle-Brydges-Federbush formula [29] and the Gawedzki-Kupiainen-Lesniewski formula [30], [31] (see eg [32] for an introduction).
1.6.Future perspectives.We have constructed the theory assuming that 1/a ≤ (λ 0 /|λ|)M , that is the cut-off is smaller than the boson mass and we have established (36) for generic values of the coupling.In this regime after the integration of the A µ the theory have scaling dimension D = 4 + n − 3n ψ /2 if n is the order and n ψ the number of fields.This requires that the "effective coupling" λ 2 /M 2 times the energy cut-off must be not too large so that the expansions are convergent.In order to reach higher cut-off one notes that the boson propagator ( 11) is composed by two terms; one which behaves as O(1/k 2 ) for k 2 >> M 2 and the other which is O(1) for k 2 >> M 2 .If the second term does not contribute the scaling dimension improves and it corresponds to a renormalizable theory D = 4 − 3n ψ /2 − n A , so in principle one can consider cut-offs higher than M and up to an exponentially large values |λ 2 log a| ≤ ε 0 .In order to have that the second term does not contribute full gauge invariance (broken in our case by the mass and gauge fixing term) is not necessarily required but is sufficient the gauge invariance in the external fields, expressed in the form of Ward Identities.
It is indeed known that renormalizability is preserved in QED, at the perturbative level, even if a mass is added to photon, see e.g.[33], [34] ; if one restricts to gauge invariant observables the contribution of the not-decaying term of the propagator is vanishing as consequence of the current conservation.To get exponentially high cut-off in d = 4 QED at a non-perturbative level is technically demanding, as it would require a simultaneous decomposition in the bosons and fermions, but the analogous statement can be rigorously proven in in d = 2 vector models [35].
In the absence of the Wilson term r = 0 we get the conservation of the chiral current in the form of a WI given by the first of (35), if Π µ,ν 1 ,...,νn is obtained replacing J µ in G ± µ,j,s with b i,s J µ,x .As a consequence the averages of invariant observables are ξ independent.This Therefore if r = 0 in invariant observables one can set ξ = 0 and the theory is perturbatively renormalizable.One expects to be able to reach exponentially high cut-off.
The Wilson term r = 0, physically necessary to avoid fermion doubling [14], breaks the WI and the conservation of chiral current for generic values of the charges, according to (36).Therefore generically the theory is non-renormalizable at scales greater than M and one cannot expect in general to be able to reach exponentially high cut-offs.However choosing the charges so that [ = 0 the contribution of the non decaying term vanishes up to subdominant terms, making possible in principle to reach exponentially high cut-offs.The anomaly cancellation for 1/a ≤ M is therefore a prerequisite for reaching higher cut-offs.In the case of the U (1) sector of the Standard Model, one has also to introduce an Higgs boson to generate the fermion mass; one can distinguish a region higher than the boson mass generated by the Higgs, where the second term of the boson propagator does not contribute due to the anomaly cancellation and the WI; and a lower one, when the infinite volume limit can be taken using the infrared freedom of QED and the massive nature of weak forces.Further challenging problems arise considering the anomaly associated to the SU (2) sector.

Proof of Theorem 1.1
In the following we denote by C or by C 1 , C 2 .. generic λ, L, a-independent constants.We integrate the bosonic variables A µ in (7), obtaining where, by ( 12) Lemma 2.1.The kernels in (40) the following bound, for n ≥ 2, m ≤ 3 and uniformly in L Proof of Lemma 2.1 We write the truncated expectations in (41) by the Battle-Brydges-Federbush formula, see e.g.Theorem 3.1 in [29] (for completeness a sketch of the proof is in Appendix 1), n ≥ 2 is the set of connected tree graphs on {1, 2, . . ., n}, the product {i,j}∈T runs over the edges of the tree graph T , V (X; t) is obtained by taking a sequence of convex linear combinations, with parameters t, of the energies V (Y ) of suitable subsets Y ⊆ X, defined as (45) and dp T (t) is a probability measure, whose explicit form is recalled in the Appendix.We use the bounds hence V (X; t) ≥ 0 and e −V (X;t) ≤ 1 so that dp T (t)e −V (X;t) < 1 therefore and finally using that T ∈T n 1 ≤ C n 4 n! by Cayley' formula [38] we finally get After the integration of A µ the generating function can be written as a Grassmann integral e W(J,J 5 ,φ) = P (dψ)e V (N+1) (ψ,J,J 5 ,φ) (51) with The fermionic propagator is massless, that is it has a power law decay at large distances and this requires a multiscale analysis based on Wilson Renormalization Group.
We introduce parameters γ > 1 and N ∈ N such that 1 γ N ≡ π/(16a) ; moreover we introduce f (t); R + → R a C ∞ non-decreasing function = 0 for 0 ≤ t ≤ γ N −1 and = 1 for t ≥ γ N ; we define also χ N (t) = 1 − f (t) which is therefore non-vanishing for t ≤ γ N .We introduce the propagator with |k − k ′ | T the distance on the 4-dimensional torus [−π/a, π/a) 4 .Therefore, for any K ∈ N we have where |x − y| T is the distance on the [−L, L) 4 torus.The above bound is derived by (discrete) integration by parts, see e.g.§3.3 of [32], using that 2 in the support of f (|k| T ) one has We can write therefore, using the addition property of gaussian Grassmann integrals, see e.g.§2.4 of [32] e W(J,J 5 ,φ) = P (dψ (≤N ) ) P (dψ where and E T N +1 is the truncated expectation with respect to the integration P (dψ (N +1) ).Using the linearity of the truncated expectations, one gets, if γ = ε, s, i, µ, β (57) with ε = ±, and J β x j is J x j or J 5 x j for β = (0, 1).Note that the W (N ) are a series in the kernels H n,m .In the l b = 0 case (the presence of φ is briefly discussed in the Appendix 1) calling Proof of Lemma 2.2.We rewrite V (N +1) (52) in a more compact way as 1 Any choice for γ N ensuring that in the support of 1 − f does not include the doubled poles, that is the poles of g(k) with r = 0 different from k = 0, could be done. 2The bound (54) follows from the presence of the Wilson term; if r = 0 a power law is found due to the presence of poles in the support of f (k).
with P set of field labels and ψ(P x(f ) .We get therefore, inserting (59) in (56 We use the Gawedzki-Kupiainen-Lesniewski [30], [31] (a sketch of the proof is in App.1; see also (see e.g.§A.3 of [36], §2 of [32] or App D of [37]) where T n denotes the set of all the 'spanning trees' on x P 1 , ..., x Ps , that is a set of lines which becomes a tree graph on {1, 2, . . ., s} if one contracts in a point all the point in x P = ∪ f ∈P x(f ), the product {i,j}∈T runs over the unordered edges of the is a probability measure (whose form is specified in the Appendix) with support on a set of t such that t i,i ′ = u i • u i ′ for some family of vectors u i ∈ R s of unit norm and G N +1,T (t) is a (n − s + 1) × (n − s + 1) matrix, whose elements are given by > then the matrix element can be written as a scalar product with The determinants are bounded by the Gram-Hadamard inequality, see e.g.§2 of [32], stating that, if M is a square matrix with elements M ij of the form M ij =< A i , B j >, where A i , B j are vectors in a Hilbert space with scalar product < We get, setting and , see e,g, lemma A3.3 of [36], Lemma 2.4 of [32] or Lemma D.4 of [32], so that As i m i = m ≤ 3 the sum over n i is bounded by so that for λ small enough In order integrate P (dψ (≤N ) )e V (N) (ψ (≤N) ,J,J 5 ) (55) we need to take into account the presence of terms with positive or negative scaling dimension D = 4 − 3l/2 − m, as can be read from (58).
In order to do that we extract from V (N ) the terms with non negative dimension.This is done defining an L (localization) linear operation acting on the kernels of W N l,m (the Fourier transform of W N l,m in(55)) in the following way; We write therefore e W(J,J 5 ,0) = P (dψ (≤N ) )e LV (N) (ψ (≤N) ,J,J 5 )+RV (N) (ψ (≤N) ,J,J 5 ) ( l,m ; the R operation produce an improvement in the bound, see eg §4.2 of [32]; for instance R W N 2,0 (k) will admit, by interpolation, a bound similar to the one for W N 2,0 (k) times a factor O(γ −2N ) due to the derivatives and an extra O(γ 2h ), with h the scale associated to the external fields due to the k 2 .Hence the R operation produces on such terms an improvement O(γ 2(h−N ) ).In coordinate space, the action consists in producing a derivative in the external field and a "zero", that is the difference of two coordinates, see e.g.§3 of [21].
Using symmetry considerations, see the Appendix 3, we get 0, 0) respectively with J and J 5 .It is possible to include the marginal quadratic terms in the fermionic gaussian integration in the following way P (dψ (≤N ) )e i,s z h,i,s Z h,s,i a 4 where ∂ is the discrete derivative and with Z N,s,i (k) = 1 + χ −1 N (k)z N,s,i , and we set Z N,s,i ≡ 1 + z N,s,i .We can write therefore e W(J,J 5 ,0) = P Z N (dψ (≤N ) )e LV (N) ( where we have rescaled the fields writing with ν N,s,i Z N,L,i Z N,R,i = n N,s,i and Z J i,s,N /Z i,s,N = Z J i,s,N , Z 5 i,s,N /Z i,s,N = Z i,s,N .We choose χ N (t) ≡ χ 0 (γ −N t) with χ 0 (t); R + → R a C ∞ non-increasing function = 1 for 0 ≤ t ≤ γ −1 and = 0 for t ≥ 1; and we write with f h (t) with support in γ h−1 ≤ t ≤ γ h+1 .We can write χ N (t) = χ N −1 (t) + f N (t) and given by (71) with χ N (k) replaced by f N (k) and Z i,s,N (k) replaced Z i,s,N .We write therefore e W(J,J 5 ,0) = P Z N (dψ where with V N given by (60); a graphical representation is in fig. 2. Using more compact notation V (N −1) = P, P ψ (≤N −1) (P ) J (P )W (N −1) (P ) (79) By using the linearity of the truncated expectations and expressing RV N by (56) we can write, calling E T (V ; ...; V ) = E T (V ; n) (78) as, see Fig. 3 V From (80) we see that N .The procedure can be iterated in a similar way writing and 1) with L acting on the kernels W (N −1) as (68), so that, after modifying the wave function renormalization and rescaling, we get to (82) Therefore, after integrating in the same way ψ (N −1) , ψ (N −2) , ..., ψ (h+1) e W(J,J 5 ,0 with P Z h (dψ (≤h) ) with propagator and and finally, if γ = α, s, i, µ, β and The ν k,i is a relevant running coupling constant representing the the renormalization of the mass of the fermion of type i; Z k,i,s = (Z k,i,s , Z J k,i,s , Z 5 k,i,s ) are the marginal couplings and represent respectively the wave function renormalization of the fermion of type i and chirality s, and the renormalization of the current and of the axial current.By construction W (h−1) is a function of the kernels W (N +1) in V N +1 and of the running coupling constants ν N , Z N , ..., ν h , Z h ; moreover, the running coupling constants verify recursive equations of the form As should be clear from the previous pictures, the W h and the β h can be conveniently represented in terms as a sum of labeled trees, called Gallavotti trees, , see Fig. 4, defined in the following way (for details see e.g.§3 of [32]) .Let us consider the family of all trees which can be constructed by joining a point r, the root, with an ordered set of n ≥ 1 points, the endpoints of the unlabeled tree, so that r is not a branching point.n will be called the order of the unlabeled tree and the branching points will be called the non trivial vertices.The unlabeled trees are partially ordered from the root to the endpoints in the natural way; we shall use the symbol < to denote the partial order.The number of unlabeled trees is ≤ 4 n , see eg §2.1 of [32].The set of labeled (or Gallavotti) trees T h,n are defined adding the above labels (1) We associate a label h ≤ N − 1 with the root and we introduce a family of vertical lines, labeled by an an integer taking values in [h, N +1] intersecting all the non-trivial vertices, the endpoints and other points called trivial vertices.The set of the vertices v of τ will be the union of the endpoints, the trivial vertices and the non trivial vertices.The scale label is h v and, if v 1 and v 2 are two vertices and is the number of subtrees with root v.Moreover, there is only one vertex immediately following the root, which will be denoted v 0 and can not be an endpoint; its scale is h + 1. (2) To the end-points v of scale h v ≤ N is associated LV (hv) ; there is the constraint that the vertex v ′ immediately preceding v, that is The end-points with h v ≤ N can be of type ν or Z. (3) To the end-points v of scale h v = N + 1 is associated one of the terms in V (N +1) (4) Among the end-points, one distinguish between the normal ones, associated to terms not containing J µ , J 5 µ , whose number is n = n − m, and the others which are called special.
(5) There is an R operation associated to each vertex except the end-points and v 0 ; if the tree contributes to RV h it is associated R while if it contributes to β h is associated L and s v 0 ≥ 2. (6) A subtree with root at scale k is called trivial if contains only the root and an endpoint of scale k + 1 The effective potential can be written as where, if v 0 is the first vertex of τ and τ 1 , .., τ sv 0 are the subtrees of τ with root v 0 , V (h) is defined inductively by the relation, h ≤ N − 1 where E T h is the truncated expectation with propagator g (h) i and is equal to one of the terms in LV (h) if h < N , or to the one of the terms in V (N +1) if h = N .We can write therefore the kernels in (87) as It is also convenient to write with T 1 h,n is the subset of T h,n containing all the trees with only end-points associated to LV k , while T 2 h,n contains the trees with at least one end-point associated to V N +1 .We define with i = 1, 2 and l,m .A similar decomposition can be done for In this case by the compact support of the propagator only trees contributing to T 2 h,n are present; the contribution from T 1 are "chain graphs" and the localization correspond in momentum space to setting k = 0, and g h (0) = 0. Finally we can write with Π with θ 1 = 0 and θ 2 = θ for a constant θ = 1/2; moreover The bound is proven showing the convergence of the expansion in ν k , λ under a smallness condition which is independent from h.Note that if we perform a multiscale integration setting L = 0 then the condition would be that λ ≤ ε h with ε h going to zero a h → −∞.The bound is similar to the one in Lemma 1.2, with the same "dimensional" factor γ (4−(3/2)l−m)h .
A crucial point is that the contributions from trees T 2 , that is the terms obtained by the contraction of the irrelevant terms, have a gain γ θ(h−N ) with respect to the dimensional bounds.This fact, and the bound (98) with that is the wave function and the vertex renormalization is bounded uniformly in h.In addition we can rewrite (100) as We consider the system We can regard the right side of (101) as a function of the whole sequence ν k,i , which we can denote by ν = {ν k } k≤N so that (101) can be read as a fixed point equation ν = T (ν) on the Banach space of sequences . By a standard proof, see e.g.App A5 of [39], it is possible to prove that there is a choice of ν i such that the sequence is bounded for any h.With this choice This means that the ν h,i is bounded so that the condition required in Lemma 2.3 are fulfilled; moreover is an easy consequence of the proof of Lemma 2.3 and of (102) that the limit L → ∞ can be taken; the proof is standard, see App.E of [39].Finally we can choose ) We finally to apply the above results and get bounds for the three current function.By (96) and the bound (97) with l = 0, m = 3, s = 0 we get hence the Fourier transform Π 5 h,µ,ν,ρ (p 1 , p 2 ) is continuous; in addition (97) with l = 0, m = 3, s = 1 + θ/2 and j = 2 hence N h=−∞ Π 5,2 h,µ,ν,ρ has continuous derivative.Note that N h=−∞ Π 5,1 h,µ,ν,ρ has a part from trees containing ν h end-points verifying (102), which by the above argument is again differentiable.We remain then with the contribution from trees with three end.pointsassociated to Z 5 , Z J , Z J .We can write the propagator as where r h (x, y) is defined by the above equation as the difference; one can verify, again by integration by parts, that fpr any K The above decomposition says that the lattice propagator is equal to the continuum one up to a term with a similar decay with an extra γ h−N .Again the contribution of such terms is differentiable and finally we can replace the Z h terms in N h=−∞ Π 5,1 h,µ,ν,ρ with Z −∞ up again to differentiable terms, by (99).In conclusion we get, see Fig. 5 Π (108) says that the Fourier transform of the 3-current correlation can be decomposed in the sum of two terms; the first Π a µ,ρ,σ (p 1 , p 2 ) is continuous and is a sum of triangle graphs equal to the its analogue in the non-interacting continuous case with momentum regularization, with vertex and wave function renormalizations depending on the species and chirality.The second R µ,ρ,σ (p 1 , p 2 ) is a complicate series of terms which is differentiable.
The renormalizations in Π a µ,ρ,σ (p 1 , p 2 ) are however the same appearing in the 2-point and vertex correlations so that we can use the Ward Identities; we can write, see App.2 and the Ward Identities (35) we get exact relations between the wave and vertex renormalizations, that is Note the crucial fact that the contribution from the terms r i,µ , coming from the trees T 2 , is subleading.In conclusion, we get with R with Holder continuous derivative and Note that I µ,ρ,σ (p 1 , p 2 ) is the anomaly for non-interacting relativistic continuum fermions with a momentum regularization which violates the vector current conservation, see [27], §3.6 for the explicit computation In contrast with I µ,ρ,σ , we have that R µ,ρ,σ has not a simple explicit expression, being expressed in terms of a convergent series depending on all the lattice and interaction details.However we use the differentiability of R µ,ρ,σ (p 1 , p 2 ) to expand it at first order obtaining, again up to O(a θ |p| 2+θ ) corrections, using the Ward Identity Finally using such values we get and the second term in the r.h.s. is which implies the Theorem 1.1 3. Appendix 1: truncated expectations 3.1.The Brydges-Battle-Federbush formula.The starting point is the formula Let us define e −V ≡ e − 1 2 j,j ′ ∈X Vj,j ′ (121) with X = (1, 2, .., n) and i,j∈X Vi,j = i≤j V i,j Vi,i = V i,i and V i,j = ( Vi,j + Vj,i )/2.
The connected part e −V (X) | T (corresponding to the truncated expectation) verify where the sum is over π are the partitions of where ℓ = (j, j ′ ) is a pair of elements j, j ′ ∈ X and t 1 (l) = t 1 if l crosses the boundary of X 1 (∂X 1 ), that is if it connect 1 with j = 1; t 1 (ℓ) = 1 otherwise.More explicitely and We have therefore expressed e −V (X) as the sum of two terms; in the first there is a bond (1, k) between X 1 and the rest is found, in the second X 1 is decoupled.If n = 2 the first term is the connected part.
We can reverse the sum over T and X where in the l.h.s. the sets have to be compatible with T .If n ′ (ℓ) is the minimal k such that ℓ crosses X k we have By calling we get where ℓ ∈ X means j, j ′ ∈ (1, .., n).

3.2.
The Gawedzki-Kupiainen-Lesniewski formula.We can write the simple expectations as is a set of Grassmann variables.Again we can write e − j,j ′ V jj ′ as in (131) obtaining E T ( ψ(P 1 )... ψ(P r )) = (138) For each tree T we divide the η in the ones appearing in T , called η, and the rest, called η so that, if and with v j orthonormal, and u 1 u 2 = t 1 , u 1 u 3 = t 1 t 2 , u 2 u 3 = t 2 and so on.

Appendix 2: proof of lemma 2.3
The proof is a generalization of the proof of lemma 2.2 adapted to the tree structure.We define P v as the set of field labels of the external fields of v and if v 1 , . . ., v sv are the s v vertices immediately following v, we denote by Q v i the intersection of P v and P v i .This definition implies that P v = ∪ i Q v i .The union of the subsets P v i \Q v i are the internal fields of v.The set of all P v , v ∈ τ is called P, and the set of all P v with v ≥ τ i is called P i .From (91) we get, if n v 0 is the number of coordinate By definition we have a truncated expectation associated to each v in the tree τ non associated to an end-point; we can write each of them by the Gawedzki-Kupiainen-Lesniewski formula.The R operation is applied and by an iterative procedure and the number of zeros associated to propagators of T and and the derivative on the fields are bounded by a constant; see e.g.§3 of [21].
The bound is done using the Gram bound for the determinant; to each vertex is therefore associated a spanning tree T v which is used to perform the sum over the coordinate difference, and T = ∪ v T v .The sum over coordinates of the propagators in T and the estimates of the determinants give a factor γ −4hv(sv−1) γ 3/2hv( i |Pv i |−|Pv|) , if S v is the number of subtrees with root v.The renormalization produces a factor v γ −zv(hv−h v ′ ) is produced by the R operation and and there is a single J field, z v = 0 otherwise.To the end-points with i ψ fields and j J fields is associated by lemma 2.1 a factor γ (4−3iv /2−jv)N (λ iv /2 (aM ) 2−iv ) with (4 − 3i v /2 − j v ) < 0 and i v ≥ 4 and (aM ) 2−iv < (aM ) −2 .We get therefore where m i,j v is the number of end-points following v with i ψ fields and j J fields , we get We use now that γ h i,j m i,j v 0 v not e.p.
where v * is the first non trivial vertex following v; this implies γ h i,j (3i/2−4)m i,j v 0 v not e.p.
Finally we use the relation v not e.p.
and using that i,j jm i,j v = n J v we finally get (j v = 0 if v is a ν-e.p.) In conclusion  [36] .In order to bound the sums over the scale labels and P we first use the inequality where v are the non trivial vertices, and v ′ is the non trivial vertex immediately preceding v or the root.The factors γ − 1 2 (h v −h v ′ ) in the r.h.s.allow to bound the sums over the scale labels by C n and P v γ − 3|P v | 4 ≤ C n , see §3.7 of of [32] .Let us consider the improvement of the bound.If T * is the set of trees with at least an end-point not of ν, Z type then, for 0 < θ < 1 To prove (150) let be v the non trivial vertex following an end-point not of ν, Z type; hence we can rewrite in (148) and γ θ(h−h v ) [ v e.p. not ν,Z γ (4−3iv /2−jv)(N −h v * ) ] ≤ γ θ(h−N ) (152) as v e.p. not ν,Z γ (4−3iv /2−jv)(N −h v * ) ≤ γ −θ(N −h v ) as there is at least an end-point not ν, Z.
Noting that d v − θ > 0 one can perform the sum as above, and the same bound is obtained with an extra γ θ(h−N ) .
In presence of a φ term there is a new relevant coupling proportional to ψφ, whose local part is vanishing again by the compact support of the propagator.We can compare the bound from the one of a term of the effective potential with l = 2 with two ν end-points.On each tree there is a vertex which is the root of the subtree to which belong both the end-points associated with (ψφ); there is an integral missing giving an extra factor γ 2 h reproducing the similar factor associated to the ν end-points.There is a decay factor proportional to x − y at scale γ h and, from the trees beloging to T * , an extra γ θ(h−N ) ; see e.g.§3.D of [24].A similar argument holds for the vertex function.Finally the proof of the L → ∞ limit is an easy corollary of the proof of lemma 2.3, see e.g.App D of [24].

Appendix 3: symmetries
By symmetry there are no quadratic contributions with i ′ = i.There is invariance under the transformation ψ ± k,s → ε s ψ ∓ k,s

with R = 1
− L (renormalization) and RV (N ) is equal to (87) with W (N ) l,m replaced by RW (N )