1 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, and 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-order 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, labelled by an index \(i=1,\ldots ,2N\); we also define the indices \(i_1=1,\ldots ,N\) and \(i_2=N+1,\ldots ,2N\). If the gamma matrices are

$$\begin{aligned} \gamma _0= \begin{pmatrix} 0 &{} \quad I \\ I &{}\quad 0 \end{pmatrix}\quad \gamma _j= \begin{pmatrix} 0 &{} \quad i\sigma _j \\ -i\sigma _j &{}\quad 0 \end{pmatrix}, \quad \gamma _5=\begin{pmatrix}&{}I&{}\quad 0\\ &{}0\quad &{}-I\end{pmatrix} \end{aligned}$$
(1)

and \(\sigma _\mu ^L=(\sigma _0,i \sigma ) \), \(\sigma _\mu ^R=(\sigma _0,-i \sigma ) \),

$$\begin{aligned} \sigma _1=\begin{pmatrix}&{}0&{}\quad 1\\ &{}1&{}\quad 0\end{pmatrix} \quad \sigma _2=\begin{pmatrix} &{}0\quad &{}-i\\ {} &{}i&{}\quad 0 \end{pmatrix} \quad \sigma _3=\begin{pmatrix}&{}\quad 1&{}\quad 0\\ &{}\quad 0&{}\quad -1\end{pmatrix} \end{aligned}$$
(2)

the formal continuum action is given by the following expression:

$$\begin{aligned}{} & {} \int dx F_{\mu ,\nu }F_{\mu ,\nu } \nonumber \\{} & {} \qquad + \sum _{i_1} \int dx \left[ \psi ^+_{i_1, L,x} \sigma _\mu ^L (\partial _\mu +\lambda Q_{i_1} A_\mu ) \psi ^-_{i_1, L,x}+\psi ^+_{i_1, R,x} \sigma _\mu ^R \partial _\mu \psi ^-_{i_1, L,x}\right] \nonumber \\{} & {} \quad \sum _{i_2} \int dx \left[ \psi ^+_{i_2, R,x} \sigma _\mu ^R (\partial _\mu +\lambda Q_{i_2} A_\mu ) \psi ^-_{i_2, R,x}+\psi ^+_{i_2, L,x} \sigma _\mu ^L \partial _\mu \psi ^-_{i_2, L,x}\right] \end{aligned}$$
(3)

with \(\mu =(0,1,2,3)\) and \(F_{\mu \nu }=\partial _\mu A_\nu -\partial _\nu A_\mu \). 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 e.g. [12,13,14]. The total current coupled to \(A_\mu \) is

$$\begin{aligned} j^T_\mu =\sum _{i_1} Q_{i_1} \psi ^+_{i_1, L,x} \sigma _\mu ^L \psi ^-_{i_1, L,x}+\sum _{i_2} Q_{i_2} \psi ^+_{i_2, R,x} \sigma _\mu ^R \psi ^-_{i_2, R,x} \end{aligned}$$
(4)

and the axial and vector part of the current is

$$\begin{aligned} j^{T,V}_\mu ={1\over 2}\sum _{i} Q_i j_{\mu ,i,x}\quad \quad j^{T,A}_\mu ={1\over 2}\sum _{i} Q_i {\widetilde{\varepsilon }}_i j^5_{\mu ,i,x} \end{aligned}$$
(5)

with \({\widetilde{\varepsilon }}_{i_1}=-{\widetilde{\varepsilon }}_{i_2}=1\), \(j_{\mu ,i,x}=\bar{\psi }_{i,x}\gamma _\mu \psi _{i,x}\), \( j^5_{\mu ,i,x}=\bar{\psi }_{i,x}\gamma _5\gamma _\mu \psi _{i,x}\) and \(\psi _{i,x}=(\psi ^-_{i,L,x},\psi ^-_{i,R,x})\), \(\bar{\psi }_{i,x}=(\psi ^+_{i,L,x},\psi ^+_{i,R,x})\gamma _0\). Note the chiral nature of the theory, as in the current the fermion with different chiralities has 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 \(N=4,\) \(i_1=(\nu _1,e_1,u_1,d_1)\) are the left-handed components and \(i_2=(\nu _2,e_2,u_2,d_2)\) 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 \(\partial _\mu j^T_\mu =0\). 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 \(j^T_{\mu ,i,x}\) unless

$$\begin{aligned} \sum _{i_1=1}^N Q_{i_1}^3-\sum _{i_2=1}^N Q_{i_2}^3 =0 \end{aligned}$$
(6)

In the elecroweak sector, the physical values \(Q_{\nu _1}=Q_{e_1}=-1\), \(Q_{u_1}=Q_{d_1}=1/3\), \(Q_{\nu _2}=0\), \(Q_{e_2}=-2\), \(Q_{u_2}=4/3\), \(Q_{d_2}=-2/3\) verify (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 \(A_\mu \) 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

$$\begin{aligned} e^{{\mathcal W}(J,J^5,\phi )}= \int P(d A) \int P(d\psi ) e^{V(\psi , A, J)+V_c(\psi )+ {\mathcal B}(J^5, \psi )+(\psi ,\phi )} \end{aligned}$$
(7)

where \(A_{\mu ,x}: \Lambda \rightarrow \mathbb {R}\), \(\Lambda =[0,L]^4\cap a \mathbb {Z}^4\), \(L=K a\), \(K\in \mathbb {N}\) \(e_\mu \), \(\mu =0,1,2,3\) an orthonormal basis, \(A_{\mu ,x}=A_{\mu , x+L e_\mu }\) (periodic boundary conditions) and the bosonic integration is

$$\begin{aligned} P(dA)={1\over {\mathcal N}_A} \left[ \prod _{x\in \Lambda } \prod _{\mu =0}^3 d A_{\mu ,x}\right] e^{-S_G(A)} \end{aligned}$$
(8)

with

$$\begin{aligned} S_G= a^4 \sum _{x}\left[ {1\over 4}F_{\mu ,\nu ,x} F_{\mu ,\nu ,x}+{M^2\over 2} A_{\mu ,x} A_{\mu ,x}+(1-\xi ) (d_\mu A_\mu )^2\right] \end{aligned}$$
(9)

is the action of a non-compact lattice U(1) gauge field with a gauge fixing and a mass term, \(F_{\mu ,\nu }=d_\nu A_\mu -d_\mu A_\nu \) and \(d_\nu A_\mu =a^{-1}(A_{\mu ,x+e_\nu a}-A_{\mu ,x})\), \({\mathcal N}_A\) is the normalization. The bosonic simple expectation

$$\begin{aligned} {\mathcal E}_A(A_{\mu _1,x_1}... A_{\mu _n,x_n})=\int P(dA) A_{\mu _1,x_1}... A_{\mu _n,x_n} \end{aligned}$$
(10)

is expressed by the Wick rule with covariance

$$\begin{aligned} g^A_{\mu ,\nu }(x,y)=\delta _{\mu ,\nu } {1\over L^4}\sum _k { e^{i k (x-y)}\over |\sigma |^2+M^2}\left( \delta _{\mu ,\nu }+ {\xi \bar{\sigma }_\mu \sigma _\nu \over (1-\xi ) |\sigma |^2+M^2}\right) \end{aligned}$$
(11)

with \(\sigma _\mu (k)=(e^{i k_\mu a}-1)a^{-1}\), \(k=2\pi n/L\), \(n\in \mathbb {N}^4\) and \(k\in [-\pi /a,\pi /a)^4\). The bosonic truncated expectation

$$\begin{aligned} {\mathcal E}^T_A (F; \cdots ; F )=\frac{\partial ^n}{\partial \lambda ^n}\log \int P(dA)\, e^{ F(A)}\Big |_{\lambda \equiv 0} \end{aligned}$$
(12)

is expressed by the Wick rule restricted to the connected terms.

We denote by \(\psi ^\pm _{i,s,x}\) the Grassmann variables, with \(i=1,..,2N\) the particle index; \(s=L,R\) the chiral index; anti-periodic boundary conditions are imposed and

$$\begin{aligned} \{\psi ^+_{i, s,x},\psi ^+_{i',s',x'}\}=\{\psi ^+_{i, s,x},\psi ^-_{i', s',x'}\}=\{\psi ^-_{i,s,x},\psi ^-_{i',s',x'}\}=0 \end{aligned}$$
(13)

We define \(\psi ^\pm _{i,s,x}={1\over L^4}\sum _k e^{\pm i k x}\widehat{\psi }^\pm _{i,s,k}\), with \(\widehat{\psi }^\pm _{i,s,k}\) another set of Grassmann variable, \(k=2\pi /L(n+1/2)\), \(n\in \mathbb {N}^4\) and \(k\in [-\pi /a,\pi /a)^4\). The fermionic Gaussian measure is defined as, \(i=1,..,2N\), \(s=L,R\)

$$\begin{aligned} P(d\psi )={1\over {\mathcal N}_\psi } \left[ \prod _{i,s,x} d\psi ^+_{i,s,x} d\psi ^-_{i,s,x}\right] e^{-S_F} \end{aligned}$$
(14)

where \({\mathcal N}_\psi \) a normalization and, if \(\psi ^\pm _{i,x}=(\psi ^\pm _{i,L,x},\psi ^\pm _{i,R,x})\)

$$\begin{aligned} S_F= & {} {1\over 2 a}\sum _{i=1}^{2N} a^4 \sum _x \Bigg [\sum _\mu (\psi ^+_{i,x}\gamma _0\gamma _\mu \psi ^-_{i, x+e_\mu a}-\psi ^+_{i,s, x+e_\mu a}\gamma _0\gamma _\mu \psi ^-_{i, x})\nonumber \\{} & {} +\quad r ( \psi ^+_{i, x}\gamma _0 \psi ^-_{i, x+e_\mu a}+\psi ^+_{i, x+e_\mu a}\gamma _0 \psi ^-_{i, x} -\psi ^+_{i, x}\gamma _0\psi ^-_{i, x}) \Bigg ] \end{aligned}$$
(15)

We can write therefore

$$\begin{aligned} S_F= & {} {1\over 2 a} \sum _{i=1}^{2N} a^4 \sum _x \Bigg [\sum _\mu \sum _{s=L,R}(\psi ^+_{i,s, x}\sigma _\mu ^s \psi ^-_{i,s, x+e_\mu a}-\psi ^+_{i,s, x+e_\mu a}\sigma ^s_\mu \psi ^-_{i,s, x})\nonumber \\{} & {} +r ( \psi ^+_{i,L, x}\psi ^-_{i,R, x+e_\mu a} +\psi ^+_{i,L, x+e_\mu a} \psi ^-_{i,R, x} -\psi ^+_{i,L, x}\psi ^-_{i,R, x}\nonumber \\{} & {} +\psi ^+_{i,R, x}\psi ^-_{i,L, x+e_\mu a}+\psi ^+_{i,R, x+e_\mu a} \psi ^-_{i,L, x} -\psi ^+_{i,R, x}\psi ^-_{i,L, x}) \Bigg ] \end{aligned}$$
(16)

The fermionic simple expectation

$$\begin{aligned} {\mathcal E}_\psi (\psi ^{\varepsilon _1}_{i_1,x_1}...\psi ^{\varepsilon _n}_{i_n,x_n})=\int P(d\psi ) \psi ^{\varepsilon _1}_{i_1,x_1}...\psi ^{\varepsilon _n}_{i_n,x_n} \end{aligned}$$
(17)

is expressed by the anticommutative Wick rule with covariance

$$\begin{aligned} g^\psi _i(x,y) =\int P(d\psi ) \psi _{i, x} \bar{\psi }_{i, y}= {1\over L^4}\sum _k e^{i k(x-y)} \widehat{g}^\psi _{i}(k) \end{aligned}$$
(18)

with

$$\begin{aligned} {\widehat{g}}_{i,k}= \left( \sum _{\mu }i \gamma _0\gamma _\mu a^{-1} \sin (k_\mu a)+ a^{-1}\gamma _0 r \sum _\mu (1-\cos k_\mu a)\right) ^{-1} \end{aligned}$$
(19)

The interaction is

$$\begin{aligned} V(A,\psi ,J)= & {} V_1(A,\psi ,J)+V_2(A,\psi ,J)\nonumber \\ V_1(A,\psi ,J)= & {} a^4\sum _{i,s,x} \left[ O^+_{\mu ,i,s,x} G_{\mu ,i,s,x}^+ + O^-_{\mu ,i,s,x} G_{\mu ,i,s}^ -\right] \nonumber \\ V_2(A,\psi ,J)= & {} {r\over 2} a^4\sum _{i,x} \left[ \psi ^+_{i,L, x}H_{\mu ,i,x}^+ \psi ^-_{i,R, x+e_\mu a} \right. \end{aligned}$$
(20)
$$\begin{aligned}{} & {} +\left. \psi ^+_{i,L, x+e_\mu a} H_{\mu ,i,x}^-\psi ^-_{i,R, x} +\psi ^+_{i,R, x}H_{\mu ,i,x}^+\psi ^-_{i,L, x+e_\mu a}\right. \nonumber \\{} & {} \left. +\psi ^+_{i,R, x+e_\mu a} H_{\mu ,i,x}^-\psi ^-_{i,L, x}\right] \end{aligned}$$
(21)

with

$$\begin{aligned} G^{\pm }_{\mu ,i,s}(x)= & {} a^{-1} (:e^{\mp i a Q_i(\lambda b_{i,s} A_{\mu ,x}+ J_{\mu ,x})}:-1)\nonumber \\ H_{\mu ,i,x}^\pm= & {} a^{-1} (e^{\mp i a Q_i J_{\mu ,x}}-1) \nonumber \\ O^+_{\mu ,i,s,x}= & {} {1\over 2 }\psi ^+_{i,s, x}\sigma _\mu ^s \psi ^-_{i,s, x+e_\mu a}\nonumber \\ O^-_{\mu ,i,s,x}= & {} -{1\over 2 }\psi ^+_{i,s, x+e_\mu a}\sigma ^s_\mu \psi ^-_{i,s, x} \end{aligned}$$
(22)

with, if \(i_1=1,N\) and \(i_2=N+1,...,2N\)

$$\begin{aligned} b_{i_1,L}=b_{i_2,R}=1;\quad \quad b_{i_1,R}= b_{i_2,L}=0 \end{aligned}$$
(23)

and \(:e^{\pm i a \lambda Q_i A_\mu (x) }:=e^{\pm i \lambda Q_i a A_\mu (x) } e^{{1\over 2} (\lambda Q_i)^2 a^2 g^A_{\mu ,\mu } (0,0) }\).

The mass counterterm is

$$\begin{aligned} V_c= \sum _{i} a^{-1} \nu _i a^4\sum _x (\psi ^+_{i,L,x}\psi ^-_{i,R,x}+\psi ^+_{i,R,x}\psi ^-_{i,L,x}) \end{aligned}$$
(24)

Finally, the source term is

$$\begin{aligned} {\mathcal B}=a^4 \sum _{\mu ,x} J^5_{\mu ,x} j^5_{\mu ,x} \quad j^5_{\mu ,x}=\sum _{i,s} {\widetilde{\varepsilon }}_i \varepsilon _s Q_j Z_{i,s}^5 \psi ^+_{x,i,s}\sigma _\mu ^s \psi ^+_{x,i,s} \end{aligned}$$

with \({\widetilde{\varepsilon }}_{i_1}=-{\widetilde{\varepsilon }}_{i_2}=1\) and \(\varepsilon _{L}=-\varepsilon _{R}=1\). \(\nu _i\) and \(Z_{i,s}^5 \) 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 \(\widehat{g}_{i,k}\) has, in the \(L\rightarrow \infty \) 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 \(r\not =0\) has the effect that only the physical pole \(k=0\) is present but the chiral symmetry is broken [15].

1.3 Physical observables

The fermionic 2-point function is

$$\begin{aligned} S^\Lambda _{i,s,s'}(x,y) ={\partial ^2 \over \partial \phi ^+_{i,s,x}\partial \phi ^-_{i,s',y} } {\mathcal W}_\Lambda (J,J^5,\phi ) |_0\end{aligned}$$
(25)

and the Fourier transform is

$$\begin{aligned} \widehat{S}^\Lambda _{i,s,s'}(k)=a^4 \sum _x S^\Lambda _{i,s,s'}(x,0) e^{-i k x} \end{aligned}$$
(26)

The vertex functions are

$$\begin{aligned} \Gamma ^\Lambda _{\mu ,i',s}(z,x,y)= & {} {\partial ^3 \over \partial J_{\mu , z} \partial \phi ^+_{i',s,x}\partial \phi ^-_{i',s,y} } {\mathcal W}(J,J^5,\phi )|_0\nonumber \\ \Gamma ^{5,\Lambda }_{\mu , i' s}(z,x,y)= & {} {\partial ^3 \over \partial J^5_{\mu , z} \partial \phi ^+_{i',s,x}\partial \phi ^-_{i',s,y} } {\mathcal W}(J,J^5,\phi )|_0 \end{aligned}$$
(27)

The Fourier transform is

$$\begin{aligned} \widehat{\Gamma }^\Lambda _{\mu ,i',s}(k,p) = a^4 \sum _z a^4\sum _y e^{-i p z-i k y} \Gamma ^\Lambda _{\mu ,i',s}(z,0,y) \end{aligned}$$
(28)

and similarly is defined \(\widehat{\Gamma }^{5,\Lambda }_{\mu ,i' s}(k,p) \). The three current vector VVV and axial AVV correlations are:

$$\begin{aligned} \Pi ^{\Lambda }_{\mu ,\nu , \rho }(z,y,x)= {\partial ^3 {\mathcal W}_\Lambda \over \partial J_{\mu ,z} \partial J_{\nu ,y}\partial J_{\rho ,x}}|_0;\quad \Pi ^{5,\Lambda }_{\mu ,\nu , \rho }(z,y,x)={\partial ^3 {\mathcal W}_\Lambda \over \partial J^5_{\mu ,z} \partial J_{\nu ,y}\partial J_{\rho ,x}}|_0 \end{aligned}$$
(29)

and

$$\begin{aligned} \widehat{\Pi }^{\Lambda }_{\mu ,\nu , \rho }(p_1,p_2)= & {} a^4 \sum _y a^4 \sum _x e^{-i p_1 y-ip_2 x}\Pi ^{\Lambda }_{\mu ,\nu , \rho }(0,y,x)\nonumber \\ \Pi ^{5,\Lambda }_{\mu ,\nu , \rho }(p_1,p_2)= & {} a^4 \sum _y a^4 \sum _x e^{-i p_1 y-ip_2 x}\Pi ^{5,\Lambda }_{\mu ,\nu , \rho }(0,y,x) \end{aligned}$$
(30)

1.4 Ward Identities

The correlations are connected by relations known as Ward identities. They can be obtained by performing the change of variables

$$\begin{aligned} \psi ^\pm _{i,s,x}\rightarrow \psi ^\pm _{i,s,x} e^{\pm i Q_i \alpha _{x} } \end{aligned}$$
(31)

with \(\alpha _x\) is a function on \(a \mathbb {Z}^4\), with the periodicity of \(\Lambda \). Let \(Q(\psi ^{+}, \psi ^{-})\) be a monomial in the Grassmann variables and \(Q_{\alpha }(\psi ^{+}, \psi ^{-})\) be the monomial obtained performing the replacement (31) in \(Q(\psi ^{+}, \psi ^{-})\). It holds that

$$\begin{aligned} \int \left[ \prod _{i,s,x} d\psi ^+_{i,s,x} d\psi ^-_{i,s,x}\right] Q(\psi ^{+}, \psi ^{-}) = \int \left[ \prod _{i,s,x} d\psi ^+_{i,s,x} d\psi ^-_{i,s,x}\right] Q_{\alpha }(\psi ^{+}, \psi ^{-})\;. \end{aligned}$$
(32)

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 \(\psi ^+_{i,s,x}\), \(\psi ^-_{i,s,x}\) come in pairs and the \(\alpha \) dependence cancels. By linearity of the Grassmann integration, the property (32) implies the following identity, valid for any function f on the finite Grassmann algebra:

$$\begin{aligned} \int \left[ \prod _{i,s,x} d\psi ^+_{i,s,x} d\psi ^-_{i,s,x}\right] f(\psi ) = \int \left[ \prod _{i,s,x} d\psi ^+_{i,s,x} d\psi ^-_{i,s,x}\right] f_{\alpha }(\psi )\;, \end{aligned}$$
(33)

with \(f_{\alpha }(\psi )\) the function obtained from \(f(\psi )\), 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

$$\begin{aligned} W(J,J^5,\phi )= W(J+d_\mu \alpha ,J^5, e^{i Q \alpha } \phi ) \end{aligned}$$
(34)

where \(J+d_\mu \alpha \) is a shorthand for \(J_{\mu ,x}+d_\mu \alpha _{x}\) and \(e^{i Q\alpha }\phi \) is a shorthand for \(e^{\pm i Q_i\alpha _{x}} \phi _{i,s, x}^{\pm }\); by differentiating, we get the Ward identities (WI)

$$\begin{aligned} \sum _\mu \sigma _\mu (p) \widehat{\Pi }^\Lambda _{\mu ,\nu _1,..,\nu _n}(p_1,.,p_{n})= & {} 0\quad p=p_1+..p_n\nonumber \\ \sum _\mu \sigma _\mu (p) \widehat{\Gamma }^\Lambda _{\mu ,i,s}(k,p)= & {} Q_i (\widehat{S}^\Lambda _{i,s,s}(k)-\widehat{S}^\Lambda _{i,s,s}(k+p))\nonumber \\ \sum _\nu \sigma _\nu (p_1) \widehat{\Pi }^{5,\Lambda }_{\mu ,\nu ,\rho }(p_1,p_2)= & {} \sum _\rho \sigma _\rho (p_2) \widehat{\Pi }^{5,\Lambda }_{\mu ,\nu , \rho }(p_1,p_2)=0 \end{aligned}$$
(35)

Remark

The above Ward identities represent the conservation of the vector part of the current coupled to the gauge field \(A_\mu \); in particular, the first is the lattice counterpart of \(\partial _\mu <j_{\mu }^{T,V};j_{\nu _1}^{T,V};\ldots j_{\nu _n}^{T,V} >_T=0\), see (5).

1.5 Main result

Our main result is the following, denoting by \(\lim _{L\rightarrow \infty } \widehat{S}^\Lambda _{i,s,s'}(k)\) and similarly the other correlations.

Theorem 1.1

Let us fix \(r=1\) and \(Ma\ge 1\). There exists \(\lambda _0,C\) independent on L, a, M such that, for \(|\lambda |\le \lambda _0 (Ma)\), it is possible to find \(\nu _i\), \(Z_{i,s}^5\) continuous functions in \(\lambda \) such that

  1. (1)

    The limits \(L\rightarrow \infty \) of \(\widehat{S}^\Lambda _{i,s,s'}(k)\), \(\widehat{\Gamma }^\Lambda _{\mu ,i',s}(k,p) \), \(\widehat{\Gamma }^{5,\Lambda }_{\mu ,i',s}(k,p) \) \(\Pi ^{\Lambda }_{\mu ,\nu , \rho }(p_1,p_2)\),\(\Pi ^{5,\Lambda }_{\mu ,\nu , \rho }(p_1,p_2)\) exist and \(\lim _{k\rightarrow 0} \widehat{S}^\Lambda _{i,s}(k)=\infty \) and \(\lim _{k,p\rightarrow 0} {\widehat{\Gamma }^{5,\Lambda }_{\mu ,i',s}(k,p)\over \widehat{\Gamma }^\Lambda _{\mu ,i',s}(k,p)}=\varepsilon _s I\) where \(\varepsilon _L=-\varepsilon _R=1\).

  2. (2)

    The AVV correlation verifies

    $$\begin{aligned}{} & {} \sum _\mu \sigma _\mu (p_1+p_2) \widehat{\Pi }^{5}_{\mu ,\rho ,\sigma }(p_1,p_2)\nonumber \\{} & {} \quad = \sum _{\mu ,\nu } \varepsilon _{\mu ,\nu ,\rho ,\sigma } {1\over 2 \pi ^2}\sigma _\mu ( p^1) \sigma _\nu (p^2) \left[ \sum _{i_1} Q_{i_1}^3-\sum _{i_2} Q_{i_2}^3\right] +r_{\rho ,\sigma }(p_1,p_2)\nonumber \\ \end{aligned}$$
    (36)

    with \(|r(p_1,p_2)|\le C a^\vartheta \bar{p}^{2+\vartheta }\), \(\bar{p}=\max (|p_1|, |p_2|)\) and \(\vartheta =1/2\).

Remarks

 

  1. (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. (2)

    The counterterms \(\nu _i\) are chosen so that the fermions remain massless in the presence of interactions; the parameters \(Z_{i,s}^5\) are fixed so that the charge associated with the vector and axial current are the same, a condition present also at a perturbative level [7].

  3. (3)

    Under the condition \(\left[ \sum _{i_1} Q_{i_1}^3-\sum _{i_2} Q_{i_2}^3\right] =0\) we have \(\sum _\mu \sigma _\mu (p) \widehat{\Pi }^\Lambda _{\mu ,\nu ,\sigma }(p_1,p_{2})=0\) and \(\sum _\mu \sigma _\mu (p) \widehat{\Pi }^{5,\Lambda }_{\mu ,\rho ,\sigma }(p_1,p_2)=O(a^\vartheta \bar{p}^{2+\vartheta })\) 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 [15] on which the perturbative cancellation was 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 [16].

  4. (4)

    Anomalies are strongly connected with transport properties in condensed matter [17,18,19], and we use indeed techniques recently developed for the proof of universality properties in metals to the anomaly cancellation on a lattice [20,21,22,23,24,25,26,27,28,29]. Such methods have their roots in the Gallavotti tree expansion [30], the Battle–Brydges–Federbush formula [31] and the Gawedzki–Kupiainen–Lesniewski formula [32, 33] (see e.g. [34] for an introduction).

1.6 Future Perspectives

We have constructed the theory assuming that \(1/a\le (\lambda _0/|\lambda |) M\), 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 \(A_\mu \) the theory have scaling dimension \(D=4+n-3 n^\psi /2\) if n is the order and \(n_\psi \) the number of fields. This requires that the “effective coupling” \(\lambda ^2/M^2\) 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 \(D=4-3 n^\psi /2-n^A\), so in principle one can consider cut-offs higher than M and up to an exponentially large values \(|\lambda ^2 \log a |\le \varepsilon _0\). 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. [35, 36]; if one restricts to gauge-invariant observables, the contribution of the not-decaying term of the propagator is vanishing as a 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 \(d=2\) vector models [37].

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 \(\widehat{\Pi }_{\mu ,\nu _1,\ldots ,\nu _n}\) is obtained replacing \(J_\mu \) in \(G^{\pm }_{\mu ,j,s}\) with \(b_{i,s} {\widetilde{J}}_{\mu ,x}\). As a consequence, the averages of invariant observables are \(\xi \) independent. This follows from \(\partial _\xi \int P(dA) \int [\prod _{i,s,x} d\psi ^+_{i,s,x} d\psi ^-_{i,s,x}] O=0\), with \(O(A,\psi )\) invariant; indeed,

$$\begin{aligned}{} & {} \partial _\xi \int P(dA)\int \left[ \prod _{i,s,x} d\psi ^+_{i,s,x} d\psi ^-_{i,s,x}\right] O\nonumber \\{} & {} \quad ={1\over L^4 }\sum _p \partial _\xi \widehat{g}^{-1}_{\mu ,\nu }(p) \int P(dA) \widehat{A}_{\mu ,p} \widehat{A}_{\nu ,-p} \int \left[ \prod _{i,s,x} d\psi ^+_{i,s,x} d\psi ^-_{i,s,x}\right] O \nonumber \\ \end{aligned}$$
(37)

from which we get, using that \(\widehat{A}_{\mu ,p}=\widehat{g}^A_{\mu ,\rho } {\partial \over \partial A_{\rho ,-p}}\)

$$\begin{aligned}{} & {} \widehat{g}^A_{\mu ,\rho ' }(p) \partial _\xi (\widehat{g}^A(p))^{-1}_{\mu ,\nu } \widehat{g}^A_{\nu ,\rho }(p) {\partial ^2\over \partial \widehat{J}_{\rho ,p}\partial \widehat{J}_{\rho ',-p}} \int P(dA)\nonumber \\{} & {} \quad \int \left[ \prod _{i,s,x} d\psi ^+_{i,s,x} d\psi ^-_{i,s,x}\right] O(A+{\widetilde{J}},\psi )|_0 \end{aligned}$$
(38)

By noting that \(\partial (\widehat{g}^A)^{-1}=- (\widehat{g}^A)^{-1} \partial _\xi \widehat{g}^A( \widehat{g}^A)^{-1}\) and \(\partial _\xi \widehat{g}^A\) is proportional to \(\bar{\sigma }_\mu \sigma _\nu \), by using \(\partial _\alpha \int P(dA) \int [\prod _{i,s,x} d\psi ^+_{i,s,x} d\psi ^-_{i,s,x}] O(A+d \alpha ,\psi )|_0=0\); then, \(\partial _\xi W\) is vanishing. Therefore, if \(r=0\) in the average of invariant observables one can set \(\xi =0\) and the theory is perturbatively renormalizable. One expects to be able to reach exponentially high cut-off.

The Wilson term \(r\not =0\), physically necessary to avoid fermion doubling [15], 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 \([\sum _{i_1} Q_{i_1}^3-\sum _{i_2} Q_{i_2}^3]=0\) 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 \(1/a\le M\) 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.

2 Proof of Theorem 1.1

In the following, we denote by C or by \(C_1,C_2..\) generic \(\lambda ,L,a\)-independent constants. We integrate the bosonic variables \(A_\mu \) in (7), obtaining

$$\begin{aligned} V_F(\psi ,J)= \log \int P(dA) e^{V_1(\psi , A, J)} \end{aligned}$$
(39)

where, by (12)

$$\begin{aligned} V_F(\psi ,J)= & {} a^4\sum _x \sum _{i,\varepsilon =\pm } a^{-1} \left( e^{-i a \varepsilon Q_{i} J_{\mu ,x}}-1\right) O^\varepsilon _{\mu ,s,i}\\{} & {} + \sum _{n=2}^\infty \ a^{4 n} \sum _{x_1,...,x_n} \sum _{\begin{array}{c} \underline{\varepsilon },\underline{i}\\ \underline{\mu },\underline{s} \end{array}} \left[ \prod _{j=1}^n O^{\varepsilon _j}_{i_j,\mu _j,s_j,x_j}e^{i\varepsilon _j a Q_{i_j} J_{\mu _j,x_j}}\right] \\{} & {} {1\over n!} a^{-n}{\mathcal E}^T_A(: e^{i\varepsilon _1 b_{i_1,s_1}\lambda a Q_{i_1} A_{\mu _1,x_1} }:;...;:e^{i \varepsilon _n b_{i_n,s_n} \lambda a Q_{i_n} A_{\mu _n, x_n}}:) \end{aligned}$$

which can be rewritten as, if \(\underline{x}=x_1,..,x_n\), \(\underline{i}=i_1,...,i_n\), \(\underline{\mu }=\mu _1,...,\mu _n\),\(\underline{m}=m_1,..,m_n\)

$$\begin{aligned} V_F(\psi ,J)= & {} a^4\sum _x \sum _{i,\varepsilon =\pm } a^{-1} \left( e^{-i a \varepsilon Q_{i} J_{\mu ,x}}-1\right) O^\varepsilon _{\mu ,s,i}\nonumber \\{} & {} +\sum _{n=2}^\infty \sum _{m=0}^\infty a^{4n} \sum _{x_1,...,x_n} \sum _{\begin{array}{c} \underline{\varepsilon },\underline{i},\underline{\mu },\underline{s},\underline{m} \\ \sum _j m_j=m \end{array}} {1\over n!}\nonumber \\{} & {} \left[ \prod _{j=1}^n O^{\varepsilon _j}_{i_j,\mu _j,s_j,x_j} (J_{\mu _j,x_j})^{m_j} \right] H_{n,m}(\underline{x},\underline{\varepsilon },\underline{i},\underline{\mu },\underline{s},\underline{m}) \end{aligned}$$
(40)

with

$$\begin{aligned} H_{n,m}(\underline{x},\underline{\varepsilon },\underline{i},\underline{\mu },\underline{s},\underline{m})= & {} {a^{-n} \over n!} {(i a \varepsilon _{j} Q_{i_j})^{m_{j}} \over m_{j}!}]\nonumber \\{} & {} \quad {\mathcal E}^T_A(: e^{i\varepsilon _1 b_{i_1,s_1}\lambda a Q_i A_{\mu _1,x_1} }:;...;:e^{i \varepsilon _n b_{i_n,s_n} \lambda a Q_{i_n} A_{\mu _n,x_n} } )\nonumber \\ \end{aligned}$$
(41)

and \( ||H_{n,m}||=L^{-4} a^{4 n}\sup _{\begin{array}{c} \underline{\varepsilon },\underline{i},\underline{\mu },\underline{s},\underline{m} \\ \sum _j m_j=m \end{array}} \sum _{x_1,...,x_n} |H_{n,m}|\).

Lemma 2.1

The kernels in (40) the following bound, for \(n\ge 2\), \(m\le 3\) and uniformly in L

$$\begin{aligned} ||H_{n,m}|| \le C^{n} a^{-(4-3 n-m)N} (|\lambda |/(M a))^{2(n-1)} \end{aligned}$$
(42)

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 [31] (for completeness a sketch of the proof is in Appendix 1), \(n\ge 2\)

$$\begin{aligned}{} & {} {\mathcal E}^T_A( e^{i\varepsilon _1 b_{i_1,s_1}\lambda Q_{i_1} A_{\mu _1,x_1} };...; e^{i \varepsilon _n b_{i_n,s_n} \lambda a Q_{i_n} A_{\mu _n,x_n} })\nonumber \\{} & {} \quad = \sum _{T\in \varvec{T}_n}\prod _{\{j,j'\}\in T} {\widetilde{g}}^A_{\mu _j\mu _{j'} }\left( x_j, x_{j'}\right) \int dp_T(\underline{t}) e^{-V(X;\underline{t})}, \end{aligned}$$
(43)

where \(X=((x_1,\varepsilon _1,i_1,\mu _1,s_1,m_1);..; (x_n,\varepsilon _n,i_n,\mu _n,s_n,m_n))\), \(\textbf{T}_n\) is the set of connected tree graphs on \(\{1,2,\ldots ,n\}\), the product \(\prod _{\{i,j\}\in T}\) runs over the edges of the tree graph T,

$$\begin{aligned} {\widetilde{g}}^A_{\mu _j\mu _{j'}}(x_{j} , x_{j'})=\lambda ^2 a^2 \varepsilon _j b_{i_j,s_j} Q_{i_j} \varepsilon _{j'} b_{i_{j'},s_{j'}} Q_{i_{j'}} g^A_{\mu _j\mu _{j'}}(x_{j} , x_{j'}), \end{aligned}$$
(44)

\(V(X;\underline{t})\) is obtained by taking a sequence of convex linear combinations, with parameters \(\underline{t}\), of the energies V(Y) of suitable subsets \(Y\subseteq X\), defined as

$$\begin{aligned} V(Y)= & {} \sum _{j,j'\in Y}\lambda ^2 \varepsilon _j\varepsilon _{j'} b_{i_j,s_j} Q_{i_j} b_{i_{j'},s_{j'}} Q_{i_{j'}} a^2 g^A_{\mu _j,\mu _{j'} }(x_j, x_{j'}) \nonumber \\= & {} {\mathcal E}_A\left( \left[ \sum _{j\in Y} \lambda b_{i_j,s_j} Q_{i_j} a \varepsilon _j A_{\mu _j}(x_j)\right] ^2\right) \end{aligned}$$
(45)

and \(dp_T(\underline{t})\) is a probability measure, whose explicit form is recalled in the Appendix 1. We use the bounds

$$\begin{aligned} |g^A_{\mu ,\nu }(x,y)|_1=a^4 \sum _x |g^A_{\mu ,\nu }(x,y)|\le C M^{-2}\quad \quad |g^A_{\mu ,\mu } (x,y)|\le C a^{-2} \end{aligned}$$
(46)

so that

$$\begin{aligned} ||H_{n,m}||\le & {} C_1^n L^{-4} \sup _{\begin{array}{c} \underline{\varepsilon },\underline{i},\underline{\mu },\underline{s},\underline{m} \\ \sum _j m_j=m \end{array}} a^{4 n}\sum _{x_1,...,x_n} {a^{-n+m}\over n!}\sum _{T\in \varvec{T}_n}\nonumber \\{} & {} \quad \prod _{\{j,j'\}\in T} |{\widetilde{g}}^A_{\mu _j\mu _{j'} }(x_j, x_{j'})|\int dp_T(\underline{t}) e^{-V(X;\underline{t})}| \end{aligned}$$
(47)

Moreover, V(Y) is stable, that is

$$\begin{aligned} V(Y)={\mathcal E}\left( \left[ \sum _{j\in Y} \lambda b_{i_j,s_j} Q_{i_j} a \varepsilon _j A_{\mu _j}(x_j)\right] ^2\right) \ge 0 \end{aligned}$$
(48)

hence \( V(X;\underline{t})\ge 0\) and \(e^{-V(X;\underline{t})}\le 1\) so that \(\int dp_T(\underline{t}) e^{-V(X;\underline{t})}<1\) therefore

$$\begin{aligned} ||H_{n,m}||\le & {} C_2^n {a^{-n+m}\over n!}\sum _{T\in \varvec{T}_n}\prod _{\{j,j'\}\in T} a^2 | g^A_{\mu _j\mu _{j'} }(x_j, x_{j'})|_1\nonumber \\\le & {} C_3^n {a^{-n+m}\over n!}\sum _{T\in \varvec{T}_n} (a M^{-1})^{2(n-1)} \end{aligned}$$
(49)

and finally using that \(\sum _{T\in \varvec{T}_n} 1\le C_4^n n!\) by Cayley’ formula [40] we finally get

$$\begin{aligned} ||H_{n,m}||\le & {} C_5^n a^{-n+m} (a M^{-1})^{2(n-1)}\nonumber \\= & {} C_5^{n} a^{-(4-3 n-m)N} (|\lambda |/(M a))^{2(n-1)} \end{aligned}$$
(50)

\(\square \)

After the integration of \(A_\mu \), the generating function can be written as a Grassmann integral:

$$\begin{aligned} e^{{\mathcal W}(J,J^5,\phi )}=\int P(d\psi ) e^{V^{(N+1)}(\psi , J, J^5,\phi )} \end{aligned}$$
(51)

with

$$\begin{aligned} V^{(N+1)}(\psi , J, J^5,\phi ) =V_F(\psi ,J)+V_2(\psi , J)+V_c(\psi )+ {\mathcal B}(J^5, \psi )+(\psi ,\phi )\qquad \end{aligned}$$
(52)

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 \(\gamma >1\) and \(N\in \mathbb {N}\) such thatFootnote 1\(\gamma ^N\equiv \pi /(16 a)\); moreover, we introduce \({\widetilde{f}}(t); \mathbb {R}^+\rightarrow \mathbb {R}\) a \(C^\infty \) non-decreasing function \(=0\) for \(0\le t\le \gamma ^{N-1}\) and \(=1\) for \(t\ge \gamma ^N\); we define also \(\chi _{N}(t)=1-{\widetilde{f}}(t)\) which is therefore non-vanishing for \(t\le \gamma ^N\). We introduce the propagator

$$\begin{aligned} g^{(N+1)}_i(x,y) = {1\over L^4}\sum _k e^{i k(x-y)} {\widetilde{f}}(|k|_T) \widehat{g}^\psi _{i}(k) \end{aligned}$$
(53)

with \(|k-k'|_T\) the distance on the 4-dimensional torus \([-\pi /a,\pi /a)^4\). Therefore, for any \(K\in \mathbb {N}\) we have

$$\begin{aligned} |g^{(N+1)}_i(x,y)|\le \gamma ^{3(N+1)} {C_K \over 1+( \gamma ^{N+1} |x-y|_{{\widetilde{T}}})^K} \end{aligned}$$
(54)

where \(|x-y|_{{\widetilde{T}}}\) 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 [34], using thatFootnote 2 in the support of \({\widetilde{f}}(|k|_T)\) one has \(\sum _\mu (1-\cos k_\mu a)^2/a^2\ge C/a^2\) and the volume of the support of \({\widetilde{f}}\) is \(\le C /a^4\).

We can write therefore, using the addition property of Gaussian Grassmann integrals, see e.g. §2.4 of [34]

$$\begin{aligned} e^{{\mathcal W}(J,J^5,\phi )}= & {} \int P(d\psi ^{(\le N)})\int P(d\psi ^{(N+1)}) e^{V^{(N+1)}(\psi ^{(\le N)}+\psi ^{(N+1)}, J, J^5,\phi ) }\nonumber \\= & {} \int P(d\psi ^{(\le N)}) e^{V^{(N)} (\psi ^{(\le N)},J,J^5,\phi )} \end{aligned}$$
(55)

where

$$\begin{aligned} V^{(N)}(\psi ^{(\le N)},J,J^5,\phi )= & {} \sum _{n=1}^\infty {1\over n!} {\mathcal E}^T_{N+1}(V^{(N+1)}(\psi ^{(\le N)}\nonumber \\{} & {} \quad +\psi ^{(N+1)},J,J^5,\phi );...; V^{(N+1)}(\psi ^{(\le N)}+\psi ^{(N+1)},J,J^5,\phi ) )\nonumber \\ \end{aligned}$$
(56)

and \({\mathcal E}^T_{N+1}\) is the truncated expectation with respect to the integration \(P(d\psi ^{(N+1)} )\) (Fig. 1).

Fig. 1
figure 1

Graphical representation of (56); the first term represents \({\mathcal E}^T_{N+1}(V^{(N+1)})\), the second \({1\over 2}{\mathcal E}^T_{N+1}(V^{(N+1)};V^{(N+1)})\) and so on

Full size image

Using the linearity of the truncated expectations, one gets, if \(\underline{\gamma }=\underline{\varepsilon },\underline{s},\underline{i},\underline{\mu },\underline{\beta }\)

$$\begin{aligned} V^{(N)}(\psi ^{(\le N)},J,J^5,\phi )= & {} a^{4 (l_a+l_b+m)}\sum _{\underline{x},\underline{y},\underline{z},\underline{\gamma }} W^{(N)}_{l_a,l_b,m}(\underline{x},\underline{y},\underline{z},\underline{\gamma })\nonumber \\{} & {} \quad \left[ \prod _{j=1}^{l_a} \psi ^{\le N, \varepsilon _j}_{x_j, i_j, s_j}\right] \left[ \prod _{j=1}^{l_b} \phi ^{\varepsilon _j}_{y_j, i_j, s_j}\right] \left[ \prod _{j=1}^m J^{\beta _j}_{\mu _j,z_j}\right] \nonumber \\ \end{aligned}$$
(57)

with \(\varepsilon =\pm \), and \(J^{\beta }_{x_j}\) is \(J_{x_j}\) or \(J^5_{x_j}\) for \(\beta =(0,1)\). Note that the \(W^{(N)}\) are a series in the kernels \(H_{n,m}\). In the \(l_b=0\) case (the presence of \(\phi \) briefly discussed in the Appendix 2) calling \(W^{(N)}_{l_a,0,m}\equiv W^{(N)}_{l_a,m}\), we define \( ||W^{(N)}_{l,m}||=L^{-4} \sup _{\underline{\gamma }} a^{4\,l+4\,m}\sum _{\underline{x},\underline{z}} |W^{(N)}_{l,m}(\underline{x},\underline{z},\underline{\gamma })|\).

Lemma 2.2

The kernels in (55) verify, for \(|\lambda |\le \lambda _0 (M a)\), \(|\nu _i|\le C (|\lambda |/(Ma))^2\), \(\lambda _0,C,C_1\) independent on a, L, N, \(l\le 2, m\le 3\), |d| is the distance between any coordinate in \(\underline{x}, \underline{z}\)

$$\begin{aligned} || d^s W^{(N)}_{l,m}|| \le C_1 \gamma ^{D N} \end{aligned}$$
(58)

with \(D=4-3 l/2 -m-s\).

Proof of Lemma 2.2

We rewrite \(V^{(N+1)}\) (52) in a more compact way as

$$\begin{aligned} V^{(N+1)}=\sum _{P}{\widetilde{\psi }}^{(\le N+1)}(P){\widetilde{J}}(P) W^{(N+1)}(P) \end{aligned}$$
(59)

with P set of field labels and \( {\widetilde{\psi }}(P)=\prod _{f\in P} \psi ^{(\le N)\varepsilon (f)}_{i(f),s(f),x(f)}\), \({\widetilde{J}}(P) =\prod _{f\in P} J^{\beta (f)}_{x(f)}\). We get therefore, inserting (59) in (56) if \(P=Q_1\cup Q_2...\cup Q_n\)

$$\begin{aligned} V^{(N)}(\psi ^{\le N},J,J^5)= & {} \sum _n {1\over n!}\sum _P \sum _{\begin{array}{c} P_1,...P_n \\ Q_1,...Q_n \end{array}} {\widetilde{\psi }}^{(\le N)}(P) \nonumber \\{} & {} \quad {\mathcal E}^T_{N+1}( {\widetilde{\psi }}^{(N+1)}(P_{1}/Q_{1}); \ldots ;{\widetilde{\psi }}^{(N+1)}(P_{n}/Q_n)) \left[ \prod _{i=1}^n W^{(N+1)}(P_i){\widetilde{J}}(P_i)\right] \nonumber \\ \end{aligned}$$
(60)

We use the Gawedzki–Kupiainen–Lesniewski [32, 33] (a sketch of the proof is in Appendix 1; see also (see e.g. §A.3 of [38], §2 of [34] or Appendix D of [39])

$$\begin{aligned}{} & {} {\mathcal E}^T_{N+1}({\widetilde{\psi }}^{(N+1)}(P_1);...;{\widetilde{\psi }}^{(N+1)}(P_s))\nonumber \\{} & {} \quad = \sum _{T\in {\mathcal T}_n}\prod _{\{i,j\}\in T} g^{(N+1)}(x_i,y_j) \int dP_{T}(\underline{t}) \det G^{N+1,T}(\underline{t}) \end{aligned}$$
(61)

where \({\mathcal T}_n\) denotes the set of all the ‘spanning trees’ on \(x_{P_1},\ldots ,x_{P_s}\), that is a set of lines which becomes a tree graph on \(\{1,2,\ldots ,s\}\) if one contracts in a point all the point in \(x_P=\cup _{f\in P} x(f)\), the product \(\prod _{\{i,j\}\in T}\) runs over the unordered edges of the T, \(\underline{t}=\{t_{i,i'}\in [0,1], 1\le i,i' \le s\}\), \(dP_{T}(\underline{t})\) is a probability measure (whose form is specified in the Appendix 1) with support on a set of \(\underline{t}\) such that \(t_{i,i'}=u_i\cdot u_{i'}\) for some family of vectors \(u_i\in \mathbb {R}^s\) of unit norm and \(G^{N+1,T}(\underline{t})\) is a \((n-s+1)\times (n-s+1)\) matrix, whose elements are given by \(G^{N+1,T}_{ij,i'j'}=t_{i,i'} g^{(N+1)}(x_{ij},y_{i'j'})\) such that if =\(<u_i\otimes A^{(N+1)}_{x(f^-_{ij}) }, u_{i'}\otimes B^{(N+1)}_{x(f^+_{i'j'})}>\) then the matrix element can be written as a scalar product

$$\begin{aligned} G^{N+1,T}_{ij,i'j'}= & {} <u_i\otimes A^{(N+1)}_{x(f^-_{ij}) }, u_{i'}\otimes B^{(N+1)}_{x(f^+_{i'j'})}>\nonumber \\= & {} (u_i\cdot u_{i'}) {1\over L^d}\sum _k \bar{A}^{(N+1)}_{x(f^-_{ij}),k)} B^{(N+1)}_{x(f^+_{i'j'}),k)} \end{aligned}$$
(62)

with \(A^{(N+1)}_{x(f^-_{ij},k)}=e^{i k x(f^-_{ij})} \sqrt{f^N(k) {\bar{g}}^ {N}(k)}\) and \(B^{(N+1)}_{x(f^+_{i'j'}),k)}=e^{i k x(f^+_{i'j'}))} \sqrt{f^N(k) g^ {N}(k)}\).

The determinants are bounded by the Gram–Hadamard inequality, see e.g. §2 of [34], 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 \(<\cdot ,\cdot>\), then \( |\det M|\le \prod _i<A_i,A_i>^{\frac{1}{2}} <B_i,B_i>^{{\frac{1}{2}}}\). Therefore,

$$\begin{aligned} |\det G^{N+1,T}|\le C_1^{\sum _i |P_i|} \gamma ^{3N \sum _ i [|P_i|-(n-1))/2] } \end{aligned}$$
(63)

We get, setting \(|P_i|\equiv n_i\), \(|Q_i|\equiv l_i\), \(0\le l_i\le n_i\),\(\sum _i m_i=m\), \(l=\sum _i l_i\) and \(| |x|^s g^{(N+1)}(x)|_1\le C_2 \gamma ^{- N-s}\)

$$\begin{aligned} ||W^{(N)}_{l,m}||\le & {} \sum _{n=1}^\infty C_3^n {1 \over n!}\sum _{n_1,\ldots n_n}\sum _{\begin{array}{c} l_1,\ldots l_n\\ \sum _i l_i=l \end{array}} \left[ \prod _{i=1}^n {n_i!\over l_i! (n_i-l_i)!} \right] \nonumber \\{} & {} \sum _T C_2^s \gamma ^{-N ((n-1)+s) } C_1^{\sum _i n_i} \gamma ^{3N \left[ \sum _ i (n_i-l_i)/2-(n-1)\right] }\nonumber \\{} & {} \qquad \left[ \prod _{i=1}^n ||W^{N+1}(P_i)||\right] \end{aligned}$$
(64)

and \(\sum _{T\in {\mathcal T}_n}\le n! C_4^{\sum _i n_i}\), see e.g. Lemma A3.3 of [38], Lemma 2.4 of [34] or Lemma D.4 of [34], so that

$$\begin{aligned} ||W^{(N)}_{l,m}||\le & {} \sum _{n=1}^\infty C_6^{n} \sum _{n_1,\ldots n_n} \sum _{\begin{array}{c} l_1,\ldots l_n \\ \sum _i l_i=l \end{array}} \left[ \prod _{i=1}^n (C_5)^{\sum _i n_i} {n_i!\over l_i! (n_i-l_i!)} \right] \\{} & {} \qquad \gamma ^{-4 N (n-1)} \gamma ^{-Ns} \gamma ^{3N \sum _i ( n_i -l_i)/2 } \left[ \prod _{i=1}^n \gamma ^{N(4-3 n_i/2-m_i)}\right] \\{} & {} \qquad \left[ \prod _{i} (|\lambda |/Ma)^{\max (2(n_i/2-1),1-m_i)}\right] \end{aligned}$$

Note that \(\sum _{l\le n} [(C_5)^{n-l} C_5^l {n!\over l! (n-l)!} ]= (2C_5)^n\) we get

$$\begin{aligned} ||W^{(N)}_{l,m}||\le \sum _{n=1}^\infty C_8^{n} \gamma ^{N(4-3 l/2-m-s)} \prod _{i}\left[ \sum _{n_i} C_7^{n_i} \left( |\lambda |/Ma\right) ^{\max \left( 2\left( n_i/2-1\right) ,1-m_i\right) }\right] \end{aligned}$$
(65)

As \(\sum _i m_i=m\le 3\) the sum over \(n_i\) is bounded by

$$\begin{aligned} \prod _{i}\left[ \sum _{n_i} C_7^{n_i} (|\lambda |/Ma)^{\max (2(n_i/2-1),1-m_i)}\right] \le C_8^n (|\lambda |/Ma)^{2\max (n-3,1)} \end{aligned}$$
(66)

so that for \(\lambda \) small enough

$$\begin{aligned} ||W^{(N)}_{l,m}||\le & {} \gamma ^{N \left( 4-3l-m \right) } C_9 \left[ 1+\sum _{n=4}^\infty C_8^n \left( |\lambda |/Ma\right) ^{2(n-3)}\right] \nonumber \\\le & {} C_1 \gamma ^{N \left( 4-3l/2 -m \right) } \end{aligned}$$
(67)

\(\square \)

In order to integrate \(\int P(d\psi ^{(\le N)}) e^{V^{(N)} (\psi ^{(\le N)},J,J^5)}\) (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 \({\mathcal L}\) (localization) linear operation acting on the kernels of \(\widehat{W}^N_{l,m}\) (the Fourier transform of \(W^N_{l,m}\) in (55)) in the following way; \({\mathcal L}\widehat{W}^N_{l,m}(\underline{k})=W^N_{l,m}(\underline{k})\) for \((n,m)\not =(2,0), (2,1)\) and

$$\begin{aligned} {\mathcal L}{\widehat{W}}^N_{2,0}(k)=\widehat{W}^N_{2,0}(0)+{\sin k_\mu a\over a}\partial _\mu \widehat{W}^N_{2,0}(0)\quad \quad {\mathcal L}\widehat{W}^N_{2,1}(k,k+p)=\widehat{W}^N_{2,1,\mu }(0,0) \end{aligned}$$
(68)

We write therefore

$$\begin{aligned} e^{{\mathcal W}(J,J^5,0)}= \int P(d\psi ^{(\le N)}) e^{{\mathcal L}V^{(N)} (\psi ^{(\le N)},J,J^5)+{\mathcal R}V^{(N)} (\psi ^{(\le N)},J,J^5)} \end{aligned}$$
(69)

with \({\mathcal R}=1-{\mathcal L}\) (renormalization) and \({\mathcal R}V^{(N)}\) is equal to (57) with \(W^{(N)}_{l,m}\) replaced by \({\mathcal R}W^{(N)}_{l,m}\); the \({\mathcal R}\) operation produce an improvement in the bound, see e.g. §4.2 of [34]; for instance, \({\mathcal R}\widehat{W}^N_{2,0}(k)\) admits, by interpolation, a bound similar to the one for \(\widehat{W}^N_{2,0}(k)\) times a factor \(O(\gamma ^{-2 N})\) due to the derivatives and an extra \(O(\gamma ^{2\,h})\), with h the scale associated with the external fields due to the \(k^2\). Hence, the \({\mathcal R}\) operation produces on such terms an improvement \(O(\gamma ^{2(h- N)})\). 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 [22].

Using symmetry considerations, see Appendix 3, we get

$$\begin{aligned} {\mathcal L}{\mathcal V}^{(N)}(\psi ,J,J^5)= & {} a^4 \sum _x \sum _{i,s} \left[ n_{N,s,i} \gamma ^N (\psi ^+_{i,L,x}\psi ^-_{i,R,x}+\psi ^+_{i,R,x}\psi ^-_{i,L,x})\right. \\{} & {} \left. + z_{N,i,s} \sigma _\mu ^s\psi ^+_{i,s,x}{\widetilde{\partial }}_\mu \psi ^+_{i,s,x}\right. \\{} & {} \left. + {\widetilde{Z}}_{i,s,N}^{J} J_{\mu ,x} \psi ^+_{i,s,x}\sigma _\mu ^s \psi ^-_{i,s,x}+\varepsilon _s {\widetilde{\varepsilon }}_i {\widetilde{Z}}_{i,s,N}^{5} J^5_{\mu ,x} \psi ^+_{i,s,x}\sigma _\mu ^s \psi ^-_{i,s,x}\right] \end{aligned}$$

with \(\varepsilon _L=-\varepsilon _R=1\), \({\widetilde{\varepsilon }}_{i_1}=-{\widetilde{\varepsilon }}_{i_2}=1\), \(n_{N,s,i} \gamma ^{-N}=\widehat{W}^N_{2,0}(0)\), \(z_{N, s,i} =\partial _\mu \widehat{W}^N_{2,0}(0)\) and \({\widetilde{Z}}_{i,s,N}^{J}=\widehat{W}^N_{2,1}(0,0)\), \({\widetilde{Z}}_{i,s,N}^{5}=\widehat{W}^N_{2,1}(0,0)\), respectively, with J and \(J^5\).

It is possible to include the marginal quadratic terms in the fermionic Gaussian integration in the following way

$$\begin{aligned} P(d\psi ^{(\le N)}) e^{\sum _{i,s} z_{h,i,s}Z_{h,s,i} a^4 \sum _x \sigma _\mu ^s\psi ^+_{i,s,x}{\widetilde{\partial }}_\mu \psi ^+_{i,s,x}}\equiv P_{Z_N} (d\psi ^{(\le N)}) \end{aligned}$$
(70)

where \({\widetilde{\partial }}\) is the discrete derivative and

$$\begin{aligned} \widehat{g}_{i}^{(\le N)}(k)= & {} \chi _N(k) \left( \sum _{\mu } \gamma _0{\widetilde{\gamma }}^N_\mu a^{-1} i \sin (k_\mu a)\right. \nonumber \\{} & {} \left. + a^{-1}\widehat{\gamma }^N_0 \sum _\mu (1-\cos k_\mu a)\right) ^{-1} \end{aligned}$$
(71)
$$\begin{aligned} {\widetilde{\gamma }}^N_0= & {} \begin{pmatrix} 0&{}\quad Z_{N,L,i}(k) I \\ Z_{N,R,i}(k)I &{} \quad 0 \end{pmatrix}\nonumber \\ {\widetilde{\gamma }}_j^N= & {} \begin{pmatrix} 0&{}\quad i Z_{N,L,i}(k) \sigma _j \\ -i Z_{N,R,i}(k)\sigma _j&{}\quad 0 \end{pmatrix} \end{aligned}$$
(72)

with \(Z_{N,s,i}(k)=1+\chi ^{-1}_N(k) z_{N,s,i}\), and we set \(Z_{N,s,i}\equiv 1+z_{N,s,i}\). We can write therefore

$$\begin{aligned}{} & {} e^{{\mathcal W}(J,J^5,0)}\nonumber \\{} & {} \quad =\int P_{Z_N}(d\psi ^{(\le N)}) e^{{\widetilde{\mathcal L}}V^{(N)} ( \sqrt{Z_N} \psi ^{(\le N)},J,J^5)+{\mathcal R}V^{(N)} (\sqrt{Z_N}\psi ^{(\le N)},J,J^5)} \end{aligned}$$
(73)

where we have rescaled the fields writing

$$\begin{aligned}{} & {} {\widetilde{\mathcal L}}{\mathcal V}^{(N)}(\sqrt{Z_N}\psi ,J,J^5)\nonumber \\{} & {} \quad =a^4 \sum _x \sum _{i,s} \left[ \nu _{N,s,i}\gamma ^{N} \sqrt{Z_{N,L,i}Z_{N,R,i}} (\psi ^+_{i,L,x}\psi ^-_{i,R,x}+\psi ^+_{i,R,x}\psi ^-_{i,L,x})\right. \nonumber \\{} & {} \left. \qquad + Z_{i,s,N}^{J} J_{\mu ,x} \psi ^+_{i,s,x}\sigma _\mu ^s \psi ^-_{i',s,x}+\varepsilon _s {\widetilde{\varepsilon }}_i Z_{i,s,N}^{5} J^5_{\mu ,x} \psi ^+_{i,s,x}\sigma _\mu ^s \psi ^-_{i,s,x}\right] \end{aligned}$$
(74)

with \(\nu _{N,s,i} \sqrt{Z_{N,L,i}Z_{N,R,i}} =n_{N,s,i}\) and \({\widetilde{Z}}_{i,s,N}^{J}/Z_{i,s,N}= Z_{i,s,N}^{J}\), \({\widetilde{Z}}_{i,s,N}^{5}/Z_{i,s,N}= Z_{i,s,N}\).

We choose \(\chi _{N}(t)\equiv \chi _0(\gamma ^{-N} t )\) with \(\chi _0(t); \mathbb {R}^+\rightarrow \mathbb {R}\) a \(C^\infty \) non-increasing function \(=1\) for \(0\le t\le \gamma ^{-1}\) and \(=0\) for \(t\ge 1\); we write

$$\begin{aligned} \chi _N(t)=\sum _{h=-\infty }^N f_h(t)\quad \quad f_h(t)=\chi _0(\gamma ^{-h} t )-\chi _0(\gamma ^{-h+1} t ) \end{aligned}$$
(75)

with \(f_h(t)\) with support in \(\gamma ^{h-1}\le t \le \gamma ^{h+1}\). We can write \(\chi _{N}(t)=\chi _{N-1}(t)+ f_N(t)\) and

$$\begin{aligned} \widehat{g}_{i}^{(\le N)}(k)=\widehat{g}_{i}^{(\le N-1)}(k)+\widehat{g}_{i}^{(N)}(k) \end{aligned}$$
(76)

with \(\widehat{g}_{i}^{(N)}(k)\) given by (71) with \(\chi _N(k)\) replaced by \(f_N(k)\) and \(Z_{i,s,N}(k)\) replaced \(Z_{i,s,N}\). We write therefore

$$\begin{aligned} e^{{\mathcal W}(J,J^5,0)}= & {} \int P_{Z_N}(d\psi ^{(\le N-1)})\int P_{Z_N}(d\psi ^{(N)}) \nonumber \\{} & {} e^{{\widetilde{\mathcal L}}V^{(N)} ( \sqrt{Z_N}\psi ^{(\le N)},J,J^5) +{\mathcal R}V^{(N)}(\sqrt{Z_N}\psi ^{(\le N)},J,J^5)}\nonumber \\= & {} \int P_{Z_N}(d\psi ^{(\le N-1)}) e^{ V^{(N-1)} ( \sqrt{Z_N} \psi ^{(\le N-1)},J,J^5)} \end{aligned}$$
(77)

where

$$\begin{aligned} V^{(N-1)}=\sum _n {1\over n!}{\mathcal E}^T_N({\widetilde{\mathcal L}}V^{(N)}+{\mathcal R}V^{(N)}, ...;{\widetilde{\mathcal L}}V^{(N)}+{\mathcal R}V^{(N)} ) \end{aligned}$$
(78)

with \({\mathcal V}^N\) given by (60); a graphical representation is shown in Fig. 2. Using more compact notation

$$\begin{aligned} V^{(N-1)}=\sum _{P,{\widetilde{P}}}{\widetilde{\psi }}^{(\le N-1)}(P){\widetilde{J}}(P) W^{(N-1)}(P) \end{aligned}$$
(79)
Fig. 2
figure 2

Graphical representation of (78); the first term represent \({\mathcal E}^T_{N}(({\mathcal L}+{\mathcal R})V^{(N+1)})\), the second \({1\over 2}{\mathcal E}^T_{N}(({\mathcal L}+{\mathcal R}) V^{(N+1)};({\mathcal L}+{\mathcal R})V^{(N+1)})\) and so on

Full size image

By using the linearity of the truncated expectations and expressing \({\mathcal R}V^N\) by (56) we can write, calling \({\mathcal E}^T(V;...;V)={\mathcal E}^T(V;n)\) (78) as, see Fig. 3

$$\begin{aligned} V^{(N-1)}=\sum _n {1\over n!}{\mathcal E}^T_N({\widetilde{\mathcal L}}V^{(N)}+ \sum _m {1\over m!}{\mathcal R}{\mathcal E}^T_ {N+1}(V^{(N+1)};m);n) \end{aligned}$$
(80)

From (80), we see that \(W^{(N-1)}\) is a function of \(W^{(N+1)},\nu _N, Z_N Z^J_N,Z^5_N\).

Fig. 3
figure 3

Graphical representation of some term in(80)

Full size image

The procedure can be iterated in a similar way writing

$$\begin{aligned} P_{Z_N}(d\psi ^{(\le N-1)}) =P_{Z_N-1}(d\psi ^{(\le N-2)})P_{Z_N}(d\psi ^{(N-1)}) \end{aligned}$$
(81)

and \(V^{(N-1)}={\mathcal L}V^{(N-1)}+{\mathcal R}V^{(N-1)}\) with \({\mathcal L}\) acting on the kernels \(W^{(N-1)}\) as (68), so that, after modifying the wave function renormalization and rescaling, we get to

$$\begin{aligned}{} & {} \int P_{Z_{N-1}}(d\psi ^{(\le N-2)}) \int P_{Z_{N-1}}(d\psi ^{(N-1)}) \nonumber \\{} & {} \quad e^{{\widetilde{\mathcal L}}V^{(N-1)} ( \sqrt{Z_{N-1}} \psi ^{(\le N-1)},J,J^5) + {\mathcal R}V^{(N-1)} (\sqrt{Z_{N-1}}\psi ^{(\le N-1)},J,J^5)} \end{aligned}$$
(82)

Therefore, after integrating in the same way \(\psi ^{(N-1)},\psi ^{(N-2)},...,\psi ^{(h+1)}\)

$$\begin{aligned} e^{{\mathcal W}(J,J^5,0)}=\int P_{Z_h}(d\psi ^{(\le h)}) e^{ V^{(h)} ( \sqrt{Z_h} \psi ^{(\le h)},J,J^5)} \end{aligned}$$
(83)

with \(P_{Z_h}(d\psi ^{(\le h)})\) with propagator

$$\begin{aligned} \widehat{g}_{i}^{(\le h)}(k)= & {} \chi _{h}(k) \left( \sum _{\mu } \gamma _0{\widetilde{\gamma }}^{h}_\mu a^{-1} i \sin (k_\mu a)\right. \nonumber \\{} & {} \left. +a^{-1}\widehat{\gamma }^{h}_0 \sum _\mu (1-\cos k_\mu a)\right) ^{-1} \end{aligned}$$
(84)
$$\begin{aligned} {\widetilde{\gamma }}^{h}_0= & {} \begin{pmatrix} 0&{} \quad Z_{h,L,i}(k) I \\ Z_{h,R,i}(k)I &{}\quad 0 \end{pmatrix}\nonumber \\ {\widetilde{\gamma }}^h_j= & {} \begin{pmatrix} 0&{} \quad i Z_{h,L,i}(k) \sigma _j \\ -i Z_{h,R,i}(k)\sigma _j&{}\quad 0 \end{pmatrix} \end{aligned}$$
(85)

and

$$\begin{aligned}{} & {} {\widetilde{\mathcal L}}{\mathcal V}^{(h)}(\sqrt{Z_h}\psi ,J,J^5)\nonumber \\{} & {} \quad =a^4 \sum _x \sum _{i,s} \left[ \nu _{h,s}\gamma ^{h} \sqrt{Z_{h,L,i}Z_{h,R,i}} (\psi ^+_{i,L,x}\psi ^-_{i,R,x}+\psi ^+_{i,R,x}\psi ^-_{i,L,x})\right. \nonumber \\{} & {} \left. \qquad + Z_{i,s,h}^{J} J_{\mu ,x} \psi ^+_{i,s,x}\sigma _\mu ^s \psi ^-_{i',s,x}+\varepsilon _s {\widetilde{\varepsilon }}_i Z_{i,s,h}^{5} J^5_{\mu ,x} \psi ^+_{i,s,x}\sigma _\mu ^s \psi ^-_{i,s,x}\right] \end{aligned}$$
(86)

and finally, if \(\underline{\gamma }=\underline{\alpha },\underline{s}, \underline{i},\underline{\mu },\underline{\beta }\)

$$\begin{aligned}{} & {} V^{(h-1)} (\sqrt{Z_h} \psi ^{(\le h-1)},J,J^5)\nonumber \\{} & {} \quad =\sum _{l,m} a^{4 l+4m}\sum _{\underline{x},\underline{z}} \sum _{\underline{\gamma }} W^{(h-1)}_{l,m}(\underline{x},\underline{z},\underline{\gamma })\left[ \prod _{j=1}^{l} \psi ^{\le h-1, \varepsilon _j}_{x_j, i_j, s_j}\right] \left[ \prod _{j=1}^m J^{\beta _j}_{\mu _j,z_j}\right] \end{aligned}$$
(87)

and

$$\begin{aligned} ||W^{h-1)}_{l,m}||=L^{-4} \sup _{\underline{\gamma }} a^{4 l+4 m}\sum _{\underline{x},\underline{z}} |W^{(h-1)}_{l,m}(\underline{x},\underline{z})| \end{aligned}$$
(88)

The \(\nu _{k,i}\) is a relevant running coupling constant representing the the renormalization of the mass of the fermion of type i; \({\mathcal Z}_{k,i,s}=(Z_{k,i,s},Z^J_{k,i,s},Z^5_{k,i,s})\) 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 \(\nu _N, {\mathcal Z}_N,...,\nu _h,{\mathcal Z}_h\); moreover, the running coupling constants verify recursive equations of the form

$$\begin{aligned} \nu _{h-1,i}= & {} \gamma \nu _{h,i}+\beta ^h_{\nu ,i}(\nu _N,...,\nu _h, W^{(N+1)})\nonumber \\ {\mathcal Z}_{h-1,i,s}= & {} {\mathcal Z}_{h,i,s}+\beta ^h_{\nu ,i,s}(\nu _N,{\mathcal Z}_N...,\nu _h,{\mathcal Z}_h, W^{(N+1)}) \end{aligned}$$
(89)

As should be clear from the previous pictures, the \(W^h\) and the \(\beta ^h\) can be conveniently represented in terms as a sum of labelled trees, called Gallavotti trees, see Fig. 4, defined in the following way (for details, see, e.g. §3 of [34]).

Fig. 4
figure 4

A Gallavotti tree

Full size image

Let us consider the family of all trees which can be constructed by joining a point r, the root, with an ordered set of \(n\ge 1\) points, the endpoints of the unlabelled tree, so that r is not a branching point. n will be called the order of the unlabelled tree and the branching points will be called the non-trivial vertices. The unlabelled 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 unlabelled trees is \(\le 4^n\), see, e.g. §2.1 of [34]. The set of labelled (or Gallavotti) trees \({\mathcal T}_{h,n}\) are defined adding the above labels

  1. (1)

    We associate a label \(h\le N-1\) with the root and we introduce a family of vertical lines, labelled by 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 \(\tau \) 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 \(v_1<v_2\), then \(h_{v_1}<h_{v_2}\). \(s_v\) 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 cannot be an endpoint; its scale is \(h+1\).

  2. (2)

    To the end-points v of scale, \(h_v\le N\) is associated \({\widetilde{\mathcal L}}V^{(h_v)}\); there is the constraint that the vertex \(v'\) immediately preceding v, that is \(h_{v'}=h_v-1\) is non-trivial (as \({\mathcal R}{\mathcal L}=0\)). The end-points with \(h_v\le N\) can be of type \(\nu \) or Z.

  3. (3)

    To the end-points v of scale \(h_v= N+1\) is associated one of the terms in \(V^{(N+1)}\)

  4. (4)

    Among the end-points, one distinguish between the normal ones, associated with terms not containing \(J_\mu , J_\mu ^5\), whose number is \(\bar{n}=n-m\), and the others which are called special.

  5. (5)

    There is an \({\mathcal R}\) operation associated with each vertex except the end-points and \(v_0\); if the tree contributes to \({\mathcal R}V^h\), it is associated \({\mathcal R}\) while if it contributes to \(\beta _h\) is associated \({\mathcal L}\) and \(s_{v_0}\ge 2\).

  6. (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:

$$\begin{aligned} V^{(h)}(\sqrt{Z_h}\psi ^{(\le h)},J,J^5) = \sum _{n=1}^\infty \sum _{\tau \in {\mathcal T}_{h,n}} V^{(h)}(\tau ,\sqrt{Z_h}\psi ^{(\le h)},J,J^5)\;, \end{aligned}$$
(90)

where, if \(v_0\) is the first vertex of \(\tau \) and \(\tau _1,..,\tau _{s_{v_0}}\) are the subtrees of \(\tau \) with root \(v_0\), \(V^{(h)}\) is defined inductively by the relation, \(h\le N-1\)

$$\begin{aligned}{} & {} V^{(h-1)}(\tau ,\sqrt{Z_h}\psi ^{(\le h)},J,J^5)= {(-1)^{{s_{v_0}}+1}\over {s_{v_0}}!} {\mathcal E}^T_{h}\nonumber \\{} & {} \quad \left[ \bar{V}^{(h)}(\tau _1,\sqrt{Z_h}\psi ^{(\le h)},J,J^5);..; \bar{V}^{(h)}(\tau _{{s_{v_0}}},\sqrt{Z_h}\psi ^{(\le h)},J,J^5)\right] \end{aligned}$$
(91)

where \({\mathcal E}^T_{h}\) is the truncated expectation with propagator \(g_{i}^{(h)}\) and

  • if \(\tau _i\) is non-trivial \(\bar{V}^{(h)}(\tau _i,\sqrt{Z_h}\psi ^{(\le h)},J,J^5)= {\mathcal R}V^{(h)}(\tau _i,\sqrt{Z_h}\psi ^{(\le h)},J,J^5)\)

  • if \(\tau \) is trivial it is equal to one of the terms in \({\widetilde{\mathcal L}}V^{(h)}\) 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

$$\begin{aligned} W^{(h)}_{l,m}(\underline{x},\underline{z})=\sum _{n=1}^\infty \sum _{\tau \in {\mathcal T}_{h,n}} W^{(h)}_{l,m}(\tau ,\underline{x},\underline{z}) \end{aligned}$$
(92)

It is also convenient to write

$$\begin{aligned} {\mathcal T}_{h,n}={\mathcal T}^1_{h,n}\cup {\mathcal T}^2_{h,n} \end{aligned}$$
(93)

with \({\mathcal T}^1_{h,n}\) is the subset of \({\mathcal T}_{h,n}\) containing all the trees with only end-points associated with \({\widetilde{\mathcal L}}V^k\), while \({\mathcal T}^2_{h,n}\) contains the trees with at least one end-point associated with \(V^{N+1}\). We define

$$\begin{aligned} W^{i (h)}_{l,m}(\underline{x},\underline{z},\underline{\gamma }) =\sum _{n=1}^\infty \sum _{\tau \in {\mathcal T}^i_{h,n}} W^{(h)}_{l,m}(\tau ;\underline{x},\underline{z},\underline{\gamma }) \end{aligned}$$
(94)

with \(i=1,2\) and \(W^{ (h)}_{l,m}=W^{1 (h)}_{l,m}+ W^{2 (h)}_{l,m}\). A similar decomposition can be done for

$$\begin{aligned} \beta ^h_\nu =\sum _{n=1}^\infty \sum _{\tau \in {\mathcal T}^2_{h,n}} \beta ^h_\nu (\tau )\quad \quad \beta ^h_{\mathcal Z}=\sum _{n=1}^\infty \sum _{\tau \in {\mathcal T}^2_{h,n}}\beta ^h_{\mathcal Z}(\tau ) \end{aligned}$$
(95)

In this case by the compact support of the propagator only trees contributing to \({\mathcal T}^2_{h,n}\) are present; the contribution from \({\mathcal T}^1\) are “chain graphs” and the localization corresponds in momentum space to setting \(k=0\), and \(\widehat{g}^h(0)=0\). Finally, we can write

$$\begin{aligned} \Pi ^5_{\mu ,\nu ,\rho }=\sum _{h=-\infty }^N \Pi ^{5,1}_{h,\mu ,\nu ,\rho }+\sum _{h=-\infty }^N \Pi ^{5,2}_{h,\mu ,\nu ,\rho } \end{aligned}$$
(96)

with \(\Pi ^{5,i}_{h,\mu ,\nu ,\rho }= \sum _{n=1}^\infty \sum _{\tau \in {\mathcal T}^i_{h,n}} W^h_{0,3}(\tau )\). The following lemma holds, see Appendix 2.

Lemma 2.3

There exists a constant \(\varepsilon \) such that, for \(|{\mathcal Z}_{k}|\le e^{\varepsilon ({a M})^{2} }\), \(\max (|\nu _N|,\ldots ,|\nu _h|,(\lambda / Ma)^2\le \varepsilon \) than if \(m\le 3\), d is the distance between any two coordinate

$$\begin{aligned} ||d^s W^{j(h)}_{l,m}|| \le C^{l+m} \gamma ^{(4-(3/2)l-m-s) h}\gamma ^{\vartheta _j(h-N)} \varepsilon ^{\max (l/2-1,1)} \end{aligned}$$
(97)

with \(\vartheta _1=0\) and \(\vartheta _2=\vartheta \) for a constant \(\vartheta = 1/2\); moreover,

$$\begin{aligned} |\beta ^h_\nu |\le \varepsilon \gamma ^{\vartheta (h-N)}\quad |\beta ^h_{\mathcal Z}|\le \varepsilon \gamma ^{\vartheta (h-N)} \end{aligned}$$
(98)

The bound is proven showing the convergence of the expansion in \(\nu _k,\lambda \) under a smallness condition which is independent from h. Note that if we perform a multiscale integration setting \({\mathcal L}=0\) then the condition would be that \(\lambda \le \varepsilon _h\) with \(\varepsilon _h\) going to zero a \(h\rightarrow -\infty \). The bound is similar to the one in Lemma 1.2, with the same “dimensional” factor \(\gamma ^{(4-(3/2)l-m) h}\).

A crucial point is that the contributions from trees \({\mathcal T}^2\), that is the terms obtained by the contraction of the irrelevant terms, have a gain \(\gamma ^{\vartheta (h-N)}\) with respect to the dimensional bounds. This fact, and the bound (98) with \({\mathcal Z}_{h-1}=1+\sum _{k=h}^N \beta ^k_{\mathcal Z}\) implies

$$\begin{aligned} |{\mathcal Z}_{-\infty }-1|\le C \varepsilon \quad \quad |{\mathcal Z}_{-\infty }-{\mathcal Z}_h|\le C \varepsilon \gamma ^{\vartheta (h-N)} \end{aligned}$$
(99)

that is the wave function and the vertex renormalization is bounded uniformly in h. In addition, we can rewrite (100) as

$$\begin{aligned} \nu _{h-1,i}=\gamma ^{-h}\left( \nu _{N,i}+\sum _{k=h}^N \gamma ^k \beta ^{(h)}_{\nu ,i}\right) \end{aligned}$$
(100)

We consider the system

$$\begin{aligned} \nu _{h-1,i}=\gamma ^{-h}\left( -\sum _{k\le h} \gamma ^k \beta ^{(h)}_{\nu ,i}\right) \end{aligned}$$
(101)

We can regard the right side of (101) as a function of the whole sequence \(\nu _{k,i}\), which we can denote by \(\underline{\nu }=\{\nu _k\}_{k\le N}\) so that (101) can be read as a fixed point equation \(\underline{\nu }=T(\underline{\nu })\) on the Banach space of sequences \(\nu \) such that \(||\nu ||=\sup _{k\le N} \gamma ^{\vartheta (k-N)} |\nu _k|\le C \lambda ^2 (M a)^{-2} \). By a standard proof, see, e.g. Appendix A5 of [41], it is possible to prove that there is a choice of \(\nu _i\) such that the sequence is bounded for any h. With this choice

$$\begin{aligned} |\nu _h|\le C \gamma ^{\vartheta (h-N)} \varepsilon \end{aligned}$$
(102)

This means that the \(\nu _{h,i}\) is bounded so that the condition required in Lemma 2.3 is fulfilled; moreover, it is an easy consequence of the Proof of Lemma 2.3 and of (102) that the limit \(L\rightarrow \infty \) can be taken; the proof is standard, see Appendix E of [41]. Finally, we can choose \(Z^5_{i,s}=1+O(\varepsilon )\) so that

$$\begin{aligned} Z^5_{i,s,-\infty }=Z^{J}_{i,s,-\infty } \end{aligned}$$
(103)

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

$$\begin{aligned} \left| \sum _{h=-\infty }^N \Pi ^{5}_{h,\mu ,\nu ,\rho }(x,y,0)\right| _{1}\le C \sum _{h=-\infty }^N \gamma ^{h}<C_N \end{aligned}$$
(104)

hence the Fourier transform \(\widehat{\Pi }^{5}_{h,\mu ,\nu ,\rho }(p_1,p_2)\) is continuous; in addition (97) with \(l=0, m=3,s=1+\vartheta /2\) and \(j=2\)

$$\begin{aligned} \left| \sum _{h=-\infty }^N \left( |x|^{1+\vartheta /2}+|y|^{1+\vartheta /2}\right) \Pi ^{5,2}_{h,\mu ,\nu ,\rho }(x,y,0)\right| _{1}\le C \sum _{h=-\infty }^N \gamma ^{-\vartheta /2 h}\gamma ^{\vartheta (h-N)}<\bar{C}_N \end{aligned}$$
(105)

hence \(\sum _{h=-\infty }^N\widehat{\Pi }^{5,2}_{h,\mu ,\nu ,\rho }\) has continuous derivative.

Note that \(\sum _{h=-\infty }^N \widehat{\Pi }^{5,1}_{h,\mu ,\nu ,\rho }\) has a part from trees containing \(\nu _h\) end-points verifying (102), which by the above argument is again differentiable. We remain then with the contribution from trees with three end points associated with \(Z^5,Z^J,Z^J\). We can write the propagator as

$$\begin{aligned} g^{(h)}_{i,s,s'}(x,y)= \delta _ {s,s'}{1\over L^4}\sum _k {f_h(|k|_T)\over -i \sigma ^s_\mu k_\mu } e^{i k(x-y)}+r^h_{i,s,s'}(x,y) \end{aligned}$$
(106)

where \(r^h(x,y)\) is defined by the above equation as the difference; one can verify, again by integration by parts, that for any K

$$\begin{aligned} |g^{(h)}_{i,s,s'}(x,y)|\le & {} {1\over Z_{h,i,s}}\gamma ^{3(h+1)} {C_K \over 1+\left( \gamma ^{h+1} |x-y|_{{\widetilde{T}}}\right) ^K}\nonumber \\ |r^{(h)}_{i,s,s'}(x,y)|\le & {} \gamma ^{3(h+1)}\gamma ^{h-N} {C_K \over 1+\left( \gamma ^{h+1} |x-y|_{{\widetilde{T}}}\right) ^K} \end{aligned}$$
(107)

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 \(\gamma ^{h-N}\). Again the contribution of such terms is differentiable and finally we can replace the \({\mathcal Z}_h\) terms in \(\sum _{h=-\infty }^N \widehat{\Pi }^{5,1}_{h,\mu ,\nu ,\rho }\) with \({\mathcal Z}_{-\infty }\) up again to differentiable terms, by (99). In conclusion, we get, see Fig. 5

$$\begin{aligned} \widehat{\Pi }_{\mu ,\rho ,\sigma }\left( p_1,p_2\right) =\widehat{\Pi }^a_{\mu ,\rho ,\sigma }(p_1,p_2) +\widehat{R}_{\mu ,\rho ,\sigma }\left( p_1,p_2\right) \end{aligned}$$
(108)

with, \(p=p_1+p_2\),

$$\begin{aligned} \widehat{\Pi }^a_{\mu ,\rho ,\sigma }(p_1,p_2)= & {} \sum _{\begin{array}{c} h_1\\ h_2,h_3 \end{array}} \sum _{i,s} {\widetilde{\varepsilon }}_i \varepsilon _s Q_{i}^3 { Z^{5}_{-\infty ,i,s} \over Z_{-\infty ,i,s}} {Z^{J}_{-\infty , i,s}\over Z_{-\infty ,i,s}} {Z^{J}_{-\infty , i, s}\over Z_{-\infty ,i,s}}\nonumber \\{} & {} \quad \int {dk \over (2\pi )^4}{\textrm{Tr}}{f_ {h_1}(k)\over i \sigma ^{s}_\mu k_\mu } i\sigma ^{s}_\mu { f_{h_2}\over i \sigma ^{s}_\mu (k_\mu +p_\mu )} i \sigma ^{s}_\nu {f_ {h_3}\over i \sigma ^{s}_\mu (k_\mu +p^2_\mu )}(i \sigma ^{s}_\rho )\nonumber \\ \end{aligned}$$
(109)

(108) says that the Fourier transform of the 3-current correlation can be decomposed in the sum of two terms; the first \(\widehat{\Pi }^a_{\mu ,\rho ,\sigma }(p_1,p_2)\) 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 \(\widehat{R}_{\mu ,\rho ,\sigma }(p_1,p_2)\) is a complicate series of terms which is differentiable.

The renormalizations in \(\widehat{\Pi }^a_{\mu ,\rho ,\sigma }(p_1,p_2)\) are, however, the same appearing in the 2-point and vertex correlations so that we can use the Ward identities; we can write, see Appendix 2

$$\begin{aligned} \widehat{S}_{i,s}(k)={1\over (i \sigma ^s_\mu k_\mu )} \left( {I \over Z_{i, s,-\infty }}+r_1(k)\right) \end{aligned}$$
(110)

and

$$\begin{aligned} \widehat{\Gamma }_{\mu ,i,s}(k,p)={1\over (i \sigma ^s_\mu k_\mu )} {Z^J_{i,s,-\infty } \over Z^2_{i,s,-\infty }} (i \sigma _\mu ^s +r_{2,\mu }(k,p)) {1\over (i \sigma ^s_\mu (k_\mu +p_\mu ))} \end{aligned}$$
(111)

with \(|r_1(k)|\le C (a |k|)^\vartheta \) and \(|r_{2,\mu }(k,p)|\le C (a |k|)^\vartheta \) with \(|p|\le |k|\).

Fig. 5
figure 5

Graphical representation of (108)

Full size image

By inserting (110), (111) in the Ward identities (35), we get exact relations between the wave and vertex renormalizations, that is

$$\begin{aligned} {Z_{-\infty ,i,s}^{J} \over Z_{-\infty ,i,s}}=1 \end{aligned}$$
(112)

Note the crucial fact that the contribution from the terms \(r_{i,\mu }\), coming from the trees \({\mathcal T}^2\), is subleading. In conclusion, we get

$$\begin{aligned} \widehat{\Pi }^5_{\mu ,\rho ,\sigma }=\widehat{I}_{\mu ,\rho ,\sigma }+\widehat{{\mathcal R}}_{\mu ,\rho ,\sigma } \end{aligned}$$
(113)

with \(\widehat{{\mathcal R}}\) with Holder continuous derivative and

$$\begin{aligned} \widehat{I}_{\mu ,\rho ,\sigma }(p_1,p_2)= \left( \sum _i {\widetilde{\varepsilon }}_i Q_i^3\right) \int {dk \over (2\pi )^4}{\textrm{Tr}} {\chi (k)\over \not k }\gamma _\mu \gamma _5 {\chi (k+p)\over \not k+ \not p}\gamma _\nu {\chi (k+p^2)\over \not k+\not p^2}\gamma _\sigma \end{aligned}$$
(114)

Note that \(\widehat{I}_{\mu ,\rho ,\sigma }(p_1,p_2)\) is the anomaly for non-interacting relativistic continuum fermions with a momentum regularization which violates the vector current conservation, see [29], §3.6 for the explicit computation

$$\begin{aligned} \sum _\mu (p_{1,\mu } + p_{2,\mu }) \widehat{I}_{\mu ,\nu ,\sigma }= & {} {(\sum _i {\widetilde{\varepsilon }}_i Q_i^3)\over 6\pi ^{2}} p_{1,\alpha } p_{2,\beta } \varepsilon _{\alpha \beta \nu \sigma }\nonumber \\ \sum _\nu p_{1,\nu } \widehat{I}_{\mu ,\nu ,\sigma }= & {} {(\sum _i {\widetilde{\varepsilon }}_i Q_i^3)\over 6\pi ^{2}} p_{1,\alpha } p_{2,\beta } \varepsilon _{\alpha \beta \mu \sigma } \end{aligned}$$
(115)

up to \(O(a^\vartheta |\bar{p}|^{2+\vartheta })\) corrections. In contrast to \(\widehat{I}_{\mu ,\rho ,\sigma }\), we have that \(\widehat{{\mathcal R}}_{\mu ,\rho ,\sigma }\) 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 \(\widehat{{\mathcal R}}_{\mu ,\rho ,\sigma }(p_1,p_2)\) to expand it at first order obtaining, again up to \(O(a^\vartheta |\bar{p}|^{2+\vartheta })\) corrections, using the Ward identity

$$\begin{aligned}{} & {} \frac{1}{6\pi ^{2}} \left( \sum _i {\widetilde{\varepsilon }}_i Q_i^3\right) p_{1,\alpha } p_{2,\beta } \varepsilon _{\alpha \beta \mu \sigma } +\sum _\nu p_{1,\nu } \Bigg (\widehat{{\mathcal R}}_{\mu ,\nu ,\sigma }({0}, { 0})\nonumber \\{} & {} \quad +\sum _{a=1,2} \sum _{\rho } p_{a,\rho } \frac{\partial \widehat{{\mathcal R}}_{\mu ,\nu ,\sigma }}{\partial p_{a,\rho }}({0}, {0}) \Bigg )=0 \end{aligned}$$

This implies that

$$\begin{aligned} \widehat{{\mathcal R}}_{\mu ,\nu ,\sigma }({0}, { 0})=0 \end{aligned}$$
(116)

and

$$\begin{aligned} {\partial \widehat{{\mathcal R}}_{\mu ,\nu ,\sigma }\over \partial p_{2,\beta }}= & {} - {1\over 6\pi ^{2}}\varepsilon _{\nu \beta \mu \sigma } \left( \sum _i {\widetilde{\varepsilon }}_i Q_i^3\right) \nonumber \\ {\partial \widehat{{\mathcal R}}_{\mu ,\nu ,\sigma }\over \partial p_{1,\beta }} (0, {0})= & {} {1\over 6\pi ^{2}}\varepsilon _{\nu \beta \mu \sigma } \left( \sum _i {\widetilde{\varepsilon }}_i Q_i^3\right) \end{aligned}$$
(117)

Finally, using such values we get

$$\begin{aligned}{} & {} \sum _\mu (p_{1,\mu } + p_{2,\mu })\widehat{\Pi }^5_{\mu ,\nu ,\sigma }(p_1,p_2)=\sum _{\alpha ,\beta }\frac{(\sum _i {\widetilde{\varepsilon }}_i Q_i^3)}{6\pi ^{2}} p_{1,\alpha } p_{2,\beta } \varepsilon _{\alpha \beta \nu \sigma }\nonumber \\{} & {} \quad +\sum _{\mu ,\beta } (p_{1,\mu } + p_{2,\mu })\left( {\widehat{{\mathcal R}}_{\mu ,\nu ,\sigma }\over \partial p_{2,\beta }}(0,0) p_{2,\beta } +{\widehat{{\mathcal R}}_{\mu ,\nu ,\sigma }\over \partial p_{1,\beta }} (0,0) p_{1,\beta }\right) \end{aligned}$$
(118)

and the second term in the r.h.s. is

$$\begin{aligned}{} & {} -\frac{1}{6\pi ^2} \left( p_{1,\mu } + p_{2,\mu }\right) \sum _{a=1,2}(-1)^a p_{a,\beta }\varepsilon _{\nu \beta \mu \sigma }\left( \sum _i {\widetilde{\varepsilon }}_i Q_i^3\right) \nonumber \\{} & {} \quad =\frac{1}{3\pi ^2} p_{1,\mu } p_{2,\beta }\varepsilon _{\nu \beta \mu \sigma }\left( \sum _i {\widetilde{\varepsilon }}_i Q_i^3\right) \end{aligned}$$
(119)

which implies Theorem 1.1 \(\square \)