# Vanishing of the anomaly in lattice chiral gauge theory

Vieri Mastropietro  Università degli Studi di Milano Dipartimento di
Matematica Via Saldini 50 20133 Milano Italy<EMAIL_ADDRESS>
###### Abstract.

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.

##  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, 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

$\gamma_{0}=\begin{pmatrix}0&I\\\
I&0\end{pmatrix}\quad\gamma_{j}=\begin{pmatrix}0&i\sigma_{j}\\\
-i\sigma_{j}&0\end{pmatrix},\quad\gamma_{5}=\begin{pmatrix}&I&0\\\
&0&-I\end{pmatrix}$ (1)

and $\sigma_{\mu}^{L}=(\sigma_{0},i\sigma)$,
$\sigma_{\mu}^{R}=(\sigma_{0},-i\sigma)$,

$\sigma_{1}=\begin{pmatrix}&0&1\\\
&1&0\end{pmatrix}\quad\sigma_{2}=\begin{pmatrix}&0&-i\\\
&i&0\end{pmatrix}\quad\sigma_{3}=\begin{pmatrix}&1&0\\\ &0&-1\end{pmatrix}$
(2)

the formal continuum action is given by the following expression:

$\displaystyle\int dxF_{\mu,\nu}F_{\mu,\nu}+\sum_{i_{1}}\int
dx[\psi^{+}_{i_{1},L,x}\sigma_{\mu}^{L}(\partial_{\mu}+\lambda
Q_{i_{1}}A_{\mu})\psi^{-}_{i_{1},L,x}+\psi^{+}_{i_{1},R,x}\sigma_{\mu}^{R}\partial_{\mu}\psi^{-}_{i_{1},L,x}]$
$\displaystyle\sum_{i_{2}}\int
dx[\psi^{+}_{i_{2},R,x}\sigma_{\mu}^{R}(\partial_{\mu}+\lambda
Q_{i_{2}}A_{\mu})\psi^{-}_{i_{2},R,x}+\psi^{+}_{i_{2},L,x}\sigma_{\mu}^{L}\partial_{\mu}\psi^{-}_{i_{2},L,x}]$
(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 eg [12], [13].The total
current coupled to $A_{\mu}$ is

$j^{T}_{\mu}=\sum_{i_{1}}Q_{i_{1}}\psi^{+}_{i_{1},L,x}\sigma_{\mu}^{L}\psi^{-}_{i_{1},L,x}+\sum_{i_{2}}Q_{i_{2}}\psi^{+}_{i_{2},R,x}\sigma_{\mu}^{R}\psi^{-}_{i_{2},R,x}$
(4)

and the axial and vector part of the current is

$j^{T,V}_{\mu}={1\over 2}\sum_{i}Q_{i}j_{\mu,i,x}\quad\quad
j^{T,A}_{\mu}={1\over
2}\sum_{i}Q_{i}{\widetilde{\varepsilon}}_{i}j^{5}_{\mu,i,x}$ (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
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 $N=4,$ $i_{1}=(\nu_{1},e_{1},u_{1},d_{1})$ 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

$\sum_{i_{1}=1}^{N}Q_{i_{1}}^{3}-\sum_{i_{2}=1}^{N}Q_{i_{2}}^{3}=0$ (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

$e^{{\cal W}(J,J^{5},\phi)}=\int P(dA)\int
P(d\psi)e^{V(\psi,A,J)+V_{c}(\psi)+{\mathcal{B}}(J^{5},\psi)+(\psi,\phi)}$ (7)

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

$P(dA)={1\over{\cal
N}_{A}}[\prod_{x\in\Lambda}\prod_{\mu=0}^{3}dA_{\mu,x}]e^{-S_{G}(A)}$ (8)

with

$S_{G}=a^{4}\sum_{x}[{1\over 4}F_{\mu,\nu,x}F_{\mu,\nu,x}+{M^{2}\over
2}A_{\mu,x}A_{\mu,x}+(1-\xi)(d_{\mu}A_{\mu})^{2}]$ (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
$d_{\nu}A_{\mu}=a^{-1}(A_{\mu,x+e_{\nu}a}-A_{\mu,x})$, ${\cal N}_{A}$ is the
normalization. The bosonic simple expectation

${\cal E}_{A}(A_{\mu_{1},x_{1}}...A_{\mu_{n},x_{n}})=\int
P(dA)A_{\mu_{1},x_{1}}...A_{\mu_{n},x_{n}}$ (10)

is expressed by the Wick rule with covariance

$g^{A}_{\mu,\nu}(x,y)={1\over
L^{4}}\sum_{k}{e^{ik(x-y)}\over|\sigma|^{2}+M^{2}}(\delta_{\mu,\nu}+{\xi\bar{\sigma}_{\mu}\sigma_{\nu}\over(1-\xi)|\sigma|^{2}+M^{2}})$
(11)

with $\sigma_{\mu}(k)=(e^{ik_{\mu}a}-1)a^{-1}$, $k=2\pi n/L$,
$n\in\hbox{\msytw N}^{4}$ and $k\in[-\pi/a,\pi/a)^{4}$. The bosonic truncated
expectation

${\cal E}^{T}_{A}(F;\cdots;F)=\frac{\partial^{n}}{\partial\lambda^{n}}\log\int
P(dA)\,e^{F(A)}\Big{|}_{\lambda\equiv 0}$ (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

$\\{\psi^{+}_{i,s,x},\psi^{+}_{i^{\prime},s^{\prime},,x^{\prime}}\\}=\\{\psi^{+}_{i,s,x},\psi^{-}_{i^{\prime},s^{\prime},x^{\prime}}\\}=\\{\psi^{-}_{i,s,x},\psi^{-}_{i^{\prime},s^{\prime},x^{\prime}}\\}=0$
(13)

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

$P(d\psi)={1\over{\cal
N}_{\psi}}[\prod_{i,s,x}d\psi^{+}_{i,s,x}d\psi^{-}_{i,s,x}]e^{-S_{F}}$ (14)

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

$\displaystyle S_{F}={1\over
2a}\sum_{i=1}^{2N}a^{4}\sum_{x}[\sum_{\mu}(\psi^{+}_{i,x}\gamma_{0}\gamma_{\mu}\psi^{-}_{i,x+e_{\mu}a}-\psi^{+}_{i,s,x+e_{\mu}a}\gamma_{0}\gamma_{\mu}\psi^{-}_{i,x})+$
$\displaystyle
r(\psi^{+}_{i,x}\gamma_{0}\psi^{-}_{i,x+e_{\mu}a}+\psi^{+}_{i,x+e_{\mu}a}\gamma_{0}\psi^{-}_{i,x}-\psi^{+}_{i,x}\gamma_{0}\psi^{-}_{i,x})]$
(15)

We can write therefore

$\displaystyle S_{F}={1\over
2a}\sum_{i=1}^{2N}a^{4}\sum_{x}[\sum_{\mu}\sum_{s=L,R}(\psi^{+}_{i,s,x}\sigma_{\mu}^{s}\psi^{-}_{i,s,x+e_{\mu}a}-\psi^{+}_{i,s,x+e_{\mu}a}\sigma^{s}_{\mu}\psi^{-}_{i,s,x})+r(\psi^{+}_{i,L,x}\psi^{-}_{i,R,x+e_{\mu}a}+$
$\displaystyle\psi^{+}_{i,L,x+e_{\mu}a}\psi^{-}_{i,R,x}-\psi^{+}_{i,L,x}\psi^{-}_{i,R,x}+\psi^{+}_{i,R,x}\psi^{-}_{i,L,x+e_{\mu}a}+\psi^{+}_{i,R,x+e_{\mu}a}\psi^{-}_{i,L,x}-\psi^{+}_{i,R,x}\psi^{-}_{i,L,x})]$
(16)

The fermionic simple expectation

${\cal
E}_{\psi}(\psi^{\varepsilon_{1}}_{i_{1},x_{1}}...\psi^{\varepsilon_{n}}_{i_{n},x_{n}})=\int
P(d\psi)\psi^{\varepsilon_{1}}_{i_{1},x_{1}}...\psi^{\varepsilon_{n}}_{i_{n},x_{n}}$
(17)

is expressed by the anticommutative Wick rule with covariance

$g^{\psi}_{i}(x,y)=\int P(d\psi)\psi_{i,x}\bar{\psi}_{i,y}={1\over
L^{4}}\sum_{k}e^{ik(x-y)}\widehat{g}^{\psi}_{i}(k)$ (18)

with

$\widehat{g}_{i,k}=(\sum_{\mu}i\gamma_{0}\gamma_{\mu}a^{-1}\sin(k_{\mu}a)+a^{-1}\gamma_{0}r\sum_{\mu}(1-\cos
k_{\mu}a))^{-1}$ (19)

The interaction is

$\displaystyle V(A,\psi,J)=V_{1}(A,\psi,J)+V_{2}(A,\psi,J)$ (20)
$\displaystyle
V_{1}(A,\psi,J)=a^{4}\sum_{i,s,x}[O^{+}_{\mu,i,s,x}G_{\mu,i,s,x}^{+}+O^{-}_{\mu,i,s,x}G_{\mu,i,s}^{-}]$
$\displaystyle V_{2}(A,\psi,J)={r\over
2}a^{4}\sum_{i,x}[\psi^{+}_{i,L,x}H_{\mu,i,x}^{+}\psi^{-}_{i,R,x+e_{\mu}a}+$
$\displaystyle\psi^{+}_{i,L,x+e_{\mu}a}H_{\mu,i,x}^{-}\psi^{-}_{i,R,x}+\psi^{+}_{i,R,x}H_{\mu,i,x}^{+}\psi^{-}_{i,L,x+e_{\mu}a}+\psi^{+}_{i,R,x+e_{\mu}a}H_{\mu,i,x}^{-}\psi^{-}_{i,L,x}]$
(21)

with

$\displaystyle G^{\pm}_{\mu,i,s}(x)=a^{-1}(:e^{\mp iaQ_{i}(\lambda
b_{i,s}A_{\mu,x}+J_{\mu,x})}:-1)\quad H_{\mu,i,x}^{\pm}=a^{-1}(e^{\mp
iaQ_{i}J_{\mu,x}}-1)$ $\displaystyle O^{+}_{\mu,i,s,x}={1\over
2}\psi^{+}_{i,s,x}\sigma_{\mu}^{s}\psi^{-}_{i,s,x+e_{\mu}a}\quad\quad
O^{-}_{\mu,i,s,x}=-{1\over
2}\psi^{+}_{i,s,x+e_{\mu}a}\sigma^{s}_{\mu}\psi^{-}_{i,s,x}$ (22)

with, if $i_{1}=1,,N$ and $i_{2}=N+1,...,2N$

$b_{i_{1},L}=b_{i_{2},R}=1;\quad\quad b_{i_{1},R}=b_{i_{2},L}=0$ (23)

and $:e^{\pm ia\lambda Q_{i}A_{\mu}(x)}:=e^{\pm i\lambda
Q_{i}aA_{\mu}(x)}e^{{1\over 2}(\lambda Q_{i})^{2}a^{2}g^{A}_{\mu,\mu}(0,0)}$.

The mass counterterm is

$V_{c}=\sum_{i}a^{-1}\nu_{i}a^{4}\sum_{x}(\psi^{+}_{i,L,x}\psi^{-}_{i,R,x}+\psi^{+}_{i,R,x}\psi^{-}_{i,L,x})$
(24)

Finally the source term is

${\mathcal{B}}=a^{4}\sum_{\mu,x}J^{5}_{\mu,x}j^{5}_{\mu,x}\quad
j^{5}_{\mu,x}=\sum_{i,s}{\widetilde{\varepsilon}}_{i}\varepsilon_{s}Q_{j}Z_{i,s}^{5}\psi^{+}_{x,i,s}\sigma_{\mu}^{s}\psi^{+}_{x,i,s}$

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 [14].

###  1.3. Physical observables

The fermionic 2-point function is

$S^{\Lambda}_{i,s,s^{\prime}}(x,y)={\partial^{2}\over\partial\phi^{+}_{i,s,x}\partial\phi^{-}_{i,s^{\prime},y}}{\cal
W}_{\Lambda}(J,J^{5},\phi)|_{0}$ (25)

and the Fourier transform is

$\widehat{S}^{\Lambda}_{i,s,s^{\prime}}(k)=a^{4}\sum_{x}S^{\Lambda}_{i,s,s^{\prime}}(x,0)e^{-ikx}$
(26)

The vertex functions are

$\displaystyle\Gamma^{\Lambda}_{\mu,i^{\prime},s}(z,x,y)={\partial^{3}\over\partial
J_{\mu,z}\partial\phi^{+}_{i^{\prime},s,x}\partial\phi^{-}_{i^{\prime},s,y}}{\cal
W}(J,J^{5},\phi)|_{0}$
$\displaystyle\Gamma^{5,\Lambda}_{\mu,i^{\prime}s}(z,x,y)={\partial^{3}\over\partial
J^{5}_{\mu,z}\partial\phi^{+}_{i^{\prime},s,x}\partial\phi^{-}_{i^{\prime},s,y}}{\cal
W}(J,J^{5},\phi)|_{0}$ (27)

The Fourier transform is

$\widehat{\Gamma}^{\Lambda}_{\mu,i^{\prime},s}(k,p)=a^{4}\sum_{z}a^{4}\sum_{y}e^{-ipz-
iky}\Gamma^{\Lambda}_{\mu,i^{\prime},s}(z,0,y)$ (28)

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

$\Pi^{\Lambda}_{\mu,\nu,\rho}(z,y,x)={\partial^{3}{\cal
W}_{\Lambda}\over\partial J_{\mu,z}\partial J_{\nu,y}\partial
J_{\rho,x}}|_{0};\quad\Pi^{5,\Lambda}_{\mu,\nu,\rho}(z,y,x)={\partial^{3}{\cal
W}_{\Lambda}\over\partial J^{5}_{\mu,z}\partial J_{\nu,y}\partial
J_{\rho,x}}|_{0}$ (29)

and

$\displaystyle\widehat{\Pi}^{\Lambda}_{\mu,\nu,\rho}(p_{1},p_{2})=a^{4}\sum_{y}a^{4}\sum_{x}e^{-ip_{1}y-ip_{2}x}\Pi^{\Lambda}_{\mu,\nu,\rho}(0,y,x)$
$\displaystyle\Pi^{5,\Lambda}_{\mu,\nu,\rho}(p_{1},p_{2})=a^{4}\sum_{y}a^{4}\sum_{x}e^{-ip_{1}y-ip_{2}x}\Pi^{5,\Lambda}_{\mu,\nu,\rho}(0,y,x)$
(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

$\psi^{\pm}_{i,s,x}\rightarrow\psi^{\pm}_{i,s,x}e^{\pm iQ_{i}\alpha_{x}}$ (31)

with $\alpha_{x}$ is a function on $a\hbox{\msytw Z}^{4}$, 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

$\int[\prod_{i,s,x}d\psi^{+}_{i,s,x}d\psi^{-}_{i,s,x}]Q(\psi^{+},\psi^{-})=\int[\prod_{i,s,x}d\psi^{+}_{i,s,x}d\psi^{-}_{i,s,x}]Q_{\alpha}(\psi^{+},\psi^{-})\;.$
(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 fhe following identity, valid for any function $f$ on the finite
Grassmann algebra:

$\int[\prod_{i,s,x}d\psi^{+}_{i,s,x}d\psi^{-}_{i,s,x}]f(\psi)=\int[\prod_{i,s,x}d\psi^{+}_{i,s,x}d\psi^{-}_{i,s,x}]f_{\alpha}(\psi)\;,$
(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

$W(J,J^{5},\phi)=W(J+d_{\mu}\alpha,J^{5},e^{iQ\alpha}\phi)$ (34)

where $J+d_{\mu}\alpha$ is a shorthand for $J_{\mu,x}+d_{\mu}\alpha_{x}$ and
$e^{iQ\alpha}\phi$ is a shorthand for $e^{\pm
iQ_{i}\alpha_{x}}\phi_{i,s,x}^{\pm}$; by differentiating we get the Ward
Identities (WI)

$\displaystyle\sum_{\mu}\sigma_{\mu}(p)\widehat{\Pi}^{\Lambda}_{\mu,\nu_{1},..,\nu_{n}}(p_{1},.,p_{n})=0\quad
p=p_{1}+..p_{n}$
$\displaystyle\sum_{\mu}\sigma_{\mu}(p)\widehat{\Gamma}^{\Lambda}_{\mu,i,s}(k,p)=Q_{i}(\widehat{S}^{\Lambda}_{i,s,s}(k)-\widehat{S}^{\Lambda}_{i,s,s}(k+p))$
(35)
$\displaystyle\sum_{\nu}\sigma_{\nu}(p_{1})\widehat{\Pi}^{5,\Lambda}_{\mu,\nu,\rho}(p_{1},p_{2})=\sum_{\rho}\sigma_{\rho}(p_{2})\widehat{\Pi}^{5,\Lambda}_{\mu,\nu,\rho}(p_{1},p_{2})=0$

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
$\partial_{\mu}<j_{\mu}^{T,V};j_{\nu_{1}}^{T,V};...j_{\nu_{n}}^{T,V}>_{T}=0$,
see (5).

###  1.5. Main result

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

######  Theorem 1.1.

Let us fix $r=1$ and $Ma\geq 1$. There exists $\lambda_{0},C$ independent on
$L,a,M$ such that, for $|\lambda|\leq\lambda_{0}(Ma)$, it is possible to find
$\nu_{i}$, $Z_{i,s}^{5}$ continuous functions in $\lambda$ such that 1) The
limits $L\rightarrow\infty$ of $\widehat{S}^{\Lambda}_{i,s,s^{\prime}}(k)$,
$\widehat{\Gamma}^{\Lambda}_{\mu,i^{\prime},s}(k,p)$,
$\widehat{\Gamma}^{5,\Lambda}_{\mu,i^{\prime},s}(k,p)$
$\Pi^{\Lambda}_{\mu,\nu,\rho}(p_{1},p_{2})$,$\Pi^{5,\Lambda}_{\mu,\nu,\rho}(p_{1},p_{2})$
exist and $\lim_{k\rightarrow 0}\widehat{S}^{\Lambda}_{i,s}(k)=\infty$ and
$\lim_{k,p\rightarrow
0}{\widehat{\Gamma}^{5,\Lambda}_{\mu,i^{\prime},s}(k,p)\over\widehat{\Gamma}^{\Lambda}_{\mu,i^{\prime},s}(k,p)}=\varepsilon_{s}I$
where $\varepsilon_{L}=-\varepsilon_{R}=1$. 2) The AVV correlation verifies

$\sum_{\mu}\sigma_{\mu}(p_{1}+p_{2})\widehat{\Pi}^{5}_{\mu,\rho,\sigma}(p_{1},p_{2})=\sum_{\mu,\nu}\varepsilon_{\mu,\nu,\rho,\sigma}{1\over
2\pi^{2}}\sigma_{\mu}(p^{1})\sigma_{\nu}(p^{2})[\sum_{i_{1}}Q_{i_{1}}^{3}-\sum_{i_{2}}Q_{i_{2}}^{3}]+r_{\rho,\sigma}(p_{1},p_{2})$
(36)

with $|r(p_{1},p_{2})|\leq Ca^{\theta}\bar{p}^{2+\theta}$,
$\bar{p}=\max(|p_{1}|,|p_{2}|)$ and $\theta=1/2$.

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
presence of interactions; the parameters $Z_{i,s}^{5}$ 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. (3)

Under the condition $[\sum_{i_{1}}Q_{i_{1}}^{3}-\sum_{i_{2}}Q_{i_{2}}^{3}]=0$
we have
$\sum_{\mu}\sigma_{\mu}(p)\widehat{\Pi}^{\Lambda}_{\mu,\nu,\sigma}(p_{1},p_{2})=0$
and
$\sum_{\mu}\sigma_{\mu}(p)\widehat{\Pi}^{5,\Lambda}_{\mu,\rho,\sigma}(p_{1},p_{2})=O(a^{\theta}\bar{p}^{2+\theta})$
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. (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
$1/a\leq(\lambda_{0}/|\lambda|)M$, 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 $D=4+n-3n^{\psi}/2$ 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 $D=4-3n^{\psi}/2-n^{A}$, so in
principle one can consider cut-offs higher than $M$ and up to an exponentially
large values $|\lambda^{2}\log a|\leq\varepsilon_{0}$. 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.
[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
$\widehat{\Pi}_{\mu,\nu_{1},...,\nu_{n}}$ 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

$\partial_{\xi}\int
P(dA)\int[\prod_{i,s,x}d\psi^{+}_{i,s,x}d\psi^{-}_{i,s,x}]O={1\over
L^{4}}\sum_{p}\partial_{\xi}\widehat{g}^{-1}_{\mu,\nu}(p)\int
P(dA)\widehat{A}_{\mu,p}\widehat{A}_{\nu,-p}\int[\prod_{i,s,x}d\psi^{+}_{i,s,x}d\psi^{-}_{i,s,x}]O$
(37)

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

$\widehat{g}^{A}_{\mu,\rho^{\prime}}(p)\partial_{\xi}(\widehat{g}^{A}(p))^{-1}_{\mu,\nu}\widehat{g}^{A}_{\nu,\rho}(p){\partial^{2}\over\partial\widehat{J}_{\rho,p}\partial\widehat{J}_{\rho^{\prime},-p}}\int
P(dA)\int[\prod_{i,s,x}d\psi^{+}_{i,s,x}d\psi^{-}_{i,s,x}]O(A+{\widetilde{J}},\psi)|_{0}$
(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 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
[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 $[\sum_{i_{1}}Q_{i_{1}}^{3}-\sum_{i_{2}}Q_{i_{2}}^{3}]=0$ 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 $1/a\leq M$ 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

$V_{F}(\psi,J)=\log\int P(dA)e^{V_{1}(\psi,A,J)}$ (39)

where, by (12)

$\displaystyle
V_{F}(\psi,J)=a^{4}\sum_{x}\sum_{i,\varepsilon=\pm}a^{-1}(e^{-ia\varepsilon
Q_{i}J_{\mu,x}}-1)O^{\varepsilon}_{\mu,s,i}+\sum_{n=2}^{\infty}\
a^{4n}\sum_{x_{1},...,x_{n}}\sum_{\underline{\varepsilon},\underline{i}\atop\underline{\mu},\underline{s}}$
$\displaystyle[\prod_{j=1}^{n}O^{\varepsilon_{j}}_{i_{j},\mu_{j},s_{j},x_{j}}e^{i\varepsilon_{j}aQ_{i_{j}}J_{\mu_{j},x_{j}}}]{1\over
n!}a^{-n}{\cal E}^{T}_{A}(:e^{i\varepsilon_{1}b_{i_{1},s_{1}}\lambda
aQ_{i_{1}}A_{\mu_{1},x_{1}}}:;...;:e^{i\varepsilon_{n}b_{i_{n},s_{n}}\lambda
aQ_{i_{n}}A_{\mu_{n},x_{n}}}:)$

which can be rewritten as, if $\underline{x}=x_{1},..,x_{n}$,
$\underline{i}=i_{1},...,i_{n}$,
$\underline{\mu}=\mu_{1},...,\mu_{n}$,$\underline{m}=m_{1},..,m_{n}$

$\displaystyle
V_{F}(\psi,J)=a^{4}\sum_{x}\sum_{i,\varepsilon=\pm}a^{-1}(e^{-ia\varepsilon
Q_{i}J_{\mu,x}}-1)O^{\varepsilon}_{\mu,s,i}+$ (40)
$\displaystyle\sum_{n=2}^{\infty}\sum_{m=0}^{\infty}a^{4n}\sum_{x_{1},...,x_{n}}\sum_{\underline{\varepsilon},\underline{i},\underline{\mu},\underline{s},\underline{m}\atop\sum_{j}m_{j}=m}{1\over
n!}[\prod_{j=1}^{n}O^{\varepsilon_{j}}_{i_{j},\mu_{j},s_{j},x_{j}}(J_{\mu_{j},x_{j}})^{m_{j}}]H_{n,m}(\underline{x},\underline{\varepsilon},\underline{i},\underline{\mu},\underline{s},\underline{m})$

with

$H_{n,m}(\underline{x},\underline{\varepsilon},\underline{i},\underline{\mu},\underline{s},\underline{m})={a^{-n}\over
n!}{(ia\varepsilon_{j}Q_{i_{j}})^{m_{j}}\over m_{j}!}]{\cal
E}^{T}_{A}(:e^{i\varepsilon_{1}b_{i_{1},s_{1}}\lambda
Q_{i_{1}}A_{\mu_{1},x_{1}}}:;...;:e^{i\varepsilon_{n}b_{i_{n},s_{n}}\lambda
aQ_{i_{n}}A_{\mu_{n},x_{n}}})$ (41)

and
$||H_{n,m}||=L^{-4}a^{4n}\sup_{\underline{\varepsilon},\underline{i},\underline{\mu},\underline{s},\underline{m}\atop\sum_{j}m_{j}=m}\sum_{x_{1},...,x_{n}}|H_{n,m}|$.

######  Lemma 2.1.

The kernels in (40) the following bound, for $n\geq 2$, $m\leq 3$ and
uniformly in $L$

$||H_{n,m}||\leq C^{n}a^{-(4-3n-m)N}(|\lambda|/(Ma))^{2(n-1)}$ (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 [29] (for completeness a
sketch of the proof is in Appendix 1), $n\geq 2$

${\cal E}^{T}_{A}(e^{i\varepsilon_{1}b_{i_{1},s_{1}}\lambda
Q_{i_{1}}A_{\mu_{1},x_{1}}};...;e^{i\varepsilon_{n}b_{i_{n},s_{n}}\lambda
aQ_{i_{n}}A_{\mu_{n},x_{n}}})=\sum_{T\in\boldsymbol{T}_{n}}\prod_{\\{j,j^{\prime}\\}\in
T}{\widetilde{g}}^{A}_{\mu_{j}\mu_{j^{\prime}}}(x_{j},x_{j^{\prime}})\int
dp_{T}(\underline{t})e^{-V(X;\underline{t})},$ (43)

where:
$X=((x_{1},\varepsilon_{1},i_{1},\mu_{1},s_{1},m_{1});..;(x_{n},\varepsilon_{n},i_{n},\mu_{n},s_{n},m_{n}))$,
${\bf T}_{n}$ 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 $T$,

${\widetilde{g}}^{A}_{\mu_{j}\mu_{j^{\prime}}}(x_{j},x_{j^{\prime}})=\lambda^{2}a^{2}\varepsilon_{j}b_{i_{j},s_{j}}Q_{i_{j}}\varepsilon_{j^{\prime}}b_{i_{j^{\prime}},s_{j^{\prime}}}Q_{i_{j^{\prime}}}g^{A}_{\mu_{j}\mu_{j^{\prime}}}(x_{j},x_{j^{\prime}}),$
(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

$V(Y)=\sum_{j,j^{\prime}\in
Y}\lambda^{2}\varepsilon_{j}\varepsilon_{j^{\prime}}b_{i_{j},s_{j}}Q_{i_{j}}b_{i_{j^{\prime}},s_{j^{\prime}}}Q_{i_{j^{\prime}}}a^{2}g^{A}_{\mu_{j},\mu_{j^{\prime}}}(x_{j},x_{j^{\prime}})={\cal
E}_{A}([\sum_{j\in Y}\lambda
b_{i_{j},s_{j}}Q_{i_{j}}a\varepsilon_{j}A_{\mu_{j}}(x_{j})]^{2})$ (45)

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

$|g^{A}_{\mu,\nu}(x,y)|_{1}=a^{4}\sum_{x}|g^{A}_{\mu,\nu}(x,y)|\leq
CM^{-2}\quad\quad|g^{A}_{\mu,\mu}(x,y)|\leq Ca^{-2}$ (46)

so that

$||H_{n,m}||\leq
C_{1}^{n}L^{-4}\sup_{\underline{\varepsilon},\underline{i},\underline{\mu},\underline{s},\underline{m}\atop\sum_{j}m_{j}=m}a^{4n}\sum_{x_{1},...,x_{n}}{a^{-n+m}\over
n!}\sum_{T\in\boldsymbol{T}_{n}}\prod_{\\{j,j^{\prime}\\}\in
T}|{\widetilde{g}}^{A}_{\mu_{j}\mu_{j^{\prime}}}(x_{j},x_{j^{\prime}})|\int
dp_{T}(\underline{t})e^{-V(X;\underline{t})}|$ (47)

Moreover $V(Y)$ is stable, that is

$V(Y)={\cal E}([\sum_{j\in Y}\lambda
b_{i_{j},s_{j}}Q_{i_{j}}a\varepsilon_{j}A_{\mu_{j}}(x_{j})]^{2})\geq 0$ (48)

hence $V(X;\underline{t})\geq 0$ and $e^{-V(X;\underline{t})}\leq 1$ so that
$\int dp_{T}(\underline{t})e^{-V(X;\underline{t})}<1$ therefore

$||H_{n,m}||\leq C_{2}^{n}{a^{-n+m}\over
n!}\sum_{T\in\boldsymbol{T}_{n}}\prod_{\\{j,j^{\prime}\\}\in
T}a^{2}|g^{A}_{\mu_{j}\mu_{j^{\prime}}}(x_{j},x_{j^{\prime}})|_{1}\leq
C_{3}^{n}{a^{-n+m}\over n!}\sum_{T\in\boldsymbol{T}_{n}}(aM^{-1})^{2(n-1)}$
(49)

and finally using that $\sum_{T\in\boldsymbol{T}_{n}}1\leq C_{4}^{n}n!$ by
Cayley’ formula [38] we finally get

$||H_{n,m}||\leq
C_{5}^{n}a^{-n+m}(aM^{-1})^{2(n-1)}=C_{5}^{n}a^{-(4-3n-m)N}(|\lambda|/(Ma))^{2(n-1)}$
(50)

After the integration of $A_{\mu}$ the generating function can be written as a
Grassmann integral

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

with

$V^{(N+1)}(\psi,J,J^{5},\phi)=V_{F}(\psi,J)+V_{2}(\psi,J)+V_{c}(\psi)+{\mathcal{B}}(J^{5},\psi)+(\psi,\phi)$
(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 $N\in\hbox{\msytw N}$ such that111Any
choice for $\gamma^{N}$ ensuring that in the support of $1-{\widetilde{f}}$
does not include the doubled poles, that is the poles of $\widehat{g}(k)$ with
$r=0$ different from $k=0$, could be done. $\gamma^{N}\equiv\pi/(16a)$ ;
moreover we introduce ${\widetilde{f}}(t);\hbox{\msytw
R}^{+}\rightarrow\hbox{\msytw R}$ a $C^{\infty}$ non-decreasing function $=0$
for $0\leq t\leq\gamma^{N-1}$ and $=1$ for $t\geq\gamma^{N}$; we define also
$\chi_{N}(t)=1-{\widetilde{f}}(t)$ which is therefore non-vanishing for
$t\leq\gamma^{N}$. We introduce the propagator

$g^{(N+1)}_{i}(x,y)={1\over
L^{4}}\sum_{k}e^{ik(x-y)}{\widetilde{f}}(|k|_{T})\widehat{g}^{\psi}_{i}(k)$
(53)

with $|k-k^{\prime}|_{T}$ the distance on the 4-dimensional torus
$[-\pi/a,\pi/a)^{4}$. Therefore, for any $K\in\hbox{\msytw N}$ we have

$|g^{(N+1)}_{i}(x,y)|\leq\gamma^{3(N+1)}{C_{K}\over
1+(\gamma^{N+1}|x-y|_{{\widetilde{T}}})^{K}}$ (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
[32], using that222The 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 ${\widetilde{f}}(k)$. in the support of ${\widetilde{f}}(|k|_{T})$
one has $\sum_{\mu}(1-\cos k_{\mu}a)^{2}/a^{2}\geq C/a^{2}$ and the volume of
the support of ${\widetilde{f}}$ is $\leq C/a^{4}$.

We can write therefore, using the addition property of gaussian Grassmann
integrals, see e.g. §2.4 of [32]

$\displaystyle e^{{\cal W}(J,J^{5},\phi)}=\int P(d\psi^{(\leq N)})\int
P(d\psi^{(N+1)})e^{V^{(N+1)}(\psi^{(\leq N)}+\psi^{(N+1)},J,J^{5},\phi)}=$
$\displaystyle\int P(d\psi^{(\leq N)})e^{V^{(N)}(\psi^{(\leq
N)},J,J^{5},\phi)}$ (55)

where

$\displaystyle V^{(N)}(\psi^{(\leq
N)},J,J^{5},\phi)=\sum_{n=1}^{\infty}{1\over n!}$ $\displaystyle{\cal
E}^{T}_{N+1}(V^{(N+1)}(\psi^{(\leq
N)}+\psi^{(N+1)},J,J^{5},\phi);...;V^{(N+1)}(\psi^{(\leq
N)}+\psi^{(N+1)},J,J^{5},\phi))$ (56)

and ${\cal E}^{T}_{N+1}$ is the truncated expectation with respect to the
integration $P(d\psi^{(N+1)})$.

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

$V^{(N)}(\psi^{(\leq
N)},J,J^{5},\phi)=a^{4(l_{a}+l_{b}+m)}\sum_{\underline{x},\underline{y},\underline{z},\underline{\gamma}}W^{(N)}_{l_{a},l_{b},m}(\underline{x},\underline{y},\underline{z},\underline{\gamma})[\prod_{j=1}^{l_{a}}\psi^{\leq
N,\varepsilon_{j}}_{x_{j},i_{j},s_{j}}][\prod_{j=1}^{l_{b}}\phi^{\varepsilon_{j}}_{y_{j},i_{j},s_{j}}][\prod_{j=1}^{m}J^{\beta_{j}}_{\mu_{j},z_{j}}]$
(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 $\phi$ is briefly
discussed in the Appendix 1) calling $W^{(N)}_{l_{a},0,m}\equiv
W^{(N)}_{l_{a},m}$ we define
$||W^{(N)}_{l,m}||=L^{-4}\sup_{\underline{\gamma}}a^{4l+4m}\sum_{\underline{x},\underline{z}}|W^{(N)}_{l,m}(\underline{x},\underline{z},\underline{\gamma})|$.

######  Lemma 2.2.

The kernels in (55) verify, for $|\lambda|\leq\lambda_{0}(Ma)$, $|\nu_{i}|\leq
C(|\lambda|/(Ma))^{2}$, $\lambda_{0},C,C_{1}$ independent on $a,L,N$, $l\leq
2,m\leq 3$, $|d|$ is the distance between any coordinate in
$\underline{x},\underline{z}$

$||d^{s}W^{(N)}_{l,m}||\leq C_{1}\gamma^{DN}$ (58)

with $D=4-3l/2-m-s$.

Proof of Lemma 2.2. We rewrite $V^{(N+1)}$ (52) in a more compact way as

$V^{(N+1)}=\sum_{P}{\widetilde{\psi}}^{(\leq
N+1)}(P){\widetilde{J}}(P)W^{(N+1)}(P)$ (59)

with $P$ set of field labels and ${\widetilde{\psi}}(P)=\prod_{f\in
P}\psi^{(\leq N)\varepsilon(f)}_{i(f),s(f),x(f)}$,
${\widetilde{J}}(P)=\prod_{f\in P}J^{\beta(f)}_{x(f)}$. We get therefore,
inserting (59) in (56) if $P=Q_{1}\cup Q_{2}...\cup Q_{n}$

$\displaystyle V^{(N)}(\psi^{\leq N},J,J^{5})=\sum_{n}{1\over
n!}\sum_{P}\sum_{P_{1},...P_{n}\atop Q_{1},...Q_{n}}{\widetilde{\psi}}^{(\leq
N)}(P)$ $\displaystyle{\cal
E}^{T}_{N+1}({\widetilde{\psi}}^{(N+1)}(P_{1}/Q_{1});...;{\widetilde{\psi}}^{(N+1)}(P_{n}/Q_{n}))[\prod_{i=1}^{n}W^{(N+1)}(P_{i}){\widetilde{J}}(P_{i})]$
(60)

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])

${\cal
E}^{T}_{N+1}({\widetilde{\psi}}^{(N+1)}(P_{1});...;{\widetilde{\psi}}^{(N+1)}(P_{s}))=\sum_{T\in{\cal
T}_{n}}\prod_{\\{i,j\\}\in T}g^{(N+1)}(x_{i},y_{j})\int
dP_{T}(\underline{t})\det G^{N+1,T}(\underline{t})$ (61)

where ${\cal T}_{n}$ denotes the set of all the ‘spanning trees’ on
$x_{P_{1}},...,x_{P_{s}}$, 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 $T$, $\underline{t}=\\{t_{i,i^{\prime}}\in[0,1],1\leq
i,i^{\prime}\leq s\\}$, $dP_{T}(\underline{t})$ is a probability measure
(whose form is specified in the Appendix) with support on a set of
$\underline{t}$ such that $t_{i,i^{\prime}}=u_{i}\cdot u_{i^{\prime}}$ for
some family of vectors $u_{i}\in\hbox{\msytw R}^{s}$ 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
$G^{N+1,T}_{ij,i^{\prime}j^{\prime}}=t_{i,i^{\prime}}g^{(N+1)}(x_{ij},y_{i^{\prime}j^{\prime}})$
such that if =$<u_{i}\otimes A^{(N+1)}_{x(f^{-}_{ij})},u_{i^{\prime}}\otimes
B^{(N+1)}_{x(f^{+}_{i^{\prime}j^{\prime}})}>$ then the matrix element can be
written as a scalar product

$G^{N+1,T}_{ij,i^{\prime}j^{\prime}}=<u_{i}\otimes
A^{(N+1)}_{x(f^{-}_{ij})},u_{i^{\prime}}\otimes
B^{(N+1)}_{x(f^{+}_{i^{\prime}j^{\prime}})}>=(u_{i}\cdot
u_{i^{\prime}}){1\over
L^{d}}\sum_{k}\bar{A}^{(N+1)}_{x(f^{-}_{ij}),k)}B^{(N+1)}_{x(f^{+}_{i^{\prime}j^{\prime}}),k)}$
(62)

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

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 $<\cdot,\cdot>$, then $|\det
M|\leq\prod_{i}<A_{i},A_{i}>^{1\over 2}<B_{j},B_{j}>^{1\over 2}$. Therefore

$|\det G^{N+1,T}|\leq
C_{1}^{\sum_{i}|P_{i}|}\gamma^{3N\sum_{i}[|P_{i}|-(n-1))/2]})$ (63)

We get, setting $|P_{i}|\equiv n_{i}$, $|Q_{i}|\equiv l_{i}$, $0\leq l_{i}\leq
n_{i}$,$\sum_{i}m_{i}=m$, $l=\sum_{i}l_{i}$ and $||x|^{s}g^{(N+1)}(x)|_{1}\leq
C_{2}\gamma^{-N-s}$

$\displaystyle||W^{(N)}_{l,m}||\leq\sum_{n=1}^{\infty}C_{3}^{n}{1\over
n!}\sum_{n_{1},...n_{n}}\sum_{l_{1},..l_{n}\atop\sum_{i}l_{i}=l}[\prod_{i=1}^{n}{n_{i}!\over
l_{i}!(n_{i}-l_{i})!}]$
$\displaystyle\sum_{T}C_{2}^{s}\gamma^{-N((n-1)+s)}C_{1}^{\sum_{i}n_{i}}\gamma^{3N[\sum_{i}(n_{i}-l_{i})/2-(n-1)]})[\prod_{i=1}^{n}||W^{N+1}(P_{i})||]$
(64)

and $\sum_{T\in{\cal T}_{n}}\leq n!C_{4}^{\sum_{i}n_{i}}$, see e,g, lemma A3.3
of [36], Lemma 2.4 of [32] or Lemma D.4 of [32], so that

$\displaystyle||W^{(N)}_{l,m}||\leq\sum_{n=1}^{\infty}C_{6}^{n}\sum_{n_{1},...n_{n}}\sum_{l_{1},..l_{n}\atop\sum_{i}l_{i}=l}[\prod_{i=1}^{n}(C_{5})^{\sum_{i}n_{i}}{n_{i}!\over
l_{i}!(n_{i}-l_{i}!)}]$
$\displaystyle\gamma^{-4N(n-1)}\gamma^{-Ns}\gamma^{3N(\sum_{i}n_{i}-l_{i})/2}[\prod_{i=1}^{n}\gamma^{N(4-3n_{i}/2-m_{i})}][\prod_{i}(|\lambda|/Ma)^{\max(2(n_{i}/2-1),1-m_{i})}]]$

Note that $\sum_{l\leq n}[(C_{5})^{n-l}C_{5}^{l}{n!\over l!(n-l)!}]=2C_{5}$ we
get

$||W^{(N)}_{l,m}||\leq\sum_{n=1}^{\infty}C_{8}^{n}\gamma^{N(4-3l/2-m-s)}]\prod_{i}[\sum_{n_{i}}C_{7}^{n_{i}}(|\lambda|/Ma)^{\max(2(n_{i}/2-1),1-m_{i})}]]$
(65)

As $\sum_{i}m_{i}=m\leq 3$ the sum over $n_{i}$ is bounded by

$\prod_{i}[\sum_{n_{i}}C_{7}^{n_{i}}(|\lambda|/Ma)^{\max(2(n_{i}/2-1),1-m_{i})}]]\leq
C_{8}^{n}(|\lambda|/Ma)^{2\max(n-3,1)}$ (66)

so that for $\lambda$ small enough

$||W^{(N)}_{l,m}||\leq\gamma^{N(4-3l-m)}C_{9}(1+\sum_{n=4}^{\infty}C_{8}^{n}(|\lambda|/Ma)^{2(n-3)}]\leq
C_{1}\gamma^{N(4-3l/2-m)}$ (67)

In order integrate $\int P(d\psi^{(\leq N)})e^{V^{(N)}(\psi^{(\leq
N)},J,J^{5})}$ (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 ${\cal L}$ (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; ${\cal
L}\widehat{W}^{N}_{l,m}(\underline{k})=W^{N}_{l,m}(\underline{k})$ for
$(n,m)\not=(2,0),(2,1)$ and

${\cal L}\widehat{W}^{N}_{2,0}(k)=\widehat{W}^{N}_{2,0}(0)+{\sin k_{\mu}a\over
a}\partial_{\mu}\widehat{W}^{N}_{2,0}(0)\quad\quad{\cal
L}\widehat{W}^{N}_{2,1}(k,k+p)=\widehat{W}^{N}_{2,1,\mu}(0,0)$ (68)

We write therefore

$e^{{\cal W}(J,J^{5},0)}=\int P(d\psi^{(\leq N)})e^{{\cal
L}V^{(N)}(\psi^{(\leq N)},J,J^{5})+{\cal R}V^{(N)}(\psi^{(\leq N)},J,J^{5})}$
(69)

with ${\cal R}=1-{\cal L}$ (renormalization) and ${\cal R}V^{(N)}$ is equal to
(87) with $W^{(N)}_{l,m}$ replaced by ${\cal R}W^{(N)}_{l,m}$; the ${\cal R}$
operation produce an improvement in the bound, see eg §4.2 of [32]; for
instance ${\cal R}\widehat{W}^{N}_{2,0}(k)$ will admit, by interpolation, a
bound similar to the one for $\widehat{W}^{N}_{2,0}(k)$ times a factor
$O(\gamma^{-2N})$ due to the derivatives and an extra $O(\gamma^{2h})$, with
$h$ the scale associated to the external fields due to the $k^{2}$. Hence the
${\cal R}$ 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 [21].

Using symmetry considerations, see the Appendix 3, we get

$\displaystyle{\cal L}{\cal
V}^{(N)}(\psi,J,J^{5})=a^{4}\sum_{x}\sum_{i,s}[n_{N,s,i}\gamma^{-N}(\psi^{+}_{i,L,x}\psi^{-}_{i,R,x}+\psi^{+}_{i,R,x}\psi^{-}_{i,L,x})+$
$\displaystyle
z_{N,i,s}\sigma_{\mu}^{s}\psi^{+}_{i,s,x}{\widetilde{\partial}}_{\mu}\psi^{+}_{i,s,x}+{\widetilde{Z}}_{i,s,N}^{J}J_{\mu,x}\psi^{+}_{i,s,x}\sigma_{\mu}^{s}\psi^{-}_{i,s,x}+\varepsilon_{s}{\widetilde{\varepsilon}}_{i}{\widetilde{Z}}_{i,s,N}^{5}J^{5}_{\mu,x}\psi^{+}_{i,s,x}\sigma_{\mu}^{s}\psi^{-}_{i,s,x}]$

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

$P(d\psi^{(\leq
N)})e^{\sum_{i,s}z_{h,i,s}Z_{h,s,i}a^{4}\sum_{x}\sigma_{\mu}^{s}\psi^{+}_{i,s,x}{\widetilde{\partial}}_{\mu}\psi^{+}_{i,s,x}}\equiv
P_{Z_{N}}(d\psi^{(\leq N)})$ (70)

where ${\widetilde{\partial}}$ is the discrete derivative and

$\widehat{g}_{i}^{(\leq
N)}(k)=\chi_{N}(k)(\sum_{\mu}\gamma_{0}{\widetilde{\gamma}}^{N}_{\mu}a^{-1}i\sin(k_{\mu}a)+a^{-1}\widehat{\gamma}^{N}_{0}\sum_{\mu}(1-\cos
k_{\mu}a))^{-1}$ (71)
${\widetilde{\gamma}}^{N}_{0}=\begin{pmatrix}0&Z_{N,L,i}(k)I\\\
Z_{N,R,i}(k)I&0\end{pmatrix}\quad{\widetilde{\gamma}}^{h}_{j}=\begin{pmatrix}0&iZ_{N,L,i}(k)\sigma_{j}\\\
-iZ_{N,R,i}(k)\sigma_{j}&0\end{pmatrix}$ (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

$e^{{\cal W}(J,J^{5},0)}=\int P_{Z_{N}}(d\psi^{(\leq N)})e^{{\widetilde{\cal
L}}V^{(N)}(\sqrt{Z_{N}}\psi^{(\leq N)},J,J^{5})+{\cal
R}V^{(N)}(\sqrt{Z_{N}}\psi^{(\leq N)},J,J^{5})}$ (73)

where we have rescaled the fields writing

$\displaystyle{\widetilde{\cal L}}{\cal
V}^{(N)}(\sqrt{Z_{N}}\psi,J,J^{5})=a^{4}\sum_{x}\sum_{i,s}[\nu_{N,s,i}\gamma^{-N}\sqrt{Z_{N,L,i}Z_{N,R,i}}(\psi^{+}_{i,L,x}\psi^{-}_{i,R,x}+\psi^{+}_{i,R,x}\psi^{-}_{i,L,x})+$
$\displaystyle+Z_{i,s,N}^{J}J_{\mu,x}\psi^{+}_{i,s,x}\sigma_{\mu}^{s}\psi^{-}_{i^{\prime},s,x}+\varepsilon_{s}{\widetilde{\varepsilon}}_{i}Z_{i,s,N}^{5}J^{5}_{\mu,x}\psi^{+}_{i,s,x}\sigma_{\mu}^{s}\psi^{-}_{i,s,x}]$
(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
$\chi_{0}(t);\hbox{\msytw R}^{+}\rightarrow\hbox{\msytw R}$ a $C^{\infty}$
non-increasing function $=1$ for $0\leq t\leq\gamma^{-1}$ and $=0$ for $t\geq
1$; and we write

$\chi_{N}(t)=\sum_{h=-\infty}^{N}f_{h}(t)\quad\quad
f_{h}(t)=\chi_{0}(\gamma^{-h}t)-\chi_{0}(\gamma^{-h+1}t)$ (75)

with $f_{h}(t)$ with support in $\gamma^{h-1}\leq t\leq\gamma^{h+1}$. We can
write $\chi_{N}(t)=\chi_{N-1}(t)+f_{N}(t)$ and

$\widehat{g}_{i}^{(\leq N)}(k)=\widehat{g}_{i}^{(\leq
N-1)}(k)+\widehat{g}_{i}^{(N)}(k)$ (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

$\displaystyle e^{{\cal W}(J,J^{5},0)}=\int P_{Z_{N}}(d\psi^{(\leq N-1)})\int
P_{Z_{N}}(d\psi^{(N)})e^{{\widetilde{\cal L}}V^{(N)}(\sqrt{Z_{N}}\psi^{(\leq
N)},J,J^{5})+{\cal R}V^{(N)}(\sqrt{Z_{N}}\psi^{(\leq N)},J,J^{5})}=$
$\displaystyle\int P_{Z_{N}}(d\psi^{(\leq
N-1)})e^{V^{(N-1)}(\sqrt{Z_{N}}\psi^{(\leq N-1)},J,J^{5})}$ (77)

where

$V^{(N-1)}=\sum_{n}{1\over n!}{\cal E}^{T}_{N}({\widetilde{\cal
L}}V^{(N)}+{\cal R}V^{(N)},...;{\widetilde{\cal L}}V^{(N)}+{\cal R}V^{(N)})$
(78)

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

$V^{(N-1)}=\sum_{P,{\widetilde{P}}}{\widetilde{\psi}}^{(\leq
N-1)}(P){\widetilde{J}}(P)W^{(N-1)}(P)$ (79)

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

$V^{(N-1)}=\sum_{n}{1\over n!}{\cal E}^{T}_{N}({\widetilde{\cal
L}}V^{(N)}+\sum_{m}{1\over m!}{\cal R}{\cal E}^{T}_{N+1}(V^{(N+1)};m);n)$ (80)

From (80) we see that $W^{(N-1)}$ is a function of
$W^{(N+1)},\nu_{N},Z_{N}Z^{J}_{N},Z^{5}_{N}$.

The procedure can be iterated in a similar way writing

$P_{Z_{N}}(d\psi^{(\leq N-1)})=P_{Z_{N}-1}(d\psi^{(\leq
N-2)})P_{Z_{N}}(d\psi^{(N-1)})$ (81)

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

$\int P_{Z_{N-1}}(d\psi^{(\leq N-2)})\int
P_{Z_{N-1}}(d\psi^{(N-1)})e^{{\widetilde{\cal
L}}V^{(N-1)}(\sqrt{Z_{N-1}}\psi^{(\leq N-1)},J,J^{5})+{\cal
R}V^{(N-1)}(\sqrt{Z_{N-1}}\psi^{(\leq N-1)},J,J^{5})}$ (82)

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

$e^{{\cal W}(J,J^{5},0)}=\int P_{Z_{h}}(d\psi^{(\leq
h)})e^{V^{(h)}(\sqrt{Z_{h}}\psi^{(\leq h)},J,J^{5})}$ (83)

with $P_{Z_{h}}(d\psi^{(\leq h)})$ with propagator

$\widehat{g}_{i}^{(\leq
h)}(k)=\chi_{h}(k)(\sum_{\mu}\gamma_{0}{\widetilde{\gamma}}^{h}_{\mu}a^{-1}i\sin(k_{\mu}a)+a^{-1}\widehat{\gamma}^{h}_{0}\sum_{\mu}(1-\cos
k_{\mu}a))^{-1}$ (84)
${\widetilde{\gamma}}^{h}_{0}=\begin{pmatrix}0&Z_{h,L,i}(k)I\\\
Z_{h,R,i}(k)I&0\end{pmatrix}\quad{\widetilde{\gamma}}^{h}_{j}=\begin{pmatrix}0&iZ_{h,L,i}(k)\sigma_{j}\\\
-iZ_{h,R,i}(k)\sigma_{j}&0\end{pmatrix}$ (85)

and

$\displaystyle{\widetilde{\cal L}}{\cal
V}^{(h)}(\sqrt{Z_{h}}\psi,J,J^{5})=a^{4}\sum_{x}\sum_{i,s}[\nu_{h,s}\gamma^{h}\sqrt{Z_{h,L,i}Z_{h,R,i}}(\psi^{+}_{i,L,x}\psi^{-}_{i,R,x}+\psi^{+}_{i,R,x}\psi^{-}_{i,L,x})+$
$\displaystyle+Z_{i,s,h}^{J}J_{\mu,x}\psi^{+}_{i,s,x}\sigma_{\mu}^{s}\psi^{-}_{i^{\prime},s,x}+\varepsilon_{s}{\widetilde{\varepsilon}}_{i}Z_{i,s,h}^{5}J^{5}_{\mu,x}\psi^{+}_{i,s,x}\sigma_{\mu}^{s}\psi^{-}_{i,s,x}]$
(86)

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

$V^{(h-1)}(\sqrt{Z_{h}}\psi^{(\leq
h-1)},J,J^{5})=\sum_{l,m}a^{4l+4m}\sum_{\underline{x},\underline{z}}\sum_{\underline{\gamma}}W^{(h-1)}_{l,m}(\underline{x},\underline{z},\underline{\gamma})[\prod_{j=1}^{l}\psi^{\leq
h-1,\varepsilon_{j}}_{x_{j},i_{j},s_{j}}][\prod_{j=1}^{m}J^{\beta_{j}}_{\mu_{j},z_{j}}]$
(87)

and

$||W^{h-1)}_{l,m}||=L^{-4}\sup_{\underline{\gamma}}a^{4l+4m}\sum_{\underline{x},\underline{z}}|W^{(h-1)}_{l,m}(\underline{x},\underline{z})|$
(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

$\nu_{h-1,i}=\gamma\nu_{h,i}+\beta^{h}_{\nu,i}(\nu_{N},...,\nu_{h},W^{(N+1)})\quad{\mathcal{Z}}_{h-1,i,s}={\mathcal{Z}}_{h,i,s}+\beta^{h}_{\nu,i,s}(\nu_{N},{\mathcal{Z}}_{N}...,\nu_{h},{\mathcal{Z}}_{h},W^{(N+1)})$
(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 labeled trees, called
Gallavotti trees, , see Fig.4, defined in the following way (for details see
e.g. §3 of [32]) .

$v_{0}$

$v$

$v^{\prime}$

$h_{v}$

$N$

$N+1$

Figure 1.  A Gallavotti tree

Let us consider the family of all trees which can be constructed by joining a
point $r$, the root, with an ordered set of $n\geq 1$ 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 $\leq 4^{n}$, see eg §2.1 of
[32]. The set of labeled (or Gallavotti) trees ${\cal T}_{h,n}$ are defined
adding the above labels

  1. (1)

We associate a label $h\leq N-1$ 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 $\tau$ 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 can not be an endpoint; its scale is
$h+1$.

  2. (2)

To the end-points $v$ of scale $h_{v}\leq N$ is associated ${\widetilde{\cal
L}}V^{(h_{v})}$; there is the constraint that the vertex $v^{\prime}$
immediately preceding $v$, that is $h_{v^{\prime}}=h_{v}-1$ is non trivial (as
${\cal R}{\cal L}=0$). The end-points with $h_{v}\leq N$ 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 to
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 ${\cal R}$ operation associated to each vertex except the end-
points and $v_{0}$; if the tree contributes to ${\cal R}V^{h}$ it is
associated ${\cal R}$ while if it contributes to $\beta_{h}$ is associated
${\cal L}$ and $s_{v_{0}}\geq 2$.

  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

$V^{(h)}(\sqrt{Z_{h}}\psi^{(\leq
h)},J,J^{5})=\sum_{n=1}^{\infty}\sum_{\tau\in{\cal
T}_{h,n}}V^{(h)}(\tau,\sqrt{Z_{h}}\psi^{(\leq h)},J,J^{5})\;,$ (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, $h\leq N-1$

$\displaystyle V^{(h-1)}(\tau,\sqrt{Z_{h}}\psi^{(\leq h)},J,J^{5})=$ (91)
$\displaystyle{(-1)^{{s_{v_{0}}}+1}\over{s_{v_{0}}}!}{\cal
E}^{T}_{h}[\bar{V}^{(h)}(\tau_{1},\sqrt{Z_{h}}\psi^{(\leq
h)},J,J^{5});..;\bar{V}^{(h)}(\tau_{{s_{v_{0}}}},\sqrt{Z_{h}}\psi^{(\leq
h)},J,J^{5})]$

where ${\cal E}^{T}_{h}$ is the truncated expectation with propagator
$g_{i}^{(h)}$ and

  * •

if $\tau_{i}$ is non trivial $\bar{V}^{(h)}(\tau_{i},\sqrt{Z_{h}}\psi^{(\leq
h)},J,J^{5})={\cal R}V^{(h)}(\tau_{i},\sqrt{Z_{h}}\psi^{(\leq h)},J,J^{5})$

  * •

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

$W^{(h)}_{l,m}(\underline{x},\underline{z})=\sum_{n=1}^{\infty}\sum_{\tau\in{\cal
T}_{h,n}}W^{(h)}_{l,m}(\tau,\underline{x},\underline{z})$ (92)

It is also convenient to write

${\cal T}_{h,n}={\cal T}^{1}_{h,n}\cup{\cal T}^{2}_{h,n}$ (93)

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

$W^{i(h)}_{l,m}(\underline{x},\underline{z},\underline{\gamma})=\sum_{n=1}^{\infty}\sum_{\tau\in{\cal
T}^{i}_{h,n}}W^{(h)}_{l,m}(\tau;\underline{x},\underline{z},\underline{\gamma})$
(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

$\beta^{h}_{\nu}=\sum_{n=1}^{\infty}\sum_{\tau\in{\cal
T}^{2}_{h,n}}\beta^{h}_{\nu}(\tau)\quad\quad\beta^{h}_{\mathcal{Z}}=\sum_{n=1}^{\infty}\sum_{\tau\in{\cal
T}^{2}_{h,n}}\beta^{h}_{\mathcal{Z}}(\tau)$ (95)

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

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

with $\Pi^{5,i}_{h,\mu,\nu,\rho}=\sum_{n=1}^{\infty}\sum_{\tau\in{\cal
T}^{i}_{h,n}}W^{h}_{0,3}(\tau)$. The following lemma holds, see App. 2.

######  Lemma 2.3.

There exists a constant $\varepsilon$ such that, for $|{\mathcal{Z}}_{k}|\leq
e^{\varepsilon({aM})^{2}}$,
$\max(|\nu_{N}|,..,|\nu_{h}|,(\lambda/Ma)^{2}\leq\varepsilon$ than if $m\leq
3$, $d$ is the distance between any two coordinate

$||d^{s}W^{j(h)}_{l,m}||\leq
C^{l+m}\gamma^{(4-(3/2)l-m-s)h}\gamma^{\theta_{j}(h-N)}\varepsilon^{\max(l/2-1,1)}$
(97)

with $\theta_{1}=0$ and $\theta_{2}=\theta$ for a constant $\theta=1/2$;
moreover

$|\beta^{h}_{\nu}|\leq\varepsilon\gamma^{\theta(h-N)}\quad|\beta^{h}_{\mathcal{Z}}|\leq\varepsilon\gamma^{\theta(h-N)}$
(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 ${\cal L}=0$ then the
condition would be that $\lambda\leq\varepsilon_{h}$ 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 ${\cal T}^{2}$, that is
the terms obtained by the contraction of the irrelevant terms, have a gain
$\gamma^{\theta(h-N)}$ 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

$|{\mathcal{Z}}_{-\infty}-1|\leq
C\varepsilon\quad\quad|{\mathcal{Z}}_{-\infty}-{\mathcal{Z}}_{h}|\leq
C\varepsilon\gamma^{\theta(h-N)}$ (99)

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

$\nu_{h-1,i}=\gamma^{-h}(\nu_{N,i}+\sum_{k=h}^{N}\gamma^{k}\beta^{(h)}_{\nu,i})$
(100)

We consider the system

$\nu_{h-1,i}=\gamma^{-h}(-\sum_{k\leq h}\gamma^{k}\beta^{(h)}_{\nu,i})$ (101)

We can regard the right side of (101) as a function of the whole sequence
$\nu_{k,i}$, which we can denote by $\underline{\nu}=\\{\nu_{k}\\}_{k\leq N}$
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 $||\nu||=\sup_{k\leq N}\gamma^{\theta(k-N)}|\nu_{k}|\leq
C\lambda^{2}(Ma)^{-2}$. By a standard proof, see e.g. App A5 of [39], 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

$|\nu_{h}|\leq C\gamma^{\theta(h-N)}\varepsilon$ (102)

This means that the $\nu_{h,i}$ 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 $L\rightarrow\infty$ can be taken; the proof
is standard, see App. E of [39]. Finally we can choose
$Z^{5}_{i,s}=1+O(\varepsilon)$ so that

$Z^{5}_{i,s,-\infty}=Z^{J}_{i,s,-\infty}$ (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

$|\sum_{h=-\infty}^{N}\Pi^{5}_{h,\mu,\nu,\rho}(x,y,0)|_{1}\leq
C\sum_{h=-\infty}^{N}\gamma^{h}<C_{N}$ (104)

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

$|\sum_{h=-\infty}^{N}(|x|^{1+\theta/2}+|y|^{1+\theta/2})\Pi^{5,2}_{h,\mu,\nu,\rho}(x,y,0)|_{1}\leq
C\sum_{h=-\infty}^{N}\gamma^{-\theta/2h}\gamma^{\theta(h-N)}<\bar{C}_{N}$
(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 to $Z^{5},Z^{J},Z^{J}$. We can
write the propagator as

$g^{(h)}_{i,s,s^{\prime}}(x,y)=\delta_{s,s^{\prime}}{1\over
L^{4}}\sum_{k}{f_{h}(|k|_{T})\over-i\sigma^{s}_{\mu}k_{\mu}}e^{ik(x-y)}+r^{h}_{i,s,s^{\prime}}(x,y)$
(106)

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 $K$

$\displaystyle|g^{(h)}_{i,s,s^{\prime}}(x,y)|\leq\delta_{s,s^{\prime}}{1\over
Z_{h,i,s}}\gamma^{3(h+1)}{C_{K}\over
1+(\gamma^{h+1}|x-y|_{{\widetilde{T}}})^{K}}$
$\displaystyle|r^{(h)}_{i,s,s^{\prime}}(x,y)|\leq\gamma^{3(h+1)}\gamma^{h-N}{C_{K}\over
1+(\gamma^{h+1}|x-y|_{{\widetilde{T}}})^{K}}$ (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

$\widehat{\Pi}_{\mu,\rho,\sigma}(p_{1},p_{2})=\widehat{\Pi}^{a}_{\mu,\rho,\sigma}(p_{1},p_{2})+\widehat{R}_{\mu,\rho,\sigma}(p_{1},p_{2})$
(108)

with, $p=p_{1}+p_{2}$,

$\displaystyle\widehat{\Pi}^{a}_{\mu,\rho,\sigma}(p_{1},p_{2})=\sum_{h_{1}\atop
h_{2},h_{3}}\sum_{i,s}{\widetilde{\varepsilon}}_{i}\varepsilon_{s}Q_{i}^{3}{Z^{5}_{-\infty,i,s}\over
Z_{-\infty,i,s}}{Z^{J}_{-\infty,i,,s}\over
Z_{-\infty,i,s}}{Z^{J}_{-\infty,i,s}\over Z_{-\infty,i,s}}$
$\displaystyle\int{dk\over(2\pi)^{4}}{\rm Tr}{f_{h_{1}}(k)\over
i\sigma^{s}_{\mu}k_{\mu}}i\sigma^{s}_{\mu}{f_{h_{2}}\over
i\sigma^{s}_{\mu}(k_{\mu}+p_{\mu})}i\sigma^{s}_{\nu}{f_{h_{3}}\over
i\sigma^{s}_{\mu}(k_{\mu}+p^{2}_{\mu})}(i\sigma^{s}_{\rho})$ (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 App.2

$\widehat{S}_{i,s}(k)={1\over(i\sigma^{s}_{\mu}k_{\mu})}({I\over
Z_{i,s,-\infty}}+r_{1}(k))$ (110)

and

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

with $|r_{1}(k)|\leq C(a|k|)^{\theta}$ and $|r_{2,\mu}(k,p)|\leq
C(a|k|)^{\theta}$ with $|p|\leq|k|$.

$=$

$+$

Figure 2.  Graphical representation of (108)

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

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

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

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

with $\widehat{{\cal R}}$ with Holder continuous derivative and

$\widehat{I}_{\mu,\rho,\sigma}(p_{1},p_{2})=(\sum_{i}{\widetilde{\varepsilon}}_{i}Q_{i}^{3})\int{dk\over(2\pi)^{4}}{\rm
Tr}{\chi(k)\over\not k}\gamma_{\mu}\gamma_{5}{\chi(k+p)\over\not k+\not
p}\gamma_{\nu}{\chi(k+p^{2})\over\not k+\not p^{2}}\gamma_{\sigma}$ (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 [27], §3.6 for the
explicit computation

$\sum_{\mu}(p_{1,\mu}+p_{2,\mu})\widehat{I}_{\mu,\nu,\sigma}={(\sum_{i}{\widetilde{\varepsilon}}_{i}Q_{i}^{3})\over
6\pi^{2}}p_{1,\alpha}p_{2,\beta}\varepsilon_{\alpha\beta\nu\sigma}\quad\sum_{\nu}p_{1,\nu}\widehat{I}_{\mu,\nu,\sigma}={(\sum_{i}{\widetilde{\varepsilon}}_{i}Q_{i}^{3})\over
6\pi^{2}}p_{1,\alpha}p_{2,\beta}\varepsilon_{\alpha\beta\mu\sigma}$ (115)

up to $O(a^{\theta}|\bar{p}|^{2+\theta})$ corrections. In contrast with
$\widehat{I}_{\mu,\rho,\sigma}$, we have that $\widehat{{\cal
R}}_{\mu,\rho,\sigma}$ 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 $\widehat{{\cal
R}}_{\mu,\rho,\sigma}(p_{1},p_{2})$ to expand it at first order obtaining,
again up to $O(a^{\theta}|\bar{p}|^{2+\theta})$ corrections, using the Ward
Identity

$\frac{1}{6\pi^{2}}(\sum_{i}{\widetilde{\varepsilon}}_{i}Q_{i}^{3})p_{1,\alpha}p_{2,\beta}\varepsilon_{\alpha\beta\mu\sigma}+\sum_{\nu}p_{1,\nu}\Big{(}\widehat{{\cal
R}}_{\mu,\nu,\sigma}({0},{0})\hfill\break+\sum_{a=1,2}\sum_{\rho}p_{a,\rho}\frac{\partial\widehat{{\cal
R}}_{\mu,\nu,\sigma}}{\partial p_{a,\rho}}({0},{0})\Big{)}=0$

This implies that

$\widehat{{\cal R}}_{\mu,\nu,\sigma}({0},{0})=0$ (116)

and

${\partial\widehat{{\cal R}}_{\mu,\nu,\sigma}\over\partial
p_{2,\beta}}=-{1\over
6\pi^{2}}\varepsilon_{\nu\beta\mu\sigma}(\sum_{i}{\widetilde{\varepsilon}}_{i}Q_{i}^{3})\quad\quad{\partial\widehat{{\cal
R}}_{\mu,\nu,\sigma}\over\partial p_{1,\beta}}(0,{0})={1\over
6\pi^{2}}\varepsilon_{\nu\beta\mu\sigma}(\sum_{i}{\widetilde{\varepsilon}}_{i}Q_{i}^{3})$
(117)

Finally using such values we get

$\displaystyle\sum_{\mu}(p_{1,\mu}+p_{2,\mu})\widehat{\Pi}^{5}_{\mu,\nu,\sigma}(p_{1},p_{2})=\sum_{\alpha,\beta}\frac{(\sum_{i}{\widetilde{\varepsilon}}_{i}Q_{i}^{3})}{6\pi^{2}}p_{1,\alpha}p_{2,\beta}\varepsilon_{\alpha\beta\nu\sigma}$
(118) $\displaystyle+\sum_{\mu,\beta}(p_{1,\mu}+p_{2,\mu})({\widehat{{\cal
R}}_{\mu,\nu,\sigma}\over\partial p_{2,\beta}}(0,0)p_{2,\beta}+{\widehat{{\cal
R}}_{\mu,\nu,\sigma}\over\partial p_{1,\beta}}(0,0)p_{1,\beta})$

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

$-\frac{1}{6\pi^{2}}(p_{1,\mu}+p_{2,\mu})\sum_{a=1,2}(-1)^{a}p_{a,\beta}\varepsilon_{\nu\beta\mu\sigma}(\sum_{i}{\widetilde{\varepsilon}}_{i}Q_{i}^{3})=\frac{1}{3\pi^{2}}p_{1,\mu}p_{2,\beta}\varepsilon_{\nu\beta\mu\sigma}(\sum_{i}{\widetilde{\varepsilon}}_{i}Q_{i}^{3})$
(119)

which implies the Theorem 1.1

##  3\. Appendix 1: truncated expectations

###  3.1. The Brydges-Battle-Federbush formula

The starting point is the formula

${\cal
E}_{A}(\prod_{i=1}^{n}e^{i\varepsilon_{i}\alpha_{i}A_{\mu_{i}}(x_{i})})=e^{-{1\over
2}\sum_{i,j}\varepsilon_{i}\varepsilon_{j}\alpha_{i}\alpha_{j}g^{A}_{\mu_{i},\mu_{j}}(x_{i},x_{j})}$
(120)

Let us define

$e^{-V}\equiv e^{-{1\over 2}\sum_{j,j^{\prime}\in X}\bar{V}_{j,j^{\prime}}}$
(121)

with $X=(1,2,..,n)$ and $\sum_{i,j\in X}\bar{V}_{i,j}=\sum_{i\leq j}V_{i,j}$
$\bar{V}_{i,i}=V_{i,i}$ and $V_{i,j}=(\bar{V}_{i,j}+\bar{V}_{j,i})/2$.

The connected part $e^{-V(X)}|_{T}$ (corresponding to the truncated
expectation) verify

$e^{-V(X)}=\sum_{\pi}\prod_{Y\in\pi}e^{-V(Y)}|_{T}$ (122)

where the sum is over $\pi$ are the partitions of $X$, that is
$Y_{1},Y_{2},...$ with $Y_{1}\cup Y_{2}\cup..=X$.

If $X_{1}=\\{1\\}$ we can define

$W_{X}(X_{1};t_{1})=\sum_{\ell}t_{1}(l)V_{l}$ (123)

where $\ell=(j,j^{\prime})$ is a pair of elements $j,j^{\prime}\in X$ and
$t_{1}(l)=t_{1}$ if $l$ crosses the boundary of $X_{1}$ ($\partial X_{1}$),
that is if it connect $1$ with $j\not=1$; $t_{1}(\ell)=1$ otherwise. More
explicitely

$\displaystyle W_{X}(X_{1},t_{1})=V_{1,1}+t_{1}\sum_{k\geq
2}V_{1,k}+\sum_{2\leq k\leq k^{\prime}}V_{k,k^{\prime}}=$ $\displaystyle
t_{1}(V_{1,1}+\sum_{k\geq 2}V_{1,k}+\sum_{2\leq k\leq
k^{\prime}}V_{k,k^{\prime}})+(1-t_{1})(V_{1,1}+\sum_{2\leq k\leq
k^{\prime}}V_{k,k^{\prime}})=$ $\displaystyle
t_{1}V(X)+(1-t_{1})(V(X_{1})+V(X/X_{1}))$ (124)

We get $W_{X}(X_{1},0)=V(X_{1})+V(X/X_{1})$, that is if $t_{1}=0$ $X_{1}$ is
disconnected from the rest. Therefore, using that
$\partial_{1}W(X_{1},t_{1})=\sum_{k\geq 2}V_{1,k}=\sum_{l_{1}}V_{l_{1}}$ we
can write

$e^{-V(X)}=\int_{0}^{1}dt_{1}\partial_{1}e^{-W_{X}(X_{1},t_{1})}+e^{-W_{X}(X_{1},0)}$
(125)

and

$e^{-V(X)}=\int_{0}^{1}dt_{1}\sum_{k\geq
2}V_{1,k}e^{-W_{X}(X_{1},t_{1})}+e^{-V(X_{1})}e^{-V(X/X_{1})}$ (126)

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.

If $n\not=2$ we further decompose the first term in the r.h.s of (126); we
write $X_{2}=\\{1,k\\}$ and

$\displaystyle\int_{0}^{1}dt_{1}\sum_{k\geq 2}V_{1,k}e^{-W_{X}(X_{1},t_{1})}=$
(127) $\displaystyle\int_{0}^{1}dt_{1}\sum_{k\geq
2}V_{1,k}\int_{0}^{1}dt_{2}\partial_{t_{2}}e^{-W_{X}(X_{1},X_{2};t_{1},t_{2})}+\int_{0}^{1}dt_{1}\sum_{k\geq
2}V_{1,k}e^{-W_{X}(X_{1},X_{2};t_{1},0)}$

where

$W_{X}(X_{1},X_{2},t_{1},t_{2})=(1-t_{2})[W_{X_{2}}(X_{1},t_{1})+V(X/X_{2})]+t_{2}W_{X}(X_{1},t_{1})$
(128)

and for $X_{2}=(1,2)$

$W_{X}(X_{1},X_{2},t_{1},t_{2})=V_{1,1}+V_{2,2}+t_{1}t_{2}\sum_{k\geq
3}V_{1,k}+t_{1}V_{1,2}+t_{2}\sum_{k\geq 3}V_{2,k}+\sum_{3\leq k\leq
k^{\prime}}V_{k,k^{\prime}}$ (129)

Suppose that $X=\\{1,2,3\\}$ and $X_{2}=\\{1,2\\}$, then
$W_{X_{3}}(X_{1},X_{2},t_{1},t_{2})=V_{1,1}+V_{2,2}+t_{1}t_{2}V_{1,3}+t_{1}V_{1,2}+t_{2}V_{2,3}+V_{3,3}$
and

$\displaystyle\int_{0}^{1}dt_{1}V_{1,2}e^{-W_{X}(X_{1},t_{1})}=$ (130)
$\displaystyle\int_{0}^{1}dt_{1}V_{1,2}\int_{0}^{1}dt_{2}(t_{1}V_{1,3}+V_{2,3})e^{-W_{X}(X_{1},X_{2};t_{1},t_{2})}+[\int_{0}^{1}dt_{1}V_{1,2}e^{-W_{X_{2}}(X_{1};t_{1})}]e^{-V(X/X_{2})}$

and the first term is connected; similar expressions for $X_{2}=\\{1,3\\}$.

Proceeding in this way

$\displaystyle e^{-V(X)}=\sum_{r=1}^{n}\sum_{X_{r}\subset
X}\sum_{X_{1},..,X_{r-1}}\sum_{T}[\prod_{\ell\in T}V_{l}]$
$\displaystyle[\sum_{X_{1},..,X_{r-1}}\int_{0}^{1}dt_{1}...\int_{0}^{1}dt_{r-1}\prod_{\ell\in
T}{\prod_{k=1}^{r-1}t_{k}(\ell)\over
t_{n(\ell)}}e^{-W_{X_{r}}(X_{1},..,X_{r-1};t_{1},..,t_{r-1})}]e^{-V(X/X_{r})}$
(131)

where $X_{1}\subset X_{2}\subset...X_{r-1}$ are sets such that $|X_{i}|=i$,
$T$ is a tree composed by $r-1$ lines $\ell=(j,j^{\prime})$ such that all the
boundaries $\partial X_{k}$ are intersected at least by a line
$\ell=(j,j^{\prime})$,

$W_{X}(X_{1},..,X_{r};t_{1},..,t_{r})=\sum_{l}t_{1}(l)t_{2}(l)...t_{r}(l)V_{l}$
(132)

with $t_{i}(l)=t_{i}$ if $l$ crosses $\partial X_{i}$ and $t_{i}(l)=1$
otherwise, $n(l)$ is the max over $k$ such that $l$ crosses $\partial X_{k}$.
For instance in the case (130) the trees are $l_{1}=(1,2),l_{2}=(2,3)$ so that
$t_{1}(l_{1})=t_{1}$, $t_{1}(l_{2})=1$, $t_{2}(l_{2})=t_{2}$; and
$l_{1}=(1,2)$, $l_{2}=(1,3)$ so that $t_{1}(l_{1})=t_{1}$ and
$t_{1}(l_{2})=t_{1}$,$t_{2}(l_{2})=t_{2}$.

We can reverse the sum over $T$ and $X$

$\sum_{T}\sum_{X_{1},..,X_{r-1}}=\sum_{X_{1},..,X_{r-1}}\sum_{T}$ (133)

where in the l.h.s. the sets have to be compatible with $T$. If
$n^{\prime}(\ell)$ is the minimal $k$ such that $\ell$ crosses $X_{k}$ we have
${\prod_{k=1}^{r-1}t_{k}(\ell)\over
t_{n(\ell)}}=t_{n^{\prime}(\ell)}...t_{n(\ell)-1}$ and, see e.g. Lemma 2.3 in
[32]

$\sum_{X_{1},..,X_{r-1}\atop
fixedT}\int_{0}^{1}dt_{1}...\int_{0}^{1}dt_{r-1}t_{n^{\prime}(\ell)}...t_{n(\ell)-1}=1$
(134)

By calling

$dp_{T}(t)=\sum_{X_{1},..,X_{r-1}\atop fixedT}{\prod_{k=1}^{r-1}t_{k}(l)\over
t_{n(l)}}$ (135)

we get

$e^{-V(X)}|_{T}=\sum_{T}[\prod_{\ell\in
T}V_{l}]\int_{0}^{1}d\underline{t}dp_{T}(t)\varepsilon^{-\sum_{\ell\in
X^{\prime}\atop\ell\not T}t_{n^{\prime}(\ell)}...t_{n(\ell)-1}V_{\ell}}$ (136)

where $\ell\in X$ means $j,j^{\prime}\in(1,..,n)$.

###  3.2. The Gawedzki-Kupiainen-Lesniewski formula

We can write the simple expectations as

${\cal
E}({\widetilde{\psi}}(P_{1})...{\widetilde{\psi}}(P_{r}))=\int\prod_{i,j}d\eta_{i,j}e^{-\sum_{j,j^{\prime}}V_{jj^{\prime}}}$
(137)

with
$V_{jj^{\prime}}=\sum_{i=1}^{|P_{j}|}\sum_{i^{\prime}=1}^{|P_{j}^{\prime}|}\eta^{+}_{x_{ij}}g(x_{ij},x_{i^{\prime}j^{\prime}})\eta^{-}_{x_{i^{\prime}j^{\prime}}}$
and $\eta^{\pm}_{i,j}$ is a set of Grassmann variables. Again we can write
$e^{-\sum_{j,j^{\prime}}V_{jj^{\prime}}}$ as in (131) obtaining

$\displaystyle{\cal
E}^{T}({\widetilde{\psi}}(P_{1})...{\widetilde{\psi}}(P_{r}))=$ (138)
$\displaystyle\int\prod d\eta^{+}_{i,j}d\eta^{-}_{i,j}\sum_{T}[\prod_{l\in
T}V_{l}]\int_{0}^{1}\underline{d}\bar{t}dp_{T}(\underline{t})\varepsilon^{-\sum_{\ell\in
X^{\prime}}t_{n^{\prime}(\ell)}...t_{n(\ell)-1}V_{\ell}}$ (139)

with
$V_{\ell}=\sum_{i}\sum_{i^{\prime}}\eta^{+}_{i,j}g(x_{ij},x_{i^{\prime}j^{\prime}}\eta^{-}_{i,j}$,
$\ell=(j,j^{\prime})$. For each tree $T$ we divide the $\eta$ in the ones
appearing in $T$, called ${\widetilde{\eta}}$, and the rest, called
$\bar{\eta}$ so that, if $\sum_{\ell\in
X^{\prime}}t_{n^{\prime}(\ell)}...t_{n(\ell)-1}V_{\ell}={\widetilde{V}}(\underline{t})+\bar{V}(\underline{t})$
with $\bar{V}(\underline{t})$ obtained setting ${\widetilde{\eta}}=0$

${\cal
E}^{T}({\widetilde{\psi}}(P_{1})...{\widetilde{\psi}}(P_{r}))=\sum_{T}[\prod_{l\in
T}g_{\ell}]\int_{0}^{1}\underline{d}\bar{t}dp_{T}(\underline{t})\int\prod
d\bar{\eta}^{+}_{i,j}d\bar{\eta}^{-}_{i,j}\varepsilon^{-\bar{V}(\underline{t})}$
(140)

and $\prod
d\bar{\eta}^{+}_{i,j}d\bar{\eta}^{-}_{i,j}\varepsilon^{-\bar{V}(\underline{t})}=\det
G_{T}$ with $G_{T}$ with elements
$t_{n^{\prime}(j,j^{\prime})}...t_{n(jj^{\prime})-1}g(x_{ij},x_{i^{\prime}j^{\prime}})$.
Fixed $T$ we can relabel the $X_{k}$ so that
$t_{j}...t_{j^{\prime}-1}=u_{j}u_{j^{\prime}}$ with $u_{1}=v_{1}$,
$u_{j}=t_{j-1}u_{j-1}+v_{j}\sqrt{1-t^{2}_{j-1}}$ 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.

##  4\. 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 $v_{1},\ldots,v_{s_{v}}$ 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 $P_{v}=\cup_{i}Q_{v_{i}}$. The union
of the subsets $P_{v_{i}}\backslash Q_{v_{i}}$ are the internal fields of $v$.
The set of all $P_{v}$, $v\in\tau$ is called ${\bf P}$, and the set of all
$P_{v}$ with $v\geq\tau_{i}$ is called ${\bf P}_{i}$. From (91) we get, if
$n_{v_{0}}$ is the number of coordinate

$V^{(h)}(\tau)=\sum_{{\bf P}}a^{4n_{v_{0}}}\sum_{x_{v_{0}}}W^{(h)}_{\tau,{\bf
P}}(x_{v_{0}})[\prod_{f\in P_{v_{0}}}\sqrt{Z_{h}}\psi^{\varepsilon(f)(\leq
h)}_{x(f),i(f),s(f)}][\prod_{f}J(x_{f})]$ (141)

By definition we have a truncated expectation associated to each $v$ in the
tree $\tau$ non associated to an end-point; we can write each of them by the
Gawedzki-Kupiainen-Lesniewski formula. The ${\cal R}$ 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 $T=\cup_{v}T_{v}$. The sum over
coordinates of the propagators in $T$ and the estimates of the determinants
give a factor
$\gamma^{-4h_{v}(s_{v}-1)}\gamma^{3/2h_{v}(\sum_{i}|P_{v_{i}}|-|P_{v}|)}$, if
$S_{v}$ is the number of subtrees with root $v$. The renormalization produces
a factor $\prod_{v}\gamma^{-z_{v}(h_{v}-h_{v^{\prime}})}$ is produced by the
${\cal R}$ operation and $z_{v}=2$ if $|P_{v}|=2$ and there are no $J$ fields,
$z_{v}=1$ if $|P_{v}|=2$ and there is a single $J$ field, $z_{v}=0$ otherwise.
To the end-points with $i$ $\psi$ fields and $j$ $J$ fields is associated by
lemma 2.1 a factor
$\gamma^{(4-3i_{v}/2-j_{v})N}(\lambda^{i_{v}/2}(aM)^{2-i_{v}})$ with
$(4-3i_{v}/2-j_{v})<0$ and $i_{v}\geq 4$ and $(aM)^{2-i_{v}}<(aM)^{-2}$. We
get therefore

$\displaystyle a^{4n_{v_{0}}}\sum_{x_{v_{0}}}|W_{\tau,{\bf P}}(x_{v_{0}})|\leq
L^{4}\sum_{T}\prod_{v\,not\,e.p.}{1\over
s_{v}!}C^{\sum_{i=1}^{s_{v}}|P_{v_{i}}|-|P_{v}|}\gamma^{-4h_{v}(s_{v}-1)}$
(142)
$\displaystyle\gamma^{3/2h_{v}(\sum_{i}|P_{v_{i}}|-|P_{v}|)}[\prod_{v}\gamma^{-z_{v}(h_{v}-h_{v^{\prime}})}][\prod_{v\;e.p.\;not\;\nu,Z}\gamma^{(4-3i_{v}/2-j_{v})N}][\prod_{v\;e.p.\;\nu}\gamma^{h_{v}}]\varepsilon^{\bar{n}}$

By using that

$\displaystyle\sum_{v}(h_{v}-h)(s_{v}-1)=\sum_{v}(h_{v}-h_{v^{\prime}})(\sum_{i,j}m^{i,j}_{v}-1)$
$\displaystyle\sum_{v}(h_{v}-h)(\sum_{i}|P_{v_{i}}|-|P_{v}|)=\sum_{v}(h_{v}-h_{v^{\prime}})(\sum_{i,j}im^{i,j}_{v}-|P_{v}|)$
(143)

where $m^{i,j}_{v}$ is the number of end-points following $v$ with $i$ $\psi$
fields and $j$ $J$ fields , we get

$\displaystyle a^{4n_{v_{0}}}\sum_{x_{v_{0}}}|W_{\tau,{\bf
P},T}(x_{v_{0}})|\leq L^{4}\gamma^{-h[-4+{3|P_{v_{0}}|\over
2}-\sum_{i,j}(3i/2-4)m^{i,j}_{v_{0}}]}\varepsilon^{\bar{n}}$
$\displaystyle\prod_{v\;not\;e.p.}\left\\{{1\over
s_{v}!}C^{\sum_{i=1}^{s_{v}}|P_{v_{i}}|-|P_{v}|}\gamma^{-(-4+{3|P_{v}|\over
2}-\sum_{i,j}(3i/2-4)m^{i,j}_{v}+z_{v})(h_{v}-h_{v^{\prime}})}\right\\}$
$\displaystyle[\prod_{v\;e.p.\;not\;\nu}\gamma^{(4-3i_{v}/2-j_{v})N}][\prod_{v\;e.p.\;\nu}\gamma^{h_{v}}]$

We use now that

$\gamma^{h\sum_{i,j}m^{i,j}_{v_{0}}}\prod_{v\;not\;e.p.}\gamma^{\sum_{i,j}(h_{v}-h_{v^{\prime}})m^{i,j}_{v}}=\prod_{v\;e.p.}\gamma^{h_{v^{*}}}$
(144)

where $v^{*}$ is the first non trivial vertex following $v$; this implies

$\gamma^{h\sum_{i,j}(3i/2-4)m^{i,j}_{v_{0}}}\prod_{v\;not\;e.p.}\gamma^{\sum_{i,j}(3i/2-4)m^{i,j}_{v}(h_{v}-h_{v^{\prime}})}=\prod_{v\;e.p\;not\;\nu}\gamma^{h_{v^{*}}(3i_{v}/2-4)}\prod_{v\;e.p.\;\nu}\gamma^{-h_{v}}$
(145)

so that

$\displaystyle a^{4n_{v_{0}}}\sum_{x_{v_{0}}}|W_{\tau,{\bf
P},T}(x_{v_{0}})|\leq L^{4}\gamma^{-h[-4+{3|P_{v_{0}}|\over
2}]}\prod_{v\;e.p\;not\;\nu}\gamma^{h_{v^{*}}(3i_{v}/2-4)}\varepsilon^{\bar{n}}$
$\displaystyle\prod_{v\;not\;e.p.}\left\\{{1\over
s_{v}!}C^{\sum_{i=1}^{s_{v}}|P_{v_{i}}|-|P_{v}|}\gamma^{-(-4+{3|P_{v}|\over
2}+z_{v})(h_{v}-h_{v^{\prime}})}\right\\}[\prod_{v\;e.p.\;not\;\nu}\gamma^{(4-3i_{v}/2-j_{v})N}]$
(146)

Finally we use the relation

$[\prod_{v\;e.p.}\gamma^{h_{v^{*}}j_{v}}][\prod_{v\;e.p.}\gamma^{-h_{v^{*}}j_{v}}]=[\prod_{v\;e.p.}\gamma^{h_{v^{*}}j_{v}}]\gamma^{-h\sum_{i,j}jm^{i,j}_{v_{0}}}\prod_{v\;not\;e.p.}\gamma^{-\sum_{i,j}(h_{v}-h_{v^{\prime}})jm^{i,j}_{v}}$
(147)

and using that $\sum_{i,j}jm^{i,j}_{v}=n^{J}_{v}$ we finally get ($j_{v}=0$ if
$v$ is a $\nu$-e.p.)

$\displaystyle a^{4n_{v_{0}}}\sum_{x_{v_{0}}}|W_{\tau,{\bf
P},T}(x_{v_{0}})|\leq L^{4}\gamma^{-h[-4+{3|P_{v_{0}}|\over
2}+n^{J}_{v_{0}}]}\varepsilon_{h}^{\bar{n}}$
$\displaystyle\prod_{v\,\hbox{\ottorm not e.p.}}\left\\{{1\over
s_{v}!}C^{\sum_{i=1}^{s_{v}}|P_{v_{i}}|-|P_{v}|}\gamma^{-(-4+{3|P_{v}|\over
2}+z_{v}+n^{J}_{v})(h_{v}-h_{v^{\prime}})}\right\\}[\prod_{v\;e.p.\;not\;\nu,Z}\gamma^{(4-3i_{v}/2-j_{v})(N-h_{v^{*}})}]$

In conclusion

$a^{4n_{v_{0}}}\sum_{x_{v_{0}}}|W_{\tau,{\bf P},T}({\bf x}_{v_{0}})|\leq
L^{4}\gamma^{-hd_{v_{0}}}C^{n}\varepsilon^{\bar{n}}[\prod_{{\widetilde{v}}}{1\over
s_{{\widetilde{v}}}!}\gamma^{-d_{{\widetilde{v}}}(h_{{\widetilde{v}}}-h_{{\widetilde{v}}^{\prime}})}][\prod_{v\;e.p.\;not;\nu,Z}\gamma^{(4-3i_{v}/2-j_{v})(N-h_{v^{*}})}]$
(148)

where: ${\widetilde{v}}\in{\widetilde{V}}$ are the vertices on the tree such
that $\sum_{i}|P_{v_{i}}|-|P_{v}|\not=0$, ${\widetilde{v}}^{\prime}$ is the
vertex in ${\widetilde{V}}$ immediately preceding ${\widetilde{v}}$ or the
root; $d_{v}=-4+{3|P_{v}|\over 2}+n^{J}_{v}+z_{v}$. Finally the number of
addenda in $\sum_{T\in{\bf T}}$ is bounded by
$\prod_{v}s_{v}!\;C^{\sum_{i=1}^{s_{v}}|P_{v_{i}}|-|P_{v}|}$, see e.g.§2.1 of
[36] . In order to bound the sums over the scale labels and ${\bf P}$ we first
use the inequality

$\prod_{{\widetilde{v}}}\gamma^{-d_{{\widetilde{v}}}(h_{{\widetilde{v}}}-h_{{\widetilde{v}}^{\prime}})}\leq[\prod_{{\widetilde{v}}}\gamma^{-{1\over
2}(h_{{\widetilde{v}}}-h_{{\widetilde{v}}^{\prime}})}][\prod_{{\widetilde{v}}}\gamma^{-{3|P_{{\widetilde{v}}}|\over
4}}]$ (149)

where ${\widetilde{v}}$ are the non trivial vertices, and
${\widetilde{v}}^{\prime}$ is the non trivial vertex immediately preceding
${\widetilde{v}}$ or the root. The factors $\gamma^{-{1\over
2}(h_{{\widetilde{v}}}-h_{{\widetilde{v}}^{\prime}})}$ in the r.h.s. allow to
bound the sums over the scale labels by $C^{n}$ and $\sum_{\bf
P}\prod_{{\widetilde{v}}}\gamma^{-{3|P_{{\widetilde{v}}}|\over 4}}\leq C^{n}$,
see §3.7 of of [32] .

Let us consider the improvement of the bound. If ${\cal T}^{*}$ is the set of
trees with at least an end-point not of $\nu,Z$ type then, for $0<\theta<1$

$\sum_{\tau\in{\cal T}^{*}}\sum_{{\bf
P},T}a^{4n_{v_{0}}}\sum_{x_{v_{0}}}|W_{\tau,{\bf P},T}(x_{v_{0}})|\leq
L^{4}\gamma^{(4-(3/2)l-m)h}\gamma^{\theta(h-N)}\varepsilon^{\max(l/2-1,1)}$
(150)

To prove (150) let be $\widehat{v}$ the non trivial vertex following an end-
point not of $\nu,Z$ type; hence we can rewrite in (148)

$[\prod_{{\widetilde{v}}}\gamma^{-d_{{\widetilde{v}}}(h_{{\widetilde{v}}}-h_{{\widetilde{v}}^{\prime}})}]=[\prod_{{\widetilde{v}}}\gamma^{-(d_{{\widetilde{v}}}-\theta)(h_{{\widetilde{v}}}-h_{{\widetilde{v}}^{\prime}})}]\gamma^{\theta(h-h_{\widehat{v}})}$
(151)

and

$\gamma^{\theta(h-h_{\widehat{v}})}[\prod_{v\;e.p.\;not\;\nu,Z}\gamma^{(4-3i_{v}/2-j_{v})(N-h_{v^{*}})}]\leq\gamma^{\theta(h-N)}$
(152)

as
$\prod_{v\;e.p.\;not\;\nu,Z}\gamma^{(4-3i_{v}/2-j_{v})(N-h_{v^{*}})}\leq\gamma^{-\theta(N-h_{\widehat{v}})}$
as there is at least an end-point not $\nu,Z$. Noting that
$d_{{\widetilde{v}}}-\theta>0$ one can perform the sum as above, and the same
bound is obtained with an extra $\gamma^{\theta(h-N)}$.

In presence of a $\phi$ term there is a new relevant coupling proportional to
$\psi\phi$, 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 $\nu$ 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 $(\psi\phi)$; there is an integral missing giving an extra
factor $\gamma^{2\bar{h}}$ reproducing the similar factor associated to the
$\nu$ end-points. There is a decay factor proportional to $x-y$ at scale
$\gamma^{\bar{h}}$ and, from the trees beloging to ${\cal T}^{*}$, an extra
$\gamma^{\theta(h-N)}$; see e.g. §3.D of [24]. A similar argument holds for
the vertex function. Finally the proof of the $L\rightarrow\infty$ limit is an
easy corollary of the proof of lemma 2.3, see e.g. App D of [24].

##  5\. Appendix 3: symmetries

By symmetry there are no quadratic contributions with $i^{\prime}\not=i$.
There is invariance under the transformation
$\psi^{\pm}_{k,s}\rightarrow\varepsilon_{s}\psi^{\mp}_{{\widetilde{k}},s}\sigma_{1}$,
$J_{{\bf k}}\,A_{\bf k}\rightarrow J_{{\widetilde{\bf k}}},A_{{\widetilde{\bf
k}}}$ invariant, if ${\widetilde{k}}$ is equal to $k$ with $k_{0},k_{1}$
replaced with $-k_{0},-k_{1}$ and $k_{2},k_{3}$ invariant. As $j=(2,3)$
$\sigma_{1}\sigma_{j}\sigma_{1}=-\sigma_{j}$ hence $\sum_{k}\sin
k_{j}\psi^{+}_{k,s}\sigma_{j}\psi^{-}_{k,s}\rightarrow\sum_{k}\sin
k_{j}\psi^{-}_{{\widetilde{k}},s}\sigma_{1}\sigma_{j}\sigma_{1}\psi^{+}_{{\widetilde{k}},s}=\sum_{k}\sin
k_{j}\psi^{+}_{k,s}\sigma_{0}\psi^{-}_{k}$ and for $j=0,1$
$\sigma_{1}\sigma_{j}\sigma_{1}=\sigma_{j}$ hence $\sum_{k}\sin
k_{j}\psi^{+}_{k,s}\sigma_{0}\psi^{-}_{k}\rightarrow\sum_{k}\sin
k_{j}\psi^{-}_{{\widetilde{k}},s}\sigma_{1}\sigma_{j}\sigma_{1}\psi^{+}_{{\widetilde{k}},s}=\sum_{k}\sin
k_{j}\psi^{+}_{k,s}\sigma_{j}\psi^{-}_{k,s}$; and $\sum_{k}\cos
k_{j}\psi^{+}_{k,L}\sigma_{0}\psi^{-}_{k,R}\rightarrow\sum_{k}-\cos
k_{i}\psi^{-}_{{\widetilde{k}},L}\sigma_{1}\sigma_{0}\sigma_{1}\psi^{+}_{{\widetilde{k}},R}$.
Similarly there is invariance under the trasformation
$\psi^{\pm}_{k,s}\rightarrow\varepsilon_{s}\psi^{\mp}_{{\widetilde{k}},s}\sigma_{2}$,
$J_{{\bf k}}\,A_{\bf k}\rightarrow J_{{\widetilde{\bf k}}},A_{{\widetilde{\bf
k}}}$ invariant, if ${\widetilde{k}}$ is equal to $k$ with $k_{0},k_{2}$
replaced with $-k_{0},-k_{2}$ and $k_{1},k_{3}$ invariant. As
$\sigma_{2}\sigma_{j}\sigma_{2}=-\sigma_{j}$ $j=(1,3)$ hence $\sum_{k}\sin
k_{j}\psi^{+}_{k,s}\sigma_{j}\psi^{-}_{k,s}\rightarrow\sum_{k}\sin
k_{j}\psi^{-}_{{\widetilde{k}},s}\sigma_{1}\sigma_{j}\sigma_{1}\psi^{+}_{{\widetilde{k}},s}=\sum_{k}\sin
k_{j}\psi^{+}_{k,s}\sigma_{0}\psi^{-}_{k}$ and for $j=0,2$
$\sigma_{2}\sigma_{j}\sigma_{2}=\sigma_{j}$ hence $\sum_{k}\sin
k_{j}\psi^{+}_{k,s}\sigma_{0}\psi^{-}_{k}\rightarrow\sum_{k}\sin
k_{j}\psi^{-}_{{\widetilde{k}},s}\sigma_{1}\sigma_{j}\sigma_{1}\psi^{+}_{{\widetilde{k}},s}=\sum_{k}\sin
k_{j}\psi^{+}_{k,s}\sigma_{j}\psi^{-}_{k,s}$; and $\sum_{k}\cos
k_{j}\psi^{+}_{k,L}\sigma_{0}\psi^{-}_{k,R}\rightarrow-\sum_{k}\cos
k_{i}\psi^{-}_{{\widetilde{k}},L}\sigma_{1}\sigma_{0}\sigma_{1}\psi^{+}_{{\widetilde{k}},R}$.

We can write
$\sum_{k}k_{2}\psi^{+}_{{\widetilde{k}},s}A_{2}\psi^{-}_{{\widetilde{k}},s}=\sum_{k}k_{2}[a\sigma_{0}+b_{\mu}\sigma_{1}+c_{\mu}\sigma_{2}+d_{\mu}\sigma_{3}]$.
We apply the first transformation to
$\sum_{k}k_{2}\psi^{+}_{k,s}\psi^{-}_{k,s}(a\sigma_{0}+b\sigma_{1}+c\sigma_{2}+d\sigma_{3})\rightarrow-\sum_{k}k_{2}\psi^{+}_{{\widetilde{k}}}\sigma_{1}(a\sigma_{0}+b\sigma_{1}+c\sigma_{2}+d\sigma_{3})\sigma_{1}\psi^{-}_{{\widetilde{k}}}=-\sum_{k}k_{2}\psi^{+}_{k,s}\psi^{-}_{k,s}(a\sigma_{0}+b\sigma_{1}-c\sigma_{2}-d\sigma_{3})\sigma_{1}\psi^{-}_{k}$
hence $a=b=0$. Now we apply the second transformation then
$\sum_{k}k_{2}\psi^{+}_{k,s}\psi^{-}_{k,s}\sigma_{2}(c\sigma_{2}+d\sigma_{3})\sigma_{2}\rightarrow-\sum_{k}k_{2}\psi^{+}_{{\widetilde{k}},s}\psi^{-}_{{\widetilde{k}},s}(c\sigma_{2}-d\sigma_{3})=\sum_{k}k_{2}\psi^{+}_{{\widetilde{k}},s}\psi^{-}_{{\widetilde{k}},s}(c\sigma_{2}-d\sigma_{3})$
hence $d=0$. Then
$\sum_{k}k_{2}\psi^{+}_{{\widetilde{k}},s}A\psi^{-}_{{\widetilde{k}},s}=\sum_{k}k_{2}b\psi^{+}_{k,s}\sigma_{2}\psi^{-}_{k,s}$,
and the geeral relation follows from isotropy. Proceeding in a similar way
with the terms with different chirality
$\sum_{k}k_{2}{\widetilde{b}}\psi^{+}_{{\widetilde{k}},L}\sigma_{2}\psi^{-}_{{\widetilde{k}},R}\rightarrow-\sum_{k}k_{2}{\widetilde{b}}\psi^{+}_{{\widetilde{k}},L}\sigma_{2}\psi^{-}_{{\widetilde{k}},R}$
hence ${\widetilde{b}}=0$.

Finally by the first tranformation
$\sum_{k}\psi^{+}_{k,L}\psi^{-}_{k,R}(a\sigma_{0}+b\sigma_{1}+c\sigma_{2}+d\sigma_{3})\rightarrow\sum_{k}\psi^{+}_{{\widetilde{k}},L}\sigma_{1}(a\sigma_{0}+b\sigma_{1}+c\sigma_{2}+d\sigma_{3})\sigma_{1}\psi^{-}_{{\widetilde{k}},R}=\sum_{k}\psi^{+}_{k,L}(a\sigma_{0}+b\sigma_{1}-c\sigma_{2}-d\sigma_{3})\sigma_{1}\psi^{-}_{k,R}$
so that $c=d=0$; by the second
$\sum_{k}\psi^{+}_{k,L}\psi^{-}_{k,R}(a\sigma_{0}+b\sigma_{2})\rightarrow\sum_{k}\psi^{+}_{{\widetilde{k}},L}\sigma_{2}(a\sigma_{0}+b\sigma_{1})\sigma_{2}\psi^{-}_{{\widetilde{k}},R}=\sum_{k}\psi^{+}_{\,L}\sigma_{2}(a\sigma_{0}-b\sigma_{1})\sigma_{2}\psi^{-}_{k,R}$
hence $b=0$.

Acknowledgements.  This work has been done partly at the Institute for
Advanced Study, Princeton. We got support also from MUR, project MaQuMA,
PRIN201719VMAST01, and INDAM-GNFM.

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