# Nonperturbative renormalization of the lattice Sommerfield vector model

Vieri Mastropietro  Institute for Advanced Study, Princeton, USA  University
of Milan, Italy

###### Abstract

The lattice Sommerfield model, describing a massive vector gauge field coupled
to a light fermion in 2d, is an ideal candidate to verify perturbative
conclusions. In contrast with continuum exact solutions, we prove that there
is no infinite field renormalization, implying the reduction of the degree of
the ultraviolet divergence, and that the anomalies are non renormalized. Such
features are the counterpart of analogue properties at the basis of the
Standard Model perturbative renormalizability. The results are non-
perturbative, in the sense that the averages of invariant observables are
expressed in terms of convergent expansions uniformly in the lattice and
volume.

##  I Introduction.

Most properties of the Standard Model are known only at a perturbative level
with series expansions expected to be generically diverging; in particular its
perturbative renormalizability [1],[2] relies on two crucial properties, the
reduction of the degree of divergence with respect to power counting and the
cancellation of the anomalies [3] ensured by the Adler-Bardeen theorem [4].
Such properties are essential to maintain the renormalizability present with
massless bosons. The phenomenon of the reduction of the degree of divergence
can be already seen in a $U(1)$ gauge theory like QED. Adding a mass to the
photon breaks gauge invariance and produces a propagator of the form ${1\over
k^{2}+M^{2}}(\delta_{\mu\nu}+{k_{\mu}k_{\nu}\over M^{2}})$; due to the lack of
decay of the second term the theory becomes dimensionally non renormalizable.
However the transition in a $U(1)$ gauge theory like QED from a $M=0$ to a
$M\not=0$ case is soft and the theory remains perturbatively renormalizable
[5]; the photons are coupled to a conserved current
$k_{\mu}\widehat{j}_{\mu}=0$ so that the contribution of the non-decaying part
of the propagator is vanishing. A similar reduction happens in the electroweak
sector, but the fermion mass violates the chiral symmetry and leads to the
Higgs introduction; again the renormalizability proof relies on the fact that
the $k_{\mu}k_{\nu}$ term in the propagator does not contribute [2]. The
chiral symmetry is generically violated by anomalies which need to cancel out,
and such cancellation is based on the Adler-Bardeen property.

All the above arguments are valid in perturbation theory and non-perturbative
effects could be missed. This issue would be solved by a non-perturbative
lattice anomaly-free formulation of gauge theory, which is still out of reach,
see for instance [6]-[8]. In particular one needs to get high values of cut-
off, exponential in the inverse coupling, a property which is the non-
perturbative analogue of renormalizabiliy. The implementation of the Adler-
Bardeen theorem and of the reduction of the degree of divergence in a non-
perturbative context is however a non trivial issue, as their perturbative
derivation uses dimensional regularizations, and functional integral
derivations [9] are essentially one loop results [10].

It is convenient therefore to investigate such properties in a simpler
context, and the Sommerfield model [11], describing a massive vector gauge
field coupled to a light fermion in 2d, appears to be the ideal candidate, see
also [12],[13],[14]. More exactly, we consider a version of this model with
non zero fermionic mass, but our results are uniform in the mass. The model
can be seen as a $d=2$ QED with a massive photon; as in 4d, at the level of
perturbation theory the transition from $M=0$ to a $M\not=0$ is soft and the
theory remains super-renormalizable. Again this follows from the conservation
of currents, which is ensured at the level of correlations by dimensional
regularization; the same regularization provides the anomaly non-
renormalization [12]. In this case however we have access to non-perturbative
information and we can check such conclusions. Exact solutions are known in
the continuum version of the Sommerfield model [15], [11], [16], [17].
Remarkably the above perturbative features are not verified; there is an
infinite wave function renormalization incompatible with the
superrenormalizability, and anomalies have a value depending on the
regularization.

In this paper we consider the Sommerfield model on the lattice, and we analyze
it using the methods of constructive renormalization. The lattice preserves a
number of symmetries, in the form of Ward Identities. Our main result is that
there is no infinite field renormalization, which is the counterpart of
superrenormalizability, and that the Adler-Bardeen theorem holds with finite
lattice. Non perturbative violation of the above perturbative conclusions is
therefore excluded. Other 2d models previously rigorously constructed, see
[18]-[25], lack of these features. Quantum simulations of 2d models [26]-[28]
have been also considered in the literature, but they regard mostly the
Schwinger model, to which the Sommerfield model reduces when the boson and
fermion mass is vanishing. Our results are non-perturbative, in the sense that
the averages of gauge invariant observables are expressed in terms of
convergent expansions uniformly in the lattice and volume.

The paper is organized in the following way. In §II we define a lattice
version of the Sommerfield model. In §III we derive exact Ward Identities for
the model. In §IV we integrate the boson field and in §V we perform a nn-
perturbative multiscale analysis for the fermionic fields. In §VI we prove the
validity of the Adler-Bardeen theorem and in §VII the conclusions are
presented.

##  II The lattice Sommerfield model

If $\gamma_{0}=\sigma_{1}$, $\gamma_{1}=\sigma_{2}$, we define

$\left\langle O\right\rangle={1\over
Z}\int\prod_{x}d\bar{\psi}_{x}d\psi_{x}\int_{R^{2|\Lambda|}}\prod_{x,\mu}dA_{\mu,x}e^{-S(A,\psi)}O$
(1)

where $Z$ the normalization, $x\in\Lambda$, with $\Lambda$ is a square lattice
with step $a$ with antiperiodic boundary conditions and

$S(A,\psi)=S_{A}(A)+S_{\psi}(A,\psi)$ (2)

with

$\displaystyle S_{A}(A)=a^{2}\sum_{x}[{1\over
4}F_{\mu,\nu,x}F_{\mu,\nu,x}+{M^{2}\over 2}A_{\mu,x}A_{\mu,x}]$ $\displaystyle
S_{\psi}(A,\psi)=a^{2}\sum_{x}[{\widetilde{m}}\bar{\psi}_{x}\psi_{x}+$ (3)
$\displaystyle
a^{-1}Z_{\psi}(\bar{\psi}_{x}\gamma^{+}_{\mu}e^{iaeA_{\mu,x}}\psi_{x+a_{\mu}}-\bar{\psi}_{x+a_{\mu}}\gamma^{-}_{\mu}e^{-iaeA_{\mu,x}}\psi_{x})]$

with $a_{\mu}=ae_{\mu}$, $e_{0}=(1,0),e_{1}=(0,1)$,
$\gamma^{\pm}_{\mu}=\gamma_{\mu}\mp r$
$F_{\mu,\nu}=d_{\nu}A_{\mu}-d_{\mu}A_{\nu}$ and
$d_{\nu}A_{\mu}=a^{-1}(A_{\mu,x+e_{\nu}a}-A_{\mu,x})$,
${\widetilde{m}}=(m+4r/a)$ and $r=1$ is the Wilson term. Note that if $1/a$
and $L$ are finite the integral is finite dimensional.

We generalize the model adding a term
$(1-\xi)a^{2}\sum_{x}\sum_{\mu}(d_{\mu}A_{\mu})^{2}$, $\xi\leq 1$ so that the
bosonic action is given by ${1\over
2}a^{2}\sum_{x}(\sum_{\mu,\nu}(d_{\mu}A_{\nu})^{2}+\xi\sum_{\mu}(d_{\mu}A_{\mu})^{2})$.
The original model is recovered with $\xi=1$.

The correlations can be written as derivatives of the generating function,

$\displaystyle e^{W_{\xi}(J,B,\phi)}=$ (4) $\displaystyle\int
P(d\psi)P(dA)e^{-V(A+J,\psi)+(\psi,\phi)+a^{2}\sum_{x}B_{x}O}$

with $O=O(A+J,\psi)$ an observable, and $P(dA)$ the gaussian measure with
covariance

$\widehat{g}^{A}_{\mu,\nu}(k)={1\over|\sigma|^{2}+M^{2}}(\delta_{\mu,\nu}+{\xi\bar{\sigma}_{\mu}\sigma_{\nu}\over(1-\xi)|\sigma|^{2}+M^{2}})$
(5)

with $\sigma_{\mu}(k)=(e^{ik_{\mu}a}-1)a^{-1}$.

$P(d\psi)$ is the fermionic integration with propagator
$\widehat{g}^{\psi}(k)=Z_{\psi}^{-1}({\widetilde{k}}_{\mu}\gamma_{\mu}+a^{-1}m(k)I)^{-1}$
with ${\widetilde{k}}_{\mu}={\sin(k_{\mu}a)\over a}$ and $m(k)=m+ra^{-1}(\cos
ak_{0}+\cos ak_{1}-2)$; finally

$V(A,\psi)=a^{2}\sum_{x}[O^{+}_{\mu,x}G_{\mu,x}^{+}(A)+O^{-}_{\mu,x}G_{\mu,x}^{-}(A)]$
(6)

with $O^{+}_{\mu}=Z_{\psi}\bar{\psi}_{x}(\gamma_{\mu}-r)\psi_{x+a_{\mu}}$ and

$O^{-}_{\mu}=-Z_{\psi}\bar{\psi}_{x+a_{\mu}}(\gamma_{\mu}+r)\psi_{x}$ (7)

and $G_{\mu}^{\pm}=a^{-1}(e^{\pm ieaA_{\mu,x}}-1)$.

If $M=0$ the model (1) invariant under the gauge transformation
$A_{\mu,x}\rightarrow A_{\mu,x}+d_{\mu}\alpha_{x}$ and
$\psi_{x}\rightarrow\psi e^{-ie\alpha_{x}}$; if $M\not=0$ the invariance is
lost.

##  III Ward Identities and $\xi$-independence

If we restrict to observables such that $O(A,\psi)=O(A+d\alpha,\psi
e^{-ie\alpha})$ (which we call invariant observables) there is gauge
invariance in the external fields also for $M\not=0$, that is

$W_{\xi}(J+d\alpha,e^{-ie\alpha}\phi,B)=W_{\xi}(J,\phi,B)$ (8)

This follows by performing in (II) the change of variables
$\psi_{x}\rightarrow\psi e^{ie\alpha_{x}}$, with Jacobian equal to $1$ (the
integral is finite-dimensional) and noting that
$(e^{ie\alpha}\psi,\phi)=(\psi,\phi e^{-ie\alpha})$ and

$S_{\psi}(A+J,\psi e^{ie\alpha})=S_{\psi}(A+J+d\alpha,\psi)$ (9)

(8) implies that $\partial_{\alpha}W_{\xi}(J+d\alpha,e^{-ie\alpha}\phi,B)=0$.
We define $\Gamma_{\mu_{1},..,\mu_{n},\nu_{1},..\nu_{m}}$ as the derivatives
of $W_{\xi}$ with respect to $J_{\mu_{1},x_{1}},..,B_{\nu_{n},x_{n}}$. By
performing in (8) derivatives with respect to $\alpha$ and the external fields
we get the Ward Identities (expressing current conservation)

$\sum_{\mu_{1}}\sigma_{\mu_{1}}(p_{1})\widehat{\Gamma}_{\mu_{1},..,\nu_{n}}(p_{1},..,p_{n-1})=0$
(10)

and

$\sigma_{\mu}(p)\widehat{\Gamma}_{\mu}(p.k)=\widehat{S}(k)-\widehat{S}(k+p)$
(11)

where
$\widehat{\Gamma}_{\mu}(p,k)={\partial^{3}W\over\partial\widehat{J}_{\mu,p}\partial\widehat{\phi}_{k}\partial\bar{\phi}_{k-p}}|_{0}$
is the vertex function and
$\widehat{S}(k)={\partial^{2}W\over\partial\widehat{\phi}_{k}\partial\bar{\phi}_{k}}|_{0}$
is the 2-point function.

The conservation of current expressed by the above WI implies that for
invariant observables

$\partial_{\xi}W_{\xi}(J,0,B)=0$ (12)

that is the averages are $\xi$ independent. This follows from
$\partial_{\xi}\int P(dA)\int\prod_{x}d\psi_{x}d\bar{\psi}_{x}O=0$, with
$O(A,\psi)$ invariant; indeed

$\displaystyle\partial_{\xi}\int P(dA)\int\prod d\psi_{x}d\bar{\psi}_{x}O=$
(13) $\displaystyle{1\over
L^{2}}\sum_{p}\partial_{\xi}\widehat{g}^{-1}_{\mu,\nu}(p)\int
P(dA)A_{\mu,p}A_{\nu,-p}\int\prod d\psi_{x}d\bar{\psi}_{x}O$

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

$\displaystyle\widehat{g}^{A}_{\mu,\rho^{\prime}}(p)\partial_{\xi}(\widehat{g}^{A}(p))^{-1}_{\mu,\nu}\widehat{g}^{A}_{\nu,\rho}(p)$
(14)
$\displaystyle{\partial^{2}\over\partial\widehat{J}_{\rho,p}\partial\widehat{J}_{\rho^{\prime},-p}}\int
P(dA)\int\prod_{x}d\psi_{x}d\bar{\psi}_{x}O(A+J,\psi)|_{0}$

By noting that

$\partial(\widehat{g}^{A})^{-1}=-(\widehat{g}^{A})^{-1}\partial_{\xi}\widehat{g}^{A}(\widehat{g}^{A})^{-1}$
(15)

and $\partial_{\xi}\widehat{g}^{A}$ is proportional to
$\bar{\sigma}_{\mu}\sigma_{\nu}$, by using

$\partial_{\alpha}\int
P(dA)\int\prod_{x}d\psi_{x}d\bar{\psi}_{x}O(A+d\alpha,\psi)|_{0}=0$ (16)

then (13) is vanishing.

(12) ensures that the averages does not depend on $\xi$, so that one can set
$\xi=0$ in the boson propagator, that is the non decaying part of the
propagator does not contribute. In perturbation theory the scaling dimension
with $\xi=0$ ($z=2$) and $\xi=1$ ($z=0$) is, if $n$ is the order, $n_{A}$ the
number of $A$ fields and $n_{\psi}$ the number of $\psi$ fields

$d+(d-z-2)n/2-(d-1)n_{\psi}/2-(d-z)n_{A}/2$ (17)

hence in $d=2$ the theory is dimensionally renormalizable with $\xi=1$ and
superrenormalizable with $\xi=0$ (in $d=4$ one pass from non-renormalizability
to renormalizability). The lattice regularization ensures that i the theory
remains perturbatively superrenormalizable, as with dimensional
regularization. We will investigate the validity of this property at a non-
perturbative level.

Finally, we define the axial current as
$j_{\mu}^{5}=Z^{5}\bar{\psi}_{x}\gamma_{\mu}\gamma_{5}\psi_{x}$, where $Z^{5}$
is a constant to be chosen so that the electric charge of the chiral and e.m.
current are the same, defined as the amputated part of the 3-point correlation
at zero momenta (see [10]), that is

$\lim_{k,p\rightarrow
0}{\partial^{3}W\over\partial\widehat{B}^{5}_{\mu,p}\partial\widehat{\phi}_{k}\partial\bar{\phi}_{k-p}}|_{0}/{\partial^{3}W\over\partial\widehat{J}_{\mu,p}\partial\widehat{\phi}_{k}\partial\bar{\phi}_{k-p}}|_{0}=1$
(18)

where the source term is $(B_{\mu}^{5},j_{\mu}^{5})$. The axial current is non
conserved even for $m=0$, due to the presence of Wilson term, and one has

$\sigma_{\mu}(p)\widehat{\Gamma}^{5}_{\mu,\nu}(p)=H_{\nu}(p)$ (19)

with $\Gamma^{5}_{\mu,\nu}$ the derivative of $W$ with respect to
$B_{\mu,x_{1}},J_{\nu,x_{2}}$. $H_{\nu}$ is called the anomaly and in the non-
interacting case $V=0$ one gets if $m=0$ $H_{\mu}={1\over
2\pi}\varepsilon_{\mu,\nu}p_{\nu}+O(ap^{2})$ (lattice or dimensional
regularization [12] produce the same result) and $Z^{5}=1$. In the interacting
case $H_{\nu}(p)$ is a series in $e$ and the non renormalization property
means that all higher orders corrections vanishes.

##  IV Integration of the boson fields

We can integrate the boson field

$\int P(dA)e^{-V}=e^{\sum_{n=0}^{\infty}{(-1)^{n}\over n!}{\cal
E}^{T}_{A}(V;n)}\equiv e^{V^{N}(\psi,J)}$ (20)

where ${\cal E}^{T}_{A}$ is the truncated expectation, that is the sum of
connected diagrams, and $V^{N}=$

$\displaystyle a^{2}\sum_{x}\sum_{\varepsilon}a^{-1}e^{-{1\over
2}e^{2}a^{2}g^{A}_{\mu,\mu}(x,x)}e^{iaeJ_{\mu,x}}O^{\varepsilon}_{\mu}+$ (21)
$\displaystyle\sum_{n,m}a^{n+m}\sum_{\underline{x},\underline{y}}\sum_{\underline{\mu},\underline{\varepsilon}}[\prod_{j=1}^{n}O^{\varepsilon_{j}}_{\mu_{j},x_{j}}][\prod_{k=1}^{m}G^{\varepsilon_{j}}_{\mu_{j},y_{j}}(J)]W_{n,m}(\underline{x},\underline{y})$

Note that $a^{2}g^{A}_{\mu,\mu}(x,x)\leq C$. We call $a=\gamma^{-N}$, where
$\gamma>1$ is a scaling parameter.

###### Theorem 1

The kernels in (21) for $n\geq 2$ verify,

$|W_{n,m}|\leq C^{n+m}e^{2(n-1)}\gamma^{N(2-n-m)}(|g^{A}|_{1})^{n}$ (22)

Proof. A convenient representation for ${\cal E}^{T}_{A}$ is given by the
following formula [29]

${\cal
E}^{T}_{A}(\prod_{k=1}^{n}e^{i\varepsilon_{k}aA_{\mu_{k},x_{k}}})=\sum_{T\in{\cal
T}}\prod_{i,j\in T}V_{i,j}\int dp_{T}(s)e^{-V_{T}(s)}$ (23)

where $V_{i,j}=e^{2}a^{2}{\cal E}_{A}(A_{\mu_{i},x_{i}}A_{\mu_{j},x_{j}})$,
${\cal T}$ is the set of tree graphs $T$ on $X=(1..,n)$, $s\in(0,1)$ is an
interpolation parameter, $V_{T}(s)$ is a convex linear combination of
$V(Y)=\sum_{i,j\in Y}\varepsilon_{i}\varepsilon_{j}V_{i,j}$, $Y$ subsets of
$X$ and $dp_{T}$ is a probability measure. The crucial point is that $V(Y)$ is
stable, that is

$V(Y)=\sum_{i,j\in Y}V_{i,j}=a^{2}e^{2}{\cal E}_{A}([\sum_{i\in
Y}\varepsilon_{i}A_{\mu_{i},x_{i}}]^{2})\geq 0$ (24)

Therefore one can bound the exponential $e^{-V_{T}(s)}\leq 1$ finding

$\displaystyle|W_{n,0}|\leq C^{n}a^{-n}{1\over n!}\sum_{T\in{\cal
T}}\prod_{(i,j)\in T}a^{2}e^{2}|g^{A}(x_{i},x_{j})|_{1}\leq$ (25)
$\displaystyle\sum_{T\in{\cal T}}C^{n}e^{2(n-1)}a^{n-2}(|g^{A}|_{1})^{n}\leq
C^{n}e^{2(n-1)}\gamma^{N(2-n)}(|g^{A}|_{1})^{n}$

With $m\not=0$ we get an extra $a^{-Nm}$, so that one recovers the dimensional
factor $\gamma^{N(2-l/2-m)}$.

For $\xi=0$ $|g^{A}|_{1}\leq CM^{-2}$ and $|g^{A}|_{\infty}\leq C|\log a|$
while for $\xi=1$ $|g^{A}|_{1}\leq C|\log a|$ and $|g^{A}|_{\infty}\leq
Ca^{-2}$. We write $W_{n,m}=\lambda^{n-1}\bar{W}_{n}$ with $\lambda=e^{2}$ and
$\bar{W}_{n}$ bounded. The normalization $Z_{\xi}$ in the analogue of (1) is
intere and $\xi$ independent; our strategy is to prove that
$Z_{0}=1+O(\lambda)$ and is analytic together with correlations for
$|\lambda|\leq\lambda_{0}$ with $\lambda_{0}$ $a,L$ independent; the same
therefore is true for $Z_{1}$, and as the numerator (1) is intere, than (1) is
analytic in $|\lambda|\leq\lambda_{0}$ and equal to the $\xi=0$ case. It
remains then to prove that the correlations with $\xi=0$ are analytic for
$|\lambda|\leq\lambda_{0}$ and $Z_{0}=1+O(\lambda)$.

The factor $D=2-n-m$ is the scaling dimension, and the terms with $D<0$ are
irrelevant. The marginal term for $\xi=0$ is ${\cal E}^{T}_{A}(V;2)=$

$\sum_{\varepsilon_{1},\varepsilon_{2}}a^{4}\sum_{x_{1},x_{2}}e^{i\varepsilon_{1}aJ_{\mu,x_{1}}}O^{\varepsilon_{1}}_{\mu,x_{1}}e^{i\varepsilon_{2}aJ_{\mu,x_{2}}}O^{\varepsilon_{2}}_{\mu,x_{2}}\lambda
v_{\mu,\varepsilon_{1},\varepsilon_{2}}$ (26)

where $\lambda v_{\mu,\varepsilon_{1},\varepsilon_{2}}={\cal
E}^{T}_{A}(e^{i\varepsilon_{1}eaA_{\mu,x_{1}}};e^{i\varepsilon_{2}eaA_{\mu,x_{2}}})$
and, $\lambda=e^{2}$

$\displaystyle v_{\mu,\varepsilon_{1},\varepsilon_{2}}(x,y)=e^{-{1\over
2}e^{2}a^{2}g^{A}_{\mu,\mu}(x_{1},x_{1})}$ (27) $\displaystyle e^{-{1\over
2}e^{2}a^{2}g^{A}_{\mu,\mu}(x_{2},x_{2})}(e^{-a^{2}\varepsilon_{1}\varepsilon_{2}g^{A}_{\mu,\mu}(x_{1},x_{2})}-1)e^{-2}a^{-2}$

which can be rewritten as

$\int_{0}^{1}dtg^{A}_{\mu,\mu}(x_{1},x_{2})e^{-{\widetilde{V}}(t)}$ (28)

with

$\displaystyle
2{\widetilde{V}}(t)=a^{2}t\left\langle(\varepsilon_{1}A_{\mu}(x_{1})+\varepsilon_{2}A_{\mu}(x_{2}))^{2}\right\rangle$
(29)
$\displaystyle+a^{2}(1-t)(g^{A}_{\mu,\mu}(x_{1},x_{1})+g^{A}_{\mu,\mu}(x_{2},x_{2}))$

in agreement with (23). For definiteness we keep only the dimensionally non
irrelevant terms considering

$e^{W_{1}(J,B,\phi)}=\int P(d\psi)e^{{\cal V}+G(B)+(\psi,\phi)}$ (30)

with $G(B)$ is a generic source term for gauge invariant observables and
${\cal V}=$

$a^{2}\sum_{x}\sum_{\varepsilon}a^{-1}(e^{-{1\over
2}e^{2}a^{2}g^{A}_{\mu,\mu}(x,x)}e^{iaeJ_{\mu,x}}-1)O^{\varepsilon}_{\mu}+{\cal
E}_{A}^{T}(V;2)$ (31)

Note that $a^{2}g^{A}_{\mu,\mu}$ vanishes as $a\rightarrow 0$. In the case of
the chiral current

$G(B^{5})=a^{2}\sum_{x}Z^{5}B^{5}_{x,\mu}\bar{\psi}_{x}\gamma_{\mu}\gamma_{5}\psi_{x}$
(32)

##  V Integration of the fermionic fields.

Our main result is the following

###### Theorem 2

For $|\lambda|\leq\lambda_{0}M^{2}$, with $\lambda_{0}$ independent on $a,m,M$
and $Z_{\psi}=1$ the correlations of (30) are analytic in $\lambda$; when the
fermion mass is vanishing the anomaly is
$H_{\mu}={\varepsilon_{\mu\nu}p_{\nu}\over 2\pi}+O(ap^{2})$.

In order to integrate the fermionic fields we introduce a decomposition of the
propagator

$g^{\psi}(x)=\sum_{h=-\infty}^{N}g^{(h)}(x)$ (33)

$\widehat{g}^{h}(k)=f^{h}(k)\widehat{g}(k)$ with $f^{h}(k)$ with support in
$\gamma^{h-1}\leq|k|\leq\gamma^{h+1}$. One has to distinguish two regimes, the
ultraviolet high energy scales $h\geq h_{M}$ with $h_{M}\sim\log M$ the mass
scale, and the infrared regime $h\leq h_{M}$. In the first regime, one uses
the non locality of the quartic interation [19],[30],[31]. After the
integration of the fields $\psi^{N},\psi^{N-1},..,\psi^{h}$, $h\geq h_{M}$ one
gets an effective potential with kernels $W^{h}_{l,m}$ with $l$ fields
($l=2n$) similar to (21), which can be written as an expansion in $\lambda$
and in the kernels $W^{k}_{2,0}$, $W^{k}_{4,0}$,$W^{k}_{2,1}$ with $k\geq
h+1$. Assuming that, for $k\geq h+1$ one has
$|W^{k}_{2,0}|_{1}\leq\gamma^{h}\lambda/M^{2}$,
$|W^{k}_{4,0}-v\lambda|_{1}\leq\lambda^{2}/M^{2}$ and
$|W^{k}_{2,0}-1|_{1}\leq\lambda^{2}/M^{2}$ then we get

$|W^{h}_{l,m}|_{1}\leq C^{l+m}(\lambda/M^{2})^{d_{l,m}}\gamma^{h(2-{l\over
2}-m)h}$ (34)

for $d_{l,m}=max(l/2-1,1)$ if $m=0$, and $d_{l,m}=max(l/2-1,0)$ if $m=1$. The
proof of (34) is based on the analogous of formula (23) for Grassmann
expectations

${\cal E}^{T}_{\psi}(\prod_{k=1}^{n}{\widetilde{\psi}}(P_{i}))=\sum_{T\in{\cal
T}}\prod_{i,j\in T}V_{i,j}\int dp_{T}(s)\det G$ (35)

and the use of Gram bounds for get an estimate on $\det G$; in addition one
uses that $|v|_{1}\leq CM^{-2}$, $|g^{h}|_{1}\leq C\gamma^{-h}$,
$|g^{h}|_{\infty}\leq C\gamma^{h}$. We proceed by induction to prove the
assumption. One needs to show that there is an improvement in the bounds due
to the non locality of the boson propagator. The kernel of the 2-point
function $W_{2,0}^{h}(x,y)$ which can be written as sum over $n$ of truncated
expectations and, if ${\cal E}^{T}_{h,N}$ is the truncated expectation with
respect to $P(d\psi^{[h,N]})$

$\lambda{\partial\over\partial\phi^{+}_{x}}{a^{4}\over(n-1)!}\sum_{x_{1},x_{2}}v_{\mu,\varepsilon_{1},\varepsilon_{2}}{\cal
E}^{T}_{h,N}({\partial\over\partial\phi^{-}_{y}}O^{\varepsilon_{1}}_{\mu,x_{1}}O^{\varepsilon_{2}}_{\mu,x_{2}};V;...)$
(36)

By using the property, if ${\widetilde{\psi}}(P)=\prod_{f\in P}\psi_{x_{f}}$

$\displaystyle{\cal E}^{T}_{h,N}({\widetilde{\psi}}(P_{1}\cap
P_{2})...{\widetilde{\psi}}(P_{n}))={\cal
E}^{T}_{h,N}({\widetilde{\psi}}(P_{1}){\widetilde{\psi}}(P_{2})...{\widetilde{\psi}}(P_{n}))+$
$\displaystyle\sum_{K_{1},K_{2}\atop K_{1}\cup K_{2}=3,..,n}{\cal
E}^{T}_{h,N}({\widetilde{\psi}}(P_{1})\prod_{j\in
K_{1}}{\widetilde{\psi}}(P_{i})){\cal
E}^{T}_{h,N}({\widetilde{\psi}}(P_{2})\prod_{j\in
K_{2}}{\widetilde{\psi}}(P_{i}))$

we get, omitting the $\varepsilon,\mu$ dependence, $W_{2,0}^{h}(x,y)=$

$\displaystyle\lambda
a^{4}\sum_{z_{1},z_{2}}v(y,z_{1})g^{[h,N]}(y+a_{\mu},z_{2})W^{h}_{0,2}(z_{2},x)W_{1,0}(z_{1})$
$\displaystyle\lambda a^{-1}(e^{-{1\over
2}e^{2}a^{2}g^{A}_{\mu,\mu}(0))}-1)a^{2}\sum_{z}g^{[h,N]}(x,z)W_{2,0}^{h}(z,y)$
$\displaystyle+\lambda
a^{4}\sum_{z_{1},z_{2}}v(y,z_{2})g^{[h,N]}(y+a_{\mu},z_{1})W^{h}_{2,1}(z_{2};z_{1},x)$
(37)

The second term is bounded by

$C\lambda\gamma^{h}\gamma^{-h}a\log a\leq\lambda/M^{2}\gamma^{h}/2$ (38)

for $a$ small enough. The first term contains $\widehat{W}_{1,0}(0)=0$.
Regarding the last term we get a bound

$\sup_{z_{1},z_{2}}|a^{2}\sum_{y}v(y,z_{2})g^{[h,N]}(y+a_{\mu},z_{1})|a^{2}\sum_{z_{2},z_{1}}|W^{h}_{2,1}(z_{2};z_{1},0)|$
(39)

$W^{h}_{2,0}$

$=$

$W^{h}_{0,1}$

$W^{h}_{2,0}$

$W^{h}_{2,0}$

$W^{h}_{2,1}$

$+$

$+$

Figure 1:  Graphical representation of (V)

By using the inductive hypothesis
$a^{2}\sum_{z_{2},z_{1}}|W^{h}_{2,1}(z_{2};z_{1},0)|\leq C$ we get for (39)
the bound

$\displaystyle\lambda C_{1}|a^{2}\sum_{y}|vg^{[h,N]}|\leq\lambda
C_{1}[a^{2}\sum_{y}|v|^{3}]^{1\over 3}\times$ (40)
$\displaystyle[a^{2}\sum_{y}(g^{[h,N]})^{3\over 2}]^{2\over 3}\leq\lambda
M^{-2}C_{2}\gamma^{h}\gamma^{-{4\over 3}(h-M)}\leq\lambda M^{-2}\gamma^{h}/2$

for $h\geq h_{M}$, for $h_{M}=C\log M$ and $C$ large enough. Note that the
above estimates uses crucially that $\xi=0$; for $\xi=1$
$[a^{2}\sum_{y}|v|^{3}]^{1\over 3}$ would be non bounded uniformly in $N$.

A similar computation can be repeated for $W_{2,1}$; in particular for the
quartic term one uses that the boubble graph is finite $A=\int dk{\rm
Tr}g(k)\gamma_{\mu}g(k)\gamma_{\nu}$ so that $|W_{1,2}|_{1}\leq
C\lambda/M^{2}(\gamma^{h-M}+A|W_{1,2}|_{1})$. The above estimates work for
$h\geq h_{M}$ and it says that the theory is superenormalizable up to that
scale.

In the infrared regime $h_{m}\leq h\leq h_{M}$, where $h_{m}=\log_{\gamma}m$
is the fermion mass scale, the multiscale integration procedure is the same as
in the Thirring model with a finite cut-off [23]. The theory is renormalizable
in this regime and there is wave function renormalization at each scale
$Z_{h}\sim\gamma^{-\eta h}$, $\eta=O(\lambda^{2})>0$ and an effective coupling
with asymptotically vanishing function. The expansions converge therefore
uniformly in $a,L,M$ and the limit $a\rightarrow 0,L\rightarrow\infty$ can be
taken.

$+$

$=$

$p_{\mu}$

$(1\pm\tau)$

Figure 2:  Graphical representation of (45)

##  VI Anomaly non-renormalization

The average of the chiral current
$\Gamma^{5}_{\mu,\nu}={\partial^{2}W\over\partial B_{\mu}\partial
J_{\nu}}|_{0}$ for $m=0$ is expressed by a series in $\lambda$. It is
convenient to introduce a continuum relativistic model
$e^{{\widetilde{W}}(J,B,\phi)}=$

$\int
P_{{\widetilde{Z}}}(d\psi)e^{-V+{\widetilde{Z}}^{+}(J,j)+{\widetilde{Z}}^{-}(B,j^{5})+(\psi,\phi)}$
(41)

where $P_{{\widetilde{Z}}}(d\psi)$ has propagator
${1\over{\widetilde{Z}}}{\chi(k)\over\gamma_{\mu}k_{\mu}}$, with $\chi$ a
momentum cut-off selecting momenta $\leq\gamma^{{\widetilde{N}}}$, and

$V={\widetilde{Z}}^{2}{\widetilde{\lambda}}\int dxdyv(x,y)j_{\mu,x}j_{\mu,y}$
(42)

with $v$ exponentially decaying with rate rate $M^{-1}$ with quartic coupling
${\widetilde{\lambda}}$; finally $j^{+}_{\mu,x}\equiv
j_{\mu,x}=\bar{\psi}_{x}\gamma_{\mu}\psi_{x}$ and
$j^{-}=\bar{\psi}_{x}\gamma_{\mu}\gamma_{5}\psi_{x}$.

The infrared scales $h\leq h_{M}$ of the two models differs by irrelevant
terms and one can choose ${\widetilde{\lambda}}$ and
${\widetilde{Z}},{\widetilde{Z}}^{-},{\widetilde{Z}}^{+}$ as function of
$\lambda$ so that the corresponding running couplings flow to the same fixed
point for $h\rightarrow-\infty$. As a result, defining

${\widetilde{\Gamma}}^{5}_{\mu,\nu}={\partial^{2}{\widetilde{W}}\over\partial
B_{\mu}\partial J_{\nu}}|_{0}$ (43)

we get

$\widehat{\Gamma}^{5}_{\mu,\nu}(p)=Z_{5}{\widetilde{\Gamma}}^{5}_{\mu,\nu}(p)+R_{\mu,\nu}(p)$
(44)

where $R_{\mu,\nu}(p)$ is a continuous function at $p=0$, while
${\widetilde{\Gamma}}^{5}_{\mu,\nu}(p)$ is not; this provide a relation
between the lattice and the continuum model.

The model (41) has two global symmetries, that is $\psi\rightarrow
e^{i\alpha}\psi$ and $\psi\rightarrow e^{i\alpha\gamma^{5}}\psi$, but the WI
acquires extra terms associated with the momentum regularization [30]. In
particular, if $\tau={\widetilde{\lambda}}\widehat{v}(0)/4\pi$, in the limit
of removed cut-off ${\widetilde{N}}\rightarrow\infty$

$(1\mp\tau)p_{\mu}{\widetilde{\Gamma}}^{\pm}_{\mu}(k,p)={{\widetilde{Z}}^{\pm}\over{\widetilde{Z}}}\gamma^{\pm}({\widetilde{S}}(k)-{\widetilde{S}}(k+p))$
(45)

where ${\widetilde{\Gamma}}^{\pm}_{\mu}$ is the vertex function of are the
vertex correlations of (41) of the current $(+)$ and chiral current $(-)$ and
$\gamma^{+}=I$, $\gamma^{-}=\gamma_{5}$. In the same way the WI for the
current is

$p_{\mu}{\widetilde{\Gamma}}^{5}_{\mu,\nu}={{\widetilde{Z}}^{+}{\widetilde{Z}}^{-}\over
4\pi{\widetilde{Z}}^{2}}{\varepsilon_{\mu\nu}p_{\mu}\over(1+\tau)}\quad
p_{\nu}{\widetilde{\Gamma}}^{5}_{\mu,\nu}={{\widetilde{Z}}^{+}{\widetilde{Z}}^{-}\over
4\pi{\widetilde{Z}}^{2}}{\varepsilon_{\nu\mu}p_{\nu}\over(1-\tau)}$ (46)

By comparing (45) with the Ward Identity (11), and using that the vertex and
the 2-point correlations of lattice and continuum model coincide up to
subleading term in the momentum, we get a relation between the parameters
$\tau,{\widetilde{Z}}^{+},{\widetilde{Z}}$

${{\widetilde{Z}}^{+}\over{\widetilde{Z}}(1-\tau)}=1$ (47)

Moreover the condition on $Z^{5}$ (18) and (45) imply

${{\widetilde{Z}}^{+}\over{\widetilde{Z}}(1-\tau)}=Z_{5}{{\widetilde{Z}}^{-}\over{\widetilde{Z}}(1+\tau)}=1$
(48)

from which $Z_{5}=(1+\tau){{\widetilde{Z}}\over{\widetilde{Z}}^{-}}$. By the
Ward Identity (10) we get

$p_{\nu}\widehat{\Gamma}^{5}_{\mu,\nu}(p)={{\widetilde{Z}}^{+}{\widetilde{Z}}^{-}\over
2\pi{\widetilde{Z}}^{2}}{\varepsilon_{\nu\mu}p_{\nu}\over(1-\tau)}+p_{\nu}R_{\mu,\nu}(p)=0$
(49)

so that

$R_{\mu,\nu}(0)=-{{\widetilde{Z}}^{+}{\widetilde{Z}}^{-}\over 2\pi
Z^{2}}{\varepsilon_{\mu\nu}\over(1-\tau)}=-(1+\tau)\varepsilon_{\nu\mu}/Z_{5}$
(50)

Finally

$\displaystyle
p_{\mu}\widehat{\Gamma}^{5}_{\mu,\nu}(p)=Z_{5}p_{\mu}[{\widetilde{\Gamma}}^{5}_{\mu,\nu}(p)+R_{\mu,\nu}(p)]=$
$\displaystyle[(1-\tau)\varepsilon_{\mu,\nu}-(1+\tau)\varepsilon_{\nu,\mu}]p_{\mu}/4\pi=1/2\pi\varepsilon_{\mu,\nu}p_{\nu}$
(51)

that is all the dependence of the coupling disappears.

##  VII Conclusions

We have analyzed a lattice version of the Sommerfield model. Both the
reduction of the degree of ultraviolet divergence, manifesting in the
finiteness of the field renormalization, and the Adler-Bardeen theorem hold at
a non-perturbative level, in contrast with exact solutions in the continuum.
Non perturbative violation of perturbative results are therefore excluded.
This provides support to the possibility of a rigorous lattice formulation of
the electroweak sector of the Standard Model with step exponentially small in
the inverse coupling, which requires an analogous reduction of degree of
divergence. New problems include the fact that a multiscale analysis is
necessary also for the boson sector, and the fact that the symmetry is chiral
and anomaly cancellation is required; Adler-Bardeen theorem on a lattice is
exact for non chiral theories [33] but has subdominant corrections for chiral
ones [34].

Aknowledments This material is based upon work supported by a grant from the
Institute for Advanced Study School of Mathematics. I thank also GNFM and MUR
through PRIN MAQUMA.

## References

  * [1] S. Weinberg A Model of Leptons Phys. Rev. Lett. 19, 1264 (1967)
  * [2] G. ’t Hooft, Renormalizable Lagrangians for massive Yang-Mills fields Nuclear Physics 167-188 (1971)
  * [3] Bouchiat, C., Iliopoulos, J., Meyer, P. An anomaly-free version of Weinberg’s model : Phys. Lett. B 38, 519 (1972)
  * [4] Adler S.L., Bardeen W.A: Absence of Higher-Order Corrections in the Anomalous Axial-Vector Divergence Equation. Phys.Rev. 182 15171536 (1969)
  * [5] L.B. Okun Leptons and Quarks Elsevier 1985
  * [6] M. Luscher, Abelian chiral gauge theories on the lattice with exact gauge invariance Nucl. Phys. B 549, 295 (1999)
  * [7] D. M. Grabowska, D. B. Kaplan Nonperturbative Regulator for Chiral Gauge Theories? Phys. Rev. Lett.. 116, 211602 (2016)
  * [8] M. Golterman Lattice Chiral Gauge Theories Nucl.Phys.Proc.Suppl. 94 , 189-203 (2001)
  * [9] Fujikawa, K., Comment on Chiral and Conformal Anomalies, Phys. Rev. Lett. 44, pp. 1733-1736 (1980).
  * [10] S. L. Adler Fifty Years of Yang-Mills Theory, G. ’t Hooft editor, World Scientific
  * [11] Sommerfield, CM ”On the definition of currents and the action principle in field theories of one spatial dimension”. Annals of Physics. 26 (1): 1–43. (1964)
  * [12] H. Georgi J. M. Rawls. Anomalies of the Axial-Vector Current in Two Dimensions Phys. Rev. D 3, 874 (1971)
  * [13] H. Georgi, Y. Kats Unparticle Example in 2D Phys. Rev. Lett. 101, 131603 (2008)
  * [14] Georgi, H., Warner, B.Generalizations of the Sommerfield and Schwinger models Journal of High Energy Physics, 01 47 (2020)
  * [15] Brown L.S.: Gauge Invariance and Mass in a Two-Dimensional Model. N.Cim. 29 617-643 (1963).
  * [16] Hagen C.R.: Current Definition and Mass Renormalization in a Model Field Theory. N.Cim.A 51 1033-1052 (1967).
  * [17] Das A., Mathur V.S.: Path-integral solubility of two-dimensional models. Phys.Rev.D 33 489495 (1985).
  * [18] Balaban, T; Imbrie, J; Jaffe, A; Brydges, D The Mass Gap for Higgs Models on a Unit Lattice Annals of physics 158, 281-319 (1984)
  * [19] A. Lesniewski Effective action for the Yukawa2 quantum field theory Comm. Math. Phys. 108(3): 437-467 (1987)
  * [20] C. Kopper, J. Magnen, V. Rivasseau Mass generation in the large N Gross-Neveu-model Comm. Math. Phys. 169(1): 121-180 (1995).
  * [21] M. Salmhofer and E. Seiler Proof of Chiral Symmetry Breaking in Strongly Coupled Lattice Gauge Theory 139(2): 395-432 (1991)
  * [22] T Hurd. J. Dimock A renormalization group analysis of infrared QED J. Math. Phys. 33, 814 (1992);
  * [23] G.Benfatto, P.Falco, V.Mastropietro. Functional Integral Construction of the Thirring model: axioms verification and massless limit. Comm.Math. Phys. 273, 1, 67–118 (2007)
  * [24] A. Giuliani, V. Mastropietro S. Rychkov Gentle introduction to rigorous Renormalization Group: a worked fermionic example. Journal of High Energy Physics 26 (2021)
  * [25] Antti Kupiainen, Rémi Rhodes Vincent Vargas The DOZZ formula from the path integral Journal of High Energy Physics 94 (2018)
  * [26] Hamer C, Kogut J, Crewther D, Mazzolini M. The massive Schwinger model on a lattice: background field, chiral symmetry and the string tension. Nucl. Phys. B 208, 413-438 (1982)
  * [27] Byrnes TMR, Sriganesh P, Bursill RJ, Hamer CJ. 2002Density matrix renormalization group approach to the massive Schwinger model. Phys. Rev. D 66, 013002
  * [28] P. Hauke, D. Marcos,1 . Dalmonte, and P. Zoller Quantum Simulation of a Lattice Schwinger Model in a Chain of Trapped Ions Phys. Rev. X 3, 041018 (2013)
  * [29] D. Brydges . A short course on Cluster Expansions, Les Houches 1984, K. Osterwalder, R. Stora eds., North Holland Press, 1986.r
  * [30] V. Mastropietro: Nonperturbative Adler-Bardeen theorem. J.Math.Phys. 48 022302 (2007).
  * [31] P. Falco Vector and axial anomaly in the Thirring–Wess model J. Math. Phys. 51, 082306 (2010)
  * [32] V. Mastropietro. Emergent Adler-Bardeen theorem. JHEP 95 (2020)
  * [33] A.Giuliani, V.Mastropietro, M.Porta. Anomaly non-renormalization in interacting Weyl semimetals. Comm. Math. Phys. 384 (2), 997-1060 (2021)
  * [34] V. Mastropietro Anomaly cancellation condition in an effective nonperturbative electroweak theory Phys. Rev. D 103, 013009 (2021)