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.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction and Main Results
1.1 Chiral Gauge Theory
The perturbative consistence (renormalizability) of the Standard Model relies on the vanishing of the anomalies, achieved under certain algebraic conditions [1] severely constraining the elementary particles charges and providing a partial explanation of the charge quantization. In order to go beyond a purely perturbative framework in terms of diverging series [2], one needs a lattice formulation with functional integrals with cut-off much higher than the experiments scale; due to triviality [3, 4], the cut-off cannot be completely removed, at least in the electroweak sector, and hence, the theory can be seen as an effective one.
One expects a relation between the perturbative renormalizability properties and the size of the cut-off. The electroweak theory is renormalizable [5, 6] so that a construction up to exponentially large cut-off could be in principle possible, and such cut-off is much higher than the scales of experiments. However, this requires as a crucial prerequisite that the anomalies cancel, at least to a certain extent. This rises the natural question: does the anomaly cancel at a non-perturbative level with finite lattice, under the same condition as in the continuum?
In the continuum, the cancellation is based on compensations at every order [7] based on dimensional regularizations and symmetries, but finite lattice cut-off produce corrections and the question is if they cancel or not. Jacobian arguments are used to support vanishing of higher-order contributions to anomalies but are essentially one loop results, as shown in [8]. Topological arguments explain the anomaly cancellation on a lattice [9] with classical gauge fields, but in the quantum case they work only at lowest order (one loop). The cancellation would be obtained if a non-perturbative regulator for lattice chiral gauge theories could be found, but this is a long-standing unsolved problem and only order by order results are known [10, 11].
We consider a lattice chiral gauge theory, given by 2N massless fermions in four dimensions, labelled by an index \(i=1,\ldots ,2N\); we also define the indices \(i_1=1,\ldots ,N\) and \(i_2=N+1,\ldots ,2N\). If the gamma matrices are
and \(\sigma _\mu ^L=(\sigma _0,i \sigma ) \), \(\sigma _\mu ^R=(\sigma _0,-i \sigma ) \),
the formal continuum action is given by the following expression:
with \(\mu =(0,1,2,3)\) and \(F_{\mu \nu }=\partial _\mu A_\nu -\partial _\nu A_\mu \). Note that the R fermions of kind \(i_1\) and the L fermions of kind \(i_2\) decouple and are fictitious, non-interacting degrees of freedom, which are convenient to introduce in view of the lattice regularization, see e.g. [12,13,14]. The total current coupled to \(A_\mu \) is
and the axial and vector part of the current is
with \({\widetilde{\varepsilon }}_{i_1}=-{\widetilde{\varepsilon }}_{i_2}=1\), \(j_{\mu ,i,x}=\bar{\psi }_{i,x}\gamma _\mu \psi _{i,x}\), \( j^5_{\mu ,i,x}=\bar{\psi }_{i,x}\gamma _5\gamma _\mu \psi _{i,x}\) and \(\psi _{i,x}=(\psi ^-_{i,L,x},\psi ^-_{i,R,x})\), \(\bar{\psi }_{i,x}=(\psi ^+_{i,L,x},\psi ^+_{i,R,x})\gamma _0\). Note the chiral nature of the theory, as in the current the fermion with different chiralities has different charges. An example of chiral theory is obtained setting \(Q_{i_2}=0\); in such a case one is describing N fermions with the same chirality interacting with a gauge field. A physically more important example is given by the U(1) sector of the Standard Model with no Higgs and massless fermions; in this case \(N=4,\) \(i_1=(\nu _1,e_1,u_1,d_1)\) are the left-handed components and \(i_2=(\nu _2,e_2,u_2,d_2)\) the right-handed of the leptons and quarks. A formal application of Noether theorem with classical fermions and bosons says that the invariance under phase and chiral symmetry, implying the current conservation \(\partial _\mu j^T_\mu =0\). If the fermions are quantum (and the bosons classical), the conservation of current is reflected in Ward identities, and it turns out that anomalies generically break the conservation of \(j^T_{\mu ,i,x}\) unless
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
where \(A_{\mu ,x}: \Lambda \rightarrow \mathbb {R}\), \(\Lambda =[0,L]^4\cap a \mathbb {Z}^4\), \(L=K a\), \(K\in \mathbb {N}\) \(e_\mu \), \(\mu =0,1,2,3\) an orthonormal basis, \(A_{\mu ,x}=A_{\mu , x+L e_\mu }\) (periodic boundary conditions) and the bosonic integration is
with
is the action of a non-compact lattice U(1) gauge field with a gauge fixing and a mass term, \(F_{\mu ,\nu }=d_\nu A_\mu -d_\mu A_\nu \) and \(d_\nu A_\mu =a^{-1}(A_{\mu ,x+e_\nu a}-A_{\mu ,x})\), \({\mathcal N}_A\) is the normalization. The bosonic simple expectation
is expressed by the Wick rule with covariance
with \(\sigma _\mu (k)=(e^{i k_\mu a}-1)a^{-1}\), \(k=2\pi n/L\), \(n\in \mathbb {N}^4\) and \(k\in [-\pi /a,\pi /a)^4\). The bosonic truncated expectation
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
We define \(\psi ^\pm _{i,s,x}={1\over L^4}\sum _k e^{\pm i k x}\widehat{\psi }^\pm _{i,s,k}\), with \(\widehat{\psi }^\pm _{i,s,k}\) another set of Grassmann variable, \(k=2\pi /L(n+1/2)\), \(n\in \mathbb {N}^4\) and \(k\in [-\pi /a,\pi /a)^4\). The fermionic Gaussian measure is defined as, \(i=1,..,2N\), \(s=L,R\)
where \({\mathcal N}_\psi \) a normalization and, if \(\psi ^\pm _{i,x}=(\psi ^\pm _{i,L,x},\psi ^\pm _{i,R,x})\)
We can write therefore
The fermionic simple expectation
is expressed by the anticommutative Wick rule with covariance
with
The interaction is
with
with, if \(i_1=1,N\) and \(i_2=N+1,...,2N\)
and \(:e^{\pm i a \lambda Q_i A_\mu (x) }:=e^{\pm i \lambda Q_i a A_\mu (x) } e^{{1\over 2} (\lambda Q_i)^2 a^2 g^A_{\mu ,\mu } (0,0) }\).
The mass counterterm is
Finally, the source term is
with \({\widetilde{\varepsilon }}_{i_1}=-{\widetilde{\varepsilon }}_{i_2}=1\) and \(\varepsilon _{L}=-\varepsilon _{R}=1\). \(\nu _i\) and \(Z_{i,s}^5 \) are parameters to be fixed by the renormlization conditions, see below.
Remark
The term proportional to r in \(S_F\) (16) is called Wilson term. If \(r=0\), the fermionic propagator \(\widehat{g}_{i,k}\) has, in the \(L\rightarrow \infty \) limit, several poles; this has the effect that the low energy behaviour of the lattice theory would not correspond to the continuum target theory (3); the presence of the Wilson term \(r\not =0\) has the effect that only the physical pole \(k=0\) is present but the chiral symmetry is broken [15].
1.3 Physical observables
The fermionic 2-point function is
and the Fourier transform is
The vertex functions are
The Fourier transform is
and similarly is defined \(\widehat{\Gamma }^{5,\Lambda }_{\mu ,i' s}(k,p) \). The three current vector VVV and axial AVV correlations are:
and
1.4 Ward Identities
The correlations are connected by relations known as Ward identities. They can be obtained by performing the change of variables
with \(\alpha _x\) is a function on \(a \mathbb {Z}^4\), with the periodicity of \(\Lambda \). Let \(Q(\psi ^{+}, \psi ^{-})\) be a monomial in the Grassmann variables and \(Q_{\alpha }(\psi ^{+}, \psi ^{-})\) be the monomial obtained performing the replacement (31) in \(Q(\psi ^{+}, \psi ^{-})\). It holds that
as both the left-hand side and the right-hand side of (32) are zero unless the same Grassmann field appears once in the monomial; hence, the fields \(\psi ^+_{i,s,x}\), \(\psi ^-_{i,s,x}\) come in pairs and the \(\alpha \) dependence cancels. By linearity of the Grassmann integration, the property (32) implies the following identity, valid for any function f on the finite Grassmann algebra:
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
where \(J+d_\mu \alpha \) is a shorthand for \(J_{\mu ,x}+d_\mu \alpha _{x}\) and \(e^{i Q\alpha }\phi \) is a shorthand for \(e^{\pm i Q_i\alpha _{x}} \phi _{i,s, x}^{\pm }\); by differentiating, we get the Ward identities (WI)
Remark
The above Ward identities represent the conservation of the vector part of the current coupled to the gauge field \(A_\mu \); in particular, the first is the lattice counterpart of \(\partial _\mu <j_{\mu }^{T,V};j_{\nu _1}^{T,V};\ldots j_{\nu _n}^{T,V} >_T=0\), see (5).
1.5 Main result
Our main result is the following, denoting by \(\lim _{L\rightarrow \infty } \widehat{S}^\Lambda _{i,s,s'}(k)\) and similarly the other correlations.
Theorem 1.1
Let us fix \(r=1\) and \(Ma\ge 1\). There exists \(\lambda _0,C\) independent on L, a, M such that, for \(|\lambda |\le \lambda _0 (Ma)\), it is possible to find \(\nu _i\), \(Z_{i,s}^5\) continuous functions in \(\lambda \) such that
-
(1)
The limits \(L\rightarrow \infty \) of \(\widehat{S}^\Lambda _{i,s,s'}(k)\), \(\widehat{\Gamma }^\Lambda _{\mu ,i',s}(k,p) \), \(\widehat{\Gamma }^{5,\Lambda }_{\mu ,i',s}(k,p) \) \(\Pi ^{\Lambda }_{\mu ,\nu , \rho }(p_1,p_2)\),\(\Pi ^{5,\Lambda }_{\mu ,\nu , \rho }(p_1,p_2)\) exist and \(\lim _{k\rightarrow 0} \widehat{S}^\Lambda _{i,s}(k)=\infty \) and \(\lim _{k,p\rightarrow 0} {\widehat{\Gamma }^{5,\Lambda }_{\mu ,i',s}(k,p)\over \widehat{\Gamma }^\Lambda _{\mu ,i',s}(k,p)}=\varepsilon _s I\) where \(\varepsilon _L=-\varepsilon _R=1\).
-
(2)
The AVV correlation verifies
$$\begin{aligned}{} & {} \sum _\mu \sigma _\mu (p_1+p_2) \widehat{\Pi }^{5}_{\mu ,\rho ,\sigma }(p_1,p_2)\nonumber \\{} & {} \quad = \sum _{\mu ,\nu } \varepsilon _{\mu ,\nu ,\rho ,\sigma } {1\over 2 \pi ^2}\sigma _\mu ( p^1) \sigma _\nu (p^2) \left[ \sum _{i_1} Q_{i_1}^3-\sum _{i_2} Q_{i_2}^3\right] +r_{\rho ,\sigma }(p_1,p_2)\nonumber \\ \end{aligned}$$(36)with \(|r(p_1,p_2)|\le C a^\vartheta \bar{p}^{2+\vartheta }\), \(\bar{p}=\max (|p_1|, |p_2|)\) and \(\vartheta =1/2\).
Remarks
-
(1)
The correlations are written in the form of expansions which are convergent in the limit of infinite volume, provided that the lattice cut-off is smaller than the boson mass.
-
(2)
The counterterms \(\nu _i\) are chosen so that the fermions remain massless in the presence of interactions; the parameters \(Z_{i,s}^5\) are fixed so that the charge associated with the vector and axial current are the same, a condition present also at a perturbative level [7].
-
(3)
Under the condition \(\left[ \sum _{i_1} Q_{i_1}^3-\sum _{i_2} Q_{i_2}^3\right] =0\) we have \(\sum _\mu \sigma _\mu (p) \widehat{\Pi }^\Lambda _{\mu ,\nu ,\sigma }(p_1,p_{2})=0\) and \(\sum _\mu \sigma _\mu (p) \widehat{\Pi }^{5,\Lambda }_{\mu ,\rho ,\sigma }(p_1,p_2)=O(a^\vartheta \bar{p}^{2+\vartheta })\) expressing the conservation of the chiral current in the sense of correlations and up to subdominant terms for momenta far from the cut-off. The vanishing of the anomaly, obtained up to now only at a purely perturbative level, is proved with a finite lattice cut-off, even if the cut-off breaks important symmetries [15] on which the perturbative cancellation was based, like the Lorentz or the chiral one, and excluding non-perturbative effects. The anomaly cancellation condition is the same as in the continuum case. The lattice regularization plays an essential role; with momentum one a much weaker result holds [16].
-
(4)
Anomalies are strongly connected with transport properties in condensed matter [17,18,19], and we use indeed techniques recently developed for the proof of universality properties in metals to the anomaly cancellation on a lattice [20,21,22,23,24,25,26,27,28,29]. Such methods have their roots in the Gallavotti tree expansion [30], the Battle–Brydges–Federbush formula [31] and the Gawedzki–Kupiainen–Lesniewski formula [32, 33] (see e.g. [34] for an introduction).
1.6 Future Perspectives
We have constructed the theory assuming that \(1/a\le (\lambda _0/|\lambda |) M\), that is the cut-off is smaller than the boson mass and we have established (36) for generic values of the coupling. In this regime after the integration of the \(A_\mu \) the theory have scaling dimension \(D=4+n-3 n^\psi /2\) if n is the order and \(n_\psi \) the number of fields. This requires that the “effective coupling” \(\lambda ^2/M^2\) times the energy cut-off must be not too large so that the expansions are convergent. In order to reach higher cut-off one notes that the boson propagator (11) is composed by two terms: one which behaves as \(O(1/k^2)\) for \(k^2>>M^2\) and the other which is O(1) for \(k^2>>M^2\). If the second term does not contribute the scaling dimension improves and it corresponds to a renormalizable theory \(D=4-3 n^\psi /2-n^A\), so in principle one can consider cut-offs higher than M and up to an exponentially large values \(|\lambda ^2 \log a |\le \varepsilon _0\). In order to have that the second term does not contribute full gauge invariance (broken in our case by the mass and gauge fixing term) is not necessarily required but is sufficient the gauge invariance in the external fields, expressed in the form of Ward identities. It is indeed known that renormalizability is preserved in QED, at the perturbative level, even if a mass is added to photon, see e.g. [35, 36]; if one restricts to gauge-invariant observables, the contribution of the not-decaying term of the propagator is vanishing as a consequence of the current conservation. To get exponentially high cut-off in \(d=4\), QED at a non-perturbative level is technically demanding, as it would require a simultaneous decomposition in the bosons and fermions, but the analogous statement can be rigorously proven in \(d=2\) vector models [37].
In the absence of the Wilson term \(r=0\), we get the conservation of the chiral current in the form of a WI given by the first of (35), if \(\widehat{\Pi }_{\mu ,\nu _1,\ldots ,\nu _n}\) is obtained replacing \(J_\mu \) in \(G^{\pm }_{\mu ,j,s}\) with \(b_{i,s} {\widetilde{J}}_{\mu ,x}\). As a consequence, the averages of invariant observables are \(\xi \) independent. This follows from \(\partial _\xi \int P(dA) \int [\prod _{i,s,x} d\psi ^+_{i,s,x} d\psi ^-_{i,s,x}] O=0\), with \(O(A,\psi )\) invariant; indeed,
from which we get, using that \(\widehat{A}_{\mu ,p}=\widehat{g}^A_{\mu ,\rho } {\partial \over \partial A_{\rho ,-p}}\)
By noting that \(\partial (\widehat{g}^A)^{-1}=- (\widehat{g}^A)^{-1} \partial _\xi \widehat{g}^A( \widehat{g}^A)^{-1}\) and \(\partial _\xi \widehat{g}^A\) is proportional to \(\bar{\sigma }_\mu \sigma _\nu \), by using \(\partial _\alpha \int P(dA) \int [\prod _{i,s,x} d\psi ^+_{i,s,x} d\psi ^-_{i,s,x}] O(A+d \alpha ,\psi )|_0=0\); then, \(\partial _\xi W\) is vanishing. Therefore, if \(r=0\) in the average of invariant observables one can set \(\xi =0\) and the theory is perturbatively renormalizable. One expects to be able to reach exponentially high cut-off.
The Wilson term \(r\not =0\), physically necessary to avoid fermion doubling [15], breaks the WI and the conservation of chiral current for generic values of the charges, according to (36). Therefore generically the theory is non-renormalizable at scales greater than M and one cannot expect in general to be able to reach exponentially high cut-offs. However, choosing the charges so that \([\sum _{i_1} Q_{i_1}^3-\sum _{i_2} Q_{i_2}^3]=0\) the contribution of the non-decaying term vanishes up to subdominant terms, making possible in principle to reach exponentially high cut-offs. The anomaly cancellation for \(1/a\le M\) is therefore a prerequisite for reaching higher cut-offs. In the case of the U(1) sector of the Standard Model, one has also to introduce an Higgs boson to generate the fermion mass; one can distinguish a region higher than the boson mass generated by the Higgs, where the second term of the boson propagator does not contribute due to the anomaly cancellation and the WI; and a lower one, when the infinite volume limit can be taken using the infrared freedom of QED and the massive nature of weak forces. Further challenging problems arise considering the anomaly associated to the SU(2) sector.
2 Proof of Theorem 1.1
In the following, we denote by C or by \(C_1,C_2..\) generic \(\lambda ,L,a\)-independent constants. We integrate the bosonic variables \(A_\mu \) in (7), obtaining
where, by (12)
which can be rewritten as, if \(\underline{x}=x_1,..,x_n\), \(\underline{i}=i_1,...,i_n\), \(\underline{\mu }=\mu _1,...,\mu _n\),\(\underline{m}=m_1,..,m_n\)
with
and \( ||H_{n,m}||=L^{-4} a^{4 n}\sup _{\begin{array}{c} \underline{\varepsilon },\underline{i},\underline{\mu },\underline{s},\underline{m} \\ \sum _j m_j=m \end{array}} \sum _{x_1,...,x_n} |H_{n,m}|\).
Lemma 2.1
The kernels in (40) the following bound, for \(n\ge 2\), \(m\le 3\) and uniformly in L
Proof of Lemma 2.1
We write the truncated expectations in (41) by the Battle–Brydges–Federbush formula, see e.g. Theorem 3.1 in [31] (for completeness a sketch of the proof is in Appendix 1), \(n\ge 2\)
where \(X=((x_1,\varepsilon _1,i_1,\mu _1,s_1,m_1);..; (x_n,\varepsilon _n,i_n,\mu _n,s_n,m_n))\), \(\textbf{T}_n\) is the set of connected tree graphs on \(\{1,2,\ldots ,n\}\), the product \(\prod _{\{i,j\}\in T}\) runs over the edges of the tree graph T,
\(V(X;\underline{t})\) 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
and \(dp_T(\underline{t})\) is a probability measure, whose explicit form is recalled in the Appendix 1. We use the bounds
so that
Moreover, V(Y) is stable, that is
hence \( V(X;\underline{t})\ge 0\) and \(e^{-V(X;\underline{t})}\le 1\) so that \(\int dp_T(\underline{t}) e^{-V(X;\underline{t})}<1\) therefore
and finally using that \(\sum _{T\in \varvec{T}_n} 1\le C_4^n n!\) by Cayley’ formula [40] we finally get
\(\square \)
After the integration of \(A_\mu \), the generating function can be written as a Grassmann integral:
with
The fermionic propagator is massless, that is it has a power law decay at large distances, and this requires a multiscale analysis based on Wilson Renormalization Group.
We introduce parameters \(\gamma >1\) and \(N\in \mathbb {N}\) such thatFootnote 1\(\gamma ^N\equiv \pi /(16 a)\); moreover, we introduce \({\widetilde{f}}(t); \mathbb {R}^+\rightarrow \mathbb {R}\) a \(C^\infty \) non-decreasing function \(=0\) for \(0\le t\le \gamma ^{N-1}\) and \(=1\) for \(t\ge \gamma ^N\); we define also \(\chi _{N}(t)=1-{\widetilde{f}}(t)\) which is therefore non-vanishing for \(t\le \gamma ^N\). We introduce the propagator
with \(|k-k'|_T\) the distance on the 4-dimensional torus \([-\pi /a,\pi /a)^4\). Therefore, for any \(K\in \mathbb {N}\) we have
where \(|x-y|_{{\widetilde{T}}}\) is the distance on the \([-L,L)^4\) torus. The above bound is derived by (discrete) integration by parts, see e.g. §3.3 of [34], using thatFootnote 2 in the support of \({\widetilde{f}}(|k|_T)\) one has \(\sum _\mu (1-\cos k_\mu a)^2/a^2\ge C/a^2\) and the volume of the support of \({\widetilde{f}}\) is \(\le C /a^4\).
We can write therefore, using the addition property of Gaussian Grassmann integrals, see e.g. §2.4 of [34]
where
and \({\mathcal E}^T_{N+1}\) is the truncated expectation with respect to the integration \(P(d\psi ^{(N+1)} )\) (Fig. 1).
Using the linearity of the truncated expectations, one gets, if \(\underline{\gamma }=\underline{\varepsilon },\underline{s},\underline{i},\underline{\mu },\underline{\beta }\)
with \(\varepsilon =\pm \), and \(J^{\beta }_{x_j}\) is \(J_{x_j}\) or \(J^5_{x_j}\) for \(\beta =(0,1)\). Note that the \(W^{(N)}\) are a series in the kernels \(H_{n,m}\). In the \(l_b=0\) case (the presence of \(\phi \) briefly discussed in the Appendix 2) calling \(W^{(N)}_{l_a,0,m}\equiv W^{(N)}_{l_a,m}\), we define \( ||W^{(N)}_{l,m}||=L^{-4} \sup _{\underline{\gamma }} a^{4\,l+4\,m}\sum _{\underline{x},\underline{z}} |W^{(N)}_{l,m}(\underline{x},\underline{z},\underline{\gamma })|\).
Lemma 2.2
The kernels in (55) verify, for \(|\lambda |\le \lambda _0 (M a)\), \(|\nu _i|\le C (|\lambda |/(Ma))^2\), \(\lambda _0,C,C_1\) independent on a, L, N, \(l\le 2, m\le 3\), |d| is the distance between any coordinate in \(\underline{x}, \underline{z}\)
with \(D=4-3 l/2 -m-s\).
Proof of Lemma 2.2
We rewrite \(V^{(N+1)}\) (52) in a more compact way as
with P set of field labels and \( {\widetilde{\psi }}(P)=\prod _{f\in P} \psi ^{(\le N)\varepsilon (f)}_{i(f),s(f),x(f)}\), \({\widetilde{J}}(P) =\prod _{f\in P} J^{\beta (f)}_{x(f)}\). We get therefore, inserting (59) in (56) if \(P=Q_1\cup Q_2...\cup Q_n\)
We use the Gawedzki–Kupiainen–Lesniewski [32, 33] (a sketch of the proof is in Appendix 1; see also (see e.g. §A.3 of [38], §2 of [34] or Appendix D of [39])
where \({\mathcal T}_n\) denotes the set of all the ‘spanning trees’ on \(x_{P_1},\ldots ,x_{P_s}\), that is a set of lines which becomes a tree graph on \(\{1,2,\ldots ,s\}\) if one contracts in a point all the point in \(x_P=\cup _{f\in P} x(f)\), the product \(\prod _{\{i,j\}\in T}\) runs over the unordered edges of the T, \(\underline{t}=\{t_{i,i'}\in [0,1], 1\le i,i' \le s\}\), \(dP_{T}(\underline{t})\) is a probability measure (whose form is specified in the Appendix 1) with support on a set of \(\underline{t}\) such that \(t_{i,i'}=u_i\cdot u_{i'}\) for some family of vectors \(u_i\in \mathbb {R}^s\) of unit norm and \(G^{N+1,T}(\underline{t})\) is a \((n-s+1)\times (n-s+1)\) matrix, whose elements are given by \(G^{N+1,T}_{ij,i'j'}=t_{i,i'} g^{(N+1)}(x_{ij},y_{i'j'})\) such that if =\(<u_i\otimes A^{(N+1)}_{x(f^-_{ij}) }, u_{i'}\otimes B^{(N+1)}_{x(f^+_{i'j'})}>\) then the matrix element can be written as a scalar product
with \(A^{(N+1)}_{x(f^-_{ij},k)}=e^{i k x(f^-_{ij})} \sqrt{f^N(k) {\bar{g}}^ {N}(k)}\) and \(B^{(N+1)}_{x(f^+_{i'j'}),k)}=e^{i k x(f^+_{i'j'}))} \sqrt{f^N(k) g^ {N}(k)}\).
The determinants are bounded by the Gram–Hadamard inequality, see e.g. §2 of [34], stating that, if M is a square matrix with elements \(M_{ij}\) of the form \(M_{ij}=<A_i,B_j>\), where \(A_i\), \(B_j\) are vectors in a Hilbert space with scalar product \(<\cdot ,\cdot>\), then \( |\det M|\le \prod _i<A_i,A_i>^{\frac{1}{2}} <B_i,B_i>^{{\frac{1}{2}}}\). Therefore,
We get, setting \(|P_i|\equiv n_i\), \(|Q_i|\equiv l_i\), \(0\le l_i\le n_i\),\(\sum _i m_i=m\), \(l=\sum _i l_i\) and \(| |x|^s g^{(N+1)}(x)|_1\le C_2 \gamma ^{- N-s}\)
and \(\sum _{T\in {\mathcal T}_n}\le n! C_4^{\sum _i n_i}\), see e.g. Lemma A3.3 of [38], Lemma 2.4 of [34] or Lemma D.4 of [34], so that
Note that \(\sum _{l\le n} [(C_5)^{n-l} C_5^l {n!\over l! (n-l)!} ]= (2C_5)^n\) we get
As \(\sum _i m_i=m\le 3\) the sum over \(n_i\) is bounded by
so that for \(\lambda \) small enough
\(\square \)
In order to integrate \(\int P(d\psi ^{(\le N)}) e^{V^{(N)} (\psi ^{(\le N)},J,J^5)}\) (55), we need to take into account the presence of terms with positive or negative scaling dimension \(D=4-3l/2-m\), as can be read from (58).
In order to do that, we extract from \(V^{(N)}\) the terms with non-negative dimension. This is done defining an \({\mathcal L}\) (localization) linear operation acting on the kernels of \(\widehat{W}^N_{l,m}\) (the Fourier transform of \(W^N_{l,m}\) in (55)) in the following way; \({\mathcal L}\widehat{W}^N_{l,m}(\underline{k})=W^N_{l,m}(\underline{k})\) for \((n,m)\not =(2,0), (2,1)\) and
We write therefore
with \({\mathcal R}=1-{\mathcal L}\) (renormalization) and \({\mathcal R}V^{(N)}\) is equal to (57) with \(W^{(N)}_{l,m}\) replaced by \({\mathcal R}W^{(N)}_{l,m}\); the \({\mathcal R}\) operation produce an improvement in the bound, see e.g. §4.2 of [34]; for instance, \({\mathcal R}\widehat{W}^N_{2,0}(k)\) admits, by interpolation, a bound similar to the one for \(\widehat{W}^N_{2,0}(k)\) times a factor \(O(\gamma ^{-2 N})\) due to the derivatives and an extra \(O(\gamma ^{2\,h})\), with h the scale associated with the external fields due to the \(k^2\). Hence, the \({\mathcal R}\) operation produces on such terms an improvement \(O(\gamma ^{2(h- N)})\). In coordinate space, the action consists in producing a derivative in the external field and a “zero”, that is the difference of two coordinates, see, e.g. §3 of [22].
Using symmetry considerations, see Appendix 3, we get
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
where \({\widetilde{\partial }}\) is the discrete derivative and
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
where we have rescaled the fields writing
with \(\nu _{N,s,i} \sqrt{Z_{N,L,i}Z_{N,R,i}} =n_{N,s,i}\) and \({\widetilde{Z}}_{i,s,N}^{J}/Z_{i,s,N}= Z_{i,s,N}^{J}\), \({\widetilde{Z}}_{i,s,N}^{5}/Z_{i,s,N}= Z_{i,s,N}\).
We choose \(\chi _{N}(t)\equiv \chi _0(\gamma ^{-N} t )\) with \(\chi _0(t); \mathbb {R}^+\rightarrow \mathbb {R}\) a \(C^\infty \) non-increasing function \(=1\) for \(0\le t\le \gamma ^{-1}\) and \(=0\) for \(t\ge 1\); we write
with \(f_h(t)\) with support in \(\gamma ^{h-1}\le t \le \gamma ^{h+1}\). We can write \(\chi _{N}(t)=\chi _{N-1}(t)+ f_N(t)\) and
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
where
with \({\mathcal V}^N\) given by (60); a graphical representation is shown in Fig. 2. Using more compact notation
By using the linearity of the truncated expectations and expressing \({\mathcal R}V^N\) by (56) we can write, calling \({\mathcal E}^T(V;...;V)={\mathcal E}^T(V;n)\) (78) as, see Fig. 3
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
and \(V^{(N-1)}={\mathcal L}V^{(N-1)}+{\mathcal R}V^{(N-1)}\) with \({\mathcal L}\) acting on the kernels \(W^{(N-1)}\) as (68), so that, after modifying the wave function renormalization and rescaling, we get to
Therefore, after integrating in the same way \(\psi ^{(N-1)},\psi ^{(N-2)},...,\psi ^{(h+1)}\)
with \(P_{Z_h}(d\psi ^{(\le h)})\) with propagator
and
and finally, if \(\underline{\gamma }=\underline{\alpha },\underline{s}, \underline{i},\underline{\mu },\underline{\beta }\)
and
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
As should be clear from the previous pictures, the \(W^h\) and the \(\beta ^h\) can be conveniently represented in terms as a sum of labelled trees, called Gallavotti trees, see Fig. 4, defined in the following way (for details, see, e.g. §3 of [34]).
Let us consider the family of all trees which can be constructed by joining a point r, the root, with an ordered set of \(n\ge 1\) points, the endpoints of the unlabelled tree, so that r is not a branching point. n will be called the order of the unlabelled tree and the branching points will be called the non-trivial vertices. The unlabelled trees are partially ordered from the root to the endpoints in the natural way; we shall use the symbol < to denote the partial order. The number of unlabelled trees is \(\le 4^n\), see, e.g. §2.1 of [34]. The set of labelled (or Gallavotti) trees \({\mathcal T}_{h,n}\) are defined adding the above labels
-
(1)
We associate a label \(h\le N-1\) with the root and we introduce a family of vertical lines, labelled by an integer taking values in \([h,N+1]\) intersecting all the non-trivial vertices, the endpoints and other points called trivial vertices. The set of the vertices v of \(\tau \) will be the union of the endpoints, the trivial vertices and the non-trivial vertices. The scale label is \(h_v\) and, if \(v_1\) and \(v_2\) are two vertices and \(v_1<v_2\), then \(h_{v_1}<h_{v_2}\). \(s_v\) is the number of subtrees with root v. Moreover, there is only one vertex immediately following the root, which will be denoted \(v_0\) and cannot be an endpoint; its scale is \(h+1\).
-
(2)
To the end-points v of scale, \(h_v\le N\) is associated \({\widetilde{\mathcal L}}V^{(h_v)}\); there is the constraint that the vertex \(v'\) immediately preceding v, that is \(h_{v'}=h_v-1\) is non-trivial (as \({\mathcal R}{\mathcal L}=0\)). The end-points with \(h_v\le N\) can be of type \(\nu \) or Z.
-
(3)
To the end-points v of scale \(h_v= N+1\) is associated one of the terms in \(V^{(N+1)}\)
-
(4)
Among the end-points, one distinguish between the normal ones, associated with terms not containing \(J_\mu , J_\mu ^5\), whose number is \(\bar{n}=n-m\), and the others which are called special.
-
(5)
There is an \({\mathcal R}\) operation associated with each vertex except the end-points and \(v_0\); if the tree contributes to \({\mathcal R}V^h\), it is associated \({\mathcal R}\) while if it contributes to \(\beta _h\) is associated \({\mathcal L}\) and \(s_{v_0}\ge 2\).
-
(6)
A subtree with root at scale k is called trivial if contains only the root and an endpoint of scale \(k+ 1\)
The effective potential can be written as:
where, if \(v_0\) is the first vertex of \(\tau \) and \(\tau _1,..,\tau _{s_{v_0}}\) are the subtrees of \(\tau \) with root \(v_0\), \(V^{(h)}\) is defined inductively by the relation, \(h\le N-1\)
where \({\mathcal E}^T_{h}\) is the truncated expectation with propagator \(g_{i}^{(h)}\) and
-
if \(\tau _i\) is non-trivial \(\bar{V}^{(h)}(\tau _i,\sqrt{Z_h}\psi ^{(\le h)},J,J^5)= {\mathcal R}V^{(h)}(\tau _i,\sqrt{Z_h}\psi ^{(\le h)},J,J^5)\)
-
if \(\tau \) is trivial it is equal to one of the terms in \({\widetilde{\mathcal L}}V^{(h)}\) if \(h<N\), or to the one of the terms in \(V^{(N+1)}\) if \(h=N\).
We can write therefore the kernels in (87) as
It is also convenient to write
with \({\mathcal T}^1_{h,n}\) is the subset of \({\mathcal T}_{h,n}\) containing all the trees with only end-points associated with \({\widetilde{\mathcal L}}V^k\), while \({\mathcal T}^2_{h,n}\) contains the trees with at least one end-point associated with \(V^{N+1}\). We define
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
In this case by the compact support of the propagator only trees contributing to \({\mathcal T}^2_{h,n}\) are present; the contribution from \({\mathcal T}^1\) are “chain graphs” and the localization corresponds in momentum space to setting \(k=0\), and \(\widehat{g}^h(0)=0\). Finally, we can write
with \(\Pi ^{5,i}_{h,\mu ,\nu ,\rho }= \sum _{n=1}^\infty \sum _{\tau \in {\mathcal T}^i_{h,n}} W^h_{0,3}(\tau )\). The following lemma holds, see Appendix 2.
Lemma 2.3
There exists a constant \(\varepsilon \) such that, for \(|{\mathcal Z}_{k}|\le e^{\varepsilon ({a M})^{2} }\), \(\max (|\nu _N|,\ldots ,|\nu _h|,(\lambda / Ma)^2\le \varepsilon \) than if \(m\le 3\), d is the distance between any two coordinate
with \(\vartheta _1=0\) and \(\vartheta _2=\vartheta \) for a constant \(\vartheta = 1/2\); moreover,
The bound is proven showing the convergence of the expansion in \(\nu _k,\lambda \) under a smallness condition which is independent from h. Note that if we perform a multiscale integration setting \({\mathcal L}=0\) then the condition would be that \(\lambda \le \varepsilon _h\) with \(\varepsilon _h\) going to zero a \(h\rightarrow -\infty \). The bound is similar to the one in Lemma 1.2, with the same “dimensional” factor \(\gamma ^{(4-(3/2)l-m) h}\).
A crucial point is that the contributions from trees \({\mathcal T}^2\), that is the terms obtained by the contraction of the irrelevant terms, have a gain \(\gamma ^{\vartheta (h-N)}\) with respect to the dimensional bounds. This fact, and the bound (98) with \({\mathcal Z}_{h-1}=1+\sum _{k=h}^N \beta ^k_{\mathcal Z}\) implies
that is the wave function and the vertex renormalization is bounded uniformly in h. In addition, we can rewrite (100) as
We consider the system
We can regard the right side of (101) as a function of the whole sequence \(\nu _{k,i}\), which we can denote by \(\underline{\nu }=\{\nu _k\}_{k\le N}\) so that (101) can be read as a fixed point equation \(\underline{\nu }=T(\underline{\nu })\) on the Banach space of sequences \(\nu \) such that \(||\nu ||=\sup _{k\le N} \gamma ^{\vartheta (k-N)} |\nu _k|\le C \lambda ^2 (M a)^{-2} \). By a standard proof, see, e.g. Appendix A5 of [41], it is possible to prove that there is a choice of \(\nu _i\) such that the sequence is bounded for any h. With this choice
This means that the \(\nu _{h,i}\) is bounded so that the condition required in Lemma 2.3 is fulfilled; moreover, it is an easy consequence of the Proof of Lemma 2.3 and of (102) that the limit \(L\rightarrow \infty \) can be taken; the proof is standard, see Appendix E of [41]. Finally, we can choose \(Z^5_{i,s}=1+O(\varepsilon )\) so that
We finally to apply the above results and get bounds for the three current function. By (96) and the bound (97) with \(l=0, m=3,s=0\), we get
hence the Fourier transform \(\widehat{\Pi }^{5}_{h,\mu ,\nu ,\rho }(p_1,p_2)\) is continuous; in addition (97) with \(l=0, m=3,s=1+\vartheta /2\) and \(j=2\)
hence \(\sum _{h=-\infty }^N\widehat{\Pi }^{5,2}_{h,\mu ,\nu ,\rho }\) has continuous derivative.
Note that \(\sum _{h=-\infty }^N \widehat{\Pi }^{5,1}_{h,\mu ,\nu ,\rho }\) has a part from trees containing \(\nu _h\) end-points verifying (102), which by the above argument is again differentiable. We remain then with the contribution from trees with three end points associated with \(Z^5,Z^J,Z^J\). We can write the propagator as
where \(r^h(x,y)\) is defined by the above equation as the difference; one can verify, again by integration by parts, that for any K
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
with, \(p=p_1+p_2\),
(108) says that the Fourier transform of the 3-current correlation can be decomposed in the sum of two terms; the first \(\widehat{\Pi }^a_{\mu ,\rho ,\sigma }(p_1,p_2)\) is continuous and is a sum of triangle graphs equal to the its analogue in the non-interacting continuous case with momentum regularization, with vertex and wave function renormalizations depending on the species and chirality. The second \(\widehat{R}_{\mu ,\rho ,\sigma }(p_1,p_2)\) is a complicate series of terms which is differentiable.
The renormalizations in \(\widehat{\Pi }^a_{\mu ,\rho ,\sigma }(p_1,p_2)\) are, however, the same appearing in the 2-point and vertex correlations so that we can use the Ward identities; we can write, see Appendix 2
and
with \(|r_1(k)|\le C (a |k|)^\vartheta \) and \(|r_{2,\mu }(k,p)|\le C (a |k|)^\vartheta \) with \(|p|\le |k|\).
By inserting (110), (111) in the Ward identities (35), we get exact relations between the wave and vertex renormalizations, that is
Note the crucial fact that the contribution from the terms \(r_{i,\mu }\), coming from the trees \({\mathcal T}^2\), is subleading. In conclusion, we get
with \(\widehat{{\mathcal R}}\) with Holder continuous derivative and
Note that \(\widehat{I}_{\mu ,\rho ,\sigma }(p_1,p_2)\) is the anomaly for non-interacting relativistic continuum fermions with a momentum regularization which violates the vector current conservation, see [29], §3.6 for the explicit computation
up to \(O(a^\vartheta |\bar{p}|^{2+\vartheta })\) corrections. In contrast to \(\widehat{I}_{\mu ,\rho ,\sigma }\), we have that \(\widehat{{\mathcal R}}_{\mu ,\rho ,\sigma }\) has not a simple explicit expression, being expressed in terms of a convergent series depending on all the lattice and interaction details. However, we use the differentiability of \(\widehat{{\mathcal R}}_{\mu ,\rho ,\sigma }(p_1,p_2)\) to expand it at first order obtaining, again up to \(O(a^\vartheta |\bar{p}|^{2+\vartheta })\) corrections, using the Ward identity
This implies that
and
Finally, using such values we get
and the second term in the r.h.s. is
which implies Theorem 1.1 \(\square \)
Notes
Any 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.
The 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 \({\widetilde{f}}(k)\).
References
Bouchiat, C., Iliopoulos, J., Meyer, P.: An anomaly-free version of Weinberg’s model: Phys. Lett. B 38, 519 (1972)
Fredenhagen, K., Rehren, K., Seiler, E.: Quantum field theory: Where we are an assessment of current paradigms in the physics of fundamental phenomena. Lect. Notes Phys 721, 61–87 (2007)
Fröhlich, J.: On the triviality of \(\lambda {\phi }^{4}\) theories and the approach to the critical point in d\(>\)4 dimensions. Nuclear Phys. B 200(2), 281–296 (1982)
Aizenman, M., Duminil-Copin, H.: Marginal triviality of the scaling limits of critical 4D Ising and \(\lambda {\phi }\) models. Ann. Math. (2) 194(1), 163–235 (2021)
Weinberg, S.: A Model of Leptons. Phys. Rev. Lett. 19, 1264 (1967)
Hooft, G.’t.: Renormalizable Lagrangians for massive Yang–Mills fields. Nuclear Phys. 167–188 (1971)
Adler, S.L., Bardeen, W.A.: Absence of higher-order corrections in the anomalous axialvector divergence equation. Phys. Rev. 182, 1517 (1969)
Kopper, C., Leveque, B.: Regularized path integrals and anomalies: U(1) chiral gauge theory. J. Math. Phys 53, 022305 (2012)
Zinn-Justin, J.: Chiral anomalies and topology,topology and geometry in physics. In: Bick, E., Steffen, F.D. (Eds.) Springer (2005)
Neuberger, H.: Noncompact chiral U(1) gauge theories on the lattice. Phys. Rev. D63: 014503 (2001)
Luscher, M.: Lattice regularization of chiral gauge theories JHEP 0006:028 (2000)
Borrelli, A., Maiani, L., Rossi, G.C., Sisto, R., Testa, M.: Phys. Lett. B 221, 360 (1989)
Borrelli, A., Maiani, L., Rossi, G.C., Sisto, R., Testa, M.: Nucl. Phys. B 333, 335 (1990)
Testa, M.: The rome approach to chirality. In: PCTP - ICTP Joint International Conference (AIJIC 97) on Recent Developments in Nonperturbative Quantum Field Theory, pp. 114–127
Nielsen, H.B., Ninomiya, M.: The Adler–Bell–Jackiw anomaly and Weyl fermions in a metal. Phys. Lett. B 130, 389 (1983)
Mastropietro, V.: Anomaly cancellation condition in an effective non-perturbative electroweak theory. Phys. Rev. D 103, 013009 (2021)
Fröhlich, J., Kerler, T.: Universality in quantum Hall systems. Nucl. Phys. B 354, 369–417 (1991)
Fröhlich, J., Zee, A.: Large-scale physics of the quantum Hall fluids. Nucl. Phys. B 364, 517–540 (1991)
Fröhlich, J., Studer, U.M.: Gauge invariance and current algebra in non-relativistic many-body theory. Rev. Mod. Phys. 65, 733–802 (1993)
Mastropietro, V.: Nonperturbative Adler-Bardeen theorem. J. Math. Phys 48, 022302 (2007)
Benfatto, G., Falco, P., Mastropietro, V.: Universal relations for nonsolvable statistical models. Phys. Rev. Lett. 104, 075701 (2010)
Benfatto, G., Mastropietro, V.: Renormalization group, hidden symmetries and approximate Ward identities in the XY Z model. Rev. Math. Phys. 13, 1323–1435 (2001)
Benfatto, G., Falco, P., Mastropietro, V.: Functional integral construction of the massive thirring model: Verification of axioms and massless limit. Commun. Math. Phys. 273, 67–118 (2007)
Benfatto, G., Falco, P., Mastropietro, V.: Massless Sine-Gordon and massive thirring models: Proof of Coleman’s equivalence. Commun. Math. Phys. 285, 713–762 (2009)
Giuliani, A.l., Mastropietro, V.: The two-dimensional Hubbard model on the honeycomb lattice. Commun. Math. Phys. 293, Article number: 301 (2010)
Giuliani, A., Mastropietro, V., Porta, M.: Absence of interaction corrections in graphene conductivity. Phys. Rev. B 83, 195401 (2011)
Giuliani, A., Mastropietro, V., Porta, M.: Universality of conductivity in interacting graphene Comm. Math. Phys. 311(2), 317–355 (2012)
Mastropietro, V.: Emergent Adler Bardeen theorem. JHEP 95, (2020)
Giuliani, A., Mastropietro, V.: Porta anomaly non-renormalization in interacting Weyl semimetals. Comm. Math. Phys 384, 997–1060 (2021)
Gallavotti, G.: Renormalization theory and ultraviolet stability for scalar fields via renormalization group methods. Rev. Mod. Phys. 57, 471 (1985)
Brydges, D.: A short course on Cluster Expansions, Les Houches
Gawedzki, K., Kupiainen, E.A.: Gross–Neveu model through convergent perturbation expansions. Commun. Math. Phys. 102, 1 (1985)
Lesniewski, A.: Effective action for the Yukawa 2 quantum field theory. Comm. Math. Phys. 108(3), 437–467 (1987)
Mastropietro, V.: Non Perturbative Renormalization. World Scientific, Singapore (2008)
Okun, L.P.: Quarks and Leptons, North Holland (1985)
Mandl, F., Shaw, G.: Quantum Field Theory. Wiley, New York (2010)
Mastropietro, V.: Nonperturbative renormalization of the lattice Sommerfield vector model Vieri Mastropietro. Phys. Rev. D 105, 114502 (2022)
Gentile, G., Mastropietro, V.: Renormalization group for one-dimensional fermions: A review on mathematical results. Phys. Rep. 352, 273 (2001)
Giuliani, A., Mastropietro, V., Rychkov, S.: Gentle introduction to rigorous renormalization group: a worked fermionic example, JHEP 2021, 26 (2021)
Cayley, A.: A theorem on trees. Quart. J. Pure Appl. Math. 23, 376–378 (1989)
Giuliani, A., Mastropietro, V.: Anomalous universality in the anisotropic Ashkin–Teller model. Comm. Math. Phys. 256, 681–735 (2005)
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.
Funding
Open access funding provided by Universitá degli Studi di Milano within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Antti Kupiainen.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix 1: Truncated Expectations
1.1 The Brydges–Battle–Federbush Formula
The starting point is the formula
Let us define
with \(X=(1,2,..,n)\) and \( \sum _{i,j\in X} \bar{V}_{i,j}=\sum _{i\le j}V_{i,j}\) \(\bar{V}_{i,i}=V_{i,i}\) 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
where \(\pi \) 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
where \(\ell =(j,j')\) is a pair of elements \(j,j'\in X\) 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 explicitly,
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 \( \partial _1 W(X_1,t_1)=\sum _{k\ge 2} V_{1,k}=\sum _{l_1} V_{l_1}\) we can write
and
We have therefore expressed \(e^{-V(X)}\) as the sum of two terms; in the first there is a bond (1, k) between \(X_1\) and the rest is found, in the second \(X_1\) is decoupled. If \(n=2\), the first term is the connected part.
If \(n\not =2\), we further decompose the first term in the r.h.s of (126); we write \(X_2=\{1,k\}\) and
where
and for \(X_2=(1,2)\)
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
and the first term is connected; similar expressions for \(X_2=\{1,3\}\).
Proceeding in this way
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 \(\ell =(j,j')\) such that all the boundaries \(\partial X_k\) are intersected at least by a line \(\ell =(j,j')\),
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
where in the l.h.s. the sets have to be compatible with T. If \(n'(\ell )\) is the minimal k such that \(\ell \) crosses \(X_k\) we have \({\prod _{k=1}^{r-1} t_k(\ell )\over t_{n(\ell )}}= t_{n'(\ell )}\ldots t_{n(\ell )-1}\) and, see, e.g. Lemma 2.3 in [34]
By calling
we get
where \(\ell \in X\) means \(j,j'\in (1,..,n)\).
1.2 The Gawedzki–Kupiainen–Lesniewski Formula
We can write the simple expectations as
with \(V_{jj'}=\sum _{i=1}^{|P_j|}\sum _{i'=1}^{|P_j'|} \eta ^+_{x_{ij} } g(x_{ij}, x_{i'j'}) \eta ^-_{x_{i'j'}}\) and \(\eta ^\pm _{i,j}\) is a set of Grassmann variables. Again, we can write \(e^{-\sum _{j,j'} V_{jj'}}\) as in (131) obtaining
with \(V_{\ell }=\sum _{i}\sum _{i'} \eta ^+_{i,j} g(x_{ij},x_{i'j' }) \eta ^-_{i,j}\), \(\ell =(j,j')\). 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 \(\sum _{\ell \in X'} t_{n'(\ell )}...t_{n(\ell )-1 }V_{\ell }={\widetilde{V}}(\underline{t})+\bar{V}(\underline{t})\) with \(\bar{V}(\underline{t})\) obtained setting \({\widetilde{\eta }}=0\)
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 \(t_{n'(j,j')}...t_{n(jj')-1 }g(x_{ij},x_{i'j' })\). Fixed T we can relabel the \(X_k\) so that \(t_{j}...t_{j'-1 }=u_j u_{j'}\) 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.
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 \({\mathcal P}\), and the set of all \(P_v\) with \(v\in \tau _i\) is called \({\mathcal P}_i\). From (91) we get, if \(n_{v_0}\) is the number of coordinate
By definition, we have a truncated expectation associated with each v in the tree \(\tau \) non-associated with an end-point; we can write each of them by the Gawedzki–Kupiainen–Lesniewski formula. The \({\mathcal R}\) operation is applied and by an iterative procedure one can show that the number of zeros associated with propagators of T and the derivative on the fields is bounded by a constant; see e.g. §3 of [22].
The bound is obtained 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 \(\gamma ^{-4 h_v (s_v-1)} \gamma ^{3/2 h_v (\sum _i |P_{v_i}|-|P_v|)}\), if \(S_v\) is the number of subtrees with root v. The renormalization produces a factor \( \prod _v \gamma ^{-z_v(h_v-h_{v'} ) }\) is produced by the \({\mathcal R}\) 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 not \(\nu ,Z\) is with i \(\psi \) fields and j J fields is associated by lemma 2.1 a factor \( \gamma ^{(4-3 i_v/2-j_v)N} (\lambda ^{i_v/2} (a M)^{2-i_v})\) with \((4-3 i_v/2-j_v)<0\) and \(i_v\ge 4\) and \((a M)^{2-i_v}<(a M)^{-2}\). We get therefore
By using that
where \(m^{i,j}_v\) is the number of end-points following v with i \(\psi \) fields and j J fields, we get
We use now that
where \(v^*\) is the first non-trivial vertex following v; this implies
so that
Finally, we use the relation
and using that \(\sum _{i,j} j m^{i,j}_v =n^J_v\), we finally get (\(j_v=0\) if v is a \(\nu \)-e.p.)
In conclusion,
where \({\widetilde{v}}\in {\widetilde{V}}\) are the vertices on the tree such that \(\sum _i |P_{v_i}|-|P_v|\not =0\), \({\widetilde{v}}'\) 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 \(\sum _{T\in {\textbf{T}}}\) is bounded by \(\prod _{v} s_v!\; C^{\sum _{i=1}^{s_v}|P_{v_i}|-|P_v|}\), see e.g. §2.1 of [38]. In order to bound the sums over the scale labels and \(\textbf{P}\), we first use the inequality
where \({\widetilde{v}}\) are the non-trivial vertices, and \({\widetilde{v}}'\) is the non-trivial vertex immediately preceding \({\widetilde{v}}\) or the root. The factors \(\gamma ^{-{1\over 2}(h_{{\widetilde{v}}}-h_{{\widetilde{v}}'})}\) in the r.h.s. allow to bound the sums over the scale labels by \(C^n\) and \(\sum _\textbf{P} \prod _{{\widetilde{v}}}\gamma ^{-{3|P_{{\widetilde{v}}}|\over 4}}\le C^n\), see §3.7 of of [34].
Let us consider the improvement of the bound. If \({\mathcal T}^*\) is the set of trees with at least an end-point not of \(\nu , Z\) type then, for \(0<\vartheta <1\)
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)
and
as \(\prod _{v\;e.p. \; not\; \nu , Z}\gamma ^{(4-3 i_v/2-j_v)(N-h_{v^*})}\le \gamma ^{-\vartheta (N-h_{\widehat{v}}) }\) as there is at least an end-point not \(\nu ,Z\). Noting that \(d_{{\widetilde{v}}}-\vartheta >0\) one can perform the sum as above, and the same bound is obtained with an extra \(\gamma ^{\vartheta (h-N)}\). \(\square \)
In the 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 with scale \(\bar{h}\) 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 \(\gamma ^{4\bar{h}}\) and a \(\gamma ^{-2\bar{h} }\) from the lack of the \(\nu \) end-points. There is a decay factor proportional to \(x-y\) at scale \(\gamma ^{\bar{h}}\) and, from the trees belonging to \({\mathcal T}^*\), an extra \(\gamma ^{\vartheta (h-N)}\); see e.g. §3.D of [25]. 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. Appendix D of [25].
Appendix 3: Symmetries
By symmetry, there are no quadratic contributions with \(i'\not =i\). There is invariance under the transformation \(\psi ^\pm _{k,s}\rightarrow \varepsilon _s\psi ^\mp _{{\widetilde{k}},s}\sigma _1\), \(J_{\textbf{k}}\,A_\textbf{k}\rightarrow J_{{\widetilde{\textbf{k}}}},A_{{\widetilde{\textbf{k}}}}\) 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 transformation \(\psi ^\pm _{k,s}\rightarrow \varepsilon _s\psi ^\mp _{{\widetilde{k}},s}\sigma _2\), \(J_{\textbf{k}}\,A_\textbf{k}\rightarrow J_{{\widetilde{\textbf{k}}}},A_{{\widetilde{\textbf{k}}}}\) 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, \(b=0\).
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Mastropietro, V. Vanishing of the Anomaly in Lattice Chiral Gauge Theory. Ann. Henri Poincaré 25, 1047–1081 (2024). https://doi.org/10.1007/s00023-023-01274-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-023-01274-5