Massless Sine-Gordon and Massive Thirring Models: Proof of the Coleman’S Equivalence

Massless Sine-Gordon and Massive Thirring Models: proof of the Coleman’s equivalence

G. Benfatto1 P. Falco2 V. Mastropietro1

1 Dipartimento di Matematica, Universit`adi Roma “Tor Vergata” via della Ricerca Scientiﬁca, I-00133, Roma

2 Mathematics Department, University of British Columbia, Vancouver, BC Canada, V6T 1Z2

Abstract We prove the Coleman’s conjecture on the equivalence between the massless Sine-Gordon model with ﬁnite volume interaction and the Thirring model with a ﬁnite volume mass term.

1 Introduction

1.1 Coleman’s Equivalence One of the most fascinating aspects of QFT in d = 1+1 is the phenomenon of bosonization; fermionic systems can be mapped in bosonic ones and viceversa. The simplest example is provided by the equivalence between free massless Dirac fermions and free massless bosons with the identiﬁcations (see for instance [ID]):

iσ√4πφx µ 1 µν ψ¯x(1 + σγ )ψx b :e : , ψ¯xγ ψx ε ∂ φx (1) 5 ∼ 0 ∼ −√π ν where σ = 1 and b0 is a suitable constant, depending on the precise deﬁni- tion of the± Wick product. Such equivalence can be extended to interacting theories; Coleman [C] showed the equivalence, in the zero charge sector, between the massive Thirring model, with Lagrangian (with our conventions)

λ 2 µ = iZψ¯x ∂ ψx Z µψ¯xψx Z j xj (2) L 6 − 1 − 4 µ, x 1 µ where Z and Z1 are (formal) renormalization constants, jµ,x = ψ¯xγ ψx and the massless Sine-Gordon model, with Lagrangian

1 µ = ∂ ϕx∂ ϕx + ζ : cos(αφx): (3) L 2 µ with the identiﬁcations

iασφx µ µν Z ψ¯x(1 + σγ )ψx b :e : , Zψ¯xγ ψx b ε ∂ φx (4) 1 5 ∼ 0 ∼ − 1 ν where b0, b1 are two suitable constants, depending on λ and the details of the ultraviolet regularization. Moreover, this equivalence is valid if certain relations between the Thirring parameters λ, µ and the Sine-Gordon parameters α, ζ are assumed. The case α2 = 4π is special, as it corresponds to free fermions (λ = 0); the choice ζ = 0 (free bosons) corresponds to massless fermions (µ = 0). In order to establish such equivalence, Coleman considered a fixed infrared regularizations of the models (2) and (3), replacing µ in (2) with µχΛ(x) and ζ with ζχΛ(x), with χΛ(x) a compact support function; this means that the mass term in the Thirring model, and the interaction in the Sine-Gordon is concentrated on a finite volume Λ. Such regularization makes possible a perturbative expansion, respectively in µ for the Thirring model and ζ for the Sine-Gordon model; it turned out that the coefficients of such series expansions can be explicitly computed (in the case of the Thirring coefficients this was possible thanks to the explicit formulas for the correlations of the massless Thirring model given first in [Ha, K]) and they are order by order identical if the identification (4) is done and provided that suitable relations between the parameters are imposed. The identification of the series expansions coefficients would give a rigorous proof of the equivalence provided that the series are convergent. The issue of convergence, which was mentioned but not addressed in [C], is technically quite involved and crucial; there are several physical examples in which order by order arguments without convergence lead to uncorrect prediction. The search for a rigorous proof of Coleman equivalence was the subject of an intense investigation in the framework of constructive QFT, leading to a number of impressive results. In [FS] it was rigorously proven the equivalence between the massive Sine-Gordon model (with mass M large enough) at α2 < 4π and a Thirring model with a large long-range interaction; similar ideas were also used in [SU]. The properties of the massive Sine-Gordon model for α2 4π were later on deeply investigated. In [BGN] and [NRS] it was proved≥ that the model is stable if one adds a finite number, increasing with α, of vacuum counterterms, while the full construction, through a cluster expansion, of the model was partially realized in [DH]. In [DH] it was also proved that the correlation functions are analytic in ζ, for any α2 < 8π.A

2 proof of analyticity, only based on a multiscale analysis of the perturbative expansion, was first given in [B], for α2 < 4π, and then extended in [BK] up to α2 < 16/3π. Using the results in [DH] for a fixed finite volume, Dimock [D] was able finally to achieve a proof of Coleman’s equivalence, in the Euclidean version of the models, for the case α2 = 4π; such a value is quite special as it corresponds to λ = 0, that is the equivalence is with a free massive fermionic system, without current-current interaction. Such limitation was mainly due to the fact that the constructive analysis of interacting fermionic systems was much less developed at that time: indeed a rigorous construction of the massive Thirring model in a functional integral approach has been achieved only quite recently [BFM]. A more physically oriented research on Coleman’s equivalence was fo- cused in recovering bosonization in the framework of the (formal) path- integral approach, [N, FGS]. The idea is to introduce a vector field Aµ and to use the identity λ 2 √ exp dxjµ,xjµ,x = DA exp dx Aµ,x + λAµ,xjµ,x) (5) (− 4 ) − Z Z ½Z h i¾ By parameterizing Aµ in terms of scalar fields ξx, φx

Aµ = ∂µξx + εµ,ν∂νφx (6) it turns out that the massive Thirring model can be expressed in terms of the boson fields ξx and φx: the first is a massless free field, while the second one has an exponential interaction when µ = 0. In the expectations 6 of the operators ψ¯x(1 + σγ5)ψx and jµ,x, the ξx field has no role and it can be integrated out; the resulting correlations imply the identification (4). Such computations are however based on formal manipulations of functional integrals (with no cut-offs, hence formally infinite) and it is well known that such arguments can lead to incorrect result (see for instance the discussion in 1 in [BFM]). §In this paper we will give the first proof of Coleman’s equivalence between the Euclidean massive Thirring model with a small interaction and the Eu- clidean massless Sine-Gordon model with α around 4π. We will follow the Coleman strategy, but an extension of the multiscale techniques developed in [B] for the Sine-Gordon model and in [BM, BFM] for the Thirring model allow us to achieve the convergence of the expansion.

1.2 Main results We start from a suitable regularization of the Sine-Gordon and Thirring models via the introduction of infrared and ultraviolet cut-oﬀs, to be removed at the end.

3 Let us consider first the (Euclidean) Sine-Gordon model. Let γ > 1, h be a large negative integer (γh is the infrared cutoff) and N be a large positive in- N teger (γ is the ultraviolet cutoff). Moreover, let ϕx a 2-dimensional bosonic def field and Ph,N (dϕ) be the Gaussian measure with covariance Ch,N (x) = N j j=h C0(γ x), for P def 1 dk k2 (γk)2 ikx C (x) = e− e− e (7) 0 (2π)2 k2 − Z h i Given the two real parameters ζ, the coupling, and α (related with the in- verse temperature β, in the Coulomb gas interpretation of the model, by the relation β = α2), the Sine-Gordon model with finite volume interaction and ultraviolet and infrared cutoffs is defined by the interacting measure P (dϕ) exp ζ V (ϕ) , with h,N { N } α2 C0,N (0) V (ϕ)= dx cos(αϕx) , ζN = e 2 ζ (8) ZΛ where Λ is a fixed volume of size 1. Note that ζN V (ϕ)= ζ Λ dx : cos(αϕx):, where 2 R iaϕ def iaϕ a C (0) :e x: = e x e 2 0,N (9) is the Wick product of eiαϕx , a R, with respect to the measure with ∈ covariance C0,N (x) (for any h); hence ζN has the role of the bare strength. We consider now the Thirring model. The precise regularization of the path integral for fermions was already described in [BFM], 1.2, therefore § we only remind the main features. We introduce in ΛL [ L/2, L/2] [ L/2, L/2] a lattice Λ whose sites represent the space-time≡ points.− We also× − a consider the lattice a of space-time momenta k = (k, k0). We introduce D ¯ a set of Grassmann spinors ψk, ψk, k a, such that ψk = (ψk−,+, ψk−, ), ¯ + 0 + + + ∈ D − ψ = ψ γ and ψk =(ψk,+, ψk, ). The γ matrices are explicitly given by − 0 1 0 i 1 0 γ0 = , γ1 = − , γ5 = iγ0γ1 = . 1 0 i 0 − 0 1 µ ¶ µ ¶ µ − ¶ We also define a Grassmann field on the lattice Λa by Fourier transform, according to the following convention:

def 1 ψ[h,N]σ = eiσkxψ[h,N]σ , x Λ . (10) x,ω L2 k,ω ∈ a kXa ∈D b [h,N]σ σ Sometimes ψx,ω will be shorten into ψx,ω. Moreover, since the limit a 0 [h,N]σ → is trivial [BFM], we shall consider in the following ψx,ω as defined in the continuous box ΛL. In order to introduce an ultraviolet and an infrared cutoff, we could use a gaussian cut-off as in (21), but for technical reason,and to use the results of

4 [BFM], we find more convenient to use a compact support cut-off. We define the function χh,N (k) in the following way; let χ C∞(R+) be a Gevrey function of class 2, non-negative, non-increasing smooth∈ function such that

1 if 0 t 1 χ(t)= (11) 0 if ≤t ≤γ , ½ ≥ 0 for a fixed choice of γ : 1 < γ γ; then we define, for any h j N, 0 0 ≤ ≤ ≤ j (j 1) f (k)= χ γ− k χ γ− − k (12) j | | − | | ³ ´ ³ ´ N and χh,N (k) = j=h fj(k); hence χh,N (k) acts as a smooth cutoff for mo- N+1 h 1 menta k γ and k γ − . | | ≥ P | | ≤ Given two real parameters, the bare coupling λ and the bare mass µ, the Thirring model with finite volume mass term and ultraviolet and infrared cutoffs is defined by the interacting measure P (dψ) exp (ψ) , with h,N {V } λ 2 ¯ µ 2 (1) ¯ (ψ)= ZN dx ψxγ ψx + ZN µ dx ψxψx + Eh,N ΛL (13) V − 4 ΛL Λ | | Z ³ ´ Z and

def 4 2 2 2 1 P (dψ) = dψ L− Z ( k C (k) − h,N N | −| | h,N · k Y[h,N] h i b ∈D 1 Dω(k) + exp Z ψ ψ− ′ , (14)  N 2 k,ω k,ω  − L ω= k [h,N] χh,N (k)  X± X  ∈D b b def   where Dω(k) = ik0 + ωk1 and Eh,N is constant chosen so that, if µ = 0, P (dψ) exp −(ψ) = 1. We will prove the following theorem. h,N {V } R Theorem 1.1 Assume ζ , λ , µ small enough, α2 < 16π/3; then there | | |2 | | | 3 2 exist two constants η = aλ + O(λ ) and η+ = bλ + O(λ ), with a,b > 0, − independent of µ and analytic in λ, such that, if we put

η−N (1) η+N ZN = γ− , ZN = γ− (15) then, if r = 0 and q 2 or r 1, for any choice of the non coinciding points (x ,..., x , y ,..., y≥), and of≥σ = 1, i = 1,...,q, ν = 0, 1, j = 1,...,r, 1 q 1 r i ± j q r T iσiαϕxi νj µ µ lim :e : ( 1)ε ∂ φyj SG = (16) h,N h" #" − #i − →∞ iY=1 jY=1 q r 1+ σiγ5 (1) q r ¯ ¯ νj T = lim (b0ZN ) (b1ZN ) ψxi ψxi ψyj γ ψyj T h h,N h" 2 #" #i − →∞ iY=1 µ ¶ jY=1

5 where T and T denote the truncated expectations in the Thirring h·iT h h·iSG (in the limit L ) and Sine-Gordon models, respectively, b0 and b1 are bounded functions→ of ∞λ and the following relations between the parameters of the two models have to be verified: α2 =1+ η η+ , ζ = b0µ (17) 4π − − If q = 1 and r = 0 both the r.h.s. and the l.h.s. of (16) are diverging for λ 0, while the equality still holds for λ> 0. A divergence also appears, for λ ≤ 0, in the pressure, but only for the second order term in ζ or µ; however, if≤ we add a suitable vacuum counterterm, also the pressures are equal. This Theorem proves Coleman’s equivalence (4). We remark that the relations between the Sine-Gordon parameters and the Thirring parameters in (17) are slightly different with respect to those in [C], for λ = 0; this is true in particular for the first equation, involving only quantitie6s which have a physical meaning in the removed cutoff limit, if we express them in terms of λ, as Coleman does. This is not surprising, as the relations between the physical quantities, like the critical indices η , and the bare coupling depend on the ± details of the regularization, and in our Renormalization Group analysis the running coupling constants have a bounded but non trivial flow from the ultraviolet to the infrared scales. Indeed, with a different regularization of the Thirring model (that is starting from a non local current-current interaction and performing the local limit after the limit N ), as in [M1, M2], one would get a simple relation between α and λ. This→ new ∞ relation again is not equal to that of [C], but is in agreement with the regularization procedure of [J], see footnote 7 of [C]. Another important remark concerns the limit Λ . In the case of Sine-Gordon model, one expects that, in this limit, there→ ∞ is exponential decrease of correlations (implying the screening phenomenon in the Coulomb gas interpretation), which is not compatible with convergence of perturbative expansion (in this case the correlations would have a power decay as in the free theory). Up to now, screening has been proved only for α2 << 4π [Y], but it is expected to be verified in all range of validity of the model (α2 < 8π), hence even around α2 = 4π. However, if the interaction is restricted to a fixed finite volume, convergence is possible and we could indeed prove it, for α2 < 6π; in this paper, for simplicity, we give the proof only for α2 < 16π/3, which is sufficient to state the main result. The situation for the massive Thirring model is slightly different, because it has been shown [BFM] that it is well defined in the limit Λ and that → ∞ its correlations decay at least as exp( c µ 1+O(λ) x ). Hence, even if the − | | | | power expansion in the mass can be convergentq only if we fix the volume, the proof of Coleman’s conjecture strongly supports the related conjecture that

6 even the Sine-Gordon model is well defined around α2 = 4π in the infinite volume limit and has exponential decrease of correlations. The proof is organized in the following way. In 2 we analyze the massless Sine-Gordon model with finite volume interaction and§ α2 < 16π/3, extending the proof of analyticity in ζ given in [B] for the massive case in the infinite volume limit and α2 < 4π. With respect to the technique used in [D], where only the case α2 = 4π was analyzed, our method has the advantage that an explicit expression of the coefficients can be easily achieved; this is probably possible even with the other method, but the proof was given only for α2 < 4π and, as a consequence, the correlations in the model with α2 = 4π were defined as the limit α2 4π of those with α2 < 4π. In 3 we use the methods developed→ in [BFM, M1, M2] to prove the analyticity§ in µ of the Thirring model; the explicit expressions of the coefficients are obtained in 4, by using the explicit expression of the field correlation functions given in§ the Appendix (through the solution of a Schwinger-Dyson equation, based on a rigorous implementation of Ward Identities) and by a rigorous implementation, in a RG context, of the point spitting procedure used in theoretical physics.

2 The Massless Sine-Gordon Model with a ﬁnite volume interaction

We want to study the measure defined in 1.2 in the limit of removed cutoff, h,N . To this purpose, we consider§ the Generating functional, −(J, A, ζ→), defined ∞ by the equation Kh,N ζN V (ϕ) h,N (J, A, ζ) = log Ph,N (dϕ) e K Z · exp dxJ σ:eiασϕx:+ dyAν (∂νϕ ) (18)  x y y  · σ= 1 Z ν=0,1 Z  X± X  σ µ where Jz and Ay are two-dimensional, external bosonic fields. Then, given two non negative integers q and r, as well as two sets of labels σ =(σ1, . . . , σq) and ν = (ν1, . . . , νr), together with two sets of two by two distinct points z = (z1,..., zq) and y = (y1,..., yq), we consider the Schwinger functions, defined by the equation q+r (q,r;ζ) def ∂ h,N Kh,N (z, y; σ, ν) = σ σq K ν ν (0, 0, ζ) (19) ∂J 1 ∂Jz ∂A 1 ∂A r z1 ··· q y1 ··· yr Theorem 2.1 If ζ is small enough, α2 < 16π/3 and q 2, if r = 0, or | | ≥ q 0, if r 1, the limit ≥ ≥ (q,r;ζ) def (q,r;ζ) K (z, y; σ, ν) = lim Kh,N (z, y; σ, ν) (20) h,N + − → ∞ 7 exists and is analytic in ζ. In the case q = r = 0 (the pressure), the limit does exist and is analytic, up to a divergence in the second order term, present only for α2 4π. ≥ For clarity’s sake, we prefer to give the proof of the above theorem in the special cases (q, r)=(k, 0) and (q, r) = (0, k) separately; the proof in the general case is a consequence of the very same ideas that will be discussed for the special ones, but it needs a more involved notation, so we will not report its details.

2.1 The free measure By the deﬁnitions given in 1.2, the regularized free measure is the two– dimensional boson Gaussian§ measure with covariance

N 1 dk (γ−N k)2 (γ−h+1k)2 ikx j C (x)= e− e− e = C (γ x) (21) h,N (2π)2 k2 − 0 Z h i jX=h The two–dimensional massless boson Gaussian measure is obtained by taking the limits h and N . It is easy to prove that → −∞ → ∞

log γ q0 q1 κ x C (0) = , ∂ ∂ C (x) A e− | | (22) 0 2π x0 x1 0 ≤ q0,q1,κ ¯ ¯ ¯ ¯ where q0, q1 are non negative integers¯ and κ0 is¯ an arbitrary positive constant. Let us now consider the function

h Ch, (x) = lim Ch,N (x)= C0, (γ x) (23) ∞ N ∞ →∞ It is easy to show, by a standard calculation, that there exists a constant c such that 1 2 2 C0, (x)+ log(c x ) C x (24) | ∞ 4π | | | ≤ | | 1 h Hence, Ch, (x) diverges for h as (2π)− log(γ x ). However, if we ∞ → −∞ − | | deﬁne 1 1 2h ∆h,− (x)= Ch, (x)+ log(cγ ) (25) ∞ ∞ 4π we have, by (24):

1 def 1 1 ∆− (x) = lim ∆h,− (x)= log x (26) h ∞ →−∞ −2π | | Then it is natural to deﬁne the Coulomb potential with ultraviolet cutoﬀ by

1 def 1 1 def 1 2h ∆N− (x) = lim ∆h,N− (x) , ∆h,N− (x) = Ch,N (x)+ log(cγ ) (27) h →−∞ 4π 8 Since Ch,N (x)= Ch, (x) CN, (x), by using (22) and (26), we see that ∞ − ∞ 1 1 κγN x N ∆− (x)+ log x Ce− | | , γ x 1 (28) N 2π | | ≤ | | ≥ ¯ ¯ ¯ ¯ and, by using¯ (24), we see that ¯

1 1 2N 2N 2 N ∆− (x) log(cγ ) Cγ x , γ x 1 (29) | N − 4π | ≤ | | | | ≤

We deﬁne h,N and j to be the expectation with respect to the Gaussian E E j measures with covariance Ch,N (x) and Cj(x) = C0(γ x), respectively; a su- perscript T in the expectation will indicate a truncated expectation. Recall that, for a generic probability measure with expectation , and any family of random variables (f ,...,f ), T is deﬁned as E 1 s E

T Π 1 [f1; . . . ; fs]= ( 1)| |− ( Π 1)! fi (30) E Π − | | − X Π E "i X # X Y∈ Y∈ where Π denotes the sum over the partitions of the set (1,...,s). Finally iσβϕx iσβϕx we remindP that :e : is the Wick normal ordering of e always taken with respect to the measure with covariance C (x) (see deﬁnitions in 1.2). 0,N § def Lemma 2.2 Let σ 1, +1 , i = 1,...,n, and α R. If Q = σ , i ∈ {− } ∈ r r then P n α2 2 −1 iασrϕxr n α σrσs∆ (xr xs) lim h,N :e : = δQ,0c− 8π e− r

If is the expectation h,N in the limit h,N , by taking the limit N E in the r.h.s. of (31),E we get, in the− case Q→= ∞ 0, → ∞ n α2 α2 iασrϕxr n σrσs :e : = δQ,0c− 8π xr xs 2π (32) E " # r

~~9 2.2 The case q = r = 0 (the pressure) To begin with we analyze the pressure:~~

def def ζN V (ϕ) p(ζ) = lim log Zh,N (ζ) , Zh,N (ζ) = Ph,N (dϕ) e (33) h,N − →∞ Z We proceed as in [B], by studying the multiscale expansion associated with the following decomposition of the covariance: N def j Ch,N (x)= Cj(x)+ Ch, 1(x) , Cj(x) = C0(γ x) (34) − jX=0 In comparison with [B], where the case h = 0 - the “Yukawa gas” - was considered, here we are collecting in a single integration step all scales below h = 0: as we shall see, this is eﬀective since the volume size is ﬁxed to be 1. To simplify the notation, from now on 1 will denote the expectation w.r.t. E− Ch, 1(x), while j will have the previous meaning for j 0. − (N) E ≥ Let n , n 2, be the family of labelled trees with the following properties: T ≥

~~1) there is a root r and n ordered endpoints ei, i = 1,...,n, which are connected by the tree; the tree is ordered from the root to the endpoints;~~

2) each vertex v carries a frequency label hv, which is an integer taking values between 1 and N + 1, with the condition that hu < hv, if u precedes v in the− order of the tree; moreover, the root has frequency 1 and the endpoint ei has frequency hi + 1, if hi is the frequency of −the higher vertex preceding it.

3) The endpoint ei carries two other labels, the charge σi and the position xi. These trees differ from those used in [BFM] for the Thirring model, because there are no “trivial vertices” on the lines of the tree. (N) (N+1) ( ) (N) (N) Since n n , then n ∞ = limN n is obtained from n →∞ by lettingT the frequency⊂ T indicesT free to vary betweenT 1 and . T We shall also use the following definitions: − ∞ a) Given a tree τ, we shall call non trivial (n.t. in the following) the tree vertices different from the root and from the endpoints. If v τ is a n.t. vertex, s 2 will denote the number of lines branching∈ from v ≥ v in the positive direction, v′ τ is the higher non trivial vertex ∈ preceding v, if it does exist, or the root, otherwise. Moreover, Xv will be the set of endpoints following v along the tree; Xv will be called the cluster of v and nv will denote the number of its elements. If v is an endpoint, Xv will denote the endpoint itself. Finally we define

Φ(Xv)= i : e Xv σiϕxi . i∈ P 10 b) Given a n.t. vertex v and an integer j [ 1,N], we shall denote ∈ − def U (v) = σ σ C (x x )= Φ2(X ) 0 (35) j r m j r − m Ej v ≥ r,m : er,em Xv X ∈ h i the (double of the) total energy on scale j associated with its cluster. If k′ + 1 k 1, we shall also deﬁne ≤ − k 1 def − Uk′,k(v) = Uj(v) (36) ′ j=Xk +1 c) If X and Y are two disjoint clusters and j is an integer contained in [ 1,N], we denote − def W (X,Y ) = σ σ C (x x )= [Φ(X)Φ(Y )] (37) j r m j r − m Ej r,m : er X, em Y X∈ ∈ the interaction energy on scale j between X and Y .

~~d) Given a n.t. vertex v,(v1,...,vsv ), will denote the set of vertices following it along the tree; moreover we deﬁne~~

~~T iαΦ(X ) iαΦ(X ) G (v ,...,v )= e v1 ; . . . ; e vsv (38) j 1 sv Ej h i By proceeding as in [B], it is easy to see that~~

~~∞ n (N) ζV (ϕ)+ ζ Vn (ϕ) Zh,N (ζ)= Ph, 1(ϕ) e n=2 (39) − Z P where~~

~~1 n (N) iα σrϕxr V (ϕ)= dx1 dxn e r=1 Vτ (σ, x) (40) n n n (N) 2 σ ,...,σ Λ ··· τ 1 n Z P ∈TXn X~~

n 2 α2 G (v ,...,v ) α he hv 1 sv Uh ′ ,hv (v) Vτ (σ, x)= γ 4π i e− 2 v (41) Ãi=1 ! n.t. v τ sv! Y Y∈ We note that Vτ (σ, x) is independent of N and h. In order to prove that the pressure, see (33), is well deﬁned, the main step is to verify that, uniformly in h and N, Zh,N (ζ)=1+ O(ζ). As we will discuss later in this section, since the only dependence on h in (39) is through the measure Ph, 1(dϕ), which has support on smooth functions for − n any h, the wanted bound for Zh,N (ζ) is an easy consequence of a uniform C (N) bound of Vn (ϕ). Since

(N) def 1 V (ϕ) bτ , bτ = dx1 dxn Vτ (σ, x) (42) n n n | | ≤ (N) 2 σ ,...,σ Λ ··· | | τ 1 n Z ∈TXn X 11 and the number of trees is of order Cn, we shall look for a “good” bound of bτ . The main ingredients in this task are the positivity of Uj(v), see (35), and the Battle-Federbush formula for the truncated expectations (see [Br]):

2 2 α U (v,t) G (v ,...,v )= α W (X , X ) dp (t) e− 2 j (43) j 1 s − j vr vm T T ¯s r,m T Z X∈T h Yi∈ h i where s = sv, ¯s is the family of connected tree graphs on the set of integers 1 T 1,...,s , 2 Uj(v,t) is obtained by taking a sequence of convex linear combi- {nation, with} parameters t, of the energies of suitable subsets of X = X v ∪i vi (hence Uj(v,t) 0) and dpT (t) is a probability measure. By using (41),≥ (43) and (37), we get

n α2 he Vτ (σ, x) γ 4π i | | ≤ Ã ! · iY=1 2(sv 1) α − ′ Chv (xe xe ) (44) · sv! e∈X | − | n.t. v τ T ¯s r,m T vr ∈ v ′ Y X∈T h Yi∈ e ∈XXvm On the other hand, for any given ε> 0, we can use the bound

~~s 2(s 1) 2εnv n n s! ε− − e i (45) vr vm ≤ T ¯s r,m T r=1 X∈T h Yi∈ Y Moreover, by (34) and (22), for h 0, v ≥~~

~~2hv dx Chv (x) Cγ− . (46) ZΛ | | ≤~~

~~The trees, as deﬁned after (34), satisfy the following identity: w v(sw 1) = n 1; as a consequence, if v is the ﬁrst non trivial vertex of τ≥, − v − 0 P~~

′ hv(sv 1) = hr(nv0 1) + (hv hv )(nv 1) n.t. v τ − − n.t. v τ − − X∈ X∈ where v′ is the n.t. vertex immediately preceding v or the root, if v = v0. This allows us to write: n α2 n 4π hei 2hv(sv 1) 2εnv bτ Cε γ γ− − e (47) ≤ Ãi=1 ! n.t. v τ ≤ Y Y∈ n (hv hv′ )[D(nv) 2εnv] Cε γ− − − ≤ n.t. v τ Y∈ where the dimension D(n) is given by

α2 D(n)=2(n 1) n . (48) − − 4π 12 Let us consider first the case α2 < 4π. This condition implies that D(n) > 0 for any n 2; hence, the bound (47) implies in the usual way that ≥n 2 V (σ, x) C , for a constant C , which diverges as α (4π)−; since | τ | ≤ α α → Λ = 1, this bound is valid also for bτ . By a little further effort, see below, |one| can then prove that the pressure p(ζ) is an analytic function of ζ, for ζ small enough. If 4π α2 < 16π/3, D(n) > 0 only for n 3, so that the previous bound ≤ ≥ is divergent for all trees containing at least one vertex with nv = 2. In (N) particular, V2 (ϕ) diverges as N ; this divergence is related with the 2 → ∞ fact that the term of order ζ and σ1 = σ2 in the perturbative expansion of the pressure is really divergent, as one− can easily check. The only way to cure this specific divergence is to renormalize the model by subtracting a suitable constant of order ζ2 from the potential, as we shall see below. However, all other terms, even those associated with a tree containing at least one vertex with nv = 2, are indeed bounded uniformly in N; in order to prove this claim, we need to improve the bound (47) by the two following lemmas.

~~Lemma 2.3 If nv = 2 and v1,v2 are the two endpoints following v with positions x1, x2 respectively and equal charges σ1 = σ2, then~~

2 α α2 h Uh ,h (v) (hv h ′ ) γ v x1 x2 G (v ,v )e− 2 v′ v Cγ− π − v e− | − | (49) | hv 1 2 | ≤ Proof. Since h ′ + 1 0, it is easy to check that v ≥ 2 α 2 2 Uh ,h (v) α C (x1 x2) α C0(0) G (v ,v )e− 2 v′ v = e− hv − 1 e− hv 1 2 − · 2 hv−1 h i α [C0(0)+Ck(x1 x2)] def k=h ′ +1 e− v − = F (x1 x2) · P − Hence, by using (22) with κ> 1 and since C (x) C (0), we get 0 ≤ 0 hv−1 h 2 α2 C (γkz) C (0) κγ v z 2α C0(0)(hv h ′ ) k=h +1 [ 0 0 ] F (z) Ce− | |e− − v e− v′ − | | ≤ ≤ α2 2 h P (hv h ′ ) 2α RC0(0) (κ κ )γ v z Cγ− π − v e e− − R | | ≤ def 2 r where κ = α B ∞ γ− , the constant B is such that C (x) C (0) R r=R | 0 − 0 | ≤ B x (it exists byP (22) for (q0,q1) = (1, 0), (0, 1)) and R is an arbitrary positive| | integer, that we can choose so that κ κ 1. − R ≥

Lemma 2.4 For j 0, if the cluster X is made of two endpoints with ≥ positions x1, x2 and opposite charges, and Y is another arbitrary cluster, then 1 j j γ x2+t(x1 x2) y Wj(X,Y ) Cγ x1 x2 dt e− | − − | (50) | | ≤ | − | y Y Z0 X∈ 13 Moreover, if also the cluster Y is made of two endpoints with opposite charge and positions y1, y2, then

~~1 j 2j γ x2+t(x1 x2) y2 s(y1 y2) Wj(X,Y ) Cγ x1 x2 y1 y2 dtds e− | − − − − | | | ≤ | − | | − | 0 Z (51) Proof. By using the identity~~

~~1 C0(x1 y) C0(x2 y)= (x1 x2)a dt (∂aC0)[x2+t(x1 x2) y] (52) − − − − 0 − − aX=0,1 Z together with (22) (with κ 1), we get the bound ≥~~

1 j j γ x2+t(x1 x2) y Cj(x1 y) Cj(x2 y) Cγ x1 x2 dt e− | − − | (53) | − − − | ≤ | − | Z0 which immediately implies (50). The bound (51) is proved in a similar way, by using the identity

C (x y ) C (x y ) C (x y )+ C (x y )= (x x ) 0 1 − 1 − 0 2 − 1 − 0 1 − 2 0 2 − 2 1 − 2 a · a,bX=0,1 1 (y1 y2)b dtds (∂a∂bC0)[x2 + t(x1 x2) y2 s(y1 y2)] (54) · − Z0 − − − −

Let us now consider a tree with n 3 endpoints. By using Lemma 2.3, ≥ 2 we can improve the bound (47) by replacing D(nv) with D(nv)+ α /π in all 2 2 vertices with nv = 2 and σ1 = σ2. Since D(2) + α /π =2+ α /(2π) > 0, this is sufficient to make the corresponding sum over h h ′ convergent. It v − v follows that the sum over all trees with n 3 and no vertex with nv = 2 and Q = 0 is finite, uniformly in h, if α2 < 16≥ π/3. A similar argument can be used to control the vertices with nv = 2 and Q = 0. In fact, if v is a vertex with nv = 2, then v′ is certainly a n.t. vertex, otherwise n would be equal to 2 and we are supposing n 3. ≥ Hence, we can use Lemma 2.4 in (43) for the vertex v′, which allows us to hv′ improve the bound of (43) for the vertex v: since γ x1 x2 Ghv (v1,v2) h (hv h ′ ) γ v x1 x2 /2 | − | | | ≤ Cγ− − v e− | − | , if v1 and v2 are the two endpoints following v, we can modify the bound (47) by adding 1 to the dimension D(nv) of v; this is sufficient, since D(2)+1 = 3 α2/(2π) is positive for α2 < 6π. It follows n − 2 that Vτ (σ, x) Cα holds for all n 3, with C(α) as α (16π/3)−. By a| further| effort, ≤ one could prove≥ that C(α) can→ be ∞ substituted→ with a new constant, which is indeed finite up to 6π, but we do not need here this stronger property.

~~14 v1 v~~

~~v′ v2~~

~~hv′ . . . hv hv + 1~~

Figure 1: A subtree of τ with nv = 2 and Q = 0. While v1 and v2 are endpoints, therefore their scale has to be hv + 1, v′ is the higher non trivial vertex of τ preceding v, hence the only constraint is that h h ′ N. v − v ≤

~~Let us now come back to the terms of order two. It is easy to see that~~

~~2 (N) ζ (N) V2 (ϕ)= V2,σ (ϕ) (55) 2 σ= 1 X±~~

(N) (N) V2,σ (ϕ)= dxdy cos[αϕx + σαϕy]W (x y) (56) 2 σ ZΛ − N α2 2 2 j−1 (N) 1 (j 1) σα C (x y) α [Cr(0)+σCr(x y)] W (x y)= γ 2π − e− j − 1 e− r=0 − (57) σ − 2 − jX=0 h i P (N) By proceeding as in Lemma 2.3, it is easy to see that V2,+ (ϕ) is bounded (N) uniformly in N for any α. This is not true for V2, (ϕ); in fact, if we define − (N) cN = h, 1(V2, (ϕ)) (58) E − − 2 one can easily check that cN diverges for N and that ζ cN /2 is equal 2 → ∞ to the term of order ζ and σ1 = σ2 in the perturbative expansion of the − ζ2 cN pressure. However, if we define Zh,N (ζ)= Zh,N (ζ)e− 2 , we can show that the renormalized pressure (in presence of the cutoffs) ph,N (ζ) = log Zh,N (ζ) has a power expansion uniformlye convergent as h,N , for α2 < 16π/3 (the result is indeed true for α2 < 6π). − →e ∞ e It is easy to see that

α2h ∞ 4π n (h) ph,N (ζ)= ζγ + ζ pτ (59) n=2 (N) X τ Xn e ∈T where (N) is a family of trees deﬁned as (N), withe the following diﬀerences: Tn Tn 1) thee root has scale 2; − 15 2) there is no tree which has only two endpoint with opposite charge.

Moreover (h) 1 (h) p = dx1 dxn V (σ, x) (60) τ n n τ 2 σ ,...,σ Λ ··· 1X n Z n e 2 α2 α Ghv (v1,...,vsv ) U (v) (h,N) 4π (hi+1) 2 h ′ ,hv Vτ (σ, x)= γ e− v (61) Ãi=1 ! n.t. v τ sv! Y Y∈ e e where G (v ,...,v ) = G (v ,...,v ), if h 0, while, if h = 1 and hv 1 sv hv 1 sv v ≥ v − sv = s, e T G 1(v1,...,vs)= h, 1 [F (ϕ, Xv1 ); . . . ; F (ϕ, Xvs )] (62) − E − with, given a clustere X,

cos[αΦ(X)] 1 , if X = 2 and σ = σ F (ϕ, X)= − | | 1 − 2 (63) ( cos[αΦ(X)] , otherwise where we subtracted a 1 in the terms with X = 2 and σ1 = σ2 (without changing the value of the− truncated expectation,| | since s 2),− in order to improve the bound in the corresponding vertex, with an argume≥ nt similar to that used before. In fact, in order to bound (62), we shall use the deﬁnition (30) and the bound

m (i) (i) 2 h, 1 F (ϕ, Xi) x1 x2 ¯E − " #¯ ≤  | − |  · ¯ i=1 ¯ i: Xi =2 ¯ Y ¯ | Y| ¯ α2 m2 ¯   ¯ sup ¯ ∂ϕ 2 ∂ϕ 2 (64) h, 1 y1 ym2 · Ã 2 ! y1,...,ym Λ E − | | · · · | | 2 ∈ h i where m2 m is the number of clusters with 2 endpoints and, for each cluster ≤ (i) (i) 2 def 2 of this type, x1 and x2 are the two endpoint positions; ∂ϕy = (∂0ϕy) + 2 | | (∂1ϕy) . On the other hand, it is easy to see that there is a constant c0, independent of h, such that h, 1 [∂a1 ϕx1 ∂a2 ϕx2 ] c0. It follows, by using − |E 2 2| ≤ q q q the Wick Theorem, that h, 1 ∂ϕy1 ∂ϕyq 2 c0(2q 1)!! C q!, E − | | · · · | | ≤ − ≤ so that, if we choose C 1 (whichh allows us to substitutei m2 with m) and use (30), we obtain the following≥ bound

(i) (i) 2 G 1(v1,...,vs) x1 x2 (65) | − | ≤  | − |  · i: Xv =2 | Yi | e   s 1 s! k (k 1)! (Cmr m !) k! m ! m ! r · k=1 − m1,...,mk≥1 1 k r=1 X k X ··· Y m =s r=1 r P 16 s s 1 s 1 s 1 s The sum in the second line is equal to C s! k=1 − 2 − C s!, so k Ã k 1 ! ≤ that P −

s (i) (i) 2 G 1(v1,...,vs) C s! x1 x2 (66) | − | ≤  | − |  i: Xv =2 | Yi | e   (i) (i) 2 The factors x1 x2 can be used control the sum over the scale labels of the vertices| with− X| = 2, by the same argument used in the discus- | vi | sion following (54). Hence, if we deﬁne the function Wn,h,N (x) so that (h) τ (N) pτ = Λn dx1 dxn Wn,h,N (x), the previous arguments imply that ∈Tn ··· thereP exist positiveR functions fτ (x), independent of h and N, and a constant C, suche that

def n Wn,h,N (x) fτ (x) = Hn,N (x) , dx Hn,N (x) C (67) | | ≤ Λn ≤ τ (N) Z ∈XTn Since (N) (N+1), H (x) is monotone in N. Hence, by the Monotone n n e n,N T ⊂ T 1 Convergence Theorem, Hn,N (x) has a L limit Hn(x), as N ; by (67), W e(x) He (x). On the other hand, by deﬁnition we have→ ∞ | n,h,N | ≤ n 1 1 T iασ1ϕx1 iασnϕxn Wn,h,N (x)= n h,N :e :; . . . ;:e : (68) n! 2 σ ,...,σ E 1X n h i and Lemma 2.2, (28) and (30) imply that Wn,h,N (x) is almost everywhere convergent as h,N . Then, by the Dominated Convergence Theorem, − → ∞ (59) and (67), p(ζ) = lim h,N ph,N (ζ) does exist and is an analytic func- − →∞ n tion of ζ, for ζ small enough; moreover, p(ζ)= n∞=2 pnζ and, if n 3, by e e ≥ (32) P 2 e α n Q=0 c− 8π 1 pn = dx1 dxn (69) n n n! 2 σ ,...,σ Λ ··· · 1X n Z

~~2.3 The case r = 0 (the charge correlation functions)~~

Let ξi = (zi, σi), i = 1,...,k, a family of ﬁxed positions and charges, such that z = z for i = j and µ , i = 1,...,k, a set of real numbers. If i 6 j 6 i k iασrϕ(zr) ζN V (ϕ)+ µr:e : Zh,N (ζ, ξ, µ)= P[h,N](dϕ)e r=1 (70) Z P 17 the charge correlation function of order k, k 1, deﬁned by (19), is given by ≥ k (k,ζ) ∂ Kh,N (z, σ)= log Zh,N (ζ, ξ, µ) (71) ∂µ ∂µ ¯ 1 k ¯µ=0 ··· ¯ ¯ By proceeding as in Sect. 2.2, one can show that ¯

V (N)(ζ,ϕ)+B(N)(ζ,ϕ,ξ,µ)+R(N)(ζ,ϕ,ξ,µ) Zh,N (ζ, ξ, µ)= P[h, 1](dϕ)e eff (72) Z − (N) n (N) (N) where Veff (ζ, ϕ)= ζV (ϕ)+ n∞=2 ζ Vn (ϕ), B (ζ, ϕ, ξ, µ) is the sum over (N) the terms of order at most 1P in each of the µr, and R (ζ, ϕ, ξ, µ) is the rest. (71) implies that

k (k,ζ) ∂ Kh,N (z, σ)= log Zh,N (ζ, ξ, µ) (73) ∂µ ∂µ ¯ 1 k ¯µ=0 ··· ¯ e ¯ where ¯ (N) 2 (N) V (ζ,ϕ) ζ cN /2+B (ζ,ϕ,ξ,µ) Zh,N (ζ, ξ, µ)= Ph, 1(dϕ) e eff − (74) − Z In ordere to describe the functional B(N)(ζ, ϕ, ξ, µ), we need to introduce (N) a new definition. We shall call n,k the family of labelled trees whose prop- T (N) erties are very similar to those of n , with the only difference that there are n + m endpoints, n 0, 1 mT k; n endpoints, to be called normal, are associated as before≥ to the≤ interaction,≤ while the others, to be called special, are associated with m different variables ξi, whose set of indices we shall denote Iτ , while ξτ will denote the set of variables itself. It is easy to see that (N) ∞ n (N) B (ζ, ϕ, ξ, µ)= ζ Bn (ϕ, ξ, µ) (75) nX=0

k 1 (N) iασϕzr B (ϕ, ξ, µ)= δn,0 µr:e :+ dx1 dxn n n n (N) 2 ′ ′ Λ ··· · r=1 τ∈T σ ,...,σn Z X Xn,k 1X n+m≥2 n ′ iα[ σrϕxr + σsϕzs ] µ e r=1 s∈Iτ V (σ′, x, ξ ) (76) · s τ τ s Iτ Y∈ P P where Vτ (σ′, x, ξτ ) is deﬁned exactly as in (41), with (σ′, στ , x, zτ ) in place of (σ, x). One can easily check that, if α2 4π, the terms with n = k = 1 and ≥ σ1 = σ¯1 in the r.h.s. of (76) have a divergent bound as N . This is − (1,ζ) → ∞ related to the fact that the function Kh,N (z; σ) is indeed divergent at the ﬁrst order in ζ. However, if we regularize these terms by subtracting their

18 (k,ζ) value at ϕ = 0, the counterterms give no contribution to Kh,N (z; σ), for k 2. Hence, we can proceed as in the bound of the pressure and we get similar≥ results. There are however a few diﬀerences to discuss. Given a tree τ (with root of scale 2) contributing to K(k,ζ)(z; σ), we − h,N call τ ∗ the tree which is obtained from τ by erasing all the vertices which are not needed to connect the m k special endpoints. The endpoints of τ ∗ ≤ are the m special endpoints of τ, which we denote ei∗, i = 1,...,m. Given a vertex v τ ∗, we shall call z the subset of the positions associated with ∈ v the endpoints following v in τ ∗; moreover, we shall call sv∗ the number of branches following v in τ ∗. The positions in zv are connected in our bound by a spanning tree of propagators of scales j h ; hence, if we use the bound ≥ v h 2γh x γh x c γj x ∞ j/2 e− | | e− | | e− j=0 | | , c = γ− (77) ≤ · P jX=0 and deﬁne δ = min1 i

∗ h 2hv(s 1) cγ v δ 2(m 1) γ v− e− (cδ)− − (2m 2)! (78) v τ ∗ ≤ − Y∈ where we used the identity v τ (sv∗ 1) = m 1. Since m k, the sum ∈ ∗ − − ≤ over the scale labels can beP done exactly as in the pressure case, up to a Ck(2k)! overall factor. There is another diﬀerence to analyze, related with the fact that, in the analogue of (62), the function F (ϕ, Xv) corresponding to a cluster with two endpoints of opposite charge, one normal and one special, is bounded 2 2 by x z supy ∂ϕy , rather than bounded by x z supy ∂ϕy . The fact| that− | the zero| in| the positions is of order| one− has| no| consequ| ence, since such a zero is suﬃcient to regularize the bound over a cluster with two endpoints of opposite charges. The fact that ∂ϕy appears, instead of its square, is also irrelevant, since the only consequence| | is that, in the 2 2 bound analogue to (64), one has to substitute h, 1 ∂ϕy1 ∂ϕym with E − | | · · · | 2 | h i h, 1 ∂ϕy1 ∂ϕym′ , with m2′ 2m2. However, by Schwartz inequality, E − | | · · · | 2 | ≤ · ¸ 2 2 h, 1[ ∂ϕy1 ∂ϕym′ ] h, 1[ ∂ϕy1 ∂ϕym′ ] and we can still use E − | | · · · | 2 | ≤ E − | | · · · | 2 | the Wick Theorem to get anq even better bound. (k,ζ) The previous arguments allow us to prove that K (z; σ) = lim h,N + (k,z) − → ∞ Kh,N (z, σ) does exist and is an analytic function of ζ around ζ = 0, with a

19 radius of convergence independent of δ (the minimum distance between two points in z). On the other hand, it is easy to check the well known identity n (k,ζ) ∞ ζ 1 Kh,N (z; σ)= n dx1 dxn n! 2 ′ ··· nX=0 Xσ Z T iασ ϕ iασ ϕ iασ′ ϕ iασ′ ϕ :e 1 z1:; . . . ;:e k zk:;:e 1 x1:; . . . ;:e n xn: (79) Eh,N An argument similarh to that used at the end of 2.2 allows usi to prove that the power expansion of K(k,ζ)(z; σ) is obtained by§ the previous equation, by T T substituting in the r.h.s h,N with . Hence, by using (32), we get that (k,ζ) n E E K (z; σ)= n∞=0 ζ gk,n(z; σ), with 2 P α (n+k) c− 8π 1 gk,n(z; σ)= dx1 dxn (80) n n n! 2 σ′ ,...,σ′ Λ ··· · 1 n Z n Xk σ′+ σ =0 i=1 i r=1 r P P α2 Π 1 σ¯rσ¯s  ( 1)| |− ( Π 1)! yr ys 2π   Π − | | − Y Π r,s∈Y | − |  X Y∈ r

2.4 The case r > 0 (the ∂ϕ correlation functions) Let y = (y ,..., y ) a set of k 1 distinct ﬁxed positions, ν = (ν , . . . , ν ) 1 k ≥ 1 k a set of derivative indices and µ =(µ1, . . . , µk) a set of real numbers. If

k νr ζN V (ϕ)+ µr∂ ϕ(yr) Zh,N (ζ, y, ν, µ)= P[h,N](dϕ)e r=1 (81) Z P the ∂ϕ correlation function of order k, k 1, is given by ≥ k (k,ζ) ∂ Kh,N (y; ν)= log Zh,N (ζ, y, ν, µ) (82) ∂µ ∂µ ¯ 1 k ¯µ=0 ··· ¯ ¯ We can proceed as in 2.3 and we can represent K(k,ζ¯)(y; ν) as in (73), § h,N that is we can substitute in (82) Zh,N (ζ, y, ν, µ) with

(N) 2 (N) V (ζ,ϕ) 2ζ cN +B (ζ,ϕ,y,ν,µ) Zh,N (ζ, y, ν, µ)= P[h, 1](dϕ)e eff − (83) Z − e (N) n (N) It is not hard to see that B (ζ, ϕ, y, ν, µ)= n∞=0 ζ Bn (ϕ, y, ν, µ), with k P (N) νr 1 B (ϕ, y, ν, µ)= δn,0 µr∂ ϕ(yr)+ dx1 dxn n n n (N) 2 σ ,...,σ Λ ··· · r=1 τ∈T 1 n Z X Xn,k X n+m≥2 n iα σ ϕ(x ) e r=1 r r V (σ, ν, x, y ) µ (84) · τ τ r P rYIτ e ∈ 20 (N) where n,k is defined exactly as in (76), except for the fact that the m specialT endpoints (1 m k) are associated with the ∂ϕ terms; moreover ≤ ≤ Vτ (σ, x, yτ ) is defined in a way similar to Vτ (σ, x, ξτ ), but, before giving its expression, we need a few new definitions. e If v is a non trivial vertex, we shall call Iv Iτ the set of special endpoints immediately following v (that is the set of ∂ϕ⊂endpoints which are contracted in v),s ¯v the number of vertices immediately following v, which are not special endpoints, and s∗ = I (hence s =s ¯ + s∗). Moreover, we shall use X v | v| v v v v to denote the set of normal endpoints (instead of all endpoints) following v. Then we can write

n 2 α2 α h Ghv (v1,...,vsv ) U (v) 4π ei − 2 hv′ ,hv Vτ (σ, ν, x, yτ )= γ e (85) Ãi=1 ! n.t. v τ sv! Y Y∈ e e T ν with G (v ,...,v ) = [F (ϕ); . . . ; F (ϕ)], where F (ϕ) = ∂ ϕy, if the j 1 s Ej v1 vs v vertex v is a special endpoint with position y and label ν, otherwise Fv(ϕ)= e exp(iαΦ(Xv)). We can always rearrange the order of the arguments so that the ﬁrst possibility happens for i = 1,...,m. If m = 0, we can use the identity (43), otherwise we can write

~~∂m Gj(v1,...,vs)= Hj(λ1, . . . , λm) (86) ∂λ ∂ ¯ 1 λm ¯λ=0 ··· ¯ e ¯ where ¯~~

~~T λ ∂ν1 ϕ λ ∂νm ϕ iαΦ(X ) iαΦ(X ) H (λ)= (e 1 y1 ; . . . ; e m ym ; e vm+1 ; . . . ; e vs ) (87) j Ej is a quantity which satisﬁes an identity similar to (43), that is~~

~~1 Uj (v,t,λ) Hj(λ)= ca,b dpT (t) e− 2 (88) T ¯s a,b T Z X∈T h Yi∈ e where~~

def 2 ca,b = α Wj(Xva , Xvb ) if a,b>m def def−  νa ca,b = λaca,b = iαλa r : er Xv σr (∂ Cj)(ya xr) if a m < b (89)  e ∈ b − ≤  def λ λ c = Pλ λ (∂νa ∂νb C )(y y ) if a, b m . aeb a,b a b j a b  − − ≤  It follows that e

1 Uj (v,t,0) Gj(v1,...,vs)= ca,b dpT (t)e− 2 (90) T ¯ ′ a,b T Z X∈Ts h Yi∈ e e e where ¯ ′ is the set of T ¯ , such that all special endpoints are leaves of T . Ts ∈ Ts Note that Uj(v,t, 0) is a positive quantity, since it is a convex combination

e 21 of “interaction energies” which do not involve the special vertices; hence we can safely bound the r.h.s. of (90), as in the previous sections. Let us deﬁne 1 bτ (y)= dx Vτ (σ, ν, x, y ) (91) n n τ 2 ν,σ Λ | | X Z e e The bound of bτ (y) diﬀers from the r.h.s. of (47) for the following reasons:

~~1) there is a eγki factor more, coming from the ﬁeld derivative, for the i- th special endpoint, if ki is the scale label of the higher n.t. vertex preceding it (the vertex where it is contracted);~~

~~∗ hv(s 1) 2) there is a factor γ v− more, which takes into account the fact that the special endpoints positions are not integrate, for each n.t. vertex~~

~~v such that sv∗ > 0;~~

~~3) if δ = min1 i 0, coming from the same argument used in the case of the charge correlation functions.~~

Hence, if m 1 is the number of special endpoints in τ, we get τ ≥ n mτ α2 n+mτ 4π hei 2hv(sv 1) 2εnv ki bτ (y) Cε γ γ− − e γ ≤ Ãi=1 !Ãn.t. v τ i=1 ! · Y Y∈ Y ∗ h e 2hv(s 1) cγ v δ γ v− e− (92) · ∗ n.t.vY:sv>0 The last product can be bounded as in (78), so that, by “distributing along the tree” the other factors, we get

n+k 2(mτ 1) (hv hv′ )(D(nv,mv) 2εnv) bτ (y) Cε (δ)− − (2mτ 2)! γ− − − (93) ≤ − n.t. v τ Y∈ e e where nv and mv denote the number of normal and special endpoints following v, respectively, and α2 D(n, m)=2(n 1) n + m (94) − − 4π f Let us consider ﬁrst the case α2 < 4π. Since n + m 2, D(n ,m ) v v ≥ v v is always positive, except if nv = 0 and mv = 2. However, no tree may have a non trivial vertex of this type, except the trees with only twof special (N) endpoints and no normal endpoint, that is the trees belonging to 0,2 , and it is very easy to see that T

N V (ν , ν , y , y )= γ2j (∂ν1 ∂ν2 C ) γj(y y ) (95) τ 1 2 1 2 − 0 1 − 2 τ X(N) jX=0 ³ ´ ∈T0,2 e 22 By (22), this quantity has a finite limit as N , if y1 = y2, as we are supposing. Hence there is no ultraviolet divergence→ ∞ in the expa6 nsion (84) of (N) Bn (ϕ, y, ν, µ) and we have only to check that there is no infrared problem related with the integration over the ϕ field in (83). This follows as in 2.2, by using the identity (30); it is sufficient to observe that §

m s m νi iαΦ(Xv ) 2 h, 1 (∂ ϕy ) e j h, 1 ∂ϕy (96) ¯ −  i ¯ v − i ¯E Ãi=1 ! j=1 ¯ ≤ uE "i=1 | | # ¯ Y Y ¯ u Y ¯  ¯ t ¯ ¯ and then¯ apply the arguments used in 2.2¯ to bound the sum over the partitions. § Let us now suppose that 4π α2 < 16π/3. In this case D(n ,m ) can be ≤ v v non positive only if either mv = 0 and nv =2 or mv = nv = 1. The vertices satisfying the first condition can be regularized as before, forf the others we 2 α α2 Uh ,h (v) (hv h ′ 1) can use the factor e− 2 v′ v = γ− 4π − v − to make their dimension positive; in fact D(1, 1) + α2/(4π) = 1. The integration over the ϕ field in (83) can now be done by an obvious modification of the argument used for the charge correlationf functions. (k,ζ) It is now easy to prove, as in the previous sections, that Kh,N (y; ν) has a finite limit, as h,N , if δ > 0, and that this limit is an analytic function of ζ around− ζ =→ 0, ∞ with a radius of convergence independent of δ. On the other hand,

n (k,ζ) ∞ ζ 1 Kh,N (y; ν)= n dx1 dxn n! 2 σ ··· nX=0 X Z T ν1 νk iασ1ϕx iασnϕx ∂ ϕy ; . . . ; ∂ ϕy ;:e 1:; . . . ;:e n: (97) Eh,N 1 k h i An argument similar to that used at the end of 2.2 allows us to prove that the power expansion of K(k,ζ)(y; ν) is obtained by§ the previous equation, by T T substituting in the r.h.s h,N with . Moreover, it is not hard to check that, if n > 0, Q n Eσ and h E(x, y; σ, ν) is the limiting value of the ≡ i=1 i k,n truncated expectationP in (97), we have

~~2 k α n hk,n(x, y; σ, ν)= δQ,0 c− 8π W (yr, x; νr, σ) (98) Ã ! · rY=1~~

~~P n n ν ν 1 iα (xi y) − W (y, x; σ, ν)= iα σi ∂ ∆ (y xi)= σi − 2 (99) − 2π y xi Xi=1 ³ ´ Xi=1 | − | 23 while, if n = 0,~~

T ν1 ν2 ν1,ν2 h (y, ν)= δ (∂ ϕy );(∂ ϕy ) = δ h (y y ) (100) k,0 k,2E 1 2 k,2 1 − 2 h i with 1 yν1 yν2 hν1,ν2 (y)= δν1,ν2 2 (101) 2π y 2 " − y 2 # | | | | (k,ζ) n¯ Hence, we get that K (y; ν)= n∞=0 ζ hk,n(y; ν), with 1 1 P h¯k,n(y; ν)= dx1 dxn hk,n(x, y; σ, ν) (102) n n n! 2 σ1,...,σn Λ ··· n Z Xσ =0 i=1 i P Note that h¯1,n(y, ν) = 0 for any n, since W (y, x, σ, ν) is odd in σ and the sum in (102) is restricted to the σ such that Q = 0; hence K(1,ζ)(y, ν) = 0.

~~3 The Thirring model with a ﬁnite volume mass term~~

~~The Generating Functional, h,N (J, A, µ), of the Thirring model with cutoff and with a mass term in finiteW volume is defined by the equation~~

def 2 (1) ¯ h,N (J, A, µ) = log Ph,N (dψ) exp λZN VL(ψ)+ µZN dx ψxψx + W − Λ Z n Z (1) σ ¯ σ ν ¯ ν +ZN dx Jx ψxΓ ψx + ZN dx Ax ψxγ ψx (103) σ= 1 Z ν=0,1 Z X± ³ ´ X ³ ´ o (1) where the free measure Ph,N (dψ) is defined by (14), ZN and ZN are defined σ µ in (15), Jz and Ay are two-dimensional, external bosonic fields and

5 2 def 1 ¯ µ 2 σ def I + σγ ZN VL(ψ) = dx ZN ψxγ ψx + Eh,N ΛL , Γ = (104) 4 ΛL | | 2 Z ³ ´ Eh,N being the vacuum counterterm introduced in (13); it is chosen so that (0, 0, 0) = 0. Wh,N Given the set of non coinciding points x = (x1,..., xq) and the set σ = (σ , . . . , σ ), σ = 1, we want to study the Schwinger functions 1 q i ± q (q,r;µ) def ∂ h,N Gh,N (x, y; σ, ν) = lim σ σWq ν ν (0, 0, µ) . (105) a−1,L ∂J 1 ∂Jx ∂A 1 ∂A r →∞ x1 ··· q y1 ··· yr Theorem 3.1 If µ and λ are small enough and q 2, if r = 0, or q 0, if r 1, the limit ≥ ≥ ≥ (q,r;ζ) def (q,r;ζ) G (z, w; σ, ν) = lim Gh,N (z, w; σ, ν) (106) h,N + − → ∞ 24 exists and is analytic in µ. In the case q = r = 0 (the pressure), the limit does exist and is analytic, up to a divergence in the second order term, present only for λ 0. ≤ As in 2, we shall give the proof of the above theorem only in the special cases (§q, r)=(k, 0) and (q, r)=(0, k) separately. In order to prove Theorem 3.1, we note first that definition (105) and the ¯ ¯ σ identity ψxψx = σ ψxΓ ψx imply that P p (q,r;µ) µ (2n,r) Gh,N (z, y; σ, ν)= dx χΛ(x)Sh,N (zx, y; σσ′, ν) (107) p=2n−q σ′ p! Z Xn≥0 X where zx =(z1,..., zq, x1,..., xp), σσ′ =(σ1, . . . , σq, σ1′ , . . . , σp′ ), we defined

def def χ (x) = χ (x ) χ (x ) , S(m,r)(x, y; σ, ν) = G(m,r;0)(x, y; σ, ν) . Λ Λ 1 ··· Λ p h,N h,N (108) (m,r;0) and we used the fact that Gh,N (x, y; σ, ν) can be diﬀerent from 0 only if m i=1 σi = 0, implying in particular that m is even. (m,r) P In the following we shall give a bound for the functions Sh,N (x, y; σ, ν), uniform in the cutoﬀs and implying (by an argument similar to that used for the Sine-Gordon model, that we shall skip here) that the limit exists, is integrable and is exchangeable with the integral in (107). It follows that

~~p (q,r;µ) µ (2n,r) G (z, y; σ, ν)= dx χΛ(x)S (zx, y; σσ′, ν) (109) p=2n−q σ′ p! Z Xn≥0 X~~

We remark that χΛ(x) is not a regular test function since it is not vanishing for coinciding points, and hence we could encounter divergences caused by the ultraviolet problem. Indeed, as we shall see, the integration of G(2,0;0) will be finite only for λ > 0 (and small in absolute value), so that the pressure G(0,0;µ) and the, if x Λ, “density” G(1,0;µ)(x, σ) are really divergent for λ 0, since this is true∈ for the terms with 2n = 2 and r = 0 in the r.h.s. of (109).≤ As announced in the introduction, we first consider the case q, r = 0 then we discuss the case q > 0 and r = 0; and finally the case q = 0 and r> 0.

3.1 Case q = r = 0 (the pressure) Our deﬁnitions imply that S(0,0) = 0. If m 2 and even (otherwise it is 0 by symmetry), the m-points Schwinger function≥ S(m,0)(x, σ) is obtained as the m-th order functional derivative of the generating function h,N (J, 0, 0) σ1 σm W with respect to Jx1 ,...,Jxm at J = 0. We can proceed as in [BFM] and we get an expansion similar to eq. (2.28) of that paper, which we refer to

25 for the notation. The only difference is that the special endpoints of type (1) ¯ σ (1) + J are associated with the terms Zj σ ψxΓ ψx = Zj σ ψx, σψx−,σ instead + − of Zj σ ψx,σψx−,σ, but this does notP change the structureP of the expansion; (1) we onlyP have to add, for each special endpoint of scale hi, a factor Zhi /Zhi , (1) which can be controlled by studying the flow of Zj . It turns out that there 2 are two constants η+(λ) = bλ + O(λ ), b > 0, and c+(λ)=1+ O(λ), such (1) η+j that, in the limit N , Zj = c+(λ)γ− ; this result is obtained by an argument similar to that→ ∞ used in [BFM] to prove that there are two constants η (λ) = aλ2 + O(λ3), a > 0, and c (λ)=1+ O(λ), such that, in the limit − − η−j N , Zj = c (λ)γ− (in [BFM] c (λ) = 1, since the definition of ZN − − differs→ ∞ by a constant chosen so to get this result). In analogy to eq. (2.40) of [BFM], we can write

~~N 1 (m,0) ∞ − S (x, σ)= m! lim S0,m,τ,σ(x) , (110) h ,N | | →∞ n=0 j0= τ 0,m P X X−∞ ∈TXj0,n X∈P~~

Given a tree τ contributing to the r.h.s. of (110), we call τ ∗ the tree which is obtained from τ by erasing all the vertices which are not needed to connect the m special endpoints (all of type J). The endpoints of τ ∗ are the m special endpoints of τ, which we denote vi∗, i = 1,...,m; with each of them a space- time point x and a label σ are associated. Given a vertex v τ ∗, we shall i i ∈ call uv the set of the space-time points associated with the normal endpoints of τ that follow v in τ (in [BFM] they were called internal points); xv will denote the subset of x made of all points associated with the endpoints of τ ∗ following v.

Furthermore, we shall call sv∗ the number of branches of τ ∗ following ,1 ,2 v τ ∗, sv∗ the number of branches containing only one endpoint and sv∗ = ∈ ,1 s∗ s∗ . For each n.t. vertex or endpoint v τ ∗, shortening the notation v − v ∈ of sv∗ into s, we choose one point in xv, let it be called wv, with the only constraint that, if v1,...,vs are the n.t. vertices or endpoints following v, then w is one among w w ,..., w . v v ≡ { v1 vs } The bound of S0,m,τ,σ(x) will be done as in [BFM], by comparing it with the bound of its integral over x, given by eq. (2.36) of that paper. However, we shall slightly modify the procedure, to get an estimate more convenient for our actual needs. Given the space-time points v = (v1,..., vp) connected by the tree T , we shall deﬁne, if v and v denote the endpoints of the line l T , l,i l,f ∈ def def DT (v) = T = vl,i vl,f (111) k k l T | − | X∈ q Now we want to show that, from the bounds of the propagators associated with the lines l of the spanning tree Tτ = v Tv, we can extract a factor S 26 ′ h c √γ v DC (w ) e− v v for each n.t. vertex v τ ∗, where C is a chain of segments ∈ v that only depends on τ and Tτ , and connects the space-time points wv. Indeed, given a n.t. v τ ∗, there is a subtree T ∗ of T connecting the ∈ v τ points w together with a subset of x u . Since T ∗ is made of lines of v v ∪ v v scale j hv, the decaying factors in the bounds of the propagator in Tv∗ can be written≥ as

h c√γh x c √γh x 2c′ √γj x e− | | = e− 2 | | e− j=−∞ | | , (112) · P j/2 for c′ = c/ 4 j∞=0 γ− . Hence, collecting the latter factor for each of the ′ hv ∗ h P 2c √i γ Tv lines Tv∗ we obtain e− k k. We ﬁnally would like to replace, in the previous bound, T ∗ with C , k v k k vk up to a constant, for a Cv which does not depend on the position of the internal points of Tv∗. This is possible as a consequence of the following lemma.

l Lemma 3.2 Let T be a tree graph connecting the points wj j=1 together q { } with other “internal points”, uj j=1. Then there exists a chain C connecting {l } all and only the points wj j=1 such that 2 T C and C only depends on T . { } k k ≥ k k q Proof. Suppose that the points uj j=1 are ﬁxed in an arbitrary way and let us consider the oriented closed path{ }¯ obtained by “circumnavigating” T , for example in the clockwise direction;C this path contains twice each branch of T , with both possible orientations. We shall call the oriented closed path obtained by continuous deformation of ¯, as the pointsC u q vary in R2. C { j}j=1 The path allows us to reorder the points w1,..., wl into wt(1),..., wt(l), by puttingCt(1) = 1 and by choosing t(i + 1), 1 i l 1, so that w ≤ ≤ − t(i+1) is the point following wt(i) on . The chain C is obtained by joining with a segment w and w , forCi = 1,...,l 1; the condition 2 T C t(i) t(i+1) − k k ≥ k k then easily follows from the triangle inequality for the function x x 1/2. → | | As a consequence of the above lemma and (112), we can extract from ′ hv c √γ DCv (w ) the propagator bounds, for each choice of Tv, a factor v τ ∗ e− v , ∈ which does not depend on the internal points positions,Q by leaving a factor (c/2)√γj x y e− | − | for each propagator of Tτ , to be used for bounding the integral over the internal points. The ﬁnal bound of S0,m,τ,σ(x) will be obtained by “undoing”, in the r.h.s of eq. (2.36) of [BFM], the sum over T for any v τ ∗ (note that C depends v ∈ v on Tv∗ and hence on Tv), then adding the factors coming from the previous considerations, together with a factor taking into account that there are

1+ v τ ∗ (sv∗ 1) = m integrations less to do. By suitably choosing them, the ∈ − P 27 ∗ 2hv(sv 1) 2 lacking integrations produce in the bound an extra factor v τ ∗ γ − L− , ∈ so that we get Q

Pv /2 | | m n j0( 2+m) Zhv dv S (x) C (Cλ¯ ) γ− − γ− 0,m,τ,σ j0   | | ≤ v not e.p. ÃZhv 1 ! · Y − m (1)   Z 1 ∗ ′ h hi 2hv(s 1) c √γ v DC (w ) γ v− e− v v (113)     i=1 Zhi n.t.v τ ∗ sv! T Y Y∈ Xv     where hi is the scale of the i-th endpoint of type J and

def d = 2+ m + P /2+ z , (114) v − v | v| v with mv = Xv and zv equal to the parameter z(Pv) defined by eq. (2.38) of [BFM]. | | e We have now to bound the integral of S0,m,τ,σ(x)χΛ(x), let us call it Im,τ,σ. In order to exploit the improvement related with the restriction of the integration variables to a fixed volume of size 1, we shall proceed in a way different with respect to that followed in [BFM], that is we bound the integral before the sums over the trees Tv. We use the bound: 2h c′√γh x y γ− if h> 0 dx χ (x)e− | − | C (115) Λ ≤ 1 if h 0 Z ( ≤ The sum over the tree graphs is done in the usual way and we get

m (1) ∗ Z m n j0( 2+m) 2hv(s 1) hi I C (Cλ¯ ) γ− − γ v− m,τ,σ j0   ≤ Ãn.t.v τ ∗ ! i=1 Zhi · Y∈ Y Pv /2 hv>0   | | ∗ Zhv dv 2hv(s 1) γ− γ− v− (116)     · v not e.p. ÃZhv 1 ! n.t.v τ ∗ Y − Y∈     Let us now call Ei the family of trivial vertices belonging to the branch of τ ∗ which connects vi∗ with the higher non trivial vertex of τ ∗ preceding (1) hiη¯ it and note that, by the remark preceding (110), Zhi /Zhi Cγ− , with ¯2 ,1 ≤ η¯ = c0λ + O(λj0 ), c0 > 0. Hence, the deﬁnition of sv∗ implies that, if E = iEi, ∪ m (1) Z ∗,1 hi m η¯ hvηs¯ C γ− γ− v . (117) i=1 Zhi ≤ Ãv E ! n.t.v τ ∗ Y Y∈ Y∈ Let v0∗ be the ﬁrst vertex with sv∗ 2 following v0 (recall that v0 is the vertex immediately following the root≥ of τ, of scale j + 1); since m 2, this 0 ≥ vertex is certainly present. Then, since mv = m for v0 v v0∗, we have the identity ≤ ≤

~~j0( 2+m) dv hv∗ ( 2+mv∗ ) dv γ− − γ− = γ− 0 − 0 γ− , (118) ∗ ∗ v0 vv . v ∗ e0 ≥By inserting (117)∈ and (118) in the r.h.s. of (116), we get e~~

m n dv dv η¯ dv I C (Cλ¯ ) γ− γ− − γ− m,τ,σ j0     ≤ v∈ /τ∗ " # ∗ · v E v0 v0 | | ∗ Zhv 2hv(s 1) γ− v− F , (119)     τ · v not e.p. ÃZhv 1 ! n.t.v τ ∗ Y − Y∈ where     ∗ v v0 h ∗ ( 2+m ∗ ) ∗ ∗,1 ≥ v v hv[2(sv 1) ηs¯ v ] dv F = γ− 0 − 0 γ − − γ− . (120) τ   "n.t.v τ ∗ # v (τ ∗ E) Y∈ ∈ Y\ ,1   Given a n.t. vertex v τ ∗, let s = sv∗, s1 = sv∗ , s2 = s s1 and v1,...,vs2 ∈ − s2 the n.t. vertices immediately following v in τ ∗. Since mv = s1 + i=1 mvi , we can write P s2 ( 2+ m ) + [2(s 1) ηs¯ ]= s (1 η¯) ( 2+ m ) . (121) − − v − − 1 1 − − − vi Xi=1

This identity, applied to the vertex v0∗, implies that, if v1,...,vs2 , s2 = ,1 sv∗∗ sv∗∗ , are the n.t. vertices immediately following v0∗ in τ ∗, then 0 − 0 ∗,1 h ∗ 2(s∗ 1) ηs¯ hv∗ ( 2+mv∗ ) v v∗ v∗ γ− 0 − 0 γ 0 0 − − 0 = h i s ′ 2 α v∗ hv∗ hv ( 2+mv ) 2+mv = γ 0 0 γ− i − i γ− , (122)   i=1 · v Y Y∈Ci   where i is the path connecting v0∗ with vi in τ ∗ (not including vi) and we used theC deﬁnition ,1 α′ = s∗ (1 η¯) . (123) v v − hv ( 2+mv ) The presence of the factor γ− i − i for each vertex vi in the r.h.s. of (122) implies that an identity similar to (122) can be used for each n.t. vertex v τ ∗. It is then easy to show that ∈ ∗ v v0 ′ ≥ α vhv dv F = γ γ− . (124) τ   "n.t.v τ ∗ # v (τ ∗ E) Y∈ ∈ Y\  e  By inserting this equation in (119), we get

hv>0 ′ ∗ m n α vhv 2hv(s 1) I C (Cλ¯ ) γ γ− v− m,τ,σ j0   ≤ Ãn.t.v τ ∗ ! n.t.v τ ∗ · Y∈ Y∈ Pv /2   | | ′ Zhv d v γ− , (125)   · v not e.p. ÃZhv 1 ! Y −   29 where d if v τ ∗, v v / E v ∈ 0 ≤ ∈ d′v =  dv +η ¯ if v E (126)  e ∈  dv otherwise and it is always strictly greater than zero, ifη ¯ (hence λ) is small enough. This allows us to control the sum over the τ scale labels in the usual way, by keeping ﬁxed h ∗ . However, before doing that, it is necessary to extract v0 from the r.h.s. of (125) some factors needed to control the sum over h ∗ too. v0 First of all, in order to control the sum over the negative values of h ∗ , we try v0 to replace the non-negative quantity α′v with another non-negative one, αv, α ∗ s.t. v0 is strictly positive, paying a price in the dimension of the vertices; this can be easily achieved by ﬁxing ε> 0 and using the inequality

~~∗,2 εhv∗ ε(s 1)hv ε 1 γ 0 γ v − γ (127) ≤ Ãn.t.v τ ∗ ! v τ ∗,v /E Y∈ ∈ Y∈ which allows us to replace (125) with~~

hv>0 ∗ m n εhv∗ αvhv 2hv(s 1) I C (Cλ¯ ) γ 0 γ γ− v− m,τ,σ j0   ≤ Ãn.t.v τ ∗ ! n.t.v τ ∗ · Y∈ Y∈ Pv /2   | | Zhv d¯v γ− , (128)   · v not e.p. ÃZhv 1 ! Y −   ,2 ¯ ¯ with αv = α′v + ε(sv∗ 1) and dv = dv′ ε, if v0∗ v E; and dv = dv′ otherwise. − − ≤ 6∈ Let us now deﬁne χ = 1 if h > 0 and χ = 0 for h 0. If we put v v v v ≤ w = v0∗, we can write

hv>0 ∗ εhv∗ αvhv 2hv(s 1) γ 0 γ γ− v− = (129)   "n.t.v τ ∗ # n.t.v τ ∗ Y∈ Y∈ ∗  ∗  [αw+ε 2χw(s 1)]hw [αv 2χv(s 1)]hv = γ − w− γ − v− n.t.v∈τ∗ vY6=w and, if λ << ε < 1/2, we use the two straightforward inequalities | | | | ,1 αv = ε(sv∗ 1) + sv∗ (1 η¯ ε) ε − − −,2 ≥ α 2(s∗ 1) = (2 ε) (1 +η ¯)s∗ s∗ (1 ε η¯) v − v − − − v − v − − (2 ε) (1 +η ¯)s∗ < 0 (130) ≤ − − v Hence, the two terms in square brackets in the r.h.s. of (129) can be bounded by cm, irrespective of the sign of η¯, that is the sign of λ.

~~30 Thanks to these arguments, we can replace (128) with~~

Pv /2 ∗ | | m n [αw+ε 2χw(s 1)]hw Zhv d¯v I C (Cλ¯ ) γ − w− γ− (131) m,τ,σ j0   ≤ v not e.p. ÃZhv 1 ! Y −   Now the sum over the tree scale labels, as well as the sum over the trees with a ﬁxed value of hw, can be performed in the usual way, using the fact that d¯v is always strictly positive and bounded below by a quantity proportional m to d¯v itself; this gives a C bound. We ﬁnally have to bound the sum over hw; note that

~~+ ∞ ∗ ∗ [αw+ε 2χw(s 1)]hw [αw+ε 2(s 1)]h [αw+ε]h γ − w− = γ − w− + γ (132) hw= h>0 h 0 X−∞ X X≤~~

~~The second sum is always ﬁnite since αw + ε εsw∗ 2ε. Regarding the ﬁrst sum, we note that ≥ ≥~~

,2 α + ε 2(s∗ 1) = 2 (1 +η ¯)s∗ s∗ (1 ε η¯) w − w − − w − w − − 2 (1 +η ¯)s∗ (133) ≤ − w Hence, the sum is always bounded, except in the caseη ¯ 0 (that is λ 0) ,1 ≤ ≤ with sw∗ = sw∗ = 2. It follows, by (110), that

(m) m dx χΛ(z) S (z, σ) m!C , m 3 (134) Z | | ≤ ≥ so that, by (109), the pressure can be deﬁned only by subtracting from G(0,µ) the term with m = 2. The renormalized pressure is analytic in µ, for µ small enough.

3.2 Case q 2, r = 0 ≥ (m) We have for S (zx; σ, σ′) an expansion analogous to (110), but now the special endpoints are associated with two different types of space-time points, those which have to be integrated as before (x) and those which are fixed (z). We denote by xv and zv the points following v of the two types and we slightly modify the definition of the point w to be one point in z , if z = , v v v 6 ∅ or one point in xv, otherwise; we still require that wv wv. We want to mimic the strategy used for the Sine-Gordon∈ correlations o functions. Therefore we introduce a new tree τ , that is obtained from τ ∗ by erasing all the vertices which are not needed to connect the q special endpoints carrying a space-time point of type z. (We remark that the roles o of the trees τ and τ ∗ of the bosonic theory here are played by τ ∗ and τ o respectively). Correspondingly, we define sv the number of the branches

31 of τ o following v τ o; mote that the space-time points associated with the o ∈ o o endpoints of τ following v τ are those in zv, hence w v(sw 1) = zv 1. ∈ ≥ − | |− A bound similar to (113) holds. In this case, anyway,P we prefer to have a separate decaying factor in the distance of the points z: for each nontrivial vertex v of τ o 1 1 DCv (wv) D (zv wv)+ DCv (wv) . (135) ≥ 2 Cv ∩ 2 where C denotes the ordered pathe connecting the points in (z w ), made v v ∩ v of lines which connect a point with that following it in the ordered path Cv, see Lemmae 3.2. Therefore, in place of (113), we have:

Pv /2 | | m n j0( 2+m) Zhv dv S (z, x) C (Cλ¯ ) γ− − γ− 0,m,τ,σ j0   | | ≤ v not e.p. ÃZhv 1 ! · Y − m (1)   Z ∗ hv hi 1 2hv(s 1) c′ γ v− exp γ 2 D (w )   Cv v · i=1 Zhi n.t.v τ ∗ sv! T (− 2 ) · Y Y∈ Xv   c′ hv exp γ 2 D (z w ) (136) Cv v v · n.t.v τ o (− 2 ∩ ) Y∈ We can repeat, with no essentiale modiﬁcation, the steps that from (113) have led to (116). Hence, if we call Im,τ,σ(z) the integral over x of S0,m,τ,σ(zx)χΛ(x), we get the bound:

∗ m ¯ n j0( 2+m) 2hv(sv 1) Im,τ,σ(z) C (Cλj0 ) γ− − γ − ≤ Ãn.t.v τ ∗ ! · Y∈ m (1) Pv /2 hv>0 Z | | ∗ hi Zhv dv 2hv(s 1) γ− γ− v−       · i=1 Zhi v not e.p. ÃZhv 1 ! n.t.v τ ∗ · Y Y − Y∈  hv>0      o c′ hv 2hv(sv 1) γ − exp γ 2 D (zv wv) (137)   Cv · n.t.v τ o n.t.v τ o (− 2 ∩ ) Y∈ Y∈   e Indeed, we observe that the chain Cv is a spanning tree of propagators with root in one of the zv points (if any, see the deﬁnition of wv). Hence, inte- grating down the position of the vertices xv from the endpoints of such a tree to the root, in the case at hand there are, with respect to the procedure o for q = 0, sv 1 missing integration for each nontrivial vertex v of the tree o o − 2hv(s 1) τ . By (115), this means a factor γ− v− less, if hv > 0, and a constant factor less, if hv 0; this explains the last line of (137). Going on in parallel with 3.1, we obtain≤ the analogous of (131); recalling that w is the lowest §

~~32 n.t. vertex of the tree τ ∗,~~

Pv /2 ∗ | | m n [αw+ε 2χw(s 1)]hw Zhv d¯v I (z) C (Cλ¯ ) γ − w− γ− m,τ,σ j0   ≤ v not e.p. ÃZhv 1 ! · Y − hv>0   o c′ hv 2hv(sv 1) γ − exp γ 2 D (zv wv) (138)   Cv · n.t.v τ o n.t.v τ o (− 2 ∩ ) Y∈ Y∈   At this point, in contrast with the pressure bound,e we want to take advantage of the exponential fall oﬀ in the diameter of zv wv to prove the convergence of the correlations (with q 2) for any sign of∩η¯. ≥ Note that our deﬁnitions imply that n.t.v τ 0 zv wv = z and that ∪ ∈ ∩ n.t.v τ 0 Cv is a tree connecting all the points in z. This remark, together ∪ ∈ with the trivial bound hv hv∗ , implies that e ≥ 0

h c′ v c′ hv∗ exp γ 2 D (z w ) exp γ 0 diam(z) (139) Cv v v n.t.v τ o (− 4 ∩ ) ≤ (− 4 q ) Y∈ e 0 On the other hand, since zv wv 2 for any n.t. v τ , if we deﬁne def | ∩ | ≥ ∈ δ = min z z , we have i,j | i − j| 0 2(sv 1) o hv − 0 2hv(s 1) c′ C 0 4(s 1) γ v− exp γ 2 D (z w ) (s 1) v− (140) Cv v v v (− 4 ∩ ) ≤ µ δ ¶ − e 0 so that, by using also the identity v τ 0 (sv 1) = q 1, ∈ − −

hv>0 P o c 2hv(s 1) ′ h γ v− exp γ v diam(z w )   v v n.t.v τ o n.t.v τ o (− 2 ∩ ) Y∈ Y∈ q   2(q 1) 4 C − c′ hv∗ [(q 1)!] exp γ 0 diam(z) (141) ≤ − µ δ ¶ (− 4 q ) Let us now remark that the quantity

+ ∞ ∗ (αw+ε) [αw+ε 2χw(sw 1)]h h 1 + diam(z) γ − − exp c0 γ diam(z) (142) h= ½− ¾ h i X−∞ q is bounded by a constant. In fact, the series is convergent also without the exponential, as shown before, and this is suﬃcient, if diam(z) 1; if diam(z)= ≤ h0 + (αw+ε)h h γ− , h0 0, we can bound the series by 2 h=∞ γ exp[ c0 γ ], ≤ −∞ − which is convergent, sice αw + ε 2ε. Hence weP get, by using 2ε aw +qε q(1 + ε η¯), that there is a constant≥ C , such that ≤ ≤ − q

~~(m) 2(q 1) Cq dx χ (x) S (z, x, σ) m! 1+ δ− − (143) Λ | | ≤ 1 + diam(z)2ε Z ³ ´~~

~~33 3.3 Case r 1, q = 0 ≥ This case is very similar to the previous one; therefore we limit ourself to the discussion of the diﬀerences.~~

Formula (136) still holds, with yv in place of zv (to be consistent with no- m (1) p (1) tation in (109)) and with the replacement i=1(Zhi /Zhi ) i=1(Zhi /Zhi ), −→ ¯ µ following from the fact that the strengthQ renormalization ofQthe field ψγ ψ is equal to Zh. It is easy to go along the developments of 3.2 again, up to a couple of differences. The minor one is that in formulas§ (117) and (126) the set E has to be replaced with the set E Y , where Y is the family of \ trivial vertices of τ ∗ belonging to the branches ending up with an endpoint of type y; but this is not a problem, since the dimensions of all the vertices remain strictly positive. The major difference is that in (117), in the case ,1 ,1 ,1 ,1 at hand, there is hvη¯(sv∗ t∗v ) in place of hvηs¯ v∗ , if tv∗ is the number of branches departing from v−and ending up with one endpoint of type y (hence ,1 ,1 0 t∗ s∗ ). At the end of the developments, the latter fact generates a ≤ v ≤ v new αv, that we have to prove to be positive in order to control the bound in the vertices v = w such that hv 0 (as done in (130) for the old one). With simple computations6 we find: ≥

~~,1 ,1 ,1 α = ε(s∗ 1)+(s∗ t∗ )(1 η¯ ε)+ t∗ (1 ε) ε (144) v v − v − v − − v − ≥~~

Also, we need to prove that αv 2(sv∗ 1) is negative, in order to to control the bound in the vertices v = w−such that− h > 0; and indeed: 6 v ,1 ,1 ,2 α 2(s∗ 1) = (2 ε) s∗ (s∗ t∗ )¯η s∗ (1 ε) v − v − − − v − v − v − v − (2 ε) (1 η¯ )s∗ < 0 (145) ≤ − − − | | v Finally, the summation on the scale of w is controlled by the the exponential fall oﬀ in the diameter of y, as in (142).

~~4 Explicit expression of the coeﬃcients in the mass expansion and proof of Theorem 1.1~~

4.1 The case r = 0 As explained in the remark preceding (109), in order to get an explicit expression for the coefficients of the expansion (109), it is sufficient to calculate the correlations S(m,0)(x, σ). We now show how to get this result by com- puting the correlations of the ψ field at non coinciding points. We consider the following generating function

2 ¯(1) σ ¯ σ λZ V (ψ)+Z dxdyJx δε(x y)ψxΓ ψy N,ε(J) = lim log Ph,N (dψ)e− N N σ − (146) W h →−∞ Z P R 34 where δε(x) is a smooth approximation of the delta function, rotational invariant, whose support does not contain the point x = 0; for deﬁniteness we 2 1 1 will choose δε(x) = ε− v(ε− x ), v(ρ) being a function on R with support | | 1 in [1, 2], such that dρρv(ρ)=(2π)− (so that dxδε(x) = 1). We deﬁne

~~R m R (m) ∂ S¯ (x, σ)= N,ε(J) J=0 (147) N,ε σ1 σm ∂Jx1 ...∂Jxm W |~~

~~(m) while SN,0 (x, σ) will denote the analogous quantity with δε(x y) δ(x y). (m,0) (m) − → − Note that S (x, σ) = limN SN,0 (x, σ). →∞~~

Lemma 4.1 If λ is small enough, there exists a constant c1 = 1+ O(λ), (1) ¯ η+ such that, if we put ZN = c1ε , then, for any set x of m distinct points, ¯(m) (m) lim lim SN,ε (x, σ) = lim SN,0 (x, σ) (148) ε 0 N N → →∞ →∞ Dim. - The proof of the Lemma is based on a multiscale analysis of the functional N,ε(J), performed by using the techniques explained in sect. 2 of [BFM]. WeW shall not give here the detailed proof, but we shall stress only the relevant differences with respect to the case studied there. First of all, the external field ϕ is zero and the free measure has mass zero. Moreover the terms linear in J and quadratic in ψ contains the monomial + ¯ σ + ψx, σψy−,σ = ψxΓ ψy, instead of ψx,σψy−,σ. This difference is unimportant from− the point of view of the dimensional analysis, so that, in the case ε = 0, we can essentially repeat the analysis of [BFM] with obvious minor changes. The situation is different for ε> 0, since in this case these terms (which are marginal) are not local on the scale N, so that they need a more accurate discussion. (j) Let us call BJ (ψ) the contribution to the effective potential on scale j, [h,j]+ [h,j] which is linear in J and has as external fields ψx,ω and ψy, ω− and let hε hε − be the largest integer such that γ− ε and let N >h . We want to show ≥ ε that, if N j hε, this term, which is dimensionally marginal, is indeed irrelevant, so≥ there≥ is no need to localize it. This follows from the observation (j) that BJ (ψ) is of the form

(j) ¯(1) ω [h,j]+ [h,j] BJ (ψ)= ZN dxdyJz δε(x y)ψx, ω ψy,ω − + (149) ω − − X Z ω [h,j]+ [h,j] + dzJz dz¯dxdyδε(z z¯)Wj(z, z¯, x, y)ψx, ω ψy,ω − ω − − X Z Z where Wj(z, z¯, x, y) is the kernel of the sum over all graphs containing at least one λ vertex. It is easy to see that it is of the form

~~Wj(z, z¯, x, y)= Wj(z, x)Wj(z¯, y)+ W¯ j(z, z¯, x, y) (150)~~

f f 35 where the second term is given by the sum over the graphs which stay connected after cutting the line δε, while the first term is associated with the other graphs. The first term do not need a localization, even for j j, the other one of scale h f j. The momentumf conservation and the compact support prop- 2 ≤ erties of the single scale propagators imply that h1 = j + 1, so that there is no diverging sum associated with h1, as one could expect since the first term has a bound C λ . On the other hand, it easy to see that the second term satisfies the bound| |

2(j hε) dz¯dxdyδε(z z¯) W¯ j(z, z¯, x, y) C λ γ− − (151) Z − | | ≤ | | This immediately follows by comparing this bound with the analogous one for ε = 0, which is C λ for dimensional reasons. With respect to the case | | ε = 0, we have a new vertex z¯, which is linked to the graph by the line δε and a propagator of scale j′ > j. The bound (151) is obtained by using the decaying properties of this propagator to integrate over z¯ and by bounding 2 δε by Cε− . Note that this procedure is convenient only because j h , otherwise it ≥ ε would be convenient to integrate over z¯ by using δε and we should get the dimensional bound C λ of the case ε = 0. It follows that, starting from | | (j) j = hε, we have to apply to BJ (ψ) the localization procedure; then we define, if j h , ≥ ε (j) ¯(1) ω [h,j]+ [h,j] BJ (ψ)= Zj dzJz ψz, ω ψz,ω − (152) L ω − X Z ¯(1) and we perform the limit N . In this limit, Zj can be represented as an expansion in terms of→ trees, ∞ which have one special vertex (the J vertex) and an arbitrary number of normal vertices, the normal vertices being associated with the limiting value λ of the running coupling (whose (1) −∞ (1) ¯ ¯ hεη+ flow is independent of the Zj flow). It follows that Zhe = c1γ− [1+O(λ)] and that, if j

¯(1) ¯(1) η+ hεη+ (hε j)/2 Zj 1 = Zj γ + O( λ γ− γ− − ) (153) − | | where the first term comes from the trees with the special vertex of scale hε; it is exactly equal to the term one would get in the theory with ε = 0, in≤ the limit N . The second term is the contribution of the trees with → ∞ the special vertex of scale > hε (these trees must have at least one normal vertex); it is of course proportional to εη+ and takes into account the “short memory property” (exponential decrease of the irrelevant terms influence). The flow (153) immediately implies that, for any fixed j and η < 1/2, | +| 36 (1) (1) (1) (1) (1) (1) ¯ ¯ η+ ¯ limε 0(Zj 1/Zj ) = γ = (Zj 1/Zj ) and that Zj = c1[1 + O(λ)]Zj . → − − ¯(1) (1) Hence, by suitably choosing c1, we can get limε 0 Zj = Zj . →

(m) Note that S (x,ω) is diﬀerent from 0 only if m is even and i ωi = 0; moreover the truncated correlations can be written as sums overP the non truncated ones. Hence, in order to get an explicit formula for S(m)(x,ω), it is suﬃcient to calculate the correlation n (n) (1) 2n ¯ + ¯ K (x, u) = lim (ZN ) ψxj Γ ψxj ψuj Γ−ψuj (154) h,N h i − →∞ jY=1 ³ ´ ³ ´ where denotes the expectation with respect to the zero mass Thirring measure.h·i By using Lemma 4.1, we have K(n)(x, u)= c2n lim ε2nη+ (155) 1 ε 0 · n → dydv[ δ (x y )δ (u v )]K(2n)(x, y, u, v) · ε i − i ε i − i Z iY=1 f where n (2n) + K (x, y, u, v)= ψ¯y Γ ψx ψ¯v Γ−ψu (156) h j j j j i jY=1 ³ ´ ³ ´ On the otherf hand, by using the results of [BFM], see Theorem A.2 below, def one can prove that, if is the mean value for λ = 0 and ψ− = ψ− , h·i0 i xi,ωi + def + ψi = ψ ′ , yi,ωi

~~+ + λA(a a¯)n + + ψn− ψ1−ψ1 ψn = c0 − ψn− ψ1−ψ1 ψn 0 h ··· ···λA i h ··· λA ··· ′ i′ · s~~

~~λA(a a¯) 2n c − F (x, y, u, v) K(2n)(x, y, u, v)= 0 (161) λA (a+¯a)  2π  ( x y u v ) 4π s | s − s|| s − s| f   where F (x, y, u, v) is a continuous functionQ such that~~

~~2(1 λA a¯) 2(1 λA a¯) s~~

~~38 If we compare (165) with (80) and use the remark at the beginning of this section, we get the formal equivalence~~

(1) ¯ σ iασϕx lim ZN ψxΓ ψx b0 :e : (166) h,N − →∞ ∼ with λA(a a¯) 2 α c3(η+)c1c0 − b = c 8π (167) 0 2π if the following relation between α and λ is satisﬁed: α2 λA = 1 a¯ =1+ η η+ (168) 4π − 2π − − where we also used (164). This completes the proof of Theorem 1.1 for r = 0.

4.2 The case r > 0 Let us deﬁne µ ¯ µ + + jx = ψxγ ψx , ρx,ω = ψx,ωψx−,ω , Πx,ω = ψx, ωψx−,ω (169) − We have 0 + 1 + jx = ψx,ωψx−,ω , jx = i ωψx,ωψx−,ω (170) ω ω X X (2n,r) Hence, in order to calculate S (zx, y; ωω′, ν), it is suﬃcient to calculate the correlation function

k++k− (1) 2n Dk+,k−,n(a, c, y, v) lim (ZN ) (ZN ) h,N ≡ − →∞ · k+ k− n n (h,N) ( ρa ,+)( ρc , )( Πy ,+)( Πv , ) (171) h i i − i i − i iY=1 iY=1 iY=1 iY=1 where < >(h,N) denotes the expectation w.r.t. the massless Thirring measure. · By an obvious extension of Lemma 4.1, we know that there are two constants c1 and c2 (smooth functions of λ, equal to 1 for λ = 0), such that

η− (k++k−) η+ 2n Dk+,k−,n(a, c, y, v) = lim (c2ε2 ) (c1ε1 ) db dd dx du ε1,ε2 0 → Z · δ (b a)δ (d b)δ (x y)δ (u v)Ω(a, b, c, d, y, x, v, u) (172) · ε2 − ε2 − ε1 − ε1 − where, if x =(x ,..., x ), δ (x)= k δ (x ), with δ (x) deﬁned as in 4.1; 1 k ε i=1 ε i ε § moreover, Q k+ k− + + − − Ω(a, b, c, d, y, x, v, u) = ( ψai,+ψbi,+)( ψci, ψdi, ) h i=1 i=1 − − · nY nY + + ( ψy , ψx− ,+)( ψv ,+ψu− , ) (173) · i − i i i − i iY=1 iY=1 39 By using the identities (157) and (159) and by doing some simple algebra, one can see that, if z =(a, b, c, d) and w =(y, x, v, u),

λA(a a¯)(k++k−) F1(z, w) Ω(z, w) = c − 0 k+ 1 k− 1 i=1 g+− (ai bi) i=1 g− (ci di) · − − − Q F2(z, w) Q 2n k k K (w) (174) + η− − η− · i=1 bi ai i=1 di ci | − | | − | f Q Q where K2n(w) is deﬁned as in (161), and

~~(+) ( ) f F (z, w) = h (a , b , a , b ) h − (c , d , c , d ) 1 s,t s s t t s,t s s t t · s~~

~~( ) ( ) F (z, w) = h − (a , b , a , b ) h − (c , d , c , d ) 2 s,t s s t t s,t s s t t · s~~

~~η− (k++k−) Dk+,k−,n(a, c, y, v) = lim (c2ε2 ) db dd dx du ε2 0 → Z · δ (b a)δ (d b)Ω (a, b, c, d, y, v) (179) · ε2 − ε2 − 0 where, if we put w0 =(y, v),~~

λA(a a¯)(k++k−) F1(z, y, y, v, v) Ω (z, w ) = c − (180) 0 0 0 k+ 1 k− 1 i=1 g+− (ai bi) i=1 g− (ci di) · − − − FQ2(z, y, y, v, v) Q (n) k k K (w0) · + b a η− − d c η− i=1 | i − i| i=1 | i − i| (n) Q Q K (w0) being deﬁned as in (154); its explicit expression is given by (165).

~~40 Let us now perform the limit ε2 0. We use the identity, following from obvious symmetry arguments, →~~

εη−(k++k−) lim db dd δε2 (b a)δε2 (d b) k k ε2 0 − − + b a η− − d c η− · → Z i=1 | i − i| i=1 | i − i| F (a, b, c, d) Q = Q k+ 1 k− 1 · i=1 g+− (bi ai) i=1 g− (di ci) − − − k+ η− k− η− Q Q ε2 ε2 = lim db dd δε2 (b a)δε2 (d b) ε2 0 − − Ã bi ai ! Ã di ci ! · → Z iY=1 | − | iY=1 | − | k+ k− [(b a ) ∂b ] [(d c ) ∂d ]F (a, a, c, c) i=1 i − i · i i=1 i − i · i = k+ 1 k− 1 ·Q i=1 g+− (bi Q ai) i=1 g− (di ci) · − − − k++k− k+ k+ c3(η ) Q + Q = − Db− Dd F (a, a, c, c) (181) Ã 4π ! i i iY=1 iY=1 where c (η) is deﬁned as in 4.1 and Dω ∂ + iω ∂ . 3 x ∂x0 ∂x1 To complete the calculation,§ note that,≡ up to terms which are of the second order in at least one of the diﬀerences bs as or ds cs (these terms give no contribution in the limit ε 0), − − 2 →

~~(σ) as yt as vt hs,t (as, bs, vt, yt) 1+ ησ(bs as) − − (182) ≃ − · " a y 2 − a v 2 # | s − t| | s − t|~~

~~(σ) (bs as) (bt at) hs,t (as, bs, at, bt) 1+ ησ − · − + ≃ " − a a 2 | s − t| [(b a ) (a a )][(b a ) (a a )] +2 s − s · s − t t − t · s − t (183) a a 4 # | s − t|~~

~~(σ) (bs as) (dt ct) hs,t (as, bs, ct, dt) 1+ ησ − · − + ≃ " − a c 2 | s − t| [(b a ) (a c )][(d c ) (a c )] +2 s − s · s − t t − t · s − t (184) a c 4 # | s − t|~~

(σ) 1 1 1 hs,t (as, bs, vt, yt) 1+ gσ− (bs as) (185) ≃ − "g 1(a y ) − g 1(a v )# σ− s − t σ− s − t e (σ) 1 1 1 h (a , b , a , b ) 1+ g− (b a )g− (b a ) (186) s,t s s t t ≃ σ s − s σ t − t [g 1(a a )]2 σ− s − t In ordere to get the ﬁnal result, we have to substitute these expression in the r.h.s of (175) and (176), expand their product, keep the terms which are of

41 the first order in all the differences bi ai and di ci and, finally, apply to k+ − k+ + − − them the differential operator i=1 Dbi i=1 Ddi , whose effect can be easily obtained by using the trivial identitiesQ Q ω 1 D− g− (b a ) = 4π (187) bs ω s − s ω (bs as) z Db − · = 2πg ω(z) (188) s z 2 − | | Dω Dω (b a ) (b a ) = 0 (189) bs bt s − s · t − t ω ω D D− (b a ) (b a ) = 2 (190) bs bt s − s · t − t Let us consider, for example, the case n = 0. Then we see immediately that Dk+,k−,0(a, c) is different from 0 only if k+ +k = 2m and that, if we put − z = (a, c) and ωi = +1 if zi a, ωi = 1 if zi c, Dk+,k−,0(a, c) satisfies ∈ − ∈ (2) 2 the Wick Theorem with covariance Cω1,ω2 (z1 z2) = lim h,N (ZN ) < (2) − − →∞ ρzi,ωi ρzj ,ωj >, that is ZN ρx,ω, in the removed cutoffs limit, is a Gaussian field in the massless Thirring model. It is easy to check that

λA(a−a¯) 2 c0 c2c3 2π (1 + η /2) C (z z )= δ · ¸ − (191) ω1,ω2 1 − 2 ω1,ω2 [(z z )+ iω(z z )]2 1,0 − 2,0 1,1 − 2,1 T It follows that, if we call Dk+,k−,n(a, c, y, v) the truncated expectation corresponding to Dk+,k−,n(a, c, y, v), we have DT (a, c)= δ C (z z ) (192) k+,k−,0 k,2 ω1,ω2 1 − 2 Hence, by using (170) and the definition (101), we get S(0,k)(z , z ; ν , ν )= δ b2hν1,ν2 (z z ) (193) 1 2 1 2 − k,2 1 − 2 with 2 1 λA(a a¯) 2 η b = [c − c c ] (1 + − ) (194) π 0 2 3 2 Let us now consider the case n> 0; in this case we shall give the explicit T expression of Dk+,k−,n(a, c, y, v). This quantity can be obtained, by expand- (n) ing K (w0) in terms of products of connected expectations in the usual way and then trying to get a connected quantity by using the terms which sur- vive to the limit ε2 0, see discussion above. It is obvious that a connected contribution can be→ obtained only by keeping the products of different first (σ) (σ) order zeros coming from the functions hs,t (as, bs, vt, yt), hs,t (as, bs, vt, yt) and the analogous with (cs, ds) in place of (as, bs), together with the trun- (n) e (n) cated expectation in the expansion of K (w0), that we shall call KT (w0). It follows that

k++k− T (n) Dk+,k−,n(a, c, y, v)= KT (w0) W (zr,ωr, w0, σ) (195) rY=1 42 where we deﬁned σ so that σ = +1 if w y, σ = 1 if w v and i i ∈ i − i ∈ n λA(a a¯) η η+ W (z, ω, w , σ)= ωc − c c 1+ − − σ g (z w ) (196) 0 − 0 2 3 2 j ω − j µ ¶ Xj=i By using (170), we easily get the ﬁnal result

~~k (2n,k) (2n,0) νr S (w, y; σ, ν)= S (w; σ) W (yr, w, σ) (197) rY=1 f with~~

n µ ν i λA(a a¯) η η+ εν,µ(y wj) − − W (y, w0, σ)= c0 c2c3 1+ − σj − 2 (198) π 2 y wj µ ¶ Xj=i | − | f It follows that (197) has the same structure than (98), so that, by comparing (198) with (99), as well as (193) with (100), we get, in agreement with the considerations after (6), the equivalence

~~(2) ν µ µ lim ZN jx εν,µ(b1∂ ϕx + b2∂ ξx) (199) h,N − →∞ ∼ − where ξ is a free boson ﬁeld of zero mass, independent of φ, and~~

2 λA(a a¯) η η+ b1 = c0 − c2c3 1+ − − (200) α µ 2 ¶ 1 2 1 One can check, by using the relations a− = 1 λ/(4π)+ O(λ ),a ¯− = 1+ λ/(4π)+ O(λ2) and A =1+ O(λ2), that, if b2−is the constant deﬁned in (194), b2 = b2 b2 = O( λ 3) (201) 2 − 1 | | However, one can prove that b2 = 0. This follows from the remark that

2 η− η+ b2 1+ − 1 = 2 (202) 2 ³ ´ η− b (1 + η η+) 1+ − − 2 ³ ´ where we used (200), (194) and (168). Hence, in order to prove that b2 = 0, it is sufficient to prove that the r.h.s. of (202) is equal to 1; by a simple calculation one can check that this condition is equivalent, since η > 0, to − the condition 2 2 (1 + η ) =1+ η+ (203) − Our solution of the Thirring model allows us to represent η and η+ as well − defined power series in the physical value λ of the running coupling (see eq. −∞ (2.35) of [BFM] for η ). This representation is not convenient to verify the − 43 identity (203); however, (203) is independent of the details of the ultraviolet regularization of the model, hence it can be checked also by using the explicit (rigorous) representations of η and η+ in terms of the bare coupling, which − were found in [M1] and [M2] with a different ultraviolet regularization (by the way, they are also in agreement with the heuristic procedure proposed in [J] and [K]). In this approach, if we call λ the bare constant and put x = λ/(4π), one gets e 2 e 2x 2x η = , η+ = (204) − 1 x2 1 x2 − − and one can check that (203) is indeed satisfied. This completes the proof of Theorem 1.1 for r> 0.

~~A The explicit formula for the ﬁeld correlation functions~~

A.1 The Schwinger-Dyson equation In this appendix we will derive the explicit expression of the n-point Schwinger functions (157), by extending the arguments used in [BFM], to which we refer for details, to analyze the 2-point function. Let us define (J, ϕ) = (N) W log Ph,N (dψ) exp (√ZN ψ, J, ϕ) , where the free measure Ph,N (dψ) is (N{V) } + + definedR by (14), = λV √ZN ψ + ω dx[Jx,ωZN ψx,ωψx−,ω +ϕx,ωψx−,ω + + V − ψx,ωϕx−,ω] and the fields ϕ± are³ anticommuting´ P R between themselves and with ψ±. We shall introduce the Fourier transform of various fields. In doing that, + + + we shall consider the fields ρ = ψ ψ−, ψ e ϕ as incoming fields, while α, J, ψ− e ϕ− will be outcoming fields. First of all, we note that the Schwinger-Dyson equations are generated by the identity

2 ∂e ϕ− eW dp ∂ e D (k) W = χ (k) k,ω λ W (205) ω + h,N  2 +  ∂ϕk,ω ZN − (2π) ∂Jp, ω∂ϕk+p,ω b Z −   Indeed, givenb any F (ψ) which is a power series in theb ﬁeld,b we have, by the Wick Theorem, [h,N] gω (k) ∂F (ψ) − ψk,ωF (ψ) 0 = + 0. (206) h i ZN h ∂ψk i b ,ω b [h,N] where 0 is the mean value with respect to Ph,N andb gω (k)= χh,N (k)/Dω(k). Then,h·i (205) is a consequence of the identity b [h,N] (N) (N) ∂eW √ZN ψ,ϕ gω (k) ∂ √ZN ψ,ϕ − V ( ) V ( ) + = ψk,ωe 0 = + e 0 (207) ∂ϕk,ω h i ZN h∂ψk,ω i b b b 44 b + + and the remark that V (ψ)= dxψx,+ψx−,+ψx, ψx−, . − − R A.2 The approximate Ward–Takahashi identities

We consider a new generating functional, (α, J, ϕ) = log Ph,N (dψ) A (N) W (σ) ( ) exp (√ZN ψ, J, ϕ) + [ZN 0 + ZN σ= νN σ](α, ψ) whereR νN± are · {V A ± A } two suitable constants, to be chosen belowP as functions of λ, and def dq dp + 0(α, ψ) = Cω(q, p)αq p,ωψq,ωψp−,ω , (208) 4 − A ω= (2π) X± Z b b def dq dp b + σ(α, ψ) = Dσω(p q)αq p,ωψq,σωψp−,σω (209) 4 − A ω= (2π) − X± Z b b b 1 having deﬁned, as in [BFM], Dω(k)= ik0 +ωk1 and Cω(q, p) = [χh,N− (p) 1 − − 1]Dω(p) [χh,N− (q) 1]Dω(q). − − iαx,ω By doing the transformation ψx±,ω e ψx±,ω, see [BM] for a rigorous deﬁnition, we get →

~~∂ dk + ∂ ∂ − Dµ(p) W (J, ϕ) = 2 ϕk+p,µ W+ W ϕk,µ ∂Jp,µ Z (2π) " ∂ϕk,µ − ∂ϕk−+p,µ # −~~

∂ 0 b b b WA (0, J, ϕ);b b (210) − ∂αp,µ where 0 is the same as but neglecting the interactions σ(α, ψ). A Ab The lastW term in the r.h.s. ofW (210) is not negligible in the removedA cutoﬀs limit, but we can extract its leading contribution by introducing suitable counterterms [BFM], so that the rest will vanish, by putting

(+) ∂ ( ) ∂ (1 νN )Dµ(p) W (J, ϕ) νN− D µ(p) W (J, ϕ) = (211) − ∂Jp,µ − − ∂Jp, µ − (h) dk + b∂ ∂ b ∂ − A = 2 ϕk+p,µ W+ (J, ϕ) W (J, ϕ)ϕk,µ W (0, J, ϕ) Z (2π) " ∂ϕk,µ − ∂ϕk−+p,µ # − ∂αp,µ If we deﬁne b b b b b ( ) (+) 1 ( ) (+) 1 a = [1 ν − ν ]− , a¯ = [1+ ν − ν ]− (212) N − N − N N N − N by some simple algebra we obtain the identity

∂e A ′ ∂e A W (0, ϕ)+ N,σσ W (0, 0, ϕ) = (213) ∂Jp,σ ′ Dσ(p) ∂αp,σ′ Xσ AN,σσ′ dk + ∂eW ∂eW b b ′ − ′ = 2 ϕk+p,σ + (0, ϕ) (0, ϕ)ϕk,σ ′ Dσ(p) (2π) " ∂ϕk,σ′ − ∂ϕk−+p,σ′ # Xσ Z b b where AN,σ = (aN + σa¯N )/2. With theb argument explainedb in [BFM], it would be easy to prove that the term in is vanishing in the limit of A removed cutoﬀ; anyway this is not our currentW objective.

45 A.3 A Closed Equation for the field correlation functions By doing an arbitrary number of functional derivatives with respect to the ϕ external field in (205) and then putting ϕ = 0, one can obtain an infinite number of relations between the field correlation functions and other correlations involving several fields and one current, integrated over the current momentum. We want now to show that, by using the identity (213), it is possible to get a closed equation for the field correlation functions, in the ω limit of removed cutoffs. Let us define ∂x = ∂0 + iω∂1.

Lemma A.1 For λ small enough, there exists a constant A =1+ O(λ), such that the equations| | of motion for the truncated Schwinger functions - except the two point Schwinger function- in the limit of removed cutoﬀs are generated, at non coinciding points, by the identity:

2 2 ω1 ∂ + ∂ ∂ ∂ = λA A dz g (x z) ϕ ϕ− x1 +W ω1µ ω1 1 z,µ + W+ + W z,µ ∂ϕx ,ω µ − − − " ∂ϕz,µ∂ϕx ,ω − ∂ϕx ,ω ∂ϕz−,µ # 1 1 X Z 1 1 1 1 + ∂ ∂ ∂ ∂ − +λA A ω1µ dz g ω1 (x1 z) ϕz,µ W+ +W +W W ϕz,µ (214) µ − − − " ∂ϕz,µ ∂ϕx ,ω − ∂ϕx ,ω ∂ϕz−,µ # X Z 1 1 1 1 + Proof. If we make a derivative with respect to ϕk+p,ω in both sides of (213), with ω = σ, and then integrate over p (which is meaningful, since the correlation functions can not have a singularity at pb= 0 and have compact support in p for h and N ﬁnite), we get

2 2 A dp ∂ eW dp AN, ωµ ∂ eW − 2 + = 2 + + (215) (2π) ∂Jp, ω∂ϕk+p,ω − µ (2π) D ω(p) ∂αp,µ∂ϕk+p,ω Z − X Z − 2 2 dp dq AN, ωµ + ∂ eW ∂ eW b b − b b − + 4 ϕq+p,µ + + + ϕq,µ µ (2π) D ω(p) " ∂∂ϕq,µϕk+p,ω − ∂ϕk+p,ω∂ϕq−+p,µ # X Z − b b where both sides are calculated at J =b αb= 0 and web usedb the fact that Dω(p) is odd in p to cancel one term in the r.h.s. of (215). We introduce the generating functionals ,µ(β, ϕ), for µ = , deﬁned T as W ±

def 2 T ,µ(β,ϕ) λN ZN V (ψ) + + eW = Ph,N (dψ) e− exp ϕx,ωψx−,ω + ψx,ωϕx−,ω Z ½Z Z ¾ · (µ) (σ) (µ) exp 0 + νN σ ZN ψ, ZN β (216) · ("T σ= T # µ ¶) · X± q q (µω) (3) (µω) (1) exp α λ ω ρ ω ZN ψ, ZN β · (ω= − B − B µ ¶) X± h i q q

~~46 (µ) (µ) with α µ= , ρ µ= , four real parameters to be ﬁxed later and { } ± { } ±~~

(µ) def dk dp dq Cµ(q, q p) + − − 0 (ψ, β) = 6 − βk,ωψk+p,ωψq,µψ p+q,µ , T ω= (2π) D ω(p) − X± Z − b b b b (µ) def dk dp dq Dσµ( p) + − − σ (ψ, β) = 6 − βk,ωψk+p,ωψq,σµψ p+q,σµ , T ω= (2π) D ω(p) − X± Z − (3) def dk dp dq + b b b b − − ω (ψ, β) = 6 βk,ωψk+p,ωψq, ωψ p+k, ω , B Z (2π) − − − (1) def dk b b b b − ω (β, ψ) = 2 βk,ωDω(k)ψk,ω (217) B Z (2π) b b We remark that ,µ(β, ϕ) diﬀers from the analogous generating functional WT (1) introduced in [BFM] because of the presence of the interactions ω (β, ψ) (3) B and ω (β, ψ), that in the cited paper were - in a sense - reconstructed a posteriori;B here we describe a faster way to implement the same procedure of [BFM]. We have the following identity:

2 dp 1 ∂ eWA 1 ∂eWT ,µ 2 + (0, 0, ϕ)= (0, ϕ) + (218) (2π) D ω(p) ∂αp,µ∂ϕk+p,ω ZN ∂βk,ω Z − 2 dp ∂ eW ∂eW (µω) b b (µω) b +α λ 2 + (0, ϕ)+ ρ Dω(k) + (0, ϕ) (2π) ∂Jp, ω∂ϕk+p,ω ∂ϕk,ω Z − which, plugged into (215), givesb b b

2 (µ) dp ∂ eW 1+ λ AN, µ α = − 2 + Ã µ ! (2π) ∂Jp, ω∂ϕk+p,ω X Z − T ,µ AN, ωµ ∂eW b (µ) ∂eW = − AN, µρ b Dω(k) + + (219) − − µ ZN ∂βk,ω − Ã µ ! ∂ϕk,ω X X 2 2 dp dq AN, ωµ + ∂ eW ∂ eW b− b − + 4 ϕq+p,µ + + + ϕq,µ . µ (2π) D ω(p) " ∂ϕq,µ∂ϕk+p,ω − ∂ϕk+p,ω∂ϕq−+p,µ # X Z − b b This equation, together with (205), theb identityb δµ,ω =b (1+ µωb )/2 and the remark that ,µ(0, ϕ)= (0, ϕ), implies that WT W ∂ BN λAN ∂ ,µ Dω(k) W+ = χh,N (k) ϕk−,ω + χh,N (k) AN, ωµ WT (0, ϕ) + (220) − ∂ϕk,ω ZN ZN µ ∂βk,ω X 2 2 dq dp AN,b ωµ + ∂ eW ∂ eW b − −W b − +λAN χh,N (k) 4 e ϕq,µ + + + ϕq+p,µ µ (2π) D ω(p) " ∂ϕq+p,µ∂ϕk p,ω − ∂ϕk p,ω∂ϕq−,µ # X Z − − − b b (µ) (µ) 1 (µ) where AN = [1+λ µ AN, µ(α ρ )]− andbBN = [1+b λ µ AN,b µα ]Ab N . − − − Before doing theP limit h,N , we can rewrite theP previous identity in the space coordinates, by− doing→ the ∞ Fourier transform in both sides. Since

47 we want to get an identity involving only the correlations with at least four points, the first term in the r.h.s. gives no contribution. Hence, it is easy to see that we get the identity (214), with A = limN AN , if the correlations →∞ obtained from derivatives of the last term are proved to be vanishing in the limit of removed cutoffs and if, in this limit, we can safely substitute χh,N (k) with 1. Let us first consider this problem, without giving the technical details. If we make a certain number of derivatives with respect to the field ϕ at non coinciding points and put ϕ = 0, we are faced with the problem of calculating the limit of expressions of the type

dz δh,N (z)gω(x1 x2 z)Gh,N (x1 z, y2,..., yn) (221) Z − − − where δh,N (z) is the Fourier transform of χh,N (k) and the points x1, x2, y2,..., yn are all different, except x2 and y2, which can be equal. The function Gh,N (y) is a (truncated) Schwinger function, which was proved in [BFM] ε′ (eq. 2.58) to decay as yi − , 0 < ε′ < 1, if yi , while the other points are fixed. By using| | the bound 2.52 of [BFM]| | → it ∞ is also possible to ε′′ prove that it diverges as yi yj − , 0 < ε′′ < 1, if yi yj 0, while the other points stay constant.| − On| the other hand, it| is easy− to| → prove that 2N 2 1 δ (z) C(γ− + z )− . These properties and the good convergence | h,N | ≤ | | properties of Gh,N (y) as h,N (uniform if the points vary in non in- tersecting neighborhoods− of the→ arguments) ∞ imply that one can make without any problem in (221) the limit h and substitute G (y) with → −∞ h,N G(y) = lim h,N Gh,N (y). The previous remarks imply also that the func- − →∞ tion g (x x z) G(x z, y ,..., y ) is a L1 function of z with a finite ω 1 − 2 − 1 − 2 n number of singularities; moreover, δ ,N (z) δ(z) as N . It follows −∞ that the limit of (221) does exist and is given by→g (x x ) G→(x ∞, y ,..., y ); ω 1− 2 1 2 n one can also see that the limit is uniform, if the xi vary in small non intersect- ing neighborhoods, so implying that one can exchange the derivative with the removed cutoff limit in the l.h.s. of (220), written in the space coordinates. We still have to discuss the main point, that is the fact that the correlations obtained from derivatives of the last term of (220) vanish in the removed cutoff limit. Since we assume some familiarity with [BM], we shall do that very briefly, by studying the flow of the marginal terms proportional to the field β in the effective potential related with the generating functional (216). (j′) After the integration of the fields ψ , j′ > j, we obtain an expression of the type:

(h) (j) T ,µ(β,ϕ) (j) eW = P[h,j](dψ) exp ϕ, Zjψ + ,irr β, ϕ, Zjψ V WT · Z n ³ q ´ ³ q ´o Z 2 Z exp N (µ) + N ν(σ) (µ) Z ψ, Z β  0 j σ j j  · Ã Zj ! T Zj σ= T · h X± i ³q q ´  48  N (3,µω) (3) ZN (1,µω) (1) exp ζj + ζk Zjψ, Zjβ (222) · " B Zj B #   kX=j ³q q ´ e e (j) (j)  where is the eﬀective potential for β = 0, ,irr is the irrelevant part of the termsV of order at least 1 in β, while the restW representsT the corresponding marginal terms, written in terms of two running coupling constants:

(µ) αN λ for j = N (3,µ) def − ζj =   (µ) (µ)  λ α λj for j N 1 ; e j − N ≤ −   e (µ) ρN for j = N (1,µ) def − ζj =  (223)  (µ) (µ) Zj  zj αN zj for j N 1 ; e − ZN ≤ − ³ ´ µ µ where λ and z  aree exactly the coupling studied in [BM]; while λ { j } { j } { j} and zj are exactly the effective coupling and the field renormalization of the original{ }e generatinge functional, . In [BM] (equation (144)) it was proved µ W that there exist αN such that the following two bounds are both satisfied

(µ) (µ) (N j)/2 (µ) (µ) (N j)/2 λ α λ Cγ− − , z α z Cγ− − (224) | j − N j| ≤ | j − N i| ≤ e (µ) (1,µ) (1) Thereby, if we put ρN = j N 1 ζj ,e the factor in front of in (222) is ≤ − B NP e N 1 ZN (1,µ) ZN − (1,µ) (µ) ζk = ζk ρN Zj Zj  −  kX=j kX=j e  e 

ZN (1,µ) (µ) (µ) Zk = ζk = (zk αN zk) (225) − Zj k j 1 k j 1 − Zj ≤X− ≤X− e e (N j)/4 and the last term, by the second (224), can be bounded by Cγ− − . This remark, together with the ﬁrst of (224), allows us to prove that contribution to the correlation functions of the last term in (222) vanishes in the limit of removed cutoﬀs, as a consequence of the short memory property, see [BM] for details.

~~A.4 Solution of the closed equations~~

1 1 y | | Let us deﬁne a = limN aN ,a ¯ = limN a¯N , ∆− (x y)= 2π ln x and →∞ →∞ | | | ³ ´ 2n (2n) ( , ) ∂ eW ′ −W ′ ′ Gω,ω (x, y)= e + + (0, 0) (226) ′ ′ ∂ϕxn,ωn ∂ϕx ,ω ∂ϕy− ,ω ∂ϕy− ,ω ··· 1 1 1 1 ··· n n

49 Theorem A.2 For λ small enough and x = y, | | 6 (2) (2) λA(a−a¯) ∆−1(x y z) S ′ (x, y) G ′ (x, y)= δ ′ g (x y)e 2 − | (227) ω,ω ≡ ω,ω ω,ω ω − where z is a ﬁxed, non-zero position, whose arbitrariness reﬂects the ar- (2) bitrariness of a factor in front of Sω,ω′ (x, y). Furthermore, if n > 1 and (x1,..., xn, y1,..., yn) is a family of two by two distinct points,

~~(2n) π (2n) Gω,ω′ (x, y)= ( 1) ω,π(ω′)(x, π(y)) (228) π P − G X∈ X def where X = 1,...,n , PX is the set of the permutations of the elements of X and { }~~

n (2n) (2) ω,ω′ (x, y)= Sω ,ω′ (xj, yj) G  j j  · jY=1 s 1. The reason for this procedure is that, as we have discussed A.3, we were able to get a closed equation, in the limit of removed cutoﬀs,§ only for the truncated correlation functions with n > 1. However, it is worthwhile to give ﬁrst the heuristic argument which allows us to conjecture that (228) is the right expression for the not truncated correlation functions. In the limit N , if we put Z = limN and we ignore the fact that →∞ Z = 0, the identity→ (220) ∞ can be written in terms of the space coordinates as

x1 ∂eW B ∂ = eW ϕ− + λA A ω1 + x1,ω1 ω1ε ∂ϕx ,ω Z ε − · 1 1 X W W + ∂e ∂e dz g ω1 (x1 z) ϕz,ε ϕz−,ε (230) · − − ∂ϕ+ ∂ϕ+ − ∂ϕ+ ∂ϕ Z " z,ε x1,ω1 x1,ω1 z−,ε #

This implies, if η = λAA and Z = limN ZN (we will ignore then fact (2) − →∞ that Z = 0), that S ′ (x, y)= δ ′ S (x y), with ω,ω ω,ω ω − B ∂ωSω(x)= δ(x) ηg ω(x)Sω(x) (231) Z − − 1 Hence, since, for any value of z, g ω(x)= ∂ω∆− (x z), we get − − | B η[∆−1(x z) ∆−1(0 z)] S (x)= e | − | g (x) (232) ω Z ω

~~50 1 where ∆− (0 z)=+ , which should balance, in this formal calculation, the fact that Z =| 0. In∞ fact, this equation implies the correct value (227) of Sω(x), if we choose z so that~~

B η∆−1(0 z) = e | (233) Z Hence, we are encouraged to pursue this formal procedure. If we take 2n 1 − suitable functional derivatives in both sides of (230) and we call xj the vector x without the element xj, we ﬁnd the following equation:

n x1 (2n) B k 1 (2n 2) ′ − ∂ω Gω,ω′ (x, y)= ( 1) − δω1,ω δ(x1 yk)Gω ,ω′ (x1, y ) (234) 1 Z − k − 1 k k kX=1 n n (2n) ′ +λA A ω1ωk g ω1 (x1 xk) A ω1ω g ω1 (x1 yk) Gω,ω′ (x, y) " − − − − − k − − # kX=2 kX=1 By using (233), this equation can be written as

n n −1 x −1 x x z λAA−ω ω′ ∆ (x1 yh z) (2n) 1 λAA−ω1ωh ∆ ( 1 h ) − 1 − | ∂ω e − | e h Gω,ω′ (x, y) = 1 "Ã !Ã ! # hY=2 hY=1 n k 1 = ( 1) − δω ,ω′ δ(x1 yk) (235) − 1 k − · kX=1

n n −1 −1 y x z λAA−ω ω′ ∆ (yk yh z) (2n 2) λAA−ω1ωh ∆ ( k h ) − 1 − | e − | e h G −′ (x1, y )   ω1,ωk k · Ãh=2 ! h=1 Y  hY6=k    (2n) (2n 2) and hence we arrive at a formula for G in terms of G − :

~~n (2n) k 1 (2) (2n 2) − Gω,ω′ (x, y)= ( 1) − Sω ,ω′ (x1 yk)Gω ,ω′ (x1, y ) (236) − 1 k − 1 k k · kX=1~~

n n −1 −1 λAA ′ ∆ (yk yh x1 yh) λAA−ω ω ∆ (yk xh x1 xh) −ω1ω e 1 h − | −  e− h − | −  · Ãh=2 ! h=1 Y  hY6=k    where all Z factors disappeared. Such an iterative relation is clearly solved by (228) together with (229). Let us now assume that the expression (228) is correct; we want to check if the corresponding truncated correlation functions satisfy the identity (214), for n > 1. First of all, we remind the connection between the two kind + def + of functions. In order to abridge the notation, we put ϕj = ϕxj ,ωj and def ε def ε ϕj− = ϕy− ,ω′ ; furthermore, if Xj X = 1,...,n , we deﬁne ϕX = k Xj ϕk, j j ⊂ { } j ∈ with the factors ordered with decreasing k, if ε = +, and with increasingQ k

51 otherwise. Expanding eW (0, ϕ) in powers of , we find: W 2n n ∂ eW 1 ∗ π + (0, 0) = ( 1) (237) ∂ϕX ∂ϕX− m! X ,...,X − · m=0 X1,...,Xm π 1 m X X ∈PXX 2 X1 2 Xm ∂ | | ∂ | | + W (0, 0) . . . + W (0, 0) ∂ϕ ∂ϕ− ∂ϕ ∂ϕ− X1 π(X1) Xm π(Xm) where X∗ 1,...,Xm denotes the sum over all the possible partitions of X into m, X1,...,Xm distinguishableP and non empty subsets X1,...,Xm. Furthermore, X is the quotient set of the permutations of the elements of X, , whereP two PX elements are identified if they differ only for a permutation in X1 Xm . To find out the explicit expression of the truncated functions,P ⊗···⊗P we define

def 1 ′ V (xs Pt) = λAA ωsω ∆− (xs yt xs xt) | − t − | − def V = V (x P ) V (y P ) (238) s,t s| t − s| t and, exploiting the analogy of the expression (229) with the partition function of a lattice gas, we perform the Mayer expansion: s

(2n) Vs,t (2) Sω,ω′ (x, y)= e 1 Sω ,ω′ (xk, yk) (241) − k k g X(X) s,tY g ³ ´ kYX e ∈C h i∈ ∈ We now perform some manipulations. From the previous expression we get s=1 6 −1 V (x1 Ps) λAA−∆ (x1 y1 z) (2n) e− | e− − | S ′ (x, y)=   ω,ω s X Y∈   n e (2) ′ = δω1,ω gω1 (x1 y1) Sω ,ω′ (xk yk) (242) 1 − Ã k k − ! · kY=2 s,1 /g s,1 g s,t=1 h i∈ h i∈ 6 V (x1 Ps) V (y1 Ps) V (x1 Ps) Vs,t e− | e− | e− | e 1   · g (X) s X s X − s,t g − ∈CX Y∈ ³ ´ Y∈ h Yi∈ ³ ´   52 and therefore, by taking a derivative w.r.t. x1, we ﬁnd

s=1 6 −1 x1 V (x1 Ps) λAA−∆ (x1 y1 z) (2n) ∂ e− | e− − | S ′ (x, y) = ω1   ω,ω  s X Y∈   n e  (2) ′ = δω1,ω gω1 (x1 y1) Sω ,ω′ (xk yk) (243) 1 − Ã k k − ! · kY=2 s6=h s6=h h=1 hs,1i∈/g hs,1i∈g 6 V (x1 Ps) V (y1 Ps) V (x1 Ps) e− | e− | e− | ·   − · h X g (X) s X s X ³ ´ X∈ ∈CX  Y∈  Y∈ s,t=1   6 Vs,t Θ[ h,1 /g] V (x1 Ph) e 1 ( 1) h i∈ e− | (∂1V )(x1 Ph) · s,1 g − − | h Yi∈ ³ ´ where Θ[ ] is equal to 1 if the relation [ ] is true; Θ[ ] is zero otherwise. If the · · · def graph g (X) does not contain the link h, 1 , then also the graph g′ = g ∈ C h Θ[ h,i1 /g] ∪ h, 1 is in (X); because of the factor ( 1) h i∈ their contribution cancel h i C − each other. We call h,1 (X) the remaining set of graphs: it is made of the h i graphs in (X) whichC are no longer in (X) if the link h, 1 is erased. Clearly theyC can be also constructed byC joining with theh linki h, 1 two graphs g (X ) and g (X ), for any choice of disjoint X andh Xi s.t. 1 ∈ C 1 h ∈ C h 1 h 1 X1, h Xh and X1 Xh = X. Because of these considerations we arrive at∈ the expression∈ ∪

s=1 6 −1 x1 V (x1 Ps) λAA−∆ (x1 y1 z) (2n) ∂ e− | e− − | S ′ (x, y) = (244) ω1   ω,ω  s X Y∈ s=1  en  6 −1 V (x1 Ps) λAA−∆ (x1 y1 z) (2) = e− | e− − | S ′ (xk yk)   ωk,ωk s X Ãk=1 − ! · Y∈ Y h=1  6 ∗∗ (∂ V )(x P ) eVs,t 1 eVs,t 1 · 1 1| h  −   −  h X X1,Xh g1∈C(X1) (s,t) g1 (s,t) gh X∈ X g ∈CX(X ) Y∈ Y∈ h h     where ∗∗ is the same of ∗ , with the further constraint that 1 X1 X1,Xh X1,Xh ∈ and h PXh (such a notationP is abusive, but quite clear): therefore X1, Xh is ∈ (2n) ordered, and there is no factor (1/2!). As consequence, for n> 1, Sω,ω′ (x, y) satisfy (214), that, after suitable derivatives in the ﬁelds, reads

n ∗∗ x1 (2n) π n,h ∂ S ′ (x, y)= λA ( 1) M (x, y ) + (245) ω1 ω,ω X1,Xh π X ,X − ◦ h=2 X1,Xh π P 1 h X X ∈ XX n n (2n) ′ +λA A ω1ωh g ω1 (x1 xh) A ω1ω g ω1 (x1 yh) Sω,ω′ (x, y) " − − − − − h − − # hX=2 hX=1

~~53 with~~

n,h def ′ MX ,X (x, y) = A ω1ωh g ω1 (x1 xh) A ω1ω g ω1 (x1 yh) 1 h − − − − − h − − · (2 X1 ) h (2 Xh ) i S | |′ (x , y )S | |′ (x , y ) (246) ω ,ω X1 X ω ,ω Xh X · X1 X1 1 Xh Xh h

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