Mass generation in the large N Gross-Neveu model: a constructive proof without intermediate eld

We give a new constructive proof of the infrared behavior of the Euclidean Gross-Neveu model in two dimensions with small coupling and large component number N . Our agument does not rely on the use of an intermediate (auxiliary bosonic) ﬁeld. Instead bubble series are resummed by hand, and determinant bounds replaced by a control of local factorials relying on combinatorial arguments and Pauli’s principle. The discrete symmetry-breaking is ensured by considering the model directly with a mass counterterm chosen in such a way as to cancel tadpole diagrams. Then the fermion two-point function is shown to decay (quasi-)exponentially as in [12].


Introduction
The Gross-Neveu model is a theory of N interacting charged fermions (ψ a ,ψ a ), a = 1, . . . , N , two-component spinor fields on R 2 . We study the model in imaginary time, i.e. after a Wick rotation, so that the theory is rotation-invariant in the coordinate plane (x 0 , x 1 ). The field index a is called flavour index (by reference to QCD). The covariance of the free fermion field is chosen to be C(p) = χ(|p|) p , where χ is a scale 1, rotation-invariant UV cut-off in momentum space (see later on in the text for details), and p = γ µ p µ (Eistein's summation intended, with µ = 0, 1), where γ-matrices are chosen as (ψ a ψ aψb ψ b )(x), (1.1) withψ · ψ = i=0,1 N a=1ψ a i ψ a i , where λ is a small enough, positive coupling constant, and N is large enough (depending on λ).
The model is interesting per se from a theoretical point of view, as it is one of the simplest QFT models exhibiting chiral symmetry breaking. This was understood long ago by D. Gross and A. Neveu themselves [10]. Briefly said, the model can be rewritten in terms of a coupled fermion/boson model with an auxiliary, scalar, bosonic field σ, featuring a "QED"-like (non-derivative) cubic vertex λ N σ(ψ · ψ). Integrating out fermion fields yields a purely bosonic theory whose effective potentiel V (σ) has the form of a "Mexican hat", i.e. is symmetric under the inversion σ → −σ, and has global minimum at σ = ±σ * with λ N σ * ≈ e −π/λ , zero being a local maximum. As in the low-temperature analysis of the Ising model, it may be proved that the model exhibits two pure phases, which can be selected by using suitable boundary conditions. These statements were proved at the level of mathematical rigor by C. Kopper, J. Magnen and V. Rivasseau more than twenty years ago [12], using a single-scale cluster expansion and a detailed analysis of the bosonic action functional L(σ) = Tr log( p + λ N σ) after an appropriate translation of the σ-field, σ → σ − N λ m ψ , where m ψ ≈ e −π/λ is chosen such that N λ m ψ ≈ σ * . Their method also allows a computation of the connnected fermion two-point function, which is shown to be massive, i.e. exponentially decaying at large distances with a decay rate ≈ m ψ .
The aim of this article is to prove discrete symmetry breaking and massiveness of the fermion field by an alternative method, avoiding the introduction of the auxiliary field σ. Generally speaking, this participates in an effort to discuss rigorously QFT theories with discrete or continuous symmetry breaking, for which auxiliary fields are gauge fields. These fields have several lowest-energy configurations related by the symmetries, which makes them difficult to handle in combination with cluster expansions and multi-scale decompositions, in particular in the case of continuous symmetry breaking. So the present article may be seen as a first attempt in this direction.
In order to motivate our strategy, let us first discuss the lowest-order terms in the bosonic action functional L(σ). The first-order term is calculated in terms of the tadpole diagram T , where fermion lines represent the fermionic propagator C * (p) = 1 p+m dressed by the mass parameter m coming from the translation of the σ-field. The second-order term is calculated in terms of the bubble diagram Υ, and (in function of the transfer momentum q) A q (Υ) = − 1 (2π) 2 Tr 2 dp χ(|p|) p + m ψ χ(|p + q|) p+ q + m ψ (1. 3) for some UV cut-off function χ. Note our particular spinor component convention; A(T ) has been chosen to be one of the two (equal) diagonal components 1 (2π) 2 dp χ(|p|) p+m ψ i,i , i = 0, 1, while A(Υ) is evaluated as a trace Tr 2 over spinor components, taking into account the (−1) fermionic loop factor. In practice, these diagrams come with an extra prefactor λ N due to the coupling constant, and with a flavor-counting factor O(N ) which is computed by direct inspection.
1. We choose a smooth UV cut-off function χ : = 0. Contrary to [12], we do not bother to take an analytic cut-off. The price to pay is that we only prove quasi-exponential decay of fermion two-point functions in the end (see Theorem 1.1 below).
2. We choose m ψ ≈ e −π/λ to be solution of the above gap equation (see Lemma 4.6).
3. We make the substitutions a 2N × 2N matrix with 2 × 2 flavor components C * ab (p) = δ a,b C * a,a (p); Except for terms interacting with the UV cut-off, the model is the same. However, the latter terms would have the effect of breaking the discrete σ ↔ −σ symmetry, should one introduce an auxiliary field σ as in [KMR]. It turns out that discrete symmetry-breaking in our model is a consequence of (1.5,1.6), which spares us the need to carefully choose boundary conditions.
The gap equation then ensures the exact cancellation of tadpole diagrams by the counterterm δL.
4. Schwinger functions with N ext external points are completely expanded in terms of the free fermionic measure · 0 : and rewritten as a sum of Feynman diagrams using Wick's formula. Then bubble chains are resummed by hand and represented as dashed wiggling lines connecting two field pairsψ a ψ a ,ψ b ψ b . Resumming the bubble series at transfer momentum q yields the Σ-kernel, where A q (Υ) is given exactly by (1.3). In particular, computations (see Lemma 4.3) show that A 0 (Υ) ∼ λ→0 1/λ (1.9) at zero transfer momentum, calling for a detailed analysis of the denominator of (1.7) at small momenta. Letting π(q) := λA q (Υ), (1.10) Σ(q) = λ N (1 − π(q)) −1 is (up to normalization) the covariance kernel of the auxiliary field σ in [12]. It is proved in Lemma 4.5 that where "O((1 + m ψ |x|) −∞ )" means: ≤ C n (1 + m ψ |x|) −n for every n ≥ 0, where C n is a constant independent from the parameters λ and N . Thus the kernel Σ(x) exhibits a quasi-exponential decay with rate ≈ m ψ , in general agreement with the estimates proved in [12]. The prefactor in (1.11) deteriorates for |x| smaller, but the relevant quantity is the L 1 -norm of Σ, for which we prove Note the absence of logarithmic prefactor log(1/m ψ ) ≈ 1/λ in (1.12), as opposed to (1.9).
At the end of this stage, we have a series of Feynman diagrams with delocalized vertices, thereafter called: partially resummed Feynman diagrams or simply (if no ambiguity can arise) Feynman diagrams. Feynman diagrams in our sense make up a set FD which is precisely defined in Definition 2.2. Identifying (by formal analogy) delocalized vertices Σ(x, y) with a photon-photon propagator σ(x)σ(y) 0 , the set FD may be identified as a subset of the set of Feynman diagrams of QED with N fermions. *************************************** We pause to comment on large orders of the theory. The power-counting for just renormalizable theories with a 1/N -expansion is well-known; let us redo computations here. Consider a Feynman diagram with V vertices, I internal lines, N ext external lines, and L loops. Graph topology yields L = I − V + 1 and 4V = 2I + N ext . Generally speaking, one expects (i) that the effective volume of integration Vol for a vertex is O(m −2 ψ ) because of the quasi-exponential decay rate m ψ ; Taking into account the coupling constant, the power-counting for an integrated vertex is ≈ λ N m −2 ψ . Finally, one must count N per fermionic loop (sum over flavor indices).
Disregarding N ext (which is fixed), the naive power-counting for a large Feynman diagram is therefore expected to be roughly which looks convergent. However, by substituting in (ii) an averaged quantity to C * (x, y) one has overlooked the fact that the prefactor in front of C * (x, y) (see (4.2)) is not m ψ but 1/(1 ∨ |x − y|) for |x − y| < m −1 ψ , where: 1 ∨ |x − y| := max(1, |x − y|). The square of this function has logarithmic divergence when m ψ → 0. For small but fixed m ψ , this translates into the logarithmic behavior in O(1/λ) = O(log(1/m ψ )) of the bubble at zero transfer momentum (see (4.20)).
From the above it may be conjectured that logarithms appear only due to bubbles, and that they disappear by resumming bubble chains into the integrable kernel Σ. This is precisely what we do. The main result of the article is the following. Theorem 1.1 (main result) There exists λ max > 0 such that the following holds. Assume λ < λ max and N ≫ m 120 ψ , where m ψ ≈ e −π/λ is the solution of the gap equation (4.48). Then the (connected) two-point function ψ a i (x)ψ a j (x ′ ) of the model defined by 1., 2., 3. above has quasi-exponential decay at large distances, where C n is a constant independent from the parameters λ and N . *************************************** We may now resume our general presentation, and discuss the main elements of the proof of Theorem 1.1.

5.
A well-known problem in constructive field theory is that of local factorials, which we set about to explain briefly. Let us partition coordinate space into ⊎ ∆∈D ∆, where ∆ are square boxes of size m −1 ψ , corresponding to the inverse of the decay rate of the kernels C * and Σ. Forgetting about signs (so that our remark also holds for bosonic theories), the sum of the absolute values of the amplitudes of individual Feynman diagrams with a fixed number n a ∆ of fields ψ a per box ∆ ∈ D (and therefore, the same number of conjugate fieldsψ a ) involves the combinatorial factor ∆∈D a O(n a ∆ !), as a consequence of Wick's formula, a product over boxes ∆ of so-called local factorials.
There are several known approaches to cure this problem in the case of a fermionic theory, that can all go under the name of determinant bounds. (i) One is to note that the sum over all diagrams obtained by Wick's rule from may be written as a determinant of the general form N a=1 det C * (y a i , (y ′ ) a j ) i,j , where y a i , resp. (y ′ ) a j , are the locations of the fields ψ a , resp.ψ a , and to use for example Gram determinant bounds, see e.g. [14,15], and also e.g. [2,3,4,8] for some applications to important condensed matter models. (ii) Another argument, which is due to Feldman and can be found in the appendix of Iagolnitzer-Magnen [11], consists in Taylor expanding to a certain fixed order k the fields ψ a (y a i ) located in a cell δ around the center x δ of the cell (which is equivalent to Taylor expanding the covariance functions C * ). Expanding by multilinearity and keeping in mind that determinants of a matrix with two identical lines vanish, we see that all but a finite number of lines i with y a i ∈ δ must feature a Taylor remainder of the form (y a Choosing carefully the order k (large enough) and the diameters of the cells (small enough) produces extra inverse local factorials to the desired power, compensating the above-mentioned local factorials.
The problem now is that the partial resummation (step 4.) destroys the determinantal structure of Schwinger functions (the Σ-kernel is a bosonic two-point function). However, we have been able to modify Feldman's argument to make it compatible with the bubble resummation; see section 2. The core of the modified argument is the following. Determinants with two equal lines vanish as a consequence of the fact that they are alternating multilinear forms, namely, permuting fields, (ψ(y a i )) i −→ (ψ(y a σ(i) )) i according to a permutation of indices σ leaves the determinant invariant up to a sign ε(σ) (signature of the permutation). Then summing over all permutations yields zero because σ (−1) σ ε(σ) = 0. Each individual term in the determinant (arising from Leibniz's expansion into a sum over all permutations) is a Feynman diagram, so permutations define a mapping on the set of Feynman diagrams. Now, we prove that a (large, even if partial) group of permutations leaves our set of partially resummed Feynman diagrams FD invariant. This is the key to our generalized determinant bound.
6. The last point is to produce bounds for the sum of terms produced by the above expansions. Leaving aside the field translations and gradients produced by the Taylor expansion in 5., Feynman diagrams still have the same topological structure as Feynman diagrams of QED, namely, they present themselves as a set of single-flavored fermion loops connected by Σ-kernels. The general aim is to prove that all diagrams not reduced to a single propagator have extra 1/N factors. The first step is to choose a loop spanning tree and distinguish between connecting vertices (edges of the tree) and non-connecting vertices. The power-counting of connecting vertices is computed in such a way as to include the sum over the flavors of the loops. On the other hand, there is no flavor sum involved with non-connecting vertices, i.e. there is a 1/N factor involved with each of them. We conclude by proving that the number of non-connecting vertices increases linearly with the total number of vertices, and that the associated 1/N -factors ensure the convergence of the series of perturbations for N large enough.
Here is a very brief outline of the article. Our generalized determinant bound (see 5.) is presented in section 2. Constructive bounds (see 6.), in particular, a proof of Theorem 1.1, are presented in section 3. Finally, diagram and kernel estimates used in the proofs, together with a fixed-point argument to solve the gap equation, are presented in Appendix.
The ultra-violet limit of the model is not discussed in the present article. It was understood perturbatively in the original work by Gross and Neveu, and later on proved at the level of mathematical rigor in [6,7], that the model is asymptotically free at high energies. It would be interesting to couple our analysis with that of [6,7], and also to be able to overcome the strong, λ-dependent condition on N , in such a way as to 'construct' the model without any cut-off for N large enough, and an effective coupling constant λ small enough at momentum scale 1. We plan to do this in a near future.
Acknowledgements. Many warm thanks go to J. Magnen (even though he did not work actively on this project) for the long-standing collaboration which gave rise to this project.
Notations. We frequently use the notation f (λ, N ) g(λ, N ) when there exists a constant c > 0 independent of λ, N (possibly depending on the cut-off function χ) such that f (λ, N ) ≤ cg(λ, N ); the inequality holds in a (λ, N ) region (λ small enough, N large enough) which will be clear in the context. Similarly,

The generalized determinant bound
The section (see step 5. in the Introduction) is divided into two parts. In §2.1, we discuss the set FD of partially resummed Feynman graphs (which we simply call: set of Feynman graphs thereafter), and the action of a group of permutations S on FD. In §2.2, we Taylor expand covariance kernels of diagrams and show that averaging w.r. to the action of S kills leading order terms when the number of fields in a given cell of side m −1 ψ exceeds a certain finite value, leaving small factors that compensate local factorials.
Boxes are cells of side m −1 ψ (later on, in §2.2, we introduce smaller cells δ ⊂ ∆; only cells ∆ of side m −1 ψ are called boxes). All computations below are done assuming that all vertices are enclosed in a square volume Ω of the type The proof of Theorem 1.1 is given assuming implicitly that n → ∞, i.e. in the thermodynamic limit.

Action of permutations on the set of Feynman graphs
We fix in the ensuing discussion: -a finite set of external fields -for each ∆ ∈ ∆ and each flavor a, an integer number n a ∆ (the number of vertices of type (ψ a ψ a )(x) with x located in ∆, or equivalently, the number of fields ψ a (x) with x located in ∆; see Definition below). By assumption a n a ∆ > 0. Let n := (n a ∆ ) ∆∈∆,1≤a≤N and |n| := ∆,a n a ∆ . Below, we define an integer m a ∆ which is ≥ ⌈n a ∆ /5⌉ (= min{m ∈ N | m ≥ n a ∆ /5}). Delocalized vertices (see Fig. 3 In the remainder of the section, we shall say vertex instead of half-vertex. Fermion propagators (full lines) are oriented (following usual convention in QFT) fromψ to ψ.
Following fermion lines, one obtains single-flavored, oriented fermion loops. Fermion loops are connected by Σ-kernels. The precise structure of such diagrams will now be described.
-each vertex v ∈ V int has a flavor a(v) and a cell localization ∆(v) ∈ ∆; letting V a ∆ be the subset of internal vertices with flavor a and localized in ∆, |V a ∆ | = n a ∆ ; -E = ⊎ 1≤a≤n E a ⊎ E bub , where elements of E a are oriented edges (drawn as simple lines with an arrow giving the orientation), and elements of E bub non-oriented edges (drawn as dashed wiggling lines); a vertex v ∈ V int has two oriented edges with same flavor index a attached to it (one incoming, the other outgoing), and one non-oriented; -following oriented edges, one obtains closed loops, and open loops originated from and ending in external vertices; -non-oriented edges connect (closed or open) loops; -a closed loop γ is connected to ≥ 3 (closed or open) loops, i.e. there are ≥ 3 vertices along γ.
The length n(γ) of a loop γ is the number of vertices along γ. If n ≥ 0, we let be the set of Feynman diagrams with n vertices. Finally, FD := ⊎ n≥0 FD n . The evaluation A detailed example. Below, a Feynman diagram with 14 internal vertices numbered from 1 to 14 (see Definition 2.2), two external vertices (located at x, x ′ ), and three loops (to which we have given here arbitrary labels 1, 2, 3 for convenience) (called: distance of v i and v j along γ) is the minimal number of oriented edges along γ If v, v ′ are internal vertices which do not belong to the same loop, one defines d(v, v ′ ) = +∞.
Let Γ ∈ FD n . Fix ∆ ∈ ∆ and a = 1, . . . , N . Assume n a ∆ ≥ 1. We define inductively a shortlist of m a The algorithm stops when the set in (2.3) is empty, and m a ∆ is the last value of k. Then the ordering of internal vertices in Γ is permuted by shuffling them in such a way (For more readability we skipped the (∆, a)-indices, but v i , i = 1, . . . , n a ∆ are rewritten v a ∆,i , i = 1, . . . , n a ∆ below when (∆, a) are allowed to vary). We must still prove that m a We define inductively a sequence of non-empty sets of loops γ 1 , γ 2 , . . . , γ k , . . . , γ m a ∆ . Initially (k = 1), γ 1 is the set of loops containing at least one of the vertices in V a ∆ . Cutting out N (v i,1 ) from γ 1 , we get a new set of loops γ 2 . Then i 2 is the first index such that v i 2 belongs to γ 2 ; one cuts out N (v i 2 ) from γ 2 , and so on. The process may go on as long as γ k is non-empty. Since the total number of vertices along Consider now the following diagram This graphical operation on Feynman diagrams is called Π ∆,a (i, j). There are two different cases from a topological point of view; we letd be the new distance function between vertices. We use the fact that closed loops have ≥ 3 vertices along them.
If v a ∆,i and v a ∆,j were originally on the same loop, then the loop has split into two parts, and they are now on two different loops, sõ In the next subsection §2.2), we shall consider cells δ which are obtained by partitioning a box ∆ ∈ D, and fields ψ a i 1 , . . . , ψ a i m a ∆ will actually be gradient fields ∇ κ 1 ψ a i 1 , . . . , ∇ Then the same conclusion holds provided there exist 1 ≤ j = k ≤ i m a ∆ such that multi-indices κ i j , κ i k coincide.
A detailed example (continued). See Fig. 2 The effect has been to split the fermion loop labeled 1 into two loops along the edges 7 → 1 and 8 → 2 leading to 1 and 2 along the oriented loop.
The action of σ 13 yields instead The effect has been to merge the fermions loops labeled 1 and 2 along the edges 7 → 1 and 11 → 3.
Note that splitting/merging operations do not necessarily preserve diagram connectedness.
On the other hand, it preserves box-connectedness: if the diagramΓ obtained from Γ by gluing together vertices located in the same box is connected, then the same holds for the image of Γ by σ ∈ S n , in other words, σ(Γ) is box-connected.

Generalized bound
As explained in the Introduction, we adapt an argument from [11]. Fix k ∈ N large enough, but independently of all parameters N, λ, · · · (the value of k will be fixed in the end of the argument). For a given box ∆ and color a = 1, . . . , N , propagators C * a (x, ·) = ψ a (x)ψ a (·) 0 with x ∈ ∆ (∆ ∈ D) can be replaced using a Taylor expansion by where x δ is the center of a sub-box δ ⊂ ∆ containing x. This is equivalent to displacing the field ψ(x) to the location x δ . Terms on the first line (2.5) are called fully expanded terms, there are 2(1 + 2 + 2 2 + . . . + 2 k−1 ) = 2(2 k − 1) < 2 k+1 of them; terms on the second line (2.6) are called Taylor remainders.
Let n a δ be the number of flavor a fields located in δ, and (after reshuffling of the list of ψ a fields located in δ, see (2.3)) ψ 1 , . . . , ψ ⌈n a δ /5⌋ the ⌈n a δ /5⌉ ≤ m a δ first ψ a -fields, forming what we call the shortlist. The key argument is the following: by Property (P) (see below (2.4)), contributions with two fields ψ a (x δ ) in the shortlist with same (flavor, spinor and gradient) indices and located at the same point vanish after averaging over the action of the permutation group S δ,a . Therefore, at most 2 k+1 fields can be fully expanded; if ⌈n a δ /5⌉ ≥ 2 k+1 , then fully expanded terms associated to remaining ⌈n a δ /5⌉ − 2 k+1 fields give zero contribution, so remaining fields can be replaced by their Taylor remainders.

Constructive bounds
We prove here Theorem 1.1. We separate the two main issues. We first solve the problem of summing over topologies of Feynman diagrams (see §3.1); it is a power-counting argument, characteristic of 1/N expansions met in all theories with N flavors and O(1/N )-coupling constant, based in particular on the construction of a loop spanning tree. In the second part (see §3.2), we expound a classical, general constructive argument implying convergence of the sum over the box locations of vertices.

Loop spanning tree and 1/N expansion
Applying Taylor's formula ( §2.2) has 'displaced' the fields, ψ a i (x) ∇ κ ψ a i (x δ ), but not changed the topological structure of diagrams: we have a theory with delocalized vertices made up of two half-vertices v =ψ a ψ a , v ′ =ψ b ψ b connected by a Σ-kernel; following half-vertices, one gets monochromatic fermionic loops. The first step is to choose a loop spanning tree, i.e. a tree connecting loops, 4 3 2 The key point is that the number n γ of half-vertices along a given loop γ is ≥ 3: γ v = 1 is excluded because tadpoles are compensated by the mass counterterm (as a consequence of the gap equation), and γ v = 2 is excluded because bubble chains have been resummed.

1
A mild complication comes from the fact that kernels C * (x) and Σ(x) have a much worse behavior when |x| < m −1 ψ than the behavior expected from the large-distance estimates, (see Lemmas 4.1 and 4.5 for all these), but bounds for |x| varying from 1 to m −1 ψ have a characteristic inverse polynomial decrease in (m ψ |x|) −1 or |x| −2 (log(1/m ψ |x|)) −2 . A multiscale expansion -or, more simply perhaps, a detailed analysis of the convolution of C * and Σ-kernels in a diagram -would do justice to these factors in a proper way. Here we shall be content with using the less-than-optimal bounds (3.2), in which a loss of one or two powers of m ψ compared to (3.1) is manifest.
Let us redo the power-counting of (1.13). The result is generally expressed as a product of contributions coming from connecting vertices (explored inductively from the root of the loop spanning trees to its leaves by any search algorithm) and of contributions coming from non-connecting vertices. Let n be the total number of vertices of the graph, n ′ the number of connecting vertices, and n ′′ = n − n ′ the number of non-connecting vertices. Because (as emphasized above) there are at least 3 half-vertices along a given loop, L ′ ≤ 2 3 n. Now, n ′ = L ′ − 1 by construction, so and (assuming N ≥ m −4 ψ ) the total weight of the above graph is at most of order

Sum over cell locations
We prove in this section Theorem 1.1. We fix a flavor a; two spin indices i, j; two boxes ∆ ext , ∆ ′ ext ∈ D; x ∈ ∆ ext , x ′ ∈ ∆ ′ ext , and compute ψ a i (x)ψ a j (x ′ ) using the expansion (1.7). We resum by hand bubble chains as indicated in the Introduction, and apply the Taylor expansion of §2.2. Expanding produces a sum over permutations of an (unordered) set of n vertices, with n = 0, 1, . . .. We consider only box-connected contributions (see last paragraph of §2.1). Thus we must find a way to generate all box-connected diagrams (with vertices located in the same box glued together, and up to permutations of edges connecting the same pair of boxes (∆, ∆ ′ ), ∆, ∆ ′ ∈ D, where possibly ∆ = ∆ ′ ) containing boxes ∆ ext , ∆ ′ ext ; and bound the sum of all these terms. The combinatorial factor resulting from the sum over permutations of fields located in the same box has been taken into account in §2.2, and powers of N carefully unraveled in §3.1; as a result, we are left with O(N −1/6 ) per vertex, times the product of the decay factors for each kernel (see below) to help us sum over box-connected diagrams. This is a standard argument in statistical mechanics and constructive field theory; see e.g. [13], Corollary 5.3. Let q ≥ 1. The following algorithm generates all box-connected diagrams containing ∆ ext (and possibly ∆ ′ ext ) and a total number of q covariance kernels (counting both C * and Σ kernels). Start from ∆ 1 := ∆ ext . Sum at step 1 over all possible boxes ∆ ′ 1 ∈ D (including ∆ 1 ), and add a link between ∆ 1 and ∆ ′ 1 , i.e. a pairing ψ a (x 1 )ψ a (x ′ 1 ) 0 or ψ a (x 1 )ψ a (x ′ 1 ) 0 between fields located at is a scaled distance between two boxes. Summing over all possible cases yields a factor O(C p ), with By ordering boxes by their distance to ∆ 1 , one obtains provided p ≥ 3. Continue in a second step by picking a second link between ∆ 1 and a box ∆ ′′ 1 , and so on, until all pairings between fields in ∆ 1 and fields either in ∆ 1 or in any other box have been exhausted, each time producing a new multiplicative factor O(C p ) for the sum over all possibilities. Now one must look for all possible pairings between a field located in ∆ 2 defined as ∆ ′ i , where i := min{i ′ ≥ 1 | ∆ ′ i = ∆ 1 }, and a field located in ∆ ′ 2 = ∆ 1 , and so on. After q steps, the procedure stops.
The outcome is as follows. First, the sum over all possibilities has produced a factor (O(C p )) q . Then (since a delocalized vertex as on Fig. 3.1 contributes three kernels: two C * and one Σ) we have a small factor (O(N −1/6 )) ⌊q/3⌋ (see after (3.5)) left from the procedure in §3.1. Summing over q yields q≥0 (O(C p )) q (O(N −1/6 )) ⌊q/3⌋ = O(1) for N large enough.
We split the proof into several points. Using polar coordinates p = (ρ, θ), we obtain
Proof. We use the following identities, where err. denotes O(1) error terms coming from the UV cut-off. Now, γ-identities yield Other terms contain an odd number of γ-matrices and therefore vanish. Hence (letting (4.31) with error terms O(1) coming from the UV cut-off.
We start with a somewhat informal discussion, whose outcome (4.34, 4.36) shows the connection to the auxiliary field approach in [12], but is not needed for our estimates. From Lemma 4.3 1. and Lemma 4.4, to leading order as |q| → 0 and λ → 0, could be interpreted as the mass of an equivalent intermediate field as in [12]. Fourier inversion yields asymptotically The above computations are simply based on the second-order Taylor expansion of the πkernel around 0, so they should not be taken at face value, and in any case, they give no indication on the behavior of Σ(x) for |x| 1/m ψ , nor on the L 1 -norm of Σ. This is the purpose of our next Lemma. 1. Assume |q| ≫ m ψ , then 1−π(q) λ log(|q|/m ψ ) and and then (by induction on |κ|, using (4.23)) |∇ κ Σ(q)| 1/N |q| |κ| log 2 (|q|/m ψ ) . 3. (L 1 -bound) We let j φ := ⌊ π/λ ln(2) ⌋ be the gap energy scale, so that