The Low Lying Energy-Momentum Spectrum for the Lattice Four-Fermi Model

We obtain the low-lying energy-momentum spectrum for the imaginary-time lattice four-Fermi or GrossNeveu model in d + 1 space-time dimensions (d = 1, 2, 3) and with N -component fermions. Let 0 < κ 0 be the hopping parameter, λ > 0 the four-fermion coupling, m > 0 the bare fermion mass and take s × s spin matrices (s = 2, 4). Our analysis of the one and the two-particle spectrum is based on spectral representation for suitable twoand four-fermion correlations. The one-particle energy-momentum spectrum is obtained rigorously and is manifested by sN 2 isolated and identical dispersion curves, and the mass of particles has asymptotic value order − lnκ. The existence of two-particle bound states above or below the two-particle band depends on whether Gaussian domination does hold or does not, respectively. Two-particle bound states emerge from solutions to a lattice Bethe-Salpeter equation, in a ladder approximation. Within this approximation, the ( sN 2 − 1) sN 4 identical bound states have O(κ) binding energies at zero system momentum and their masses are all equal, with value ≈ −2 lnκ. Our results can be validated to the complete model as the Bethe-Salpeter kernel exhibits good decay properties. MSC 2010: 81Qxx, 81Txx, 81Vxx


Introduction
The determination of elementary excitations in physical models is of great importance, as they determine the time evolution of quantum systems, and are related to the falloff rate of correlation functions (CFs). Furthermore, in models of classical statistical mechanics and quantum field theory, the spectra of the associated Hamiltonian and momentum operators can be simultaneously determined and are associated with a particle interpretation.
In this work, we determine the low-lying particle spectrum of an imaginary-time (Euclidean) lattice functional integral formulation of a four-Fermi or Gross-Neveu (GN) model, with N flavor-component fermion fields in d + 1 dimensions (d = 1, 2, 3 is the spatial dimension) and a global U(N ) flavor symmetry. This is an interesting model, both from theoretical and experimental point of view, because it presents a set of nice features and has a nontrivial behavior. For example, in d = 1, the GN model is perturbatively renormalizable, asymptotically free in the ultraviolet scale (if N > 1) and also exhibits the phenomenon of dynamical symmetry breaking with mass generation and dimensional transmutation (see [16]). On the other hand, the GN model is simple enough to be, in principle, accessible via purely analytical approaches (see [6,1]). Moreover, being a model with pure fermion fields, and due to the fact that the Pauli principle implies that fermion fields are associated with bounded operators, the GN model is placed in a special position to apply both perturbative and non-perturbative techniques to obtain the rigorous control of the ultraviolet and infrared limits, as well as some important correlation properties, going far beyond the level of the most common analysis employing simply a few perturbation orders. In this way, the ultraviolet limit of the complete model was constructed in [13,14] for d = 1 and N > 1, and asymptotic completeness has been verified up to the two-particle threshold in [19]. In [23], the existence of the infrared limit and mass generation was proved for the complete model, for d = 1 and N 1. For d = 2, this was done in [30] and, though the model is perturbatively non-renormalizable, its ultraviolet limit was constructed for large but finite N and small values of the coupling constant in [7], using non-perturbative analytical techniques. This corresponds to one of the few examples of a theory which is not trivial but is asymptotically safe, with a stable renormalization group fixed point, and has become a source of inspiration to the search of a good quantum model for Einstein's gravity (see e.g. [3,5,17,24,25] and references therein). The existence of the ultraviolet limit of the model for d = 3 is an open question. However, the question of the existence of the ultraviolet limit or how to adapt the previous constructions to our cutoff model does not bother us here since we keep the unitary lattice step fixed throughout this work. Furthermore, it is worth mentioning that the GN model plays an important role in testing properties of more sophisticated field theoretical models [41] and in the description of interesting physical phenomena such as superconductivity and strong interactions (see e.g. [21,26]).
Let 0 < κ 1 denote the lattice hopping parameter, M > 0 the fermion bare mass, λ > 0 the four-fermion coupling and let us consider s × s Euclidean spin matrices. Here, we consider the lattice GN model defined by the action with summation over repeated indices and whereψα,a(x) and ψ β,b (y) are Grassmann variables, where Is is the s-dimensional identity matrix. For d = 2, 3, we take s = 4 and γ μ , μ = 0, 1, . . . , d are the 4 × 4 Euclidean Dirac matrices where σ j , j = 1, 2, 3, are hermitian, traceless and anti-commuting Pauli spin matrices. For d = 1 and alternatively for d = 2, we can consider another choice of spin matrices by taking s = 2, From now on, we will concentrate our text only on the d = 3 case, with Dirac matrices. The analysis for the other dimensions is very similar and can be easily reproduced from the one given here.
To be precise, we start in a compact box of finite-volume Λ ⊂ Z d+1 o , of size |Λ|. Using fermionic (Berezin) integration (for details see [2]), the normalized statistical mechanical expectation or averages of a function F (ψ, ψ) of the fields is formally given by where dψ dψ = x,α,a dψα,a(x)dψα,a(x) and Z Λ = Λ e −S(ψ,ψ) dψ dψ. The 2n-point CFs are given by choosing F (ψ, ψ) =ψ(y 1 ) · · ·ψ(yn)ψ(z 1 ) · · · ψ(zn), for external points y's and z's. The CFs vanish if the number ofψ and ψ are different, and are all real, for κ = 0. The choice of the shifted lattice for the time direction, avoiding the zero-time coordinate, follows what happens for bosonic models. In this case, in the continuum limit, two-sided equal time limits of field correlations can be accommodated. This is also the case here, at least for d = 1. Besides, the above choice of parameters guarantees, with the extra condition | κ M | e −5 4(d+1)sN |1 + 2λ M 2 | 1−sN , that the thermodynamic limit Λ → Z d+1 o of the model exists, and the physical positivity holds [29]. For M = 0, the lattice thermodynamic limit (if any!) cannot be easily reached, at least in the same way as before, and taking M > 0 guarantees that the one-particle free fermion dispersion curve is monotonically increasing in each momentum component, which is physically desirable.
To show the existence of the thermodynamic limit, we use cluster expansion techniques, such as a polymer expansion (see [38,39]). In the infinite-volume limit |Λ| → ∞, denoted by · , are lattice translation invariant and joint analytic in κ and λ for | κ M | < e −5 4sN (d+1) . As a consequence, the truncated CFs have exponential tree decay with rate of at least (1 − ε)| ln(O(1)κ)|, where ε → 0 as κ → 0. From now on, we work only in the infinite-volume limit.
For completeness, we give a brief and intuitive idea on how the polymer expansion is generated. Like a standard virial expansion in statistical mechanics, in the region of parameters where the coupling between different lattice sites is weak, we write e κψ(x)Γ ψ(y) = 1 + ϑ xy and expand the expectation value of an observable O in the small parameter ϑ xy . By this expansion, we obtain decoupled expectations of the form O <xy>∈ς ϑ xy Λ | κ=0 , where ς is a set of lattice links. Note that the expectation AB Λ | κ=0 factorizes into A Λ | κ=0 B Λ | κ=0 whenever A and B depend on disjoint sets of field variables. A polymer ςp is a set of field variables that does not break up into disjoint subsets. So, we obtain an expansion for Oe −S Λ | κ=0 in terms of the quantities z(γp) = O <xy>∈ςp ϑ xy Λ | κ=0 , called the activity of the polymer ςp. Writing this expansion for both the numerator and the denominator in (1.2), we can take the quotient and show that the resulting expansion has a finite domain of convergence (see the convergence condition above) uniformly in the volume Λ of the finite box Λ, so that infinite-volume limit can be taken.
For κ 0, the action of (1.1) preserves positivity. Thus, by the fundamental work of [27,28], the CFs can be analytically extended to a quantum filed theory in the Minkowski space. The idea consists of using the CFs to define (through a Feynman-Kac formula [15]) an inner product in the underlying physical quantum mechanical Hilbert space of physical states H (see [29,38]). This construction provides an imaginary-time lattice quantum field theory. Besides, associated with this model, we can show there is a positive metric Hilbert space, an inner product (F, G) H , a vacuum vector Ω and field operatorsψ a α (x), ψ a α (x). Thus, we can construct in H commuting self-adjoint energy-momentum (E-M) operators H 0, P = (P 1 , . . . , P d ) (with spectral points p ∈ T d ≡ (−π, π] d ). These operators are the generators of lattice semi-group of time translations T 0 , with T 2 0 > 0, and the unitary group of space translations T = e i P · x , respectively. Therefore, by the spectral theorem, the operators H and P can be diagonalized in a common basis, and so points in the spectrum of H can be written as functions of the points in the spectrum of P .
A straightforward consequence of the above construction is that the squared transfer matrix T 2 0 = e −2H , in the infinite volume limit, is a well-defined self-adjoint contraction operator in the Hilbert space H if we assume that T 0 does not have a 0 eigenvalue.
The methods we apply here allow us to access the low-lying energy-momentum spectrum and were introduced and established in [8,34,35,36,37], for lattice spin chains, and in [9,11,12,33], for treating other field theoretical models. We analyze the one-and two-particle sectors of the energy-momentum spectrum. Up to near the threeparticle threshold, the energy-momentum spectrum exhibits isolated dispersion curves that are identified as particles and bound states. In the one-particle sector, using our method and the U(N ) flavor symmetry, we show that there are sN 2 particles corresponding to identical and isolated dispersion curves w( p) ≡ w( p, κ, λ). We thus provide a rigorous proof of both the lower and upper mass gap properties of the spectrum. Also, the particle masses m(κ) ≡ w( p = 0) are all equal, and we show that the asymptotic value of the one-particle masses are of order (− ln κ).
Above the one-particle spectrum, there are finite width free (noninteracting) multiparticle spectral bands. Using a lattice version of the Bethe-Salpeter (B-S) equation, we show that two-particle bound states may appear near the two-particle band. At least in a leading approximation, that we call the ladder approximation, the existence of these bound states, either above or below the free two-particle band, depends on whether or not a Gaussian domination condition holds, respectively. The Gaussian domination condition is characterized by the negative sign of the quantity Otherwise, if ℵ > 0, we say that we are in the Gaussian subjugation regime. Within our ladder approximation, the ( sN 2 − 1) sN 2 bound states we detect in any case have masses which are all equal and of order (−2 ln κ), with an O(1) (κ-independent) binding energy. This spectral pattern is similar to the one obtained in our previous work on the O(N ) spin model (see [8]).
We note that a discussion on the asymptotic completeness up to the two-particle threshold, for the Gross-Neveu model in the continuum space-time, can be found in [20]. Of course, to establish this result we need to know about the existence of two-particle bound states below this threshold. This paper does not solve the problem, however, so that our bound state analysis for the model on the lattice is justified. Besides this point, our spectral bound states results are relative to the two-particle band. This is a feature of the lattice model, which is absent in the continuum spacetime case. An interesting point is to see how much of the results obtained for the lattice subsist in the continuum limit.
It is also worth emphasizing that our spectral results are obtained from spectral representations for correlations which follow from the spectral theorem applied to the energy-momentum operators and a Feynman-Kac formula. This is a fundamental point in our work that guarantees we are indeed talking about the particle spectrum. Note that the commonly found analysis in the theoretical physics literature, involving only the decay rate of correlations, is not sufficient to this end. The analysis is still more delicate when various spectral points are too close and a continuous spectrum is present.
Last, we note that our method incorporates the main ingredients to validate our bound state results beyond our ladder approximation to the complete model, as it was done in the context of the stochastic Ginzburg-Landau models and for the ferromagnetic spin models (see [9,35,36]). The basic ingredient is the fast enough decay properties satisfied by the B-S kernel K (see below). This result will be established elsewhere.
The remainder of the paper is organized as follows. In Section 2, we introduce the two-and four-point CFs and obtain the corresponding spectral representations. We also discuss some symmetry properties that are used to simplify these CFs. In Section 3, we obtain the one-particle spectrum. In Section 4, we introduce the lattice B-S equation and derive our ladder approximation to the B-S kernel. In Section 5, we solve the B-S equation in this approximation and obtain the two-particle bound states and the corresponding binding energies. Finally, we draw some conclusions in Section 6.
, F and G functions of the fields with finite support on x 0 > 1 2 , and for the uppercase checkˇdenoting an operator on H, taking x 0 > 0, we have the Feynman-Kac formula [15] (recall that Ω denotes the vacuum) and Θ is an anti-linear operator which involves time reflection. More precisely, the action of Θ on the fermion fields is given by The virtue of the Feynman-Kac formula (2.1) is that it relates inner products involving functions of the field operators in the Hilbert space (left-hand side) with statistical mechanical random process (field variables) correlations (righthand side). In our method, the key idea to detect particles and bound states in the E-M spectrum consists of using the Feynman-Kac formula (2.1) to obtain spectral representations for convenient truncated two-and four-point CFs. It is these spectral representations that allow us to mathematically justify the relation between the complex momentum singularities of the Fourier transform of these CFs to points in the E-M spectrum (see [8]). This basic ingredient is missing in many works in the literature, where only the exponential decay rates of correlations are obtained. These decay rates may, of course, have no relation with the spectrum, specially when narrow spectral splittings and spectral bands are involved.
In order to obtain the one-particle spectrum, that is the spectrum associated with the subspace generated by vectors of the formψ αa(x)Ω, we analyze the following two-point CFs: where χα is the characteristic function of the set α and * denotes complex conjugation. Using the spectral theorem, we can write spectral representations for the T x 0 0 and T x operators given aš where E 0 (λ 0 ) and E j (λ j ) are the respective spectral families ofŤ 0 ,Ť j , and we define E(λ) = E 0 (λ 0 ) d Using these representations in the Feynman-Kac formula (2.1), we obtain the spectral representation for the twopoint CFs: To obtain the two-particle spectrum, that is the spectrum associated with the subspace generated by vectors of the formψα 1 a1 (x 1 = (x 0 , x 1 ))ψα 2 a2 (x 2 = (x 0 , x 2 )) Ω, we analyze the partially truncated four-point CFs: where ab (resp. αβ) means a 1 a 2 b 1 b 2 (resp. α 1 α 2 β 1 β 2 ), and we work in the equal time representation (i.e. x 0 1 = x 0 2 and y 0 1 = y 0 2 ). Using the lattice relative coordinates (see [33]) Note that vacuum contributions are automatically removed in (2.4) since they vanish by the imbalance on the number of fermion fields. Thus, D has an exponential temporal decay in τ 0 . Next, in the same way as before for the two-point CFs, we can write the spectral representation (τ 0 = 0) , using (2.3) and lattice translational invariance, the Fourier transformG ab αβ (p) of G ab (2.6) Therefore, for p fixed, the singularities in the imaginary p 0 axis are associated with the one-particle spectrum. Similarly,D ab αβ (p, q, θ 0 , τ ), the Fourier transform of D ab αβ ( ξ, η, τ ) is given by Thus, for p fixed, the singularities in the imaginary p 0 axis determine the twoparticle spectrum and so bound states.
As we stressed above, without these spectral representations, the analysis of the decay rate of suitable CFs may not lead to the identification of particles and bound states. Particles and bound states are isolated dispersion curves in the E-M spectrum. If there is by chance a relation with the spectrum, for example, decay rates may be related to points within a spectral band.
The CFs given in (2.2) and (2.4) are associated with matrix-valued operators on convenient spaces, and we can use symmetries at the level of CFs to simplify them. We follow [11] where the most usual symmetries were treated in detail. A symmetry operation Y defined on the Grassmann algebra G of the fields ψ andψ is an operation of G onto itself such that where A and B are 4 × 4 complex matrices and Y is a bijective linear map of Z d+1 0 into itself. The symbol ∼ stands for the introduction or the removal of a bar. A symmetry of the model is defined to be a symmetry operation which leaves invariant the action (1.1) and the CFs. Usual symmetries such as discrete spatial rotations, time reversal T , charge conjugation C and parity P, which can be implemented as unitary (or anti-unitary for time reversal) operators in the Hilbert space H, are symmetries of the model (see [12]). We also found a new time reflection symmetry T, which can be implemented only at the CFs level, and which is described by By applying C, the one-particle and the one anti-particle CFs are the same. So, we only need to consider the lower spin indexes (i.e. α, β = 3, 4) in the CFs given by (2.2) and (2.4). Next, using CP T, we see that the two-point CFs G defined by (2.2) is a multiple of the identity. So, we can drop the spin and flavor (due to U(N ) flavor symmetry) indices for the two-point function.
Using these properties, next we discuss the determination of the one-and the two-particle spectrum.

One-particle spectrum
Concerning the one-particle spectrum, we adapt the analysis of [34,39] for spin models. By (2.6), we just need to determine the imaginary p 0 singularities of the Fourier transform of G. To identify these singularities, we use the correspondence between the spacetime decay of the two-point CFs and the width of the analyticity strip of the two-point CFs in momentum coordinates, in accordance with the Paley-Wiener theorem (see [31,32]). Thus, we need a detailed control of the spacetime decay rates of the two-point CFs. Also, it is important to note that the polymer expansion employed here can be controlled even with the introduction of distinct hopping parameters for each lattice link. To explore these features, we use the hyperplane decoupling method (see [40]). This is done by introducing a family of new lattice link variables za µ , μ = 0, 1, . . . , d, where a 0 ∈ Z 1 2 , and a i ∈ Z, i = 1, . . . , d, by changing the hopping parameter from κ to za µ , in all action terms that connect lattice point pairs such that (x, x + e μ ) and (x + e μ , x), with x μ = a. Thus, taking za µ = 0 separates the model in two disconnected regions, whose frontier is given by the aμ hyperplane. The application of the hyperplane decoupling method corresponds to Taylor expanding the interpolated CFs up to first order in each interpolating variable za µ . By construction, we have G(x, y, za i , κ) = 0, for za i = 0 and all x i a i < y i . On the other hand, the polymer expansion allows us to extend the interpolated CFs to complex za µ , so from the Taylor expansion, we can write G x, y, za µ , κ = 1 2π where C is a circle with center in za i = 0 and radius given by the convergence radius of the polymer expansion. This integral representation allows us to apply a Cauchy estimate and shows that G(x, y) satisfies the global bound

by a generalized Paley-Wiener theorem, this global bound implies thatG(p) is analytic in a strip of width
Moreover, we note that the za µ derivative of G has the product formula which is an instrumental property in what follows.
With the introduction of the hyperplane variables, CFs are no longer invariant under translations but, by Holmgren's lemma (see [39]) and (3.1), we can show that the modified G defines a bounded operator in the space ). Thus, taking the decomposition where G d collects only local contributions to G, as a matrix operator on 2 (Z d+1 0 ), with G(x, x) = 0, and thus we can define the convolution inverse correlation Γ through a convergent Neumann series applied to the factor Using the hyperplane decoupling method again and the product formula of the za i derivative of G, we see that Also, a straightforward application of the hyperplane decoupling method allows us to show that Γ (x, y) = 0 and ∂ 2 Γ ∂z 2 a i = 0 for za i = 0 and all x i a i < y i . So, by a Cauchy estimate, we also get that the improved global bound which holds for x − y > 1. This bound shows that Γ (x, y) has a faster decay than G(x, y). Using again the Paley-Wiener theorem, this faster decay implies that the Fourier transformΓ (p) = x Γ (x)e −ip·x is analytic up to close to the three-particle threshold, which is approximately −(3 − ) ln κ. This strip of analyticity is larger than the one forG(p). Hence,Γ −1 provides an analytic continuation forG on this enlarged strip. This property is important either to prove the upper mass gap property or for the analysis of the two-particle spectrum given below.
From (2.6), the one-particle E-M spectrum is determined by the zeroes of detΓ . Using (3.2) and a Taylor expansion of G in κ = 0 allows us to write Next, letΓ 0 (p 0 , p) be the Fourier transform of Γ (x) − Ω(x). We see that, for p ∈ T d , detΓ 0 (p 0 , p) has exactly one root E( p) = ln ζ 0 , with So, by applying the analytic implicit function theorem (see [18]) and Rouché's theorem (i.e. the principle of the argument, see [4]), we can show that, since κ is small, the zeroes of detΓ (ω(p 0 , p)) are close enough to E( p). Therefore, we see that there is a lower mass gap of multiplicity sN 2 (due to spin and flavor symmetries), corresponding to the isolated one-particle dispersion curves (3.4) The one-particle mass ω( p = 0) is given by where g 2 = ψ αa(x)ψαa(x) | κ=0 . Above the one-particle spectrum there are two-, three-, . . . free particle bands, with finite width, and eventually overlapping for a large particle number.
In the next section, we inspect for the existence of two-particle bound states near the free two-particle band.

Bethe-Salpeter equation and ladder approximation
To perform our bound state analysis, we use a lattice version of the B-S equation for this model which, in operator form, reads In terms of its operator kernel, D 0 is given by  Using the exponential decay of the two-point CFs, we see that D 0 decays and has a finite norm. On the other hand, the polymer expansion implies that D T = D − D 0 , the connected (truncated) four-point function, has an exponential tree decay, so also D has a finite norm. Therefore, both operators exist as bounded operators in 2 (A), . Decomposing D and D 0 into diagonal and non-diagonal contributions, we can use again Holmgren's lemma and a Neumann series argument to show that D −1 0 , D −1 and B-S operator K also exist as bounded operators in 2 (A). With this, (4.3) is validated starting from the B-S equation (4.1).
In this paper, we restrict our analysis of the two-particle bound state spectrum to the leading approximation of the B-S operator, which we call a ladder approximation. By computing K to the leading nonvanishing κ contribution K ≡ L, and using L to solve (4.1) for D, we then look for the singularities ofD( ξ, η, k), the τ Fourier transform of D( ξ, η, τ ), to determine the two-particle bound states.
We now show that L is local and O(κ 0 ). To do this, we write where D T is the connected (truncated) four-point function. At κ = 0, D T vanishes for non-coincident points (see [8]), as seen by writing it in terms of source derivatives of the logarithm of the generating function. Using a simple Taylor expansion in κ = 0, we find where Pxy = δx 1 x2 δy 1 y2 δy 1 x2 , S ab αβ = δα 1 α2 δ β1β2 δ β1α2 δa 1 a2 δ b1b2 δ b1a2 , 8

Journal of Physical Mathematics
Ia is the identity on the anti-symmetric subspace of 2 (A), and g 4 is a function of m, λ, N defined by With all this, we have where ℵ is defined in (1.3) and with I being the 4N 2 × 4N 2 identity matrix. In order to proceed, it is worth mentioning that the B-S equation (4.1) has the same structure as an operator Schrödinger equation in the resolvent form, with D 0 being similar to the resolvent (H 0 − z) −1 for the free Schrödinger Hamiltonian operator (kinetic energy) H 0 , D the interacting resolvent (H − z) −1 for the Hamiltonian H = H 0 + V and, modulo a multiplicative constant, K is the potential V . We refer to [10] for more details. In view of this similarity, consider a Schrödinger Hamiltonian for two interacting particles with mass m > 0 in ( 2 (Z d ) × 2 (Z d )), given by where Δ 1 = Δ ⊗ I, Δ 2 = I ⊗ Δ and −Δ is minus the lattice Laplacian. The action of −Δ on f ∈ 2 (Z d ) is given by The interaction potential V 2 (x − y) has the form where, by hypothesis, the W term collects all non-local contributions to the potential V 2 , and has exponential decay of order m −2 . Thus, for λ > 0, the first term is a repulsive hardcore potential. The corresponding resolvent Schrödinger equation is given by (4.5) By identifying L to −V 2 , the resolvent equation (4.5) is analogous to our B-S equation in the ladder approximation.
Using the above decomposition of V 2 , we find where with L( x, y) = λδ( x − y).
Hence, restricting our attention to the dominant contribution to the potential, we obtain the spectral properties of H l = H 0 + λδ. Let σ(O) denote the spectrum of the operator O. If z ∈ σ(H l ) ∪ σ(H 0 ), then (4.7) yields to For z ∈ [0, 4d], σ(H l ) coincides with the z singularities of (4.7), that is the zeroes of the denominator of (4.8).
Hence, taking k = 0, we have the bound state equation (4.9) Consider the attractive case (λ < 0). For small momenta, −Δ( p) ≈ | p| 2 . Thus, for d = 1, 2, (4.9) diverges as E b → 0, and since the integrand is strictly monotone in the binding energy E b > 0, we can find a unique bound state solution for any λ < 0. For d 3, the integral in (4.9) converges absolutely and remains finite as E b → 0, therefore there is a critical value λc < 0 that allows bound states to exist. This is a consequence of the Birman-Schwinger bounds given for example in [31]. Also, (4.6) and a perturbation argument can be used to show that the singularities of (H 2 − z) −1 are close to the singularities of (H l − z) −1 .
In the next section, we rigorously determine bound states in the ladder approximation.

Bound states
In the ladder approximation, the B-S equation reads where we sum over repeated indices. Changing to relative coordinates and lettingf ( ξ, η, k) = τ f ( ξ, η, τ )e −ik 0 τ 0 denote the Fourier transform in τ only. At zero spatial momentum ( k = 0), we find Therefore, we obtain from (5.1) By inspection, there are only two distinct eigenvalues. First, for i = 1, . . . , 2N 2 + N , we have μ i (k 0 ) = 1. Thus, this eigenvalue cannot satisfy the bound state condition. Also, note that the corresponding eigenvectors lie in the symmetric subspace of R 4N 2 . Second, for i = 2N 2 + N, . . . , 4N 2 , we have the eigenvalue μ i (k 0 ) = 1 − 2ℵR(k 0 ), and the corresponding eigenvectors lie in the antisymmetric subspace of R 4N 2 . These states, in principle, can satisfy our bound state condition. For them, the bound state condition is described by R(k 0 ) = (2ℵ) −1 . We now analyze this equation, using the spectral representation established in (2.3) and the conclusions of the analysis of the two-point CFs performed in Section 3, to obtain the spectral representation (see [34,39]) where, for ω( p) given in (3.4), we use the measure decomposition dσ p (E) = Z( p, κ)δ E − ω( p) dE + dσ p (E), (5.4) with Z( p, κ) = ∂Γ ∂χ (p 0 = iχ, p)| χ=ω( p) , and whereΓ (p) = [G(p)] −1 . Both dσ p (E) and dσ p (E) are positive measures. The measure dσ p (E) has support in (m, ∞), wherem ≈ −3 ln κ approaches from below the onset of the free three-particle spectrum. The decomposition in (5.4) of dσ p (E)d p separates the one-particle contribution from the hyperplane decoupling method. In terms of the lattice relative coordinates of Section 2, we show the global bound K ab αβ ( ξ, η, τ ) O(1)|κ| 4|τ 0 |+ 1 2 ( ξ+ η+2 τ + ξ + η ) , revealing a decay with a rate greater than the two-particle one. Whether or not the degeneracy of the bound states is preserved in the complete model is an open question. Finally, we remark that the spectral pattern obtained here for the U(N ) GN model is similar to the one found for the O(2N ) spin model in [8], except that the latter has also a bound state of a different multiplicity, 2N .