3D Field Theories with Chern--Simons Term for Large $N$ in the Weyl Gauge

Three dimensional, $U(N)$ symmetric, field theory with fermion matter coupled to a topological Chern--Simons term, in the large $N$ limit is analyzed in details. We determine the conditions for the existence of a massless conformal invariant ground state as well as the conditions for a massive phase. We analyze the phase structure and calculate gauge invariant corelators comparing them in several cases to existing results. In addition to the non-critical explicitly broken scale invariance massive case we consider also a massive ground state where the scale symmetry is spontaneously broken. We show that such a phase appears only in the presence of a marginal deformation that is introduced by adding a certain scalar auxiliary field and discuss the fermion-boson dual mapping. The ground state contains in this case a massless $U(N)$ singlet bound state goldstone boson- the dilaton whose properties are determined. We employ here the temporal gauge which is at variance with respect to past calculations using the light-cone gauge and thus, a check (though limited) of gauge independence is at hand. The large $N$ properties are determined by using a field integral formalism and the steepest descent method. The saddle point equations, which take here the form of integral equations for non-local fields, determine the mass gap and the dressed fermion propagator. Vertex functions are calculated at leading order in $1/N$ as exact solutions of integral equations. From the vertex functions, we infer gauge invariant two-point correlation functions for scalar operators and a current. Indications about the consistency of the method are obtained by verifying that gauge-invariant quantities have a natural $O(3)$ covariant form. As a further verification, in several occasions, a few terms of the perturbative expansion are calculated and compared with the exact results in the appropriate order.

ABSTRACT Three dimensional, U (N ) symmetric, field theory with fermion matter coupled to a topological Chern-Simons term, in the large N limit is analyzed in details. We determine the conditions for the existence of a massless conformal invariant ground state as well as the conditions for a massive phase. We analyze the phase structure and calculate gauge invariant corelators comparing them in several cases to existing results. In addition to the non-critical explicitly broken scale invariance massive case we consider also a massive ground state where the scale symmetry is spontaneously broken. We show that such a phase appears only in the presence of a marginal deformation that is introduced by adding a certain scalar auxiliary field and discuss the fermion-boson dual mapping. The ground state contains in this case a massless U (N ) singlet bound state goldstone boson-the dilaton whose properties are determined. We employ here the temporal gauge which is at variance with respect to past calculations using the light-cone gauge and thus, a check (though limited) of gauge independence is at hand. The large N properties are determined by using a field integral formalism and the steepest descent method. The saddle point equations, which take here the form of integral equations for non-local fields, determine the mass gap and the dressed fermion propagator. Vertex functions are calculated at leading order in 1/N as exact solutions of integral equations. From the vertex functions, we infer gauge invariant two-point correlation functions for scalar operators and a current. Indications about the consistency of the method are obtained by verifying that gauge-invariant quantities have a natural O(3) covariant form. As a further verification, in several occasions, a few terms of the perturbative expansion are calculated and compared with the exact results in the appropriate order.

Introduction
It has been conjectured [1,2] that AdS/CFT correspondence implies a close relation between the singlet sector of O(N ) and U (N ) bosons vector theory in d = 3 space-time dimensions at large N and Vasiliev's higher spin gravity theory [3] on AdS 4 . Computation of correlation functions of the higher spin gauge theory [4,5] strengthen this conjecture. Since the quantum completion of the gravitational side of this duality is not known, only tree level could be considered. The AdS/CFT correspondence in this case provides a testing ground of AdS/CFT ideas. The large N limit of O(N ) κ and U (N ) κ level κ Chern-Simons gauge theories, at N, κ → ∞ with a fixed 't Hooft coupling λ = N/κ, coupled to scalar and fermion matter fields in the fundamental representation in d=3 dimensions were studied recently [6][7][8][9][10][11]. Since pure Chern-Simons gauge theories has no propagating degrees of freedom, only the matter fields in the fundamental representation provide the true independent canonical degrees of freedom in these d = 3 theories.
O(N ) and U (N ) symmetric field theories at large N are reasonably well understood [12]. In the case of Chern-Simons matter theories in d = 3 dimensions, the explicit calculations that were performed [6][7][8][9][10][11] shed extra light on the AdS/CFT correspondence. The original O(N ) model of g(φ 2 ) 2 in [1] was also deformed to include a marginal interaction term ∼ λ 6 (φ 2 ) 3 . The massless conformal invariant phase was analysed either by a generalized Hubbard-Stratanovich method [6] or in perturbation theory [7]. It was conjectured that the gravity dual of these 3D theories on the boundary is a parity broken version of Vasiliev's higher spin theory [3] on AdS 4 in the bulk with a parity breaking parameter θ which depends on the 't Hooft coupling λ = N/κ. Supersymmetric extension of this idea was introduced in [8]. It has also been conjectured that there is a duality between the boson and fermion theories [9] since the theory of bosons and fermions in the fundamental representation of U (N ) coupled to Chern-Simons gauge fields are dual to the same bulk gravity theory on AdS 4 . Calculation of the thermal free energy in large N Chern-Simons field coupled to fermions and bosons has further strengthened this duality [11]. The thermal free energy has been calculated also in [6].
It was stated in Ref. [11] that the introduction of masses through spontaneous breaking of scale invariance [13,14] is an alternative at large N . Though at finite N the breakdown of the scale symmetry is explicit, it is of order O(N −1 ), namely the same order by which the AdS/CFT correspondence is approximated in the above mentioned conjectures. Therefore it was found appealing to further analyse the Chern-Simons matter theory with bosons [15] and fermions [16] with masses introduced through spontaneous breaking of scale invariance, which ensures the existence of a massive ground state but leaves the boundary d = 3 theory conformal to order O(N −1 ). The calculations mentioned above were done using the light cone gauge fixing for the Chern-Simons gauge theory.
In this article we analyse the large N limit of three dimensional, U (N ) symmetric, field theory with fermion matter coupled by the topological Chern-Simons gauge field. Our calculational framework differs from the above mentioned calculations as we are employing the A 3 = 0 gauge rather than the light-cone gauge.
In the temporal gauge, as in the light-cone gauge, the Chern-Simons action is reduced to a quadratic form and allows integrating out the gauge field. The large N properties are determined by using then a path integral formalism and the steepest descent method in Euclidean space-time. In section 2 we start with a short discussion of the origin and properties of the topological Chern-Simons action. In section 3, the detailed description is presented of the Chern-Simons gauge field in the A 3 = 0 gauge coupled to U (N ) fermions in the fundamental representation. The large N action is discussed and the saddle point equations are determined. The saddle point equations, which take here the form of integral equations for non-local fields are then solved. In particular, the fermion dressed two point functionΓ (2) (p), the gap equation and the ground state energy in the limit N → ∞ are calculated in details. in section 3 we also derive the gauge-independent correlators ofψ α (x)ψ α (x) two-and three-point functions at zero momentum. As mentioned, these results are obtained by calculating the relevant path integral at large N and thus they are further checked in section 4 by calculating in perturbation theory the fermion propagator and the ground state energy up to two-loops and three-loops, respectively. In section 5 the vertex functionΓ 1,2 (k; p 1 , p 2 ) is calculated at leading order as solutions of integral equations.
Using the exact expression of the vertex function given in section 5, we compute in section 6 the exact expression for the two-point function of R(x) = 1 Nψ (x) · ψ(x). Several correlation functions involving the J µ current are calculated in section 8. Clearly, all these quantities are gauge-independent and can thus be compared to results obtained in other gauges. Indications about the consistency of the method are obtained by verifying that gauge-invariant quantities have a natural O(3) covariant form. As a verification, we calculate a few terms of the perturbative expansion of the same correlation functions. In section 9, we introduce a scalar field σ(x) coupled to the fermions. This deformation is analogous to the triple-trace deformation λ 6 φ 6 (x) added to the Chern-Simons boson theory [11,15]. We show that when a certain relation between the couplings constants is satisfied, the model exhibits spontaneous breaking of scale invariance, again in clear analogy to the Cherns-Simons boson case [11,15]. The dilaton massless Goldstone boson excitation is a U (N ) singlet bound state represented by an effective field which, in the large N limit, is proportional to to σ(x) − σ ∝ψ(x) · ψ(x). Summary and conclusions are found in section 10. Several technical points and useful relations are included in several appendices.

3D topological Chern-Simons action
We first shortly recall the origin of the 3D Chern-Simons action and its main properties.

Chiral anomaly, topology and instantons
We consider an Abelian axial current in the framework of a non-Abelian gauge theory in four Euclidean dimensions. The fermion fields transform non-trivially under a unitary gauge group G and A µ is the corresponding gauge field. The fermion action reads with the convention where A µ is an anti-hermitian matrix, and the curvature tensor In a gauge transformation represented by a unitary matrix g(x), the gauge field A µ and the Dirac operator become The axial current J 5 µ (x) = iψ(x)γ 5 γ µ ψ(x) is gauge invariant. The axial anomaly [17] leads to a non-vanishing of the divergence of the axial current given by (2.5) From general arguments, one knows that the expression (2.5) must be a total derivative. Indeed, one verifies that ǫ µνρσ tr F µν F ρσ = 4 ǫ µνρσ ∂ µ tr(A ν ∂ ρ A σ + 2 3 A ν A ρ A σ ). (2.6) Anomaly, topology and instantons. Gauge field configurations can be found that contribute to the chiral anomaly. An especially interesting example is provided by instantons, that is finite action solutions of Euclidean field equations.
General instanton solutions can be exhibited by considering only pure gauge theories and the gauge group SU (2), since for a Lie group containing SU (2) as a subgroup the instantons are those of the SU (2) subgroup.
Instantons can be classified by a topological charge (2.7) The expression Q(A µ ) is proportional to the integral of the chiral anomaly (2.5), here written in SO(3) notation. We know that F µν ·F µν is a pure divergence (Eq. (2.6)). In SO(3) notation, The integral thus depends only on the behaviour of the gauge field at large distances and its values are quantized.
Stokes theorem implies where dΩ is the measure on the boundary ∂D of the four-volume D andn µ the unit vector normal to ∂D. We take for D a sphere of large radius R and find for the topological charge The finiteness of the gauge action implies that classical solutions must asymptotically become pure gauges. Since SU (2) is topologically equivalent to the sphere S 3 , the pure gauge configurations on a sphere of large radius |x| = R define a mapping from S 3 to S 3 . Such mappings belong to different homotopy classes that are characterized by an integer called the winding number. Here, we identify the homotopy group π 3 (S 3 ), which again is isomorphic to the additive group of integers Z. Without explicit calculation, one knows from the analysis of the index of the Dirac operator, that the topological charge is an integer: (2.10)

Topological Chern-Simons action in 3D field theory
We now study some applications to 3D field theory of the existence of topological terms involving gauge fields in 4D quantum field theory, as recalled in section 2.1. We first discuss the pure Chern-Simons action and, then, the Chern-Simons action coupled to fermion matter in the large N limit. We first consider the 3D Euclidean action in the form of a Chern-Simons (CS) term where A µ is a gauge field associated with U (N ) gauge transformations as defined by equation (2.4): The CS action is locally gauge invariant but globally gauge invariant only up to a constant. Indeed, starting from four dimensions, using the relation (2.6) and Stokes theorem, one infers that the expression where then µ is the normal to a 3-surface, dσ 3 the corresponding surface element and , is gauge invariant up to a constant that depends only on a topological index, a property explained in section 2.1. The general result can then be applied to three-dimensional flat space.
As a consequence, strict gauge invariance of the integral over gauge fields then implies θ = 2πκ , (2.12) where κ is an integer.
Chern-Simons term and gauge transformations. As an exercise, we directly verify how CS 3 (A µ ) transforms in a gauge transformation. We first rewrite CS 3 (A µ ) as It is convenient to define in such a way that gauge transformations read For a pure gauge, CS 3 becomes The quantity, is a topological term. Let us verify this statement directly. We set Then, The variation of expression (2.14) is We integrate by parts. The integrated term vanishes because the manifold has no boundaries. Thus, Therefore, the quantity T does not depend on local changes of g(x) but only on its global properties. Using gauge transformations in the form (2.13), we find We integrate by parts, The vanishing of the boundary term implies that B µ should vanish and, thus, asymptotically g(x) should go to a constant for |x| large. Therefore, R 3 should be given the topology of S 3 , as in the discussion of section 2.1. Assuming this condition satisfied and expanding, we conclude The Chern-Simons changes by a quantity that depends on global properties of the gauge transformation. Its values have indeed been calculated in section 2.1.
The classical equation of motion in an external source. We add a source to the action and consider the new action The equation of motion obtained by varying A µ in CS 3 yields δCS 3 δA µ = ǫ µνρ F νρ and, thus, For J = 0, the stationary solutions are pure gauges. In general, the gauge field has to be coupled to a gauge-covariant conserved source.

The Chern-Simons action in the temporal gauge
In general, to be able to define integrals over gauge fields, gauge fixing is required. In a general gauge, this leads to the introduction of Faddeev-Popov ghosts and the solution of the quantum field theory in the large N limit involves the summation of planar diagrams [18], a problem whose solution is only known in lower dimensions [19]. However, in the case of the Chern-Simons action, in some special gauges like the light-cone or axial gauges, the cubic interaction term in the CS action vanishes, and the integral over gauge fields can be performed explicitly.
In contrast to previous work that mainly uses the light-cone gauge, we choose the gauge A 3 = 0. Since we use an Euclidean formalism, we may identify x 3 with the Euclidean time (and below occasionally use the notation t ≡ x 3 and ω ≡ p 3 in the Fourier representation). The CS action reduces to the quadratic form Gauge invariance restricted to gauge functions g(x 1 , x 2 ) should be maintained. Assuming some boundary conditions at Euclidean times t i and t f and integrating over time between t i and t f , one finds This condition is automatically satisfied for the quantum partition function at finite temperature. Here, we work at zero temperature and set topological issues aside.
The gauge propagator. For convenience we now set, The topological action then reads The propagator can be calculated by adding external sources. Then, The solutions of the corresponding equations of motion are Note that, in the absence of explicit boundary conditions, the inverse of ∂ 3 is only defined up to the addition of a function of x 1 , x 2 . Independence of boundary terms implies These conditions enforce the remaining gauge invariance of the action corresponding to group elements g(x 1 , x 2 ). They can only be used if the time Fourier components are quantized, for example, in the case of the partition function where fermions satisfy antiperiodic boundary conditions. The action calculated for the solution then is Using 3 Chern-Simons gauge fields coupled to U (N ) fermions We first describe a U (N ) symmetric fermion theory with Chern-Simons term and then study its large N limit.

Conventions
In this section, we assume that A µ are antihermitian matrices belonging to the adjoint representation of the group U (N ). As a basis of the Lie algebra, one can take N 2 antihermitian matrices t a orthogonal by the trace where the structure constants f abc are real and totally antisymmetric. Moreover, the orthogonality relations (3.1) imply In terms of the t a matrices, the gauge field then can be parametrized as The Chern-Simons action then reads If we add to the action the source terms and integrate over the gauge field, we obtain The non-vanishing component of the gauge propagator then is In the Fourier representation, the CS action takes the form (3.5) and the source terms become An integration over the gauge field yields the free energy The Fourier representation of the gauge field propagator is where ǫ 12 = 1 = −ǫ 21 . Later we use a simple pole notation, but the pole term always stands for a principal part: PP(1/p 3 ) as depicted in Eqs. (2.19, 2.20).

The CS action coupled to a gauge-invariant fermion action
We now add to the Chern-Simons action, quantized in the A 3 = 0 gauge, a U (N ) gauge-invariant action for an N -component spinor field ψ, where D is defined in (2.2) and we have added a mass term, which violates parity, like the CS action. The γ-matrices here reduce to the three Pauli matrices: Regularization. Power counting shows that the coupling constant is dimensionless and, thus, UV divergences are expected. Moreover, the gauge field propagator is not isotropic, which leads to additional difficulties. In a first step, we write formal expressions and postpone the regularization problem, a non-trivial issue in a non-Abelian gauge theory with a Chern-Simons term. Fortunately, we will discover that here the number of independent divergent contributions is small and we will thus deal with the problem in a rather empirical way.

Large N expansion: field integral formalism
Solving non-Abelian gauge theories in the large N limit is in general a highly non-trivial problem [18], but the problem drastically simplifies when the gauge action reduces to a Chern-Simons term, at least in some specific gauges.
To solve the field theory with the action (3.7) in the large N limit we use a field integral formalism, a standard method to generate large N expansions [12], even if here the context is different and new features can be expected [6].
We thus consider the field integral where S is the Euclidean action (3.7). In the A 3 = 0 gauge, the gauge action is quadratic and the integration over the gauge field can be performed explicitly. One then obtains an effective quartic interaction for the fermions, non-local in Euclidean time. The components of the currents coupled to the gauge field are where the lower index is the spinor index and the upper index the U (N ) vector index.
Using the result (3.4), one finds With the help of the identity (3.2), the sum over the group index a can be performed and yields . In terms of the Fourier components ψ(x) = d 3 p e ipx ψ(p),ψ(x) = d 3 p e −ipxψ (p), the action (3.10) becomes (3.11)

The large N action
To render the N -dependence explicit and be able to study the large N limit, we introduce additional bilocal (in Euclidean time) composite fields {ρ α (t ′ , t, x)} and {λ α (t, t ′ , x)}, α = 1, 2, with x ∈ R 2 , to implement the relation We then add to the action (3.11), (3.13) This is a generalization (see also Ref. [6] and references therein) of the standard method [12] and a reflection of the required gauge field integration. The total action can be written as The integration overψ and ψ can be performed and generates the factor where the operator K is represented by the kernel (3.14) The large N action then reads The integration over the ρ-field is also Gaussian and can be performed. It amounts to replacing ρ by the solution of the field equation.

Saddle point equations
A non-trivial large N limit is obtained by taking the limit with g fixed [18]. The integral is then dominated by saddle points, solution of equations obtained by varying the ρ-and λ-fields. A variation of the ρ-fields yields Varying the λ-fields, one obtains One looks for solutions that do not break time and space translation invariance and, thus, Equations (3.16) then reduce to We defineλ The operator (3.14) in the Fourier representation takes a diagonal form in ω, p space (p ∈ R 2 ), with elements the 2 × 2 matrices The action density then reads It is convenient to set Then,K reduces tõ Its inverse can be obtained in the form Symmetries implỹ After Fourier transformation, Eqs. (3.18) become Similarly, Eqs. (3.17) becomẽ .
The four equations can be summarized by the unique pair of equations (3.24b)

Solution
It is useful to introduce the function .
Here, and later, 1/(ω − p 3 ) stands for P P [1/(ω − p 3 )]. The function Θ requires some regularization but, with a suitable space symmetric regularization, has a regularization-independent limit because the integrand is globally odd: it is most easily calculated in real space (see appendix A1.1) and one finds Then, initially motivated by perturbative calculations (see section 4.1), we set where M is a free mass parameter that will be later identified with the fermion physical mass. From equation (3.24a), one then infers The integral can be performed by expanding in powers of g, symmetrizing over all integration momenta and using the identities (A2.2-A2.3), a technique that will be used systematically in many places in this work. Using the identity (A2.2), one verifies that, for n > 0 , For n = 0, an additional term is generated (identity (A2.3)) and one finds where we define more generally, for later purpose, (3.28) For n = 1, it is a divergent quantity that has to be regularized. One obtains We then choose the mass parameter M to be the solution of the gap equation and we verify later that M is then the fermion physical mass. We conclude and, thus, The verification of equation (3.24b) is then straightforward. The constants Ω n (M ). For n > 1, Ω n (M ), defined by Eq. (3.28) is given by (3.32) For n = 1, assuming some regularization, we define the UV cut-off Λ by In dimensional regularization (which we will not use) this contribution vanishes. Then, In what follows we will omit everywhere the M 2 /Λ correction.

The free energy density
We define the free energy density, which is proportional to the ground state energy, dividing by a factor N , by where Z is the partition function, Z 0 a normalization and V the volume. From Eq. (3.19), one infers, in the large N limit, whereλ andρ have to be replaced by the solutions of the saddle point equations and complex conjugate forλ 2 ,ρ 2 . Moreover, tr lnK = ln(p 2 + ω 2 + M 2 ).
Then, from the saddle point equation, Thus, .
Expanding in powers of g, replacing the functions Θ by their integral representation and evaluating each term using identities of the kind explained in section A2.1, one obtains where the first term has still to be regularized. Differentiating the first term, one finds ∂ ∂M Using the explicit form (3.34) and integrating back, one obtains We note that W is even in the change M ↔ −M , g ⇔ −g or, equivalently M 0 ↔ −M 0 , g ⇔ −g.

The fermion two-point function for N large
The fermion two-point function has the general form In the large N limit, the fermion two-point function is obtained by inverting the operator (3.14) at the saddle point. Using the expression (3.21) and replacing the functions µ α by the explicit solutions (3.30) of the saddle point equations, one finds D(p) = p 2 + M 2 , and thusK We note thatK It is convenient to define The dressed fermion fermion propagator then takes the form This expression confirms that the parameter M is the physical fermion mass. Moreover, this form suggests that, after mass renormalization and a non-local phase transformation, the fermion theory is equivalent to a free theory. Correspondingly, the two-point vertex function, inverse of W (2) , can be written asΓ (2) More explicit expressions, using the explicit form of Θ, arẽ a result consistent with equation (3.27).

Gauge-invariant observables
To be able to distinguish gauge artefacts from gauge independent properties, it is necessary to calculate gauge invariant observables. The ψ-field two-point function is not gauge invariant, except in the limit of coinciding points. Two gauge invariant operators are (no summation assumed) Below, we consider the scalar combination and the third component of the current Expectation value. We can already determine the expectation value of the operator R α (x). The equal-time expectation value of ρ 1 is given by Expanding in powers of g and replacing Θ(p 3 ) by its integral representation (3.25), symmetrizing the integrand with respect to p and all the other integration momenta, one verifies that all terms vanish except the two first ones. This result, and other similar ones, rely on the identities (A2.5). Since the result is real, J 3 = 0 and We find that the expectation value, which is gauge invariant, is indeed given by an explicitly O(3) symmetric expression, in the sense that it makes no reference to the gauge propagator and is expressed in terms of an obviously covariant integral. In terms of the explicit expression (3.34), for M > 0 it reads R expectation value and ground state energy. The normalized free energy density defined by Eq. (3.35) and calculated in section 3.7, is given by (Eq. (3.36)) which can be rewritten as (Eq. (3.29)) One then verifies the consistency between the expressions (3.47) and (3.46). Connected R correlation function at zero momentum. The R two-point function, at zero momentum is obtained by differentiating R with respect to M 0 . One then finds (the subscript c stands for connected) This quantity requires only an additive renormalization: Higher order functions can be obtained by further differentiation. here differentiating again, one obtains (still M > 0) which is finite. All other connected correlation functions then vanish for Λ → ∞.

Perturbative calculations at large N
It is interesting to see how some results that we have obtained by field integral techniques emerge from perturbative calculations. Therefore, in this section, we calculate a few orders of some of the quantities that we have determined in the preceding sections.

The fermion two-point function at two-loop order
We first expand the fermion two-point function, which is not a gauge-invariant quantity but which has been determined exactly in the large N limit (equation (3.39)).
The fermion propagator. The bare fermion propagator ( • 0 means Gaussian expectation value) is given by One-loop calculation: the two-point vertex function. Perturbative calculations of the two-point vertex function or mass operator involve the one-loop diagram of figure 1. It can be written as The evaluation is simple and yields where Θ 0 is the function (3.25) in which M is replaced by M 0 . The expression in the case of M 0 = 0 agrees with the calculation in the temporal gauge in Ref. [6] at one-loop order. The two-point vertex function is theñ After introduction of the physical fermion mass (3.29), which amounts to a mass renormalization, the expression agrees with the expansion of the result (3.40) at order g.
Two-loop order. After mass renormalization (equation (3.29)), the two-loop contribution Σ 2 to the two-point vertex function (two-loop diagram of figure 1) is proportional to which is again consistent with the expansion of the expression (3.40).

The ground state (or vacuum) energy up to three loops
We have defined the normalized, gauge-independent, free energy density by equation (3.35). We expand it as and keep only the leading terms for N large. Note that in view of the remark after Eq. (3.45), the expectation value R at leading N contains only terms up to order g 2 and thus in view of Eq. (3.47) there is no term of order O(g 3 ) in Eq. (4.4). This observation was also made in Ref. [6] where the free energy has been calculated in the light-cone gauge. We choose Z 0 such that The two-loop contribution is ( Fig. 2) Calculating the trace, one verifies that the gauge propagator cancels and one finds Three-loop order. The three-loop order calculation (order g 2 , Fig. 3) is lengthier. Setting one obtains After evaluating the trace, the expression decomposes into the sum of two terms, .
The first contribution can now symmetrized over r 3 , p 3 , q 3 . The gauge field poles then cancel and the result is The expansion of the free energy expressed in terms of the fermion physical mass (3.29) then becomes in agreement with the expression (3.47).

Correlation functions involving
We now determine the (ψ · ψ)ψψ vertex function at leading order for N large. But before a simple remark is useful.

A simple transformation
To generate insertions of R α operators, we introduce non-diagonal, space-dependent, mass terms. A simple limit corresponds to correlations of d 2 x R α (x 3 , x). They can be generated by We then make on ψ a phase transformation of the form In the action (3.8) (where M is replaced by M α (x 3 )), only the x 3 derivative is modified: Expressing the diagonal matrix M of eigenvalues M 1 , M 2 in the form we reduce the mass term (5.1) to We conclude that the expectation values of products of d 2 x R α (x 3 , x) operators are trivially related to the expectation values of d 2 x R(x 3 , x) and this provides some checks in the calculations.

Field integral formalism: expansion at the saddle point
Connected correlation functions involving the R operator at non-vanishing momenta can be calculated for N large by using the field integral formulation of section 3.3. The calculation involves expanding the large N action (3.15) at the saddle point to quadratic order and performing a Gaussian integration. The quadratic form depends on the second functional derivatives of the large N action with respect to ρ, λ at the saddle point. The less trivial part is the second functional derivative of the determinant with respect to λ.
The second functional derivative of tr ln K. Calculating the second functional derivative of the trace of the logarithm of the operator (3.14) with respect to x ′ )}, at the saddle point, we obtain the formal expression (no summation on α, β implied) Other functional derivatives. The second functional derivative of the large N action (3.15) This determines the 2 × 2 matrix of functional derivatives, which has still to be inverted and this is the non-trivial part of the calculation of R correlation functions. Fourier representation. We introduce the Fourier representation where the explicit expression is given by Eq. (3.37). Thus, In a global Fourier representation, this yields In the same way, the λρ element becomes and the ρρ element We note that all expressions are diagonal in the momentum variables.
Remarks. The calculation of the inverse operator involves solving two coupled integral equations, a problem we discuss in the coming sections. Note that if the inverse operator is expanded in powers of the coupling g, it generates a sum of ladder diagrams with the dressed fermion propagator (3.39), which sums all propagator corrections.
To calculate correlation functions of R α (t, x) =ψ α (t, x)·ψ α (t, x)/N , one needs the operator sources for the vectors ρ α (t, t, x) of the form In the Fourier representation, they become If one wants to calculate only correlation functions of R = ρ 1 (t, t, x) + ρ 2 (t, t, x), one needs a source in t, x space that takes the form In the Fourier representation, it becomes This somewhat simplifies the calculation.

The (ψψ)ψψ vertex function
Though ultimately we want to calculate the R two-point function, we have first to calculate the connected three-point correlation function with one R insertion, because, unlike the R two-point function, it satisfies an integral equation for N large, as the analysis of section 5 shows. We denote its Fourier transform byW (1,2) (k; ℓ − k/2, ℓ + k/2) and it is related to the corresponding vertex function bỹ (5.6) An important property, which we use systematically later, is the following: one verifies that the vertex functionΓ (1,2) has the following general decomposition: The R(x) =ψ(x) · ψ(x) insertion at zero momentum. A differentiation of the fermion two-point function (3.40) with respect to the bare mass M 0 yields the vertex function (5.7) with an operator insertion at zero momentum. Using the gap equation (7.1), one finds that the mass insertion is given by The result is consistent with the decomposition (5.7) and one immediately infers the values

The perturbative expansion of the (ψψ)ψψ vertex function at two loops
To gather some insight about the general structure of correlation functions involving theψψ operator, we begin with a perturbative calculation for N large of the (ψψ)ψψ vertex functionΓ (1,2) . However, we have learned in section 5 that, for N large, the perturbative expansion reduces to a sum of ladder diagrams (figure 4) with the dressed propagator (3.39).
In the decomposition (5.7) the functions E and F have a perturbative expansion of the form We introduce the compact notation Recursion relation. Since the perturbative expansion reduces to a sum of ladder diagrams, it is possible to generate them by a recursion formula (see section 5). It is given by a 2 × 2 matrix acting on the vector (E n , F n ): which corresponds to adding two dressed fermion propagators and a gauge propagator to a diagram.
The functions (E n , F n ) depend only on k and on p 3 , if we denote by p the additional integration variables. For g = 0 (i.e., with the free propagator), we can write the matrix (but a kernel in (p 3 , ℓ 3 )) as where we have set Terms proportional to (k 1 p 2 − k 2 p 1 ), with a vanishing integral due to rotation symmetry in the (1, 2) plane, have been omitted.
It is useful to decompose T into the sum of two terms, with, correspondingly, and (5.14) Note that T 1 (k, p 3 ) depends only on the component p 3 .
With the dressed propagator, T 2 is not modified. Quite generally, we define and, moreover, here we set We can then write the dressed matrix (5.14) as Definitions. The calculations that follow involve only two new basic functions. We first define which is a one-loop scalar function. Note that in three dimensions all one-loop diagrams can be reduced to elementary functions. We also define It satisfies the relations One-loop contribution. We need only the free fermion propagator and, thus, the matrix (5.10) since at one-loop order the phase χ = O(g) does not contribute.
The one-loop integrand is obtained by acting with T on the vector (1, 0). One finds Integrating, one obtains and where we have used the definitions (5.19, 5.20) and the notation (5.11). We note that in F 1 the second term comes from the matrix (5.13).
This result suggests the factorizatioñ where at one-loop orderṼ (1,2) reduces tõ 25) The connected function is then given bỹ Quite generally, we introduce the decomposition, The relation between the vectors (E, F ) and (A, B) then is We expand A and B in powers of g, and obtain Two-loop order. The two-loop order is the sum of the contribution obtained by acting with the matrix T taken at g = 0 on the vector (E 1 , F 1 ) and the contribution of order g of T acting on (1, 0).
Details about the calculation can be found in appendix A3. One obtains Then, using the relation (5.28), one verifies that all terms proportional to Θ functions cancel, justifying to two-loop order the transformation (5.24), and one finds Higher orders. With the definition a calculation to order g 6 (appendix A3) then suggests the general form Using Eq. (5.21), one verifies that the coefficients A and B for k = 0 are consistent with the expressions (5.8).

Integral equation
For N large, the vertex functions are solutions to one-dimensional coupled integral equations, which in terms of the vector V EF ≡ (E, F ) read Preliminary remark. Acting with gT 2 on both sides of the equation, one obtains with the notation (Eq. (5.15)), The usual identity Rescaled integral equation. In the integral equation, we now rescale χ → εχ, T 2 → εT 2 and determine the ε dependence of the solution. Differentiating with respect to ε, we find We now use the identity (5.32) (which is not affected by the rescaling) to eliminate T 2 . The equation becomes we obtain the equation which has X = 0 as the only solution expandable in powers of g. We infer and thus We recognize the relation (5.28) and thus The vector V = (A, B) is thus solution of Eq. (5.31) with ε = 0.

Solution of the reduced integral equation
The vector V = (A, B) is solution of the reduced integral equation or in component form, Inserting the expressions (5.30) in the equations, we note that the numerators simplify since The verification that the functions are the solutions of the integral equation is then simple and relies on a few identities derived in A2.2 concerning integrals of sine and cosine functions. The proofs of the latter identities rely on an expansion order by order in powers of the coupling g.
We have already determined the connected R two-point function R (k)R(−k) c at zero momentum (Eq. (3.49)). We now calculate it for generic momenta. We begin with a three-loop perturbative calculation in the large N limit and then derive the exact result from the known vertex function.

Perturbative calculations
Since calculations are performed using the dressed fermion propagator (3.39), only ladder diagrams have to be considered (see section 5).
One-loop result. The one-loop diagram (first diagram in Fig. 5) with the dressed propagator reads is the angle (5.15). We note that At one-loop order, χ 1 = 0 and, thus, the contribution is Two-loop result. At two-loop, one needs the second diagram of Fig. 5 with the free propagator together with the order g contribution of expression (6.1) (which corresponds to the diagram of Fig. 6). The two-loop ladder diagram yields where the (p, q) symmetry has been used. The last term is exactly cancelled by the order g of expression (6.1), which yields Adding the two contributions, one finds Three-loop contribution. Details about the direct three-and four-loop calculations are given in section A5. One finds (Eqs. (A5.6, A5.7))

R two-point function and vertex three-point function
The R two-point function is derived from the vertex three point function (5.7) by multiplying it by two additional dressed fermion propagators and taking the trace. This leads to the general relation where χ 2 ≡ χ(q 3 ) (definition (5.15)), A and B are the coefficients of 1 and iσ 3 in the vertex functionṼ 1,2) (the relation (5.28) between (E, F ) and (A, B) has been used). We first prove phase factor factorization and cancellation, in such a way that the relation (6.2) reduces to where A and B are the coefficients of 1 and iσ 3 inṼ (1,2) given by equations (5.30).
Proof at first order. The first contribution of the additional terms in Eq. (6.2) with respect to (6.4) is We transform the first term by using the integral equation (5.34): .
Integrating over q, we find and, thus, the first order difference vanishes.
A general identity. For the general proof, we need We introduce the integral representation of χ(q 3 ), The integral becomes Except for the gauge propagator, the integrand in symmetric in (q, r i ). Symmetrizing the whole integrand and using the identity (A2.2), we find General proof. We now consider the whole contribution coming from the numerator (6.3) and proportional to (Dp 2 + Dm 2 ), We again use the integral equation (5.34) both for A and B and obtain We substitute this expression into Eq. (6.2), integrate over q and find the numerator with a denominator Dp 1 Dm 1 . We have still to add the remaining part of (6.3) (replacing q by p as integration variable), [4t 1 cos(χ 1 /2) + 4M p 3 sin(χ 1 /2)] A(p 3 , k) + 4M p 3 cos(χ 1 /2) + (4t 2 + 4p 2 3 ) sin(χ 1 /2) B(p 3 , k). The χ-dependence cancels and one recovers the anticipated expression (6.4),

The exact R two-point function at large N
We now derive the connected two-point function R (k)R(−k) c from the expression (5.30) of the vertex function. In section 6.2, we have proved the reduced relation (6.4), where A and B are the coefficients of 1 and iσ 3 inṼ (1,2) given by equations (5.30). Inserting into the equation the expressions (5.30), one finds a factor − k 2 + 4M 2 cos gkB 1 (k) − 2(M/k) sin gkB 1 (k) multiplied by the integral Using the result (A2.7), one obtains the two-point function  Introducing the explicit result (3.34), we infer from Eq. (7.1) that M 0 can be written as where m, which is finite for Λ → ∞ since M is finite, provides a physical mass scale. The gap equation (7.1) then reads For g = g c = ±4π, the equation implies M = 0, that is, fermions are massless and the theory is thus conformal invariant. By contrast, for m = 0 and g = ±4π, the equation is always satisfied and M remains undetermined. We then note that if indeed M = 0 in a theory with no renormalized mass scale, m = 0 indicates a spontaneous breaking of scale invariance. As in similar cases (like in [15] and earlier works [13]) it requires the appearance of a dilaton pole at g = ±4π. This issue is discussed further below where we show that no dilaton pole appears, yet another indication that, indeed, the physical range is limited to −4π ≤ g ≤ 4π.
Returning to the definition (2.16) of the coupling constant g, one verifies that the critical values g c are acceptable since they imply the relation between integers κ = ±N .
The massless or large momentum limit. In the limit M → 0 or, equivalently, k → ∞, the expression (6.6) reduces to which, up to a constant, is simply proportional to the two-fermion phase space.
For all values of k, the correlation function diverges for g = ±4π and for 4π < |g| < 8π it violates positivity. Therefore, in the domain containing the origin, the physical range is limited to |g| ≤ 4π. The special case |g| = 4π has to be examined separately. Eq. (7.7) is in agreement with Ref. [10] where calculations were done in the light-cone gauge (See Eqs. (32) and (33) in Ref. [10]). A detailed analysis of this result in different regularization schemes in the light-cone gauge can be found in the appendix of Ref. [16].
Singularity of the two-point function and critical couplings. It is now interesting to look at the zeros of the denominator in expression (6.6). The denominator vanishes for tan g 4π arctan(k/2M ) = k 2M and, thus, g 4π arctan(k/2M ) = arctan(k/2M ) (mod π).
The denominator vanishes for all values of k for g = 4π, that is, for one critical value of g that is singled out by the fermion gap equation. More precisely, This expression shows that even after a renormalization of the R-field to remove the singularity at g = 4π, the R two-point function has no massless pole at k = 0. Other solutions are k-dependent. Another family of solutions is given by which, for k → ∞ converge toward g = −4π from below. Since in the Euclidean formulation poles with k real are unphysical, this confirms again that the admissible range of values of g is restricted to −4π ≤ g ≤ 4π. In this range, the physical fermion mass is always an increasing function of the bare mass (Eq. (7.4)), which sounds reasonable. We thus note that the case m = M 0 − M c = 0 and g = ±4π, which allowed for a possible non vanishing fermion mass M , is inconsistent. Such a spontaneously broken scale invariance phase would require the appearance of a massless pole in Eq. (6.6). However, this does not happen. After a renormalization of the R-field by a factor √ 4π − g to remove the singularity at g = 4π, one verifies from Eq. (7.8) that the correlation function has no pole at k 2 = 0. Moreover, expression (3.50) immediately shows that this is not a suitable renormalization to remove singularities at g = 4π. A more suitable renormalization would seem to be a multiplication by (4π − g) but then the two-point function, as well as the three-point function at zero-momentum, vanish in the g = 4π limit.

The current two-point function
The current associated to fermion number conservation is The current is gauge-invariant and conserved. The third component is related to quantities already defined since

The J 3 ψψ vertex
Following the same strategy as for the R two-point function, we first determine the J 3 ψψ vertex, which again has a simple decomposition of the form (5.7). It satisfies the integral equation (5.31) but with the different initial conditions The proof of section 5.5 does not depend on the explicit form of the inhomogeneous term and, therefore, the vertex function factorizes as in expression (5.24), where V

Perturbative calculations
To gain some insight about the structure of the current vertex function, we calculate a few terms of the perturbative expansion.
One-and two-loop order. At one-loop order, one finds The two-loop order contributions are All orders: the massless limit. In the limit M = 0, the form of the vertex can be guessed to all orders. In terms of the quantity (5.29), for M = 0, these expressions can be written as All orders. Calculating more terms, and using the experience gained with the scalar function, one guesses that in terms of the quantity (5.29), the exact expressions are In this form, one finds linear combinations of the same trigonometric functions as in expressions (5.30).

Integral equation
The function A and B are solutions of integral equations of the form (5.34), but with different boundary conditions, which read . (8.6b) Inserting the expressions (8.5), one obtains the numerators Using the same identities as in the scalar case (for details see A2.2), one can check that the functions (8.5) are indeed the solutions of the integral equations with the proper boundary conditions.

From the J 3 vertex to the J 3 two-point function
We follow directly the method of section 6.2. The general relation between the J 3 vertex and the J 3 two-point function is where χ 2 ≡ χ(q 3 ) (definition (5.15)), A and B are the coefficients of 1 and iσ 3 in the vertex functionṼ 1,2 We first prove that the relation reduces to We use a strategy similar to the one used in section 6.2.
We have to transform the combination We use the integral equation (8.6) both for A and B and obtain We then integrate over q and find the numerator All χ dependent terms then cancel and the remaining term is in agreement with expression (8.7).

The J 3 two-point function
Due to current conservation, the current two-point function has the general form The current two-point function can be determined by calculating only the J 3 two-point function since The J 3 two-point function is also given by where we have used the property that all functions are real and Since RR is already known, only one function remains to be determined. Equation (6.4) is modified. We have proved in section 8.4 that after subtraction of the terms that cancel the phase factors, the relation becomes (Eq. (8.7)) Leading (one-loop) order. It is given by The calculation of the remaining integral can be found in appendix A6.1. One finds (Eq. (A6.2)) Therefore, where the divergent additive constant is fixed by current conservation.
Higher-loop calculations. After some algebra, one finds .
The known terms sum in the form The result is consistent with expressions (8.5). Indeed, inserting the expressions (8.5) into equation (8.7), one finds in the numerator the combination (we have factorized −4) −2M t 2 k sin(kgB 1 ) cos(2gτ Ξ) + 2τ q 3 k sin(kgB 1 ) sin(2gτ Ξ) One then expands for q 3 → ∞ and keeps the two leading terms, proportional to q 2 3 and q 0 3 . The term proportional to q 2 3 can be integrated using the results (A2.6) and (8.10). The integration of the term proportional to q 0 3 relies on S n,n in equation (A2.5). One then recovers the expression (8.11).
The large momentum or zero mass limit. In the limit M → 0, the two-point function reduces to sin(g/4). (8.12) We note that positivity is only satisfied if sin(g/4)/g is positive. This again implies |g| ≤ 4π. More generally, one verifies that for g = 4π, the two-point function vanishes linearly, another confirmation that the field theory does not make sense for |g| > 4π.
Using Eqs. (8.8, 8.9) we have in the conformal limit of M → 0 also  (37)), which were calculated in the light-cone gauge.

The R three-point function
We have already evaluated the three-point function at zero momentum (expression (3.50)). We can now evaluate it with only one zero momentum by differentiating expression (6.6) with respect to the bare mass or, in terms of the physical fermion mass where The general connected RRR three-point at leading order. We find We define the scalar three-point vertex as . (8.16) In particular, Then, The generalization to all orders of the denominator in (8.14) is straightforward.
However, the generalization to all orders of the function R (k)R(p)R(−p − k) c is not straightforward and requires more complicate calculations.

Adding a deformation to the Chern-Simons fermion action
We now introduce a scalar field σ(x) and add two new terms to the action (3.7) of the form where the new parameters g σ and R are fixed when N → ∞.
The extra σ(x) terms in Eq. (9.1) are analogous to the triple-trace deformation λ 6 φ(x) 6 added to the Chern-Simons boson theory [11,15]. The action in Eq. (9.1) differs from the action in [11] by the term Rσ(x) added here in order to have a proper perturbative meaning to the σ term in the action.
From dimensional analysis, one concludes that σ has mass dimension 1 and that no other term of dimension 3 or less and odd in σ can be added. Moreover, g σ is dimensionless and R has dimension two.
For R = 0, in the classical limit σ(x) has a non-vanishing expectation value σ, obtained by varying σ(x), which is given by Then, setting σ(x) = σ + ς(x) we note that the expectation value σ adds to the bare mass M 0 and Since the propagator of ς(x) is a constant, the dimension of ς(x), from the viewpoint now of power counting, is 3 2 in such a way that ς 3 (x) has dimension 9 2 and, thus, is irrelevant (non-renormalizable). We show below that, by contrast, in the large N limit the situation is different.

Large N limit: gap equation and free energy
We note that the reflection symmetry g → −g, M 0 → −M 0 is now extended if g σ → −g σ and R → −R and then the saddle point value σ → −σ and also M → −M . From now on we thus again choose M ≥ 0. First, we perform the transformations of section 3.4 in the addition (9.1). In particular, By varying σ, we now obtain the additional saddle point equation In the other saddle point equations, the bare mass M 0 is simply replaced by M 0 + σ. The gap equation (7.1) is thus replaced by and σ is solution of (using Eq. (3.46)) Eliminating σ between the two equations, we transform the gap equation into or, more explicitly, an equation quadratic in M . The effective σ potential. After taking into account the λ and ρ saddle point equations, the action density for constant σ fields normalized as in Eq. (3.35) becomes where W is given by Eq. the expression for σ in Eq. (9.5) is then obtained from Furthermore, by calculating the successive derivatives of V eff. at the saddle point, one obtains the vertex (or 1PI) functions at zero momentum. Divergences and counter-terms. We now assume that g and g σ need not to be renormalized, which we have checked to some extent. Then the cancellation of divergences in the coefficient of M implies that M 0 has the form where m is finite and Similarly, the cancellation of divergences in the constant term yields the condition where η is a constant parameter and 14) The gap equation then reads Note also that the relation (9.4) becomes and, therefore, 2σ Critical limit. If m = 0 or η = 4π/g σ the gap equation has the solution M = 0, which corresponds to massless fermions. For m = 0 and η = 4π/g σ , the gap equation has then also a non-zero solution Since the solution can also be positive, checking which is the leading solution requires calculating the corresponding free energy.
Demanding that the coefficient of M also vanishes yields either g = ±4π, which here also is a singular point, or m = 0. Then, both solutions of the gap equation coalesce and M = 0.
Finally, the gap equation is satisfied for any value of M ≥ 0 if the coefficient of M 2 also vanishes, that is, for We examine this special situation in section 9.5. For M < 0, the analysis is the same; one has just to change g → −g, g σ → −g σ .

Connected scalar two-point functions at zero momentum
Since R =ψψ and σ are both scalar U (N )-invariant fields, they are coupled. To calculate the two-point functions at zero momentum, we have to differentiate the general saddle point equations twice with respect to M 0 at R fixed, with respect to M 0 and R and, finally, twice with respect to R.
The R two-point function. The two-point function is given by where Eq. (9.5) has been used. Differentiating Eq. (9.4) with respect to M 0 and combining it with Eq. (9.19) to eliminate ∂σ/∂M 0 , we obtain 1 − g 4π We note that 0 is given by the expression (3.49). Therefore, It follows More explicitly, Then, for m = 0, the equation D = 0 directly implies the relation (9.18). This result is consistent with the existence of a massless pole of the R two-point function when the relation (9.18) is satisfied but the example of the point g = 4π shows that we still have to verify that the coefficient of k 2 in the inverse of thẽ R(k) two-point function does not vanish.
The Rς two-point function.
Then, from (9.20, 9.21, 9.22), Thus, This relation will be seen to be a direct consequence of the relation (9.35).
The ς vertex functions at zero momentum. The ς two-point vertex function at zero-momentum can be directly calculated by differentiating expression (9.10) with respect to σ.
First from Eq. (9.4), one infers Then, from and Eq. (9.17), one derives Similarly, the third derivative yields the three-point function: Finally, all other n-point functions with n > 3 vanish at zero momentum.

The generic R two-point function
Expanding around the saddle point at second order in the fields, we find that the modified action generates the two additional quadratic terms To calculate the R correlation function, we can integrate over ς, which amounts to replacing ς by solution of the field equation obtained by differentiating with respect ς(x). One finds the contribution which adds to the quadratic term in ̺, An expansion in powers of g σ then leads to where R (k)R(−k) c,0 is the connected two-point function (6.6). It is convenient to rewrite the expression as The poles of the two-point function are thus solution of Using Eq. (9.16), we note that The denominator of the expression (9.27) is thus a finite quantity, as it should. Moreover, we see that the R two-point function now requires an additive but also a multiplicative renormalization σ/g σ as Eq. (9.27) explicitly shows.
The ς two-point function. To calculate the ς two-point function, it is convenient to first integrate over ̺ at R fixed. Since we have determined the R two-point function, we know the result of the integral. We introduce the notation The effective quadratic action then reads c,0 the inverse has to be understood in the sense of kernels. Note that if we integrate over ς, we recover the inverse connected R two-point function in the form which is equivalent to Eq. (9.27). However, we can now instead integrate over R s and find Thus, with (Eq. (9.28)) We see that this correlation function is finite in the infinite cut-off limit. We show in section A1.2 that the function D has a zero for k pure imaginary and |k| < 2M if the two conditions are satisfied: the first condition, equivalent to D(0) ≥ 0, being necessary for the theory to be physical. This zero corresponds to a massive scalar bound state.
Small k expansion. Concentrating on the situation where D(0) is positive and small and M = 0, we can expand the denominator for k small. Using 2M k tan (gkB 1 (k)) We find We note that the positivity of the coefficient of k 2 implies g < 4π. Then, D has a zero corresponding to a scalar particle with mass given by where M/m = f (g, g σ , η) is the solution of the gap equation (9.15).
Remark. From the correlation functions, we note that in the large cut-off limit and thus ς is the renormalized R s field.

The critical limit
In the limit M → 0 + , the expression reduces to where either m = 0 or η = 4π/g σ . In the latter case, m/g σ must be negative. In both cases, for g > 0 the positivity condition tan(g/8) > 0 implies g < 4π.
In the case m = 0, the connected σ two-point function becomes identical to the propagator of a free massless scalar particle: A special limit. A special limit is g = g σ , m = 0 where all physical masses M and M S vanish but also all bare dimensional parameters,

Situation with vanishing coefficient of M 2 in the gap equation
We are especially interested in the case where the relation (9.18) is satisfied. Then, The physical parameters correspond to m > 0 and, since g σ then is negative, also implies m/g σ < 0. When the relation (9.18) is satisfied, quite generally Eq. (9.16) leads to Inserting the value of M in D, we obtain This expression is physical only if Then, the mass M S of the scalarψψ bound state is given by where this expression assumes that If in addition m = 0, M is undetermined and the two-point function has the form A pole is found for k = 0, which corresponds to a massless particle provided the residue is positive, which again implies, starting from g = 0, that |g| < 4π. Since the mass of the fermions M is arbitrary, this pole may have the interpretation of a dilaton pole. We then expect that all vertex functions vanish at zero momentum. Indeed, one verifies that this condition is satisfied by using the relation (9.18) in the three-point function (9.25), since, as noted, other vertex functions vanish automatically.
The dilaton effective action. At large distance or small momentum limit, for M = 0 the fermions decouple and one can derive an effective action for the massless dilaton field ς(x) (which is a U (N ) singlet) in the form of a derivative expansion.
It is convenient to normalize the dilaton field, where f D is defined from Eq. (9.41), As noted above, all vertex functions vanish at zero momentum and thus the effective action contains no derivative-free terms. The action of the renormalized dilaton field D(x) has then the form If in the action one keeps only the terms with two derivatives, by a simple change of field coordinates the action can be reduced to a free massless field action, like in the example of the O(2) non-linear σ model. In order to obtain the full interacting dilaton effective action, one has then to calculate the n-point vertex (or 1PI) function of ς(x). More on the dilaton effective potential in these three dimensional Chern-Simons boson and fermion theories can be found in [15] and [16].

Spontaneously broken scale invariance in boson and fermion theories
The relation between 3D U (N ) fermion and boson theories each coupled to Chern-Simons gauge fields, at large N , has raised recently much interest [6,9,10,11]. In particular, a precise mapping between the boson and fermion theories was exhibited. Previous published results were obtained within the light-cone gauge and we compared them to our results derived within the temporal gauge in preceding sections. The comparison was done in the case of the massless conformal phase. It is interesting to compare our results also in the case of the massive phase with spontaneously broken scale invariance. The comparison will check the consistency of our results with the light-cone results and at the same time will confirm the boson-fermion duality mapping in the case of the massive phase.
The condition we found in Eq. (9.18) for the existence of a massive phase in the fermion case was, In the boson theory, the existence of a massive ground state requires [11,15] where λ b is the Chern-Simons gauge coupling in the boson theory and λ 6 is the marginal coupling of the boson λ 6 (φ † · φ) 3 /6N 2 interaction. It is now interesting to find out whether the boson and fermion conditions in Eqs. (9.45) and (9.46) are dual copies when the dual values of the couplings are taken into account. Indeed, using in Eq. (9.46) the mapping between the fermion and the boson theories of Ref. [10,11], which was derived in the light-cone gauge,

Summary and conclusions
In this article, we have presented a detailed analysis of the large N limit of three-dimensional U (N ) symmetric field theory with fermion in the fundamental representation coupled to a Chern-Simons gauge field. As mentioned in the introduction, this model rose interest in recent years [1-11, 15, 16] since its singlet sector has been conjectured through AdS/CFT correspondence to be related to Vasiliev's [3] higher spin gravity theory on AdS 4 . Previous analysis of this system were done in the light-cone gauge whereas we worked in the axial A 3 = 0 gauge, which is somewhat more difficult to analyse but lacks the peculiarity of the light-cone gauge. We determine the large N properties by using a field integral formalism and the steepest descent method [12]. The saddle point equations, which take here the form of integral equations for non-local fields, determine the ground state phase structure. The vertex functions and correlators were calculated at leading order in 1/N . We derived exact solutions to the integral equations. From the vertex functions, we have inferred two-point correlation functions for scalar and current operators. The calculation were done first in the massive phase away from criticality in the phase in which scale invariance is explicitly broken. Later, the critical massless conformal phase was considered. The results were checked by calculating several orders in perturbation theory. We have compared our results with results obtained in previous works for gauge invariant physical quantities in the conformal massless limit.
In particular, we have thoroughly investigated the conditions for the occurrence of a massive phase in which scale invariance is spontaneously broken and its detailed properties. Indeed, the issue of a possible spontaneously broken massive phase was recently debated in several papers [11,15,16]. As described in section 6.3, we do not find such a massive spontaneously broken phase in our results: the gap equation (7.6) allows the physical fermion mass M = 0 provided g = ±4π, however, Eq. (6.6) clearly shows that the range of physical values for the coupling is −4π < g < 4π leaving the solution M = 0 outside the physical range. Moreover, spontaneously broken scale symmetry would have required a massless dilaton to appear in the theory when g = ±4π. This does not happen since the R (k)R(−k) c correlator in Eq. (6.6) diverges at g = ±4π for any value of the momentum k and no massless dilaton pole appears there. Thus, in the critical theory, the conformal invariant massless ground state is the only possible ground state in the Chern-Simons fermion matter system. A massive ground state appears only away from criticality M 0 − M c = m = 0 were we have an explicit breaking of scale invariance.
The absence of a massive ground state agrees with the results of Ref. [11] calculated in the light-cone gauge. However, in the case of the bosons interacting with Chern-Simons gauge fields a massive spontaneously broken scale invariant phase exists only when a marginal deformation is added of the form (λ 6 /6N 2 )(φ † · φ) 3 . The massive phase and the dilaton massless pole appears [11,15] in this case when λ 2 + λ 6 /8π 2 = 4. The range of the Chern-Simons coupling λ is in this case −1 < λ < 1 and the added self interaction marginal operator allows a massive ground state. In the absence of the Chern-Simons gauge field, this result agrees with Ref. [13].
As noted in Ref. [11], in order for a massive, spontaneously broken scale invariance phase to appear in the fermion case, here too a marginal deformation of the type σψψ + (N/3!)λ f 6 σ 3 has to be added. In section 9, we added to the action the above mentioned self-interacting auxiliary field σ(x) coupled to the fermions, which effectively induces a marginal deformation to the Chern-Simons fermion action. Indeed, in our A 3 = 0 gauge calculations a spontaneously broken massive phase appears and a massless dilaton pole appears in the R (k)R(−k) c correlator. We found that the massive phase appears in the spectrum provided the σ self interacting coupling g σ is related to the Chern-Simons gauge coupling g by g σ /4π = −(1−g/4π) 2 /(2−g/4π) in analogy to the relation between λ 6 and λ that ensures a massive ground state in the Chern-Simons boson case [11,15]. Moreover, in section 9.6 we have shown that the conditions for spontaneous breaking of scale invariance in the boson and fermion theories are dual copies of each other.

A1 Some explicit expressions
We give some technical details about explicit expressions that appear in this work.

A1.1 The function Θ(ω) and regularization
To exhibit more clearly the problems arising from the regularization, we calculate the function (3.25) first and Fourier transform only after. Since in momentum space the function is given by a convolution, it becomes a product (M > 0) of the form e −M |x| |x| sgn(x 3 ).
The integration over p 1 , p 2 yields x 1 = x 2 = 0. Thus, We see that the singularity at x 3 = 0 requires a regularization. If we choose a reflection symmetric regularization, |x 3 | ≥ ε > 0, we can divide the integral into x 3 > 0 and x 3 < 0, change x 3 into −x 3 for x 3 < 0 and add again the two contributions. The result is The limit ε = 0 can then be taken and the integral calculated. One recovers expression (3.26). By contrast, a non-symmetric short distance cut-off yields a logarithmically divergent limit. However, we note that in a Feynman parametrization such a symmetrization is automatically achieved when the order of integrations is interchanged and the logarithmic divergence then does not show up:

A1.2 The function D and the bound state problem
We consider the function (9.30) tan gkB 1 (k) 1 − 2M tan gkB 1 (k) /k and assume that the parameters are such that D(0) is positive. We write it as We are interested in the real time region below the ψψ threshold. We thus set The function φ is monotonous decreasing between 0 and 1, which varies between 0 and −1/(1 − λ). Thus, if the condition is satisfied, one finds a massive scalar bound state.

A2 Relevant algebraic identities
We now describe the algebraic identities that have been systematically used in this work to solve the various integral equations that were encountered.

A2.1 Some elementary though useful identities
In this work, we use systematically identities based on the partial fraction decomposition of rational functions of the form where P is a polynomial. If P is of degree smaller than n, then In this form, the identity is equivalent to Lagrange interpolation formula. For higher degrees, additional terms are generated and for the relevant cases one finds A reformulation of the identities (A2.1-A2.4) where z ≡ ω n+1 , is also directly useful. We define where S symmetrizes over {ω 1 , ω 2 , . . . , ω n+1 }. One then finds S n,m = 0 for m < n , S n,n = 1 n + 1 , S n,n+1 = 1 Finally, in section 8.5 we need Symmetrizing over all integration momenta and using equations (A2.6,A2.5) and noting that the remaining factors in the integrand are even functions, we find , and, thus, the general equation The remaining integral is calculated in section A6.1.
A third relevant family of integrals is Then, for p 3 → ∞, ).
Following the same lines, one obtains Integral equations and corresponding integrals. A second set contains integrals relevant for the integral equations. They can be split into two families.
Even functions. A first family has the form .
We then replace the function Ξ by its integral representation, .
We symmetrize the integrand over the (2n + 1) integration variables p, q i and use the identities (A2.1). We infer In particular, if F (z) is an even function, expandable at z = 0, and A related integral is We write it a sum, using p 3 = ℓ 3 − (ℓ 3 − p 3 ). The first term yields ℓ 3 I n . We symmetrize the second term and use the identities (A2.6). Only the term of highest degree in p 3 contributes and one finds For an even function F , one then obtains A last integral of the family is .
We substitute this time ). The first term yields ℓ 3 J n . For the second term, after cancellation of the gauge propagator, we use the identities (A2.6). The degree of the denominator is 2n and of the numerator (2n + 1). For p 3 → ∞, Thus, only the term of highest degree contributes. But since it is odd and the remaining integrand is even, it does not contribute. We conclude that if F is an even function, (A2.12) Odd functions. We now examine integrals of odd functions. We first need Once the function Ξ is replaced by its integral representation, as a function of p 3 , numerator and denominator have the same degree 2n. We symmetrize and now use the identities (A2.1,A2.2). The result is the sum of two contributions. The first one is 1 2n τ 2n (ℓ 3 )Ξ 2n (ℓ 3 , k).
The second one comes only from the term of highest degree in p 3 of τ 2n : Then, if F is an odd function and the result is We substitute p 3 = ℓ 3 − (ℓ 3 − p 3 ). The first term yields ℓ 3 L n . The second term leads to a ratio of a polynomial of degree 2n over a polynomial of degree (2n −1). Using the identities (A2.6), we find a sum of odd terms while the integrand is even. M(F ) = ℓ 3 L(F ). (A2.14)

A3 The (ψψ)ψψ vertex: two-loop calculations
We calculate the (ψψ)ψψ vertex (or 1PI) function up to two loops for N large.
Notation. To simplify all expressions, we introduce the compact notation

A3.1 Perturbative calculations
One-loop results. We define We now calculate the corresponding vertex function in the Fourier representation, settingΓ (1,2) F n g n and the boundary conditions E 0 = 1, F 0 = 0. We use the dressed fermion propagator (3.39) in such a way that only ladder diagrams have to be summed.
The order g reduces to Integrating, one obtains We have also defined (Eq. (5.24)) We expand A = n=0 A n g n , B = n=0 B n g n .
Then A 0 = 1, B 0 = 0 and where all quantities are expressed in terms of the fermion physical mass M . The expansion ofṼ (1,2) to order g inserted into expression (5.24) then generates contributions of order g 2 in the expansion ofΓ (1,2) . They are given by 1b) The vector (A 2 , B 2 ) is then obtained by subtracting them from (E 2 , F 2 ).
Two-loop calculation. The two-loop contribution corresponds to a sum of diagrams, a vertex correction and two renormalizations of the fermion propagator, which are generated by expanding the one-loop term calculated with the dressed propagator.
Denoting by q the additional integration vector, we obtain the vertex correction by first acting with the product T (ℓ 3 , q 3 )T (q 3 , p 3 ), where T is the matrix (5.10), on the vector (1, 0).
The propagator renormalizations are inferred from the order g in the expansion of T to order g. In the latter contribution, we replace the function Θ by its integral representation. With the choice of momenta of figure 4, we substitute Θ(p 3 + k 3 /2) + Θ(p 3 − k 3 /2) = 1 (2π) 3 d 3 q (Dm 2 + Dp 2 ) (p 3 − q 3 ) Dm 2 Dp 2 .
Subtracting δE 2 and δF 2 , we obtain A5 The R two-point function First, we give now some details about the perturbative calculations of the R two-point function.

A5.1 The R two-point function at three loops
The first diagram is represented in figure 7 while the other diagrams all correspond to propagator corrections applied to one-loop and two-loop diagrams. To display the different contributions, we complete the compact notation introduced in section A3. We define Dp 3 = (r + k/2) 2 + M 2 , Dm 3 = (r − k/2) 2 + M 2 .
The product of the four integrals is the symmetric function of the remaining four integration variables, which we denote by I(ω).

A6 Two-point J 3 calculations
In this calculation, to keep the notation simple, we assume M > 0.
Moreover, the quantitiesΓ µ − iσ µ , since they are only linear combinations of the matrices 1 and σ 3 , all satisfy an integral equation of the form (5.31) but with different inhomogeneous terms.