$\epsilon$-Expansion in the Gross-Neveu Model from Conformal Field Theory

We compute the anomalous dimensions of a class of operators of the form $(\bar\psi\psi)^p$ and $(\bar\psi\psi)^p\psi$ to leading order in $\epsilon$ in the Gross-Neveu model in $2+\epsilon$ dimensions. We use the techniques developed in arXiv: 1505.00963.


Introduction
In recent work [1] (see also [2]) the techniques of conformal field theory have been used for the computation of leading order anomalous dimensions of composite operators in interacting CFTs defined in terms of epsilon expansions about d = 3, 4 spacetime dimensions. The novelty of this technique lies in using conformal symmetry judiciously without taking recourse to any perturbative methods and Feynman diagrams, which has so far been used in such calculations.
The goal of this work is to compute the leading order -in the epsilon expansionanomalous dimensions of a class of composite operators in the Gross-Neveu model in 2 + ǫ dimensions. Our analysis involves two and three point functions and the OPE of relevant operators, and uses only conformal symmetry. We thus accomplish this without relying on Feynman diagrams and conventional perturbation theory techniques. The analysis follows closely the methods of [1] who first used the method to determine anomalous dimensions of similar operators in the O(N ) vector model. This note is organised as follows. We provide the basic set up in section 2. In section 3 we use methods similar to [1] to compute the anomalous dimensions of the operators ψ and ψψ. The result of this section is in agreement with that available in the literature. After this simple illustration of the technique, we turn to the general case of higher composite operators. In the appendix we compute the required combinatorial coefficients in the free theory OPE using a recursive diagrammatic approach [2]. In section 4, the two and three point functions, as well as the OPE, of the interacting theory are used and matching with the expected free theory results ultimately leads to a pair of recursion relations involving the leading order anomalous dimensions. The final result for the leading order anomalous dimensions are given in equation (4.45). In section 5, we compute the anomalous dimensions of scalars which are not singlet under U (Ñ ). As far we know, these have not been computed before in the literature and the results of section 4 and 5 are new.

The Gross-Neveu model
The Gross-Neveu model [3] is a renormalizable field theory in two dimensions. It is described by a U (Ñ ) symmetric action forÑ massless self-interacting Dirac fermions {ψ I ,ψ I }. We will consider the Gross-Neveu model in 2 + ǫ dimensions [4] S = d 2+ǫ x ψ I / ∂ψ I + 1 2 gµ −ǫ ψ I ψ I 2 , I = 1, ....,Ñ . (2.1) Here g is the coupling constant which is dimensionless in two dimensions. This theory has a weakly coupled UV fixed point given by the non-trivial zero of the beta function, Here Tr I is the trace of identity in Dirac fermion space, and in two dimensions N = 2Ñ . The fixed point occurs at 3) The special case of N = 2 for which the β function vanishes identically corresponds to the Thirring model. In this paper we consider the case for which N > 2.
The dimensions of the fermion ψ I , ∆ 1 , and composite scalarψ I ψ I , ∆ 2 , are given by The anomalous dimensions of the fundamental fermions and the composite scalar in the ǫexpansion have been computed in perturbation theory using the standard Feynman diagram techniques and, to leading order in ǫ, are given by The purpose of this note is to derive the above expressions, and similar ones for higher dimensional composite operators, using conformal field theory techniques without doing Feynman diagram computations 1 . For this we assume that the fixed point is a conformal fixed point. In two dimensions, the fermion propagator is given by We normalise our fields In order to simplify the notation in the analysis below, we use the normalised elementary field but denoted by the old variable. In this normalisation, the two point function is In the free theory the fermions satisfy / ∂ψ I = 0, ∂ µψ I Γ µ = 0 which are the shortening conditions for the multiplets {ψ I } free and {ψ I } free . In addition all other bilinears of ψ I andψ I are primary operators. At the interacting fixed point {ψ I } fixed pt and {ψ I } fixed pt are no longer short multiplets. The primary operators in the free theory ψ I ψ J ψ J and ψ I ψ J ψ J now become descendants of the {ψ I } fixed pt. and {ψ I } fixed pt. respectively. This phenomena of multiplet recombination was observed in φ 4 -theory [1] where two conformal multiplets in the free theory join and become a single conformal multiplet at the interacting fixed point. As in [1] we assume that every operator O in the free theory has a counterpart V O at the interacting fixed point. The operators V O and their correlation functions in the interacting theory, approach, respectively, O and their free correlation function in the ǫ → 0 limit. In the Gross Neveu model at the IR free point, various operators are constructed out of products of elementary operators ψ andψ. We will denote operators in the interacting theory as V 2p , V 2p+1 andV 2p+1 such that in the limit ǫ → 0 (IR free point) We also require that the multiplet recombination is achieved by 1 See [6,7] for various aspects of the Gross-Neveu model in 2 + ǫ dimensions 2 In general, for non-integer dimensions, gamma matrices are infinite dimensional and there are infinite number of antisymmetrized products. However for the calculation of anomalous dimensions to the leading order in ǫ, for the class of operators (ψψ) n and ψ(ψψ) n , this complication will not play any role.
for some unknown function α ≡ α(ǫ) which will be determined below. As an equation of motion, this follows from the Gross-Neveu lagrangian, but in the non-lagrangian approach we follow it is to be interpreted purely as an operator relation indicating that the operator V 3 is, in the interacting theory, a descendant of the primary operator V 1 .
Let us illustrate, schematically, how multiplet recombination is used together with the OPE to determine the leading order anomalous dimensions (This is the method developed by [1] and used in later sections here). Suppose the interacting theory has an operator relation of the form ∂V O = αV O ′ (as explained above, O's denote operators in the free theory whose counterparts in the interacting theory are the V O 's ). We first find an OPE in the free theory of primary composite operators O p , O p ′ which contains, in the leading terms, O and O ′ : (2.13) The dots above denote descendant terms (derivatives acing on O) and other subleading terms contain other operators in the spectrum (we have supressed various powers of x). The leading order terms suffice for our purpose. ρ is a combinatorial coefficient, determined by Wick contractions in the free theory (see the appendix). Now in the interacting theory, the corresponding OPE would read: (2.14) Note that, using the operator relation, the second term on the right hand side above is proportonal to V O ′ . In the interacting theory, this operator (V O ′ ) is a descendant and this crucial fact, together with matching with the free theory in the ǫ → 0 limit, implies qα = ρ. The coefficient q will be determined later in terms of anomalous dimensions of the operators in the OPE and will be seen to be singular in the ǫ → 0 limit. The coefficient α is determined below (to the required leading order in ǫ) and the above relation will be seen (in section 4.3) to lead to a recursion relation for the leading order anomalous dimensions.
We turn now to the determination of α. We have Differentiating the above expression and contracting with Γ µ matrices, we get Now requiring that in the limit ǫ → 0, V I 3 (x)V J 3 (y) approaches the free theory correlation we get the expression for α 3 Anomalous dimension of ψ,ψ andψψ In this section we will compute the anomalous dimensions of the fundamental fermion and the composite scalar. The results of this section are in perfect agreement with the leading order anomalous dimension computed from Feynman diagram techniques. In the next section we will generalise this to higher dimensional operators and derive some new results for the anomalous dimensions. We consider the OPE between ψ andψψ in the free theory. 3 For this we do not need the full OPE except those terms which are sensitive to the multiplet recombination, We will compare the above expression for the free OPE with the OPE at the interacting UV fixed point. For this we need the three point function at the interacting fixed point. According to [8], we have 4 In the above f is a constant. From this we can compute the following OPE Here 4) It is important to note here that in the above OPE, we have kept only contributions coming from the conformal family of V I 1 which includes V 3 as its descendant. Although the multiplet recombination does not necessarily require a lagrangian description, here our knowledge of the shortening condition follows from the equation of motion (2.12). Even though we use different methods, we are dealing with perturbative fixed points which do have a lagrangian description. Thus our analysis is not entirely lagrangian independent 5 . A, B i , C i , .. are functions of conformal dimensions which we determine by considering x 1 → x 3 and expanding (3.2) in powers of x 13 , Comparing with (3.3) we can get all the coefficients. We list here the first few coefficients Now we consider the following free correlators in the limit |x 1 | << |x 2 |, Using the OPE (3.3), we have This will match with the free correlator if f → −1 in the limit ǫ → 0. Next we compare the correlation function with the insertion of the descendant operatorV I 3 , HereV K 3 is the descendant ofV K 1 defined in (2.12) and the derivative acts on the first insertion. It is very easy to see that the first two terms containing A, B 1 in the expansion of C on the right hand side go to zero as we take ǫ → 0, Now we see that the contribution to (3.9) will come from the term with B 2 . In fact using the expansion we get In the above we used the equation of motion for the primary field (2.12). Thus we see that it will go to the free correlator if f B 2 α ∼ O(1) in the limit ǫ → 0. Since f goes to constant and α goes to zero, B 2 must diverge. We also see from (3.6) that B 2 has a chance of blowing up. If we define . (3.13) Thus B 2 will blow up if γ 1 vanishes as at least O(ǫ 2 ) Now we write γ 1 ∼ y 1,2 ǫ 2 , γ 2 ∼ y 2,1 ǫ . (3.14) Then we get Using that f → −1, we get Also in the interacting theory, the conformal dimension ∆ 3 of the descendant V I 3 (x 1 ) is related to ∆ 1 of V I 1 (x 1 ) by We will show this by explicit computation in the next section. Now we are interested in finding the OPE between V I 3 and V 2 . This can be obtained from (3.3) by acting with a derivative and using (2.12).
In order to compare with the free correlator, we also need the following OPE Proceeding as before, we find that for |x 1 | << |x 2 |, we have Thus in order to match with the free correlator, we requirefB 2 → 1. Now using that Using (3.12) and (3.14), to leading order in ǫ, we get y 2,1 + y 2 2,1 = 4y 1,2 .
Therefore the anomalous dimensions are These results are in agreement with results in [4,5].

Anomalous dimensions of (ψψ) p and (ψψ) p ψ
In this section we will compute the leading order anomalous dimensions of a class of higher dimensional composite operators in the interacting theory described by the UV fixed point of the Gross-Neveu Model. In the free theory limit (ǫ → 0) these operators are of the form (ψψ) p and ψ(ψψ) p with p > 1. Let us denote these operators in the interacting theory as V 2p and V 2p+1 such that in the limit ǫ → 0 (axiom)

The structure of the OPEs
We will need the following OPEs in the free theory where I is an U (Ñ ) index. f 2p , f 2p+1 and ρ 2p , ρ 2p+1 are combinatorial coefficients. Counting all possible Wick contractions gives their values to be See the appendix for details of the calculation. Now let us consider the corresponding OPEs in the interacting theory. The most general structure of the OPE, in the first case where the free theory limit is eq. (4.2), is .
(4.6) The dots indicate other primary operators that can appear in the OPE. Here The differential operators C(x 12 , ∂ 2 ) and D(x 12 , ∂ 2 ) have the general form For the OPE of two generic primary operators (one bosonic and the other fermionic) both of these structures can occur. However now we will show that, for the V 2p (x 1 )V I 2p+1 (0) OPE, only the first structure in eq. (4.6) is consistent with our axiomatic requirement that in the limit ǫ → 0 correlators of the interacting theory should match with corresponding correlators in the free theory.
For this consider the 3 pt. function . Then using the OPEeq. (4.6) -we have, In the free theory the OPE, eq. (4.2) gives, Now since in the ǫ → 0 limit we require and, Hence we clearly see that only the first structure in eqn. (4.6) needs to be considered. In other words all the coefficients appearing in D(x 12 , ∂ 2 ) can be set to zero in this case.
Next consider the OPE of V I 2p+1 and V 2p+2 . Again just on grounds of conformal symmetry we can write down an expression similar to eq. (4.6). But once again it is easy to show using the free theory OPE, eq. (4.3) that our axiom allows only the second structure of eq. (4.6) for the OPE of V I 2p+1 and V 2p+2 . Note that the above distinction is important when both operators involved in the OPE are primary operators. When one of the operators is a descendant the structure of the OPE simply follows by acting with derivatives on the OPE of the primary operators. For example when p = 1 the OPE of V 2 and V I 3 can be obtained from the OPE of V I 1 and V 2 by differentiating the latter.

Determining the coefficients in the OPE
We will now obtain the expression for the coefficients in eq. (4.9). The method for doing this is simple. The form of the 3 pt. function which is fixed in the usual way by conformal invariance is matched against the form obtained by taking the OPE of the first two operators within the 3-pt. function. We start with the following 3 pt. function, The form is determined by conformal invariance which allows for both the above structures 6 . Now using the OPE . (4.17) we get, In obtaining the second line above we have used the following results: But from eqn. (4.16), Comparing the above equation with eq. (4.18) we get, Since the tensor structure of the first term in eq. (4.22) does not have any matching with the tensor structures appearing in eq. (4.18), we can set g 1 = 0. Finally we have, . (4.24) Next we consider the following 3 pt. function In this case using the OPE we get .
(4.27) Therefore, In the limit |x 1 | ≪ |x 3 | eq. (4.25) becomes (4.29) Comparing the above equation with eq. (4.28), we obtain, This gives, Here, similar arguments as above would set g ′ 2 = 0. This again shows that in the 3 pt. function of two primary fermion operators and a primary scalar operator in general one must keep both tensor structures. Which structure contributes in a specific case depends upon the particular primary operators under consideration. When one of the operators involved in the 3 pt. function is a descendant, the allowed structure is of course determined by the correlator of primary operators.

Recursion relations for the leading order anomalous dimensions
In the ǫ → 0 limit, the OPEs of the interacting theory should go over to the free theory OPEs -eqs. (4.2, 4.3) -and the corresponding 3 pt. functions must match as well. This matching gives We use the following relations Writing γ k (ǫ) = y k,1 ǫ + y k,2 ǫ 2 + .... we get, Using y 1,2 = N −1 4(N −2) 2 this gives, Similarily we get for the other case, Solving the recursion relations eqs. (4.40) and (4.44) (with σ = 1) we get our desired result, Thus we have for the scaling dimensions of these composite operators, Note, in particular, that the classically marginal operator (ψψ) 2 receives corrections to its conformal dimension only at O(ǫ 2 ), since for p = 2 the second and third terms in the expression for ∆ (ψψ) p cancel. This is analogous to the bosonic case treated in [1] where the classically marginal operator (φ.φ) 2 has the same property.

Other scalar primaries
In this section we will consider a scalar primary which is not a singlet under the symmetry group U (Ñ ) and calculate its anomalous dimension. In the free theory we consider a scalar of the form In order to calculate the OPE we need the following correlation functions in the free theory: Therefore for x ∼ 0, we get Thus we get the following OPE in the free theory, Now we proceed as before. We assume that there exists an operator V O (LM ) (x) at the fixed point corresponding to the operator O (LM ) (x). Based on the symmetries, the 3-point function involving the scalar at the fixed point is given by The OPE obtained in (3.3) should hold in the case of scalar fields. All the corresponding coefficients are given in (3.6) except that ∆ 2 is replaced by ∆ (LM ) . Thus in this case, with E(x 13 , ∂ z ) = A ′ + B ′ 1 x µ 13 + B ′ 2 / x 13 Γ µ ∂ µ + C ′ 1 x µ 13 x ν 13 + C ′ 2 x µ 13 / x 13 Γ ν + C ′ 3 x 2 13 Γ µ Γ ν ∂ µ ∂ ν +.........
We proceed as in previous cases. We find thatf should approach −1 in the limit ǫ → 0. Furthermore the 3-point function with the descendant has the form Therefore the leading order anomalous dimension is (5.14) We could not find a check for this result in the literature. It would be interesting to compare this new result against a perturbative computation of the anomalous dimension.

Discussion
In this note we have computed, to first order in the epsilon expansion, the anomalous dimensions of a class of composite operators in the Gross-Neveu model. As emphasised earlier, we have done the computation without using the usual perturbative techniques. The primary input was conformal symmetry, which fixed for us the two and three point functions and the required OPEs. The main results, which, to our knowledge, have not been known before, are given in eq. (4.45), . It is to be emphasised that the methods used here can fix only the leading order anomalous dimensions. It would be interesting to extend the computations to second order in ǫ. As discussed in [1] for the case of the O(N ) bosonic vector model, two and three point functions would not suffice for the higher order computation and one would require conformal bootstrap of the four point functions to extract further information. How conformal symmetry can be used together with OPE associativity to deduce anomalous dimensions at higher orders in ǫ remains an interesting open problem (see [10,11] for recent work in this direction).
To compute f 2p we need to contract p pairs of ψ's and finally leave ψ I . We will get the following three diagrams. In these diagrams the top row has p blocks at x and bottom row has p blocks together with one × at origin. X X