A test of bosonization at the level of four-point functions in Chern-Simons vector models

We study four-point functions in Chern-Simons vector models in the large N limit. We compute the four-point function of the scalar primary to all orders in the ‘t Hooft coupling λ = N/k in U (N )k Chern-Simons theory coupled to a fundamental fermion, in both the critical and non-critical theory, for a particular case of the external momenta. These theories cover the entire 3-parameter “quasi-boson” and 2-parameter “quasi-fermion” families of 3-dimensional quantum field theories with a slightly-broken higher spin symmetry. Our results are consistent with the celebrated bosonization duality, as we explicitly verify by calculating four-point functions in the free critical and non-critical bosonic theories.


Introduction
U (N ) k Chern-Simons theories coupled to fundamental matter provide an interesting class of interacting three-dimensional conformal field theories that are exactly solvable in the 't Hooft limit: N → ∞, k → ∞ with the 't Hooft coupling λ ≡ N k held fixed. These theories, which have been intensively studied in the past few years [1][2][3][4][5][6][7][8][9][10][11][12][13], are particularly interesting because they provide examples of non-supersymmetric dualities. For instance, they are widely believed to be dual to Vasiliev higher-spin gauge theories (see [14] for a review). They also exhibit a spectacular bosonization duality relating Chern-Simons theory coupled to fundamental fermions to critical Chern-Simons theory coupled to fundamental bosons, which can be thought of as a non-supersymmetric generalisation of the ABJ and Giveon-Kutasov dualities. [1-8, 10, 15-17] The bosonization duality has been tested via three point functions and also in thermal free energy computations, leaving little doubt to its correctness. However, it is still of independent interest to directly test the duality at the level of four-point functions; which are not determined by purely kinematic considerations.
In this paper, we calculate four-point correlation functions of the primary scalar operator J (0) in the critical and non-critical U (N ) k Chern-Simons theory coupled to fundamental JHEP12(2015)032 fermions. For a particular choice of external momenta, we are able to obtain a closed form (but highly non-trivial) expression for the four point function of the scalar primary as a function of λ -which we then compare to the free and critical bosonic theories to obtain another independent check of the bosonization duality.

Review of the Bosonization duality
The bosonization duality [3], which can be thought of as a non-supersymmetric generalization of the Giveon-Kutasov duality [16], states that a U (N f ) k f Chern-Simons theory coupled to fermions in the fundamental representation is dual to a U (N b ) k b Chern-Simons theory coupled to critical bosons in the fundamental representation. The critical theory is obtained by deforming the usual (non-critical) theory by a double trace operator λ 4 φ 2 φ 2 and taking the coupling to infinity. (The coupling λ 4 should not be confused with The conjectured duality claims that the two theories are equivalent, with the following relation between parameters: Though we present the duality in terms of k and N , the duality has only been tested in the large N , 't Hooft limit; at finite N there will be some shifts of ±1/2 in the Chern-Simons level for the fermionic theory as discussed in [3,4]. All parameters are defined in a dimensional reduction regularization scheme, used in [3]. In terms of λ = N k , the duality can be written as: 3) From these results, we have the simple relation |λ b | = 1 − |λ f | and sign(λ b ) = − sign(λ f ). As both sides of the theory are exactly solvable, the simplest way to verify the duality is to calculate correlation functions on both sides, which we illustrate below.
The two-point function of the scalar primary, which is defined as J 0 f ≡ψψ, in the fermionic theory is: In the critical bosonic theory, the two-point function is:

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These two-point functions determine the relative normalisation of the scalar operator in the two descriptions. We see that, taking J 0 b = 4πλ b J 0 f , 1 the two point functions are identical: The duality now implies that, for the three-point functions: Using the results in [3,4], we can explicitly compute that: which agrees with the duality. There are additional predictions for three-point functions; for instance for the threepoint function of vector and scalar operators, which are non-zero, and which can be tested on similar lines.
Applying the duality to four-point functions, we obtain: In what follows, we directly calculate the l.h.s. of (1.10) (for a restricted class of external momenta) and obtain a finite answer in the limit λ f → 1 (when expressed in terms of N b ). The result can then be compared to a calculation the critical bosonic theory at λ b = 0 on the r.h.s. and we find perfect agreement. As described below, and in [3,4], the non-critical bosonic theory is dual to a critical fermionic theory. We also compare the critical fermionic theory to the non-critical bosonic theory and find agreement.

The exact ladder diagram
In this paper, we will exclusively use light-cone gauge. 2 Two crucial features of lightcone gauge are that ghosts decouple and cubic vertices are absent. Therefore any planar 1 This relation makes sense because λ and J 0 f are odd under parity, while J 0 b is even. 2 The conventions used in the following sections are those of [1,3,4]. In particular, our light-cone gauge is defined in Euclidean space and described in detail in [1]. We also use the notation that γ A may be the 2 × 2 identity matrix 1 or any of the three γ µ and p 2 s = p 2 1 + p 2 2 = 2p+p−. correlation function can be evaluated to all orders in λ given the exact propagator, first evaluated in [1], and the "exact ladder diagram" defined diagrammatically in figure 1, which we denote by Γ AB (p, q, r)γ A ⊗ γ B . More precisely, we define Γ AB (p, q, r)γ A ⊗ γ B as the following four-fermion interaction term in the quantum effective action for fermions, obtained after integrating out the gauge field in light cone gauge.

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S(p), the exact large N fermion propagator, valid to all orders in λ, is defined via and is given by [1]: The gauge propagator is defined via A µ (p)A ν (−q) = δ ab (2π) 3 δ 3 (p − q)G νµ (p) and is given by 3) The self-consistent Schwinger-Dyson equation, given diagrammatically in figure 1, where all propagators are taken to be exact propagators S(p), is: In terms of S(p) = S A (p)γ A , we have: (2.5) JHEP12(2015)032

Rewriting the Schwinger-Dyson equation
There are 16 components of Γ AB , which appear to be coupled. We now obtain an alternate expression for Γ AB that "diagonalizes" the Schwinger-Dyson equation (2.5) and shows that most of the 16 components are not independent. For this purpose it is convenient to define: It is easy to see that the inverse relation is (where 'Tr' denotes a trace over the gamma matrices.) Let us also define, following [1] The Schwinger-Dyson equation (2.5) can be re-written as Because H + contains only the identity and γ + components,this means that Evaluating (2.12) explicitly, we obtain: (2.14) 2.2 Evaluating the exact ladder diagram when q ± = 0 We have not yet been able to solve this integral equation for arbitrary q. However, if we restrict ourselves to q ± = 0 it is possible to obtain a solution, which will enable us to calculate the four-point function of scalar primaries for a restricted class external momenta.
To motivate our ansatz for the solution, we note that the results of section 2.1 can also be thought of diagrammatically as follows: let f (0) (p, q, r) be any 2 × 2 matrix (with spinor indices) that is a function of p, q and r. We think of f (0) δ i j as representing an arbitrary "contraction" of the ladder diagram on the right, so that the tree level ladder diagram acting on f 0 δ l m is given by as pictured in figure 2.
We then define f (n) (p, q, r) (which can be thought of as the ladder diagram with n "rungs", contracted with f (0) on the right) recursively in terms of f (n−1) : Because G +3 = −G 3+ are the only nonzero components of G µν , only two components of f (n−1) contribute to f (n) , which are In terms of these variables, the equation (2.16) is: (2.18) JHEP12(2015)032 The infinite sum f (n) is related to A Q P defined in (2.6) of the previous subsection and f − as follows: Let us choose f (0) (which is an arbitrary matrix) such that f The parameter c is arbitrary, and introduced for convenience: when we set c = 0, equation (2.19) determines A I + and A I I and when c → +∞, equation (2.19) determines A − + and A − I . We note that it is consistent to assume f (n) (p, q, r) is independent of p 3 . Evaluating equation (2.16), including the l 3 integral, we have: (2.21) After integrating to obtain the first few terms, we find f We can sum the series to obtain: (2.26)

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We now use the identity and, following [3], introduce the dimensionless variables: to carry out the angular integrals and rewrite (2.24) and (2.25) as To solve these coupled four-variable equations (which could in principle have been obtained directly from equation (2.14) with the approporiate ansatz for A Q P ) we differentiate and the resulting differential equation decouples into two sets of two variable coupled equations.
Equating coefficients of r + p + , p + and r + , we obtain: The general solution of the set is 2e −2iλ arctan[x] 1 + e −2iλ arctan(Λ) . (2.33) To determine A I + and A I I we set c = 0 and to determine A − + and A − I we take the limit c → ∞ (i.e., equate coefficients of c on both sides of equation (2.19)). Writing the answers in the form: we have: In appendix A, we evaluate the exact vertex for the scalar primary using this result.

JHEP12(2015)032 3 Four-point correlation functions
We now proceed to calculate the (gauge-invariant and parity-invariant) four-point function of the scalar primary. In this section, we evaluate the four-point function in the free fermionic theory, the interacting fermionic theory and the critical interacting fermionic theory. In section 4, we then evaluate the four-point function in the free and critical bosonic theories to test the duality. The four-point functions depend on external momenta q (i) ; as discussed in the previous section, our calculations are only valid in the special case of q ± = 0 (i.e., only q 3 = 0). Hence, in what follows, we drop all spacetime-indices and label the four external momenta as q 1 , q 2 , q 3 and q 4 .

Interacting fermionic theory
We now proceed to calculate the four-point function in the interacting theory.
In the interacting theory, our basic ingredients are the exact propagator, the exact J (0) vertex V [4], and the ladder diagram in section 2. The correlator can be written as as a sum of Diagrams A, B and C in figure 4 (where it is understood that all propagators are exact), The diagrams B and C involve the ladder diagram. Diagram A is given by  and diagrams B and C are given by It is difficult to solve this integral in closed from for arbitrary p, q, and t. To obtain humanly readable answers that can be easily compared to the bosonic theory, we observe that the limit where two momenta (q → 0 + and t → 0 + ) vanish is relatively tractable. Let us first consider the diagrams where the two non-vanishing external momenta are "diagonal" (as depicted in figure 4 when q → 0 + and t → 0 + ). In this limit, the integral is solvable. We find the 2 "diagonal" permutations of diagram A are given by The next step is to evaluate the remaining permutations of diagrams A, B and C with two adjacent non-vanishing external momenta (i.e., permutations of the external momenta not depicted in figure 4). These integrals are more nontrivial, but can be obtained by using the substitution q → p − q in (3.5) and (3.7) which evaluates to and ψ(x) = Γ (x) Γ(x) is the Digamma function. It can be seen that the above equation goes to 0 for λ → 0 and decreases approximately as − tan 2 ( πλ 2 ). This property will be later be of use in the critical theory.
In summary, after adding all the diagrams we get the following result It is worthwhile to notice that both terms in the R.H.S of (3.13) are parity invariant. We can draw a parallel between (3.2) , (3.3) and (3.10) , (3.13) respectively.
It is natural to conjecture that this expression generalises to the following expression for general momenta: where F (q 1 , q 2 , q 3 , q 4 ) is the four-point function of the scalar operator in the free fermionic theory and H(q 1 , q 2 , q 3 , q 4 , λ) is an additional structure, which goes to zero as λ → 0.

Critical fermionic theory
We now consider the four point function of the scalar primary in the critical fermionic theory described in [3], [4], which is conjectured to be dual to the (non-critical) bosonic theory.
Let us briefly review the definition of the critical fermonic theory, which at zerocoupling is essentially the Gross-Neveu model (in three-dimensions): we introduce a field σ (without a kinetic term) that couples to the scalar primary as S σ = d 3 xσψψ and perform a path integral over σ. The equation of motion for σ isψψ = 0; therefore, instead ofψψ, the single trace scalar primary operator in the critical theory is σ. Notice that,ψψ has scaling dimension 2, so σ has scaling dimension 1, which matches the scaling dimension of the scalar primary J (0) b =φφ in the non-critical bosonic theory, as required for a duality. Because σ has mass dimension 1, there is the possibility of an additional marginal coupling of the form d 3 xN λ F 6 3! σ 3 , which is related to the marginal φ 6 coupling in the bosonic theory. In the large N limit, the exact two point function of σ is clearly related the inverse of the two point function ofψψ, via

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where two-point function of the scalar primary J (0) f in the non-critical theory is Directly using the result (3.14), it is not hard to calculate the four-point function of the scalar primary operator σ in the critical fermionic theory. The diagrams that contribute to the four-point function of σ are shown in figure 5. (3.17) The critical fermionic theory is dual to the non-critical bosonic theory in the limit λ f → 1 and λ 6 = 0. In this limit, there are two surviving terms in (3.17), the first two diagrams of figure 5. Using (3.14) and the fermionic three point function [4] we find In the last step we have applied the limit λ f → 1 and used (1.6) to express the answer in terms of |λ b |, anticipating the comparison in the next section. We have also used the fact that H(q 1 , q 2 , q 3 , q 4 ) ∼ tan 2 πλ f 2 to eliminate it from (3.19). Using equation (

Non-critical bosonic theory
In the free theory, we have (4.1) We solve (4.1) in the limit q ± = 0. The integral evaluates to N 2 (q 2 2 + q 2 3 + q 1 q 2 + q 1 q 3 + q 2 q 3 ) |q 1 |q 2 q 3 (q 1 + q 2 + q 3 )(q 1 + q 2 )(q 1 + q 3 )(q 2 + q 3 ) + (q 1 → q 2 , q 3 ) where the arrows in the numerator imply symmetric terms on replacing q 1 with q 2 and q 3 . Using the normalisation relation which follows from comparing the two point functions of the scalar primaries in both theories, it is easy to see that this result matches the R.H.S of (3.19) -i.e., that thereby verifying the duality between the critical fermionic theory and the non-critical bosonic theory (for our restricted choice of external momenta).

Critical bosonic theory
Next we turn to the four point correlator at the critical fixed point of the theory. This is accomplished by adding a double trace term to the scalar action. The vertex is given by We next flow to the IR limit with the IR scalar mass zero by tuning λ 4 to infinity. The scalar propagator does not get a finite correction from this deformation. The divergent terms can be subtracted by a mass counterterm. Two and three point correlators in the critical theory were discussed in [3,4].

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We defineJ (0) = λ 4 J (0) as the scalar operator at the critical fixed point. In the IR limit, taking λ 4 → ∞ expanding the denominator and keeping the leading term we get (4.8) Figure 6(A) turns out to be (4.9) We can use (4.2) and use the same methods to get (4.10) Adding (4.8) and (4.10) to obtain , it is easy to see that the momentum dependence equates to the non-critical free fermion correlator (3.3). Employing normalisationJ  (4.11) thereby verifying the duality between the critical bosonic theory and the non-critical fermionic theory for our restricted choice of external momenta.

Discussion
The main result of the paper is (3.14), an explicit expression for the four-point function of the scalar primary in a particular limit of external momenta for both the non-critical fermionic theory. We also calculated the four-point function in the critical fermionic theory, and compared to critical and non-critical free bosons, providing an independent confirmation of the bosonization duality introduced in section 1 at the level of four-point functions.
Our calculations crucially relied on the off-shell exact ladder diagram (2.34)-(2.37) together with (2.19). It is relatively straightforward to solve the resulting integral equations for the case when q ± = 0. However, if we could generalise the calculation above to the case q ± = 0, we would be able to calculate four-point functions with arbitrary momenta. More importantly, the off-shell ladder diagram is also required for calculating 1/N corrections (and M/N corrections in a bifundamental theory, see [18]) to all orders in λ. We hope to return to this off-shell ladder diagram in the future. We note that the on-shell four point function is calculated to all orders in [6], and its supersymmetric generalization [13].

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For J (0) , f 0 I and f 0 − can be seen from figure 2 with the contracted vertex on the right as the free scalar vertex. Subsequently, it can be written as ( Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.