A test of bosonization at the level of four-point functions in Chern-Simons vector models
AA test of bosonization at the level of four-pointfunctions in Chern-Simons vector models
Akshay Bedhotiya and Shiroman Prakash a a Dayalbagh Educational Institute, Agra, India
E-mail: [email protected] , [email protected] Abstract:
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 fundamentalfermion, in both the critical and non-critical theory, for a particular case of the externalmomenta. These theories cover the entire 3-parameter "quasi-boson" and 2-parameter"quasi-fermion" families of 3-dimensional quantum field theories with a slightly-brokenhigher 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-criticalbosonic theories. a r X i v : . [ h e p - t h ] O c t ontents q ± = 0 U ( N ) k Chern-Simons theories coupled to fundamental matter provide an interesting classof interacting three-dimensional conformal field theories that are exactly solvable in the’t Hooft limit: N → ∞ , k → ∞ with the ’t Hooft coupling λ ≡ Nk held fixed. Thesetheories, which have been intensively studied in the past few years [1–13], are particularlyinteresting 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 areview.) They also exhibit a spectacular bosonization duality relating Chern-Simons theorycoupled to fundamental fermions to critical Chern-Simons theory coupled to fundamentalbosons, which can be thought of as a non-supersymmetric generalisation of the ABJ andGiveon-Kutasov dualities. [1–8, 10, 15–17]The bosonization duality has been tested via three point functions and also in thermalfree energy computations, leaving little doubt to its correctness. However, it is still ofindependent interest to directly test the duality at the level of four-point functions; whichare not determined by purely kinematic considerations.In this paper, we calculate four-point correlation functions of the primary scalar opera-tor J (0) in the critical and non-critical U ( N ) k Chern-Simons theory coupled to fundamentalfermions. 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– 1 –unction of λ – which we then compare to the free and critical bosonic theories to obtainanother independent check of the bosonization duality. The bosonization duality [3], which can be thought of as a non-supersymmetric general-ization of the Giveon-Kutasov duality [16], states that a U ( N f ) k f Chern-Simons theorycoupled to fermions in the fundamental representation is dual to a U ( N b ) k b Chern-Simonstheory coupled to critical bosons in the fundamental representation. The critical theory isobtained by deforming the usual (non-critical) theory by a double trace operator λ φ φ andtaking the coupling to infinity. (The coupling λ should not be confused with λ b = N b /k b .)The conjectured duality claims that the two theories are equivalent, with the followingrelation between parameters: k f = − k b (1.1) N f = | k b | − N b (1.2)Though we present the duality in terms of k and N , the duality has only been tested inthe large N , ’t Hooft limit; at finite N there will be some shifts of ± / in the Chern-Simons level for the fermionic theory as discussed in [3, 4]. All parameters are defined ina dimensional reduction regularization scheme, used in [3]. In terms of λ = Nk , the dualitycan be written as: λ f = λ b − sign ( λ b ) , (1.3) N f = N b − | λ | b | λ | b , (1.4)or λ b = λ f − sign ( λ f ) , (1.5) N b = N f − | λ | f | λ | f . (1.6)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 dualityis to calculate correlation functions on both sides, which we illustrate below.The two-point function of the scalar primary, which is defined as J f ≡ ¯ ψψ , in thefermionic theory is: (cid:104) J f ( − q ) J f (cid:105) = − N f tan( πλ f / πλ f | q | . (1.7)In the critical bosonic theory, the two-point function is: (cid:104) J b ( − q ) J b (cid:105) = − N b πλ b tan( πλ b / | q | . (1.8)– 2 –hese two-point functions determine the relative normalisation of the scalar operator in thetwo descriptions. We see that, taking J b = 4 πλ b J f , the two point functions are identical: (cid:104) J b ( − q ) J b (cid:105) = − N b πλ b tan( πλ b / | q |(cid:104) J f ( − q ) J f (cid:105) = − N b cot( πλ b / πλ b | q | = N f sign ( λ b ) tan( πλ f / π (1 − | λ b | ) | q | = − N f tan( πλ f / πλ f | q | The duality now implies that, for the three-point functions: (cid:104) J f ( q ) J f ( q ) J f ( q ) (cid:105) = (cid:18) πλ b (cid:19) (cid:104) J b ( q ) J b ( q ) J b ( q ) (cid:105) (1.9)Using the results in [3, 4], we can explicitly compute that: (cid:104) J f ( q ) J f ( q ) J f ( q ) (cid:105) = 0 (cid:104) J b ( q ) J b ( q ) J b ( q ) (cid:105) = 0 which agrees with the duality,.There are additional predictions for three-point functions; for instance for the three-point function of vector and scalar operators, which are non-zero, and which can be testedon similar lines.Applying the duality to four-point functions, we obtain: (4 π ( λ f − sign ( λ f ))) (cid:10) J f ( q ) J f ( q ) J f ( q ) J f ( q ) (cid:11) = (cid:10) J b ( q ) J b ( q ) J b ( q ) J b ( q ) (cid:11) (1.10)In what follows, we directly calculate the LHS of (1.10) (for a restricted class of externalmomenta) and obtain a finite answer in the limit λ f → (when expressed in terms of N b ).The result can then be compared to a calculation the critical bosonic theory at λ b = 0 onthe RHS and we find perfect agreement.As described below, and in [3, 4], the non-critical bosonic theory is dual to a criticalfermionic theory. We also compare the critical fermionic theory to the non-critical bosonictheory and find agreement. In this paper, we will exclusively use light-cone gauge . Two crucial features of light-cone gauge are that ghosts decouple and cubic vertices are absent. Therefore any planar This relation makes sense because λ and J f are odd under parity, while J b is even. The conventions used in the following sections are those of [1, 3, 4]. In particular, our light-cone gaugeis defined in Euclidean space and described in detail in [1]. We also use the notation that γ A may be the × identity matrix or any of the three γ µ and p s = p + p = 2 p + p − . – 3 – igure 1 . The diagrammatic definition of the exact ladder diagram, which is the shaded box. Allpropagators in this diagram are exact propagators. In light cone gauge, this diagram is sufficientto calculate all planar correlation functions of single trace operators. correlation function can be evaluated to all orders in λ given the exact propagator, firstevaluated 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 interactionterm in the quantum effective action for fermions, obtained after integrating out the gaugefield in light cone gauge. S eff = − (cid:90) d pd qd r (2 π ) Γ AB ( p, q, r ) ¯ ψ i ( − p ) γ A ψ j ( r ) ¯ ψ j ( − r − q ) γ B ψ i ( p + q ) . (2.1) S ( p ) , the exact large N fermion propagator, valid to all orders in λ , is defined via (cid:104) ψ i ( p ) ¯ ψ j ( q ) (cid:105) = δ ji (2 π ) δ ( p − q ) S ( p ) and is given by [1]: S ( p ) = (cid:18) − iγ µ p µ + iλ γ + p − + λp s p (cid:19) . (2.2)The gauge propagator is defined via (cid:104) A µ ( p ) A ν ( − q ) (cid:105) = δ ab (2 π ) δ ( p − q ) G νµ ( p ) and is givenby G +3 = 4 πik p + . (2.3)The self-consistent Schwinger-Dyson equation, given diagrammatically in Figure 1, whereall propagators are taken to be exact propagators S ( p ) , is: Γ AB ( p, q, r ) γ A ⊗ γ B = (cid:18) − πik (cid:19) p − r ) + (cid:0) γ + ⊗ γ − γ ⊗ γ + (cid:1) + (cid:18) − N πik (cid:19) (cid:90) d l (2 π ) l + Γ CD ( l, q, r ) × (cid:0) γ + S ( l ) γ C ⊗ γ D S ( l + q ) γ − γ S ( l ) γ C ⊗ γ D S ( l + q ) γ + (cid:1) . (2.4)In terms of S ( p ) = S A ( p ) γ A , we have: Γ AB ( p, q, r ) γ A ⊗ γ B = (cid:18) − πik (cid:19) p − r ) + (cid:0) γ + ⊗ γ − γ ⊗ γ + (cid:1) − (2 λπi ) (cid:90) d l (2 π ) l + Γ CD ( l, q, r ) S E ( l ) S F ( l + q ) × (cid:0) γ + γ F γ C ⊗ γ D γ E γ − γ γ F γ C ⊗ γ D γ E γ + (cid:1) (2.5)– 4 – .1 Rewriting the Schwinger-Dyson equation There are 16 components of Γ AB , which appear to be coupled. We now obtain an alternateexpression for Γ AB that “diagonalizes” the Schwinger-Dyson equation (2.5) and shows thatmost of the 16 components are not independent.For this purpose it is convenient to define: πiA P Q ( p, q, r ) γ P = Γ AB ( p, q, r ) γ A γ Q γ B . (2.6)It is easy to see that the inverse relation is πi Tr (cid:0) γ A γ P γ B γ Q (cid:1) A P Q ( p, q, r ) = Γ AB ( p, q, r ) (2.7)(where ‘Tr’ denotes a trace over the gamma matrices.)Let us also define, following [1] H + ( Y ) = γ Y γ + − γ + Y γ = 2( Y I γ + − Y − I ) . (2.8)The Schwinger-Dyson equation (2.5) can be re-written as πiA P Q ( p, q, r ) γ P = Γ AB ( p, q, r ) γ A γ Q γ B (2.9) = (cid:18) − πik (cid:19) p − r ) + (cid:0) γ + γ Q γ − γ γ Q γ + (cid:1) + (cid:18) − N πik (cid:19) (cid:90) d l (2 π ) ll + Γ CD ( l, q, r ) S E ( l ) S F ( l + q ) × (cid:0) γ + γ F γ C γ Q γ D γ E γ − γ γ F γ C γ Q γ D γ E γ + (cid:1) (2.10) = (cid:18) πik (cid:19) p − r ) + H + ( γ Q )+ (cid:18) N πik (cid:19) (cid:90) d l (2 π ) l + Γ CD ( l, q, r ) S E ( l ) S F ( l + q ) × H + ( γ F γ C γ Q γ D γ E ) (2.11) A P Q ( p, q, r ) γ P = (cid:18) k (cid:19) p − r ) + H + ( γ Q )+2 πiλ (cid:90) d l (2 π ) l + A P Q ( l, q, r ) S E ( l ) S F ( l + q ) × H + ( γ F γ P γ E ) (2.12)Because H + contains only the identity and γ + components,this means that A Q ( p, q, r ) = A − Q ( p, q, r ) = 0 (2.13)for all Q . This translates into linear equations relating various of the 16 components of Γ AB . Moreover, the 8 A + Q and A I Q are the only non-vanishing components of A , and theyare independent for different values of Q . It is also consistent to set A P = 0 and A P + = 0 so we have only 4 equations, which are 2 pairs of 2 coupled integral equations.– 5 –valuating (2.12) explicitly, we obtain: A + Q ( p, q, r ) γ + + A I Q ( p, q, r ) = (cid:18) k (cid:19) p − r ) + H + ( γ Q )+2 πiλ (cid:90) d l (2 π ) l + A + Q ( l, q, r ) S E ( l ) S F ( l + q ) × H + ( γ F γ + γ E )+2 πiλ (cid:90) d l (2 π ) l + A I Q ( l, q, r ) S E ( l ) S F ( l + q ) × H + ( γ F γ E ) (2.14) q ± = 0 We have not yet been able to solve this integral equation for arbitrary q . However, ifwe restrict ourselves to q ± = 0 it is possible to obtain a solution, which will enable us tocalculate 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 alsobe thought of diagrammatically as follows: Let f (0) ( p, q, r ) be any × matrix (with spinorindices) that is a function of p , q and r . We think of f (0) δ ij as representing an arbitrary"contraction" of the ladder diagram on the right, so that the tree level ladder diagramacting on f δ lm is given by N γ ν f (0) ( p, q, r ) γ µ G µν ( p − r ) δ ij = N δ ij G +3 ( p − r ) H + (cid:16) f (0) ( p, q, r ) (cid:17) (2.15)as pictured in Figure 2. Figure 2 . A contraction of the tree-level ladder diagram corresponding to Equation (2.15). Notethat there is no integration over momenta.
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 − : γ ν f ( n ) ( p, q, r ) γ µ G µν ( p − r ) = N γ ν (cid:18)(cid:90) d l (2 π ) S ( l ) γ σ f ( n − ( l, q, r ) γ ρ G ρσ ( l − r ) S ( l + q ) G µν ( p − l ) (cid:19) γ µ (2.16)as pictured in Figure 3.Because G +3 = − G are the only nonzero components of G µν , only two componentsof f ( n − contribute to f ( n ) , which are H + (cid:16) f ( n ) (cid:17) = 2 f ( n ) I γ + − f ( n ) − I . (2.17)In terms of these variables, the equation (2.16) is: H + ( f ( n ) ( p, q, r ))( p − r ) + = 2 πiλ (cid:90) d l (2 π ) H + (cid:16) S ( l ) H + (cid:16) f ( n − ( l, q, r ) (cid:17) S ( l + q ) (cid:17) l − r ) + ( p − l ) + (2.18)– 6 – igure 3 . The diagrammatic relation for f ( n ) in terms of f ( n − . The infinite sum (cid:80) f ( n ) is related to A QP defined in (2.6) of the previous subsectionand f (0) I and f (0) − as follows: (cid:80) f ( n ) I k ( p − r ) + = A I + f (0) I + A − + f (0) − , − (cid:80) f ( n ) − k ( p − r ) + = A II f (0) I + A − I f (0) − . (2.19). Let us choose f (0) (which is an arbitrary matrix) such that f (0) − = − cf (0) I r + . Theparameter c is arbitrary, and introduced for convenience: when we set c = 0 , equation(2.19) determines A I + and A II 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 . Evaluatingequation (2.16), including the l integral, we have: f nI ( p )( p − r ) + = − iλ (cid:90) d l (2 π ) f n − I ( l ) ( q + 2 iλl s ) l + − f n − − ( l ) (cid:0) (cid:0) − λ (cid:1) l s (cid:1)(cid:0) q + 4 l s (cid:1) l s p − l ) + ( l − r ) + (2.20) f n − ( p )( p − r ) + = − iλ (cid:90) d l (2 π ) f n − I ( l )2 l +2 − f n − − ( l ) (2 iλl s − q ) l + (cid:0) q + 4 l s (cid:1) l s p − l ) + ( l − r ) + (2.21)After integrating to obtain the first few terms, we find f ( n ) I ( p ) and f ( n ) − ( p ) to be of the form: f ( n ) I ( p ) = a ( n )1 + a ( n )2 r + p + (2.22) f ( n ) − ( p ) = − b ( n )1 p + − b ( n )2 r + (2.23)We can sum the series to obtain: (cid:88) f ( n ) I ( p ) − f (0) I = − iλ (cid:90) d l (2 π ) (cid:0)(cid:80) f nI ( l ) ( q + 2 iλl s ) l + − (cid:80) f n − ( l ) (cid:0) (cid:0) − λ (cid:1) l s (cid:1)(cid:1)(cid:0) q + 4 l s (cid:1) l s ( p − r ) + ( p − l ) + ( l − r ) + (2.24) (cid:88) f ( n ) − ( p ) − f (0) − = − iλ (cid:90) d l (2 π ) (cid:16)(cid:80) f ( n ) I ( l )2 l +2 − (cid:80) f ( n ) − ( l ) (2 iλl s − q ) l + (cid:17)(cid:0) q + 4 l s (cid:1) l s ( p − r ) + ( p − l ) + ( l − r ) + (2.25)From (2.22) and (2.23) we make the ansatz (cid:88) f ( n ) I ( p ) = (cid:18) a + a r + p + (cid:19) f (0) I (cid:88) f ( n ) − ( p ) = − (cid:0) b p + + b r + (cid:1) f (0) I . (2.26)We now use the identity p − l ) + ( l − r ) + = 1( p − r ) + (cid:18) p − l ) + + 1( l − r ) + (cid:19) (2.27)– 7 –nd, following [3], introduce the dimensionless variables: y = 2 p s | q | x = 2 r s | q | t = 2 l s | q | ˆΛ = 2Λ s | q | ˆ λ = λ sign ( q ) (2.28)to carry out the angular integrals and rewrite (2.24) and (2.25) as (cid:18)(cid:18) a + a r + p + (cid:19) − (cid:19) = (cid:90) yx (cid:18) a (cid:16) i ˆ λt (cid:17) + b (cid:16) − ˆ λ (cid:17) t (cid:19) dt + (cid:90) x ( a (1 + iλt ) + b ) dt + r + p + (cid:90) y (cid:18) a (cid:16) i ˆ λt (cid:17) + b (cid:16) − ˆ λ (cid:17) t (cid:19) dt (2.29) (cid:0) b p + + b r + (cid:1) − cr + = r + (cid:18)(cid:90) Λ x (cid:16) a + b (cid:16) i ˆ λt − (cid:17)(cid:17) dt + (cid:90) yx (cid:16) a + b (cid:16) i ˆ λt − (cid:17)(cid:17) dt (cid:19) + p + (cid:90) y Λ (cid:16) a + b (cid:16) i ˆ λt − (cid:17)(cid:17) dt. (2.30)To solve these coupled four-variable equations (which could in principle have beenobtained directly from equation (2.14) with the approporiate ansatz for A QP ) we differenti-ate and the resulting differential equation decouples into two sets of two variable coupledequations.Equating coefficients of r + p + , p + and r + , we obtain: ∂a ∂y = − (cid:18) a (cid:16) i ˆ λy (cid:17) + b (cid:16) − ˆ λ (cid:17) y (cid:19) ∂b ∂y = (cid:18) a + b (cid:16) i ˆ λy − (cid:17)(cid:19) ∂a ∂y = − (cid:16) a (cid:16) i ˆ λy (cid:17) + b (cid:16) − ˆ λ (cid:17) y (cid:17) ∂b ∂y = (cid:16) a + b (cid:16) i ˆ λy − (cid:17)(cid:17) (2.31)The general solution of the set is a i = α i (cid:16) β i (1 − i ˆ λy ) − (1 + i ˆ λy ) e − i ˆ λ arctan( y ) (cid:17) b i = 2 α i (cid:16) β i + e − i ˆ λ arctan( y ) (cid:17) (2.32)where the subscript i = 1 , . Requiring that the solutions satisfy the integral equations– 8 –2.29) and (2.30) fixes the integration constants: β = − e − i ˆ λ arctan[ˆΛ] β = 1 α = c (cid:16) (1 − i ˆ λx ) − (1 + i ˆ λx ) e − i ˆ λ arctan[ x ] (cid:17) − (cid:16) e − i ˆ λ arctan[ x ] (cid:17) e − i ˆ λ arctan[ x ] (cid:16) e − i ˆ λ arctan[ˆΛ] (cid:17) α = c (cid:16) (1 − i ˆ λx ) e − i ˆ λ arctan[ˆΛ] + (1 + i ˆ λx ) e − i ˆ λ arctan[ x ] (cid:17) − (cid:16) e − i ˆ λ arctan[ˆΛ] − e − i ˆ λ arctan[ x ] (cid:17) e − i ˆ λ arctan[ x ] (cid:16) e − i ˆ λ arctan(ˆΛ) (cid:17) . (2.33)To determine A I + and A II 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 answersin the form: ˜ A BC = 2 ke − i ˆ λ arctan[ x ] (cid:16) e − i ˆ λ arctan(ˆΛ) (cid:17) ( p − r ) + A BC we have: ˜ A II = (cid:16) e − i ˆ λ arctan[ˆΛ] − e − i ˆ λ arctan[ y ] (cid:17) (cid:16) e − i ˆ λ arctan[ x ] (cid:17) (cid:18) p + q (cid:19) + (cid:16) e − i ˆ λ arctan[ x ] − e − i ˆ λ arctan[ˆΛ] (cid:17) (cid:16) e − i ˆ λ arctan[ y ] (cid:17) (cid:18) r + q (cid:19) (2.34) ˜ A − I = − (cid:16) e − i ˆ λ arctan[ˆΛ] − e − i ˆ λ arctan[ y ] (cid:17) (cid:16) ( i ˆ λx −
1) + (1 + i ˆ λx ) e − i ˆ λ arctan[ x ] (cid:17) (cid:18) p + r + (cid:19) − (cid:16) (1 + i ˆ λx ) e − i ˆ λ arctan[ x ] − ( i ˆ λx − e − i ˆ λ arctan[ˆΛ] (cid:17) (cid:16) e − i ˆ λ arctan[ y ] (cid:17) (2.35) ˜ A I + = (cid:16) (1 + i ˆ λy ) e − i ˆ λ arctan[ y ] − ( i ˆ λy − e − i ˆ λ arctan[ˆΛ] (cid:17) (cid:16) e − i ˆ λ arctan[ x ] (cid:17) + (cid:16) e − i ˆ λ arctan[ˆΛ] − e − i ˆ λ arctan[ x ] (cid:17) (cid:16) ( i ˆ λy −
1) + (1 + i ˆ λy ) e − i ˆ λ arctan[ y ] (cid:17) (cid:18) r + p + (cid:19) (2.36) ˜ A − + = − (cid:16) (1 + i ˆ λy ) e − i ˆ λ arctan[ y ] − ( i ˆ λy − e − i ˆ λ arctan[ˆΛ]] (cid:17) × (cid:16)(cid:16) (1 + i ˆ λx ) e − i ˆ λ arctan[ x ] + ( i ˆ λx − (cid:17) (cid:16) q r + (cid:17)(cid:17) − (cid:16) ( i ˆ λx − e − i ˆ λ arctan[ˆΛ] − (1 + i ˆ λx ) e − i ˆ λ arctan[ x ] (cid:17) × (cid:16) ( i ˆ λy −
1) + (1 + i ˆ λy ) e − i ˆ λ arctan[ y ] (cid:17) (cid:18) q p + (cid:19) (2.37)In Appendix A, we evaluate the exact vertex for the scalar primary using this result. We now proceed to calculate the (gauge-invariant and parity-invariant) four-point functionof the scalar primary. In this section, we evaluate the four-point function in the freefermionic theory, the interacting fermionic theory and the critical interacting fermionictheory. In section 4, we then evaluate the four-point function in the free and critical– 9 –osonic 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 caseof q ± = 0 (i.e., only q (cid:54) = 0 ). Hence, in what follows, we drop all spacetime-indices andlabel the four external momenta as q , q , q and q . The free ( λ f = 0 ) four-point function in the U ( N f ) fermionic theory is given by F ( q , q , q , q ) = (cid:104) J (0) ( − q ) J (0) ( − q ) J (0) ( − q ) J (0) ( − q ) (cid:105) = − N f (cid:90) d k (2 π ) Tr ( S ( k ) S ( k + q ) S ( k + q + q ) S ( k + q + q + q ))+ permutations (3.1)where S ( q ) = ( i/q ) − is the free fermion propagator and all external momenta are restrictedin the 3 direction.Evaluating this, we obtain F ( q , q , q , q ) = N f q + q )( q + q ) ( | q | − | q | + | q | − | q | ) + F ( q , q , q , q ) = N f q | q | + q | q | + q | q | + q | q | q + q )( q + q )( q + q ) δ ( q + q + q + q ) (3.3) 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 asum of Diagrams A , B and C in Figure 4 (where it is understood that all propagators areexact), (cid:104) J (0) ( p − q ) J (0) ( − t ) J (0) ( t − p ) J (0) ( q ) (cid:105) = [ A ] + [ B ] + [ C ] . (3.4)The diagrams B and C involve the ladder diagram.Diagram A is given by ( A ) = − N (cid:90) d k (2 π ) Tr [ V ( p − q ) S ( k ) V ( − t ) S ( k + t ) V ( t − p ) S ( k + p ) V ( q ) S ( k + p − q )]+(5 permutations ) (3.5)– 10 – igure 4 . Diagrams in the interacting theory. and diagrams B and C are given by ( B ) = N (cid:90) d k (2 π ) Tr [ S ( r + p ) V ( q ) S ( r − p + q ) V ( p − q ) S ( r ) γ µ S ( k ) V ( − t ) S ( k + t ) V ( t − p ) S ( k + p ) γ ν Γ µν ( k, p, r )] + (5 permutations ) (3.6) ( C ) = N (cid:90) d k (2 π ) Tr [( S ( r + p − q − t ) V ( p − q ) S ( r − t ) V ( − t ) S ( r ) γ µ S ( k ) V ( t − p ) S ( k + p − t ) V ( q ) S ( k + p − q − t ) γ ν Γ µν ( k, p − q − t, r )] + (5 permutations ) (3.7)It is difficult to solve this integral in closed from for arbitrary p , q , and t . To obtain humanlyreadable answers that can be easily compared to the bosonic theory, we observe that thelimit where two momenta ( q → + and t → + ) 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 → + and t → + ). In this limit, the integral issolvable. We find the 2 “diagonal" permutations of diagram A are given by − N (cid:90) d k (2 π ) Tr [ V ( p − q ) S ( k ) V ( − t ) S ( k + t ) V ( t − p ) S ( k + p ) V ( q ) S ( k + q )]= − N sec( πλ ) ( πλ + sin( πλ ))4(2 πλ | p | ) (3.8)For diagrams B and C , we use (2.19)and (2.34)-(2.35) for the ladder expressions, and obtain2 permutations of B = N sec (cid:18) πλ (cid:19) (2 πλ cos (cid:0) πλ (cid:1) − (cid:0) πλ (cid:1) + sin (cid:0) πλ (cid:1) )32 pπλ (3.9)Adding (3.8) and (3.9) for B as well as C gives − N (cid:0) πλ (cid:1) tan (cid:0) πλ (cid:1) | p | πλ (3.10)The next step is to evaluate the remaining permutations of diagrams A , B and C withtwo adjacent non-vanishing external momenta (i.e., permutations of the external momenta– 11 – igure 5 . The plot of modulus of H ( p, λ ) function and tan [ πλ ] vs λ for momenta p = 1 . It isclear that the magnitude of H ( λ ) rises slower than tan [ πλ ] . not depicted in Figure ). These integrals are more nontrivial, but can be obtained by usingthe substitution q → p − q in (3.5) and (3.7) which evaluates to − N | p | (cid:32) sec (cid:0) πλ (cid:1) tan (cid:0) πλ (cid:1) πλ + h ( λ ) (cid:33) (3.11)where h ( λ ) is h ( λ ) = i e iπλ ) (cid:16) e iπλ πλ cot ( πλ ) + λψ (cid:18) − λ (cid:19) − λψ (cid:18) − λ (cid:19) + e iπλ (cid:18) λψ (cid:18) λ (cid:19) − λψ (cid:18) λ (cid:19)(cid:19) (cid:17) (3.12)and ψ ( x ) = Γ (cid:48) ( x )Γ( x ) is the Digamma function. It can be seen that the above equation goes to0 for λ → and decreases approximately as − tan ( πλ ) . This property will be later be ofuse in the critical theory.In summary, after adding all the diagrams we get the following result (cid:104) J (0) ( p − q ) J (0) ( − t ) J (0) ( t − p ) J (0) ( q ) (cid:105) = − N (cid:0) πλ (cid:1) tan (cid:0) πλ (cid:1) | p | πλ − N h ( λ ) | p | (3.13)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 expressionfor general momenta: (cid:104) J (0) ( − q ) J (0) ( − q ) J (0) ( − q ) J (0) ( − q ) (cid:105) = 2 sec (cid:0) πλ (cid:1) tan (cid:0) πλ (cid:1) πλ F ( q , q , q , q )+ H ( q , q , q , q , λ ) (3.14)– 12 –here F ( q , q , q , q ) is the four-point function of the scalar operator in the free fermionictheory and H ( q , q , q , q , λ ) is an additional structure, which goes to zero as λ → . We now consider the four point function of the scalar primary in the critical fermionictheory described in [3], [4], which is conjectured to be dual to the (non-critical) bosonictheory.Let us briefly review the definition of the critical fermonic theory, which at zero-couplingis essentially the Gross-Neveu model (in three-dimensions): We introduce a field σ (withouta kinetic term) that couples to the scalar primary as S σ = (cid:82) d xσ ¯ ψψ and perform a pathintegral over σ . The equation of motion for σ is ¯ ψψ = 0 ; therefore, instead of ¯ ψψ , thesingle trace scalar primary operator in the critical theory is σ . Notice that, ¯ ψψ has scalingdimension 2, so σ has scaling dimension , which matches the scaling dimension of the scalarprimary J (0) b = ¯ φφ in the non-critical bosonic theory, as required for a duality. Because σ has mass dimension , there is the possibility of an additional marginal coupling of theform (cid:82) d xN λ F σ , which is related to the marginal φ coupling in the bosonic theory.In the large N limit, the exact two point function of σ is clearly related the inverse ofthe two point function of ¯ ψψ , via G ( q ) = (cid:104) σ ( − q ) σ (cid:105) = (cid:16) −(cid:104) J (0) f ( − q ) J (0) f (cid:105) non-critical (cid:17) − , (3.15)where two-point function of the scalar primary J (0) f in the non-critical theory is (cid:104) J (0) f ( − q ) J (0) f (cid:105) = − N f tan (cid:16) πλ f (cid:17) πλ f | q | . (3.16) Figure 6 . The diagrams that contribute to the four-point function of σ in the critical fermionictheory. The shaded square is the exact scalar four-point function in the non-critical theory, theshaded triangle is the exact three-point function in the non-critical theory and the dashed linewith a shaded circle is the (cid:104) σσ (cid:105) propagator. The last diagram includes the contribution from themarginal λ coupling, represented by black dot. (There is also another diagram with only one λ vertex and one exact three point function that is not pictured.) – 13 –irectly using the result (3.14), it is not hard to calculate the four-point function of thescalar primary operator σ in the critical fermionic theory. The diagrams that contribute tothe four-point function of σ are shown in Figure 6. (cid:104) ˆ σ ( − q )ˆ σ ( − q )ˆ σ ( − q )ˆ σ ( − q ) (cid:105) = N ˜ λ q.b G ( − q ) G ( − q ) G ( − q ) G ( − q )( (cid:104) J (0) ( − q ) J (0) ( − q ) J (0) ( − q ) J (0) ( − q ) (cid:105) + G ( q + q )( (cid:104) J (0) ( − q ) J (0) ( − q ) J (0) ( q + q ) (cid:105)(cid:104) J (0) ( − q ) J (0) ( − q ) J (0) ( q + q ) (cid:105) + λ ( (cid:104) J (0) ( − q ) J (0) ( − q ) J (0) ( q + q ) (cid:105) + (cid:104) J (0) ( − q ) J (0) ( − q ) J (0) ( q + q ) (cid:105) ) + λ )) + permutations(3.17)The critical fermionic theory is dual to the non-critical bosonic theory in the limit λ f → and λ = 0 . In this limit, there are two surviving terms in (3.17), the first twodiagrams of Figure 6. Using (3.14) and the fermionic three point function [4] we find (cid:104) ˆ σ ( − q )ˆ σ ( − q )ˆ σ ( − q )ˆ σ ( − q ) (cid:105) = − π λ f cot (cid:16) πλ f (cid:17) | q || q || q || q | × (cid:16) (cid:16) πλ f (cid:17) tan (cid:16) πλ f (cid:17) πλ F ( q , q , q , q ) + H ( q , q , q , q )+ N f tan (cid:16) πλ f (cid:17) πλ f (cid:18) | q + q | + 1 | q + q | + 1 | q + q | (cid:19) (cid:17) (3.18) = − π λ f | q || q || q || q | cot (cid:18) πλ f (cid:19) | λ b || λ f | (cid:18) F ( q , q , q , q ) + N f (cid:18) | q + q | + 12 | q + q | + 12 | q + q | (cid:19)(cid:19) (3.19)In the last step we have applied the limit λ f → and used (1.6) to express the answer interms of | λ b | , anticipating the comparison in the next section. We have also used the factthat H ( q , q , q , q ) ∼ tan (cid:16) πλ f (cid:17) to eliminate it from (3.19). Using equation (4.4), thisexpression can be compared to the four-point correlator of ˆ J (0) b in the bosonic theory whichwe calculate in the next section. In the free theory, we have (cid:104) J (0) ( − q ) J (0) ( − q ) J (0) ( − q ) J (0) ( − q ) (cid:105) (4.1) = N (cid:90) d k (2 π ) k ( k + q ) ( k + q + q ) ( k + q + q + q ) (4.2)– 14 – igure 7 . Correction to the critical scalar four correlator. We solve (4.2) in the limit q ± = 0 . The integral evaluates to N (cid:16) ( q + q + q q + q q + q q ) | q | q q ( q + q + q )( q + q )( q + q )( q + q ) + ( q → q , q )+ q q + q q + q q q q q | q + q + q | ( q + q )( q + q )( q + q ) − ( q + q )( q + q )( q q q ( q + q + q ) | ( q + q ) | ( q + q )( q + q )) + ( q ↔ q ↔ q )) (cid:17) (4.3)where the arrows in the numerator imply symmetric terms on replacing q with q and q .Using the normalisation relation ˆ σ = − πλ f cot (cid:18) πλ f (cid:19) J (0) b (4.4)which follows from comparing the two point functions of the scalar primaries in both theo-ries, it is easy to see that this result matches the R.H.S of (3.19) – i.e., that (cid:104) σ ( − q ) σ ( − q ) σ ( − q ) σ ( − q ) (cid:105) = ( − πλ f cot ( πλ f / (cid:104) J b ( − q ) J b ( − q ) J b ( − q ) J b ( − q ) (cid:105) , thereby verifying the duality between the critical fermionic theory and the non-criticalbosonic theory (for our restricted choice of external momenta). Next we turn to the four point correlator at the critical fixed point of the theory. This isaccomplished by adding a double trace term to the scalar action. The vertex is given by − λ N ( φ † φ ) . We next flow to the IR limit with the IR scalar mass zero by tuning λ toinfinity. The scalar propagator does not get a finite correction from this deformation. Thedivergent terms can be subtracted by a mass counterterm. Two and three point correlatorsin the critical theory were discussed in [3, 4].The four point correlator receives a correction from the λ deformation in two diagrams,one with a four point diagram Figure 7(A) and one with a double three-point diagram Figure– 15 –(B). Diagram 7(B) can be evaluated as (cid:104) J (0) ( − q ) J (0) ( − q ) J (0) ( − q ) J (0) ( − q ) (cid:105) Bλ = (cid:104) J (0) ( − q ) J (0) ( − q ) J (0) (cid:105) λ =0 (cid:104) J (0) ( − q ) J (0) ( − q ) J (0) (cid:105) λ =0 × (cid:89) q = q ,q ,q ,q ,q + q ∞ (cid:88) n =0 (cid:18) − λ N b (cid:104) J (0) ( − q ) J (0) (cid:105) λ =0 (cid:19) n + ( q + q ) → ( q + q ) , ( q + q ) (4.5)The values of (cid:104) J (0) ( − q ) J (0) (cid:105) λ =0 and (cid:104) J (0) ( − q ) J (0) ( − q ) J (0) (cid:105) λ =0 are given in [3, 4]. Weare interested in the free scalar theory hence we take the momentum dependence as (cid:104) J (0) ( − q ) J (0) (cid:105) λ =0 = N b | q | (4.6) (cid:104) J (0) ( − q ) J (0) ( − q ) J (0) ( q + q ) (cid:105) λ =0 = N b | q || q || q + q | (4.7)(4.6) and (4.7)can be substituted in (4.5) to give − N b λ | q || q || q | q || q + q | (cid:89) q ,q ,q ,q ,q + q | q | λ (cid:16) | q | λ (cid:17) + (( q + q ) → ( q + q ) , ( q + q )) (4.8)We define ˜ J (0) = λ J (0) as the scalar operator at the critical fixed point. In the IR limit,taking λ → ∞ expanding the denominator and keeping the leading term we get (cid:104) ˜ J (0) ( − q ) ˜ J (0) ( − q ) ˜ J (0) ( − q ) ˜ J (0) ( q + q + q ) (cid:105) Bλ = − N b (cid:18) | q + q | + 1 | q + q | + 1 | q + q | (cid:19) (4.9)Figure 7(A) turns out to be (cid:104) J (0) ( − q ) J (0) ( − q ) J (0) ( − q ) J (0) ( − q ) (cid:105) Aλ = (cid:104) J (0) ( − q ) J (0) ( − q ) J (0) ( − q ) J (0) ( − q ) (cid:105) λ (cid:89) q = q ,q ,q ,q ∞ (cid:88) n =0 ( − λ N b (cid:104) J (0) ( − q ) J (0) (cid:105) ) n (4.10)We can use (4.3) and use the same methods to get (cid:104) ˜ J (0) ( − q ) ˜ J (0) ( − q ) ˜ J (0) ( − q ) ˜ J (0) ( q + q + q ) (cid:105) Aλ = 8 N b ( sign [ q ]( q + q + q q + q q + q q ) + ( q → q , q )) + ( sign [ q + q + q ]( q q + q q + q q )) − ( sign [ q + q ]( q + q )( q + q ) + ( q ↔ q ↔ q ))( q + q )( q + q )( q + q ) sign [( q q q ( q + q )] (4.11)Adding (4.9) and (4.11) to obtain (cid:104) ˜ J (0) ( − q ) J (0) ( − q ) J (0) ( − q ) J (0) ( − q ) (cid:105) , it is easyto see that the momentum dependence equates to the non-critical free fermion correlator– 16 –3.3). Employing normalisation ˜ J (0) b = 4 πλ b J (0) f obtained in Section 1.1 and applying thelimit λ f → to (3.14) we find (cid:104) ˜ J (0) b ( − q ) ˜ J (0) b ( − q ) ˜ J (0) b ( − q ) ˜ J (0) b ( − q ) (cid:105) = (4 πλ b ) (cid:104) J (0) f ( − q ) J (0) f ( − q ) J (0) f ( − q ) J (0) f ( − q ) (cid:105) (4.12)thereby verifying the duality between the critical bosonic theory and the non-critical fermionictheory for our restricted choice of external momenta. The main result of the paper is (3.14), an explicit expression for the four-point functionof the scalar primary in a particular limit of external momenta for both the non-criticalfermionic 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 confirma-tion 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 equationsfor the case when q ± = 0 . However, if we could generalise the calculation above to the case q ± (cid:54) = 0 , we would be able to calculate four-point functions with arbitrary momenta. Moreimportantly, the off-shell ladder diagram is also required for calculating /N corrections(and M/N corrections in a bifundamental theory, see [18]) to all orders in λ . We hope toreturn to this off-shell ladder diagram in the future. We note that the on-shell four pointfunction is calculated to all orders in [6], and its supersymmetric generalization [19].The theories we study possess a slightly-broken higher spin symmetry. In [20, 21], twodifferent classes of large N field theories with a slightly broken higher spin symmetry werefound to exist – "quasi-boson" and "quasi-fermion" theories. The quasi-fermion theorydepends on two parameters, ˜ N and ˜ λ . The quasi-boson theories depend on three parame-ters, ˜ N and ˜ λ and a . The first two parameters essentially correspond to the rank of thegauge group and the ’t Hooft coupling λ = Nk (in a microscopic description) and the thirdparameter corresponds to the φ triple-trace coupling which is exactly marginal in the large N limit of the bosonic theory.While three point functions in conformal field theories are severely constrained bypurely kinematic considerations, four-point functions are determined only up-to an unde-termined function of two conformal cross-ratios. In particular, conformal invariance restrictsthe four point function of scalars J , with scaling dimension ∆ to the form: (cid:104) J ( x ) J ( x ) J ( x ) J ( x ) (cid:105) = 1 x x f ( u, v ) (5.1)where f is any function of the conformally invariant cross-ratios u = x x x x and v = x x x x ,satisfying f ( u, v ) = ( u/v ) ∆ f ( v, u ) . However, dynamically, all correlation functions in aconformal field theory are (in principle) uniquely determined by the three-point functions(i.e., the operator-algebra) and scaling dimensions of the primary operators in the theory– 17 –ia bootstrap arguments. It would be interesting to study the four-point functions intheories with a slightly broken higher spin symmetry via bootstrap arguments. We hopethe explicit results for Chern-Simons vector models here derived here would be useful insuch a program.
Acknowledgements:
The authors thank Shiraz Minwalla and V. Umesh for readinga draft for the paper. SP acknowledges support of a DST Inspire Faculty Award. SP wouldlike to thank the Centre for High Energy Physics, IISc Bangalore and IISER Pune forhospitality when part of this work was completed and CMS, Durham University and theRudolf Peierls Centre for Theoretical Physics, Oxford University for hospitality during thefinal stages of this work.
Appendix A: Exact vertices from the ladder diagram
As a check on our calculation, the ladder diagram can be utilised to evaluate the exact J (0) vertex derived earlier in [4].For J (0) , f I and f − can be seen from Fig. 2 with the contracted vertex on the right asthe free scalar vertex. Subsequently, it can be written as N f − ( p − r ) + = − iλ (cid:90) d k (2 π ) k + (2 iλk s − q ) k ( k + q ) p − k ) + (.2) N f I ( p − r ) + = − iλ (cid:90) d k (2 π ) k s (1 − λ ) k ( k + q ) p − k ) + (.3)Using eq.(2.34)-(2.37), this immediately yields J (0) ( q ) − I = e − iλ arctan[ pq ] − e − iλ arctan[ q ] e − iλ arctan[ q ] I + q p + ( q − iλp − ( q + 2 iλp ) e − iλ arctan[ psq ] )1 + e − iλ arctan[ q ] γ + (.4)which on the substitution (2.28) gives us the exact J (0) vertex. References [1] S. Giombi, S. Minwalla, S. Prakash, S. P. Trivedi, S. R. Wadia, et al.,
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