Four-point functions in large $N$ Chern-Simons fermionic theories

We compute all four-point functions involving the operators $J_0$ and $J_1$ in large-$N$ Chern-Simons fermionic theories, in the regime where all external momenta lie along the $z$-axis. We find that our result for $\langle J_0 J_0 J_0 J_0 \rangle$ agrees with previous computations, and that the other correlators fall in line with expectations from bootstrap arguments.


Introduction
(a) The quasi-bosonic theories (b) The quasi-fermionic theories Figure 1: The bosonisation dualities have two branches that relate two pairs of CS-matter theories: (a) The quasi-bosonic theories: the starting point is the singlet sector of the free U (N ) bosonic theory ('U (N ) free bosons'). We can then turn on the couplingλ (a monotonic function of the 't Hooft coupling λ; see Section 3), and on taking it to infinity (the strong coupling limit λ → 1), the correlators of the CS-bosonic theories go over to that of the critical fermionic (Gross-Neveu) fixed point, which is the end-point of the line. The bosonisation duality relates this (critical) fermionic theory coupled to CS theory with strength 1/λ to the free bosonic theory coupled to CS theory with strengthλ. The corresponding bulk duals are the type-A θ theories (parametrised by θ ∈ [0, π/2]). (b) The quasi-fermionic theories: starting from the free fermionic theory, we can turn on the CS coupling as in the previous case, and sending it to infinity gives the critical bosonic (Wilson-Fischer) theory. The relevant bosonisation duality relates free fermionic CS theories to critical bosonic theories coupled to CS. The bulk duals are given by the type-B θ theories. In this paper, we will be interested in this family of theories.
Up to the three-point level, conformal symmetry on its own fixes all correlators up to constant factors (the OPE coefficients). Then, provided the respective tensor structures don't vanish in the collinear limit (as was the case for [6,18]), the collinear calculation determines the full result. This is no longer the case at the four-point level, and thus the first full calculation of a four-point function J 0 J 0 J 0 J 0 by [13] relied on a more powerful technique -the OPE inversion formula [19]. The inversion formula was first used to bootstrap the correlator and fix it up to three undetermined constants: whereλ parametrises the quasi-fermionic theories and is monotonically related to the 't Hooft coupling (see Section 3 for details); and the G AdS 's are 4-pt Witten diagrams in AdS; a numerical evaluation of the four-point function (using collinear perturbation theory) for various values of the external momenta then gave linear equations for the b i 's, which were solved to set these constants to 0. The second instance of such a calculation was the recent work of [14], where they calculated the four-point functions J 0 J 0 J 0 J 0 and J 2 J 0 J 0 J 0 5 purely from the (anomalous) higher-spin Ward identities.
The aim of this paper is to probe theλ dependence of two other spinning correlators -namely J 1 J 0 J 1 J 0 and J 1 J 1 J 1 J 1 , in SU (N ) k Chern-Simons theory with a fundamental fermion, via collinear perturbation theory. We compute these correlators analytically in the collinear limit and verify that theλ dependence is exactly as anticipated from the three-point functions via bootstrap arguments (see Section 3). We note that this is the first analytic result that displays the (parity-) odd tensor structures that cannot be obtained from various soluble theories 6 , albeit in a special kinematic regime. However, we will have difficulties applying the methods of [13] to our problem, since we will not be able to construct an ansatz for the full correlator due to these odd tensor structures. That is, we will find by a direct computation that (in the collinear limit): with #'s standing in for the extra structures that do not come from soluble theories 7 ; we know them for collinear momenta, but can't compute these structures in the general case. Hence, we do not know the full analogue the first term in Eqn. (1). But we can say with some confidence that the methods of [13] should be applicable in this case as well. Specifically, in the case of J 1 J 1 J 1 J 1 , the AdS diagrams that can contribute extra terms as in (1) are known 8 , and we have verified that they do not vanish in the collinear limit. Hence, this correlator may be fully determined from the analytic-in-spin answer (the analogue of the first term in Eqn. (1)) and our result. As for J 0 J 0 J 1 J 1 , the relevant allowed diagrams are not known, but we have verified that no contact diagram (to second order in derivatives) vanishes in the collinear limit (see Section 3). We thus conjecture that in both of these cases, as in J 0 J 0 J 0 J 0 , the result is analytic in the spin of the intermediate states, with no extra contact-term-in-AdS contributions.
The arrangement of the rest of the article is as follows: Section 2 reviews some facts about CS-matter theories, gives a brief summary of the formalism of collinear perturbation theory, and gives the details of our calculation of the four-point functions; Section 3 lists our results and gives a more complete discussion of the issue ofλ-dependence that was mentioned here.
2 Computing J 1 J 1 J 0 J 0 and other correlators In this section, we describe the computation of J 1 J 1 J 0 J 0 and other four-point functions to leading order in 1/N and all orders in the 't Hooft coupling λ, for configurations where all the external momenta lie along the z-axis. For concreteness, we shall be talking about a particular correlator J + 1 (q + + δ)J − 1 (−q)J 0 (− )J 0 (−δ) (where J ± 1 are the components of J 1 , expressed in the lightcone gauge defined below) but the same steps mutatis mutandis, give all correlators of the form J 1/0 J 1/0 J 1/0 J 1/0 . As was noted earlier, J 0 J 0 J 0 J 0 was computed via similar methods in [13].

Generalities
Chern-Simons (free) fermionic theory 9 with an SU (N ) gauge group is defined by the (Euclidean) action: In our case, the fermions ψ are in the fundamental representation of SU (N ). As is well known [21,22], this action (4) is gauge-invariant if the coupling k is an integer, up to a shift of half due to the 6 The tensor structures that appear in J0J0J0J0 and J2J0J0J0 may be computed in the free fermionic / critical bosonic theories, and hence the computations described earlier don't have this feature. 7 There is a small caveat to this, in that our results point to a relation between the 'fer', 'bos', and the middle (λ 2 ) structure in J1J1J1J1 (see Section 3). However, we believe that this is an artefact of the collinear regime, and do not know if it holds in the general case. 8 We thank S. Minwalla for this information, based on [20]. 9 We use 'free fermionic theory' to denote N free fermions coupled to an SU (N ) Chern-Simons (gauge) theory. 'Free theory' is a theory of free fermions. 'Weak', 'strong', and 'perturbative' describe the 't Hooft coupling. parity anomaly [23] (which is irrelevant in the large N limit). The spectrum was derived in [8]; the primaries are classified via large-N factorisation. The single-trace primaries are denoted by J s , and there is one for each spin s ∈ 0, 1, ...; these are given (schematically) by: where the parentheses stand for symmetrisation. It was also shown in [8] that the dimensions of these operators are independent of λ (to leading order in 1/N [24,25]), and in the free fermionic theory, given by: In the large N limit, the double-trace operators are given by a product of two single-trace operators. We shall not have to worry about the other multi-trace operators at this order in 1 N . The single-trace primaries for s = 1, 2 are the usual conserved current and stress-energy tensor; for s > 2, they are the almost conserved higher-spin currents: where the right-hand side is made up of double-and triple-trace operators (to O( 1 N 2 ), as described in [8]) . In the dual gravity theory, this means that the higher-spin fields are massive but the mass is generated at the quantum level (i.e., by loop corrections). As demonstrated by [12], (10) leads to great simplifications -for instance, all the three point functions are fixed to O(1/N 2 ). As discussed in the introduction, it is known that there are at least two families of conformal theories (up to the conjectured dualities) that exhibit weakly broken higher-spin symmetry: the quasi-bosonic andfermionic CS theories, which are related via dualities to the higher-spin gravity theories of Vasiliev. Whether these are the only examples of such theories; i.e., whether weakly broken higher-spin symmetry, is powerful enough to fix all correlators (to the relevant order in 1/N ), remains to be seen.

Gauge fixing, regulation, computations
To perform computations, we first move to light-cone coordinates: x ± = x ∓ = 1 √ 2 x 1 ± i x 2 , and gauge fix the theory by setting A − = 0 as in [8] and [6] 10 . This is called the light-cone gauge, and kills the self-interactions between the gluons. The gluon propagator is then given by: We also need a prescription to regulate the theory; we use the hybrid prescription as in [7], which is a cutoff Λ in the x 1 − x 2 plane, and dimensional regularisation in the x 3 direction. The Feynman rules of the theory are: The normalisation of the generators is fixed by Tr T a T b = − 1 2 δ ab . We also note that T a T a = C 2 (N )1 = N 2 1 + O 1 N . Using these rules, one may compute most of the ingredients necessary for calculating four-point functions -namely, the exact J 0 and J 1 vertices, and the exact fermionic propagator to leading order in 1/N . This was done in [18], and we give the expressions here for completeness.
The fermionic propagator: The spin-0 vertex: The spin-1 vertex: where we've defined Λ = 2Λ |q| , and the "parity-invariant" coupling λ = sgn(q)λ. The final piece that we'll need is the four-fermion vertex. This was first computed by [26], but their derivation used conventions different from ours. Since it is important to our calculations, we give our version in Appendix A.

The diagrams
The Feynman diagrams that contribute to the correlator may be divided into two main classesthe box-type (A##) and the ladder-type diagrams (B##) 11 . Box diagrams are those with no propagators connecting non-adjacent fermion lines, and ladder diagrams those with the effective four-fermion vertex linking non-adjacent fermion lines (see Fig. 2 and 3). We may then use the ingredients discussed earlier to compute these. For example, the generic box diagram is given by: . All vertices and propagators stand for their exact (all orders in λ) versions. The vertices are labelled thus -⊕ for J + 1 , for J − 1 , and for J 0 . The 'A' in the name stands for the fact that these are box diagrams. The second letter denotes the position of the vector vertices (a/b/c for non-adjacent/adjacent horizontal/adjacent vertical ) and changing the number at the end switches the two scalar vertices ( → δ). Figure 3: The ladder diagrams that contribute to J + 1 (q + + δ)J − 1 (−q)J 0 (− )J 0 (−δ) . The 'B' in the name stands for the fact that these are ladder diagrams. The second letter stands for the exchange momentum, and changing the number at the end permutes the vertices on each side of the four-point vertex.
, and corresponds to the expression (we ignore the factors of N for now, and restore them at the end): where the O's and the S's correspond to the exact vertices and propagators (as defined earlier). Similarly, the ladder diagram: , gives the expression (see Appendix A):

p 3 -type integrals
In each diagram, there are two types of integrals -the integrals over the third (z) component of the loop momenta or the "p 3 -type" integrals, and the integrals over the planar (1 and 2) components or "p s -type" integrals. We shall discuss these separately. The box diagrams contain only one p 3 integral, while the ladder diagrams contain two. However, these are tackled the same way. We use the fact that the integrands always depend on p 3 (or in case of the ladder diagrams, on r 3 and k 3 ) in the same way 12 : .
This may be easily seen by noting that the exact vertices under consideration (J 0 and J 1 ) and also the ladder, don't depend on the third component of the loop momenta; the only p 3 dependence comes from propagator corrections, which take the form

Box diagrams
We shall do the p s integrals in two steps-the angular part and the radial part (i. e. , dp 1 dp 2 = dθ p p s dp s ). For the box diagrams, the integrand after p 3 integration is independent of θ p 13 and dθ p = 2π. The p s integral, on the other hand, puts up more of a fight. This integral is, in general, intractable 14 and we shall set it aside for now...

Ladder diagrams
The angular integrals: The ladder diagrams have two loop momenta and hence comprise four integralsdθ r r s dr s dθ k k s dk s . This time, we have a non-trivial angular integral -dθ r . To tackle this, we convert the angular integral into a contour integral via the following change of variables: where 'C' stands for the contour |r + | = rs √ 2 . Hence, doing this integral merely involves computing residues. Examining the ladder and (vector-) vertex dependences on r + shows that the full diagram generally has the structure (polynomial in r + ) . This means that residues can come from two pointsr + = 0, and also r + = k + depending on if |r + | ≶ |k + | (i.e., |r s | ≶ |k s |). In summary, (29) Once this is done, the other angular integral ( dθ k ) becomes trivial 15 and yields a factor of 2π.
The planar integrals: The first planar integral to be tackled is dr s . Going back to the residue computation, it may be seen that this integral falls into two pieces: Miraculously, these integrals turn out to be soluble 16 .
However, the residues we get using the method described above are huge, unwieldy expressions. To be able to do anything with these, we have to trim them down to a reasonable form. We choose to do this with the assistance of the various exponentials and their products that pervade our expressions. These come entirely from the exact vertices and the ladder, and are of a certain form -for example, a diagram from the class Bb contains e ±2iλ tan −1 ( 2rs δ ) , e ±2iλ tan −1 ( 2rs ) (from the vertices in the upper half -hence the loop momenta and external momenta are from the upper half), e ±2iλ tan −1 ( 2ks q+δ+ ) , e ±2iλ tan −1 ( 2ks q ) (from the vertices in the lower half), e ±2iλ tan −1 ( 2rs δ+ ) , e ±2iλ tan −1 ( 2ks δ+ ) (from the ladder -the momentum that appears is the momentum exchanged between the upper and lower halves in this subclass) and their products. Rather than simplifying the whole expression at once, we treat the full expression as a polynomial in these exponentials, extract 13 This statement applies only to correlators such as J + 1 J − 1 J0J0 , which are invariant to rotations in the 1-2 plane. For other correlators like J + 1 J0J0J0 , when necessary, we must follow a method similar to the dθr integration described later on.
14 This is no longer the case in special kinematic regimes such as the double-soft limit, which is what led [13] to consider it. 15 Again, this statement is true for J + 1 J − 1 J0J0 , but not necessarily for other correlators. In such cases, the method described above is expected to prove useful. 16 At least in the case of J0J0J0J0 , J + 1 J − 1 J0J0 , and J + their coefficients (taken individually, these contain fewer terms), and simplify then separately. Once the dust has cleared, we find that the r s integrals may be done as discussed earlier.
But the k s integrals, like the p s integrals, are not soluble via Mathematica. So we once again, store these expressions and set them aside...

Arriving at the result
After the procedure described above is complete, we are left with 18 integrals that cannot be done analytically, at least via Mathematica. However, a numerical evaluation will quickly reveal that all of these integrals are (at least in general) convergent. This lets us employ the following trick: (of course, the equation above is schematic; we have more than one box and ladder integral). The integral created thus turns out to be soluble and gives us a cut-off dependent expression for the correlator. The final results (listed in Section 3) are then obtained by sending the cut-off to infinity.
3 Results and discussion

λ dependence via bootstrap arguments
Let's first look at what we can say about the correlators from bootstrap arguments, and the bosonisation duality. The conformal block expansion (in any channel) for J 0 J 0 J 0 J 0 may be written schematically as: where the 1, 'st' and 'dt' stand for the identity operator, single-trace operators, and double-trace operators respectively. Note that we've used the appropriately normalised version of the single-trace operators:J As shown in [27], the three-point function J 0 J 0 O dt at O(1/Ñ ) (it starts at O(Ñ 0 ) in our normalisation) receives corrections from two sources -the OPE coefficient c Here, we use the known form of the single-trace three point functions at O 1/ Ñ [12]. What we'll be interested in are theirλ-dependences: 00O 's (not that we'll be doing either of those). Now we observe that sendingλ to zero means sending the right-hand side of (36) to that of the free (fermionic) theory, which is solved by the corresponding parameters of the free theory. This means that a possible solution to (36) is this produces a four-point correlator proportional to that of the free theory 17 . However, we may add to this any solution of the corresponding homogeneous equation (i.e., with 0 on the right-hand side of (36)). In fact, these solutions were analysed by [27] and correspond to contact Witten diagrams in AdS.
It was then discovered by [13] that only three out of the infinitely many of these "truncated" solutions may be added to the inhomogeneous solution without spoiling the Regge behaviour of the correlator. The most general solution to crossing which is well-behaved in the Regge limit is then: where G AdS V is the contact diagram with vertex V. Furthermore, an explicit numerical evaluation of the correlator by summing the planar diagrams [13] (alternatively, the explicit result that we obtained above) showed that b 1 = b 2 = b 3 = 0 and that the full correlator at O(1/Ñ ) was indeed proportional to the free theory answer. We note that the same result was obtained recently by [14] via an explicit solution of the partial higher-spin Ward identities, which means that modifications by truncated solutions are simply incompatible with higher-spin symmetry. The resultant object also satisfies the criterion of analyticity in spin. It seems that in the case of J 0 J 0 J 0 J 0 (and also J 2 J 0 J 0 J 0 as shown by [14]), weakly broken higher-spin symmetry culls any contact terms 18 that spoil analyticity in spin.
Similar arguments go thorough for the other correlators under our consideration -J 1 J 0 J 1 J 0 and J 1 J 1 J 1 J 1 . This time though, the three-point functions have moreλ-structures, leading to: Of course, we are playing with mixed, spinning correlators and hence have to deal with the complications that come with this (see [28] for details), but all that we need to know here is that in the end, the right-hand sides of the crossing equations (in case of J 1 J 0 J 1 J 0 , the one with the correct channels) have 3 and 5λ-structures respectively, and the analogue of (36) can be solved separately for each one. If this could be done, the resultant solutions would look like: This will be the form that we expect our solutions to take. Note thatλ → 0, ∞ turns the crossing equations into those of the free fermionic and critical scalar respectively, and hence in these limits, we expect the correlator to go over to those theories 19 . Once again, any homogeneous solution may be added to these solutions with arbitraryλ dependence as long as they don't contribute in theλ → 0, ∞ limit. However, we have verified that possible homogeneous solutions give non-vanishing answers in the collinear limit, and hence any new λ dependences would be represented in the collinear limit. In the following, we will find that our results do not exhibit such dependences and are in compliance with the conjecture that J 0 J 0 J 1 J 1 and J 1 J 1 J 1 J 1 depend onλ exactly as in Eqn. (43) and (44).

Results
We list our main results from the computations of the previous section, in the parameterisation of Maldacena and Zhiboedov [12]. We also recall that all external momenta are along the z-axis, and that This matches the result obtained in [13] up to constant factors. The result is proportional to the free theory as was discussed above.

3.2.2
We see that this is pretty much what we expected. The first and last structures may be matched with the four-point functions from the relevant limiting theories. 19 Recall that the scalar operator in the critical theory is given by J cb 0 :=λJ0 and hence J +

3.2.3
The full expression for J 1 J 1 J 1 J 1 is too unwieldy to put here, and is relegated to the appendix. We quote here the result for the special case where q > > δ > 0: Looking at the result, we find a curious coincidence: the tensor structures cancel among themselves to leave three pieces, each with a factor ofλ i 1+λ 2 , i ∈ {0, ..., 2}. This is the second equality in (48), and it also holds for the general expression; the third equality (the relation between the three different structures) doesn't happen in the general case. In any case, this expression (and also the one in the appendix) is in line with our expectations, up to possible terms that vanish in the collinear limit.
As was discussed before, we may add to the expected form (44) of the solution, any homogeneous solution to the crossing equation with an arbitrary λ-dependence; and still satisfy crossing. What we've done here is to verify that J 1 J 1 J 1 J 1 in the collinear limit has the expected λ-dependence. This means that any homogeneous solution that does not vanish in the collinear limit cannot come with an arbitrary λ dependence. In other words, the full correlator has the expected λ-dependence up to those homogeneous solutions, which are well-behaved in the Regge limit, and vanish in the collinear limit.
Like in the scalar case, there are a finite number of candidate homogeneous solutions that do not spoil the Regge behaviour of the correlator. In the J 1 J 1 J 1 J 1 case, these are 20 the 4pt Witten diagrams in AdS 4 , with four external gauge fields, and with vertices (F 2 ) 2 , F 4 , and We have determined that these diagrams give non-zero answers in the collinear limit; this means that these solutions cannot contribute to the general (with arbitrary external momenta) answer with arbitraryλ dependences, while still being compliant with our results. This indicates that the full correlator has the sameλ dependence as the collinear one.
In the J 0 J 0 J 1 J 1 case, we expect that the contributing vertices will have two scalar fields, two field strength tensors, and up to two derivatives (φ 2 F ∧F and φ 2 D µ F ρσ D µ F ρσ for instance), though this has not been proven as far as we know. We do not know exactly which of these diagrams are relevant, but have verified that any such vertex (up to total derivatives) gives a non-vanishing contribution in the collinear limit, so we can reach the same conclusion about theλ-dependence.
When J 3 1 is introduced into the mix (this is the z-component of the spin-1 current), we find that the correlator vanishes (in the collinear limit, that is). For example: 20 We thank S. Minwalla for notifying us of this result of [20]. This is not very surprising and is a consequence of conservation of J 1 and the fact that all the J's are neutral. We emphasize that that this does not mean that the correlators themselves vanish. For example, we have verified by explicit computation that J 3 1 (q + + δ)J 0 (−q)J 0 (− )J 0 (−δ) does not vanish for certain non-collinear configurations. More powerful methods are needed to compute these correlators for all λ.

Conclusions and future directions
We have computed, in the collinear limit, and at leading order in 1/N , all the four-point functions that involve J 0 and J 1 in CS fermionic theories 21 . Specifically, we have analytic expressions for J 0 J 0 J 0 J 0 , which is in agreement with the result of [13], and for J + 1 J − 1 J 0 J 0 and J + 1 J − 1 J + 1 J − 1 , that fall in line with expectations from bootstrap as discussed earlier.
There are multiple lines of inquiry that lead out from this point. We wind up by listing some of these: 1. Since we have the collinear J 0 J 0 J 0 J 0 , J 1 J 1 J 0 J 0 , and J 1 J 1 J 1 J 1 in the CS + free fermionic theory, we may immediately compute the corresponding correlators in CS + critical fermionic theory via a Legendre transform (as was done in [16] and [13]). Assuming the bosonisation dualities, we then have these correlators in all the quasi-fermionic and -bosonic theories. Note that J 1 J 1 J 1 J 1 does not change under this transformation. 2. We may also use the brute force approach discussed here to directly compute the correlators in the CS + bosonic theories. Comparing these with our results would then give more evidence toward the bosonisation dualities. However, the computation there is slightly more complicated due to the fact that the spin-1 current contains gauge bosons and hence the contributing diagrams are different. 3. As was mentioned in the introduction, [17] found a way to compute all scalar correlators in the bosonic theory, in the collinear limit. It may be possible to replicate this approach in the fermionic case and hence also compute higher-point correlators involving J 0 , and perhaps also J 1 . 4. This approach (and also perhaps the one of [17]) should also be tried on correlators involving higher spin currents. As was mentioned earlier, the main problem is that for s > 1 in fermionic theories, and s > 0 in bosonic theories, J s contains gauge bosons, and hence the necessary ingredients haven't been worked out yet. 5. Our computations are in the collinear limit and, on their own, don't fix the correlator; other tools like the inversion formula must be employed to bootstrap the full correlator. Unlike in [13] and [14], our result contains extra structures that do not appear in the limiting (λ → 0, ∞) free fermion, and critical scalar theories; and are not yet known for general values of momenta. They can, however, be computed perturbatively.
As discussed in the introduction, if an analysis akin to that of [13] is employed in other cases, the collinear result could be used to the fix resultant ambiguities and hence nail down the full result. For this approach, it is necessary that the contact diagram corrections not vanish in the collinear limit. Our computations show that this is indeed the case for J 1 J 1 J 1 J 1 , and probably also for J 0 J 0 J 1 J 1 . 6. It is also interesting to study the possibility of other correlators in our theories being analytic in spin (as was speculated on by [13]). It is not yet clear precisely what this property means, so CS matter theories could potentially become a playground for understanding the inversion formula better. 7. The work of [14] has opened up another avenue to perform computations in these theories.
Their approach uses nothing but the anomalous higher-spin Ward identities and works them into a perturbation-series-like form, but inλ. Since the arguments we gave earlier in the section lead us to expect that all four-point functions of single-trace operators (normalised suitably) have a finite expansion in thisλ, this line of attack could be especially effective. This could also provide more conclusive evidence to answer the question of whether weakly broken higher-spin symmetry could uniquely fix all the correlators in CS-matter theories. One drawback of this method is that it may be insensitive to contact terms 22 , since correlators 21 We haven't looked at J + 1 J + 1 J0J0 , etc which are charged under rotations in the x-y plane, and thus vanish when all momenta are in the z direction. 22 This time, we mean delta functions in position space.
are written in position space. 8. A long term goal would be to compute all correlators in CS matter theories and match them with those in Vasiliev theories. But computations in the bulk are very hard to do and so far, only some three-point functions have been computed [29] and compared with those in the dual theory. , Figure 5: The sum of ladder-type diagrams. The fermion lines stand for exact fermionic propagators, and the ziplines stand for gluon propagators.
In the last line, we have defined Γ jn im (k, q, r), the four-fermion effective action or the ladder. Assuming that q lies along the z-axis, the resummation may be cast into a recursive Schwinger-Dyson equation, and solved following the methods of [26].