All tree level scattering amplitudes in Chern-Simons theories with fundamental matter

We show that Britto-Cachazo-Feng-Witten (BCFW) recursion relations can be used to compute all tree level scattering amplitudes in terms of $2\rightarrow2$ scattering amplitude in $U(N)$ ${\mathcal N}=2$ Chern-Simons (CS) theory coupled to matter in fundamental representation. As a byproduct, we also obtain a recursion relation for the CS theory coupled to regular fermions, even though in this case standard BCFW deformations do not have a good asymptotic behaviour. Moreover at large $N$, $2\rightarrow 2$ scattering can be computed exactly to all orders in 't Hooft coupling as was done in earlier works by some of the authors. In particular, for ${\mathcal N}=2$ theory, it was shown that $2\rightarrow 2$ scattering is tree level exact to all orders except in the anyonic channel arXiv:1505.06571, where it gets renormalized by a simple function of 't Hooft coupling. This suggests that it may be possible to compute the all loop exact result for arbitrary higher point scattering amplitudes at large $N$.


I. INTRODUCTION
Chern Simons gauge theories coupled to matter fields have a wide variety of applications in areas as diverse as quantum hall physics, anyonic physics, topology of three manifolds, quantum gravity via the AdS/CFT correspondence, etc. These theories are conjectured to enjoy a strong-weak duality that has been tested in several intense computations at large N, κ, keeping the 't Hooft coupling λ = N κ fixed . Recently, a finite N, κ form of the duality was proposed [31][32][33][34][35][36][37][38][39]. An example of the strong-weak duality is the duality between Chern-Simons gauge theory coupled to fundamental fermions and Chern-Simons gauge theory coupled to fundamental critical bosons. Other examples include self dual theories, such as N = 1, N = 2 supersymmetric CS matter theories. Very recently, at large N it was demonstrated that the S matrix for the 2 → 2 scattering computed exactly to all orders in the 't Hooft coupling displays an unusual modified crossing relation [1,16,25]. Moreover, for N = 2 theory, the result is tree level exact [1] except in the anyonic channel, where it gets renormalized by a simple function of the 't Hooft coupling.
A natural question to ask would be, is it possible to compute arbitrary m → n scattering amplitudes at all values of the 't Hooft coupling at large N, κ? Given the simplicity of the results at least in the supersymmetric case, it is also interesting to ask if the computability of scattering amplitudes extends to finite N, κ. As a first step towards these questions, we compute all tree level amplitudes for the N = 2 theory and the regular fermionic theory. The self-dual N = 2 supersymmetric theory is particularly interesting and important since via RG flow, we can obtain non supersymmetric dual pairs such as critical bosons coupled to CS and regular fermions coupled to CS [14,21].
The letter is organized as follows. In §II we review the four point scattering amplitude in the fermionic and N = 2 theory. In §III we discuss general criteria for the BCFW recursions to hold for the N = 2 theory. In §IV, we give a formal argument using background field method to show that BCFW works for the N = 2 theory. In §V we present a recursion relation for all tree level amplitudes for the N = 2 theory. Furthermore in §VI, we discuss how to use the N = 2 results to obtain recursion relations for all tree level amplitudes in fermionic theory. We end the letter with a discussion and possible future directions.

II. FOUR POINT SCATTERING AMPLITUDE
In this letter we compute scattering amplitudes in fermion coupled to SU (N ) Chern-Simons theory (FCS) and in N = 2 Chern-Simons matter theory coupled to a Chiral multiplet given by For our purposes, it is convenient to introduce the spinor helicity basis [40] defined by (3) Below we use the notation λ α i λ j,α = ij . For a supersymmetric amplitude, the standard procedure involves introduction of on-shell grassman variables θ such that the super-creation and super-annihilation operators are given by where a † i , a i / α † i , α i create and annihilate a boson/fermion with momenta p i respectively. The two onshell supercharges for n point scattering amplitudes are given by For FCS theory in (1), the tree level 2 → 2 scattering amplitude is given by [16] For N = 2 theory in (2), the tree level 2 → 2 super amplitude is given by Here A S 4 is the super-amplitude computed using the super-creation/annihilation operators defined in (4). Any component amplitude can be obtained from (7) by picking up the coefficient of products of two θ's. For example, the four fermion amplitude is given by the coefficient of θ 2 θ 4 that coincides precisely with (6).

III. HIGHER POINT SCATTERING AMPLITUDE
BCFW recursion relations are an efficient method to compute and express arbitrary higher point scattering amplitudes in terms of product of lower point amplitudes. Standard procedure for BCFW involves the deformation of two external momenta of the particles by a complex parameter z. The deformation is such that the particles continue to remain 'on shell' and the total momentum conservation of the process continues to hold. In 3D, BCFW deformations are a little more involved than in 4D and were first discussed in [41] (We follow their notations closely). BCFW recursion relations are applicable in 3D provided that the higher point amplitudes are regular functions at z → ∞ and z → 0. In the following section we study the z → ∞ (and z → 0) behavior of the amplitudes in the theories described earlier. We find it convenient to deform color contracted (we label them as '1'and '2') external legs. In 3-dimensions, momentum deformation of particles 1 and 2 can be written in terms of the spinor-helicity variables as Under the deformation (8), any tree-level scattering amplitude for FCS in (1) is not well behaved at large z and hence doesn't obey the requirements of BCFW. However this situation is cured for the N = 2 theory defined in (2). Additionally, conservation of the super-charges in (5) require that the on-shell spinor variables θ be deformed as where the R matrix is defined by (8).
Hence it is sufficient to show that either of A 0 or A 12 are well behaved since supersymmetric ward identity guarantees the required behavior for the rest of the amplitudes. It is convenient to write the fields in pair wise contractions since they transform in the fundamental representation of the gauge group. For instance we are interested in the large z behavior of amplitudes such as FIG. 1. The diagrams that have a non-regular z → ∞ behavior. O(z) part of these two diagrams cancel against each other to give a regular z → ∞ behavior of the total amplitude. In the above diagram, the solid lines correspond to fermions and the dashed lines correspond to bosons. This amplitude appears in A 0 in (10). The blue color lines corresponds to deformed hard particle.
where . . . represent color contracted bosonic or fermionic particles allowed by interactions in (2). These amplitudes appear in A 0 , A 12 in (10) respectively.
We have checked explicitly by Feynman diagrams that the amplitude A 0 = A 6 (ψ 1 φ 2φ3 ψ 4φ5 φ 6 ) is well behaved. We discuss the large z behavior of the general 2n point amplitude using the background field method [42] in the next section.

IV. ASYMPTOTIC BEHAVIOR OF AMPLITUDES
To understand the large z behavior of various scattering amplitudes, it is extremely useful to think from the background field method point of view introduced in [42]. Here z-deformed particles are considered as hard particles propagating in a background of soft particles. The amplitude is modified due to (a) modified propagator of intermediate hard particle; (b) the modified contribution of various vertices; and, (c) modified fermion wave function, in case an external deformed particle is a fermion. Detailed analysis shows (we follow closely [41,42]) that the non-trivial z → ∞ behavior of the amplitude is due to diagrams of the kind depicted in fig. 1. The values of these diagrams are: Under the 1-2 z-deformations, (8), in the z → ∞ limit the O(z) part of the amplitude cancels and the amplitude behaves as O(1/z). Hence this amplitude has a regular z → ∞ behavior for N = 2 theory. This cancellation works even for the 4-point function ψ 1 φ 2φ3 ψ 4 , which receives contributions from the diagrams in fig. 1 with the blob removed and k 3 → p 3 , k 4 → p 4 are taken to be on-shell momenta. It is important to emphasize that we In the last section, we have demonstrated that A 0 is well behaved in large z. Hence we can apply the BCFW recursion relation directly to the super amplitude in the left hand side of (10). The recursion formula for a 2n point superamplitude can be expressed in terms of lower point superamplitudes as follows (see fig 2) A where the integration is over the intermediate grassmann variable θ and A 2n (z = 1) is the undeformed 2n-point amplitude. In the above, p f is the undeformed momentum that runs in the factorization channel f and the summation in (14) runs over all the factorization channels corresponding to different intermediate particles going onshell. Here, z a;f and z b;f are given by where the null momenta q are defined in terms of the spinor helicity variables as 1 For instance, for N = 1 theory, the Lagrangian for which can be found in [1] (Equation (2.11)), (13) is modified to −2π/κ 41 w whereas (12) remains the same. Here w is a free parameter in N = 1 theory. This implies that only at w = 1, the N = 1 theory has a good large z behavior. This is exactly the point in the w line where the supersymmetry of the theory gets enhanced to N = 2. Note that the formula (14) has a very similar form (but not quite the same as discussed below) to the one obtained in [41] for the ABJM theory 2 that enjoys N = 6 supersymmetry. It is remarkable that such recursion formulae exist in a theory with much lesser supersymmetry such as the one in discussion.
As an explicit demonstration, consider the six point 3 amplitude A 6 (λ 1 . . . λ 6 ) ≡ (φψ)(ψφ)(φφ) in the N = 2 SCS theory. This amplitude factorizes into two channels as shown in fig 3. The recursion formula can be explicitly written as where z a;f , z b;f are defined in (15). Fields with hats corresponds to deformed momenta. We have checked (18) explicitly by computing the relevant Feynman diagrams. It is a curious fact that, the total number of Feynman graphs that contribute to A 6 is 15. Of these, eleven are reproduced by the channel p f = p 234 and the remaining four in the channel p f = p 256 .
2 Although, formula (14) looks very similar to ABJM case, the details are different since the external matter particles are in fundamental representation. For example, in general there will be more factorization channels here as compared to the ABJM case. For example, in the six point function, as will be clear below, there are two factorized channels, where as for the corresponding deformation in ABJM, there is only one factorized channel. 3 A general six point super amplitude in N = 2 theory can be written in terms of two independent functions as (5).

VI. RECURSION RELATIONS IN THE FERMIONIC THEORY
In this section, we show that the BCFW recursion relations can be used to compute 2n−point amplitude A 2n = (ψ 1 ψ 2 ) · · · (ψ 2n−1 ψ 2n ) for the regular fermionic theory coupled to CS gauge field (1). If we apply (8) to this amplitude, it is easy to show that, it does not have a good large z (as well as z → 0) behavior, hence we cannot readily apply the BCFW recursion relation 4 to determine all higher point fermionic amplitudes. However, we show below that we can use the recursion relation of the N = 2 to write a recursion relation for the fermionic theory.
As a first step towards this, let us note that the Feynman diagrams for any tree-level all-fermion scattering amplitude in the N = 2 theory (2) is identical to that of the tree-level scattering amplitude in the fermionic theory (1). In the previous section we proved for the N = 2 theory that an arbitrary higher-point super-amplitude can be written only in terms of the 4-point super-amplitude. Same can be said for the component amplitudes including the purely fermionic component amplitude 5 . Let us note that for the four point super-amplitude, supersymmetry relates all the component 4-point amplitudes to one component amplitude, which for instance can be taken to be 4-fermion scattering amplitude (see (7)). Thus an arbitrary higher-point component amplitude can be written only in terms of 4-fermion amplitude. This can be recursively done for an arbitrary 2n point amplitude, however for simplicity we write the recursion relation for the six point amplitude below There will be some non-trivial boundary terms that do not vanish and in general there are no good prescriptions to compute them systematically. 5 Note that the recursion relation in the N = 2 theory (14) does not directly give (ψ 1 ψ 2 ) · · · (ψ 2n−1 ψ 2n ) in terms of the lower point fermionic amplitude. However, we can use BCFW relations recursively to write down any higher point amplitude in terms of four point amplitudes such as (ψψ)(ψψ), (φφ)(φφ), (φφ)(ψψ), (φψ)(ψφ) etc. Moreover, at the level of the four point amplitude, one can rewrite this in terms of (ψψ)(ψψ). For example, (ψψ)(φφ) = 23 24 (ψψ)(ψψ). This implies that we get a recursion relation for (ψ 1 ψ 2 ) · · · (ψ 2n−1 ψ 2n ) in terms of lower point fermionic amplitudes only. Hence this can be interpreted as a BCFW recursion relation in the regular fermionic theory coupled to CS gauge field (1).
Hence, while the amplitudes in the fermionic theory by themselves don't obey the requirements for BCFW relations, using N = 2 theory we can find out the recursion relations for the fermionic theory too.

VII. DISCUSSION
In this letter we presented recursion relations for all tree level amplitudes in N = 2 CS matter theory and CS theory coupled to regular fermions. Below we discuss some interesting open questions for future research.
It was shown in [1], that the 2 → 2 scattering amplitude in the N = 2 theory does not get renormalized except in the anyonic channel, where it gets renormalized by a simple function of the 't Hooft coupling. A natural question is, why in the N = 2 theory the scattering amplitude has such a simple form, whereas the corresponding amplitudes in the fermionic [16] and other less susy N = 1 [1] theories are quite complicated. A possible explanation is that there exists some symmetry such as dual conformal invariance that appears in the N = 2 theory and it protects the amplitude from loop corrections [43]. It is natural to ask, if the simplicity of the amplitudes continues to persist with higher point amplitudes. It is also interesting to explore an analog of the Aharonov-Bohm phase for higher point amplitudes. It may very well turn out that the Aharanov-Bohm phases of higher point amplitudes are products of the Aharonov-Bohm phases of the 2 → 2 amplitude. BCFW recursion relations provide a strong indication towards this result.
To answer the above questions, we need to compute higher scattering amplitudes to all orders in λ. A possible way is to investigate the Schwinger-Dyson equa-tion. However, the Schwinger-Dyson equation approach is quite complicated even at the 6− point level. A refined approach might be to look for a larger class of symmetries such as Yangian symmetry [43] and use the powerful formulation of [44] to obtain results. Given the fact that, these theories are exactly solvable at large-N as well as the fact that N = 2 theory is self-dual, it could turn out that the N = 2 theory may be one of the simplest play grounds to develop new techniques in computing Smatrices to all orders [44]. Furthermore exact solvability at large N indicates that these models might even be integrable. One possible way to investigate integrability is to show the existence of an infinite dimensional Yangian symmetry. Since these theories relate to various physical situations, any of the above exercises may provide insight into finite N, κ computations. stages of the project. We would like to thank Y-t. Huang for sharing a useful mathematica code with us. Special thanks S. Minwalla for very useful and critical discussions. The work of KI was supported in part by a center of excellence supported by the Israel Science Foundation (grant number 1989/14), the US-Israel bi-national fund (BSF) grant number 2012383 and the Germany Israel binational fund GIF grant number I-244-303.7-2013. S.J. would like to thank TIFR for hospitality at various stages of the work. Some part of the work in this paper was completed while SJ was a postdoc at Cornell and his research was supported by grant No:488643 from the Simons Foundation. The work of PN is supported partly by Infosys Endowment for the study of the Quantum Structure of Space Time and Indo-Israel grant of S. Minwalla. We would also like to thank people of India for their steady support in basic research.

B. A Dyson-Schwinger equation for all loop six point correlator
As we saw earlier in §II, the basic building block of higher point amplitudes in the Chern-Simons matter theories at the tree level is the four point amplitude. In this section we describe the Dyson-Schwinger construction of the all loop six point correlator 8 using the superspace Schwinger-Dyson construction developed in [1]. In the above Φ i is a complex scalar superfield in N = 1 superspace defined by where φ i is a complex scalar, ψ i is a complex fermion and F i is a complex auxiliary field. The N = 2 theory can be written in N = 1 superspace in terms of Φ i . For more details see [1]. Before presenting the central idea it is informative to understand the color structure of the tree level and one loop amplitudes in the theory. In the supersymmetric Light cone gauge these are described succinctly in fig 5 and in fig 6. It turns out that, there are six different diagrams for  a) and c) the three gauge field propagators contribute a factor of O( 1 κ 3 ) and the single color loop gives a factor of N , leading to a contribution of the order λ κ 2 . Note that this is of the same order in κ as the tree level diagram displayed in fig 5. On the other hand fig b) has three gauge fields and no color loops, rendering it to be O( 1 κ 3 ).