NORDITA-2013-15 Scattering Amplitudes of Massive \mathcal{N}=2 Gauge Theories in Three Dimensions

Nordita-2013-15
Scattering Amplitudes of Massive Gauge Theories in Three Dimensions

Abhishek Agarwal111Email: abhishek@aps.org, Arthur E. Lipstein222Email: Arthur.Lipstein@maths.ox.ac.uk, Donovan Young333Email: donovany@kth.se

Physical Review Letters, American Physical Society,
1 Research Road, Ridge, NY 11961, USA
and
Physics Department, City College of CUNY, New York, NY 10031 USA

The Mathematical Institute, University of Oxford,
29-29 St Giles’, Oxford, OX1 3LB, UK

Nordita, KTH Royal Institute of Technology and Stockholm University,
Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden
Abstract

We study the scattering amplitudes of mass-deformed Chern-Simons theories and Yang-Mills-Chern-Simons theories with supersymmetry in three dimensions. In particular, we derive the on-shell supersymmetry algebras which underlie the scattering matrices of these theories. We then compute various 3 and 4-point on-shell tree-level amplitudes in these theories. For the mass-deformed Chern-Simons theory, odd-point amplitudes vanish and we find that all of the 4-point amplitudes can be encoded elegantly in superamplitudes. For the Yang-Mills-Chern-Simons theory, we obtain all of the 4-point tree-level amplitudes using a combination of perturbative techniques and algebraic constraints and we comment on difficulties related to computing amplitudes with external gauge fields using Feynman diagrams. Finally, we propose a BCFW recursion relation for mass-deformed theories in three dimensions and discuss the applicability of this proposal to mass-deformed theories.

1 Introduction

Over the past few years, there has been a great deal of progress in understanding the scattering amplitudes of three-dimensional gauge theories. The study of scattering amplitudes of Chern-Simons-Matter theories with supersymmetry and Supersymmetric Yang-Mills theories (SYM) with supersymmetry (initiated in [1] and [2, 3, 4] respectively) shows that the -matrices of three dimensional supersymmetric gauge theories contain fascinating simplifying aspects that are not manifest in their traditional Lagrangian descriptions. For instance, it was shown in [1] that the four-particle amplitudes of a large family of Chern-Simons-Matter theories have the same formal structure as the scattering matrix of the spin chain that is the large- dilatation operator of SYM in . Furthermore, amplitudes of the superconformal Chern-Simons theory known as the BLG theory [5, 6] were studied in [7].

More recently, a BCFW recursion relation [8, 9] for three-dimensional gauge theories with massless fields was developed in [10], and used to show that an superconformal Chern-Simons theory known as the ABJM theory [11] has dual superconformal symmetry both at tree [12, 13] and loop-level. Dual superconformal symmetry [14, 15, 16] is inequivalent to ordinary superconformal symmetry and generates Yangian symmetry when combined with ordinary superconformal symmetry [17]. In 4d SYM, dual superconformal symmetry corresponds to the ordinary superconformal symmetry of null-polygonal Wilson loops that are dual to the amplitudes [18, 19, 20, 21, 22]. The Yangian symmetry of SYM can be made manifest using a Grassmannian integral formula developed in [23]. An analogous formula for the ABJM theory was proposed in [24]. This formula involves an integral over the orthogonal Grassmannian. Some evidence for an amplitude/Wilson loop duality in the ABJM theory was found in [25, 26, 27]. Recently, 1-loop amplitudes were computed in the ABJM theory and shown to exhibit new structures which do not appear in 4d SYM theory, notably sign functions of the kinematic variables [28, 29, 30].

The recursion relation proposed in [10] was also used to show that three-dimensional maximal SYM amplitudes have dual conformal covariance [4]. Note that three-dimensional SYM theories do not have ordinary superconformal symmetry because the Yang-Mills coupling constant is dimensionful in three dimensions. Three-dimensional SYM theories exhibit a number of other surprising properties. In particular, references [3, 4] showed that they have helicity structure and reference [3] showed that their 4-point amplitudes have enhanced -symmetry which originates from the duality between scalars and vectors in three dimensions. Furthermore, the loop amplitudes of three-dimensional maximal SYM theory have a similar structure to those of the ABJM theory. In particular, 1-loop corrections are finite or vanish in both theories [4]. Furthermore, the 2-loop 4-point amplitudes of both theories can be matched in the Regge-limit [31]. It was recently shown that three-dimensional supergravity amplitudes can be obtained as double copies of both three-dimensional supersymmetric Chern-Simons theories and three-dimensional SYM theories [32, 33]. All these remarkable developments provide ample motivation for further investigation into the scattering amplitudes of gauge theories in three spacetime dimensions.

It may be fair to say that most of the investigations mentioned above are largely confined to the studies of massless theories with high degrees of supersymmetry. In this paper, we explore a complementary part of the landscape of gauge theories from the point of view of scattering amplitudes. Specifically, we investigate mass-deformed gauge theories with adjoint matter fields. The two theories that span this category are mass-deformed Chern-Simons theory (hereafter referred to as CSM theory) and Yang-Mills-Chern-Simons theory with supersymmetry (YMCS), and we investigate their tree-level color-ordered scattering amplitudes in this paper. Whereas the gauge field has no propagating degrees of freedom in the Chern-Simons theory, in the Yang-Mills-Chern-Simons theory the gauge field has one propagating degree of freedom, which is massive. In particular, the Chern-Simons term provides a topological mass for the gauge field without breaking gauge invariance or locality [34, 35].

From the point of view of scattering amplitudes, these theories are interesting for a number of reasons. We find that there are two different on-shell superalgebras that can potentially arise as symmetries of these theories. The first of these is the standard superalgebra with the schematic structure ; . In the case of a flavor symmetry, one can also have a “mass-deformed” algebra where the supercharges close on the momentum as well as the symmetry generator. Or, schematically, . Such mass-deformed algebras – though rare in the list of all possible superalgebras – have been shown to arise as symmetries of three dimensional gauge theories in a number of previous investigations [1, 36, 37]. In the present work we find that the mass-deformed algebra is the underlying symmetry algebra for the CSM theory. We find a convenient single particle representation of this algebra and find that all of the 4-point tree-level amplitudes can be encoded in superamplitudes (note that the odd-point amplitudes in the Chern-Simons theory have external legs which are gauge fields and therefore vanish on-shell). While component amplitudes of massive CSM theories have been studied in great detail (for example in[1]) the CSM theory studied in this paper is distinguished from the class of models investigated in[1] by virtue of allowing the matter fields to be in the adjoint representation (which is typically not possible for higher supersymmetry). The component amplitudes obtained in this paper are not known to be obtained by a trivial truncation of the amplitudes either. Furthermore, to the best of our knowledge the superamplitude presented in this paper for the massive theory is the first concrete formulation of a superamplitude for a massive gauge theory in .

In the case of the YMCS theory, we find that the underlying supersymmetry algebra is the undeformed one where the supercharges close on momenta alone. This is to be expected as there is no flavor symmetry in the bosonic sector of the theory, since the Lagrangian has only one scalar field. We derive an on-shell representation of this algebra and use it to obtain constraints on 4-pt. amplitudes (the on-shell algebra does not constrain the 3-pt. amplitudes). We find that that the relations among the 4-pt. amplitudes of the YMCS theory are considerably more complicated than those in the CSM theory. The root of the complication has to do with the absence of the extra symmetry in the bosonic sector. Nevertheless we are able to compute a number of these amplitudes and verify that the computed amplitudes are consistent with the on-shell algebra. Although we do not compute amplitudes with external gauge fields using Feynman diagrams, we are nevertheless able to deduce the 4-point amplitudes with external gauge fields using the on-shell algebra.

We also propose a BCFW recursion relation for mass-deformed three-dimensional theories which reduces to the proposal in [10] when the mass goes to zero. This recursion relation involves deforming two external legs of on-shell amplitudes by a complex parameter . In order for the the recursion relation to be applicable, the amplitudes must vanish as . We show that four-point superamplitudes of the CSM theory have good large- behavior, so our proposed recursion relation may be applicable to this theory. However, the proposed relations do not seem to apply to the YMCS theory and we comment on the relevant issues in the corresponding section of the paper.

Three-dimensional gauge theories are also interesting from various other points of view. In particular, they exhibit Seiberg duality [38, 39, 40], F-maximization [41], and an F-theorem [42]. The gravity duals of these theories are also known and have been studied in [43, 44, 42]. Finally, three-dimensional superconformal gauge theories arise from compactifications of the 6d CFT compactified on 3-manifolds [45, 46, 47]. It would very interesting to make contact with these results from the point of view of scattering amplitudes.

The structure of this paper is as follows. In section 2 we describe some general aspects of the CSM and YMCS theories whose amplitudes we compute in this paper. We pay special attention to the derivation of the on-shell supersymmetry algebras in this section. In particular, we derive the on-shell representation of the algebra for the YMCS system in some detail following canonical quantization. This derivation relies on a careful analysis of the implementation of the Gauss-law constraints on the physical Hilbert space, which is described in some detail. In section 3, we compute the four-point amplitudes of the CSM theory and show that they can be encoded in superamplitudes. We also describe the symmetries of the four-point superamplitudes which we expect to hold for higher-point superamplitudes. In section 4, we compute various three and four-point amplitudes of the YMCS theory at tree level and use the on-shell superalgebra to deduce the remaining 4-pt. amplitudes. We also comment on the complications that arise when trying to compute amplitudes with external gauge fields in the YMCS theory using perturbative techniques. In section 5, we propose a BCFW recursion relation for mass-deformed 3d theories and discuss its applicability to the theories studied in this paper. In section 6, we present our conclusions and describe some future directions. In appendix A, we describe our conventions, Feynman rules, and various other useful formulae. In appendix B we provide more details about the calculation of various 4-pt. amplitudes.

2 Mass-deformed gauge theories

In this section we review some general aspects of the mass-deformed three-dimensional supersymmetric theories whose scattering amplitudes we study in this paper. The gauge field which appears in these theories has a Chern-Simons term

(1)

As is well-known, a Chern-Simons gauge field is not parity invariant and does not have any propagating degrees of freedom. On the other hand, a Yang-Mills gauge field respects parity and has one massless degree of freedom in three dimensions. When taken in conjunction with the Yang-Mills action, the Chern-Simons term breaks parity and gives rise to a mass for the three dimensional gluon [34, 35]. There are alternate Lorentz invariant mass-terms for gluons that one can consider in three dimensions (see [48, 49] for examples) but they typically lead to non-local terms in the action. A quadratic mass term for the gauge field could also arise via the Higgs mechanism, but this would break gauge invariance. In the present paper we consider the only known mass-term for a gauge field which is Lorentz invariant, gauge invariant, and local in three dimensions, namely . Note that admits two different supersymmetric completions leading to supersymmetric Chern-Simons and Yang-Mills-Chern-Simons theories. In the first case, the gauge field does not have propagating degrees of freedom and the physical on-shell degrees of freedom consist of matter hypermultiplets. In the latter case there are Yang-Mills kinetic terms and the Chern-Simons term provides a topological mass for the gauge field (which contributes to the on-shell degrees of freedom). We will study scattering processes in both the theories while restricting ourselves to the case of supersymmetry.

Before discussing mass-deformed gauge theories in greater detail, we briefly review the 3d spinor formalism. The three-dimensional spinor formalism can be obtained by dimensional reduction of the four-dimensional spinor formalism[4]. We begin by writing a 4d null momentum in bispinor form

(2)

where and are indices which arise from the fact that the Lorentz group is . When reducing to three dimensions, the distinction between dotted an undotted indices disappears because the Lorentz group is . Alternatively, we can reduce to three dimensions by modding out by translations along a vector field , as described in [4]. Using the vector field to change dotted indices to undotted indices in (2) and symmetrizing the indices then gives

(3)

We symmetrize the indices in order to remove the component of the momentum along the direction . The resulting momentum is a symmetric object, which has three components.

We denote inner products of the spinors using bracket notation

If we square (3), we find that

(4)

Hence, if the particle is massless, then and the momentum can be written in bispinor form as . More generally, for a massive particle in three-dimensions, the momentum is given by (3). Equations (3) and (4) can be summarized as follows

In particular, .

For later convenience, we will denote and . The two spinors and are solutions of the free massive Dirac equation and are given in (74)444An exhaustive list of the properties of these spinors can be found in [1].. They satisfy

(5)

where is given explicitly by

(6)

2.1 massive Chern-Simons-Matter (CSM) theory

The CSM theory is described by the action

(7)

where,

(8)

Note that the Chern-Simons term is odd under parity, so the theory is not parity invariant. The parameter is the Chern-Simons level. The matter couples to the gauge field with coupling constant . The parameter sets the mass-scale in the superpotential. Even though it is a dimension-full number, it does not run in the super-renormalizable theory and can be regarded as a parameter of the theory. Taking the mass to zero or infinity while holding the coupling constant corresponds to taking to zero or infinity. In the massless limit this theory reduces to a conformal Chern-Simons-matter theory. In the infinitely massive limit, the theory reduces to a pure Chern-Simons theory with no propagating degrees of freedom. The conventions underlying the above action assume that all the fields are in the adjoint representation of the gauge group. Furthermore, we assume the generators of the gauge group (which we can take to be ) to be Hermitian. We then have

(9)

In terms of real variables,

(10)

where and are real and Majorana respectively.

We can immediately see that the free (Abelian) part of the action is invariant under

(11)

where , . All other supersymmetry variations vanish. In the non-Abelian / interacting theory, the SUSY variation of the scalar fields remains as above, but the variations of the fermions and the gauge fields are given by

(12)

The variations in the non-Abelian case can be obtained from the ones given above by conjugation. The fundamental anti-commutation relation between the supercharges is

(13)

where is the symmetry generator which rotates and .

For the mass-deformed Chern-Simons theory, the on-shell asymptotic states are those of the complex scalar and fermion . In our notation, the asymptotic momentum-space states of and are denoted and respectively. Using the mode expansions for these fields, which are given by (73), in the supersymmetry algebra (11), we see that the supersymmetry variations of the on-shell states are given by

(14)

where and are spinors defined in (74)555For a detailed discussion of the on-shell representation of three dimensional massive superalgebras, we refer to [1]..

We can express these transformations in a way that makes the R-symmetry of the theory manifest by forming complex combinations of the fields and supercharges

(15)

We then obtain

(16)

It is important to emphasize that the superalgebra (13) is a non-central extension of the standard superalgebra. In particular, the anticommutator of the charges does not close onto the momentum generator alone, as it also involves the symmetry generator as part of the fundamental supersymmetry algebra. Such mass-deformed algebras frequently arise in the context of three dimensional gauge theories with mass-gaps; in particular in Chern-Simons-Matter theories [1]. It is instructive to see how the algebra described above is embedded in the supersymmetry algebra of the massive theory. In the notation of [1], the matter content of CSM theories is given by scalars and fermions , where are two different indices. The fields denoted by tilde are part of the twisted hypermultiplets, while those without the tildes form the untwisted hypermultiplets. In the case of supersymmetry, one has - as part of the full supersymmetry algebra - supercharges that transform scalars and fermions belonging to the twisted and untwisted hypermultiplets to each other, while acting trivially on the indices (see the discussion in section-2.4 of [1]). These supercharges (for a fixed value of the index) generate the massive algebra considered here666There are presumably other distinct embeddings of the superalgebra in the larger superalgebra as well.. It should be noted, however, that the theory considered here is distinguished from the class of models studied in [1] by virtue of all the matter fields being in the adjoint representation, which, typically is not possible for and higher supersymmetry.

We also note that just as the supersymmetric Chern-Simons theories are not known to be obtained as the dimensional reduction of higher dimensional gauge theories, this massive superalgebra is not what one obtains by the dimensional reduction of the free theory in four dimensions. In fact, if one takes the massive free action given by

(17)

which is invariant under

(18)

it is easy to see that the algebra closes on only the momentum generators without any extensions

(19)

The algebra retains this standard form even after dimensional reduction to , however the fermion mass-term in derived from the -invariant four dimensional mass-term would be given in the three dimensional notation by . This is different from the term we have in (7) where the mass terms for both the fermions have the same sign.

In other words, in we can choose between two different fermion mass-terms

(20)

The choice – the parity conserving option – leads to the the standard algebra without extensions while leads to a mass-deformed algebra and violates parity. However the Chern-Simons term, which is present in the gauge theories we study, violates parity. Thus it is natural that the fermionic mass terms resulting from the supersymmetric completion of the Chern-Simons term violate parity as well. It is apparently this interplay between the parity invariance of the theory and the fermionic mass term that leads to the massive nature of the on-shell algebra in this case.

2.2 Yang-Mills-Chern-Simons (YMCS) theory

The second theory of relevance to this paper is the well-known YMCS theory described off-shell by the action where777We have taken the trace using (9) and also chosen to rescale the fields by the coupling constant, in comparison to (7).

(21)

Here where is the Chern-Simons level, is an index, the scalar field and auxiliary field are real, and the fermions are Majorana, so that , where the charge conjugation matrix . For further details about our conventions, see appendix A. Note that the matter fields and have mass dimension and . If they are rescaled by a factor of , i.e. if , then they will have mass dimensions and . is an auxiliary field whose elimination generates the standard quadratic mass term of the real scalar , while the Chern-Simons term gives a topological mass to the gauge field. Both the Chern-Simons term and the fermionic mass term are odd under parity, so the theory is not parity invariant when the mass is nonzero. Taking the mass to zero or infinity while holding the Yang-Mills coupling constant corresponds to taking to zero or infinity. In the massless limit, the theory reduces to Yang-Mills theory and parity is restored. In the infinitely massive limit, the theory reduces to a pure Chern-Simons theory. What is not transparent from the action given above, but is nevertheless true, is that the asymptotic physical states also involve a second massive scalar with the same mass. This scalar is nothing but the physical gauge invariant degree of freedom encoded in the gauge field.

Note that for the free theory, the supersymmetry transformation laws are:

(22)

The anti-commutator of the supercharges in these off-shell transformation laws closes onto the momentum operator alone

(23)

Note that there is no extension as there was for the superalgebra in the mass-deformed Chern-Simons theory. This is a consequence of the fact that the YMCS theory, while enjoying a R-symmetry which rotates the two fermionic fields in the theory, does not have a corresponding symmetry acting on the two bosonic fields, i.e. the scalar and gauge field. Indeed, we will find that the on-shell amplitudes of the YMCS theory exhibit R-symmetry in the fermionic sector. We do note, however, that both the algebras collapse to the same massless algebra when in the YMCS theory and in the CSM model are set to zero, which is consistent with the four-point amplitudes of undeformed three-dimensional SYM exhibiting an enhanced symmetry[3, 4].

We now focus on the on-shell superalgebra for this theory. Assuming that the on-shell degree of freedom associated with the YMCS gauge field corresponds to a massive scalar field (which we will justify shortly) and that the superalgebra in (22) is realized on the single particle asymptotic states, the transformation laws for the scattering states can be taken to be

(24)

We denote the massive scalar corresponding to the gauge field by .

One can give an argument in favor of the algebra above being the appropriate one for the YMCS theory as follows. If one starts with the part of the algebra involving the variation of , namely , there is an ambiguity about what the spinor is. This ambiguity can be resolved by applying the oscillator expansion of the fields to the off-shell transformation . In our convention, this fixes . Once this is fixed, the closure of the algebra on fixes the transformation properties and . With this part of the on-shell supersymmetry algebra determined, one can make the following ansatz for the supersymmetry algebra

(25)

assuming that the realization is linear in the fields and that the covariance of the fermionic degrees of freedom is respected. The unknown quantities are the spinors and . Closure of the algebra on requires . The solution to this equation is given by or . Furthermore, requiring that there be no extension to the algebra requires and . Thus, given a convention of the oscillator expansion of the fermion fields, the on-shell algebra is unambiguously determined.

Comparing this with (14), we see that main difference between the two sets of transformations is that the spinors appearing on the RHS of the transformation laws of the scalars above are conjugates of each other. The two spinors were same in the transformation laws for the CSM theory. The differences in the two realizations have to do with whether or not the algebra is mass-deformed.

Rather than rely on the argument above alone, it is instructive to derive (24) using the methods of canonical quantization. To this end we revert to a Hamiltonian framework and set . We define the complex combination (and its conjugate relation) for the gauge potentials. Due to the non-commutativity induced by the Chern-Simons term on the components of the electric field the Gauss law constraints can be shown to be satisfied by wave functions of the form[50]

(26)

where is a complex matrix related to the gauge potential as that transforms under time independent local gauge transformations as , where is an element of the gauge group. is a Wess-Zumino-Witten functional defined over the spatial manifold [50]. The Gauss law constraint can be translated into the following condition on

(27)

Clearly any wave functional that is a gauge invariant combination of the gauge and matter fields satisfies this constraint.

Now, to derive the on-shell supersymmetry transformation law, our strategy would be to express the quadratic part of the supercharges in terms of the canonical variables followed by a dualization of the gauge field into a scalar. We can then read off the on-shell supersymmetry transformation by looking at the action of the dualized supercharge on the dynamical fields in momentum space. To avoid the ambiguity associated with the fermionic fields and their canonical momenta in a real representation for the three dimensional Dirac matrices, we take the gamma matrices to be for the purposes of this discussion (everywhere else in the paper we shall continue to use the real matrices mentioned previously). The fermions can be taken to be with and being canonically conjugate. The quadratic part of the top component of the supercharge in this notation (the bottom component can simply be obtained by Hermitian conjugation) can be written as

(28)

This charge, derived from the action, acts on the wave function . and differ by a pure phase, so their norms are the same. However the physical observables acting on differ from those acting on by a unitary transformation. The charge acting on = [50]. The effect of this unitary transformation is to replace

(29)

This extra term generated by the unitary transformation is what generates an effective mass-term for the gauge field in the Hamiltonian obtained from the supercharge above[50].

We can now dualize the gauge field by expressing and retaining terms to linear order in . This gives (after dropping the color indices, as we are only interested in the abelian theory) , , and . The real part of is related to the physical gauge-invariant on-shell degree of freedom as [34, 50, 51]

(30)

On gauge invariant wave functionals, the dualized supercharge can now be written as

(31)

where

(32)

The momenta and appearing in the last term above are the complex combinations of the spatial components of the three-momentum. It is important to note that the fermionic variable multiplying the momentum for the dual scalar has to be identified as the top component of a fermionic field (in our case ) so that the SUSY variation of can be written in a Lorentz invariant form in the two component notation i.e. for some Majorana fermion . In our case the dualization dictates that the top component of is . Crucially for our purposes, it can be readily seen from the Hamiltonian obtained from that the Hamiltonian for has the opposite sign for the mass term as that of . Or in other words, the spinors appearing with the positive (negative) frequency parts of the mode expansion of can be identified with those associated with the negative (positive) parts of . Since the SUSY variations involving the on-shell fields and involve fermions with the opposite mass terms, the spinors appearing in the momentum space realization of these transformations are conjugate to each other. Reverting back to our real conventions for the matrices, we see that (24) can now be justified based on the grounds of canonical quantization.

3 Mass-deformed Chern-Simons amplitudes

In this section, we will describe the scattering amplitudes of the CSM theory (7). Since the Chern-Simons gauge field has no propagating degrees of freedom, scattering amplitudes with at least one external gauge field vanish. This implies that all odd-point amplitudes vanish, so the first nontrivial amplitudes occur at four-point. In the next two subsections, we will compute the 4-point amplitudes and show that they can be encoded in superamplitudes. We also describe the symmetries of these superamplitudes.

3.1 4-point amplitudes

All of the the four point amplitudes of the CSM theory are related to each other by the supersymmetry algebra in (16). Hence, there are only two independent amplitudes involving four legs. With the definitions of the complex combinations of the real degrees of freedom described in (15), we get the following relations between the color ordered four particle amplitudes

(33)

The three independent four-fermion amplitudes are related to the other amplitudes as

(34)

In appendix B we compute 4-fermion amplitudes and find

(35)

Using these amplitudes, all the other 4-pt. amplitudes are determined from the above relations and cyclic permutations.

3.2 Superamplitudes

The natural question to ask is if these relations among the 4-point amplitudes obtained in the previous subsection can be derived from superamplitudes. To the best of our knowledge no superamplitude is known for any massive model so far. However, since the superalgebra and the kinematics constraining the -matrix of the massive three dimensional theories can be thought of as dimensional reductions of the four dimensional quantities, it is natural to expect that some of the known results for four dimensional theories can be reduced to get massive three dimensional superamplitudes. In fact, one can define two types of superamplitudes, which are analogous to the “” formalism and the “” formalisms used to obtain superamplitudes for 4d super-Yang-Mills theories with supersymmetry [52]. In the “” formalism, the superamplitudes can be expressed in terms of supercharges so one can in principle apply super-BCFW recursion relations to these amplitudes. On the other hand, the R-symmetry of the theory is not manifest in the “” formalism, so the on-shell superalgebra obtained from the supercharges which act on the superamplitudes does not have a central extension. In the “” formalism, the superamplitudes are not expressed in terms of supercharges, but the superamplitudes and superfields transform covariantly under the R-symmetry. We describe the “” and “” formalisms in greater detail below, and we describe super-BCFW for 3d mass-deformed theories in section 5.

3.2.1 Formalism

We introduce the on-shell superfields

(36)

where is a complex Grassmann variable. The 4-pt superamplitudes can then be written in terms of a supermomentum delta function

(37)

where is a prefactor and

(38)

The 4-pt. superamplitudes of the CSM theory are given by

(39)

where we have ignored the numerical prefactor . Note that the superamplitudes encode the scattering of all component fields. In particular, the component amplitudes correspond to the coefficients of the Taylor expansions of the superamplitudes in the fermionic variables. For example, the coefficient of in the Taylor expansion of is

where we noted that . This indeed matches our result for the amplitude in (35). Using a similar analysis, one sees that the amplitude in (35) corresponds to the component of . Furthermore, it is easy to reproduce the relations in (33) and (34). Hence, the superamplitudes and encode all the 4-point component amplitudes in the mass-deformed Chern-Simons theory.

In addition to the supercharge defined in (38), we can also define the following supercharge which annihilates the 4-point superamplitudes

(40)

The superalgebra which acts on the superamplitudes of the CSM theory is given by

Note that the superalgebra does not have a central extension, like the one in (13). This is because the R-symmetry of the theory is not manifest in the “” formalism. The charges acting on the superamplitude and superfields can be regarded as the subset of charges in (13) that carry the same index, and thus do not have the central term in their anti-commutator. In particular, the R-symmetry acts on the on the variables as follows:

where . Under this symmetry, fields of helicity should be multiplied by , however the superfields in (36) do not respect this symmetry. Hence, in the “” formalism, the R-symmetry is broken to , which corresponds to the little group in three-dimensions. This corresponds to multiplying bosons by +1 and fermions by -1.

The R-symmetry of the theory is realized by the superamplitudes as follows. It is easy to see that the 4-pt. superamplitude is an eigenfunction of the R-symmetry generator

with eigenvalue 2. Hence, is a symmetry of the 4-pt. superamplitude. This corresponds to a R-symmetry. We expect this symmetry to persist for higher point amplitudes. In particular, we expect that the -pt. amplitude will be annihilated by .

3.2.2 Formalism

To make contact with the formalism for four dimensional SYM theories, we first recall that in the notation of [52] four dimensional amplitudes (with negative helicity particles in the and slots of the color ordered amplitude) correspond to

(41)

where the other entries correspond to . For four point amplitudes

(42)

where for the four dimensional theory – as well as the massless three dimensional SYM theory – the prefactor is given by the famous Parke-Taylor relation:

(43)

Even though the massive theory is not known to be obtainable as a dimensional reduction of a higher dimensional gauge theory the massive supersymmetry algebra (16) can be regarded as a dimensional reduction of the four dimensional massless supersymmetry algebra (where the two ’s of the Lorentz group are identified and fourth components of all the physical momenta are fixed to be ). It is thus expected that the kinematic constraints relating the different components of the superamplitudes for the massive theory can be cast in a form as in . Indeed, after defining the adjoint superfield in our notation, it is readily seen that

(44)

correctly reproduces all the relations between the massive amplitudes given above. The prefactor can be read off once any of the known four-point component amplitudes are known. For our present purposes, they can be determined in terms of the four-fermion amplitudes computed in appendix B.

4 Yang-Mills-Chern-Simons amplitudes

In this section, we will describe various three and four-point tree-level color-ordered amplitudes of the YMCS theory. In particular, we compute all of the three and four-point amplitudes without external gauge fields, and obtain the remaining 4-point amplitudes using the on-shell superalgebra (24)888Note that the on-shell superalgebra implies constraints on the 4-point amplitudes but not on the 3-point amplitudes. This has to do with the fact that the algebra is only valid when the external momenta are real. In the case of three-point amplitudes one necessarily needs to continue the amplitudes to complex momenta.. In the end of this section, we describe the difficulties associated with computing on-shell YMCS amplitudes with external gauge fields using Feynman diagrams.

4.1 3-point amplitudes

The colour ordered 3-pt. amplitudes are defined for completely general fields by the expression

(45)

where the momenta are all in-going and is the creation operator for the associated field.

The only 3-pt. amplitude which does not have external gauge fields is

(46)

Rearrangement of the fields is achieved using

(47)

The SUSY algebra does not help us determine the remaining 3-pt. amplitudes from (46).

Note that the amplitude in (46) has R-symmetry which rotates the two fermions. This symmetry follows from the R-symmetry in the fermionic sector of the Lagrangian and should therefore hold for higher-point amplitudes, as we will demonstrate at 4-point. The form of this amplitude will be useful for deducing whether or not the BCFW recursion relations are applicable to this massive gauge theory. We shall return to this issue later.

4.2 4-point amplitudes

In this section, we compute various tree-level 4-pt. amplitudes of the YMCS theory. One may determine the remaining amplitudes using the following rearrangement rules

(48)

where indicates a general field and “f.e.” means the number of times fermions (if present) are exchanged in the reordering.

We begin by computing the 4-fermion amplitudes. Then we compute two fermion–two boson amplitudes, followed by 4-boson amplitudes.

4.2.1 Four fermion amplitudes

The calculation of the 4-fermion amplitudes of the YMCS theory is described in appendix