Recursion relations for scattering amplitudes with massive particles II: massive vector bosons

Using the recently introduced recursion relations with covariant massive-massless shift, we study tree-level scattering amplitudes involving a pair of massive vector bosons and an arbitrary number of gluons in the massive spinor-helicity formalism. In particular, we derive compact expressions for cases in which i) all gluons are of the same helicity and ii) one gluon has flipped helicity and is colour adjacent to one of the massive particles. We provide numerous consistency checks of our results including the exact match of high energy limits with well-known MHV and NMHV amplitudes in pure Yang-Mills theory. As a corollary, we obtain an alternative novel representation of the NMHV amplitude.


Introduction
Spinor-helicity formalism has revolutionized our understanding of the S-matrix theory for massless particles. By trivialising the non-linear constraints such as the Gram determinant condition that Mandelstam variables have to satisfy, expressing the external scattering data in terms of spinor-helicity variables leads to remarkably simpler and conceptually revealing expressions for scattering amplitudes. For example, the Parke-Taylor amplitude [1] took a strikingly simple form when the external momenta and polarisation data were expressed in terms of spinors [2][3][4][5]. As spinors are complex, the real power of this formalism was revealed when complex deformations of external momenta was used to derive on-shell recursion relations by [6,7]. In fact, computations of the tree-level amplitudes in gauge theories and gravity get immensely simplified by implementing the BCFW recursion relations in the spinor-helicity formalism.
The recursion relations construct higher-point amplitudes in terms of lower-point amplitudes while staying on-shell. These recursion relations were generalized to the case of massive particles in [8][9][10][11][12]. In [8,9] multiple massive momenta were complexified to study the scattering amplitudes. However, in these works, the massive momenta were written in terms of certain light-like momenta due to which the covariance (with respect to the little group action of external particles) was broken. In [10] tree-level amplitudes with a pair of massive scalars and up to four gluons were computed using the BCFW shift on a pair of massless external particles (gluons) and later this method was used to compute several lower-point tree-level amplitudes involving fermions and massive vector bosons (spin-1) scattering with gluons [11]. In an another development [12], tree-level amplitude of a pair of complex scalar and an arbitrary number of positive helicity gluons was computed using the Berends-Giele and on-shell recursion relations, and obtained an extremely compact expression. In [13] this was further extended to compute the amplitude involving a pair of massive quarks and arbitrary number of gluons.
Recently, in a remarkable work a little group covariant spinor-helicity formalism for massive particles was introduced [14]. In a beautiful paper, Ochirov combined this formalism with the recursion relations proposed in [15]. In particular using the BCFW shift on a pair of gluons, Ochirov derived formulae for two classes of n-point amplitudes involving massive quarks, consistent with the previous results in [12,13].
In [16][17][18], a new set of recursion relations were derived in the massive spinor-helicity formalism for on-shell amplitudes by complexifying one massive and one massless external states. These complex momentum shifts (involving a complex parameter z) were then realized in the spinor-helicity basis by considering little group covariant deformations of massive and massless spinor-helicity variables. We call this particular shift as the covariant massive-massless shift and refer to the resulting recursion as the covariant recursion relations. In earlier work [16], we used these recursion relations to study tree-level lowerpoint amplitudes in scalar QCD as well as amplitudes involving massive vector bosons in the Higgsed Yang-Mills theory. We also classified all of valid covariant massive-massless shifts for these theories by requiring that the amplitude does indeed vanish as the complex deformation parameter z tends to ∞. In this paper, we further use these recursion relations to compute tree-level n-point amplitudes in the Higgsed Yang-Mills theory (that includes massive spin-1 particles and gluons as quanta of the theory).
As is well known, one of the earliest and striking applications of the BCFW recursion technique was in (1) the proof of Parke-Taylor formula for n-point maximally helicity violating(MHV) amplitudes and (2) the ease with which tree-level next-to-maximally helicity violating (NMHV) amplitudes could be computed. The power of BCFW recursion technique could be seen from the fact that, the n-particle MHV and NMHV amplitudes can be obtained by using a single recursion. Completely analogously, we compute two classes of n-point amplitudes involving a pair of massive vector bosons and gluons in the external data such that in the high energy limit, these amplitudes reduce to MHV and NMHV gluon amplitudes respectively.
The paper is organised as follows. In section 2, we review the massive spinor-helicity formalism and the covariant recursion relations. In section 3, we compute the tree-level colour-ordered amplitude involving a pair of massive vector bosons and an arbitrary number of gluons of same helicity that is massive analogue of the MHV amplitude. To obtain this amplitude we first use a simple relation between the amplitude involving two massive vector bosons and (n − 2) positive helicity gluons, and the amplitude involving two massive scalars and (n − 2) positive helicity gluons. This relation is a covariantized version of a relation that has appeared in [19] for a particular choice of spin projection of the massive particles. We then prove this result by using the method of induction and the covariant recursion relation. We also check consistency of this amplitude by taking the high energy limit which exactly matches with the pure gluon MHV amplitude.
In section 4, we turn to the main focus of this paper which is the computation of the tree-level colour-ordered amplitude involving a pair of massive vector bosons, one negative helicity gluon that is colour-adjacent to one of these massive particles and an arbitrary number positive helicity gluons. We obtain this amplitude using the covariant recursion relations as proposed in [16]. We will find that the single covariant recursion involves only subamplitudes that have been previously computed. Finally we check the consistency of this result by taking the high energy limit and produce the n-point NMHV amplitude.
We conclude in section 5 with a short summary and outline some immediate future directions, and collect some technical materials in the appendices.

Review of covariant recursion relation
Scattering amplitudes are Lorentz invariant objects and transform covariantly under little group which is ISO(2) for massless particles and SU(2) for massive particles in four dimension. Hence we label massless states by the helicity (h) of the particle and use symmetric 2S representation of SU(2) to represent the massive spin-S one-particle state, instead of using the standard representation of SU(2) introducing a preferred z-direction which breaks the rotational invariance of S-matrix. Amplitude involving a massless particle with momentum p j and a massive particle with momentum p i and spin S then transforms under little group as follows [14] tum 4-vector( p µ σ µ αα = p αα ). For massless particles, this is a rank-1 matrix and can be expressed as where λ α andλα are two-component Weyl spinors, known as massless spinor-helicity variables. Since we can always rescale the spinor-helicity variables it is impossible to assign unique spinor-helicity variables to express p αα . But this scaling is exactly the little group scaling for massless particle. Thus we identify λ α andλα as objects having little group weight ±1 respectively. Using spinor-helicity variables, we define Lorentz invariant and little group covariant angle and square brackets as These brackets are the basic building blocks of scattering amplitude in spinor-helicity formalism. Massless spinor-helicity variables satisfy the Weyl equation Next we turn to the particles with mass. In this case det(p αα ) = p µ p µ = 0. Hence p αα is expressed as a linear combination of two rank-1 objects [14] p αα = 2 I,J=1 where (I, J) are SU(2) little group indices for massive particle. The variables λ I α ,λ J α are called massive spinor-helicity variables . Similar to the massless case, there is no unique way to fix these spinors, satisfying the above relation due to the following transformation Unlike the massless case, these transformations do not correspond to the little group transformation of massive particle as W can be any GL(2) matrix. But if we demand det(λ I α ) = det (λ J α ) = m, then it can be shown that W is indeed a SU(2) matrix for real momenta, reflecting the above transformations as little group transformation.
The massless spinor-helicity variables λ α ,λα, which satisfy Weyl equation are independent of each other. However, the dotted and undotted massive spinor-helicity variables are related to each other via Dirac equation Therefore the scattering amplitude involving massive particles can be expressed in terms of only either λ α I orλ Iα as opposed to amplitude with only massless particles. This feature of amplitude proves extremely useful to classify all possible three-particle amplitudes [14] involving massive as well as massless particles.

Three-particle amplitude
In this section, we briefly review all the required three-particle amplitudes which will be used as basic building blocks to construct higher-point amplitudes using recursion scheme. Due to kinematics, the three-particle amplitude involving massless particles with helicity h 1,2,3 can be expressed either in terms of angle or square brackets. Little group transformation then fixes the structure upto an overall multiplicative constant, with two distinct representations ensuring smooth vanishing limit in Minkowski signature as individual spinor products vanish in this signature for real momenta. Three-particle amplitudes involving two massive particles with mass m and spin-1 coupled with a massless particle of helicity h are given by [14] A +h (2.10) These amplitudes have well-behaved massless limit and relevant for the Higgsed Yang-Mills theory [20]. Here x 12 is a non-local factor arises due to the degeneracy of masses. It is defined as follows where ζ is a reference spinor. We denoted massive spinor-helicity variables in bold notation and omitted little group indices in the amplitude. The angle and square brackets of these bolded variables are defined as a symmetric product of spinor brackets in SU (2) indices. For example, All the amplitudes that we are going to discuss in this note are obtained by gluing threepoint amplitudes involving massive spin-1 particles. So we can use these three-particle amplitudes as basic building blocks.

Massive-massless shift and the covariant recursion
We use the two-line little group covariant massive-massless shift introduced in [16,17] to compute four-and higher-particle amplitudes involving gluons and massive vector bosons in the Higgsed Yang-Mills theory 1 . Although this particular shift of external momenta is in the similar spirit with the well known BCFW shift but involves complex deformation of massless and as well as massive momenta. Let us consider that the massive and massless momenta, denoted by p i and p j respectively, are analytically continued to the complex plane while staying on-shell (2.14) Here z is complex deformation parameter and q µ is lightlike shift vector satisfying the following conditions The momentum shift can be achieved by the following deformations of the massive and massless spinor-helicity variables [16] massive shift : In [16], Here ± indicate the helicity of the massless particle and m denotes the mass of the massive particle. In particular, it has been shown that the deformed amplitude vanishes as the deformation parameter z tends to ∞. This proof is quite general in the sense that it does not depend number of external particles as long as one can deform a single massive and massless external momenta. We use this in the following recursion relation to compute amplitudes involving gluons having arbitrary helicity and massive vector bosons The sum includes all possible scattering channels as well as spin or helicity states of the exchange particle. We consider only colour-ordered amplitudes instead of fully colourdressed tree-level amplitudes since the latter can be constructed from the former using the well known colour decomposition [5,[21][22][23][24][25].

Scattering of massive vector bosons with positive helicity gluons
One of the earliest applications of the BCFW recursion relations was to provide an extremely simple proof of the formula for the n-point MHV amplitude using the principle of induction. As alluded to in the introduction, we wish to similarly apply the covariant recursion relations for the case of amplitudes involving massive particles. In this section, we consider an n-point amplitude involving a pair of colour adjacent massive vector bosons and (n − 2) positive helicity gluons. This particular scattering amplitude serves as a massive analogue of the n-point MHV amplitude, as we will see later in the following section that it reproduces the n-point MHV amplitude in the high energy limit. We will show that this massive vector boson amplitude can also be derived inductively in the covariant recursion scheme, just as the n-point MHV amplitude was derived using BCFW. The n-point MHV amplitude was already derived in [1] before the discovery of BCFW recursion. But using the BCFW recursion, the derivation became extremely simple. In our case, the massive vector boson amplitude that we want to compute using the covariant recursion is not known. To obtain this amplitude, we adopt a different strategy: firstly, we relate this amplitude to a known amplitude involving a pair of massive scalars and (n − 2) positive helicity gluons by using the little group covariant version of a formula that first appeared in [19]. Secondly, we prove this formula in detail by making use of the covariant recursion relations and the principle of induction.
The relation between the n-particle amplitude involving a pair of massive bosons and positive helicity gluons and the n-particle amplitude involving pair of massive scalars and positive helicity gluons is the following This is a covariantization (in little group indices) of a relation that has appeared previously in [19] for a particular choice of the spin projection of massive particles 2 . Furthermore, using the expressions for massive amplitudes in [16], we have explicitly verified this covariant formula in case of lower-point amplitudes, such as four-and five-particle amplitudes. Now the n-point amplitude with a pair massive scalars and (n − 2) positive helicity gluons is already known [12]: where the Mandelstam variables and p 1,l are defined as follows s 1...l := (p 1 + · · · + p l ) 2 , p 1,l := p 1 + · · · + p l .
In equation (3.2), we have introduced short hand notation for spinor products defined as follows In order to prove that the formula appeared in [19] is same as the above relation for a specific choice of the spin projection of massive vectors, we use the following decomposition of little group covariant massive spinor-helicity variables [14] λ α where λα, ηα are massless spinor-helicity variables and satisfy λη = m and ξ ±I are suitable SU(2) basis vectors. Setting the particle with momentum p1 with sz = +1 and particle with momentum pn with sz = −1 in the amplitude, we find that 1n (+,−) −→ η1λn . Therefore, we can recast the relation (3.1) with the massive particles are being in this specific spin state as follows This is the relation that appeared in [19].
Note that we treat the momentum product / p i · / p j as SU(2) matrix valued product pα α i p jαβ when being contracted with spinor helicity variables. We follow this notation throughout this paper. The product appearing in the numerator of the formula (3.2) is defined as Substituting the scalar amplitude in (3.1), we therefore find the following simple expression for the n-point amplitude with a pair massive vector bosons and (n − 2) positive helicity gluons (for n > 3): 3

Inductive proof using covariant recursion
In this section, we present an inductive proof of the above formula in (3.6) using the covariant recursion that was reviewed in Section 2.3. To set up the induction, we first of all have to ensure that the four-and five-point amplitudes that have been calculated previously in [16] are consistent with the general expression. We perform this check in Appendix A.1.
Given the match of the lower-point amplitudes we now assume that the expression (3.6) is true for n-particle amplitude and use this to construct (n + 1)-particle amplitude. We use the [12 + shift which corresponds to the shifts of the following spinor-helicity variables: whereas the spinor-helicity variables |1 I and |2] remain unchanged. With this particular shift, all possible channels that contribute to the A n+1 amplitude are shown in Figure 1.
The first three diagrams do not contribute to the amplitude due to following reasons: a) the contribution from the first diagram vanishes due to the vanishing of the right subamplitude involving a single massive vector boson, b) the contribution from the second diagram vanishes due to the vanishing of the pure gluon subamplitude with either all positive helicity gluons or a single negative helicity gluon, c) the contribution from the third diagram vanishes because a massive vector boson cannot decay into two identical gluons. Thus we only have to compute the contribution from the fourth diagram. From the only non-vanishing diagram, we get a simple pole in the z-plane by setting the shifted propagator s 23 on-shell The (n + 1)-particle amplitude A n+1 1, 2 + , · · · , n + , n+1 is therefore assembled from the n-point and 3-point subamplitudes Here we abbreviate A n+1 1, 2 + , · · · , n + , n+1 as A n+1 . The alternative helicity configuration of the internal states does not contribute to the amplitude due to the vanishing of all-positive-helicity three-particle gluon amplitude. Using the expression for n-point amplitude in equation (3.6), we get the left subamplitude but now with shifted momenta (3.10) Here S ( P ) are the Mandelstam (momentum) variable with the shifted momenta S 1I...r = ( p 1 + p I + p 4 + · · · + p r ) 2 , P 1,r = ( p 1 + p I + · · · + p r ) . (3.11) The internal momentum p I in this channel is p 2 + p 3 . Therefore we find that these shifted variables can be simply expressed in terms of the unshifted variables as S 1...r = ( p 1 + p 2 + p 3 + · · · + p r ) 2 = s 1...r , P 1,r = ( p 1 + p 2 + p 3 + · · · + p r ) = p 1,r .
Using these simplifications and gluing the three-particle gluon amplitude along with the unshifted propagator 1 s 23 onto the left subamplitude, we obtain (3.12) It remains to simplify the terms with the shifted massless spinor-helicity variable I associated with the momentum of the exchange particle. We collect all such terms and rewrite them as we finally obtain the (n + 1)-point amplitude in the form (3.16) This completes the inductive proof of n-particle amplitude with all plus helicity gluons and a pair of massive vector bosons. The scattering amplitude with two massive vector bosons and all minus helicity gluons can be read off from the expression in (3.6) by replacing all the angle brackets with square brackets and vice-versa (3.17) It is instructive to check the high energy limit of the massive vector boson amplitude (3.6). Due to the presence of angle bracket 1n 2 , the only non-zero contribution comes from the component of the massive amplitude with both massive particles having negative helicity in the high energy limit [14].

Matching the MHV amplitude in the high energy limit
In this section, we recover the known massless amplitude from the massive vector boson amplitude with all positive helicity gluons. We show that the finite energy amplitude in equation (3.6) reproduces correct MHV amplitude in the high energy limit for negative helicity configuration of massive particles in this limit. The massless amplitude is given by [21]s 123 · · · s 12...(n−2) 1n . We simplify the product in the numerator by using n-th massless particles momentum conservation and identity (A.5) repeatedly. We start with the k = n − 2 term and use momentum conservation to write 4 s 1...n−2 − / p n−2 · / p 1,n−3 · / p n−1 |n = s 1...n−2 p n−1 |n + / p n−2 · / p n · / p n−1 |n .
Having shown that the covariant recursion relations can be used to inductively prove the formula (3.6) of the massive analogue of MHV amplitude, it is worthwhile to mention that one could do the same by using the BCFW recursion relations as well. However, the real benefit of the covariant recursion will be apparent in the next section where we will show that a similar application of covariant recursion can be achieved in the case in which one of the gluon has flipped helicity.

Scattering of massive vector bosons with a flipped helicity gluon
We consider tree-level colour-ordered amplitude involving a pair of massive vector bosons, one minus helicity gluon and arbitrary number of positive helicity gluons. For simplicity, we assume that the massive particles and the negative helicity gluon are colour adjacent to each other, as indicated in the Figure 2. We shall see later that the scattering amplitude with this specific external particle configuration leads to the NMHV pure gluon amplitude in the high energy limit.
If one had used the usual BCFW shift to compute this amplitude, one would end up with subamplitudes involving the same configuration as the one we set out to compute (i.e. involving two massive vector bosons and helicity flipped gluons). In the absence of an ansatz one would need to use the recursion relation iteratively to compute those subamplitudes that appear in a given recursion. This would make the computation technically involved.
Instead, we use the the massive-massless shift [2 − 1 of the type [−m to compute this particular n-point amplitude. This shift corresponds to the following deformation in terms of the spinor-helicity variables : With this shift and the chosen configuration of external particles, the different scattering channels that contribute to the amplitude in the covariant recursion are shown in Figure  2. As one can see, all the relevant subamplitudes have already been computed: either they involve only pure gluon amplitudes or they involve two massiee vector bosons and all positive helicity gluons. As we are considering only minimal coupling while computing the amplitudes, the exchange particles can be either massive vector boson or gluon. But the exchange particle can not be a massive vector boson as a massive spin-1 particle can not decay into two massless gluons. Due to the [2 − 1 shift, particles with momenta p 1 and p 2 are always attached to different subamplitudes in the diagramatic expansion of the colour-ordered amplitude. Now the subamplitude involving the external momentum p 2 will always have only positive helicity gluons as external states. But such pure gluon amplitudes with at most one opposite helicity vanishes, except for the three-particle amplitude. Hence the internal state attached to this subamplitude must be a negative helicity gluon. Again, due to the choice of the massive-massless shift [2 − 1 , the first diagram in Figure 2 is non-vanishing only for the helicity configuration indicated. The n-particle amplitude obtained by summing over the various diagrams can thus be written as follows Here the subamplitudes are on-shell; that is, they are functions of shifted momenta and spinor-helicity variables. The right subamplitude is a pure-gluon amplitude and is given by the Parke-Taylor formula . (4.4) This takes care of all but one diagram that appears in the covariant recursion. The last diagram in Figure 2 (which corresponds to r = n − 1), has to be treated separately and we shall come to the evaluation of this diagram towards the end of this section. Let us now simplify the expression in (4.4) and write it purely in terms of the external momenta. Using p I = p 2 + r i=3 p i , the shifted Mandelstam variables ( S) and momenta ( P ) can be expressed (for k ∈ {r + 1, . . . (n − 2)}) as follows S 1...k = ( p 1 + p I + · · · + p k ) 2 = (p 1 + p 2 + · · · + p k ) 2 = s 1...k , P 1,k−1 = ( p 1 + p I + · · · + p k−1 ) = (p 1 + p 2 + · · · + p k−1 ) = p 1,k−1 . (4.5) Substituting these into the left subamplitude (4.4) and then gluing this with the pure gluon amplitude (4.3) and taking care of the unshifted propagator 1 s 2...r , we get the contribution to the n-particle amplitude from the r-th term in the covariant recursion (4.2): . (4.6) Here r ∈ {3, 4, . . . (n − 2)}. We would like to note that the product of angle brackets in the denominator, involving the massless spinor-helicity variables do not include r(r + 1) bracket as the r-and (r + 1)-th massless external legs do not attach to same subamplitude. Next we express all the spinor products in A (r) involving the intermediate spinor-helicity variable | I in terms of the spinor-helicity variables of the external particles. In order to do that, we collect all such terms as We use the following identities to write 2| / p 1 . / p 2,r−1 |r 2| / p 1 . / p 2,r |r + 1 . (4.8) It only remains to evaluate the shifted spinor product 1n . The simple pole, associated with scattering channels (except s 1n ) in Figure 2 is obtained by setting the shifted propagator s 2···r on-shell: We then use the definition of the shifted massive spinor-helicity variable in (4.1) at z = z (r) to express the spinor product 1 I n J as Substituting the expressions (4.8) and (4.10) in (4.6), one can finally rewrite A (r) in terms of the on-shell external variables: We now analyze the last diagram in Figure 2, which corresponds to the r = n−1 term in the covariant recursion. We have to treat this term separately because the left subamplitude involving two massive vector bosons and a single positive helicity gluon cannot be read off from the formula (3.6) (we explicitly assumed n > 3 in that derivation). Instead we simply glue the three-particle amplitude in (2.10) for positive helicity gluon along with the pure gluon amplitude (4.3) for r = n − 1 and the unshifted propagator 1 s 1n s 1n 23 34 · · · (n − 2)(n − 1) I|p n | 2] (n − 1) I . (4.12) Again, we have to simplify the terms with | I and evaluate the shifted spinor products, obtained by setting the shifted propagator s 1n on-shell: Firstly, by noting the following identities 14) 2| we get rid of the internal momentum dependence of A (n−1) as follows . (4.16) Secondly, we calculate the shifted spinor products appearing in this expression and in (4.12) using (4.13) and the definition of shifted spinor-helicity variables: Finally we use the following identity to derive the contribution of the last diagram A (n−1) as a function of the on-shell external variables: A (n−1) = g (n−2) 2|p 1 |n] 21 + 2|p n |1] 2n + 2m 12 2n 2 s 1n 23 34 · · · (n − 2)(n − 1) 2| / p 1 · / p n |n − 1 + m 2 2(n − 1) . (4.20) -15 - We combine the results of (4.11) and (4.20) to obtain a compact formula of the n-particle amplitude A n [1, 2 − , 3 + , . . . , n] = g n−2 ( 2|p 1 |n] 21 + 2|pn|1] 2n +2m 12 2n ) 2 s 1n 23 34 ··· (n−2)(n−1) 2| / p 1 · / p n |n−1 +m 2 2(n−1)

Matching the NMHV amplitude in high energy limit
We now consider the high energy limit of the scattering amplitude in (4.21). This should reproduce the unique massless amplitudes for different helicity configurations since we used only the minimally coupled three-particle amplitudes (2.10) as basic building blocks to construct the finite energy amplitude [14].
The procedure of taking high energy limit of massive amplitudes is laid out in [14] and further discussed in [16]. We do not repeat the procedure again but as a general rule of thumb, we show which component of massive spinor-helicity variables survives in this limit below The components of the finite energy amplitude (4.21) in the high energy limit with opposite helicity configurations for the pair of massive particles are non-vanishing due to the presence of both angle and square brackets of massive spinor-helicity variables and reproduce the correct MHV amplitudes as expected: The component of (4.21) with positive helicity configuration for both massive particles vanishes explicitly as it should. But the negative helicity configuration for both massive particles in the high energy limit should give us the NMHV amplitude. From (4.21) we obtain the following result: We simplify the product factor appearing in the numerator using the following identity: This identity can be derived from the one we have proved in Section 3.2. Furthermore, we use momentum conservation to get 2| / p 3,r . / p 1 |n + p 2 2,r n2 = 2| / p 3,r .( / p r+1 + . . . + / p n−1 )|n . (4.27) Substituting the above simplifications in (4.25), we obtain: We have obtained in (4.28) a compact expression for the n-point NMHV amplitude that at first glance appears to be different from the standard expression in [27]. Note that, the first term of the expression in equation (4.21) (that corresponds to the last diagram in Figure  2) does not contribute to the high energy limit since this involves massive spinor-helicity variables that do not survive in this limit. It can be argued that this is a consequence of the massive-massless shift [2 − 1 which we have used to derive this amplitude. In a purely massless setup, one could use the BCFW shift [1 − 2 − , in which case the last diagram in Figure 2 would certainly contribute. Therefore, in this case, the covariant massive-massless shift leads to a novel representation of the n-point NMHV amplitude. In what follows, we will first take the soft limit of this amplitude to show that it obeys the Weinberg's soft theorem at leading order and subsequently we prove that the NMHV amplitude (4.28) matches with the expression in [27] for this specific ordering of external particles.

Soft expansion of NMHV amplitude
We begin with the NMHV expression in (4.28) and take the limit p n → 0 of the gluon with momentum p n . In order to take the limit p n → 0, we first scale the spinor-helicity variables as follows with → 0. With this scaling, we find that ther = n − 2 channel of the NMHV amplitude in (4.28) has the leading order contribution (O 1 ) and the amplitude factorizes as follows This follows from Weinberg soft theorem as well, as we will see now.
Using Weinberg soft theorem, we find that in the soft limit of the n-th gluon momentum, the n-particle NMHV amplitude factorizes as a soft factor times an (n − 1)-particle MHV amplitude as follows: where the soft factor at leading order is given by [28][29][30] Expressing the massless polarization vector in the spinor-helicity formalism and chosing the reference spinor as |q] = |1], we get the soft factor as follows .

(4.36)
This is exactly what we get by the soft expansion of the NMHV amplitude in (4.28).

Matching the NMHV amplitude
Now that the preliminary check of the soft limit has been verified, we now show that the result in (4.28) matches exactly with the NMHV amplitude computed by Dixon et al. in [27] for the specific ordering of negative helicity gluons that we have considered. The result in [27] is of course more general in the sense that the positions of the two negative helicity gluons are completely arbitrary. In order to compare with our result (4.28), we begin with the result in [27] and fix the positions of the two negative helicity particles as 1 − , 2 − . The position of the third negative helicity particle is fixed to be n − in both of the results. So the n-particle amplitude . So the n-point NMHV gluon amplitude can be written as Now by making a variable change, t = r + 1 we can write . (4.51) The spurious pole in the above amplitude is given by the following condition It is easy to check that both expressions for the six-point amplitude contain the same set of physical poles. However we see that these have different spurious poles. In particular, when the spurious pole condition is satisfied for one expression, the other one is finite. As both are representations of the same scattering amplitude, we conclude that at least in this simple example, the residues of the spurious poles indeed sum up to zero.

Summary and Outlook
In this note, we applied the covariant recursion relations introduced in [16] to compute scattering amplitudes with massive particles which hitherto were not known in the literature. The class of amplitudes we chose to focus on are massive analogues of the MHV and NMHV amplitudes in Yang-Mills theory. In the high energy limit, these two classes indeed reduce to the MHV and NMHV amplitudes respectively. Our work can thus be considered as mirroring the computation of MHV and NMHV amplitudes in gauge theory using BCFW recursion relation. The analogue of the NMHV amplitude consists of two massive vector bosons, one negative helicity gluon (that is colour adjacent to the massive bosons) and remaining positive helicity gluons. We showed that for this class of amplitudes, the massive-massless shift leads to a remarkably simple computation and we could generate a compact, little group covariant formula for the final amplitude by using a single recursion. Interestingly we have shown that given the final form for the amplitude derived using the covariant recursion techniques, one can verify that our result indeed satisfies the BCFW recursion relation. This is shown in detail in Appendix B.
It is useful to recall the two key ingredients that went into this computation. First of all we derive the scattering amplitude involving a pair of massive vector bosons and only positive helicity gluons by relating it to an amplitude involving two massive scalars and positive helicity gluons. While this relation was derived previously in [19], we work with little group covariant expressions and provide an inductive proof of the scattering amplitude by making use of the covariant recursion (in the massive spinor-helicity formalism). This was then used as an input to calculate the scattering amplitude in which we flip the helicity of the gluon adjacent to one of the massive particles. We use a specific massive-massless shift such that the resulting subamplitudes in the covariant recursion involved either pure gluon amplitudes or amplitudes involving massive vector bosons and positive helicity gluons. This led to a compact expression for the relevant scattering amplitude in (4.21), which is the main result of this work.
We checked the correctness of our result by taking the high energy limit and showing that they reduce to the expected MHV and the NMHV amplitudes. As mentioned previously we also checked that the n-point amplitude satisfies the usual BCFW recursion relation. Interestingly, our representation of the NMHV amplitude obtained in the high energy limit is not identical to the one obtained previously in [27]. We showed that the two expressions are equal and it will be interesting to study the representation for NMHV amplitude that we obtained in more detail in its own right.
In this note, we have restricted ourselves to a particular configuration in which the position of the negative helicity gluon is adjacent to the massive vector bosons. But in fact it is possible to make the position of the negative helicity gluon completely arbitrary and use the covariant massive-massless shift or the BCFW shift in combination with the amplitudes calculated in this work to derive these scattering amplitudes. One could also include additional negative helicity gluons and systematically proceed to calculate the resulting scattering amplitudes. However in order to compute amplitudes with more than two massive particles using the covariant recursion relations, one would require knowledge of a wider class of amplitudes. We hope to address these issues in the future.