Helicity amplitudes for QCD with massive quarks

The novel massive spinor-helicity formalism of Arkani-Hamed, Huang and Huang provides an elegant way to calculate scattering amplitudes in quantum chromodynamics for arbitrary quark spin projections. In this note we compute two families of tree-level QCD amplitudes with one massive quark pair and n-2 gluons. The two cases include all gluons with identical helicity and one opposite-helicity gluon being color-adjacent to one of the quarks. Our results naturally incorporate the previously known amplitudes for both quark spins quantized along one of the gluonic momenta. In the all-multiplicity formulae presented here the spin quantization axes can be tuned at will, which includes the case of the definite-helicity quark states.


Introduction
The recent advances in the analytic understanding of the scattering amplitudes are often believed to be specific to massless theories, preferably with supersymmetry. It is arguably due to the absence, until recently, of a fully satisfactory spinor-helicity formalism for massive particles. Of course, the massless spinor-helicity formalism [1][2][3][4][5][6] (popularized e.g. by ref. [7]) has been applied [8][9][10] to define massive Dirac spinors. However, that construction did not manage to dispel the notion of the on-shell amplitude methods being restricted to the massless case. Recently, however, Arkani-Hamed, Huang and Huang [11] have introduced a complete version of a massive spinor-helicity formalism and used it to reconsider an array of quantum field-theoretic results from the fully on-shell perspective.
This note is about how this massive formalism can be used in one field theory of interest -quantum chromodynamics with heavy quarks. For simplicity, here we only consider the amplitudes with one massive quark-antiquark pair, with the other particles being gluons of definite helicity. The main goals of this note are two-fold: • We provide new all-multiplicity expressions, eqs. (4.1) and (4.8), for the n-point colorordered amplitudes with two quarks in case of all gluons of identical helicity and the case of one gluon of opposite helicity color-adjacent to one of the quarks.
• We pay special attention to our conventions so that our results be consistent with the vast QCD literature. That involves flexible transitions between the presented massive formalism, its massless analogue recovered in the high-energy limit, the general Dirac spinors and their realization using the massless Weyl spinors.
In view of the second goal, in section 2 we review the spinor-helicity formalism in an effort to combine brevity with comprehensiveness. We illustrate the introduced methods in section 3, where we show two ways to derive a full color-dressed amplitude for fourparticle scattering (corresponding e.g. to non-abelian Compton scattering). We highlight the difference between the Feynman-diagrammatic approach and the on-shell construction, which deals solely with gauge-invariant quantities.
In section 4 we present and prove the aforementioned all-multiplicity amplitudes with two specific gluon-helicity configurations. For that we employ the Britto-Cachazo-Feng-Witten (BCFW) on-shell recursion [12,13]. The spins of the quark and the antiquark remain unfixed throughout the calculations, which lets us specialize to the specific quarkspin projections considered previously [14] in the massless-spinor-based formalism [8][9][10]. Hence, in section 5, we give a simple dictionary (5.4) between the two descriptions and thus compare our results with the literature. It also shows that the new formalism easily incorporates the old one, the elegance of which suffered from the loss of the explicit littlegroup SU(2) symmetry.
We hope that this note will pave the way to more tree-and loop-level calculations in the newly complete spinor-helicity formalism [11], as outlined in section 6.

Spinor-helicity review
It is well-known that particles are defined as irreducible unitary representations of the Poincare group [15,16]. Once the translation operator is diagonalized and the particles are labeled by their momentum p µ , one is left with the Lorentz SO(1, 3) subgroup of the Poincare group. The remaining labels of a one-particle state turn out to belong to a representation of its little group. This subgroup of SO(1, 3) is crucial for understanding spin. It is defined through the Lorentz transformations that preserve the momentum p µ of the particle. It corresponds to SO(2) for massless states or to SO(3) for massive ones.
An important property of the SL(2, C) transformations (and hence the SU(2) ones) is that they preserve the antisymmetric form αβ = − αβ , i.e. the spinor product: This form allows to raise and lower both the spinor and massive-little-group indices at will. Now let us explore different spinor types one by one. The massless and massive Weyl spinors comprise the spinor-helicity formalism [1][2][3][4][5][6]11], while the Dirac spinors are helpful to connect it to the more traditional approaches.

Massless Weyl spinors
In the massless case, the on-shell condition p 2 = det{p αβ } = 0 means that the degenerate matrix p αβ can be decomposed as a tensor product of two Weyl spinors. That decomposition can be written in various interchangeable ways using the spinor bra-ket notation: (2.4) This notation fits the spinor products [1][2][3][4][5][6] particularly well: The Lorentz transformations (2.2) act on the Weyl spinors λ pα ≡ |p α andλα p ≡ |p]α via S ∈ SL(2, C), but only up to the little-group U(1) rotations: 2 (2.6) These spinors also give us the building blocks for the polarization vectors of gauge bosons: we conclude that Lorentz transformations act as only up to an additional term proportional to the new momentum L µ ν p ν . However, up to this caveat, this shows that these polarization vectors can be thought of as conversion coefficients between the off-shell Lorentz transformations and the corresponding on-shell little-group rotations [11]. A similar statement for the Weyl spinors is demonstrated by eq. (2.6) and is also true for the massive case, see eq. (2.13) below.
As a concrete realization of the Weyl spinors, one could use, for instance, for a null momentum expressible as p µ = E(1, cos ϕ sin θ, sin ϕ sin θ, cos θ). A more practical implementation is given in appendix A.

Massive Weyl spinors
For a nonzero mass m, we have a non-degenerate matrix p αβ that satisfies det{p αβ } = m 2 . The Weyl spinors are then introduced [11] by expanding p αβ in terms of two explicitly degenerate matrices λ 1 pαλpβ1 and λ 2 pαλpβ2 : (2.12) Here we have already indicated that the little-group indices a, b = 1, 2 are lowered and raised by the antisymmetric form ab , preserved by SU(2) rotations. Such little-group transformations follow from the action of the Lorentz group on these spinors: where ω ∈ SU(2) correspond to the SO(3) rotations in the rest frame of the massive particle momentum. These transformations are a massive analogue of eq. (2.6). Furthermore, the momentum decomposition (2.12) implies the two-dimensional version of the Dirac equation pα α λ a pα = mλα a p , p ααλα a p = mλ a pα . (2.14) For further convenience, let us rewrite the above identities in the spinor bra-ket notation: As an explicit spinor realization, one may use [11] λ a pα = (2.16) given a massive momentum expressible as p µ = (E, P cos ϕ sin θ, P sin ϕ sin θ, P cos θ), such that E 2 − P 2 = m 2 . A more detailed implementation is given in appendix B.

Dirac spinors and spin
In this paper, we wish to study massive quarks that are traditionally described in terms of the Dirac spinors. Hence it may be illuminating to consider how the Weyl spinors (2.12) naturally unify into the Dirac spinors: 3 This choice ofū a p andv a p is consistent with the conjugation properties (u a p ) † = sgn(p 0 )ū pa γ 0 , (v a p ) † = − sgn(p 0 )v pa γ 0 , assuming that the constituent Weyl spinors are parametrized as detailed in appendix B.
We can treat these spinors as quantum-mechanical wavefunctions and compute the expectation values of the spin operator Σ/2, where Σ i ≡ i ijk γ j γ k /2. Given the spinor parametrization (2.16), we obtain the three-dimensional spin vector Therefore, the spinors (2.16) have definite ±1/2 helicities, i.e. the eigenvalues of the helicity operator h =p · Σ/2, which is a conserved quantity for a one-particle state.
To delve into the subject of spin a bit further, we rewrite the massive spinor parametrization (2.16) as which makes obvious the smooth limit of the massive spinors λ pα andλα p to their massless homonymes (2.11): where ζ a − ≡ (0, 1) and ζ a + ≡ (1, 0). To rephrase this in a more general way, we can introduce two-dimensional spinors λ pα and η pα such that λ a pα andλα a p decompose as The massive momentum is now expressed as a sum of two null momenta: which gives a link to the massive extension of the massless spinor-helicity formalism used previously in the literature [8][9][10]. We make this link precise in section 5 below. Now let us discuss a subtle point concerning spin. Traditional quantum-mechanical spin operators are thought of as acting on the SU(2) indices, which seem to correspond to the little group. The spin of the decomposition (2.16) points along the three-momentum p, whereas the little-group vectors ζ a ± describe states with spin direction along the z-axis. In other words, the massive Weyl spinors (2.16) convert the physical helicity operator h =p · Σ/2 = p· σ 0 0p· σ /2 to σ 3 /2: This should be regarded as a nice feature of the parametrization (2.16) rather than an inconsistency. Indeed, the little-group SU(2) transformations correspond to SO(3) rotations in the rest frame of the massive particle, in which p µ rest = (m, 0), whereas the spinorial matrices σ β α = σαβ generate rotations in the boosted frame where p µ = (E, p). It is therefore convenient that the spinorial (p · σ), 4 taken along the momentum direction, are converted to the simplest of the Pauli matrices, σ 3a b . In principle, one can easily break the above property by SU(2)-rotating the spin states. Apart from losing the relatively simple parametrization (2.16), this would mix the pure helicity eigenstates and produce wavefunctions with a spin quantization axis other than the momentum, and therefore undetermined helicity. The massive spinor-helicity formalism of ref. [11] reviewed here allows to easily switch that axis, and this is precisely what we do in section 5 in order to compare our results with the literature.

Four-point amplitudes
In this section, we demonstrate the use of the various spinors discussed above by dissecting one full color-dressed amplitude. It is convenient to consider the simple case of one massive quark-antiquark pair and two gluons of opposite helicity. Their scattering amplitude has three Feynman diagrams: 5 Now let us recast the above numerators in the spinor-helicity formalism by plugging in the Dirac spinors (2.17) and the polarization vectors (2.7), where for brevity we label spinors as |i ≡ |p i , etc. The numerators (3.2) may seem complicated, which is due to their explicit gauge dependence on the gluonic reference vectors q 3 and q 4 . Incidentally, one can check that for any such gauge choice they nontrivially satisfy the kinematic-algebra relation n 1 − n 2 = n 3 , which is color-dual to the commutation relation c 1 − c 2 = c 3 [17,18]. A very beneficial gauge choice is q 3 = p 4 and q 4 = p 3 , for which We can thus write simple closed-form expressions for all three color-ordered amplitudes These evidently obey the Kleiss-Kuijf relation A 1243 + A 1234 + A 1324 = 0 [19], as well as the Bern-Carrasco-Johansson (BCJ) relation [18,20,21] ( The full color-dressed amplitude can thus be constructed from a single linearly independent color-ordered amplitude as [18,22] A It is interesting to note [23] that the gluonic color-ordered amplitude (3.4c) is also the correct QED amplitude [11] (up to a factor of −2 due to the color-generator conventions). Note that the above amplitudes are gauge-invariant and could have been reduced from the numerators (3.2) to the expressions (3.4) for any choice of reference vectors q 3 and q 4 . This illustrates why in general, at least in analytic calculations, it is better to avoid dealing with gauge-dependent objects and compute gauge-invariant quantities directly. Such a way to derive the above amplitudes would be via the BCFW on-shell recursion [12,13] starting from the three-point amplitudes These make sense on complex on-shell kinematics and are independent of the gauge-boson reference vector q, despite not looking that way (this feature is explained in ref. [11]). To reduce the four-point amplitude to the three-point ones, we apply a simple masslessspinor shift |3] ≡ |3] − z|4], |4 ≡ |4 + z|3 , Here we chose the reference vectors as q 3 = p 4 , q 4 = p 3 to remove most z-dependence as early as possible. Otherwise, the spinor products of q 3 and q 4 would cancel anyway, but only after plugging in the specific on-shell solutions for |3], |4 andP µ and using various Schouten identities. In the last transition of eq. (3.9), we also reduced the sum over the spin label c of the intermediate quark using the completeness relations (2.15). As a simple check, we verify that the massless limit corresponds to the well-known Parke-Taylor MHV amplitudes [24]: (3.10) It is even simpler calculation, either with the Feynman diagrams or via the on-shell recursion, to find the amplitudes with two quarks and two positive-helicity gluons . (3.11)

All-multiplicity amplitudes
In this section we turn to the main calculations of this note -two infinite families of color-ordered amplitudes with one massive quark-antiquark pair. (4.1) It is easiest derived using the BCFW recursion [12,13]. To set up the induction, we check that for n = 4 the formula (4.1) visibly reduces to the four-point amplitude A 1243 in eq. (3.11). For the inductive step, we choose to shift the gluonic spinors | n−1] ≡ |n−1] − z|n], |n ≡ |n + z|n−1 .  . (4.7)

One-minus amplitudes with two quarks
In this section we again use the on-shell recursion to derive an all-multiplicity expression  for the amplitude with two quarks and n − 2 gluons. Here we assume negative helicity of the gluon 3 color-adjacent to the quark 1, with all other gluon helicities positive, while the quark helicities are still left arbitrary. In principle, this color-ordered amplitude is enough to reconstruct the full color-dressed one-minus amplitude via the BCJ relations [18,20], such as the four-point one in eq. (3.5). Indeed, the BCJ relations allow to fix the position of any gluon to be color-adjacent to the quark, with permutations acting on the remaining gluons, hence one may choose to fix the position of the minus-helicity gluon. Moreover, oneplus amplitudes can also be retrieved from eq. (4.8) via the conjugation rule p q ↔ [q p].
To prove the above formula, we use the same "[3 4 " shift as in eq. The two residues are evaluated on the following pole kinematics: . (4.11b) 6 Our present chiral conventions λ−p = −λp,λ−p =λp, reviewed in appendix A, imply ε ± −p = −ε ± p . Therefore, crossing each gluon in an amplitude creates an additional minus sign, A(p in 1 , p out 2 , . . . , p out n ) = −A(−p out 1 , p out 2 , . . . , p out n ). (4.9) Taking into account the completeness relation ε µ p+ ε ν p− + ε µ p− ε ν p+ = −η µν + (p µ q ν + q µ p ν )/(p·q), we conclude that gluonic poles are accounted for by the intermediate propagator factor −i/p 2 . The residue at z 13 is computed immediately for any n using the all-plus expression (4.1), Here the q-dependent factors explicitly canceled after using eq. (4.11a), and we also simplified The contribution (4.12) from z 13 in fact coincides with the k = 4 term of the n−1 k=4 sum in the full formula (4.8). To see that, we only need to rewrite (4.14) Now we turn to the residue (4.10b) at z 45 . First of all, we observe that the right-hand three-gluon amplitude is invariably We can then compute the residue at z 45 as A(1 a ,3 − ,P + , 6 + , . . . , n + , s 12 3P P 6 6 7 . . . n−1|n 3|1|1+2|n Here we were able to integrate the second term in the bracket (4.17a), which corresponds to k =P , into the n−1 k=6 sum as that for k = 5. Since the k = 4 term, missing from eq. (4.17), is provided by the residue at z 13 , this concludes the proof of the formula (4.8).

Checks
As the first simple check of our all-multiplicity formulae (4.1) and (4.8), we evaluate their massless limits. The former explicitly vanishes, as it should, whereas the latter reduces to the massless MHV amplitudes with two quarks: (5.1) This analytic check, however, is only sensitive to a single term in eq. (4.8) that is not multiplied by the mass. As another partial check, we happened to have a six-point Feynmandiagrammatic calculation at easy access, with which we found numerical agreement to ten significant digits for both helicity configurations. Needless to say, the Feynman diagrams were much lengthier before evaluation than the three-term amplitude generated by the formula (4.8). The all-plus formula (4.1) can also be independently verified via the Ref. [14] actually sets q µ to the momentum of the minus-helicity gluon 3. This allowed for BCFW shifts involving this pair of massive and massless momenta, and thus set up a recursion to compute the amplitudes in question. Let us translate the results of ref. [14] to the conventions of the present paper: To conclude, we note that ref. [14] also computed the analogue of the all-plus amplitude in the massless spinor-helicity formalism (5.3) through its relation to the massive scalar amplitude of refs. [28,29] via a supersymmetric Ward identity [30][31][32] with an unfixed reference vector q µ . The same eqs. (5.5) and (5.8) allow to easily verify that these results are incorporated in our formula (4.1).

Summary and discussion
In this note we have computed two infinite families of tree-level amplitudes with two quarks of arbitrary spin and any number of gluons with specified helicities. For that we have used the new massive spinor-helicity formalism of ref. [11]. In order to check the consistency of our results with the literature, we have also established straightforward transition rules between our approach and the more traditional ones.
We hope to have demonstrated that the new massive formalism is a analytic tool well-suited for QCD computations. It is a logical extension of the massless spinor-helicity formalism [1][2][3][4][5][6]11], which in the last decades has become indispensable for scatteringamplitude calculations. Of course, the scope of the formalism is much more general than QCD, as shown by the recent applications to gravitational scattering [33,34] and the Standard Model as a whole [11,35]. It can be used streamline the consideration of all unitarity-compliant three-point [35][36][37] and four-point [11] interactions. It can also be related to much earlier off-shell reformulations of QED [38,39] and other theories [40][41][42] using two-component spinor fields.
The presented formalism has potential to facilitate many QCD calculations, both analytically and numerically. Through its analytic simplicity, it may provide a way to explicit expressions for tree amplitudes with more general gluon helicity configurations [43][44][45][46] and more quark-antiquark lines [17,18], as already achieved [47,48] for the massless QCD amplitudes with up to three quark-antiquark pairs. For example, it would be interesting to find an analytic expression even for an amplitude with two quarks and one negativehelicity gluon in an arbitrary position, provided that it is more compact than its BCJ relation [18,20] that involves various permutations of the formula computed in this note.
New loop amplitudes could also be calculated using the presented formalism. Indeed, loops can be obtained from generalized unitarity cuts [49][50][51][52][53][54][55] that are constructed from tree amplitudes. 7 It would also be interesting to investigate, in the spirit of refs. [58,59], if the massive on-shell formalism could speed up numerical evaluation of tree-level amplitudes. This would be beneficial for computing real-emission radiative QCD corrections to a vast array of elementary-particle scattering processes.
satisfying p + p − = p ⊥ p ⊥ , an explicit solution for the Weyl spinors is (adapted from ref. [60]) In the case of a real-valued momentum with positive energy, it is just a rewrite of eq. (2.11). For negative energies, the step functions θ(−p 0 ) introduce the ±i prefactors to ensure the momentum inversion rule λ −p = −λ p ,λ −p =λ p , (A. 3) assuming the principal square roots. Moreover, for a real-valued momentum p µ the spinor conjugation property is (λ pα ) * = sgn(p 0 )λ pα . If p + happens to vanish, an equivalent solution may be used: In the complex-valued case where p ± = 0 and the momentum equals (0, p 1 , ±ip 1 , 0), a valid choice is

B Massive spinor parametrizations
Here we give the massive spinor-helicity variables that are consistent with the parametrizations (A.2) through (A.5) in the massless limit. The spinors (2.19) can be rewritten as where now we take p ± = P ± p 3 , p ⊥ = p 1 + ip 2 . Moreover, we introduce a sign function in the definition P = sgn(E) p 2 .