Helicity amplitudes for QCD with massive quarks
Abstract
The novel massive spinorhelicity formalism of ArkaniHamed, 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 treelevel QCD amplitudes with one massive quark pair and gluons. The two cases include all gluons with identical helicity and one oppositehelicity gluon being coloradjacent 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 allmultiplicity formulae presented here the spin quantization axes can be tuned at will, which includes the case of the definitehelicity quark states.
1 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 spinorhelicity formalism for massive particles. Of course, the massless spinorhelicity formalism Berends:1981rb (); DeCausmaecker:1981bg (); Gunion:1985vca (); Kleiss:1985yh (); Xu:1986xb (); Gastmans:1990xh () (popularized e.g. by ref. Dixon:1996wi ()) has been applied Kleiss:1986qc (); Dittmaier:1998nn (); Schwinn:2005pi () to define massive Dirac spinors. However, that construction did not manage to dispel the notion of the onshell amplitude methods being restricted to the massless case. Recently, however, ArkaniHamed, Huang and Huang ArkaniHamed:2017jhn () have introduced a complete version of a massive spinorhelicity formalism and used it to reconsider an array of quantum fieldtheoretic results from the fully onshell 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 quarkantiquark pair, with the other particles being gluons of definite helicity. The main goals of this note are twofold:

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 highenergy 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 spinorhelicity 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 colordressed amplitude for fourparticle scattering (corresponding e.g. to nonabelian Compton scattering). We highlight the difference between the Feynmandiagrammatic approach and the onshell construction, which deals solely with gaugeinvariant quantities.
In section 4 we present and prove the aforementioned allmultiplicity amplitudes with two specific gluonhelicity configurations. For that we employ the BrittoCachazoFengWitten (BCFW) onshell recursion Britto:2004ap (); Britto:2005fq (). The spins of the quark and the antiquark remain unfixed throughout the calculations, which lets us specialize to the specific quarkspin projections considered previously Schwinn:2007ee () in the masslessspinorbased formalism Kleiss:1986qc (); Dittmaier:1998nn (); Schwinn:2005pi (). Hence, in section 5, we give a simple dictionary (55) 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 symmetry.
We hope that this note will pave the way to more tree and looplevel calculations in the newly complete spinorhelicity formalism ArkaniHamed:2017jhn (), as outlined in section 6.
2 Spinorhelicity review
It is wellknown that particles are defined as irreducible unitary representations of the Poincare group Wigner:1939cj (); Bargmann:1948ck (). Once the translation operator is diagonalized and the particles are labeled by their momentum , one is left with the Lorentz subgroup of the Poincare group. The remaining labels of a oneparticle state turn out to belong to a representation of its little group. This subgroup of is crucial for understanding spin. It is defined through the Lorentz transformations that preserve the momentum of the particle. It corresponds to for massless states or to for massive ones.
To include fermions into consideration, one must generalize to the universal covering group of . The homomorphism between these two groups is implemented by the spinor maps
(1) 
The Pauli matrices^{1}^{1}1We use , , , , as well as and . and here translate Lorentz transformations between the spinorial and vectorial languages:
(2) 
for and . At the same time, the little groups for massless and massive particles are accordingly promoted to and .
An important property of the transformations (and hence the ones) is that they preserve the antisymmetric form , i.e. the spinor product:
(3) 
This form allows to raise and lower both the spinor and massivelittlegroup indices at will.
Now let us explore different spinor types one by one. The massless and massive Weyl spinors comprise the spinorhelicity formalism Berends:1981rb (); DeCausmaecker:1981bg (); Gunion:1985vca (); Kleiss:1985yh (); Xu:1986xb (); Gastmans:1990xh (); ArkaniHamed:2017jhn (), while the Dirac spinors are helpful to connect it to the more traditional approaches.
2.1 Massless Weyl spinors
In the massless case, the onshell condition means that the degenerate matrix can be decomposed as a tensor product of two Weyl spinors. That decomposition can be written in various interchangeable ways using the spinor braket notation:
(4) 
This notation fits the spinor products Berends:1981rb (); DeCausmaecker:1981bg (); Gunion:1985vca (); Kleiss:1985yh (); Xu:1986xb (); Gastmans:1990xh () particularly well:
(5) 
The Lorentz transformations (2) act on the Weyl spinors and via , but only up to the littlegroup rotations:^{2}^{2}2In the case that the Lorentz transformation is a pure rotation around the momentum axis by the angle , the littlegroup phases in eq. (6) are unambiguous and precisely equal to .
(6)  
These spinors also give us the building blocks for the polarization vectors of gauge bosons:
(7a)  
(7b) 
where can be any null vector such that and . Indeed, different reference vectors are equivalent up to a pure gauge, e.g.
(8) 
Now it is important to note that under a Lorentz transformation (6) the polarization vectors do not actually transform as proper vectors. For instance, comparing
(9) 
we conclude that Lorentz transformations act as
(10) 
only up to an additional term proportional to the new momentum . However, up to this caveat, this shows that these polarization vectors can be thought of as conversion coefficients between the offshell Lorentz transformations and the corresponding onshell littlegroup rotations ArkaniHamed:2017jhn (). A similar statement for the Weyl spinors is demonstrated by eq. (6) and is also true for the massive case, see eq. (13) below.
As a concrete realization of the Weyl spinors, one could use, for instance,
(11) 
for a null momentum expressible as . A more practical implementation is given in appendix A.
2.2 Massive Weyl spinors
For a nonzero mass , we have a nondegenerate matrix that satisfies . The Weyl spinors are then introduced ArkaniHamed:2017jhn () by expanding in terms of two explicitly degenerate matrices and :
(12)  
Here we have already indicated that the littlegroup indices are lowered and raised by the antisymmetric form , preserved by rotations. Such littlegroup transformations follow from the action of the Lorentz group on these spinors:
(13)  
where correspond to the rotations in the rest frame of the massive particle momentum. These transformations are a massive analogue of eq. (6). Furthermore, the momentum decomposition (12) implies the twodimensional version of the Dirac equation
(14) 
For further convenience, let us rewrite the above identities in the spinor braket notation:
(15) 
As an explicit spinor realization, one may use ArkaniHamed:2017jhn ()
(16) 
given a massive momentum expressible as , such that . A more detailed implementation is given in appendix B.
2.3 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 (12) naturally unify into the Dirac spinors:^{3}^{3}3We use the Dirac matrices in the Weyl basis, , hence and .
(17) 
This choice of is consistent with the conjugation property . The energy sign here is due to our convention
(18) 
which defines the signs of the energy and mass to be the same. As discussed in appendix B, this lets us cover the  and spinors together simultaneously by eq. (17).
We can treat these spinors as quantummechanical wavefunctions and compute the expectation values of the spin operator , where . Given the spinor parametrization (16), we obtain the threedimensional spin vector
(19) 
Therefore, the spinors (16) have definite helicities, i.e. the eigenvalues of the helicity operator , which is a conserved quantity for a oneparticle state.
To delve into the subject of spin a bit further, we rewrite the massive spinor parametrization (16) as
(20a)  
(20b) 
which makes obvious the smooth limit of the massive spinors and to their massless homonymes (11):
(21) 
where and . To rephrase this in a more general way, we can introduce twodimensional spinors and such that and decompose as
(22) 
The massive momentum is now expressed as a sum of two null momenta:
(23) 
which gives a link to the massive extension of the massless spinorhelicity formalism used previously in the literature Kleiss:1986qc (); Dittmaier:1998nn (); Schwinn:2005pi (). We make this link precise in section 5 below.
Now let us discuss a subtle point concerning spin. Traditional quantummechanical spin operators are thought of as acting on the indices, which correspond to the little group. The spin of the decomposition (16) points along the threemomentum , whereas the littlegroup vectors seem to describe states with spin direction along the axis. In other words, the massive Weyl spinors (16) convert the physical helicity operator to :
(24) 
This should be regarded as a nice feature of the parametrization (16) rather than an inconsistency. Indeed, the littlegroup transformations correspond to rotations in the rest frame of the massive particle, in which , whereas the spinorial matrices generate rotations in the boosted frame where . It is therefore convenient that the spinorial ,^{4}^{4}4In fact, the other two spatial directions corresponding to the littlegroup matrices and in the sense of eq. (24) turn out to be complex for any nonzero . The corresponding spinprojection operators are thus not hermitian, and there is no unitary intertwining operator between the two representations of the complete spin operator . Indeed, such an operator would have to involve a boost transformation to the rest frame, which lies outside the rotational . taken along the momentum direction, are converted to the simplest of the Pauli matrices, .
In principle, one can easily break the above property by rotating the spin states. Apart from losing the relatively simple parametrization (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 spinorhelicity formalism of ref. ArkaniHamed:2017jhn () 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.
3 Fourpoint amplitudes
In this section, we demonstrate the use of the various spinors discussed above by dissecting one full colordressed amplitude. It is convenient to consider the simple case of one massive quarkantiquark pair and two gluons of opposite helicity. Their scattering amplitude has three Feynman diagrams:^{5}^{5}5We normalize the group generators to obey and and regard all particle momenta as outgoing. We use slashed matrices to denote either , or , depending on the spinors surrounding them. In expressions like the slash can be omitted.
(25a)  
(25b)  
(25c) 
Here denotes the Dirac spinor corresponding to negative energy and mass . It is our energy sign convention (18) that allows us not to use a separate spinor for . Now let us recast the above numerators in the spinorhelicity formalism by plugging in the Dirac spinors (17) and the polarization vectors (7),
(26a)  
(26b)  
(26c)  
where for brevity we label spinors as , etc. We also underline the massive positiveenergy spinors and overline the negativeenergy ones. The numerators (26) may seem complicated, which is due to their explicit gauge dependence on the gluonic reference vectors and . Incidentally, one can check that for any such gauge choice they nontrivially satisfy the kinematicalgebra relation , which is colordual to the commutation relation Johansson:2014zca (); Johansson:2015oia (). A very beneficial gauge choice is and , for which
(27) 
We can thus write simple closedform expressions for all three colorordered amplitudes
(28a)  
(28b)  
(28c) 
These evidently obey the KleissKuijf relation Kleiss:1988ne (), as well as the BernCarrascoJohansson (BCJ) relation Bern:2008qj (); Johansson:2015oia (); delaCruz:2015dpa ()
(29) 
The full colordressed amplitude can thus be constructed from a single linearly independent colorordered amplitude as DelDuca:1999rs (); Johansson:2015oia ()
(30) 
It is interesting to note BjerrumBohr:2013bxa () that the gluonic colorordered amplitude (28c) is also the correct QED amplitude ArkaniHamed:2017jhn () (up to a factor of due to the colorgenerator conventions).
Note that the above amplitudes are gaugeinvariant and could have been reduced from the numerators (26) to the expressions (28) for any choice of reference vectors and . This illustrates why in general, at least in analytic calculations, it is better to avoid dealing with gaugedependent objects and compute gaugeinvariant quantities directly. Such a way to derive the above amplitudes would be via the BCFW onshell recursion Britto:2004ap (); Britto:2005fq () starting from the threepoint amplitudes