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 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.
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 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 on-shell amplitude methods being restricted to the massless case. Recently, however, Arkani-Hamed, Huang and Huang Arkani-Hamed:2017jhn () 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 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 four-particle 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 Britto:2004ap (); Britto:2005fq (). The spins of the quark and the antiquark remain unfixed throughout the calculations, which lets us specialize to the specific quark-spin projections considered previously Schwinn:2007ee () in the massless-spinor-based 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 little-group symmetry.
2 Spinor-helicity review
It is well-known 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 one-particle 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
The Pauli matrices111We use , , , , as well as and . and here translate Lorentz transformations between the spinorial and vectorial languages:
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:
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 Berends:1981rb (); DeCausmaecker:1981bg (); Gunion:1985vca (); Kleiss:1985yh (); Xu:1986xb (); Gastmans:1990xh (); Arkani-Hamed: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 on-shell 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 bra-ket notation:
The Lorentz transformations (2) act on the Weyl spinors and via , but only up to the little-group rotations:222In the case that the Lorentz transformation is a pure rotation around the momentum axis by the angle , the little-group phases in eq. (6) are unambiguous and precisely equal to .
These spinors also give us the building blocks for the polarization vectors of gauge bosons:
where can be any null vector such that and . Indeed, different reference vectors are equivalent up to a pure gauge, e.g.
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
we conclude that Lorentz transformations act as
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 off-shell Lorentz transformations and the corresponding on-shell little-group rotations Arkani-Hamed: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,
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 non-degenerate matrix that satisfies . The Weyl spinors are then introduced Arkani-Hamed:2017jhn () by expanding in terms of two explicitly degenerate matrices and :
Here we have already indicated that the little-group indices are lowered and raised by the antisymmetric form , preserved by rotations. Such little-group transformations follow from the action of the Lorentz group on these spinors:
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 two-dimensional version of the Dirac equation
For further convenience, let us rewrite the above identities in the spinor bra-ket notation:
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:333We use the Dirac matrices in the Weyl basis, , hence and .
This choice of is consistent with the conjugation property . The energy sign here is due to our convention
We can treat these spinors as quantum-mechanical wavefunctions and compute the expectation values of the spin operator , where . Given the spinor parametrization (16), we obtain the three-dimensional spin vector
Therefore, the spinors (16) have definite helicities, i.e. the eigenvalues of the helicity operator , 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 (16) as
which makes obvious the smooth limit of the massive spinors and to their massless homonymes (11):
where and . To rephrase this in a more general way, we can introduce two-dimensional spinors and such that and 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 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 quantum-mechanical 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 three-momentum , whereas the little-group 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 :
This should be regarded as a nice feature of the parametrization (16) rather than an inconsistency. Indeed, the little-group 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 ,444In fact, the other two spatial directions corresponding to the little-group matrices and in the sense of eq. (24) turn out to be complex for any nonzero . The corresponding spin-projection 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 spinor-helicity formalism of ref. Arkani-Hamed: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 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:555We 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.
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 spinor-helicity formalism by plugging in the Dirac spinors (17) and the polarization vectors (7),
where for brevity we label spinors as , etc. We also underline the massive positive-energy spinors and overline the negative-energy 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 kinematic-algebra relation , which is color-dual to the commutation relation Johansson:2014zca (); Johansson:2015oia (). A very beneficial gauge choice is and , for which
We can thus write simple closed-form expressions for all three color-ordered amplitudes
It is interesting to note Bjerrum-Bohr:2013bxa () that the gluonic color-ordered amplitude (28c) is also the correct QED amplitude Arkani-Hamed:2017jhn () (up to a factor of due to the color-generator conventions).
Note that the above amplitudes are gauge-invariant 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 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 Britto:2004ap (); Britto:2005fq () starting from the three-point amplitudes