Spin-helicity formalism for gluon-gluon amplitudes

Question

In Schwarz's QFT he introduces in chapter 27 the Spin-Helicity formalism as a way of calculating gluon-gluon interactions much easier than going through all the Feynman calculus from the beginning to the end. It seems so amazing, but I am not sure I understand what is the fundamental difference between the 2 approaches that makes one a lot easier than the other. The main difference (on which spin-helicity formalism is actually based) is the fact that momentum is treated like a bi-spinor and not a vector. Why is this approach so much simpler? Can someone give me some intuition to it? Thank you!

Answer 1

The reason why calculating amplitudes using the Feynman calculus formalism is tedious is that they stem from a perturbative treatment that is formulated by upholding, above all, manifest Lorentz-invariance. While this is extremely useful for developing and formalising the theory (e.g. to detect anomalies easily), it tends to obstruct practical calculations - an early example of this is in a theory with $\text{U}(1)$ gauge symmetry: we introduce redundancy into the theory while embedding the spin-1 particle into a Lorentz-invariant object that has too many degrees of freedom, so we have to systematically kill off the additional degrees of freedom to preserve the manifest Lorentz-invariance.

The redundancy manifests itself in the Ward identities which the Feynman diagrams must obey - this introduces tons of terms in QCD interactions like $\sim A^2 \partial A$ and $\sim A^4$ that have to be dealt with, which, despite heavy cancellation, quickly render the calculation impractical.

To circumvent this, notice that the "natural" transformation for spin-1 fields is under the irreducible $\left(\frac12, \frac12\right)$ representation of $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ - so rather than working with the induced $(1 \oplus 0) \ \mathfrak{so}(3)$ four-vector representation that we are used to, we should work with the bispinors of the $\left(\frac12, \frac12\right)$ rep: the conversion between the two is $$p_{a\dot{b}}=p_\mu(\sigma^\mu)_{a\dot b} \\ (\sigma^\mu)_{a\dot b}=(1, \sigma^i)_{a\dot b} $$

This embedding avoids the redundancy of the four-vector description (although manifest Lorentz invariance may not present itself throughout these calculations, it is always lurking within). It also has the doubly nice property that the matrix associated with $p_{a\dot b}$ is rank-1 (since its determinant vanishes), so it can be factored as an outer product between a dotted and an undotted spinor. The fact that a spinor has fewer degrees of freedom than a vector facilitates many of the speed-ups that one gets for free while working with the SH-formalism.

Another benefit is that the little group transformations that leave $p_\mu$ invariant (e.g. $\text{ISO}(2)$ on a massless $(E, 0, 0, E)$) are represented linearly on the polarisation bispinors. There is still residual freedom, which allows us to pick an arbitrary "reference spinor" while building up the polarisation bispinor - through a clever choice of reference spinor, we can set swathes of terms to zero. Thus, in a sense, it channels the gauge degrees of freedom to a more useful outlet.

Historically, Parke and Taylor noticed that for Maximally Helicity Violating (MHV) pure gluon amplitudes, the final result could be expressed analytically, purely in terms of the momentum bilinears $\langle pq \rangle$. The full force of the spinor-helicity formula enables us to calculate analogous gauge-invariant amplitudes directly and essentially non-perturbatively - this is in contrast to the Feynman diagram approach which involves calculating a rather ugly Lorentz tensor perturbatively which this then contracted with the external polarizations.

Answer 2

The most practical traditional method of calculating scattering amplitudes comes from Feynman diagrams, where one draws all the relevant diagrams up to the desired order for a given theory, then using the set of Feynman rules relevant to the theory, one can assign a mathematical quantity to each diagram and hence sum everything nicely. The problem is that while manageable for small problems, the number of diagrams and intermediate calculations rapidly grows and quickly becomes too difficult to do by hand.

The spinor-helicity formalism is the starting point for studying the structure of amplitudes. By expressing the spinor brackets and using this formalism, we can uncover newer methods which expose the simplicity; cancellations are abundant and it is much easier to do with things, particularly teamed with recursion relations.

You mention that 'the spinor formalism uses bi-spinors rather than vectors'. Even doing Feynman diagram calculations uses polarisation vectors and Dirac spinors (in QCD,QED diagrams etc). It is naturally to keep track of quantum numbers such as spin or helicity (and colour in QCD), and it is much easier to deal with in SH-formalism.