Spinor-Helicity Formalism

Prerequisites: Spinors, Feynman Diagrams. Full path (28 pages)

Why is it interesting?
Intuitive
Concrete
Abstract

Why is it interesting?

“A method is more important than a discovery, since the right method will lead to new and even more important discoveries.”

Lev Landau

the standard formalism of field theory, the amplitudes are computed using Feynman diagrams, which give us “Feynman amplitudes” that are not the real amplitudes, but are instead Lorentz tensors. We contract them with polarization vectors to get the actual amplitudes–the polarization vectors are supposed to transform as “bi-fundamentals” under the Lorentz and little groups. For massive particles of any spin, there is a canonical way of associating polarization vectors with given spin states. But this is impossible for massless particles. […] In order to guarantee that the Lorentz tensors $M_{µ_1···µn}$ arising from Feynman diagrams from a Lagrangian satisfies this on-shell Ward-identity, the Lagrangian must be carefully chosen to have an (often non-linearly completed) gauge invariance, which is then gauge-fixed. From the modern point of view, then, gauge symmetry is merely a useful redundancy for describing the physics of interacting massless particle of spin 1 or 2, tied to the specific formalism of Feynman diagrams, that makes locality and unitarity as manifest as possible.

But over the past few decades, we have seen entirely different formalisms for computing scattering amplitudes not tied to this formalism, and here gauge redundancy makes no appearance whatsoever. Instead of polarization vectors that only redundantly describe massless particle states, we can use spinor-helicity variables $\lambda_a$ , $\tilde \lambda_a$ for the a'th particle, with momentum $p_a^{\alpha\dot \alpha} = \lambda_a^\alpha \tilde \lambda^{\dot \alpha}$ . The $λ, \tilde \lambda$ 's do transform cleanly as bi-fundamentals under the Lorentz and little groups; under a Lorentz transformation $Λ$ that maps $(Λp) = p$ , we have $λ → tλ$ , $\tilde \lambda→ t^{-1} \tilde \lambda$ . Thus while the description of amplitude using polarization vectors is gauge-redundant, the amplitude is directly a function of spinor-helicity variables, with the helicities encoded in behavior under rescaling $M(t_aλ_a, t^{-1}_a \tilde \lambda_a) = t^{-2h_a} M(λa, \tilde a)$ .

With this invariant description of the fundamental symmetries and kinematics of amplitudes at hand, it becomes possible to pursue entirely new strategies for determining the amplitudes. In a first stage, one can speak of a modern incarnation of the S-matrix program, where the fundamental physics of locality and unitarity are imposed to determine the amplitudes from first principles. This has allowed the computation of amplitudes in an enormous range of theories, from Yang-Mills and gravity to goldstone bosons, revealing stunning simplicity and deep new mathematical structures that are completely hidden in the usual, gauge-redundant Feynman diagram formalism.

https://arxiv.org/pdf/1612.02797.pdf

For more on the fact that polarization vectors are not lorentz vectors, have a look at Vol. 1 of Weinberg's Quantum Field Theory book, section 5.9. A summary of Weinberg's argument with an easier notation can be found in this article.

Intuitive

Massless particles such as photons and gluons have exactly two polarization states, labelled by their helicity, the spin component along the direction of motion. The standard Feynman rules describe them with four-component polarization vectors. These vectors carry more information than the two physical states need, and the extra pieces depend on an arbitrary gauge choice. In a calculation with many gluons, the individual Feynman diagrams are therefore large expressions full of gauge-dependent terms, and most of these terms cancel in the sum. The number of diagrams also grows very fast with the number of particles.

The spinor-helicity formalism uses better variables. The momentum of a massless particle can be written as a product of two two-component spinors, roughly as if we took a square root of the momentum. These spinors carry the momentum and, through the way they transform when we rotate the particle's spin frame, its helicity. Every physical quantity is then built from two kinds of simple products between spinors of different particles, called angle brackets and square brackets. Their sizes are fixed by the invariant energies of pairs of particles, and their phases keep track of helicity.

In these variables many amplitudes that take hundreds of Feynman diagrams collapse into a single line. The best known example is the Parke-Taylor formula for gluon scattering with two negative-helicity gluons and any number of positive-helicity ones. Results of this kind showed that scattering amplitudes have much more structure than individual Feynman diagrams suggest, and they were an early motivation for the modern program of computing amplitudes directly from symmetry, locality and unitarity.

Concrete

For a nice introduction to the spinor helicity formalism, see section 3.1 in Amplitudes for Astrophysicists I: Known Knowns by Daniel J. Burger et. al.

Other nice "low-level" discussion can be found here and here.

See also A brief introduction to modern amplitude methods by Lance J. Dixon

and An Experimenter’s Guide to the Helicity Formalism by Jeffrey D. Richman

Abstract

The formalism rests on the two-to-one homomorphism $SL(2,\mathbb{C}) \to SO^+(1,3)$ (a double cover) between the spin group and the proper orthochronous Lorentz group. A four-momentum is mapped to a $2\times 2$ matrix

$p_{\alpha\dot\alpha} = p_\mu \sigma^\mu_{\alpha\dot\alpha}, \qquad \sigma^\mu = (1, \vec\sigma), \qquad \det p_{\alpha\dot\alpha} = p_\mu p^\mu$

in signature $(+,-,-,-)$ . The Lorentz group acts as $p \mapsto M p M^\dagger$ with $M\in SL(2,\mathbb{C})$ , so undotted indices carry the representation $(\frac12,0)$ and dotted indices $(0,\frac12)$ . For a massless momentum $\det p = 0$ , the matrix has rank one and factorizes,

$p_{\alpha\dot\alpha} = \lambda_\alpha \tilde\lambda_{\dot\alpha}.$

For real momenta with $p^0 > 0$ one can choose $\tilde\lambda = \bar\lambda$ ; for complex momenta $\lambda$ and $\tilde\lambda$ are independent. The factorization is not unique: $\lambda \to t\lambda$ , $\tilde\lambda \to t^{-1}\tilde\lambda$ leaves $p$ invariant. For real momenta $t = e^{i\theta/2}$ is a phase, and this rescaling is the action of the little group $U(1)$ (or $\mathbb{C}^*$ for complex momenta).

Lorentz invariants are built with the antisymmetric tensors $\epsilon^{\alpha\beta}$ and $\epsilon^{\dot\alpha\dot\beta}$ :

$\langle ij\rangle = \epsilon^{\alpha\beta}\lambda_{i\,\alpha}\lambda_{j\,\beta}, \qquad [ij] = \epsilon^{\dot\alpha\dot\beta}\tilde\lambda_{i\,\dot\alpha}\tilde\lambda_{j\,\dot\beta},$

both antisymmetric in $i,j$ , with $\langle ij\rangle[ij] = \pm 2p_i\cdot p_j$ , the sign depending on the convention for raising and lowering spinor indices. By the first fundamental theorem of invariant theory for $SL(2,\mathbb{C})$ , every Lorentz invariant function of the spinors is a function of these brackets.

Little group scaling. An amplitude for massless particles with helicities $h_i$ obeys

$A\big(\{t_i\lambda_i,\, t_i^{-1}\tilde\lambda_i\}\big) = \prod_i t_i^{-2h_i}\, A\big(\{\lambda_i, \tilde\lambda_i\}\big).$

This fixes all three-particle amplitudes. For three massless momenta with $\sum p_i = 0$ and $p_i^2 = 0$ one has $p_i\cdot p_j = 0$ , so either all $\lambda_i$ are proportional or all $\tilde\lambda_i$ are. Real momenta in Minkowski signature force both, and the three-point amplitude vanishes; for complex momenta it does not. If the $\tilde\lambda_i$ are proportional, the amplitude depends only on angle brackets, and little group scaling gives

$A_3 = g\, \langle 12\rangle^{h_3 - h_1 - h_2}\, \langle 23\rangle^{h_1 - h_2 - h_3}\, \langle 31\rangle^{h_2 - h_3 - h_1},$

applicable when $h_1 + h_2 + h_3 < 0$ (the case $>0$ uses square brackets with all exponents reversed in sign). For three gluons with helicities $(-,-,+)$ this is $\langle 12\rangle^3/(\langle 23\rangle\langle 31\rangle)$ , the color-ordered Yang-Mills vertex.

MHV amplitudes. The color-ordered tree amplitude of $n$ gluons with negative helicity on $i$ and $j$ and positive helicity on all others is, up to a coupling and normalisation factor,

$A_n(1^+ \ldots i^- \ldots j^- \ldots n^+) = \frac{\langle ij\rangle^4}{\langle 12\rangle\langle 23\rangle\cdots\langle n1\rangle}.$

One can check the little group weights directly: $\lambda_i$ appears four times in the numerator and twice in the denominator, giving $t_i^{2} = t_i^{-2h_i}$ for $h_i = -1$ .

The spinor variables also give the link to twistor theory. Fourier transforming each $\tilde\lambda$ into a conjugate variable turns the pair $(\lambda, \tilde\lambda)$ into a point of twistor space, and in these variables tree-level gluon amplitudes are supported on algebraic curves (Witten, 2003). This line of work led to the BCFW recursion relations, the Grassmannian formulation and the amplituhedron for planar $\mathcal{N}=4$ super Yang-Mills theory.

Spinor-Helicity Formalism

Why is it interesting?

Intuitive

Concrete

Abstract

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