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Nima et al.: making the amplitude minirevolution massive

Nima Arkani-Hamed (Princeton), Tzu-Chen Huang (Caltech), and Yu-tin Huang (Taiwan) released their new 79-page-long paper

Scattering Amplitudes For All Masses and Spins
a few days ago. They claim to do something that may be considered remarkable: to generalize the spinor-indices-based uprising in the scattering amplitude industry of the previous 15 years to the case of particles of any mass and spin, and to deduce some properties of all possible particle theories out of their new formalism.

Is it possible? Does it work? What can they learn?

First, they remain restricted to the case of on-shell, i.e. scattering amplitudes, not general off-shell, i.e. Green's functions. They have a cute self-motivating semi-heuristic argument why they don't lose any generality by this constraint: the actual off-shell amplitudes are being experimentally measured by the analysis of some on-shell scattering that involves the particles as well as some new very heavy particles, namely the detectors and other apparatuses.

Nice. I guess that the numbers showed on the apparatuses' displays must be considered as labeling different particle species, not just polarizations of spin. If your Geiger-Müller counter shows "5" at the beginning and measures something and shows "6" at the end, it was a scattering in which the "Geiger-Müller-counter-type-5 particle species" collided with some small particles, got annihilated, and produced a similar big "*-6 counter" particle. Cute. ;-)




Second, the massless spinor-indices-based minirevolution depended on the possibility to write a massless momentum vector as\[

p^\mu = \sigma^{\mu}_{\alpha \dot\alpha} \lambda^\alpha \tilde \lambda^{\dot \alpha}.

\] Because the \(2\times 2\) matrix is written as a tensor product of two vectors, its rank is just one and the determinant is therefore zero. But the determinant is \(p^\mu p_\mu\) by some elementary calculus so the particle has to be massless.




How do you write a more general momentum which is timelike? Well, one \(\lambda \tilde \lambda\) isn't enough but the sum of two such products is enough: a time-like vector may be written as a sum of two light-like ones. This generalization has been considered by numerous people I have met – but they never got too far. Arkani-Hamed, Huang, and Huang are mature and hard-working researchers in this area, however, so they didn't just expect all the massive generalizations of the rules to be straightforward and fall into their laps. They were intensely thinking and deriving and they did derive some things.

So they write the amplitudes in the spinor variables. A particle with spin \(S\) adds \(2S\) spinor indices to the scattering amplitude, a symmetrization may be assumed in the irreducible case. There is some ambiguity in the choice of the spinors \(\lambda^I_\alpha\) and \(\lambda^I_{\dot \alpha}\) for a particle, \(I=1,2\) represents the index needed because we sum two products of spinors. The time-like momentum (a non-singular, rank-two matrix with two 2-valued spinor indices) may be used to convert dotted and undotted indices for a single particle to each other.

OK, so their starting point is to imagine that all the scattering amplitudes in a theory with particles of general spins and masses are being rewritten in terms of various rational functions of products of the spinor-index-based variables. The Lorentz invariance constrains what is allowed. Also, the amplitudes may have various singularities and Nima and his collaborators have gained an incredible amount of experience in figuring out which singularities may be present in the amplitudes, which can't, and so on.

In this massive case, they were just going to rederive similar conclusions about some more general Ansätze which include a larger number of spinors and/or indices. At the end, they seem confident that the switch to the massive case doesn't cripple the key methods that worked in the massless case. Massive particles don't need the gauge invariance for consistency because the little group is \(SU(2)\) and only relates "positive-norm" polarizations.

But some of the constraints become more intense if you switch from the massless realm to the massive one.

Using their new formalism, they review some of the general theorems and wisdoms about the allowed spins and masses of particles. A coupling of three spin-1 particle is impossible. Yang's theorem: a massive spin-1 particle cannot decay to two photons: derived using a new formalism. With gravity (spin-two massless fields), massless higher-spin particles are impossible. The Weinberg-Witten theorem, rederived.

More cutely and beyond quantum field theory, they ask whether a massive particle of spin exceeding two may be elementary. The adjective "elementary" means that it can exist in isolation – so its mass may be parameterically separated from the masses of all other particle species. The answer turns out to be No. They get it by studying the \(E\to \infty\) or, equivalently, \(m\to 0\) limit of the scattering amplitude involving such a hypothetical elementary particle. Some singularity analogous to the massless particles' singularities materializes in the scattering amplitude and that singularity may be used to argue that an additional particle species whose mass is comparable to the same \(m\) has to exist. The original one couldn't exist in isolation.

Weakly-coupled string theory is compatible with this conclusion, of course, because it gives you a whole infinite tower of massive particles. You can't cherry-pick them one by one. You either have to accept the whole package that string theory gives you, or you have to die (or at least shut up). One may see that Arkani-Hamed and Huang squared gave a new derivation of the qualitative property of string theory. With some optimism, one could argue that by adding a few more derivations like that, one could derive that "all string theory is mathematically forced upon us" by pure mathematical thought – an analysis of the scattering amplitudes written in their spinor-based formalism.

The massless twistor minirevolution already had lots of unusual expressions, lots of indices, geometric shapes of large dimensions in spaces whose dimensions were even higher – equal to products of some numbers. It was already complicated. To play the game started by this paper, you need to swallow an additional collection of indices – \(I=1,2\) for each particle that indicate the two products of spinors you need to describe the momentum, and the corresponding increase of possible forms of the amplitudes. They make it look less terrifying than it is by using bold fonts for massive variables and suppressing some little group indices.

You shouldn't expect this industry to technically simplify, at least not in a foreseeable future. But its applicability is almost certainly being expanded and as they (and I) mentioned, there are signs that they can derive some statements that go beyond the list of known theorems in quantum field theory, that really go beyond the usual quantum field theory thinking itself. Similar methods are really applicable to the S-matrix of field theories with "infinitely many species" and string theory is a representative. The stringy S-matrix is complicated, involves the ratios of gamma-function-like entities, and it must display lots of special features relatively to "random functions of a similar type" that follow from the consistency of string theory.

Their formalism may be a tool to make some or all of these features "comprehensible" and accessible to a straightforward, albeit in no way technically easy, analysis.

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