Spin-$J$ Amplitude $A_J(s,t) = - \frac{g^2(-s)^J}{t-M^2}$?

Question

In GSW equation (1.1.2) they define the scattering amplitude for a spin $J$ particle at high energies as $$A_J(s,t) = - \frac{g^2(-s)^J}{t-M^2}$$ mentioning it is an asymptotic approximation to a formula involving Legendre polynomials, and in this stackexchange post it is also defined, and in Williams 'Intro to Elementary Particles' book it is referred to as (proportional to) the Born contribution to the amplitude (section 12.9). It is also discussed from a string perspective in Tong's string theory notes eq. 6.13. Are there any references for where this formula is derived/explained/discussed in a QFT context, where it comes from, for example the general formula in terms of Legendre polynomials and how it reduces to the above form?

Answer 1

The intuitive proof is straightforward. The group $SU(2)$ is simple, and so any representation is contained in the tensor product of sufficiently many copies of the fundamental (à la Young). More specifically, a field of spin $j$ can be constructed by tensoring $2j$ copies of a spinor: $$ \phi^{\alpha_1\cdots\alpha_{2j}}\sim\psi^{\alpha_1}\otimes\cdots\otimes\psi^{\alpha_{2j}} $$ in the sense that the representation $j$ is a subspace of the representation $(1/2)^{\otimes 2j}$: $$ \bigotimes^{2j} \frac12=j\oplus (j-1)\oplus\cdots $$ from the standard rules of addition of angular momenta¹. Thus, the propagator reads² $$ \langle \phi^{\alpha_1\cdots\alpha_{2j}}\phi^{\alpha'_1\cdots\alpha'_{2j}}\rangle(p)=\frac{p^{\alpha_1\alpha'_1}\cdots p^{\alpha_{2j}}p^{\alpha'_{2j}}}{p^2-m^2}+\text{permutations}+\text{subleading} $$

Here, the pole is fixed by unitarity (as in the standard proof of Källén-Lehman: this is the one-particle state contribution). The numerator is fixed by Lorentz invariance (here $p^{\alpha\alpha'}:=p^\mu\sigma^{\alpha\alpha'}_\mu$ is the standard map $SO(3)\to Spin(3)$). The "permutations" refers to the different ways to pair up the indices (which, in turns, is fixed by the symmetry properties of $\phi$, i.e., on the specific representation we are using), and "subleading" refers to terms where the Lorentz indices are provided by $\delta^{\alpha\alpha'}$ instead of momenta.

We thus see that $$ \Delta\sim \frac{p^{2j}}{p^2-m}=\frac{s^j}{s-m} $$ modulo angle-dependent factors.

A more involved proof requires a careful analysis of the representation theory of $SU(2)$ and its invariant tensors (which specify the precise pairing of indices). One of the canonical references is Weinberg's Phys. Rev. 133, B1318, 1964 and the follow up papers Phys. Rev. 134, B882, 1964, Phys. Rev. 181, 1893. The upshot is that the rough estimate above is in fact correct.

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^{1: This refers to reps of $SU(2)$; a rep of Lorentz $SU(2)^2$ is obtained by taking pairs of reps, which leads to lower and upper indices. This is irrelevant for the discussion herein.}

^{2: I do not pay attention to the overall sign here; it is fixed by unitarity: the residue is essentially a norm $|\langle 0|\phi|p\rangle|^2$, and so should be positive for $p$ on-shell. Again, refer to Källén-Lehman for a more detailed discussion.}