### A Soft Pion Theorem

I’ve been meaning to write a longer post about Arkani-Hamed *et al*’s paper on recursion relation and $\mathcal{N}=8$ supergravity. I keep getting distracted by other things, and besides, there’s a beautiful result (of much broader interest) buried there, that deserves to be highlighted. It’s a “soft pion theorem” that we all should have learned about in grad school, but didn’t.

Consider a theory with global symmetry, $G$, spontaneously broken to a subgroup, $H$. There are Goldstone bosons, “pions”, parametrizing the symmetric space, $G/H$. Let’s call the generators of $H$, $T^a$, and the broken generator, $X^\alpha$. The Lie algebra looks like
$\begin{aligned}
[T^a,T^b]&= \tensor{f}{^a^b_c} T^c\\
[T^a,X^\alpha]&= \tensor{f}{^a^\alpha_\beta}X^\beta\\
[X^\alpha,X^\beta]&=\tensor{f}{^\alpha^\beta_a}T^a
\end{aligned}$
It’s well-known that the Goldstone bosons decouple at zero momentum. For fixed $p_i$, the $(n+1)$-point function,
$\langle\pi^{\alpha_1}(p_1)\dots \pi^{\alpha_n}(p_n)\pi^\beta(q)\rangle$
vanishes in the limit $q\to 0$. The surprising, and beautiful, fact is that taking *two* pions to zero momentum gives a nontrivial result. The trick is to take the zero momentum limit with sufficient care.
$\lim_{q_\alpha,q_\beta\to 0}\langle\pi^{\gamma_1}(p_1)\dots \pi^{\gamma_n}(p_n)\pi^\alpha(q_\alpha)\pi^\beta(q_\beta)\rangle= \tfrac{1}{2}\sum_j \frac{p_j\cdot(q_\alpha-q_\beta)}{p_j\cdot(q_\alpha+q_\beta)}\tensor{f}{^\alpha^\beta_a}\langle\pi^{\gamma_1}(p_1)\dots(T^a\pi)^{\gamma_j}(p_j) \dots \pi^{\gamma_n}(p_n)\rangle$

More generally, I believe one can show that, for a nonlinear $\sigma$-model, with target space $M$, this (slightly delicate) soft limit of two-pion insertions measure the Riemann curvature of $M$ (at the point $p\in M$, about which our vacuum is based). $\lim_{q_\alpha,q_\beta\to 0}\langle\pi^{\gamma_1}(p_1)\dots \pi^{\gamma_n}(p_n)\pi^\alpha(q_\alpha)\pi^\beta(q_\beta)\rangle= -\tfrac{1}{2}\sum_j \frac{p_j\cdot(q_\alpha-q_\beta)}{p_j\cdot(q_\alpha+q_\beta)}\tensor{R}{^{\gamma_j}_\delta_^\alpha^\beta}\langle\pi^{\gamma_1}(p_1)\dots\pi^{\delta}(p_j) \dots \pi^{\gamma_n}(p_n)\rangle$ where we’ve chosen Riemann normal coordinates to parametrize the fluctuations about the chosen vacuum.

This is reminiscent of the formula for the second-order marginal deformation of a 2D CFT giving the Riemann curvature of the moduli space of the family of CFTs. In any case, I’m really surprised that this cute result was not known to the ancients …