A Soft Pion Theorem

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I’ve been meaning to write a longer post about Arkani-Hamed et al’s paper on recursion relation and $𝒩 = 8$ 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.

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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, $X^{α}$ . The Lie algebra looks like

\begin{aligned} [T^{a}, T^{b}] & = f^{a}^{b}_{c} T^{c} \\ [T^{a}, X^{α}] & = f^{a}^{α}_{β} X^{β} \\ [X^{α}, X^{β}] & = f^{α}^{β}_{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,

⟨ π^{α_{1}} (p_{1}) \dots π^{α_{n}} (p_{n}) π^{β} (q) ⟩

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_{α}, q_{β} \to 0} ⟨ π^{γ_{1}} (p_{1}) \dots π^{γ_{n}} (p_{n}) π^{α} (q_{α}) π^{β} (q_{β}) ⟩ = \frac{1}{2} \sum_{j} \frac{p_{j} \cdot (q_{α} - q_{β})}{p_{j} \cdot (q_{α} + q_{β})} f^{α}^{β}_{a} ⟨ π^{γ_{1}} (p_{1}) \dots (T^{a} π)^{γ_{j}} (p_{j}) \dots π^{γ_{n}} (p_{n}) ⟩

More generally, I believe one can show that, for a nonlinear $σ$ -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_{α}, q_{β} \to 0} ⟨ π^{γ_{1}} (p_{1}) \dots π^{γ_{n}} (p_{n}) π^{α} (q_{α}) π^{β} (q_{β}) ⟩ = - \frac{1}{2} \sum_{j} \frac{p_{j} \cdot (q_{α} - q_{β})}{p_{j} \cdot (q_{α} + q_{β})} R^{γ_{j}}_{δ}^{α}^{β} ⟨ π^{γ_{1}} (p_{1}) \dots π^{δ} (p_{j}) \dots π^{γ_{n}} (p_{n}) ⟩

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 …

Update (9/24/2008):

After posting this, I got to wondering whether it could really be true that this result was unknown to the ancients. After all, this two-soft-pion theorem (or, the obvious generalization thereof) would be the dominant contribution in low-energy pion-nucleon scattering. So I Googled around some more, and found this paper from 1966, which seemed to have the desired result. Yesterday, I chatted with Steve Weinberg, who confirmed my suspicions. Indeed, he was quite proud of the paper in question, in that it confirmed the essentially nonabelian nature of pions (the real ones, in nature, as opposed to the abstract Goldstone bosons, discussed here), in contradistinction to Nambu who, at the time, held that the pions could be understood from an essentially abelian approach.

Posted by distler at September 9, 2008 6:08 PM

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September 9, 2008