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The Infrared Triangle

Spin memory as one vertex of a profound triality connecting soft theorems, memory effects, and asymptotic symmetries

The Infrared Triangle

Spin memory is one vertex of a profound triality connecting soft theorems, asymptotic symmetries, and memory effects in gravitational scattering.

Subleading Soft Graviton Theorem

Weinberg 1965, Cachazo–Strominger 2014

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Spin Memory Effect

Pasterski–Strominger–Zhiboedov 2016

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BMS Super-rotation Symmetry

de Paoli–Speziale 2018

Subleading Soft Graviton Theorem

When a graviton with frequency $\omega$ is emitted in a scattering process, the amplitude factorises in the soft limit:

$$\lim_{\omega\to 0}\omega\,\mathcal{M}_{n+1}(\{p_i\};\,q) = \left[S^{(0)} + \omega\,S^{(1)} + \mathcal{O}(\omega^2)\right]\mathcal{M}_n(\{p_i\})$$

The subleading factor involves the angular momentum operator:

$$S^{(1)} = -\frac{i\kappa}{2}\sum_{k=1}^n \frac{\varepsilon^{\mu\nu} q_\mu\,J_{k,\nu\rho}\,p_k^\rho}{q\cdot p_k}, \qquad \kappa = \sqrt{32\pi G}$$

Ward Identity for Super-rotations

The super-rotation Ward identity constrains scattering amplitudes:

$$\langle\text{out}\,|\,\mathcal{Q}_Y^+\,\mathcal{S} - \mathcal{S}\,\mathcal{Q}_Y^-\,|\,\text{in}\rangle = 0$$

Inserting soft graviton states and taking $\omega\to 0$ reproduces exactly $S^{(1)}$. The spin memory effect is the classical limit of this quantum Ward identity.

Complete Triality

$$\underbrace{\Delta J(Y)\neq 0}_{\text{super-rotation imbalance}} \Longleftrightarrow \underbrace{\Delta N_A \neq 0}_{\text{spin memory}} \Longleftrightarrow \underbrace{S^{(1)}\neq 0}_{\text{subleading soft graviton}}$$

Note:

The same triangle exists at one order lower (displacement memory ↔ supertranslation ↔ leading soft graviton) and one order higher (centre-of-mass memory ↔ sub-subleading soft theorems).

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