Soft Graviton Theorem & Ward Identity
The Infrared Triangle
The three corners of the infrared triangle are equivalent statements connected by Fourier transform and mode expansion. Each row shows the leading and subleading levels:
Soft Theorem
Weinberg $S^{(0)}$
Asymptotic Symmetry
Supertranslations
Memory Effect
Displacement memory
Soft Theorem
Cachazo–Strominger $S^{(1)}$
Asymptotic Symmetry
Superrotations
Memory Effect
Spin memory
The leading row (supertranslation ↔ Weinberg ↔ displacement memory) was established by Strominger (2014). The subleading row (superrotation ↔ $S^{(1)}$ ↔ spin memory) is the focus of this page.
Weinberg Leading Soft Factor $S^{(0)}$
When a graviton with momentum $q$ becomes soft ($q\to 0$), the scattering amplitude factorises universally:
$$\mathcal{M}_{n+1}(q,p_1,\ldots,p_n) \;\overset{q\to 0}{\longrightarrow}\; S^{(0)}\,\mathcal{M}_n(p_1,\ldots,p_n)$$
with the leading soft graviton factor:
$$S^{(0)} = \kappa\sum_{k=1}^{n}\frac{p_k^{\mu}\,p_k^{\nu}\,\varepsilon_{\mu\nu}(q)}{p_k\cdot q}$$
where $\kappa = \sqrt{32\pi G}$ and $\varepsilon_{\mu\nu}$ is the graviton polarisation tensor. This is the gravitational analogue of the Weinberg soft photon theorem and is equivalent to the supertranslation Ward identity.
Subleading Soft Factor $S^{(1)}$
At subleading order in the soft expansion, Cachazo and Strominger showed:
$$\mathcal{M}_{n+1} \;\overset{q\to 0}{\longrightarrow}\; \left[\frac{1}{\omega}\,S^{(0)} + S^{(1)} + O(\omega)\right]\mathcal{M}_n$$
where $\omega = q^0$ is the soft graviton energy and the subleading factor involves the angular momentum operator $J_k^{\mu\nu}$:
$$S^{(1)} = -\frac{i\kappa}{2}\sum_{k=1}^{n}\frac{p_k^{\mu}\,\varepsilon_{\mu\nu}(q)\,q_{\lambda}\,J_k^{\nu\lambda}}{p_k\cdot q}$$
In stereographic coordinates $(z,\bar{z})$ on $S^2$, the graviton direction is parametrised as $q^{\mu} = \omega(1+z\bar{z},\, z+\bar{z},\, -i(z-\bar{z}),\, 1-z\bar{z})/2$, and the subleading soft factor acts as a differential operator on the celestial sphere, generating superrotations.
Ward Identity for Spin Memory
The superrotation Ward identity equates the action of the superrotation charge$\mathcal{Q}_Y$ on the $\mathcal{S}$-matrix to zero:
$$\langle\mathrm{out}|\,\mathcal{Q}_Y^+\,\mathcal{S} - \mathcal{S}\,\mathcal{Q}_Y^-\,|\mathrm{in}\rangle = 0$$
Here $\mathcal{Q}_Y^+$ and $\mathcal{Q}_Y^-$ are the superrotation charges at $\mathscr{I}^+$ and $\mathscr{I}^-$ respectively, related by the antipodal matching condition at spatial infinity $i^0$.
Reduction to $S^{(1)}$ via Mode Expansion
Splitting the superrotation charge into hard and soft parts,$\mathcal{Q}_Y = \mathcal{Q}_Y^{\mathrm{hard}} + \mathcal{Q}_Y^{\mathrm{soft}}$, the soft part creates/annihilates a soft graviton:
$$\mathcal{Q}_Y^{\mathrm{soft}} = \frac{1}{4\pi G}\lim_{\omega\to 0}\,\omega\int d^2 z\;\left[D_{\bar{z}}^3 Y^{\bar{z}}\,a_+(\omega\hat{x}) + D_z^3 Y^z\,a_-(\omega\hat{x})\right]$$
Inserting this into the Ward identity and using the LSZ reduction formula to relate the soft graviton insertion to the amplitude, one recovers precisely the subleading soft theorem:
$$\langle\mathrm{out}|\mathcal{Q}_Y^{\mathrm{soft},+}\mathcal{S}|\mathrm{in}\rangle = -\sum_{k}\left[\frac{(1+z_k\bar{z}_k)}{2}\frac{Y^{\bar{z}}(\bar{z}_k)}{\bar{z}-\bar{z}_k}\right]\langle\mathrm{out}|\mathcal{S}|\mathrm{in}\rangle \;\longleftrightarrow\; S^{(1)}$$
This completes the subleading infrared triangle: the superrotation Ward identity is equivalent to the Cachazo–Strominger subleading soft graviton theorem, and the corresponding memory effect is the spin memory $\Delta\Psi$ derived on the previous page.