Given particles whose interaction is described by a certain initial S matrix, by adding a soft graviton (i.e., whose energy is negligible compared to the energy of the other particles) that couples to one of the incoming or outgoing particles, the resulting S' matrix is, leaving off some kinematic factors,

${\displaystyle {\cal {S}}'={\sqrt {8\pi G}}{\frac {\eta p^{\mu }p^{\nu }\epsilon _{\mu \nu }}{p\cdot p_{G}-i\eta \varepsilon }}{\cal {S}}+O(p_{G}^{0})}$  ,^[1][5]

where p is the momentum of the particle interacting with the graviton, ϵ_μν is the graviton polarization, p_G is the momentum of the graviton, ε is an infinitesimal real quantity which helps to shape the integration contour, and the factor η is equal to 1 for outgoing particles and -1 for incoming particles.

The formula comes from a power series and the last term with the big O indicates that terms of higher order are not considered. Although the series differs depending on the spin of the particle coupling to the graviton, the lowest-order term shown above is the same for all spins.^[1]

In the case of multiple soft gravitons involved, the factor in front of S is the sum of the factors due to each individual graviton.

If a soft photon (whose energy is negligible compared to the energy of the other particles) is added instead of the graviton, the resulting matrix S' is

${\displaystyle {\cal {S}}'={\frac {\eta qp\cdot \epsilon }{p\cdot p_{\gamma }-i\eta \varepsilon }}{\cal {S}}+O(p_{\gamma }^{0})}$  ,^[1][5]

with the same parameters as before but with p_γ momentum of the photon, ϵ is its polarization, and q the charge of the particle coupled to the photon.

As for the graviton, in case of more photons, a sum over all the terms occurs.

Subleading order expansion edit

The expansion of the formula to the subleading term of the series for the graviton was calculated by Andrew Strominger and Freddy Cachazo:^[4]

${\displaystyle {\cal {S}}'={\sqrt {8\pi G}}{\frac {\eta p^{\mu }p^{\nu }\epsilon _{\mu \nu }}{p\cdot p_{G}-i\eta \varepsilon }}{\cal {S}}-i{\sqrt {8\pi G}}{\frac {\eta p^{\mu }({p_{G}}_{\rho }J^{\rho \nu })\epsilon _{\mu \nu }}{p\cdot p_{G}-i\eta \varepsilon }}{\cal {S}}+O(p_{G}^{1})}$ ,

where ${\displaystyle J^{\rho \nu }}$ represents the angular momentum of the particle interacting with the graviton.

This formula is gauge invariant under rotation and is connected to the gravitational spin memory effect.^[4]

• Pasterski–Strominger–Zhiboedov triangle