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In physics, the soft graviton theorem, first formulated by Steven Weinberg in 1965, allows calculation of the S-matrix, used in calculating the outcome of collisions between particles, when low-energy (soft) gravitons come into play.

Specifically, if in a collision between n incoming particles from which m outgoing particles arise, the outcome of the collision depends on a certain S matrix, by adding one or more gravitons to the n + m particles, the resulting S matrix (let it be S ') differs from the initial S only by a factor that does not depend in any way, except for the momentum, on the type of particles to which the gravitons couple.

The theorem also holds by putting photons in place of gravitons, thus obtaining a corresponding soft photon theorem.

The theorem is used in the context of attempts to formulate a theory of quantum gravity in the form of a perturbative quantum theory, that is, as an approximation of a possible, as yet unknown, exact theory of quantum gravity.

Formulation
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,

$${\cal S}' = \sqrt{8\pi G} \frac{\eta p^\mu p^\nu \epsilon_{\mu\nu}}{p \cdot p_G - i \eta \varepsilon}{\cal S} + O(p_G^0)$$ ,

where p is the momentum of the particle interacting with the graviton, ϵμν is the graviton polarization, pG 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.

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

$${\cal S}' = \frac{\eta q p \cdot \epsilon}{p \cdot p_\gamma - i \eta \varepsilon} {\cal S} + O(p_\gamma^0)$$ ,

with the same parameters as before but with pγ 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.

Demonstration
The theorem derives from a power series development of the propagator of the photon or graviton added to each line outside the primary and unknown interaction.

Consider the case of a graviton exiting an external (line) leg (outside the interaction area), as in the figure, of momentum pG. Exact calculation of the amplitude would require knowledge of the complete theory, i.e., quantum gravity, but at low energies a Laurent series development can be used, starting from the pole of the function relative to that momentum, considering only the first term of the development. According to the LSZ rules for calculating scattering amplitudes one can use the relevant Green's functions in time ordering by amputating (thus ignoring) the outer legs.

This in practice implies that the calculations proceed by considering only the terms related to the vertex and the propagator (according to the Feynman diagrams technique).

To derive this formula, let us take any scattering process with n incoming and

m outgoing particles and then consider adding to it one outgoing photon, denoted

by a wavy line in figure, with momentum PG. (The derivation for an incoming

photon is similar.) In the soft limit, we can write the amplitude as a sum of two

types of terms, ones in which the soft photon attaches to an external line and others

in which the soft photon attaches to an internal line. The soft photon can attach to

any one of the n + m external lines, so we must include a sum over all such terms.

The full amplitude has a Laurent expansion in q with an infinite number of terms

whose detailed form depends on what theory we are talking about. For the pole we

need not specify what theory we are studying except that it has a photon. That is

one of the beauties of this formula.

The LSZ rule for computing scattering amplitudes starts out by computing

the time-ordered Green’s functions using the Feynman iε prescription and then

amputating the external legs. The Feynman diagrams have factors for vertices and

propagators. What happens when we attach the extra photon to an external leg is,

since external legs are amputated, we need only add a vertex and propagator for

the particle to whose external leg the photon is added. The difference between the

diagram with and without the attached external soft photon is just the vertex and

propagator.

Andrew Strominger - Lectures on the Infrared Structure of Gravity and Gauge Theory, p. 35

re-formulates scattering amplitudes of a set of finite energy external particles with one or more low energy external gravitons, in terms of the amplitude without the low energy gravitons.

In the classical limit, there is a different manifestation of the same theorem : here it determines the low frequency component of the gravitational wave-form produced during a scattering process in terms of the momenta and spin of the incoming and outgoing objects, without any reference to the interactions responsible for the scattering.

https://dash.harvard.edu/bitstream/handle/1/29374083/1401.7026.pdf;jsessionid=6392FB47A36DFFDF342EC0BC22893C9E?sequence=1