Abstract
It has been shown that in larger than four space-time dimensions, soft factors that relate the amplitudes with a soft photon or graviton to amplitudes without the soft particle also determine the low frequency radiative part of the electromagnetic and gravitational fields during classical scattering. In four dimensions the S-matrix becomes infrared divergent making the usual definition of the soft factor ambiguous beyond the leading order. However the radiative parts of the electromagnetic and gravitational fields provide an unambiguous definition of soft factor in the classical limit up to the usual gauge ambiguity. We show that the soft factor defined this way develops terms involving logarithm of the energy of the soft particle at the subleading order in the soft expansion.
Highlights
It has been shown that in larger than four space-time dimensions, soft factors that relate the amplitudes with a soft photon or graviton to amplitudes without the soft particle determine the low frequency radiative part of the electromagnetic and gravitational fields during classical scattering
The radiative parts of the electromagnetic and gravitational fields provide an unambiguous definition of soft factor in the classical limit up to the usual gauge ambiguity
We show that the soft factor defined this way develops terms involving logarithm of the energy of the soft particle at the subleading order in the soft expansion
Summary
We shall see that even the usual soft theorems — valid in dimensions larger than four — develop logarithmic factors when extrapolated to four space-time dimensions. For electromagnetic radiation the radiative part of the field satisfies the constraint equation kαAα = 0 This is reflected in the invariance of Sem(ε, k) under εμ → εμ + kμ. On the other hand the radiative part of the gravitational field satisfies the constraint kμeμν = 0, reflected in the invariance of Sgr(ε, k) under εμν → εμν + ξμkν + ξνkμ This allows us to determine e0μ in terms of the spatial components eij and we can focus on the spatial components eij. Note in particular that j0(aj) diverges as |t| → ∞, making the expressions ill-defined Ignoring this for the time being, for the particle kinematics described above we can express the soft factors given in (2.3) and (2.4) as q. In the following we shall verify this by explicit computation in several examples
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