Energy and momentum spectral function of coherent bremsstrahlung radiation

We calculate nonperturbatively and for small Q the cross section (Q) for a scattering process to occur with loss of four-momentum Q to unobserved photons, a quantity which may be measured by observing the net recoil to all other particles. The calculation proceeds by means of a threshold theorem which asserts that for small Q, (Q)0P(Q), where 0 is independent of Q, and P(Q) is the spectral of the coherent state of bremsstrahlung photons defined by a(k)=i(2)-32aua(ua)-1, where ak) is the annihilation operator for a photon of four-momentum k, and ea and ua are the charges and four-velocities of the scattered charged particles. Although is not in the Fock space, the evaluation of P(Q)=4(Q-Pop), where Pop is the operator of total electromagnetic four-momentum, is straightforward. The resulting function P(Q) simplifies if Q is near the light cone, where the bulk of the probability is in fact located, P(Q)(Q0)(Q2)[(1+B)]-1B(Q22)B-1I0 (Q)exp[F(Q)], where I0(Q)=-(2)-3[aeaua (uaQ)-1]2>0, B=dkI0(Q0=1, Qk), and F(Q) is given explicitly in the text, satisfies F(Q)=F(Q)-B ln, and is a smooth function as the light cone is approached. The spectral function exhibits two scaling laws, one governing the approach to the origin along a ray limQ)=0P(Q), the other governing the approach to the light cone at fixed energy Q0=E and angle k^ lim|Q|E[(E-|Q|)1-BP(E,Q]=const×[(1+B)]-1BEB-1I0(k)exp[F(k)], where k=(E,Ek). For e+-e- annihilation at 3 on 3 Gev, exp[F(k)]=E-Bexp[F(k)] produces a 30% angular modulation.