Field-gradient effect in quantum beamstrahlung.

The correction terms to the Sokolov-Ternov radiation formula due to variation of the magnetic field strength along the electron trajectory are calculated up to the second order in the power expansion of &r/B, where r is the formation time of radiation. It is found that the field-gradient effect reduces radiation intensity in the classical regime, and enhances it in the quantum regime. This is then applied to quantum beamstrahlung with Gaussian variation in e+ebunch currents. The correction is shown to be substantial for beam parameters suggested by Himel and Siegrest. Submitted to Physical Review Letters *Work-supported by the Department of Energy, contract DE-AC03-76SF00515. For future e+elinear colliders, radiation induced by beam-beam collision is expected to be very strong. ’ This radiation, called beamstrahlung, would cause a substantial loss of energy and the degradation on energy resolution. Due to these concerns, the study of the subject has been intensive during recent years.2 In the calculations done so far, the field was typically treated as locally uniform. The following question arises: On what scale must the field be uniform for this treatment to be valid? Recently one of us (PC) initiated the investigation3 on the corrections to the uniform field treatment proportional to g2/B2 and B/B in the field variation. However, due to the deficiency of the mathematical techniques employed, the result was inconclusive. Here we present an improved calculation that gives a definitive evaluation of quantum beamstrahlung that includes the field-gradient effect. Our aim is to evaluate the average energy loss of the entire beam. To achieve this we want to derive the radiation intensity of one electron that sees the local field and its gradient arising from the oncoming positron beam. Our approach follows the spirit of Schwinger,4 and Baier and Katkov5 on quantum synchrotron radiation. It has been shown5 that, to the accuracy of l/r, the radiation intensity in the Coulomb gauge, I=+!$(+/ dtz eiW(t1-ta)M*(t2)M(tl) If) , can be approximated by I=,Jdfldtz/& [(l+u+g) (;i.~2-l)+~]eiuBr’-i1’ , (2) where Q! = l/137 is the fine structure constant, & and E’ the initial and final energies, (tl, r;) and (t2, r:) the initial and final coordinates, of the electron, respectively, (w, c) the four-momentumof the photon, u = w/E ’ = w/(&-w), r = f2 t1;c = (6 fi)/ 7, and fi = g/w. Let t = (tl + t2)/2, then $ dtldta = s drdt.