The Stimulated Breit-Wheeler Process as a source of Background e + e - Pairs at the ILC Dr Anthony Hartin JAI, Oxford University Physics, Denys Wilkinson Building, Keble Road Oxford, UK Introduction Passage of beamstrahlung photons through the bunch fields at the interaction point of the ILC determines background pair production. Analysis of the pair backgrounds is a crucial step in the R&D of the ILC. The bunch fields are intense and a potential non-linearity is added to the pair production processes. The Breit-Wheeler, Landau-Lifshitz and Bethe-Heitler pair production processes have been well considered. ILC accelerator parameters have been chosen to avoid non-linearity in 1 st order background processes. However the second order Breit-Wheeler process in the presence of the bunch fields - or Stimulated Breit- Wheeler process - has not been thoroughly considered. This 2 nd order nonlinear pair production belongs to a class of processes which are known to contain resonances. The 2 nd order cross-sections potentially exceed the first order ones and a significantly different pair background may result. A thorough QED calculation is required to settle the matter. Electron Self Energy in the bunch field Since the Stimulated Breit-Wheeler virtual particle reaches the mass shell, radiative corrections need to be made in order to render the resonance finite. A good first estimate of the resonance width is given by the one loop Electron Self Energy in the bunch field. The structure of the Self Energy reveals that the required regularisation and renormalisation procedure is the same as that of the non-external field case. Propagators are transformed using a method due to Källen. Imaginary and real parts of the Self Energy are separated by using dispersion relations. Källen’s method allows infrared divergences to be first avoided in a small shift of integration lower bounds then removed by transforming the integration variables. The imaginary part of the virtual particle energy shift can be found in terms of the number r of external field photons taking part and a total integrated over all r The QED calculation and results The matrix element of the Stimulated Breit-Wheeler process accounts for the bunch field by using fermion field operators which are solutions of the Dirac equation in the presence of the field. The full QED calculation of the cross-section is reasonably straightforward, though there are ~100,000 terms and short cuts are useful! The beamstrahlung photons are generally emitted longitudinally and a number of simplifications can be made: Angles θ f and φ f which define the direction of the created electron have to be Lorentz transformed back to the centre of mass frame in the absence of the external field. Due to symmetry, the integration over φ f yields 2π and the remaining integration is over a smooth function. The Breit-Wheeler differential cross section peaks are enhanced by several orders of magnitude! Bunch fields The electromagnetic field associated with ultrarelativistic charge bunches is a constant, crossed field of 4- momentum k, transverse to the direction of motion z and described by the 4-potential A μ (k.z) For the default ILC bunch parameter set, the external field intensity and photon energy are The interaction of synchrotron (beamstrahlung) photon with the bunch field is either an azimuthally symmetric one described by Airy functions, or a non-azimuthally symmetric one described by hybrid Bessel/Airy functions Virtual particle in the bunch field The virtual electron exchanged in the Breit-Wheeler process remains in existence for a time no greater than that given by the Uncertainty Relation. Its potentially minimum energy and maximum lifetime is at the mass shell and can be compared with the persistence of the bunch field The virtual electron is definitely influenced by the bunch field which effectively contributes r bunch field photon momenta to the virtual electron momentum. The propagator denominator has a dependency on the beamstrahlung photon energy ω 1,bunch field photon energy ω and angle between the produced electron and the propagation direction of the field θ f. The denominator reaches zero whenever a number r of bunch field photons contribute such that nω=ω 1 Conclusion  The cross section of the Stimulated Breit-Wheeler process was calculated using the Dirac equation solutions in a constant crossed field and including a 1 st order radiative correction.  For default accelerator parameters at the ILC the Breit-Wheeler cross section used in background pair calculations is underestimated by several orders of magnitude.  Background pair generators need to be reconfigured in order to include the amended cross section. Any extra pairs produced will be longitudinal and twice as energetic due to the contribution from the bunch fields  The theoretical calculation can be refined for allowing for non azimuthal symmetry in which the production takes place near the periphery of the bunch Further information +44(0) For two 511 MeV beamstrahlung photons interacting in the 0.03 MeV default ILC bunch field, the virtual particle of the Stimulated Breit-Wheeler process reaches the mass shell whenever 511/0.03=16667 bunch field photons contribute to the process. Stimulated Breit-Wheeler Feynman Diagram. Double straight lines represent the fermion in the presence of the intense bunch field The Stimulated Breit- Wheeler differential cross section peak for various values of the beamstrahlung photon energy. Comparison is made with the ordinary (no external field) Breit-Wheeler process. The vertical axes is in units of steradian -1 The imaginary part of the Electron Self Energy in the bunch field is peaked at minimal contribution from bunch field photons. Integration over the total contribution reveals a self energy that varies almost linearly with a scalar product of external field 4- momentum and virtual electron momentum Non-linear processes associated with colliding charge bunches. The first order process obtains energy from the bunch field and is suppressed by brief passage through the field. The second order process exchanges a virtual particle in the midst of the field and is not suppressed Beamsstrahlung photon Charge BunchBunch Fields 1 st order nonlinear Pair Production 2 nd order nonlinear Pair Production  assume ILC bunches not disrupted so bunch field stays constant  all photons are collinear  use centre of mass frame  only use imaginary part of the self energy The full cross section of the Stimulated Breit- Wheeler process for various beamstrahlung photon energies. The effect of the bunch field is to increase the pair production cross section by 5-6 orders of magnitude.