1. Incoherent pair background processes with full polarizations at the ILC Anthony Hartin JAI, Oxford University Physics, Denys Wilkinson Building, Keble Road Oxford, UK Introduction The International Linear Collider will collide polarized particle bunches to produce physics processes of interest and background events. The polarization states of both the bunch particles and the collective field of the bunch are important parameters affecting cross-sections and hence final particle states. There are three incoherent background pair processes considered in background pair generators such as the CAIN program. Initial photons involved in these process are either real or virtual. The present version of CAIN deals with partial polarizations for the real photons only. Most of the pairs however are produced by processes involving virtual photons. The flux of virtual photons is related to the bunch fields via the usual Weizsacker-Williams method. The polarization state of virtual photons is obtained with an expression for the bunch electric field at the point of production. An expression for the pair production cross- section containing all polarizations is also required. These modifications lead to new characteristics for the produced pairs which are essential for other background studies. Breit-Wheeler cross-section In order to take advantage of the full polarisation of initial photons, the full Breit-Wheeler cross-section is required. At present in CAIN the cross-section σ circ is written down only for the product of circular polarisations ξ 2 ξ’ 2 of initial photons k and k'. The full cross-section σ full is a sum over all polarisation states and functions of final electron energy ε and momentum p. With some algebraic manipulation the two cross-sections can be written in similar form. A numerical investigation of the two cross-sections in the above equation reveal the usual peak at low energies. However accounting for full polarizations reveals a substantially reduced cross-section for electron energies approximately less than 50 MeV (see below). It was expected that such a reduction in the Breit-Wheeler cross-section would result in less background pairs. It was also considered important to determine any effect on collision luminosity. So the modified CAIN program was run for all seven 500 GeV centre of mass collider parameter sets. There was a 10-20% overall reduction in pair numbers (see above) with no discernible effect on collision luminosity. Final pair polarization The polarizations of final states are specified by the e  polarization vector (ζ 1,ζ 2,ζ 3 ). The components of this vector can be written in terms of a sum over products of initial photon polarization states and a function F ii' jj ' of 4- vector scalar products specified in a paper by V.N. Baier and A.G. Grozin (hep-ph/0209361) The components of final e  polarization vector are strongly dependent on the extent of circular polarization of initial photons. Consequently the pairs are produced with polarization components almost zero Stokes Parameters The photon polarization vector is defined with respect to a basis vector (ê x,ê y,k z ) and can be expressed in terms of a density matrix ρ, Pauli matrices σ and stokes parameters ( ξ 1, ξ 2, ξ 3 ) For virtual photons, the stokes parameters can be written in terms of the transverse bunch electric field (E x,E y ) at the point (x,y) of pair production. Since the electric field for relativistic charge bunches is constant and crossed, there is almost no circular polarization component for the initial photon polarizations Virtual Photon Polarization The stokes parameters for the virtual photons follow from an explicit expression for the bunch electric field. The field of a single relativistic charge can be expanded in plane waves. For a Gaussian bunch of size (σ x,σ y ) of N charges, the collective field is an integration over transverse wave vector (q x,q y ) of a product of Gaussian form factor and the single charge field The integrations are performed by expanding 1/(q x 2 +q y 2 ) in a Taylor series. The integration variables can then be separated and the Fourier transforms are easily performed. Since the ILC uses flat beams (σ x >> σ y ) only the first term in the Taylor series is required. The magnitude of the y component of the bunch field is much greater than that of the x component, consistent with the beam being squeezed in y. Both components are real, meaning that there is no circular polarisation of virtual photons. Conclusion  The full polarizations of the incoherent background pair processes at the interaction point of the ILC have been investigated analytically and numerically.  Virtual photon polarization is related to the bunch electric field at the point of production and, like the real beamstrahlung photons, have almost no circular polarization component  The full Breit-Wheeler cross-section with all polarization states can be written in similar form to the cross-section with circular polarizations only  For all seven parameter sets contemplated for the ILC, with 500 GeV centre of mass collision energy, there is a 10-20% reduction in low energy pair numbers with no discernible change in collision luminosity  Produced pairs have almost no polarization  Extensions of this study to the coherent processes are envisaged Further information tony.hartin@physics.ox.ac.uk http://www-pnp.physics.ox.ac.uk/~hartin/theory +44(0)1865273381 Incoherent pair particle energy, comparing partial polarizations originally in CAIN and full polarizations. Full polarizations manifest as a reduction in low energy pairs. The number of pair particles for all seven ILC parameter sets with 500GeV centre of mass energy. Inclusion of all polarizations result in 10-20% less pairs than previously simulated by CAIN Bunch Fields 1 st order nonlinear Pair Production 2 nd order nonlinear Pair Production Comparison of the existing CAIN Breit-Wheeler cross-section and the Breit- Wheeler cross-section for full polarizations of initial states Stimulated Breit-Wheeler process The pair production processes occur in the presence of intense bunch fields and the possibility must be allowed for coherent production of pairs to all orders. The Dirac equation solutions in the presence of the bunch fields A e (k) are required and turn out to be a simple product of Volkov function E p and the usual bispinor u p The full cross-section for this process requires substantial work to calculate but it retains the same form with respect to polarization as its external field free equivalent. Much of the foregoing analysis of polarization effects can be directly applied to this stimulated Breit-Wheeler process, and will be the subject of ongoing work