Lepton Pair Production in the Non-Commutative Standard Model with Polarized \({{e}^{ + }}{{e}^{ - }}\) Beams

In this paper, we study lepton pair production with polarized electron-positron beams in the context of the non-commutative standard model, an extension of the standard model based on non-commutative geometry. By employing the Seiberg–Witten map, we compute the production cross-section up to first order in the non-commutative parameter \({{\Theta }_{{\mu \nu }}}\). In our analysis we use the same value for both space-space and time-space non-commutative parameter, and we focus on collision energies in the range 300–1000 GeV, appropriate for the future ILC collider. Our main objectives are to examine the angular distribution \({{{\text{d}}\sigma } \mathord{\left/ {\vphantom {{{\text{d}}\sigma } {{\text{d}}\Omega }}} \right. \kern-0em} {{\text{d}}\Omega }}\) and the total cross-section, aiming to explore the effects of space-time non-commutativity at an energy scale around \(\Lambda \sim {\text{TeV}}\) and the possibility of observing these effect at the ILC. Our results indicate a significant deviation in the angular distribution due to non-commutativity when \(\Lambda \sim {{10}^{3}}{\kern 1pt} {\text{GeV}}\). Moreover, the total cross-section for lepton pair production is more sensitive to space-time deformations in the polarized case compared to the unpolarized case. Specifically, we find that at center-of-mass energy \(\sqrt s = 1000{\kern 1pt} \,{\text{GeV}}\), the deviation in the polarized case is approximately eight times larger than that in the unpolarized case. We also estimate from this study a bound on the scale of the non-commutative parameter \(\Lambda \).