Computing Yukawa Couplings from Magnetized Extra Dimensions

TL;DR

This paper calculates Yukawa couplings in magnetized toroidal compactifications of super-Yang-Mills theory, providing explicit formulas and exploring their dependence on background parameters, with implications for string phenomenology.

Contribution

It derives explicit Yukawa coupling formulas in magnetized extra dimensions using field theory, applicable to various string compactifications, and relates them to intersecting D-brane models.

Findings

Yukawa couplings are expressed as Riemann theta-functions.

Couplings depend on complex structure and Wilson lines.

Results match with intersecting D-brane models.

Abstract

We compute Yukawa couplings involving chiral matter fields in toroidal compactifications of higher dimensional super-Yang-Mills theory with magnetic fluxes. Specifically we focus on toroidal compactifications of D=10 super-Yang-Mills theory, which may be obtained as the low-energy limit of Type I, Type II or Heterotic strings. Chirality is obtained by turning on constant magnetic fluxes in each of the 2-tori. Our results are general and may as well be applied to lower D=6,8 dimensional field theories. We solve Dirac and Laplace equations to find out the explicit form of wavefunctions in extra dimensions. The Yukawa couplings are computed as overlap integrals of two Weyl fermions and one complex scalar over the compact dimensions. In the case of Type IIB (or Type I) string theories, the models are T-dual to (orientifolded) Type IIA with D6-branes intersecting at angles. These theories may have phenomenological relevance since particular models with SM group and three quark-lepton generations have been recently constructed. We find that the Yukawa couplings so obtained are described by Riemann theta-functions, which depend on the complex structure and Wilson line backgrounds. Different patterns of Yukawa textures are possible depending on the values of these backgrounds. We discuss the matching of these results with the analogous computation in models with intersecting D6-branes. Whereas in the latter case a string computation is required, in our case only field theory is needed.