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Computing Yukawa Couplings from Magnetized Extra Dimensions

D. Cremades, L. E. Ibanez, F. Marchesano

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.

Computing Yukawa Couplings from Magnetized Extra Dimensions

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.

Paper Structure

This paper contains 42 sections, 264 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Yukawa couplings in Kaluza-Klein theories
  3. $D=10$ N=1 Super Yang-Mills compactifications with magnetic fluxes
  4. Models with fluxes on $D = 6,8$ dimensions
  5. Toroidal wavefunctions I: Abelian Wilson lines
  6. Eigenfunctions of the Dirac equation on $T^2$
  7. Abelian gauge field
  8. Dirac zero modes
  9. Normalization
  10. Chiral matter eigenfunctions
  11. Fermions in bifundamentals
  12. Eigenfunctions of the Laplace equation
  13. Theta functions as wavefunctions
  14. Generalization to $T^{2n}$
  15. Toroidal wavefunctions II: non-Abelian Wilson lines
  16. ...and 27 more sections

Figures (2)

  • Figure 1: Probability densities $\rho^j(z) = |\psi(z)^{j,3}|^2$ for a square $T^2$ with triplication of the chiral spectrum. The gaussian-like behaviour is now present in only one axis of the torus, namely in the coordinate $y = {\rm Im\,} z / {\rm Im\,} \tau$. The gaussians are similar to each other, and centered at the points $y = - j/3$.
  • Figure 2: Action of T-duality on chiral fields. T-duality maps the chiral fields at intersection points to wavefunctions defined on the whole compactification space. However, their probability density is delocalized in the directions of the T-duality and peaked in the transverse directions. Here we have considered the case of a horizontal T-duality acting on the wavefunctions in figure 4.