Towards a theory of quark masses, mixings and CP-violation

TL;DR

The paper investigates how quark masses, mixings, and CP violation can arise in string theory through Yukawa couplings computed in intersecting D-brane models on a toroidal compactification. Yukawas are given by worldsheet-instanton sums that depend on the brane geometry and Wilson-line phases, enabling complex phases and CP violation to emerge from open-string data. Numerical analyses show that a simple N=1 SUSY, three-generation configuration can reproduce observed quark masses, CKM angles, and the Jarlskog invariant, provided the compact radii are near the string scale and the brane setup sits close to symmetric points related to a Pati-Salam structure. The work highlights a calculable, geometry-driven route to fermion mass hierarchies, with potential extensions to charged leptons and neutrinos, and suggests a deep connection between low-energy flavor patterns and high-energy brane configurations.

Abstract

We discuss the structure of Yukawa couplings in D-brane models in which the SM fermion spectrum appears at the intersections of D-branes wrapping a compact space. In simple toroidal realistic examples one can explicitly compute the Yukawa couplings as a function of the geometrical data summing over world-sheet instanton contributions. A particular simple model with a N = 1 SUSY spectrum and three quark-lepton generations is studied in some detail. Remarkably, one can reproduce the observed spectrum of quark masses and mixings for particular choices of the compact radii and brane locations. In order to reproduce the smallness of up- and down-quark masses branes should be located in simple geometric configurations leading to some accidental global symmetries. We also find that the brane configurations able to reproduce the observed data may be considered as a deformation (by brane translation) of a configuration with Pati-Salam gauge symmetry. The origin of CP-violation in this formalism is quite elegant. It appears as a consequence of the generic presence of U(1) Wilson line backgrounds in the compact dimensions. One can reproduce the observed results for the CP-violation Jarlskog invariant J as long as the compact radii are of order of the string scale. DISCLAIMER: This paper is going to be substantially revised. Althought the physics and general concepts are still valid, the Yukawa couplings of the particular model presented in this paper have a simpler form than discussed here, as we recently pointed out in hep-th/0302105. A properly revised version will be eventually sent as the paper is appropriately corrected.

Figures (5)

• Figure 1: Yukawa coupling between two quarks of opposite chirality and a Higgs boson.
• Figure 2: Intersecting brane world setup. Consider two D6-branes filling four non-compact dimensions, to be identified with $M_4$, and wrapping three 1-cycles of $T^2 \times T^2 \times T^2$. In this example the wrapping numbers are $(2,1)(1,0)(1,1)$ (solid line) and $(0,1)(1,2)(0,1)$ (dashed line). The total intersection number is $2 \times 2 \times 1 = 4$.
• Figure 3: Scheme of the model in the text. Moving brane $d$ on top of brane $a$ one gets an enhanced $SU(4)$ Pati-Salam symmetry. If in addition brane $c$ is located on top of its mirror $c$* there is an enhanced $SU(2)_R$ symmetry.
• Figure 4: Brane configuration corresponding to the MSSM-like model described in the text, for the specific value $\rho = 1$. For simplicity, we have not depicted the leptonic brane nor the mirror $a$* brane.
• Figure 5: Different triangular instantons contributing to a Yukawa coupling. In a compact space like a $T^6$, three intersection points are connected by an infinite number of triangles, each of them giving rise to a contribution to the same Yukawa coupling. We have illustrated this effect in the simple case of a two-torus (namely, the second two-torus of figure \ref{['guay']}). In the figure at the right we have depicted a fundamental region of the torus, where in the left we have patched several copies together. There we have drawn three triangles of increasing area connecting the same intersection points, and thus contributing to the same Yukawa $Q_0 H q_j$.