Yukawa hierarchies in global F-theory models
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Springer
Received: October 10, Revised: November 21, Accepted: November 26, Published: January 8,
2019 2019 2019 2020
Mirjam Cvetiˇ c,a,b,c Ling Lin,a Muyang Liu,a Hao Y. Zhanga and Gianluca Zoccaratoa a
Department of Physics and Astronomy, University of Pennsylvania, 209 S 33rd Street, Philadelphia, PA 19104-6396, U.S.A. b Department of Mathematics, University of Pennsylvania, 209 S 33rd Street, Philadelphia, PA 19104-6396, U.S.A. c Center for Applied Mathematics and Theoretical Physics, University of Maribor, Mladinska 3, Maribor SI-2000, Slovenia
E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract: We argue that global F-theory compactifications to four dimensions generally exhibit higher rank Yukawa matrices from multiple geometric contributions known as Yukawa points. The holomorphic couplings furthermore have large hierarchies for generic complex structure moduli. Unlike local considerations, the compact setup realizes these features all through geometry, and requires no instanton corrections. As an example, we consider a concrete toy model with SU(5) × U(1) gauge symmetry. From the geometry, we ¯6 5 ¯ −4 coupling, producing a rank two Yukawa mafind two Yukawa points for the 10−2 5 trix. Our methods allow us to track all complex structure dependencies of the holomorphic couplings and study the ratio numerically. This reveals hierarchies of O(105 ) and larger on a full-dimensional subspace of the moduli space. Keywords: F-Theory, Supersymmetric Effective Theories ArXiv ePrint: 1906.10119
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP01(2020)037
JHEP01(2020)037
Yukawa hierarchies in global F-theory models
Contents 1
2 Yukawa couplings in F-theory 2.1 Counting charged matter in F-theory 2.1.1 Wave functions and holomorphic Yukawa couplings 2.2 8d gauge theory and Yukawa couplings 2.2.1 Gauge theory close to Yukawa points 2.2.2 Localized modes and Yukawa couplings 2.3 Higher rank coupling from multiple Yukawa points 2.4 Beyond holomorphic couplings
3 3 4 5 5 7 8 8
3 Compact toy model with SU(5) × U(1) symmetry 3.1 Factorized SU(5) Tate model with genus-0 matter curves 3.1.1 Factorized SU(5) Tate model 3.1.2 Explicit construction with SGUT = dP2 3.1.3 U(1)-flux and massless spectrum 3.2 Higgs bundle description 3.2.1 Localized matter fields from E8 breaking 3.3 Explicit computation of Yukawa couplings 3.4 Complex structure dependence and Yukawa hierarchies
9 9 9 11 13 14 14 17 19
4 Conclusions and outlook
21
A Details on the dP2 geometry
22
1
Introduction
The Standard Model of particle physics has many peculiar features, responsible for the rich phenomenology that ultimately shape our macroscopic universe. One of these features is the texture of Yukawa couplings between different matter generations, which leads to the observed hierarchy of fermion masses. To be a viable UV completion of our physical world, it is therefore paramount for string theory to be able to reproduce the
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