We study the problem of modulus stabilisation in the framework of the modular symmetry approach to the flavour problem. By analysing simple UV-motivated CP-invariant potentials for the modulus $τ$ we find that a class of these potentials has (non-fine-tuned) CP-breaking minima in the vicinity of the point of $\mathbb{Z}_3^{ST}$ residual symmetry, $τ\simeq e^{2πi/3}$. Stabilising the modulus at these novel minima breaks spontaneously the CP symmetry and can naturally explain the mass hierarchies of charged leptons and possibly of quarks.
Fermion mass relations in one-parameter modular models
Instituto de Física Corpuscular (IFIC), Universidad de Valencia-CSIC（瓦伦西亚大学-西班牙国家科研理事会粒子物理研究所）；Department of Physics and Astronomy, University of California, Irvine（加州大学尔湾分校物理与天文学系）；Instituto de Física, Universidad Nacional Autónoma de México（墨西哥国立自治大学物理研究所）；INFN Sezione di Roma Tre（罗马第三大学意大利国家核物理研究所）
$S^\prime_4$ Quark Flavour Model in the Vicinity of the Fixed Point $τ= i\infty$
INFN/SISSA；Kavli IPMU (WPI), UTIAS, The University of Tokyo（理化学研究所 Kavli 宇宙粒子与天体物理研究所（WPI），东京大学 UTIAS）；Department of Physics, Niigata University（新潟大学物理学系）；Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences（保加利亚科学院核研究与核能研究所）
INFN Sezione di Bologna（意大利国家核物理研究所博洛尼部分所）；INFN Sezione di Roma Tre（意大利国家核物理研究所罗马三分部）；SISSA/INFN（的里雅斯特高等研究学院/意大利国家核物理研究所）；Kavli IPMU (WPI), UTIAS, The University of Tokyo（东京大学）；Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences（保加利亚科学院核能研究所）
