Speaker
Description
The short-distance singularities and infinite self-energies in classical and quantum field theories motivate the exploration of extended geometric structures. Recently [1,2], we introduced a homothetic extension of classical Weyl-integrable geometry by generalizing the conventional linear gauge transformations to affine homothetic transformations centered at a distinguished harmonic, scale-invariant form. By re-linearizing these affine transformations, we obtain a twisted exterior calculus structurally equivalent to the Witten deformation of the de Rham complex. This construction supports a complete homothetic Hodge--de Rham theory on a doubled complex of differential forms, $\bar{\Omega}^\bullet(M) = \Omega^\bullet(M) \oplus \Omega^\bullet(M)$.
While this geometric framework inherently acts as a diffuse-interface volume-penalization method that regularizes the classical point charge, yielding a finite self-energy, we demonstrate its profound implications for Beyond the Standard Model (BSM) physics. We promote the abelian homothetic structure to the non-abelian gauge symmetry of the Standard Model, $SU(3)_c \times SU(2)_L \times U(1)_Y$.
By lifting the physical fields and their homothetic offset partners into the doubled complex, the gauge connection becomes $\hat{A} = (A, A_d)^T$. We derive the Homothetic Yang--Mills action, where the physical curvature is canonically modified by a dilaton-induced geometric penalty term, $w\,d\lambda \wedge (A - A_d)$, which regularizes the gauge field at singularities. Furthermore, we construct the extended homothetic Higgs mechanism. The Higgs doublet is promoted to a homothetic field $\hat{\Phi} = (\phi, \phi_d)^T$, and spontaneous symmetry breaking is governed by the invariant bilinear $\hat{\Phi}^\dagger \hat{\Phi} = \phi^\dagger \phi + \phi_d^\dagger \phi_d$.
Ultimately, this framework provides a mathematically rigorous, top-down geometric approach to UV-regularization in the Standard Model, offering a novel paradigm for resolving foundational singularities in particle physics.
References
[1] F. Sabetghadam, ``A homothetic Gauge Theory and the Regularization of the Point Charge,'' submitted to IJGMMP, arXiv:2507.06153 [math-ph] (2025)
[2] F. Sabetghadam, ``Homothetic Hodge–de Rham Theory and a Geometric Regularization of Elliptic Boundary Value Problems,'' submitted to Elsevier, arXiv:2603.27564 [math-ph] (2026).