Kinetic coupling and dark matter on the lattice

TL;DR

The paper constructs and analyzes a $U(1)\otimes U(1)$ gauge theory coupled to a scalar, incorporating kinetic mixing between the gauge sectors. It develops a continuum formulation with explicit diagonalization into massless and massive gauge combinations, then implements a lattice nonperturbative study to map phase diagrams, compute gauge masses, and extract fine-structure constants for the emergent sectors. Three scalar-charge models (A,B,C) are explored, revealing how the scalar couples to the two gauge fields and how phase structure evolves with the kinetic mixing parameter $\kappa$ (or $\beta_3$ on the lattice). The results demonstrate symmetric-phase decoupling into $G_u$ and $G_v$ fields with controlled couplings, while in broken phases the spectra show Yukawa-like and Coulomb-like behaviors and phase transitions that are not captured by perturbation theory. These findings have relevance for dark matter models employing a dark photon portal and illustrate how nonperturbative lattice methods illuminate the interplay between kinetic mixing, symmetry breaking, and observable gauge couplings.

Abstract

We study a $U(1)\otimes U(1)$ system coupled to scalar fields. Initially the model is studied in a novel continuum formulation and study of the appropriate diagonalizations is performed. Three models are examined, in two of which the scalar field couples with both gauge fields while in the third one the scalar field couples only to the dark photon. The model is then treated on a space-time lattice. We determine the phase diagram for various values of the kinetic coupling parameter. Then there follows the determination of masses for the scalar fields and the massive gauge fields, as well as the fine structure constants for the massless gauge fields.

Figures (13)

• Figure 1: Hysteresis loops for the quantities $<F_1^2>$ and $<F_2^2>$ in model B $(b_h\equiv \beta_h).$ The quantity $<F_1^2>$ has no hysteresis loop since, in model B, the $F_1$ field does not couple with the scalar field, so it does not feel the symmetry breaking. The quantity $<F_2^2>$ indicates a first order phase transition.
• Figure 2: Hysteresis loops for ${\rm cos(link)}$ in model B.
• Figure 3: Phase diagram for the three models for $\beta_3=0.00$$(b_g\equiv \beta_g).$ Model A is represented by the uppermost curve. Notice that $\beta_g$ should be positive, otherwise the functional integral diverges.
• Figure 4: Phase diagrams for models A, B and C for $\beta_3=0.60,$ represented by the vertical line. The lowest curve represents model B, the middle curve depicts model C, while the uppermost curve represents model A. The $B_u=\beta_g-\beta_3,$ which is the effective coupling of the $G_u$ field, should be positive (compare with figure \ref{['phd_ABC000']}); therefore the curves may not extend to the left of the vertical line.
• Figure 5: Symmetric phase, correlators $< {{F}_{1}}{{F}_{2}} >$ (upper curves) and $<{{G}_{u}}{{G}_{\upsilon}}>$ (lower curves) for all models $(b_3\equiv \beta_3).$