Probing CP-Violating Neutral Triple Gauge Couplings at Electron-Positron Colliders

TL;DR

This work develops a gauge-invariant framework for CP-violating neutral triple gauge couplings (nTGCs) arising from dimension-8 SMEFT operators, addressing deficiencies in conventional form-factor parametrizations that respect only $U(1)_{ m EM}$ while violating $SU(2)_L\otimes U(1)_Y$ symmetry. The authors identify five CPV dimension-8 operators (three Higgs-containing and two pure-gauge) and establish a matching to a revised nTGC form-factor structure with an additional parameter $h_6^V$, yielding relations like $h_2^Z=(c_W/s_W) h_2^\gamma$ and $h_2^V=2 h_6^V$. They analyze sensitivity to these CPV nTGCs at future $e^+e^-$ colliders up to $\sqrt{s}=5$ TeV, using $e^+e^-\to Z\gamma$ with $Z\to f\bar f$, angular observables, and beam polarization, and find operator scales from $O(1\,\text{TeV})$ to $O(10\,\text{TeV})$ with form-factor sensitivities in the $O(10^{-4})$ to $O(10^{-8})$ range; conventional form factors overstate sensitivity by up to $2-3$ orders of magnitude at high energies. The study also compares lepton and hadron colliders, showing the complementary reach and emphasizing the importance of the gauge-consistent SMEFT formulation for robust constraints on CP violation beyond the SM. These results highlight CPV nTGCs as a viable probe of high-scale CP-violating physics potentially connected to baryogenesis.

Abstract

We study the CP-violating (CPV) neutral triple gauge couplings (nTGCs) that can be realized via dimension-8 operators in the Standard Model Effective Field Theory (SMEFT). We present a new formulation of the CPV nTGC form factors that is compatible with spontaneous breaking of the electroweak gauge symmetry, and show how these CPV form factors can be matched consistently with the corresponding dimension-8 CPV nTGC operators in the broken phase. We then study probes of the CPV nTGCs at future high-energy $e^+e^-$ colliders with centre-of-mass energies $\sqrt{s}=(0.25, 0.5, 1, 3, 5)$TeV respectively, demonstrating that the $e^{\mp}$ beam polarizations can help to improve the sensitivities of probes of the nTGCs. We estimate that the sensitivity reaches for probing the new physics scales of nTGCs can range from ${O}(\rm{TeV})$ at a 250GeV $e^+e^-$ collider to ${O}(10\,\rm{TeV})$ at an $e^+e^-$ collider of energy $(3-5)$TeV, and that the sensitivities to the nTGC form factors vary from ${O}(10^{-4})$ to ${O}(10^{-6}-10^{-8})$ for the $e^+e^-$ collision energy from 250GeV to $(3-5)$TeV.

Figures (7)

• Figure 1: Kinematics in the $e^+e^-$ collision frame of the reaction $e^+e^-\!\!\rightarrow\!Z\gamma$ followed by $Z\!\!\rightarrow\!\!f\bar{f}$ decay. In this plot, $\theta$ denotes the polar scattering angle between the directions of the outgoing $Z$ and the initial state $e^-$, and $\phi_*$ is defined as the angle between the scattering plane and the decay plane of the final state $Z$ boson in the center-of-mass frame of $e^-e^+$.
• Figure 2: Angular distributions in $\phi_*^{}$ for the reaction $\space e^+\space e^{-}\!\!\rightarrow\! Z\gamma\space$ with $Z\!\!\rightarrow\! d\space\bar{d}\space$, as generated by the form factors $(h_1^Z,\, h_1^\gamma,\, h_2^{})$ at $e^+e^-$ colliders. The plots (a)-(d) present the angular distributions for $e^+e^-$ collider energies $\sqrt{s\,}\space =\space (0.25,\space 0.5,\space 1,\space 3)\space$TeV, respectively.
• Figure 3: Sensitivity reaches ($\space 2\space\sigma$ bounds) on the CPV nTGV form factors $(h_2^{},\space h_1^Z,\space h_1^\gamma)$ in plot (a) and on the new physics scales $\Lambda$ of the corresponding dimension-8 CPV nTGC operators in plot (b) at $e^+e^-$ colliders with collision energies $\sqrt{s\,}\!=\!(0.25,\,0.5,\,1,\,3,\,5)$ TeV, by choosing integrated luminosities of $5\space$ab$^{-1}$ in each case. For the case of $\sqrt{s\,}\!=\!250$ GeV, we also present the sensitivity reaches with an integrated luminosity of $20\space$ab$^{-1}$. The sensitivities with unpolarized (polarized) beams are shown in light (heavy) colors in plot (a) and heavy (light) colors in plot (b) respectively, where we choose $(P_L^e,\space P_R^{\bar{e}})\space =(0.9,\space 0.65)$.
• Figure 4: Correlation contours ($2\sigma$ bounds) for each pair of CPV nTGC form factors $(h_1^Z,h_2^{})$ and $(h_1^\gamma,h_2)$ at $e^+e^-\!$ colliders for either unpolarized $e^\mp$ beams (blue color) or polarized $e^\mp$ beams (red color) where we choose $(P_L^e,\space P_R^{\bar{e}})\space =(0.9,\space 0.65)$. At each collison energy, we set a sample integrated luminosity of $\space 5$ ab$^{-1}$. The plots (a)-(d) present the correlation contours for the $e^+e^-$ collider energy $\sqrt{s\,}\space =\space (0.25,\space 0.5,\space 1,\space 3)\space$TeV, respectively.
• Figure 5: Correlation contours ($2\sigma$ bounds) for each pair of the CPV nTGC form factors $(h_1^Z,\space h_1^\gamma)$ at $e^+e^-$ colliders with $\sqrt{s\,}\!=\!(0.25,\space 0.5,\space 1,\space 3)\space$TeV, by choosing either unpolarized $e^\mp$ beams (blue color) or polarized $e^\mp$ beams (red color), where we choose $(P_L^e,\space P_R^{\bar{e}})\space =(0.9,\space 0.65)$. In each plot, an integrated luminosity of 5 ab$^{-1}$ is chosen. The plots (a)-(d) present the correlation contours for the $e^+e^-$ collider energy $\sqrt{s\,}\space =\space (0.25,\space 0.5,\space 1,\space 3)\space$TeV, respectively.