Probing Neutral Triple Gauge Couplings via $ZZ$ Production at $e^+e^-$ Colliders with Machine Learning

Neutral triple gauge couplings (nTGCs) first arise from the dimension-8 operators of the Standard Model Effective Field Theory (SMEFT), rather than the dimension-4 SM Lagrangian and dimension-6 SMEFT operators, opening up a unique window for probing new physics at the dimension-8 level. In this work, we formulate the nTGC form factors of $ZZV^*$ ($V\!\!=\!Z,\gamma$) that are compatible with the spontaneous breaking of the SU(2)$\otimes$U(1) electroweak gauge symmetry and consistently match the dimension-8 nTGC operators in the broken phase. We study the sensitivities for probing both the $ZZV^*$ form factors and the corresponding new physics scales through $ZZ$ production (with visible/invisible fermionic $Z$ decays) at high energy $e^+e^-$ colliders including CEPC, FCC-ee, ILC and CLIC. In particular, we identify the dimension-8 operator that contributes to the pure triple $Z$ boson coupling $ZZZ^*$ alone, but not the mixed $ZZ\gamma^*$ coupling. We further study the correlations between probes of the $ZZZ^*$ and $ZZ\gamma^*$ couplings. Using machine learning, we show that angular distributions of the final-state fermions can play key roles in suppressing the SM backgrounds. The sensitivities can be further improved by using polarized $e^\mp$ beams. We demonstrate that machine learning is advantageous for handling the 4-body final states from $ZZ$ decays and improves significantly the sensitivity reaches of probes of nTGCs in $e^+e^-$ collisions. We find that nTGC new physics scales can be probed up to the multi-TeV scale at the proposed $e^+e^-$ colliders.