We emphasize that the conventional electroweak form factor formulation imposes only the residual U(1) gauge symmetry of QED, and is in general incompatible with the SMEFT framework, which takes into account the full $\mathrm{SU}(2)\otimes \mathrm{U}(1)$ electroweak gauge symmetry of the SM. We stress that it is important to match precisely the form factors with the corresponding SMEFT operators in the broken phase, which can place additional nontrivial constraints on the structure of the form factors as a result of the spontaneous electroweak symmetry breaking of the SM. We demonstrate this point for our correct formulation of both the CPC and CPV nTGC form factors in the second section and in the Supplemental Material [14].

We note that the CMS [4] and ATLAS [5] Collaborations also measured the CPC nTGC form factor $h_4^V$ using the conventional formula, which gives rise to unphysically large high-energy behavior [1, 14].

We observe that by matching the nTGC form factors with the corresponding dimension-8 operators of the SMEFT, the $(h_{11}^{\gamma },h_{31}^{\gamma })$ and $({\widehat{h}}_1^Z,{\widehat{h}}_3^Z)$ form factors arise as a consequence of spontaneous electroweak symmetry breaking and would vanish if $⟨H⟩=0$. We also note that ${\widehat{h}}_{1,3}^{\gamma }$ and ${\widehat{h}}_{2,4}$ arise from the electroweak rotation of the $B{W}^3{W}^3$ vertex, so the ${\widehat{h}}_{3,1}^{\gamma }$ terms in Eqs. (5a) and (7a) would vanish if $s_W=0$.

In principle, the partonic center-of-mass energy $\sqrt{\widehat{s}}$ can be determined by measuring the invariant mass of the observable final-state particles. The ATLAS measurements of $M_{\ell \ell \gamma }$ during the LHC run 2 reached around 3 TeV. Accordingly, for the analysis of $Z^{*}\gamma (\nu \overline{\nu }\gamma )$ production, the relevant energy range is $\sqrt{\widehat{s}}<3\mathrm{TeV}$ for the LHC and $\sqrt{\widehat{s}}<23\mathrm{TeV}$ for the 100 TeV $pp$ collider.

We find that the situation is different for probing the nTGCs at high-energy $e^{+}{e}^{-}$ colliders [2, 3], where the interference contribution can dominate over the squared contribution. In addition, imposing the inelastic unitarity condition [21], we have derived the perturbative unitarity bounds on the cutoff scales ${\mathrm{\Lambda }}_j$ and the form factors $h_j^V$ in the Supplemental Material [14]. We have verified that these bounds are much weaker than our current collider bounds in the third section and thus do not affect our collider analyses.

From this literature, for the given signal events $S$ and background events $B$, the statistical significance under the background-with-signal hypothesis is given by $$\mathcal{Z}=\sqrt{2(B\ln \frac{B}{B+S}+S)},$$with which we obtain the formula (12) in the main text.

We also note that no correlation exists between the CPC and CPV nTGCs because their amplitudes only differ by $\pm \mathrm{i}$. Finally, correlations among ${\widehat{h}}_2$, ${\widehat{h}}_1^Z$, ${\widehat{h}}_1^{\gamma }$, and $h_{11}^{\gamma }$ are similar to those of the corresponding CPC nTGCs, because the squared term ${\sigma }_2$ dominates the signal cross section at the LHC and at the 100 TeV $pp$ collider.