Extract the energy scale of anomalous $\gamma\gamma \to W^+W^-$ scattering in the vector boson scattering process using artificial neural networks

As a model independent approach to search for the signals of new physics~(NP) beyond the Standard Model~(SM), the SM effective field theory~(SMEFT) draws a lot of attention recently. The energy scale of a process is an important parameter in the study of an EFT such as the SMEFT. However, for the processes at a hadron collider with neutrinos in the final states, the energy scales are difficult to reconstruct. In this paper, we study the energy scale of anomalous $\gamma\gamma \to W^+W^-$ scattering in the vector boson scattering~(VBS) process $pp\to j j \ell^+\ell^-\nu\bar{\nu}$ at the large hadron collider~(LHC) using artificial neural networks~(ANNs). We find that the ANN is a powerful tool to reconstruct the energy scale of $\gamma\gamma \to W^+W^-$ scattering. The factors affecting the effects of ANNs are also studied. In addition, we make an attempt to interpret the ANN and arrive at an approximate formula which has only five fitting parameters and works much better than the approximation derived from kinematic analysis. With the help of ANN approach, the unitarity bound is applied as a cut on the energy scale of $\gamma\gamma \to W^+W^-$ scattering, which is found to has a significant suppressive effect on signal events. The sensitivity of the process $pp\to j j \ell^+\ell^-\nu\bar{\nu}$ to anomalous $\gamma\gamma WW$ couplings and the expected constraints on the coefficients at current and possible future LHC are also studied.