On-shell $WW\gamma$ vertex in the T-Parity and non T-Parity Littlest Higgs model

The static electromagnetic properties of the $W$ boson, $\Delta \kappa$ and $\Delta Q$, are calculated in the T-parity and non T-parity littlest Higgs model (LHM) including terms up to the order of $(v/f)^4$, with $v$ the standard model (SM) vacuum expectation value and $f$ the scale of the global symmetry breaking. There are contributions from the virtual effects of the new heavy particles and also from the new corrections to the SM vertices, which depend on the mixing parameter $c$ and decrease quickly as $f$ increases. Depending on the value of $c$, the partial contributions to $\Delta \kappa$ and $\Delta Q$ can add constructively or destructively. The terms of the order of $(v/f)^4$ are subdominant but they can add constructively and can enhance the $W$ form factors for $f < 1$ TeV. In general the size of the $W$ form factors is very suppressed in the LHM without T-parity as the constraints on the scale $f$ from electroweak precision measurements are very tight. The LHM with T-parity has no such constraints and it allows for values of $W$ form factors similar to those found in other weakly coupled SM extensions, though they are out of the expected sensitivity of future measurements at the LHC. We also discuss the possibility that there is some enhancement due to the interactions arising from the strongly interacting ultraviolet completion of the LHM.