Nevertheless there can be important technical differences in intermediate steps. When the scalar sector is nonlinearly-realized, scalar kinetic terms are in general not automatically canonically normalized, and there can be ‘unexpected’ vector-scalar two-point transitions which need to be taken care of. We will show in the following sections that these situations indeed occur for the case of the simplest Little Higgs (SLH) model [@Schmaltz:2004de], which is proposed as a simple solution to the Higgs mass naturalness problem.

From a more general perspective, the problem we encounter is how to diagonalize a vector-scalar system in gauge field theories. Specifically, the Lagrangian we start with might not be canonically normalized in its kinetic part, and may have some general vector-scalar two-point transitions. To do perturbation theory in the usual manner, we need to first render its kinetic part canonically normalized, which could be done via the usual complete-the-square method. To remove the vector-scalar two-point transitions, strictly speaking we need to choose appropriate gauge-fixing terms. Finally we still need to diagonalize the scalar mass matrix with contribution from both the original scalar mass terms and the gauge-fixing terms. These steps set the stage for the derivation of VSS interactions.

In Section \[sec:sys\] the 