Feynman Rules in the Type III Natural Flavour-Conserving Two-Higgs Doublet Model

We consider a two Higgs-doublet model with $S_3$ symmetry, which implies a $\frac{\pi }{2}$ rather than 0 relative phase between the vacuum expectation values $<{\mathrm{\Phi }}_1>$ and $<{\mathrm{\Phi }}_2>$. The corresponding Feynman rules are derived accordingly and the transformation of the Higgs fields from the weak to the mass eigenstates includes not only an angle rotation but also a phase transformation. In this model, both doublets couple to the same type of fermions and the flavour-changing neutral currents are naturally suppressed. We also demonstrate that the Type III natural flavour-conserving model is valid at tree-level even when an explicit $S_3$ symmetry breaking perturbation is introduced to get a reasonable CKM matrix. In the special case $\beta = \alpha$,astheratio$\tan\beta = {v_2 \over v_1}$runsfrom0to$\infty$, the dominant Yukawa coupling will change from the first two generations to the third generation. In the Feynman rules, we also find that the charged Higgs currents are explicitly left-right asymmetric. The ratios between the left- and right-handed currents for the quarks in the same generations are estimated.