Effective scalar field theory ofp-adic string

Using the effective scalar field theory of the $p$-adic string, we show the equivalence of two previously derived sets of classical Feynman rules: one by the present authors, the other by Brekke et al. We use this equivalence to demonstrate a symmetry of the $p$-adic tree amplitude $A^m\mathrm{}\left(p\right)\mathrm{}$ under $p\to \frac{1}{p}$. We then present a scalar field theory with a continuous parameter $\xi$ that reduces to the two previously discussed examples as special cases.