Effective field theory from a beta function for the p-adic string.

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We describe the calculation of a $\ensuremath{\beta}$ function for the $p$-adic string $\ensuremath{\sigma}$ model in a tachyonic background, and show that the vanishing condition of the $\ensuremath{\beta}$ function reproduces the tachyonic field equation obtained from $p$-adic amplitude calculations, up to the fifth order in the tachyonic field. By examining this $\ensuremath{\sigma}$ model in the presence of a soliton background, we show that a previously obtained soliton solution of the all-order tachyonic field equation follows from the requirement of discrete scale invariance.