# A Two-Parameter Recursion Formula For Scalar Field Theory

## TL;DR

This paper introduces a two-parameter family of recursion formulas for scalar field theory, enabling continuous interpolation between existing models and analyzing the dependence of critical exponents on these parameters.

## Contribution

It proposes a novel two-parameter recursion formula that unifies Wilson's and Dyson's models, providing a new approach to study scalar field theories.

## Key findings

- Critical exponent $b3$ varies continuously with parameter ;
- The -independence can guide the development of improved recursion formulas.
- Numerical evidence supports the continuous dependence of critical behavior on the parameters.

## Abstract

We present a two-parameter family of recursion formulas for scalar field theory. The first parameter is the dimension $(D)$. The second parameter ($\zeta$) allows one to continuously extrapolate between Wilson's approximate recursion formula and the recursion formula of Dyson's hierarchical model. We show numerically that at fixed $D$, the critical exponent $\gamma $ depends continuously on $\zeta$. We suggest the use of the $\zeta -$independence as a guide to construct improved recursion formulas.