# Cubic Interaction Vertices and One-loop Self-energy in the Stable String Bit Model

## TL;DR

This paper develops a formalism for calculating cubic interaction vertices and one-loop self-energy corrections in a stable string bit model with spin degrees of freedom, revealing how the energy scales with the number of bits and spin.

## Contribution

It introduces a new formalism for cubic interaction vertices in a string bit model with no spatial movement, enabling calculation of one-loop energy corrections.

## Key findings

- Ground state one-loop self-energy scales as M^{5-s/4} for even s and M^{4-s/4} for odd s.
- For s=24, the self-energy scales as 1/M, resembling Poincaré invariance in 1+1 dimensions.
- Analytical calculations for M=3, s=1,2 and numerical confirmation for s≤4.

## Abstract

We provide a formalism to calculate the cubic interaction vertices of the stable string bit model, in which string bits have $s$ spin degrees of freedom but no space to move. With the vertices, we obtain a formula for one-loop self-energy, i.e., the $\mathcal{O}\left(1/N^{2}\right)$ correction to the energy spectrum. A rough analysis shows that, when the bit number $M$ is large, the ground state one-loop self-energy $\Delta E_{G}$ scale as $M^{5-s/4}$ for even $s$ and $M^{4-s/4}$ for odd $s$. Particularly, in $s=24$, we have $\Delta E_{G}\sim 1/M$, which resembles the Poincar\'e invariant relation $P^{-}\sim 1/P^{+}$ in $(1+1)$ dimensions. We calculate analytically the one-loop correction for the ground energies with $M=3$ and $s=1,\,2$. We then numerically confirm that the large $M$ behavior holds for $s\leq4$ cases.