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

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.