Interpolating the Coulomb Phase of Little String Theory

TL;DR

The paper investigates the Coulomb-branch effective action of $(1,1)$ little string theory by combining perturbative 6D SYM results (up to 4 loops) with tree-level double-scaled little string theory (DSLST) amplitudes, to interpolate the 8-derivative $D^4 F^4$ coupling across the Coulomb branch. It confirms exact nonrenormalization for the $F^4$ and partial protection for $D^2 F^4$, while showing that the $f_2(r)$ term is not protected and receives all-loop contributions; the authors construct an interpolating function for $f_2(r)$ by matching the small-$r$ SYM expansion to the large-$r$ DSLST expansion, and discuss the implications for UV completions, the analytic structure (branch cuts) of the interpolating function, and possible nonperturbative corrections. The work also comments on analogous interpolations in circle-compactified $(2,0)$ LST and the resulting 5D SYM, proposing a two-parameter interpolation in the Coulomb phase and arguing that certain couplings vanish at the origin in the compactified theory. Overall, the study provides a concrete bridge between perturbative field theory and string-theoretic descriptions, yielding quantitative constraints on higher-derivative couplings and insights into UV completions and nonperturbative effects.

Abstract

We study up to 8-derivative terms in the Coulomb branch effective action of (1,1) little string theory, by collecting results of 4-gluon scattering amplitudes from both perturbative 6D super-Yang-Mills theory up to 4-loop order, and tree-level double scaled little string theory (DSLST). In previous work we have matched the 6-derivative term from the 6D gauge theory to DSLST, indicating that this term is protected on the entire Coulomb branch. The 8-derivative term, on the other hand, is unprotected. In this paper we compute the 8-derivative term by interpolating from the two limits, near the origin and near the infinity on the Coulomb branch, numerically from SU(k) SYM and DSLST respectively, for k=2,3,4,5. We discuss the implication of this result on the UV completion of 6D SYM as well as the strong coupling completion of DSLST. We also comment on analogous interpolating functions in the Coulomb phase of circle-compactified (2,0) little string theory.

Figures (4)

• Figure 1: Log-log plot of the coefficient $f_S(r)$ of $(s^2+t^2+u^2)F^4$ and $f_A(r)$ of $s^2 F^4$. The dashed line is given by the DSLST tree level superamplitude (valid for large $r$). The lower green line comes from 6D SYM one loop, the middle orange line comes from one and two loops combined, and the upper blue line combines the contributions up to three loops (valid for small $r$). We interpolate the two ends by a naive extension beyond their regimes of validity.
• Figure 2: The 1-loop scalar integral $I_4^{1-loop}(s_{12},s_{14},m_{ij})$.
• Figure 3: In (a), the planar 2-loop scalar integral. In (b), the non-planar 2-loop scalar integral.
• Figure 4: The nine 3-loop scalar integrals $I^{(x)}(s_{12},s_{14})$.