Bremsstrahlung function in $\mathcal{N}=2$ SCFTs far beyond the supergravity limit

Abstract: We study the Bremsstrahlung function in two four-dimensional $\mathcal{N}=2$ superconformal gauge theories: the E-theory, with hypermultiplets in the symmetric and antisymmetric representations, and the D-theory, with antisymmetric and fundamental matter. Using matrix model techniques, we derive exact results at arbitrary coupling for the first few orders in the large-$N$ expansion. At leading order, both theories are shown to be planar equivalent to $\mathcal{N}=4$ Super Yang-Mills, while beyond the planar limit their behavior differs significantly. For the E-theory, we prove that the subleading correction is exactly given by a derivative of the free energy with respect to the coupling. This relation enables us to determine the complete strong-coupling expansion for the Bremsstrahlung function, including non-perturbative contributions, by exploiting known results for the free energy. For the D-theory, the exact expressions have a more intricate structure. We perform a detailed analysis of the strong-coupling regime of the subleading large-$N$ corrections, showing that, in this case, there are expansions in inverse powers of the coupling which truncate after finitely many terms. We further analyze the non-perturbative effects at strong coupling and derive closed-form expressions also for these contributions. These results provide precise predictions for the holographic duals of both theories beyond the supergravity approximation, probing higher orders in the large-$N$ expansion and the full subleading corrections at strong coupling.

• The paper presents exact large-N expansions of the Bremsstrahlung function, revealing precise perturbative and nonperturbative string corrections in N=2 SCFTs.
• Matrix model techniques and localization on a squashed four-sphere enabled the reorganization of 1/N corrections through normal-ordered operators.
• It demonstrates holographic implications via worldsheet instanton effects and sector-wise relations that refine predictions beyond the planar limit.

Bremsstrahlung Function in $\mathcal{N}=2$ SCFTs Far Beyond the Supergravity Limit

Overview and Motivation

The paper investigates the Bremsstrahlung function $B$ in two distinct four-dimensional $\mathcal{N}=2$ superconformal gauge theories: the E-theory and D-theory, both based on SU($N$) gauge groups but differing in matter content and string-theoretic embedding. The Bremsstrahlung function characterizes radiative energy loss for accelerated probes and appears in the small-angle expansion of the cusp anomalous dimension. While exact results for $B$ are established in $\mathcal{N}=4$ SYM via localization, the $\mathcal{N}=2$ case is less understood, especially beyond the planar limit and at strong coupling, where stringy corrections play a crucial role.

The motivation centers on deriving exact large-$N$ expansions and their strong-coupling behavior for $B$, thereby providing precise predictions for the holographic duals of these theories that probe stringy corrections inaccessible to classical supergravity.

Matrix Model Methods and Exact Large-$N$ Expansion

Both theories are analyzed using matrix models derived from localization on the squashed four-sphere, yielding exact expressions for partition functions and Wilson loop observables. The authors perform a detailed change-of-basis, introducing normal-ordered operators $B$0 to efficiently organize the $B$1 expansion.

Planar Equivalence and Beyond

• Planar Limit: Both E-theory and D-theory are shown to be planar equivalent to $B$2 SYM for the Bremsstrahlung function and Wilson loop vevs.
• Subleading Orders:
  • E-theory: Subleading corrections at $B$3 are rigorously shown to be captured by a $B$4-derivative of the free energy; the correction to $B$5 is fully encoded by $B$6.
  • D-theory: Subleading corrections emerge already at $B$7 and $B$8 (non-vanishing), featuring single- and double-trace contributions and integrals involving Bessel functions. These corrections possess intricate exact $B$9-dependence and can be resummed.

Strong-Coupling Regime: Perturbative and Non-perturbative Contributions

The strong-coupling expansion is analyzed using asymptotic properties of Bessel functions and exact resummation techniques.

E-theory

• The strong-coupling expansion for $\mathcal{N}=2$0 is obtained by exploiting the known analytic structure of $\mathcal{N}=2$1, including its non-perturbative sector.
• Non-perturbative contributions are governed by terms like $\mathcal{N}=2$2, interpreted as worldsheet instantons in the holographic dual. These are necessary for a Borel-summable analytic solution due to singularities in the Borel plane.

D-theory

• Correction terms in the strong-coupling regime exhibit a finite series expansion in $\mathcal{N}=2$3 which terminate, and infinite sums over modified Bessel functions of the second kind allow systematic extraction of all exponentially suppressed (instanton) effects.
• The method combines exact integral representations and asymptotic expansions, generalizing previous approaches to a wider class of functions.

Sector Structure and Coupling Redefinition

• The results for D-theory reveal that sector-wise relations exist between different orders in the $\mathcal{N}=2$4 expansion, suggesting differential operator connections.
• To absorb $\mathcal{N}=2$5 terms in D-theory, a shifted 't Hooft coupling $\mathcal{N}=2$6 is introduced, mixing $\mathcal{N}=2$7 orders and yielding more compact expressions for strong-coupling corrections.

Theoretical and Practical Implications

• Holographic Duals: The results provide precise predictions for string-theoretical corrections beyond the supergravity approximation in AdS/CFT for the studied $\mathcal{N}=2$8 theories.
• Instanton Interpretation: Exponentially suppressed corrections in $\mathcal{N}=2$9 correspond to worldsheet instantons on the gravity side and offer analytic control over non-perturbative effects.
• Sector Organization: The sector-wise relations among large-$N$0 corrections may hint at underlying integrable structures linking different orders, especially in simplified Sp($N$1) models.

Future Directions

Several promising extensions are suggested:

• Analysis of the Bremsstrahlung function in $N$2 Sp($N$3) theories, where Toda lattice integrability may allow recursive determination of subleading corrections.
• Study of the very strong coupling regime ($N$4, fixed $N$5), where instanton effects become relevant and modular properties are probed, especially for extended objects such as Wilson lines.
• Application of these matrix model methods to broader classes of $N$6 SCFTs and their holographic duals, extending our understanding of quantum string corrections.

Conclusion

This paper provides a rigorous and comprehensive study of the Bremsstrahlung function in two $N$7 superconformal gauge theories using supersymmetric localization and matrix model techniques. It delivers exact large-$N$8 expressions and strong-coupling expansions, including all perturbative and non-perturbative contributions. The findings have significant implications for precision holography, probe nontrivial string effects, and suggest avenues for further analytic control in related gauge/string systems, advancing the quantitative understanding of radiative observables in gauge theory and their gravitational duals (2605.24101).