Exact absorption probabilities for the D3-brane

TL;DR

The paper derives exact absorption probabilities for a minimal scalar in the D3-brane background by mapping the radial wave equation to Mathieu's modified differential equation with parameters $q=(\omega R)^2$ and $a=(l+2)^2$, and expresses transmission and reflection in terms of the Floquet exponent $\nu$ and a parity-related quantity $\chi$. It provides an algorithm to compute the low-energy expansion of the cross-section to arbitrary order in $\omega R$, yielding explicit coefficients involving $\zeta$-values and demonstrating a simple closed form $P_l = \frac{4\pi^2}{(l+1)!^4 (l+2)^2} (\frac{\omega R}{2})^{8+4l} \sum_{n,k} b_{n,k} (\omega R)^{4n} (\log \omega \bar{R})^k$. The work connects these bulk-channel results to the world-volume theory, showing how $P_{l=0}$ relates to the discontinuity of the two-point function of a gauge-theory operator ${\cal O}$ and arguing for an effective action ${\cal L} = {\cal O}_4 + R^4 {\cal O}_8$ governing the IR/world-volume dynamics. These findings illuminate aspects of the AdS/CFT duality in nontrivial regimes and provide a concrete, computable framework for testing throat-brane correspondences via exact, all-orders low-energy expansions.

Abstract

We consider a minimal scalar in the presence of a three-brane in ten dimensions. The linearized equation of motion, which is just the wave equation in the three-brane metric, can be solved in terms of associated Mathieu functions. An exact expression for the reflection and absorption probabilities can be obtained in terms of the characteristic exponent of Mathieu's equation. We describe an algorithm for obtaining the low-energy behavior as a series expansion, and discuss the implications for the world-volume theory of D3-branes.

Figures (1)

• Figure 1: A diagram with $n$ quartic vertices contributing at order $O(R^{4n})$.