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Inter-Brane Potential and the Decay of a non-BPS-D-brane to Closed Strings

Saswat Sarangi, S. -H. Henry Tye

TL;DR

The paper analyzes the inter-brane potential and decay of non-BPS D-branes by computing a complex per-unit-volume potential $V(y)$ for $Dp-\overline{D}p$ pairs and its angle-dependent generalization $V(\theta,y)$. It develops both open-string and closed-string channel calculations, showing that the imaginary part of the one-loop open-string potential, controlled by the tachyon, is matched by a corresponding imaginary part in the closed-string channel arising from the dense massive spectrum. It then interprets the imaginary part via the optical theorem as the decay width $\Gamma = V_p\, Im\,V(0)$ of a non-BPS $Dp$-brane into on-shell closed strings, and discusses the decay dynamics through very massive to light closed-string modes. An appendix provides Hardy-Ramanujan asymptotics for the closed-string level degeneracy, enabling precise open/closed-channel matching and reinforcing the dual description of the brane instability.

Abstract

We calculate the potential for $Dp-\Dbar p$ pair and show that the coincident $Dp-\Dbar p$ system has $(11-p)$ tachyonic modes, with $(9-p)$ of them due to radiative corrections. We propose that the decay width of an unstable non-BPS-$Dp$-brane to closed strings is given by the imaginary part of the one-loop contribution to the effective potential of the open string tachyon mode.

Inter-Brane Potential and the Decay of a non-BPS-D-brane to Closed Strings

TL;DR

The paper analyzes the inter-brane potential and decay of non-BPS D-branes by computing a complex per-unit-volume potential for pairs and its angle-dependent generalization . It develops both open-string and closed-string channel calculations, showing that the imaginary part of the one-loop open-string potential, controlled by the tachyon, is matched by a corresponding imaginary part in the closed-string channel arising from the dense massive spectrum. It then interprets the imaginary part via the optical theorem as the decay width of a non-BPS -brane into on-shell closed strings, and discusses the decay dynamics through very massive to light closed-string modes. An appendix provides Hardy-Ramanujan asymptotics for the closed-string level degeneracy, enabling precise open/closed-channel matching and reinforcing the dual description of the brane instability.

Abstract

We calculate the potential for pair and show that the coincident system has tachyonic modes, with of them due to radiative corrections. We propose that the decay width of an unstable non-BPS--brane to closed strings is given by the imaginary part of the one-loop contribution to the effective potential of the open string tachyon mode.

Paper Structure

This paper contains 7 sections, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. $D{p}-\overline{D} {p}$ Potential
  3. $Dp-\overline{D} p$ Potential In the Open String Channel
  4. Branes-at-an-Angle Potential
  5. The $D{p}-\overline{D} {p}$ Potential in the Closed String Channel
  6. Decay of A non-BPS Brane to Closed Strings
  7. Appendix : Hardy-Ramanujan formula for the closed string level degeneracy

Figures (3)

  • Figure 1: The potential $V(y)$ as a function of the separation $y$ for the $Dp-\overline{D} p$-brane pair for $p=$4, where ${\alpha^{\prime}}=$1. The dashed curve is the imaginary part of $V(y)$. The thick line is the real part of $V(y)$. The Coulombic potential (the thin red curve) is shown for comparison.
  • Figure 2: The vertical dashed line on the left side indicates taking the imaginary part of the $<Dp|$ closed strings $|Dp>$ amplitude in the decay of a non-BPS $Dp$ brane to closed string modes. The right side is $|f(Dp \to$ closed string$)|^2$.
  • Figure 3: The spectral flow of the lowest two level open string modes as a function of the angle $\theta$ between the branes for $y=0$. $\theta=$0 corresponds to the 2 parallel BPS $Dp$-branes (left) while $\theta=\pi$ corresponds to the $Dp-\overline{D} p$ case (right). The red (solid) lines show the NS states, the blue (dashed) lines show the R states. The number written above a solid/dashed line is the number of states represented by the line. The numbers in boldface to the right of the diagram show the states at the given mass levels for the $\theta = \pi$ case. $+$ is for bosons and $-$ is for fermions.