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Field theory models for tachyon and gauge field string dynamics

Joseph A. Minahan, Barton Zwiebach

TL;DR

Minahan and Zwiebach study toy field theory models for tachyon condensation on unstable branes by analyzing a family of lump solutions and their ℓ→∞ limit. The ℓ→∞ model yields a Gaussian lump with a potential V_∞(φ) = -\frac{1}{4} φ^2 \ln φ^2, producing a discrete, equally spaced lump spectrum including a tachyon with m^2 = -1 and a massless translation mode, while the continuum is absent; tachyon condensation drives all lump fluctuations to infinite mass, leaving no residual spectrum. They extend the framework to higher-codimension lumps with exact profiles and universal tension relations, and they incorporate a gauge field in a solvable, gauge-fixed way that yields a massless gauge field on the lump and a modified transverse spectrum. The results shed light on how higher-level states contribute to large marginal deformations in string field theory, while also highlighting the model’s limitations, such as the absence of a fully BRST-invariant formulation and discrepancies with exact open string dynamics.

Abstract

In hep-th/0008227, the unstable lump solution of φ^3 theory was shown to have a spectrum governed by the solvable Schroedinger equation with the \ell=3 reflectionless potential and was used as a model for tachyon condensation in string theory. In this paper we study in detail an \ell\to \infty scalar field theory model whose lump solution mimics remarkably the string theory setup: the original field theory tachyon and the lump tachyon have the same mass, the spectrum of the lump consists of equally spaced infinite levels, there is no continuous spectrum, and nothing survives after tachyon condensation. We also find exact solutions for lumps with codimension \ge 2, and show that that their tensions satisfy (1/(2π)) (T_p/ T_{p+1})=e/(\sqrt{2π}) \approx 1.08. We incorporate gauge fixed couplings to a U(1) gauge field which preserve solvability and result in massless gauge fields on the lump.

Field theory models for tachyon and gauge field string dynamics

TL;DR

Minahan and Zwiebach study toy field theory models for tachyon condensation on unstable branes by analyzing a family of lump solutions and their ℓ→∞ limit. The ℓ→∞ model yields a Gaussian lump with a potential V_∞(φ) = -\frac{1}{4} φ^2 \ln φ^2, producing a discrete, equally spaced lump spectrum including a tachyon with m^2 = -1 and a massless translation mode, while the continuum is absent; tachyon condensation drives all lump fluctuations to infinite mass, leaving no residual spectrum. They extend the framework to higher-codimension lumps with exact profiles and universal tension relations, and they incorporate a gauge field in a solvable, gauge-fixed way that yields a massless gauge field on the lump and a modified transverse spectrum. The results shed light on how higher-level states contribute to large marginal deformations in string field theory, while also highlighting the model’s limitations, such as the absence of a fully BRST-invariant formulation and discrepancies with exact open string dynamics.

Abstract

In hep-th/0008227, the unstable lump solution of φ^3 theory was shown to have a spectrum governed by the solvable Schroedinger equation with the \ell=3 reflectionless potential and was used as a model for tachyon condensation in string theory. In this paper we study in detail an \ell\to \infty scalar field theory model whose lump solution mimics remarkably the string theory setup: the original field theory tachyon and the lump tachyon have the same mass, the spectrum of the lump consists of equally spaced infinite levels, there is no continuous spectrum, and nothing survives after tachyon condensation. We also find exact solutions for lumps with codimension \ge 2, and show that that their tensions satisfy (1/(2π)) (T_p/ T_{p+1})=e/(\sqrt{2π}) \approx 1.08. We incorporate gauge fixed couplings to a U(1) gauge field which preserve solvability and result in massless gauge fields on the lump.

Paper Structure

This paper contains 11 sections, 62 equations, 3 figures.

Table of Contents

  1. Introduction and Summary
  2. Finite $\ell$ models and the $\ell\to \infty$ model
  3. Introducing the models
  4. Field theory spectrum on the lumps
  5. Tachyon condensation and mass flow at $\ell= \infty$
  6. Transporting the $\ell=\infty$ lumps
  7. Higher codimension lumps and descent relations in the $\ell=\infty$ model
  8. Adding gauge fields
  9. The dynamics of the gauge field on the brane
  10. Giving a mass to the gauge field transverse to the brane
  11. Concluding Remarks

Figures (3)

  • Figure 1: The $\ell\to \infty$ potential $V_\infty(\phi) = -{1\over 4} \phi^2 \ln \phi^2$.
  • Figure 2: The second derivative ${{V_\infty}}"(\phi) = -{1\over 2} (3 + \ln \phi^2)$.
  • Figure 3: The dashed line is the behavior of $A_0 = 1+T$ as a function of the lump displacement $x_0$. The continuous line shows the marginal mode $A_1(x_0)$, which takes a maximum value for $x_0=2$. The dotted line is $A_2(x_0)$, which takes a maximum value after $A_1$ does.