A Solvable Toy Model for Tachyon Condensation in String Field Theory

Abstract

The lump solution of φ^3 field theory provides a toy model for unstable D-branes of bosonic string theory. The field theory living on this lump is itself a cubic field theory involving a tachyon, two additional scalar fields, and a scalar field continuum. Its action can be written explicitly because the fluctuation spectrum of the lump turns out to be governed by a solvable Schroedinger equation; the \ell=3 case of a series of reflectionless potentials. We study the multiscalar tachyon potential both exactly and in the level expansion, obtaining insight into issues of convergence, branches of the solution space, and the mechanism for removal of states after condensation. In particular we find an interpretation for the puzzling finite domain of definition of string field marginal parameters.

Figures (6)

• Figure 1: The cubic potential $V(\phi)$. The tachyonic vacuum is at $\phi=0$ and the locally stable vacuum is at $\phi=1$.
• Figure 2: The profile of the lump solution ${\overline\phi} (x)$. As $x\to \pm \infty$, ${\overline\phi} \to +1$, which is the expectation value for the locally stable vacuum.
• Figure 3: Plot of the function $C(x)$ defined in (\ref{['errf']}). At level zero we have the curve that extends down to about 0.8 (recall that the profile ${\overline\phi}$ went all the way down to -0.5). The level (2,4) curve is dashed, and the level (2,6) curve is continuous and close to the (2,4) curve.
• Figure 4: The solid curve shows the flow of the mass squared for the lowest mass state on the lump. The flow begins at $m^2=-5/4$. The dashed line shows the flow of the massive state.
• Figure 5: The solid curve shows the expectation value of the lump marginal parameter $\psi_0$ as a function of the displacement $a$ of the lump. Note that the marginal parameter first increases, reaches a maximum, and then decreases. The dashed line shows the expectation value of the tachyon field $\phi_0$ as the lump is displaced. Note that as the displacement is large the expectation value of $\phi_0$ reaches the critical value ${\overline\phi}_0$ associated to the stable vacuum.