Abstract
We analyze tree-level string amplitudes in a linear dilaton background, motivated by its use as a gauge-invariant tracer of string interactions in scattering experiments and its genericity among simple perturbative string theory limits. A simple case is given by a lightlike dependence for the dilaton. The zero mode of the embedding coordinate in the direction of dilaton variation requires special care. Employing Gaussian wave packets and a well-defined modification of the dilaton profile far from the dominant interaction region, we obtain finite results which explicitly reproduce the interaction timescales expected from joining and splitting interactions involving oscillating strings in simple string scattering processes. There is an interesting interplay between the effects of the linear dilaton and the $i\epsilon$ prescription. In more general circumstances this provides a method for tracing the degree of non-locality in string interactions, and it gives a basis for further studies of perturbative supercritical string theory at higher loop order.
Highlights
Background and vertex operatorsThe bosonic linear dilaton CFT has Euclidean worldsheet action S = 1 4πα′d2σ√g gab∂aXμ∂bXμ + α′R Φ(X), whereΦ(X) = VμXμ, and R is the worldsheet scalar curvature.2 The string coupling gs = g0eVμXμ varies in spacetime
In order to cancel the Weyl anomaly, giving total central charge zero. This coincides with the equation of motion for the dilaton derived from the two-derivative approximation to the spacetime effective action, but it is an exact classical solution to string theory regardless of the size of V
As with the dilaton background itself, this low energy EFT description does not make clear the exactness of the mass shell condition in the linear dilaton background, which follows from the full worldsheet CFT analysis just reviewed
Summary
In order to cancel the Weyl anomaly, giving total central charge zero This coincides with the equation of motion for the dilaton derived from the two-derivative approximation to the spacetime effective action, but it is an exact classical solution to string theory regardless of the size of V. The zero mode integral in this direction is more divergent than it is for V = 0 (where it gives the energy-momentum conserving delta function). We will address this with two additional ingredients. For a lightlike linear dilaton, this is p2 = −M 2, that is psatisfies the on-shell condition in the absence of the linear dilaton In terms of this decomposition, we can write the vertex operator for a momentum eigenstate as. As with the dilaton background itself, this low energy EFT description does not make clear the exactness of the mass shell condition in the linear dilaton background, which follows from the full worldsheet CFT analysis just reviewed
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