Multiloop world-line Green functions from string theory

Abstract

We show how the multiloop bosonic Green function of closed string theory reduces to the world-line Green function as defined by Schmidt and Schubert in the limit where the string world-sheet degenerates into a Φ 3 particle diagram. To obtain this correspondence we have to make an appropriate choice of the local coordinates defined on the degenerate string world sheet. We also present a set of simple rules that specify, in the explicit setting of the Schottky parametrization, which is the corner of moduli space corresponding to a given multiloop Φ 3 diagram.

Multiloop f3 amplitudes from bosonic string theory

“…In view of this it should be clear that the string theory derivation of the Green functions (2.3), (2.4), (2.5) and (2.6) given in ref. [9] include also the symmetric Green functions (A.3) and (A.4).…”

“…[9]: In any Φ 3 -like corner of moduli space each vertex operator insertion point z i represents an external leg inserted on some very long cylinder, and we choose…”

“…In other words, we have to approach the various Φ 3 -like corners of moduli space. In each Φ 3 -like corner of moduli space there is a one-to-one map of the 3h + N complex string moduli into the SPTs of the corresponding Φ 3 diagram (plus 3h + N phases that just give rise to empty integrations), and when one takes the limit α ′ → 0, keeping all the 3h + N SPTs fixed, one approaches the singular point on the boundary of moduli space where the world-sheet degenerates into the Φ 3 particle diagram in question [9].…”

“…[9] how, in any such corner of N-punctured genus (h + 1) moduli space, where z i describes an external line inserted on the µ'th internal propagator with SPT τ (µ) n and z j a line inserted on the ν'th internal propagator with SPT τ …”

“…These corners of moduli space were studied extensively in ref. [9], where it was shown how to map the world-sheet moduli into the Schwinger proper times (SPTs) of the corresponding Φ 3 particle diagram.…”