Multiloop World-Line Green Functions from String Theory

TL;DR

The paper derives multiloop world-line Green functions for $Φ^3$ diagrams by analyzing the field-theory limit of the bosonic closed-string Green function on degenerating Riemann surfaces. It provides a concrete prescription for choosing local coordinates on degenerate surfaces and establishes a mapping, via the Schottky parametrization, between string moduli and Schwinger proper times. The authors verify that the string Green function reduces to the Schmidt–Schubert particle Green functions for all relevant insertion configurations ($G^{(m)}_{00}$, $G^{(m)}_{ii}$, $G^{(m)}_{i0}$, $G^{(m)}_{ij}$) in the $α'\to 0$ limit, thereby connecting two complementary formalisms for multiloop amplitudes. The work highlights both the potential for a unified string-based approach to multiloop amplitudes and the open challenges in extending to Yang–Mills, including issues with tachyons, contact terms, and moduli-space integration over interpolating regions. Overall, it lays a geometrical groundwork toward Bern–Kosower–style rules for multiloop gauge theories grounded in 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.

Figures (6)

• Figure 1: Example of a tree with one side-branch. The branch endpoints are $z_{B_0}=z_{\beta_0}$ and $z_{B_1}=z_{\alpha_1}$. The vertices $P,Q,R$ and $S$ are labelled by $z_{\beta_1}, z_2, z_{\alpha_1}$ and $z_1$ respectively.
• Figure 2: How to obtain eq. (\ref{['SPTabs']}). The thick line represents the new SPT flow. See the text for details.
• Figure 3: This is a typical representative of the class of $\Phi^3$ diagrams relevant for the Green function $G_{00}^{(m)}$. It corresponds to a particular ordering of the legs $z_1$, $z_2$, $z_{\alpha_i}$ and $z_{\beta_i}$. The other legs (labelled by $z_{\alpha_j}$ and $z_{\beta_j}$, $j \neq i$) have been suppressed.
• Figure 4: The typical $\Phi^3$ diagram relevant for the Green function $G_{ii}^{(m)}$. The legs labelled by $z_{\alpha_j}$ and $z_{\beta_j}$, $j \neq i$, have been suppressed.
• Figure 5: The typical $\Phi^3$ diagram relevant for the Green function $G_{i0}^{(m)}$. The legs labelled by $z_{\alpha_j}$ and $z_{\beta_j}$, $j \neq i$, have been suppressed.