One loop mass renormalization of unstable particles in superstring theory

Abstract

Most of the massive states in superstring theory are expected to undergo mass renormalization at one loop order. Typically these corrections should contain imaginary parts, indicating that the states are unstable against decay into lighter particles. However in such cases, direct computation of the renormalized mass using superstring perturbation theory yields divergent result. Previous approaches to this problem involve various analytic continuation techniques, or deforming the integral over the moduli space of the torus with two punctures into the complexified moduli space near the boundary. In this paper we use insights from string field theory to describe a different approach that gives manifestly finite result for the mass shift satisfying unitarity relations. The procedure is applicable to all states of (compactified) type II and heterotic string theories. We illustrate this by computing the one loop correction to the mass of the first massive state on the leading Regge trajectory in SO(32) heterotic string theory.

Closed superstring field theory and its applications

“…As mentioned in §4.3, this will require multiplying the vertices by i and propagators by −i relative to the Feynman rules described in §3. Our discussion will mainly follow [26,132].…”

“…Appendix G illustrates this procedure for choosing the loop energy integration contour for a simple Feynman diagram. This procedure was used in [132] to compute the real and imaginary parts of the renormalized mass 2 of a massive particle in superstring theory at one loop order.…”

The decay width of stringy hadrons

“…The first treatments of the problem were given in [7][8][9][10]62]. Since then, it was further addressed in the context of the decay of mesons in [34,44,[63][64][65][66][67][68][69][70][71][72][73][74].…”

Highly excited strings I: Generating function

“…In a series of recent papers [65][66][67][68][69][70][71][72] Sen and collaborators have revisited various aspects of superstring theory (unitarity of string amplitudes, mass and wavefunction renormalisation [67,71,72], perturbation theory around dynamically shifted string vacua [70], offshell string amplitudes [73], Wick rotations and analytic continuations [65][66][67], one-particle ir-reducible (1PI) quantum effective actions, etc.). In a very careful and complete study [65] Pius and Sen derived Cutkosky rules for superstring field theory amplitudes to all orders in perturbation theory by providing a prescription for taking integration contours of loop energies in the complex plane (which would naively otherwise yield divergent results for the corresponding S-matrix elements).…”

“…In the approach of Witten [75] one is to deform the integration cycles over moduli space of punctured Riemann surfaces into a complexified moduli space, and this establishes consistency of the former fixed-loop momenta approach with S-matrix unitarity. Sen then recently also discussed [67] an application of the fixedloop momenta approach, building on earlier work [71,72] and in particular [65], namely mass renormalisation of unstable massive string states (where a naive computation yields divergent results for the two-point one-loop amplitude), explaining how to obtain finite results that are consistent with unitarity. The basic reason for the aforementioned divergences are ultimately due to the fact that the analogue of the 'i ' prescription of quantum field theory is somewhat subtle in string theory [75] because string amplitudes are most naturally defined in Euclidean space where they are real [81].…”