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One-loop mass corrections and decay widths of Type II heavy string states

Massimo Bianchi, Maurizio Firrotta, Lorenzo Grimaldi

Abstract

We start a systematic investigation of the one-loop mass corrections to (super-)string massive higher-spin states. While the imaginary part of the relevant amplitudes are finite, being related to the width of the decay of the states into two lower-mass states at tree level, the real part is generally IR-divergent and needs regularization and renormalization. We mostly focus on states of the first Regge trajectory in the NS-NS sector of Type-II string theories. We explicitly derive a closed-form expression for the integral over the insertion point, relying on properties of elliptic functions and lattice sums. We then regularize the IR divergent integral over the modular parameter of the torus, applying the $i\varepsilon$-prescription in string theory. As a result we compute the desired mass corrections up to level $N = 10$ and analyze their behavior at increasing $N$. Finally, we speculate on the existence of mixing among lower-spin states and conjecture that the one-loop mass matrix be governed by some random matrix theory.

One-loop mass corrections and decay widths of Type II heavy string states

Abstract

We start a systematic investigation of the one-loop mass corrections to (super-)string massive higher-spin states. While the imaginary part of the relevant amplitudes are finite, being related to the width of the decay of the states into two lower-mass states at tree level, the real part is generally IR-divergent and needs regularization and renormalization. We mostly focus on states of the first Regge trajectory in the NS-NS sector of Type-II string theories. We explicitly derive a closed-form expression for the integral over the insertion point, relying on properties of elliptic functions and lattice sums. We then regularize the IR divergent integral over the modular parameter of the torus, applying the -prescription in string theory. As a result we compute the desired mass corrections up to level and analyze their behavior at increasing . Finally, we speculate on the existence of mixing among lower-spin states and conjecture that the one-loop mass matrix be governed by some random matrix theory.
Paper Structure (15 sections, 148 equations, 2 figures, 1 table)

This paper contains 15 sections, 148 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Vertex Operators and First Regge Trajectory states
  3. Fermionic and Bosonic contractions and one-loop amplitudes
  4. Integration over the worldsheet coordinate
  5. Modular integral: Regularization and $i\varepsilon$-prescription
  6. Explicit results
  7. $N=2$ case in Type II
  8. $N=3$ case in Type II
  9. $N=4$ case in Type II
  10. Summary and plot for FRT states at level $N=5,\dots,10$
  11. Conclusions and outlook
  12. Useful definitions and series expansions
  13. Full analytic computation of the worldsheet integral
  14. Worldsheet integral at level $N=2$ using $SU(2)$ characters
  15. A generic integral at level $N$ with lattice sums

Figures (2)

  • Figure 1: Pictorial representation of the two-point torus amplitude
  • Figure 2: Fit of the real and imaginary part of $\mathcal{A}^{(N)}$ with the mass level.