Threshold expansion of the three-particle quantization condition

Abstract

We recently derived a quantization condition for the energy of three relativistic particles in a cubic box [M. T. Hansen and S. R. Sharpe, Phys. Rev. D 90, 116003 (2014); M. T. Hansen and S. R. Sharpe, Phys. Rev. D 92, 114509 (2015)]. Here we use this condition to study the energy level closest to the three-particle threshold when the total three-momentum vanishes. We expand this energy in powers of $1/L$, where $L$ is the linear extent of the finite volume. The expansion begins at $\mathcal{O}(1/{L}^{3})$, and we determine the coefficients of the terms through $\mathcal{O}(1/{L}^{6})$. As is also the case for the two-particle threshold energy, the $1/{L}^{3}$, $1/{L}^{4}$ and $1/{L}^{5}$ coefficients depend only on the two-particle scattering length $a$. These can be compared to previous results obtained using nonrelativistic quantum mechanics [K. Huang and C. N. Yang, Phys. Rev. 105, 767 (1957); S. R. Beane, W. Detmold, and M. J. Savage, Phys. Rev. D 76, 074507 (2007); S. Tan, Phys. Rev. A 78, 013636 (2008)], and we find complete agreement. The $1/{L}^{6}$ coefficients depend additionally on the two-particle effective range $r$ (just as in the two-particle case) and on a suitably defined threshold three-particle scattering amplitude (a new feature for three particles). A second new feature in the three-particle case is that logarithmic dependence on $L$ appears at $\mathcal{O}(1/{L}^{6})$. Relativistic effects enter at this order, and the only comparison possible with the nonrelativistic result is for the coefficient of the logarithm, where we again find agreement. For a more thorough check of the $1/{L}^{6}$ result, and thus of the quantization condition, we also compare to a perturbative calculation of the threshold energy in relativistic $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ theory, which we have recently presented in [M. T. Hansen and S. R. Sharpe, Phys. Rev. D 93, 014506 (2016)]. Here, all terms can be compared, and we find full agreement.