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Register a new userQuark mass dependence of on-shell and half off-shell scattering amplitudes from Bethe-Salpeter wave function inside the interaction range
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We evaluate scattering amplitudes at on-shell and half off-shell for $I=2$ S-wave two-pion system using the Bethe-Salpeter wave function inside the interaction range in the quenched QCD. The scattering length and effective range are extracted from these scattering amplitudes. Quark mass dependence of them is investigated with the pion mass ranged in $0.52$--$0.86$~GeV. We examine consistency between a result by the conventional finite volume method and our estimate, as well as the phenomenological value.
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We propose a method to calculate scattering amplitudes using the Bethe-Salpeter wave function inside the interaction range on the lattice. For an exploratory study of this method, we evaluate a scattering length of $I=2$ S-wave two pions by the use of the on-shell scattering amplitude. Our result is confirmed to be consistent with the value obtained from the conventional finite volume method. The half-off-shell scattering amplitude is also evaluated.
We evaluate $I=2$ two-pion scattering length through the scattering amplitude obtained by the Bethe-Salpeter wave function inside the interaction range. The scattering length is computed with $m_pi = 0.52-0.86$ GeV in the quenched lattice QCD. Furthermore, the half-off-shell amplitude is calculated, from which the effective range is extracted. Our results are compared with those by the conventional finite size method and by chiral perturbation theory to confirm consistency.
We observe that the ratio of the on-shell scattering amplitude to the Bethe-Salpeter (BS) wave function outside the interaction range is almost independent of time in our quenched calculation of the $I=2$ two-pion scattering with almost zero momentum. In order to discuss the time independence, we present a relation between the two-pion scattering amplitude and the surface term of the BS wave function at the boundary. Using the relation under some assumptions, we show that the ratio is independent of time if the two-pion four-point function in early time is dominated by scattering states with almost zero momentum in addition to the ground state of the two-pion scattering.
Based on lattice non-relativistic QCD (NRQCD) studies we present results for Bethe-Salpeter amplitudes for $Upsilon(1S)$, $Upsilon(2S)$ and $Upsilon(3S)$ in vacuum as well as in quark-gluon plasma. Our study is based on 2+1 flavor $48^3 times 12$ lattices generated using the Highly Improved Staggered Quark (HISQ) action and with a pion mass of $161$ MeV. At zero temperature the Bethe-Salpeter amplitudes follow the expectations based on non-relativistic potential models. At non-zero temperatures, the interpretation of Bethe-Salpeter amplitudes turns out to be more nuanced, but consistent with our previous lattice QCD study of excited Upsilons in quark-gluon plasma.
We discuss an exact relation between the two-particle scattering amplitude and the Bethe-Salpeter (BS) wave function inside the interaction range in quantum field theory. In the relation the reduced BS wave function defined by the BS wave function plays an essential role. Through the relation the on-shell and half off-shell amplitudes can be calculated. We also show that the solution of Schrodinger equation with the effective potential determined from the BS wave function gives a correct on-shell scattering amplitude only at the momentum where the effective potential is determined. Furthermore we discuss a derivative expansion of the reduced BS wave function and a condition to obtain results independent of the interpolating operators in the time-dependent HALQCD method.
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