Abstract
Abstract A formalism is given to hermitize the HAL QCD potential, which needs to be non-Hermitian except for the leading-order (LO) local term in the derivative expansion as the Nambu– Bethe– Salpeter (NBS) wave functions for different energies are not orthogonal to each other. It is shown that the non-Hermitian potential can be hermitized order by order to all orders in the derivative expansion. In particular, the next-to-leading order (NLO) potential can be exactly hermitized without approximation. The formalism is then applied to a simple case of $\Xi \Xi (^{1}S_{0}) $ scattering, for which the HAL QCD calculation is available to the NLO. The NLO term gives relatively small corrections to the scattering phase shift and the LO analysis seems justified in this case. We also observe that the local part of the hermitized NLO potential works better than that of the non-Hermitian NLO potential. The Hermitian version of the HAL QCD potential is desirable for comparing it with phenomenological interactions and also for using it as a two-body interaction in many-body systems.
Highlights
Lattice quantum chromodynamics (QCD) is a successful non-perturbative method to study hadron physics from the underlying degrees of freedom, i.e., quarks and gluons
The HAL QCD potential expressed as an energy-independent non-local potential is known to be nonHermitian due to the nature of the Nambu–Bethe–Salpeter (NBS) wave function used to extract it: While the leading-order (LO) term in the derivative expansion of the potential is local and Hermitian, the higher-order terms are in general non-Hermitian
Starting from the next-to-leading order (NLO) terms, which can be made Hermitian exactly, we have shown that the higher-order
Summary
Lattice quantum chromodynamics (QCD) is a successful non-perturbative method to study hadron physics from the underlying degrees of freedom, i.e., quarks and gluons. The HAL QCD method utilizes the NBS wave function in the non-asymptotic (interacting) region, and extracts the non-local but energy-independent potentials from the space and time dependences of the NBS wave function. The non-local potential is given by the form of the derivative expansion, which is truncated by the first few orders [12]. We propose a method to hermitize the non-Hermitian Hamiltonian order by order in terms of derivatives. We consider the other issue, the non-hermiticity of the potential in the HAL QCD method. 3, we apply our method to a non-Hermitian HAL QCD potential for in lattice QCD [12], which consists of local and second- or first-order derivative terms. In terms of the original V4 and V3 , we have ijkl ijkl ijk ijk ijkl ij ijkl ijkl i
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