Abstract
Single state saturation of the temporal correlation function is a key condition to extract physical observables such as energies and matrix elements of hadrons from lattice QCD simulations. A method commonly employed to check the saturation is to seek for a plateau of the observables for large Euclidean time. Identifying the plateau in the cases having nearby states, however, is non-trivial and one may even be misled by a fake plateau. Such a situation takes place typically for the system with two or more baryons. In this study, we demonstrate explicitly the danger from a possible fake plateau in the temporal correlation functions mainly for two baryons ($\Xi\Xi$ and $NN$), and three and four baryons ($^3{\rm He}$ and $^4{\rm He})$ as well, employing (2+1)-flavor lattice QCD at $m_{\pi}=0.51$ GeV on four lattice volumes with $L=$ 2.9, 3.6, 4.3 and 5.8 fm. Caution is given for drawing conclusion on the bound $NN$, $3N$ and $4N$ systems only based on the temporal correlation functions.
Highlights
Single state saturation of the temporal correlation function is a key condition to extract physical observables such as energies and matrix elements of hadrons from lattice QCD simulations
For multi-hadrons, the energy shift of the whole system on the lattice relative to the threshold defined by the sum of each hadron masses is of interest, since it has information on the binding energy and the scattering phase shift [1]
The asymptotic formula eq (2.1) says that signal to noise ratio (S/N) becomes worse for bigger t as well as larger numbers of baryons and/or smaller quark mass
Summary
Even though the plateau method works in principle, and in practice for, e.g., the ground state meson masses, the method sometimes suffers from difficulties, in particular, in the case of multi-baryon systems. The asymptotic formula eq (2.1) says that S/N becomes worse for bigger t as well as larger numbers of baryons and/or smaller quark mass (i.e. lighter meson) This may prevent us from taking sufficiently large t to guarantee the t independence of EAeff (t), so that we can not reliably control systematic errors from excited state contaminations. We consider δEel = 50 MeV, which is the typical lowest excitation energy of elastic two-baryon scattering states in our numerical setup with La = 4.3 fm lattice (see section 3), while we take δEinel = 500 MeV, which is roughly the order of mπ in our simulations. In the absence of the excited elastic state, the effective energy shift ∆EBeffB(t) smoothly approaches to the plateau (from above for the positive c1/b1) and t 1 fm is sufficient to reduce the systematic error from the contamination to the level of accuracy we need for ∆EBB. Due to the exponentially increasing noise in time, this task is extremely difficult, and becomes even impossible practically at physical quark masses with a larger lattice box, since δEel becomes much smaller as discussed before
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