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Register a new userTheoretical and practical progresses in the HAL QCD method
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Sinya Aoki
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In this report, we discuss some theoretical and practical progresses in the HAL QCD potential method. We first clarify the issue of the derivative expansion for the non-local potential in the HAL QCD method. As the non-local potential in the original literature is not uniquely defined, we propose a procedure to define a non-local potential from NBS wave functions in terms of the derivative expansion. We then demonstrate how this definition works by using quantum mechanics with a separable potential. Secondly we discuss an issue of Hermiticity of the HAL QCD potential. Since the NBS wav functions are not orthogonal to each other in general, the HAL QCD potential is necessary to be non-Hermitian. We consider the next-to-leading order potential, which can be made Hermitian exactly by the change of variables. In general we can also make the higher order HAL QCD potential Hermitian order by order in the derivative expansion. An explicit example on how the procedure works is given for lattice QCD calculations. Finally we discuss how we can extract the HAL QCD potential from the NBS wave function in the boosted system. An explicit formula for this is derived.
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A formalism is given to hermitize the HAL QCD potential, which needs to be non-hermitian except 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.
In this article, we review the HAL QCD method to investigate baryon-baryon interactions such as nuclear forces in lattice QCD. We first explain our strategy in detail to investigate baryon-baryon interactions by defining potentials in field theories such as QCD. We introduce the Nambu-Bethe-Salpeter (NBS) wave functions in QCD for two baryons below the inelastic threshold. We then define the potential from NBS wave functions in terms of the derivative expansion, which is shown to reproduce the scattering phase shifts correctly below the inelastic threshold. Using this definition, we formulate a method to extract the potential in lattice QCD. Secondly, we discuss pros and cons of the HAL QCD method, by comparing it with the conventional method, where one directly extracts the scattering phase shifts from the finite volume energies through the Luschers formula. We give several theoretical and numerical evidences that the conventional method combined with the naive plateau fitting for the finite volume energies in the literature so far fails to work on baryon-baryon interactions due to contaminations of elastic excited states. On the other hand, we show that such a serious problem can be avoided in the HAL QCD method by defining the potential in an energy-independent way. We also discuss systematics of the HAL QCD method, in particular errors associated with a truncation of the derivative expansion. Thirdly, we present several results obtained from the HAL QCD method, which include (central) nuclear force, tensor force, spin-orbital force, and three nucleon force. We finally show the latest results calculated at the nearly physical pion mass, $m_pi simeq 146$ MeV, including hyperon forces which lead to form $OmegaOmega$ and $NOmega$ dibaryons.
We make a detailed comparison between the direct method and the HAL QCD potential method for the baryon-baryon interactions, taking the $XiXi$ system at $m_pi= 0.51$ GeV in 2+1 flavor QCD and using both smeared and wall quark sources. The energy shift $Delta E_mathrm{eff}(t)$ in the direct method shows the strong dependence on the choice of quark source operators, which means that the results with either (or both) source are false. The time-dependent HAL QCD method, on the other hand, gives the quark source independent $XiXi$ potential, thanks to the derivative expansion of the potential, which absorbs the source dependence to the next leading order correction. The HAL QCD potential predicts the absence of the bound state in the $XiXi$($^1$S$_0$) channel at $m_pi= 0.51$ GeV, which is also confirmed by the volume dependence of finite volume energy from the potential. We also demonstrate that the origin of the fake plateau in the effective energy shift $Delta E_mathrm{eff}(t)$ at $t sim 1$ fm can be clarified by a few low-lying eigenfunctions and eigenvalues on the finite volume derived from the HAL QCD potential, which implies that the ground state saturation of $XiXi$($^1$S$_0$) requires $t sim 10$ fm in the direct method for the smeared source on $(4.3 mathrm{fm})^3$ lattice, while the HAL QCD method does not suffer from such a problem.
We investigate how the derivative expansion in the HAL QCD method works to extract physical observables, using a separable potential in quantum mechanics, which is solvable but highly non-local in the coordinate system. We consider three cases for inputs to determine the HAL QCD potential in the derivative expansion, (1) energy eigenfunctions (2) time-dependent wave functions as solutions to the time dependent Schrodinger equation with some boundary conditions (3) time-dependent wave function made by a linear combination of finite number of eigenfunctions at low energy to mimic the finite volume effect. We have found that, for all three cases, the potentials provide reasonable scattering phase shifts even at the leading order of the derivative expansion, and they give more accurate results as the order of the expansion increases. By comparing the above results with those from the formal derivative expansion for the separable potential, we conclude that the derivative expansion is not a way to obtain the potential but a method to extract physical observables such as phase shifts and binding energies, and that the scattering phase shifts from the derivative expansion in the HAL QCD method converge to the exact ones much faster than those from the formal derivative expansion of the separable potential.
A comparative study between the Luschers finite volume method and the time-dependent HAL QCD method is given for the $XiXi$($^1mathrm{S}_0$) interaction as an illustrative example. By employing the smeared source and the wall source for the interpolating operators, we show that the effective energy shifts $Delta E_{rm eff} (t)$ in Luschers method do not agree between different sources, yet both exhibit fake plateaux. On the other hand, the interaction kernels $V(vec{r})$ obtained from the two sources in the HAL QCD method agree with each other already for modest values of $t$. We show that the energy eigenvalues $Delta E(L)$ in finite lattice volumes ($L^3$) calculated by $V(vec{r})$ indicate that there is no bound state in the $XiXi(^1mathrm{S}_0)$ channel at $m_{pi}=0.51$ GeV in 2+1 flavor QCD.
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