Nambu-Bethe-Salpeter Research Articles

Building Diquark Model from Lattice QCD

A novel Lattice QCD (LQCD) method to determine the quark-diquark ($q$-$D$) interaction potential together with the diquark mass ($m_D$) is proposed. Similar to the HAL QCD method, $q$-$D$ potential is determined by demanding it to reproduce the $q$-$D$ equal-time Nambu-Bethe-Salpeter (NBS) wave function. To do this, it is necessary to use the masses of the quark and the diquark as inputs, which however are not straightforwardly obtained because of the color confinement of QCD. In this work, masses of quark and diquark are determined by demanding that the p-wave spectrums from the two-point correlators be reproduced by the potentials for $c$-$\bar{c}$ and $q$-$D$ sectors determined from the NBS wave functions. Numerical calculations are performed by using 2+1 flavor QCD gauge configurations with the pion mass $m_\pi\simeq 700$ MeV generated by PACS-CS collaboration. We apply our method to the $c$-$\bar c$ system and the charm-diquark system ($\Lambda_c$ baryon) to obtain the charm quark mass, diquark mass and the $c$-$D$ potential. Our preliminary analysis leads to the diquark mass $m_D \simeq 1.127$ GeV which is roughly consistent with a naive estimate based on the constituent quark picture, i.e., $m_{D} \simeq m_{\rho} \simeq 1.12$ GeV and $m_{D} \simeq 2m_N/3 \simeq 1.06$ GeV.

Lattice QCD and Baryon-Baryon Interactions: HAL QCD Method

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 L\"uscher's 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 $\Omega\Omega$ and $N\Omega$ dibaryons.

Partial wave decomposition on the lattice and its applications to the HAL QCD method

The approximated partial wave decomposition method to the discrete data on a cubic lattice, developed by C. W. Misner, is applied to the calculation of $S$-wave hadron-hadron scatterings by the HAL QCD method in lattice QCD. We consider the Nambu-Bethe-Salpeter (NBS) wave function for the spin-singlet $\Lambda_c N$ system calculated in the $(2+1)$-flavor QCD on a $(32a~\mathrm{fm})^3$ lattice at the lattice spacing $a\simeq0.0907$ fm and $m_\pi \simeq 700$ MeV. We find that the $l=0$ component can be successfully extracted by Misner's method from the NBS wave function projected to $A_1^+$ representation of the cubic group, which contains small $l\ge 4$ components. Furthermore, while the higher partial wave components are enhanced so as to produce significant comb-like structures in the conventional HAL QCD potential if the Laplacian approximated by the usual second order difference is applied to the NBS wave function, such structures are found to be absent in the potential extracted by Misner's method, where the Laplacian can be evaluated analytically for each partial wave component. Despite the difference in the potentials, two methods give almost identical results on the central values and on the magnitude of statistical errors for the fits of the potentials, and consequently on the scattering phase shifts. This indicates not only that Misner's method works well in lattice QCD with the HAL QCD method but also that the contaminations from higher partial waves in the study of $S$-wave scatterings are well under control even in the conventional HAL QCD method. It will be of interest to study interactions in higher partial wave channels in the HAL QCD method with Misner's decomposition, where the utility of this new technique may become clearer.

Hermitizing the HAL QCD potential in the derivative expansion

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.

Lattice quantum chromodynamics and baryon-baryon interactions

After briefly reviewing the theoretical concepts and numerical methods in lattice QCD, recent simulation results of the hadron masses and hadron interactions with nearly physical quark masses are presented. Special emphasis is placed on the baryon-baryon interactions on the basis of the HAL QCD method where the integro-differential equation for the equal-time Nambu–Bethe–Salpeter amplitude plays a key role to bridge a gap between the multi-baryon correlation and the scattering observable such as the phase shift.

Asymptotic behavior of Nambu–Bethe–Salpeter wave functions for scalar systems with a bound state

We study the asymptotic behaviors of the Nambu-Bethe-Salpeter (NBS) wave functions, which are important for the HAL QCD potential method to extract hadron interactions, in the case that a bound state exists in the system. We consider the complex scalar particles, two of which lead to the formation of a bound state. In the case of the two-body system, we show that the NBS wave functions for the bound state as well as scattering states in the asymptotic region behave like the wave functions in quantum mechanics, which carry the information of the binding energy as well as the scattering phase shift. This analysis theoretically establishes under some conditions that the HAL QCD potential can correctly reproduce not only the scattering phase shift but also the binding energy. As an extension of the analysis, we also study the asymptotic behaviors of all possible NBS wave functions in the case of the three-body systems, two of which can form a bound states.

Baryon interactions from lattice QCD with physical masses — strangeness S = -1 sector —

We present our recent results of baryon interactions with strangeness S = −1 based on Nambu-Bethe-Salpeter (NBS) correlation functions calculated fromlattice QCD with almost physical quark masses corresponding to (mk,mk) ≈ (146, 525) MeV and large volume (La)4 ≈ (96a)4 ≈ (8.1 fm)4. In order to perform a comprehensive study of baryon interactions, a large number of NBS correlation functions from NN to ΞΞ are calculated simultaneously by using large scale computer resources. In this contribution, we focus on the strangeness S = −1 channels of the hyperon interactions by means of HAL QCD method. Four sets of three potentials (the 3S1 − 3 D1 central, 3S1 − 3 D1 tensor, and the 1S0 central potentials) are presented for the ∑N − ∑N (the isospin I = 3/2) diagonal, the ∧N − ∧N diagonal, the ∧N → ∑N transition, and the ∑N − ∑N (I = 1/2) diagonal interactions. Scattering phase shifts for ∑N (I = 3/2) system are presented.

Lattice QCD studies on baryon interactions in the strangeness -2 sector with physical quark masses

We investigate baryon-baryon (BB) interactions in the strangeness S = −2 sector via the coupled-channel HAL QCD method which enables us to extract the scattering observables from Nambu-Bethe-Salpeter (NBS) wave function on the lattice. The simulations are performed with (almost) physical quark masses (mπ = 146MeV) and a huge lattice volume of La = 8.1fm. We discuss the fate of H-dibaryon state through the ΛΛ and NΞ coupled-channel scatterings

Recent developments on LQCD studies of nuclear force

Recent studies of (hyperonic) nuclear forces based on lattice QCD are presented. An energy independent nonlocal potential can be obtained from Nambu–Bethe–Salpeter wave function which is measured from the lattice QCD calculation with a set of interpolating fields of baryons, where a time-dependent Schrödinger equation in imaginary time is employed so as to avoid the exact ground state saturation in the numerical computation. The central and the tensor potentials of ΛN and ΣN with isospin I = 2/3 channels are presented as specific numerical examples. We also present an attempt to study the four-nucleon bound state problem using the lattice nuclear potential with stochastic variational method.

Spin–orbit force from lattice QCD

We present a first attempt to determine nucleon–nucleon potentials in the parity-odd sector, which appear in the P11, P03, P13, P23–F23 channels, in Nf=2 lattice QCD simulations. These potentials are constructed from the Nambu–Bethe–Salpeter wave functions for JP=0−,1− and 2−, which correspond to the A1−, T1− and T2−⊕E− representation of the cubic group, respectively. We have found a large and attractive spin–orbit potential VLS(r) in the isospin-triplet channel, which is qualitatively consistent with the phenomenological determination from the experimental scattering phase shifts. The potentials obtained from lattice QCD are used to calculate the scattering phase shifts in the P11, P03, P13 and P23–F23 channels. The strong attractive spin–orbit force and a weak repulsive central force in spin-triplet P-wave channels lead to an attraction in the P23 channel, which is related to the P-wave neutron paring in neutron stars.

Asymptotic behavior of Nambu-Bethe-Salpeter wave functions for multiparticles in quantum field theories

We derive asymptotic behaviors of the Nambu-Bethe-Salpeter (NBS) wave function at large space separations for systems with more than 2 particles in quantum field theories. To deal with $n$-particles in the center of mass flame coherently, we introduce the Jacob coordinates of $n$ particles and then combine their $3(n-1)$ coordinates into the one spherical coordinate in $D=3(n-1)$ dimensions. We parametrize on-shell $T$-matrix for $n$-particle system of scalar fields at low energy, using the unitarity constraint of the $S$-matrix. We then express asymptotic behaviors of the NBS wave function for $n$ particles at low energy, in terms of parameters of $T$-matrix, and show that the NBS wave function carry the information of $T$-matrix such as phase shifts and mixing angles of the $n$-particle system in its own asymptotic behavior, so that the NBS wave function can be considered as the scattering wave of $n$-particles in quantum mechanics. This property is one of the essential ingredients of the HAL QCD scheme to define "potential" from the NBS wave function in quantum field theories such as QCD. Our results, together with an extension to systems with spin 1/2 particles, justify the HAL QCD's definition of potentials for 3 or more nucleons(baryons) in terms the NBS wave functions.

Coupled channel approach to hyperonic interactions from lattice QCD

Abstract We studied hyperonic interactions by lattice QCD simulations through a coupled channel formalism. Our approach to baryon–baryon interactions derives a potential from inverting the Schrodinger equation using a Nambu–Bethe–Salpeter (NBS) wave function simulated on the lattice. The quark mass dependences and flavor SU(3) breaking effects of the potential matrix are also discussed by comparing with results of gauge configurations obtained with different quark masses. Our numerical results are obtained from three ensembles of 2 + 1 -flavor QCD gauge configurations, which correspond to m π ∼ 700 , 570 and 410 MeV , provided by the PACS-CS Collaboration.

Coupled Channel Approach to S-Wave Hyperonic Interactions from Lattice QCD

We studied hyperonic interactions by lattice QCD simulation throuth coupled channel formalism. Our approach to baryon-baryon interactions is deriving a potential from inverting Schrodinger equation using Nambu-Bethe-Salpeter (NBS) wave function simulated on the lattice. The quark mass dependences and flavor SU(3) breaking effects of the potential matrix are also discussed by comparing with results of gauge configurations with different quark masses. Our numerical results are obtained from three ensembles of 2+1 flavor QCD gauge configurations, which corresponds to mπ ~ 700, 570 and 410 MeV, provided by the PACS-CS Collaboration.

SHORT DISTANCE REPULSION AMONG BARYONS

We review recent investigations on the short distance behaviors of potentials among baryons, which are formulated through the Nambu–Bethe–Salpeter (NBS) wave function. After explaining the method to define the potentials, we analyze the short distance behavior of the NBS wave functions and the corresponding potentials by combining the operator product expansion (OPE) and a renormalization group (RG) analysis in the perturbation theory (PT) of QCD. These analytic results are compared with numerical results obtained in lattice QCD simulations.

Nuclear Forces in the Odd Parity Sector and the LS Forces from Lattice QCD

The first lattice QCD calculation for the spin-orbit potential as well as the central and the tensor potentials in parity odd sector is presented for the two-nucleon system. These potentials are extracted from the Nambu-Bethe-Salpeter (NBS) wave functions for total angular momenta \({J^{P}=A_1^{-}(0^-),T_1^{-}(1^-),T_2^{-}\oplus E^{-}(2^-)}\) based on the representation theory of the cubic group.

ΛN and ΣN Interactions from Lattice QCD

We present our recent study on ΛN and ΣN (isospin I = 3/2) interactions by measuring Nambu–Bethe–Salpeter wave functions on the Lattice QCD. The lattice QCD calculation is performed by using the Nf = 2 + 1 gauge configurations generated by PACS-CS collaboration together with employing an improved method to obtain potentials in lattice QCD simulations. For the 1S0 channel, the central ΣN (I = 3/2) potential and the central ΛN (1S0) potential are found to be very similar. For the spin triplet (3S1−3D1) channels, the central ΛN(3S1−3D1) potential is attractive while the central ΣN(I = 3/2, 3S1−3D1) potentials is repulsive. Tensor potentials, on the other hand, are rather weak in both ΛN and ΣN(I = 3/2) systems.

Few-Baryon Interactions from Lattice QCD

We report the recent progress on the determination of three-nucleon forces (3NF) in lattice QCD. We utilize the Nambu-Bethe-Salpeter (NBS) wave function to define the potential in quantum field theory, and extract two-nucleon forces (2NF) and 3NF on equal footing. The enormous computational cost for calculating multi-baryon correlators on the lattice is drastically reduced by developing a novel contraction algorithm (the unified contraction algorithm). Quantum numbers of the three-nucleon (3N) system are chosen to be (I, J^P)=(1/2,1/2^+) (the triton channel), and we extract 3NF in which three nucleons are aligned linearly with an equal spacing. Lattice QCD simulations are performed using N_f=2 dynamical clover fermion configurations at the lattice spacing of a = 0.156 fm on a 16^3 x 32 lattice with a large quark mass corresponding to m(\pi)= 1.13 GeV. Repulsive 3NF is found at short distance.

Quark-Anti-Quark Potentials from Nambu-Bethe-Salpeter Amplitudes on Lattice

Quark--anti-quark (anti-q-q) potentials with finite quark masses are studied from the anti-q-q Nambu-Bethe-Salpeter (NBS) wave functions in quenched lattice QCD. With the use of a method which has been recently developed in the derivation of nuclear forces from lattice QCD, we derive the anti-q-q potentials from the NBS wave functions. We calculate the anti-q-q NBS wave functions in pseudo-scalar and vector channels for several quark masses. The derived potentials at each quark mass in both channels show linear plus Coulomb form. We also discuss the quark-mass and channel dependence of the anti-q-q potentials.

Toward an Understanding of Short Distance Repulsions among Baryons in QCD: NBS Wave Functions and Operator Product Expansion

We report on our recent attempts to determine the short distance behaviors of general 2-baryon and 3-baryon forces, which are defined from the Nambu-Bethe-Salpeter(NBS) wave function, by using the operator product expansion and a renormalization group analysis in QCD. We have found that the repulsion at short distance increases as the number of valence quarks increases or when the number of different flavors involved decreases. This global tendency suggests a Pauli suppression principle among quark fields at work.

Charmonium potentials: Matching perturbative and lattice QCD

Central and spin–spin potentials for charmonium, constructed from Nambu–Bethe–Salpeter amplitudes in lattice simulations of full QCD, are matched with results from perturbative QCD at an appropriate distance scale. This matching is made possible by defining the perturbative potentials through Fourier transforms with a low-momentum cutoff. The central (spin-independent) potential is compared with potentials derived from an expansion in powers of the inverse quark mass. A well-controlled continuation of the charmonium spin–spin potential from lattice QCD to short distances is performed. The mass splittings of the charmonium singlet and triplet states below the open charm threshold, obtained from the matched spin–spin potential, are in good agreement with the experimental values.