Extension of the HAL QCD approach to inelastic and multi-particle scatterings in lattice QCD
We extend the HAL QCD approach, with which potentials between two hadrons can be obtained in QCD at energy below inelastic thresholds, to inelastic and multi-particle scatterings. We first derive asymptotic behaviors of the Nambu-Bethe-Salpeter (NBS) wave function at large space separations for systems with more than 2 particles, in terms of the one-shell -matrix consrainted by the unitarity of quantum field theories. We show that its asymptotic behavior contains phase shifts and mixing angles of particle scatterings. 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. We next construct energy independent but non-local potentials above inelastic thresholds, in terms of these NBS wave functions. We demonstrate an existence of energy-independent coupled channel potentials with a non-relativistic approximation, where momenta of all particles are small compared with their own masses. Combining these two results, we can employ the HAL QCD approach also to investigate inelastic and multi-particle scatterings.
Extension of the HAL QCD approach to inelastic and multi-particle scatterings in lattice QCD
for HAL QCD Collaboration
1 Introduction: HAL QCD approach to nuclear force
To understand hadronic interactions such as nuclear forces from the fundamental theory, Quantum Chromodynamics (QCD), non-perturbative methods such as the lattice QCD combined with numerical simulations are required, since the running coupling constant in QCD becomes large at hadronic scale. Conventionally the finite size method has been employed to extract the scattering phase shift in lattice QCD, but the method is mainly applied to two-particle systems below the inelastic threshold so far.
Recently an alternative method has been proposed and employed to extract the potential between nucleons below inelastic thresholds[2, 3, 4, 5]. In the method, called the HAL QCD method, a potential between nucleons is defined in QCD, through an equal-time Nambu-Bethe-Salpeter (NBS) wave function in the center of mass system, which is defined as
where is the QCD vacuum (bra-)state, is a two-nucleon asymptotic in-state at the total energy with the nucleon mass and a relative momentum , is the time-ordered product, and with is a nucleon operator. As the distance between two nucleon operators, , becomes large, the NBS wave function satisfies the free Schrödinger equation,
at (the lowest inelastic threshold), for the partial wave with the orbital angular momentum . Form this property, it is natural to define a non-local but energy-independent potential as
where does not depend on the energy of a particular NBS wave function. It is not so difficult to construct such explicitly as
where is the inverse of
in the space spanned by . Indeed we see
for with . This construction is formal, while in practice we introduce the derivative expansion that , and the leading order potential can be obtained simply as
Once the potential is obtained, we can calculate physical observables such as scattering phase shifts and binding energies, by solving the corresponding Schrödinger equation in the infinite volume. As such an example, we give the NN potential in the iso-triplet channel obtained in 2+1 flavor QCD at fm, fm and MeV, and the scattering phase shift in Fig. 1, together with experimental data of the phase shift. The potential as well as the scattering phase shift well reproduce the qualitative feature of the nuclear force, though the attraction at low energy is still weaker than that in nature, probably due to the heaver pion mass.
In the above proposal so far, considerations are restricted to elastic scatterings of two particles. In order to extend the HAL QCD method to inelastic and/or multi-particle scatterings, we have to derive two important properties, one is an asymptotic behavior of NBS wave functions for more than 2 particles, and the other is an existence of energy-independent potentials above inelastic thresholds. In this report, the first property is shown in Sec. 2, while the second is discussed in Sec. 3. Our conclusion is given in Sec. 4.
2 NBS wave functions for multi-particles
In this section, for simplicity, we consider scalar particles, all of which have the same mass but different flavors, denoted by . We also assume that no bound state exists in this system.
2.1 Unitarity constraint
In order to derive an asymptotic behavior of NBS wave functions for multi-particles, we first determine structures of matrix using the unitarity constraint that
where are momenta of particles, is a total energy with , and is a total momentum for , while we introduce modified Jacobi coordinates and momenta as
for , and . Regarding as a momentum in dimensions, we expand as
where is the hyper-spherical harmonic function with a set of "quantum" number , which satisfies for the orbital angular momentum and the solid angle of the vector in -dimensions.
With the non-relativistic approximation that and the orthogonal property of such that
a solution to the unitarity constraint is given by
whose diagonal part becomes
with a "phase shift" . At , the above result reproduces the expression of for the two particle scattering in the non-relativistic limit that
2.2 Asymptotic behavior of NBS wave functions
The Lippmann-Schwinger equation in quantum field theories reads
where the asymptotic state satisfies , while the free state satisfies , and off-shell matrix is related to the on-shell -matrix as .
Using the Lippmann-Schwinger equation, the NBS wave function, defined by with , can be expressed as
where is a renormalization factor due to the difference between and , and we can write .
Expressing and in terms of modified Jacobi coordinates and momenta , the above equation can be written in the -dimensional hyper-coordinate as
As in the previous subsection, in terms of the hyper-spherical harmonic function, we can write
where is the hyper-spherical Bessel function. In the large limit, since integral in eq. (2.2) is dominated at the pole from the denominator, an asymptotic behavior of the NBS wave function now becomes
which demonstrates that we can obtain the particle scattering "phase shift" , determined by the unitarity constraint of the -matrix, from the NBS wave function at large . Thus the fist property necessary to generalize the HAL QCD method for multi-particle scatterings has been shown. This property can be shown also for more general inelastic scatterings with .
3 Energy-independent potential above inelastic thresholds
At energy , inelastic scatterings can occur in addition to the elastic scattering. To make our discussion simpler, let us consider the case, which corresponds to an energy interval that . In this case, 4 NBS wave functions can be defined as
where , are renormalization factors, and denote quantum numbers other than the energy . For simplicity we express these NBS wave functions as with , where denotes a number of ’s in the operator(state), and ,
In Ref. , we have shown an existence of non-local but energy-independent potential matrix which satisfies the following coupled channel Schrödinger equation:
for , where and are the kinetic energy and the free Hamiltonian for , respectively. Note that, to show the existence of , we have used a non-relativistic approximation that
in the center of mass frame. See Ref.  for an explicit construction of .
The HAL QCD approach is a promising method to extract hadronic interaction in lattice QCD. In this conference, there appear several new results in this approach, which include a LS nuclear force, an anti-symmetric LS force between octet baryons in the flavor SU(3) limit, interactions of DD, D and D, the Omega-Omega interactions. Comparisons between the HAL QCD method and the Lüscher’s finite volume method have been also reported for the NN case and the case.
In this report, we have attempted an extension of the HAL QCD method to inelastics and/or multi-particle scatterings, by determining asymptotic behaviors of NBS wave functions for scalar particles in terms of the generalized scattering phase of the -matrix, and by constructing non-local but energy-independent coupled channel potentials for inelastic scatterings. Our results are relevant for the 3 nuclear force and the coupled channel potentials between hyperons.
One of remaining issues in this study is a treatment of bound-states. For example, in the three nucleon scattering, two of them could become a bound deuteron. Whether a bound state is treated as an independent state or not is an open question in the extension of the HAL QCD approach.
We will address this issue in our future publications.
This work is supported in part by the Grant-in-Aid for Scientific Research (25287046), the Grant-in-Aid for Scientific Research on Innovative Areas(No.2004: 20105001, 20105003) and SPIRE (Strategic Program for Innovative Research).
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