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Register a new userEmergence of the $rho$ resonance from the HAL QCD potential in lattice QCD
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We investigate the $I=1$ $pi pi$ interaction using the HAL QCD method in lattice QCD. We employ the (2+1)-flavor gauge configurations on $32^3 times 64$ lattice at the lattice spacing $a approx 0.0907$ fm and $m_{pi} approx 411$ MeV, in which the $rho$ meson appears as a resonance state. We find that all-to-all propagators necessary in this calculation can be obtained with reasonable precision by a combination of three techniques, the one-end trick, the sequential propagator, and the covariant approximation averaging (CAA). The non-local $I=1$ $pi pi$ potential is determined at the next-to-next-to-leading order (N$^2$LO) of the derivative expansion for the first time, and the resonance parameters of the $rho$ meson are extracted. The obtained $rho$ meson mass is found to be consistent with the value in the literature, while the value of the coupling $g_{rho pi pi}$ turns out to be somewhat larger. The latter observation is most likely attributed to the lack of low-energy information in our lattice setup with the center-of-mass frame. Such a limitation may appear in other P-wave resonant systems and we discuss possible improvement in future. With this caution in mind, we positively conclude that we can reasonably extract the N$^2$LO potential and resonance parameters even in the system requiring the all-to-all propagators in the HAL QCD method, which opens up new possibilities for the study of resonances in lattice QCD.
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The $XiXi$ interaction in the $^1$S$_0$ channel is studied to examine the convergence of the derivative expansion of the non-local HAL QCD potential at the next-to-next-to-leading order (N$^2$LO). We find that (i) the leading order potential from the N$^2$LO analysis gives the scattering phase shifts accurately at low energies, (ii) the full N$^2$LO potential gives only small correction to the phase shifts even at higher energies below the inelastic threshold, and (iii) the potential determined from the wall quark source at the leading order analysis agrees with the one at the N$^2$LO analysis except at short distances, and thus, it gives correct phase shifts at low energies. We also study the possible systematic uncertainties in the HAL QCD potential such as the inelastic state contaminations and the finite volume artifact for the potential and find that they are well under control for this particular system.
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 calculate the parameters describing elastic $I=1$, $P$-wave $pipi$ scattering using lattice QCD with $2+1$ flavors of clover fermions. Our calculation is performed with a pion mass of $m_pi approx 320::{rm MeV}$ and a lattice size of $Lapprox 3.6$ fm. We construct the two-point correlation matrices with both quark-antiquark and two-hadron interpolating fields using a combination of smeared forward, sequential and stochastic propagators. The spectra in all relevant irreducible representations for total momenta $|vec{P}| leq sqrt{3} frac{2pi}{L}$ are extracted with two alternative methods: a variational analysis as well as multi-exponential matrix fits. We perform an analysis using Luschers formalism for the energies below the inelastic thresholds, and investigate several phase shift models, including possible nonresonant contributions. We find that our data are well described by the minimal Breit-Wigner form, with no statistically significant nonresonant component. In determining the $rho$ resonance mass and coupling we compare two different approaches: fitting the individually extracted phase shifts versus fitting the $t$-matrix model directly to the energy spectrum. We find that both methods give consistent results, and at a pion mass of $am_{pi}=0.18295(36)_{stat}$ obtain $g_{rhopipi} = 5.69(13)_{stat}(16)_{sys}$, $am_rho = 0.4609(16)_{stat}(14)_{sys}$, and $am_{rho}/am_{N} = 0.7476(38)_{stat}(23)_{sys} $, where the first uncertainty is statistical and the second is the systematic uncertainty due to the choice of fit ranges.
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 paper, we investigate the HAL QCD potential in the $I=1$ $pi pi$ scattering using the hybrid method for all-to-all propagators, in which a propagator is approximated by low-eigenmodes and the remaining high-eigenmode part is stochastically estimated. To verify the applicability of the hybrid method to systems containing quark creation$/$annihilation contributions such as the $rho$ meson, we calculate the $I=1$ $pipi$ potential with the 2+1 flavor gauge configurations on $16^3 times 32$ lattice at the lattice spacing $a approx 0.12$ fm and $(m_{pi},m_{rho}) approx (870, 1230)$ MeV, in which the $rho$ meson appears as a deeply-bound state. While we find that the naive stochastic evaluations for quark creation$/$annihilation contributions lead to extremely large statistical fluctuations, additional noise reduction methods enable us to obtain a sufficiently precise potential, which shows a strong attractive force. We also confirm that the binding energy and $k^3 cot delta$ obtained from our potential are roughly consistent with an existing $rho$ meson bound state, within a large systematic error associated with our calculation, whose possible origin is also discussed.
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