Excited-state hadron masses using the stochastic LapH method
John Bulava, Justin Foley, You-Cyuan Jhang, Keisuke J. Juge, David Lenkner, Colin Morningstar, Chik Him Wong
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8 pages, 4 figures, contribution to The XXIX International Symposium on Lattice Field Theory - Lattice 2011, July 10-16, 2011, Squaw Valley, Lake Tahoe, California, USA
Progress in computing the spectrum of excited baryons and mesons in lattice QCD is described. Large sets of spatially-extended hadron operators are used. The need for multi-hadron operators in addition to single-hadron operators is emphasized, necessitating the use of a new stochastic method of treating the low-lying modes of quark propagation which exploits Laplacian Heaviside quark-field smearing. A new glueball operator is tested and computing the mixing of this glueball operator with a quark-antiquark operator and multiple two-pion operators is shown to be feasible. Some of our initial results show warning signs about extracting high-lying resonance energies using only single-hadron operators.
Progress in computing the spectrum of excited baryons and mesons in lattice QCD is described. Large sets of spatially-extended hadron operators are used. A new method of stochastically estimating the low-lying effects of quark propagation is utilized which allows reliable determinations of temporal correlations of both single-hadron and multi-hadron operators. The method is tested on the eta, sigma, omega mesons.
Elastic $I=1/2$, $s$- and $p$-wave $Kπ$ scattering amplitudes are simultaneously calculated using a Lüscher style analysis on a single ensemble of dynamical Wilson-clover fermions at $m_π\sim 230$ MeV. Partial wave mixing due to the reduced rotational symmetries of the finite volume is included up to $\ell=2$. We also present finite-volume QCD spectra on two large anisotropic lattices ($32^3 \times 256$, $24^3 \times 128$) with $m_π\sim 230,\ 390$ MeV respectively. In each symmetry channel, a large basis of one- and two-hadron interpolating operators is employed with all-to-all quark propagati…
Progress in computing the spectrum of excited baryons and mesons in lattice QCD is described. Results in the zero-momentum bosonic I=1/2, S=1, T1u symmetry sector of QCD using a correlation matrix of 58 operators are presented. All needed Wick contractions are efficiently evaluated using a stochastic method of treating the low-lying modes of quark propagation that exploits Laplacian Heaviside quark-field smearing. Level identification using probe operators is discussed.
Progress in computing the hadron spectrum in lattice QCD using stochastic LapH quark propaga- tors is described. The stochastic LapH algorithm is a particular quark smearing algorithm that also allows the computation of all-to-all quark propagators. All-to-all quark propagators are required in our approach of using a large set of spatially extended hadron operators and explicit multi- particle operators to access excited states. We report on the progress made in the various isospin channels on 2+1 dynamical, anisotropic lattices generated by the Hadron Spectrum Collaboration.
Recent results in computing excited-state energies and meson-meson scattering phase shifts in lattice QCD are presented. A stochastic method of treating the low-lying modes of quark propagation that exploits Laplacian Heaviside quark-field smearing makes such studies possible now on large 32^3 x 256 and 48^3 x 128 lattices at near physical pion masses. Levels are identified using a variety of probe interpolating operators, which include both single-hadron and a large number of two-hadron operators.
The nuclear forces are determined by combining the HAL QCD method and a new type of source smearing technique. The new smearing is a projection to a space spanned by the lowest-lying eigenvectors of the free Laplacian operator on a lattice, which enables efficient calculation of hadron correlators at an affordable cost by utilizing the hadron-level momentum conservations. We find that this new approach reduces the statistical and systematic errors in the resultant nuclear forces.
Progress in computing the spectrum of excited baryons and mesons in lattice QCD is described. Our first results in the zero-momentum bosonic I=1, S=0, T1u+ symmetry sector of QCD using a correlation matrix of 56 operators are presented. In addition to a dozen spatially-extended meson operators, 44 two-meson operators are used, involving a wide variety of light isovector, isoscalar, and strange meson operators of varying relative momenta. All needed Wick contractions are efficiently evaluated using a stochastic method of treating the low-lying modes of quark propagation that exploits Laplacian …
The spectrum of excited isovector mesons is studied using a 32^3 x 256 anisotropic lattice with u,d quark masses set to give a pion mass near 240 MeV. Results in the bosonic isovector nonstrange symmetry channels of zero total momentum are presented using correlation matrices of unprecedented size. In addition to spatially-extended single-meson operators, large numbers of two-meson operators are used, involving a wide variety of light isovector, isoscalar, and strange meson operators of varying relative momenta. All needed Wick contractions are efficiently evaluated using a stochastic method o…
The stochastic Laplacian Heaviside (LapH) method has proven to be successful in hadronic calculations. In this work, with charm-light spectroscopy in mind, we set up and optimise the LapH procedure limiting ourselves to the evaluation of two-point mesonic correlation functions. The calculations are performed on CLS ensembles with $N_f=2+1$ Wilson-Clover fermions on a $32^3\times64$ lattice with open boundary conditions. We analyse the interplay between the LapH parameters and the boundary effects, and implement a fitting procedure to isolate excitations coming from the border.
We report on progress applying the stochastic LapH method to estimate all-to-all propagators required in correlation functions of multi-hadron operators relevant for pion-pion scattering. Large-volume results for $I=2$ and $I=1$ pion-pion scattering phase shifts with good statistical precision are obtained from an $N_{\rm f} = 2+1$ anisotropic Wilson clover ensemble with $m_π = 240\mathrm{MeV}$. We also present a preliminary determination of the $I=1$ pion-pion scattering phase shift and timelike pion form factor on an isotropic $N_{\rm f}=2+1$ flavour ensemble generated by the Coordinated Lat…
We report on our progress in computing the excitation spectrum in Lattice QCD. We focus on the isospin 0, 1 and 2 channels using the stochastic LapH algorithm for the quark propagators. For the isospin-0 channel, a new glueball operator constructed from the stochastic LapH operator is included in the variational basis along with the isoscalar meson and pi-pi operators. A representative signal for each channel is presented and the feasibility of extending the calculations to larger lattices is discussed.
Progress in calculating the spectrum of excited baryons and mesons in lattice QCD is described. Correlation matrices of sets of spatially-extended hadron operators have been studied and their effectiveness in facilitating the extraction of excited-state energies is demonstrated. First applications of the stochastic LapH method, a new method of estimating the low-lying effects of quark propagation, are presented.
We present results for the spectrum of excited mesons obtained from temporal correlations of spatially-extended single-hadron and multi-hadron operators computed in lattice QCD. The stochastic LapH algorithm is implemented on anisotropic, dynamical lattices for isovectors for pions of mass $390$ MeV. A large correlation matrix with single-particle and two-particle probe operators is diagonalized to identify resonances. The masses of excited states in the $I=1, S=0, T_{1u}^+$ channel as well as the mixing of single and multi-particle probe operators are presented.
We demonstrate the efficacy of the stochastic LapH method to treat all-to-all quark propagation on a $N_f = 2+1$ CLS ensemble with large linear spatial extent $L = 5.5$ fm, allowing us to obtain the benchmark elastic isovector p-wave pion-pion scattering amplitude to good precision already on a relatively small number of gauge configurations. These results hold promise for multi-hadron spectroscopy at close-to-physical pion mass with exponential finite-volume effects under control.
Our progress in computing the spectrum of excited baryons and mesons in lattice QCD is described. Sets of spatially-extended hadron operators with a variety of different momenta are used. A new method of stochastically estimating the low-lying effects of quark propagation is utilized which allows reliable determinations of temporal correlations of both single-hadron and multi-hadron operators. The method is tested on the isoscalar mesons in the scalar, pseudoscalar, and vector channels, and on the two-pion system of total isospin I=0,1,2.
Progress in computing the spectrum of excited baryons and mesons in lattice QCD is described. Large sets of spatially-extended hadron operators are used. The need for multi-hadron operators in addition to single-hadron operators is emphasized, necessitating the use of a new stochastic method of treating the low-lying modes of quark propagation which exploits Laplacian Heaviside quark-field smearing. A new glueball operator is tested, and computing the mixing of this glueball operator with a quark-antiquark operator and multiple two-pion operators is shown to be feasible.
We review the techniques of lattice QCD calculations for excited hadrons with light quarks and outline the future challenges that are faced in calculations with fully dynamical fermions.
A new method of stochastically estimating the low-lying effects of quark propagation is proposed which allows accurate determinations of temporal correlations of single-hadron and multi-hadron operators in lattice QCD. The method is well suited for calculations in large volumes. Contributions involving quark propagation connecting hadron sink operators at the same final time can be handled in a straightforward manner, even for a large number of final time slices. The method exploits Laplacian Heaviside (LapH) smearing. ZN noise is introduced in a novel way, and variance reduction is achieved u…
We present a new approach for determining spatially optimized operators that can be used for lattice spectroscopy of excited hadrons. Jacobi smeared quark sources with different widths are combined to construct hadron operators with different spatial wave functions. We use the variational method to determine those linear combinations of operators that have optimal overlap with ground and excited states. The details of the new approach are discussed and we demonstrate the power of the method using examples from quenched baryon and meson spectroscopy. In particular we study the Roper state and r…
We present results for masses of excited baryons from a quenched calculation with Chirally Improved quarks at pion masses down to 350 MeV. Our analysis of the correlators is based on the variational method. In order to provide a large basis set for spanning the physical states, we use interpolators with different Dirac structures and Jacobi smeared quark sources of different width. Our spectroscopy results for a wide range of ground state and excited baryons are discussed.
We overview two interconnected topics: possible effective restoration of chiral symmetry in highly excited hadrons and possible existence of confined but chirally symmetric matter at low temperatures and high densities.
Progress in extracting the spectrum of excited hadron resonances is reviewed and the key issues and challenges in such computations are outlined. The importance of multi-hadron states as simulations are done with lighter pion masses is discussed, and the need for all-to-all quark propagators is emphasized.
Progress in determining the spectrum of excited baryons and mesons in lattice QCD is described. Large sets of carefully-designed hadron operators have been studied and their effectiveness in facilitating the extraction of excited-state energies is demonstrated. A new method of stochastically estimating the low-lying effects of quark propagation is proposed which will allow reliable determinations of temporal correlations of single-hadron and multi-hadron operators.
The calculation of quark propagators for Ginsparg-Wilson-type Dirac operators is costly and thus limited to a few different sources. We present a new approach for determining spatially optimized operators for lattice spectroscopy of excited hadrons. Jacobi smeared quark sources with different widths are combined to construct hadron operators with different spatial wave functions. We study the Roper state and excited rho and pion mesons.
Restoration of chiral and $U(1)_A$ symmetries in excited hadrons is reviewed. Implications of the OPE as well as of the semiclassical expansion for this phenomenon are discussed. A solvable model of the 't Hooft type in 3+1 dimensions is presented, which demonstrates a fast restoration of both chiral and $U(1)_A$ symmetries at larger spins and radial excitations.
We report the hadron mass spectrum obtained on a $16^3 \times 40$ lattice at $β= 5.7$ using two flavors of staggered fermions with $m a = 0.01$. We calculate the masses of excited states that have the same quantum numbers as the $π$, $ρ$ and $N$. They are obtained by a combined analysis of the hadron correlators from sources of size $16^3$ and $8^3$. We also report on the hadron spectrum for a wide range of valence quark masses.
Restoration of chiral and $U(1)_A$ symmetries in excited hadrons is reviewed. A connection of these restorations with the semiclassical regime in highly excited hadron is discussed. A solvable confining field-theoretical toy model that exhibits chiral restoration is presented. Implications of the string description of the highly excited hadrons that suggests an additional dynamical symmetry of the spectra on the top of $U(2)_L \times U(2)_R$ are presented.
Amongst the bound states produced by the strong interaction, radially excited meson and nucleon states offer an important phenomenological window into the long-range behavior of the coupling constant in Quantum Chromodynamics. We here report on some technical details related to the computation of the bound state's eigenvalue spectrum in the framework of Bethe-Salpeter and Faddeev equations.
This is a second paper describing the calculation of spectroscopy for orbitally excited states from lattice simulations of Quantum Chromodynamics. New features include higher statistics for P-wave systems and first results for the spectroscopy of D-wave mesons and baryons, for relatively heavy quark masses. We parameterize the Coulomb gauge wave functions for P-wave and D-wave systems and compare them to those of their corresponding S-wave states.
Excited-state uncertainties in lattice-QCD calculations of hadron masses and scattering phase shifts
Massachusetts Institute of Technology（麻省理工学院）；University of Maryland, College Park（马里兰大学帕克分校）；Fermi National Accelerator Laboratory（费米国家加速器实验室）；University of Washington（华盛顿大学）；Pacific Northwest National Laboratory（太平洋西北国家实验室）
Quenched QCD hadron spectrum is calculated with Wilson's quark action at $β=5.85$ and 6.0 on a $24^3 \times 54$ lattice. We discuss the problem of whether we have extracted the mass of the ground state at these $β$'s without contamination of the excited states. We show that the masses of the first excited states turn out to be consistent with experiment when we are able to obtain the propagators up to the large time slice region where the propagators take the asymptotic forms of the ground states.
A large symmetry group is perhaps experimentally observed in excited hadrons which includes the chiral group U(2)_L x U(2)_R as a subgroup. To possess this large symmetry a dynamical model for excited hadrons, presumably a string model, should explain formation of chiral multiplets and, at the same time, predict coinciding slopes of the angular and radial Regge trajectories. This is possible only if both the dynamics of the string and the chirality of the quarks at the ends of the string are considered together. We construct a model-independent unitary transformation from the relativistic chir…
The study of excited hadron spectra using Lattice QCD is currently evolving. An important step toward obtaining resonance parameters from Lattice QCD is the calculation of finite volume energy spectra. Somewhat more rigorous studies of finite volume spectra are currently possible and should be completed in the near future. The inclusion of disconnected diagrams is increasingly commonplace and the simplest systems which involve mixing between single- and multi-hadron interpolating fields are being studied. Advances in all-to-all algorithms which have been crucial in this progress are reviewed a…
The chirally improved (CI) fermion action allows us to obtain results for pion masses down to 320 MeV on (in lattice units) comparatively small lattices with physical extent of 2.4 fm. We use differently smeared quarks sources to build sets of several interpolators. The variational method then leads to excellent ground state masses for most mesons and baryons. The excited state signals weaken in quality towards smaller quark masses. In particular the excited baryons come out too high.
Physics of the low-lying and high-lying hadrons in the light flavor sector is reviewed. While the low-lying hadrons are strongly affected by the spontaneous breaking of chiral symmetry, in the high-lying hadrons the chiral symmetry is restored. A manifestation of the chiral symmetry restoration in excited hadrons is a persistence of the chiral multiplet structure in both baryon and meson spectra. Meson and baryon chiral multiplets are classified. A relation between the chiral symmetry restoration and the string picture of excited hadrons is discussed.
Whereas ground state spectroscopy for quenched QCD is well understood, it is still a challenge to obtain results for excited hadron states. In our study we present results from a new approach for determining spatially optimized operators for lattice spectroscopy of excited hadrons. In order to be able to approach physical quark masses we work with the chirally improved Dirac operator, i.e., approximate Ginsparg-Wilson fermions. Since these are computationally expensive we restrict ourselves to a few quark sources. We use Jacobi smeared quark sources with different widths and combine them to co…
Progress in computing various meson-baryon scattering amplitudes is presented on a single ensemble from the Coordinated Lattice Simulations (CLS) consortium with $m_π=200$ MeV and $N_f=2+1$ dynamical fermions. The finite-volume Lüscher approach is employed to determine the lowest few partial waves from ground- and excited-state energies computed from correlation matrices rotated in a single pivot using a generalized eigenvector solution. This analysis requires evaluating matrices of correlation functions between single- and two-hadron interpolating operators which are projected onto definite s…
We present the first Dyson-Schwinger equation calculation of the light hadron spectrum that simultaneously correlates the masses of meson and baryon ground- and excited-states within a single framework. At the core of our analysis is a symmetry-preserving treatment of a vector-vector contact interaction. In comparison with relevant quantities the root-mean-square-relative-error/degree-of freedom is 13%. Notable amongst our results is agreement between the computed baryon masses and the bare masses employed in modern dynamical coupled-channels models of pion-nucleon reactions. Our analysis prov…
Employing the maximum entropy method we extract the spectral functions from meson correlators at four lattice spacings in quenched QCD with the Wilson quark action. We confirm that the masses and decay constants, obtained from the position and the area of peaks, agree well with the results from the conventional exponential fit. For the first excited state, we obtain $m_{π_1} = 660(590)$ MeV, $m_{ρ_1} = 1540(570)$ MeV, and $f_{ρ_1} = 0.085(36)$ in the continuum limit.
A novel method for the precise identification and determination of the energies that contribute in the spectral decomposition of lattice correlators is presented. The method is based on statistical concepts and it relies heavily on simulation techniques. The $η_c$ correlator is analyzed within this method and the results obtained are compared to a previous analysis based on Bayesian statistics. An analysis of the nucleon local two-point correlators leads to the identification of the excited states in the positive and negative channels. A discussion on the Roper is included.
We present results for masses of excited mesons from quenched calculations using chirally improved quarks at pion masses down to 350 MeV. The key features of our analysis are the use of a matrix of correlators from various source and sink operators and a basis which includes quark sources with different spatial widths, thereby improving overlap with states exhibiting radial excitations.
Results for excited light and strange hadrons from the lattice with two flavors of Chirally Improved sea quarks are presented. We perform simulations at several values of the pion mass ranging from 250 to 600 MeV and extrapolate to the physical pion mass. The variational method is applied to extract excited energy levels but also to discuss the content of the states. Among others, we explore the flavor singlet/octet content of Lambda states. In general, our results agree well with experiment, in particular we confirm the Lambda(1405) and its dominant flavor singlet structure.
We highlight Hermiticity issues in bound-state equations whose kernels are subject to a highly asymmetric mass and momentum distribution and whose eigenvalue spectrum becomes complex for radially excited states. We trace back the presence of imaginary components in the eigenvalues and wave functions to truncation artifacts and suggest how they can be eliminated in the case of charmed mesons. The solutions of the gap equation in the complex plane, which play a crucial role in the analytic structure of the Bethe-Salpeter kernel, are discussed for several interaction models and qualitatively and …
A detailed study of orbital and radial excited states in D, Ds, B and B_s systems is performed. The chiral quark model provides the framework for the calculation of pseudoscalar meson (pi, K, ...) hadronic transitions among heavy-light excited and ground states. To calculate the excited states masses and wavefunctions, we must resort to a relativistic quark model. Our model includes the leading order corrections in 1/m_(c,b) (e.g. mixing). Numerical results for masses and light hadronic transition rates are compared to existing experimental data. The effective coupling of the chiral quark mode…
Chiral symmetry breaking in Quantum Chromodynamics is associated with the low lying spectral modes of the Dirac operator according to the Banks-Casher relation. Here we study how removal of a variable number of low lying modes from the valence quark sector affects the masses of the ground states and first excited states of baryons and mesons in two flavor lattice QCD.
In this paper, we discuss the transition to the semiclassical regime in excited hadrons, and consequently, the restoration of chiral symmetry for these states. We use a generalised Nambu-Jona-Lasinio model with the interaction between quarks in the form of the instantaneous Lorentz-vector confining potential. This model is known to provide spontaneous breaking of chiral symmetry in the vacuum via the standard selfenergy loops for valence quarks. It has been shown recently that the effective single-quark potential is of the Lorentz-scalar nature, for the low-lying hadrons, while, for the high-l…
Recent developments for chiral and $U(1)_A$ restorations in excited hadrons are reviewed. We emphasize predictions of the chiral symmetry restoration scenario for axial charges and couplings to Goldstone bosons. Using very general chiral symmetry arguments it is shown that strict chiral restoration in a given excited nucleon forbids its decay into the $N π$ channel. We confront this prediction with the $N^*Nπ$ coupling constants extracted from the decay widths and observe a 100 % correlation of these data with the spectroscopic parity doublet patterns. These results suggest that the lowest app…
Lattice QCD has matured to a degree where it is now possible to study excited hadrons as they truly appear in nature, as short-lived resonant enhancements decaying into multiple possible final states. Through variational analysis of matrices of correlation functions computed with large bases of interpolating fields it has proven possible to extract many excited state energy levels, and these can be used to constrain the hadron-hadron scattering amplitudes in which hadron resonances can be observed. Recent progress is illustrated with several examples including coupled-channel scattering in the…
Excited-state uncertainties in lattice-QCD calculations of multi-hadron systems
Massachusetts Institute of Technology（麻省理工学院）；University of Maryland, College Park（马里兰大学帕克分校）；Fermi National Accelerator Laboratory（费米国家加速器实验室）；University of Washington（华盛顿大学）；Pacific Northwest National Laboratory（太平洋西北国家实验室）
Results of hadron spectroscopy with two dynamical mass-degenerate chirally improved quarks are presented. Three ensembles with pion masses of 322(5), 470(4) and 525(7) MeV, lattices of size 16^3 \times 32, and lattice spacings close to 0.15 fm are investigated. We discuss the possible appearance of scattering states, considering masses and eigenvectors. Partially quenched results in the scalar channel suggest the presence of a 2-particle state, however, in most channels we cannot identify them. Where available, we compare to results from quenched simulations using the same action.
We employ a novel method to analyze Euclidean correlation functions entering the calculation of hadron energies in lattice QCD. The method is based on the sampling of all possible solutions allowed by the spectral decomposition of the hadron correlators. We demonstrate the applicability of the method by studying the nucleon excited states in the positive and negative parity channels over a pion mass range of about 400~MeV to 150~MeV. The results are compared to the standard variational approach routinely used to study excited states within lattice QCD. The main advantage of our new approach is…
The task of mapping and explaining the spectrum of baryons and the structure of these states in terms of quarks and gluons is a longstanding challenge in hadron physics, which is likely to persist for another decade or more. We review the progress made in this topic using a functional method based on Dyson-Schwinger equations. This framework provides a non-perturbative, Poincaré-covariant continuum formulation of Quantum Chromodynamics which is able to extract novel insight on baryon properties since the physics at the hadron level is directly related with the underlying quark-gluon substructu…
We study operators to create hadronic states made of light quarks in quenched lattice gauge theory. We construct non-local gauge-invariant operators which provide information about the spatial extent of the ground state and excited states. The efficiency of the operators is shown by looking at the wave function of the first excited state, which has a node as a function of the spatial extent of the operator. This allows one to obtain an uncontaminated ground state for hadrons.
We investigate modifications of hadron masses at finite quark chemical potential in two-flavor and two-color QCD, of which the data are available from lattice simulations, within a linear sigma model based on approximate Pauli-Gursey $SU(4)$ symmetry. The model describes not only ground-state scalar diquarks and pseudo-scalar mesons but also the excited pseudo-scalar diquarks and scalar mesons; each ground-state diquark (meson) has the corresponding excited diquark (hadron) with opposite parity as a chiral partner. Effects of chiral symmetry breaking and diquark condensates are incorporated by…
We review results on hadron structure using lattice QCD simulations with pion masses close or at to the physical value. We pay particular attention to recent successes on the computation of the mass of the low-lying baryons and on the challenges involved in evaluating energies of excited states and resonance parameters, as well as, in studies of nucleon structure.
High-precision calculations of hadron spectroscopy are a crucial task for Lattice QCD. State-of-the-art techniques are needed to disentangle the contributions from different energy states, such as solving the generalized eigenvalue problem (GEVP) for zero-momentum hadron correlators in an efficient way. We review the method and discuss its application in the determination of the $B_s$-meson spectrum using (quenched) nonperturbative HQET at order $1/m_b$.
