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.
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.
Using Heavy Meson Chiral Perturbation Theory (HMChPT), the $B^* π$ excited states contamination of the $B \to π$ vector form factors is computed to NLO in the chiral expansion and in the static limit. The results suggest that the excited states for $h_\parallel$ are of the order of a few percent whereas $h_\perp$ receives large negative contributions and thus might be significantly underestimated in lattice simulations.
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.
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 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.
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.
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.
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.
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 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.
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.
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.
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 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 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.
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…
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 res…
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 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 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…
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.
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…
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.
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…
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.
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.
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.
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.
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.
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.
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.
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…
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…
We compute the ground-state energies of a heavy-light K-Lambda like system as a function of the relative distance r of the hadrons. The heavy quarks, one in each hadron, are treated as static. Then, the energies give rise to an adiabatic potential Va(r) which we use to study the structure of the five-quark system. The simulation is based on an anisotropic and asymmetric lattice with Wilson fermions. Energies are extracted from spectral density functions obtained with the maximum entropy method. Our results are meant to give qualitative insight: Using the resulting adiabatic potential in a Schr…
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）
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…
There is an extensive history of form factor calculations on the lattice, primarily with ground states for both initial and final states. However, there have never been any radially excited transition form factor calculations. Furthermore, the lattice faces difficulty in extracting signal from noise at large transfer momenta ($Q^2$). These measurements could give important theoretical input to experiments, such as those of JLab's 12 GeV program and studies of deformation of the nucleon. In this work, I will present a simple technique to resolve both of these difficulties and present results fr…
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…
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 …
We analyze the overlap of color-octet meson operators with the $Υ$ and the $η_b$ and their excited states, especially the first radial excitations. Our analysis is based on NRQCD and includes all terms up to order $v^4$. We use a variety of source and sink operators as a basis for the variational method, which enables us to clearly separate the mass eigenstates and hence to extract the desired amplitudes. The results show the usefulness of the variational method for determining couplings to excited hadronic states.
We analyze the overlap of local color-octet meson operators with the $Υ$ and the $η_b$ and their low-lying excited states, especially the first radial excitations. Our analysis is based on NRQCD and includes all terms up to order $v^4$. We use a variety of source and sink operators as a basis for the variational method, which enables us to clearly separate the mass eigenstates and hence to extract the desired amplitudes. The results show the usefulness of the variational method for determining couplings to excited hadronic states.
We propose the study of non-local gauge invariant operators to obtain an uncontaminated ground state for hadrons. 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 spatial extent of the operator.
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 $1/m_b$.
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…
Excited state contributions represent a formidable challenge for hadron structure calculations in lattice QCD. For physical systems that exhibit an exponential signal-to-noise problem they often hinder the extraction of ground state matrix elements, introducing a major source of systematic error in lattice calculations of such quantities. The development of methods to treat the contribution of excited states and the current status of related lattice studies are reviewed with focus on nucleon structure calculations that are notoriously affected by excited state contamination.
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）
Lattice QCD determinations appropriate to hadron spectroscopy are reviewed with emphasis on the glueball and hybrid meson states in the quenched approximation. Hybrids are discussed for heavy and for light quarks. The effects of sea quarks (unquenching) are explored. Heavy-light systems are presented - particularly excited B mesons.
The properties of the $B_c$-meson family ($c\bar b$) are still not well determined experimentally because the specific mechanisms of formation and decay remain poorly understood. Unlike heavy quarkonia, i.e. the hidden heavy quark-antiquark sectors of charmonium ($c\bar c$) and bottomonium ($b\bar b$), the $B_c$-mesons cannot annihilate into gluons and they are, consequently, more stable. The excited $B_c$ states, lying below the lowest strong-decay $BD$-threshold, can only undergo through radiative decays and hadronic transitions to the $B_c$ ground state, which then decays weakly. As a resul…
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 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 compare axial charges of excited nucleons, as predicted by the chiral symmetry restoration picture, with the traditional, moderately successful for the ground-state baryons SU(6) symmetry. The axial charges of excited nucleons can (and will) be measured in lattice QCD simulations, and comparison of the lattice results with the two different symmetry schemes will give an insight on the origins of the excited hadron masses as well as on interrelations of chiral symmetry and confinement.
We present highly excited spectra of charmonium, charm-light and charm-strange mesons using the dynamical anisotropic lattices of the Hadron Spectrum Collaboration. The use of novel techniques has allowed us to extract these spectra with a high degree of statistical precision, while also enabling us to observe states as high as spin-4, states with exotic quantum numbers and candidate gluonic excitations. We discuss the phenomenology of these spectra, and also present preliminary results of Dπ scattering considering only the lowest partial wave in the isospin-3/2 channel.
In this paper, we discuss a decoupling of the Goldstone bosons from highly excited hadrons in relation to the restoration of chiral symmetry in such hadrons. We use a generalised Nambu-Jona-Lasinio model with the interaction between quarks in the form of an 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. For highly excited hadrons, where the typical momentum of valence quarks is large, the loop contributions represent only a small correction to the chiral…
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.
The excited baryon spectrum from a unified quark model
School of Physics and Center of High Energy Physics, Peking University（北京大学物理学院与高能物理中心）；School of Physics, Peking University（北京大学物理学院）；School of Physics, Southeast University（东南大学物理学院）
