Light hadron masses from lattice QCD

This article reviews lattice QCD results for the light hadron spectrum. An overview of different formulations of lattice QCD with discussions on the fermion doubling problem and improvement programs is given. Recent developments in algorithms and analysis techniques that render calculations with light, dynamical quarks feasible on present day computer resources are summarized. Finally, spectrum results for ground state hadrons and resonances using various actions are summarized.

Figure 1

Nonperturbative running coupling constant of four flavor QCD in units of the QCD scale $\Lambda$ from a lattice calculation of 536 compared to perturbative calculation. Figure courtesy of Rainer Sommer.

Figure 2

(a) In the fermionic case, periodicity of the lattice momentum $P_{\mu }$ requires a second root (doubler) or jump within the Brillouin zone (nonlocality). (b) In the bosonic case, only the squared lattice momentum is required to be periodic which can be fulfilled without a jump or an additional root.

Figure 3

Inverse iteration count of the fermion matrix inverter for Wilson-type fermions and a number of different ensembles with bare couplings $\beta$ and approximate pion masses $M_{\pi }$. The lower tail of the distributions shows a clear separation from 0 indicating the absence of exceptional configurations in the ensembles. From 232

Figure 4

The four possible forms of closed gauge loops on a hypercubic lattice with six or less gauge links corresponding to the members of the sets ${\mathcal{C}}_0,\dots ,{\mathcal{C}}_3$ as described in the text.

Figure 5

The principle of APE smearing displayed in the two-dimensional case. The “thin” gauge link is replaced by a weighted average over the gauge link and the staples, which is then usually backprojected onto the gauge group.

Figure 6

Locality of a fermion operator coupling to a 6-times stout smeared gauge field. The stout smearing parameter is set to $\rho =0.11$. At small distances the coupling decreases exponentially with an effective mass of $\sim 2.2a^{-1}$ that is proportional to the lattice cutoff. At Euclidean distances larger than $\sqrt{50}a$ couplings are zero due to the ultralocality of the smearing procedure. From 230.

Figure 7

The effective mass $M_{\mathrm{eff}}(t+a/2)$ vs $t/a$ Eq. (142) of a decuplet baryon from a recent lattice calculation (35). One can see clearly the onset of the plateau and the eventual loss of signal. The plateau region is indicated together with the value of the mass obtained from a fit to the correlation function. Figure courtesy of the RBC-UKQCD Collaboration.

Figure 8

Graphical representation of the contraction for a flavor nonsinglet (a) Green’s function (125, 126) and of the (b) connected and (c) disconnected contributions to a flavor singlet mesonic Green’s function (128).

Figure 9

Correlation function (left panel) and effective mass (right panel) in the $\eta$ channel using a stochastic LapH source. Connected (fwd) and in the left panel disconnected (smt) contributions are plotted additionally. From 469, with permission of K. J. Juge.

Figure 10

Plot of the masses of the ground and excited states in the nucleon channel. The extraction has been performed by fitting single state correlators that were obtained by projection of a $4\times 4$ matrix correlator onto the elements of the eigenbasis of the transfer matrix $M(t,t_0)$ for different $t$ and $t_0$. From 436, with permission of D. Leinweber.

Figure 11

The landscape of recent dynamical fermion simulations projected to the $\sqrt{2M_K^2-M_{\pi }^2}$ vs $M_{\pi }$ plane. The cross marks the physical point while shaded areas with increasingly light shade indicate physically more desirable regions of parameter space. Data points are taken from the following references: ETMC’$10(2+1+1)$ (54); MILC’10 (62); QCDSF-UKQCD’10 (91); BMWc’08 (230); BMWc’10 (232); PACS-CS’09 (32, 33); RBC-UKQCD’10 (448; 35); JLQCD/TWQCD’09 (486); and HSC’10 (413). All ensembles are from $N_f=2+1$ simulations except explicitly noted otherwise. For staggered, respectively, twisted mass ensembles, the Goldstone, respectively, charged pion masses are plotted.

Figure 12

The landscape of recent dynamical fermion simulations projected to the $M_{\pi }$ vs $a$ plane. The cross marks the physical point while shaded areas with increasingly light shade indicate physically more desirable regions of parameter space. Data points are taken from the following references: ETMC’09(2) (105); ETMC’$10(2+1+1)$ (54); MILC’10 (62); QCDSF’10(2) (520); QCDSF-UKQCD’10 (91); BMWc’08 (230); BMWc’10 (232); PACS-CS’09 (32, 33); RBC-UKQCD’10 (448; 35); JLQCD/TWQCD’09 (486); HSC’10 (413); BGR’10(2) (241); and CLS’10(2) (119). All ensembles are from $N_f=2+1$ simulations except explicitly noted otherwise. For staggered, respectively, twisted mass ensembles, the Goldstone, respectively, charged pion masses are plotted.

Figure 13

The landscape of recent dynamical fermion simulations projected to the $L$ vs $M_{\pi }$ plane as given by 348. The borders of the shaded regions are placed where the expected relative error of the pion mass is 1%, 0.3%, and 0.1% according to 151. The vertical dashed line indicates the physical pion mass. Data points are taken from the following references: ETMC’09(2) (105); ETMC’$10(2+1+1)$ (54); MILC’10 (62); QCDSF’10(2) (520); QCDSF-UKQCD’10 (91); BMWc’08 (230); BMWc’10 (232); PACS-CS’09 (32, 33); RBC-UKQCD’10 (448; 35); JLQCD/TWQCD’09 (486); HSC’10 (413); BGR’10(2) (241); and CLS’10(2) (119). All ensembles are from $N_f=2+1$ simulations except explicitly noted otherwise. For staggered, respectively, twisted mass ensembles, the Goldstone, respectively, charged pion masses are plotted.

Figure 14

The finite-volume energy levels vs box size $L$ according to 419. The solid lines roughly correspond to the $\rho -\pi \pi$ system, while the dashed lines show the behavior in the case of a much smaller coupling $g_{\rho }$.

Figure 15

Quenched lattice QCD prediction of the light hadron spectrum according to 29, 30. For both data sets plotted $M_{\rho }$ and $M_{\pi }$ were used to set the scale and the light quark mass. In order to set the strange mass, either $M_K$ (filled circles) or $M_{\varphi }$ (open circles) are used. Figure courtesy of the CP-PACS Collaboration.

Figure 16

Dynamical $N_f=2$ lattice QCD prediction of the light baryon spectrum according to 12. The continuum extrapolation of the ground state light octet and decuplet baryon masses in the case where the strange mass was set via $M_K$ are displayed. The continuum extrapolation was performed using three lattice spacings that are displayed with solid circles. Open circles represent a fourth, finer lattice which was not included in the analysis due to a too small volume. Figure courtesy of the CP-PACS Collaboration.

Figure 17

Baryon spectrum obtained by the ETM Collaboration with $N_f=2$ twisted mass fermions. From 9, with permission of the ETM Collaboration.

Figure 18

Positive and negative baryon spectrum obtained by the BGR Collaboration with $N_f=2$ chirally improved fermions. From 241, with permission of the BGR Collaboration.

Figure 19

Comparison of the $N_f=2+1$ light hadron spectrum results from the MILC Collaboration (62) with experiment. The diamonds are input quantities while the circles are predictions. Experimental masses of hadrons from 20 are indicated by squares. Note that charmonium and bottomonium masses are also included with some of the later ones used to set the scale. Figure courtesy of the MILC Collaboration.

Figure 20

The extrapolated $N_f=2+1$ light hadron spectrum results from the PACS-CS Collaboration. Experimental data are from 20. From 32, with permission of the PACS-CS Collaboration.

Figure 21

Sample chiral and continuum extrapolation of the lattice hadron masses of 230 at physical $M_K^2-M_{\pi }^2/2$ in physical units. The scale setting variable $M_{\Omega }$ and the nucleon mass are plotted vs the square of the pion mass together with a fit of the data at every lattice spacing. The vertical dashed line represents the physical pion mass.

Figure 22

Prediction of the light hadron spectrum in full $N_f=2+1$ QCD according to 230. Open circles are input quantities while filled circles are predictions. Experimental masses of hadrons that are stable in QCD are given with a vertical bar while for resonant states the box indicates the decay width. Experimental numbers are from 20.

Figure 23

Chiral behavior of the ratio of individual octet masses over the average octet mass $X_N=\frac{1}{3}(M_N+M_{\Sigma }+M_{\Xi })$ vs the ratio of the square of the pion mass over the square average of pseudoscalar meson masses $X_{\pi }^2=\frac{1}{3}(2M_K^2+M_{\pi }^2)$ as obtained by the QCDSF-UKQCD Collaboration. From 91, with permission of the QCDSF-UKQCD Collaboration.

Figure 24

Comparison of part of the excited state spectrum of nucleon (left) $\Delta$ (middle), and $\Omega$ (right) type baryons as computed by the Hadron Spectrum Collaboration at three different pion masses with experiment. More details can be found in the original paper. From 125, with permission of the Hadron Spectrum Collaboration.