Recent results on the meson and baryon spectrum from lattice QCD

Recent lattice results on the meson and baryon spectrum with a focus on the determination of hadronic resonance masses and widths using a combined basis of single-hadron and hadron-hadron interpolating fields are reviewed. These mostly exploratory calculations differ from traditional lattice QCD spectrum calculations for states stable under QCD, where calculations with a full uncertainty estimate are already routinely performed. Progress and challenges in these calculations are highlighted.


Introduction
In recent years, tremendous progress has been made in calculating QCD observables from first principles using Lattice QCD. In particular, it is now possible to perform calculations at physical light-quark masses and with multiple lattice spacings and volumes. Improved actions have been developed for simulations with both light (u,d,s) and heavy (c,b) quarks, allowing for a reliable extraction of simple observables for mesons made from these quarks. Figure 1 (from [1]) shows results for the lattice spectrum of states stable or nearly stable under the strong interaction compared to experiment. After some of the experimental masses are used as input to tune the mass parameters and coupling in the action (cyan crosses in the figure), parameter free theory postdictions and predictions were obtained.  Beyond pure QCD, recent progress has allowed the dynamical simulation of both QCD and QED on the lattice. Figure 2 (from [2]) shows the isospin splitting of various ground state hadrons, resulting from both the difference of up and down quark masses and from QED. The resulting mass splittings agree perfectly with those measured in experiment, while splittings not yet observed to date in experiment can also be calculated.
In these proceedings we address more challenging observables, taking a look at hadrons close to multiparticle thresholds and hadronic resonances. While the masses of stable states well separated from multi-particle thresholds are readily extracted in modern lattice calculations, extracting the spectrum of hadronic excitations is a challenge. In Section 2.1 readers will be pointed to literature relevant to understand theoretical aspects of extracting scattering observables from Lattice QCD. In the remainder of Section 2 the current state of the art with regard to both meson-meson and mesonbaryon scattering are briefly reviewed by appealing to specific examples from the literature. In Section 2.2 examples for meson-meson scattering studies will be presented, while 2.3 highlights some of the findings from these studies. In Section 2.4 progress in the simulation of meson-baryon scattering is discussed. Section 3 provides a brief outlook.

Lüscher's finite volume method
From Euclidean space correlation functions, scattering observables are not accessible directly. In a series of seminal papers by Lüscher [3][4][5] it was pointed out that information about the continuum scattering amplitude of elastic scattering can be inferred from the finite volume dependence of the energy levels, which appear shifted from the free energy levels due to the interaction. These energy shifts are illustrated in Figure 3, where the energies for the scattering of two bosons in a channel with a resonance at three times the boson mass are drawn as a function of the box size.
The relations first provided by Lüscher have been greatly generalized over the years, now including relations for any number of (coupled) two particle channels, for 2 ↔ 1 and 2 ↔ 2 transitions (such as ππ → πγ * ), for particles with and without spin, for moving frames, etc.. For recent reviews of the formalism including the relevant references please refer to [6,7]. In practical applications it is crucial to be able to obtain enough lattice energy levels in the region of interest. To this end both multiple volumes and multiple momentum frames are used.

The ρ meson
The simplest QCD resonance is the ρ meson with quantum numbers J PC = 1 −− , seen in isospin-1 ππ scattering. It decays nearly 100% of the time into two pions [8] and there seem to be no significant inelastic contributions from the open four pion threshold. For these reasons the ρ meson is the ideal benchmark for lattice studies of elastic scattering employing the Lüscher formalism. After a pioneering study by the CP-PACS collaboration [9], multiple groups provided proof of principle calculations demonstrating the feasibility of lattice resonance studies [10][11][12]. In the meantime a number of further studies exist [13][14][15][16][17][18].   Figure 4 shows an example of a state-of-the-art calculation by the Hadron Spectrum Collaboration [15]. The figure shows results for the p-wave phase shift δ 1 of elastic ππ scattering for two unphysically large pion masses as a function of the center of mass energy E cm . Using a Breit-Wigner parameterization, the resonance mass and coupling g ρππ can be extracted. The large number of lattice data points results from using all lattice irreps for multiple center-of-mass momenta, and multiple lattice volumes. Figure 5 shows the current world data for g ρππ from the lattice determinations [10][11][12][13][14][15][16][17][18]. While there is in general a good agreement among the lattice data, the uncertainties in the plot are (in most cases) statistical only and there is no calculation demonstrating a full control of systematic uncertainties.  Figure 6. Collection of results from various lattice determination of the ρ meson resonance mass taken from [16]. Notice that deviations are expected at non-physical pion masses as different groups use different ways to determine the physical quark masses and to set the scale in their lattice calculations.
A similar plot for the extracted resonance masses is shown in Figure 6, taken from [16]. It illustrates that the neglected systematics are likely important, as the resonance masses from different simulations tend to deviate quite a bit from each other at unphysical pion masses. Notice however that some deviations are expected as different groups use different ways to determine the physical quark masses and to set the scale in their lattice calculations. As a dimensionful quantity, the resonance mass therefore should only agree for physical parameters and in the continuum limit. Demonstrating this agreement will be a task for future lattice simulations.
With the growing dataset for the ρ-resonance the question naturally arises if there are additional insights gained from taking a more detailed look at the current set of results. In [19] the authors used Unitarized Chiral Perturbation Theory (Unitarized χPT) to analyze the lattice scattering data with 2 flavors of light (up and down) dynamical quarks. Fitting to experiment data in addition, the Unitarized χPT calculation allows to "switch on" the strange quarks. As their final result the authors claim that the low resonance masses observed in 2 flavor calculations (compared to the physical rho and the corresponding 2+1 flavor calculations) are a result of the missingKK channel. If confirmed this constitutes a striking effect from the (partial) quenching of the strange quark. Future lattice calculations with both 2 and 2+1 flavors of dynamical quarks will shed light on this issue.
Beyond the resonance mass and couplings, a lattice determination of the phase shift δ 1 and its derivative also allows for the determination of the pion timelike form factor in the elastic region 500 600 700 800 900 1000 1100 500 600 700 800 900 1000 1100 Mπ=138MeV RQCD16 Figure 7. The left side in each plot shows lattice results from 2-flavor simulations of the ρ meson from a subset of results analyzed in [19]. The right side in each plot shows the chirally extrapolated 2-flavor data (blue dashed curves and blue error bands) and the results after including theKK channel (red curves and bands) compared to the experimental data (blue circles and squares). Plot from [19]. . |F π ( √ (s))| as extracted from a lattice QCD calculation at m π = 240 MeV in [20]. The curve corresponds to a Gounaris-Sakurai parameterization with m ρ and g ρππ determined from the lattice data.
2m π ≤ √ s ≤ 4m π [21]. In this region the form factor F π (E) is given by It is of phenomenological importance for lattice determinations of the hadronic vacuum polarization (HVP) contribution to (g − 2) µ [22]. Recently first lattice calculations of the form factor have been performed in [20,23]. Figure 8 shows results at unphysical parameters from [20], compared to a Gounaris-Sakurai parameterization (not a fit). The displayed results are from a single 2+1 flavor ensemble with m π ≈ 280. These recent calculations demonstrate that the timelike pion form factor can be extracted with a reasonable precision.
2.2.2 D * s0 (2317) and D s1 (2460) and their b-quark cousins In the spectrum of positive parity D s mesons, the J P = 0 + D * s0 (2317) and 1 + D s1 (2460) have masses and properties not expected from potential models forqq mesons. A particularly peculiar fact is that their mass is essentially degenerate with the corresponding D mesons, even though the strange quark is much heavier than the light up and down quarks. This fact lead to speculations that these states have an exotic structure, such as a tetraquark or molecular structure.  Figure 9. Low-lying D s meson spectrum from [24,25] presented with respect to spin-averaged mass 1 4 (m Ds + 3m D * s ). The diamonds and crosses display our lattice results, while black full lines correspond to experiment. The magenta diamonds correspond to the pole position in the T −matrix. Masses extracted as energy levels in a finite box are displayed as blue crosses. Dotted (dashed) lines correspond to DK and D * K lattice (physical) thresholds.
While traditional lattice studies using just quark-antiquark interpolating fields tend to get too large or badly determined masses for those states, a more recent study [24,25] using both quark-antiquark and D ( * ) K structures leads to a spectrum in qualitative agreement with experiment, highlighting the role of the nearby thresholds. The results from this study are shown in Figure 9. While the LHCb experiment should be able to observe them, the corresponding 0 + and 1 + states in the B s spectrum are not yet observed in experiment. Using the finite volume formalism described in Section 2.1 and performing an effective range approximation close to threshold, a recent study [26] observed two bound states below the BK and B * K thresholds (respectively). The lattice data along with the effective range parameterization is plotted in Figure 10. Figure 11 shows the resulting spectrum for the 1S and 1P states. Notice that the known 1 + and 2 + mesons above threshold [8] are well reproduced. For more details please refer to [26].  Figure 11. Spectrum of s-wave and p-wave B s states from [26]. The blue states are naive energy levels, while the bound state energy of the states in magenta results from an effective range approximation of the phase shift data close to threshold. The black lines are the energy levels from the PDG [8]. The error bars on the blue states are statistical only, while the errors on the magenta states show the full (statistical plus systematic) uncertainties.

Coupled channel scattering and light scalar mesons
Using 3 lattice volumes and several moving frames, the Hadron Spectrum Collaboration has started perform the first coupled-channel analyses of meson-meson systems in a finite volume. So far they have investigated coupled πK-ηK scattering [27,28], πη-KK scattering [29], and extended their earlier ρ meson study by also considering coupled channel ππ-KK scattering [15].  Figure 12 shows their results for coupled channel πK-ηK scattering with J P = 0 + , 1 − and 2 + . The low-lying physical resonances [8] in these channels are the K * 0 (800) (also called κ) and the K * 0 (1430) for J P = 0 + , the K * (892) for J P = 1 − , and the K * 2 (1430) for J P = 2 + . The lattice results at an larger than physical pion mass of 391 MeV qualitatively agree with the physical spectrum. In the 0 + channel shown in the left panel of Figure 12, a virtual bound state related to the physical κ resonance is found along a much heavier and very broad K * 0 resonance. The broad resonance is qualitatively compatible with the K * 0 (1430). At such a large pion mass the K * (892) is not a resonance but a closeto-threshold bound state, and the corresponding coupling is compatible with the coupling extracted from experiment. The energy levels from which this bound state can be determined are depicted in the mid pane of the figure. An earlier single-channel lattice calculation [30,31] at a lighter pion mass of 266 MeV already observed a very narrow resonance [31], also compatible with the K * (892). The right panel of Figure 12 shows the lattice data for the J P = 2 + channel, where a narrow resonance  Figure 13. Pole positions in the complex plain for states with quantum numbers 0 + (red) 1 − (orange) and 2 + (green). For a full description please refer to [27,28].
compatible with the K * 2 (1430) is observed. Above the ηK threshold the inelasticity η shown for J P = 0 + and 1 + remains large, indicating that these channels are coupled weakly. Figure 13 summarizes the findings, showing the complex poles determined from the lattice data. For more details on the analysis please refer to [27,28]. An example for a calculation involving tightly coupled channels is given by πη-KK scattering [29] with quantum numbers I(J P ) = 1(0 + ). Here up to three coupled channels (also including πη ) were considered. The lhs panel of Figure 14 shows the corresponding results for the phase shifts and inelasticities at a pion mass of 391 MeV. The rhs panel illustrates the systematic uncertainties from varying the parameterizations used below the πη threshold. All successful parameterizations lead to a pole close to the KK threshold on a single unphysical Riemann sheet. This pole is likely related to the physical a 0 (980). For a full discussion of the results the reader is referred to [29]. In addition the lattice data for the D-wave scattering features a narrow resonance, possibly related to the physical a 2 (1320).

A lesson from meson-meson scattering
Studies of meson-meson scattering have demonstrated that a diverse basis of lattice interpolating operators is needed to reliably extract the finite volume spectrum. For hadron-hadron scattering this means in particular that to ensure good overlap with the physical states, all relevant meson-meson or meson-baryon interpolators should be included explicitly in the correlator basis. with ππ without ππ Figure 15. The lowest two energy levels from isospin-1 ππ scattering extracted using different sub-matrices (interpolators sets) of the full 16 × 16 correlation matrix. For details on the basis used please refer to [11]. To illustrate this point further it is instructive to take a look at how the simulation results change when the full basis used in a given study is artificially truncated. This is illustrated in Figures 15 and  16. Figure 15 shows results for low-lying energy in three momentum frames from an early simulation of ππ scattering in the ρ meson channel [11]. When the basis consisting of both quark-antiquark and meson-meson interpolating operators is reduced to contain just quark-antiquark operators, some of the excited-state energy levels become either ill-determined or display fake-plateaus, not representative of the physical spectrum. Figure 16 from [15] provides an illustration of the same problem: Using only the energy levels from single hadron (quark-antiquark) interpolators, the true phase-shift shown with the fit extracted from the full data set (curve) is missed and one would naively extract a much narrower resonance.

Baryon resonances in Meson-Baryon scattering
Studies of baryon-meson scattering on the lattice are complicated by several aspects. For one, the signal to noise ratio at large Euclidean times is exponentially suppressed, leading to much noisier data. Furthermore the number of possible contractions is usually larger and the objects one calculates are more complex. Frames with total nonzero momentum lead to parity mixing in the lattice data, which makes the extraction of the spectrum even more challenging.
Consequently most studies of baryon spectroscopy to date extracted the energy levels from threequark interpolators only. In most cases multi-hadron levels are absent from the resulting spectra, and the cautionary remarks from Section 2.3 also apply to these studies. While showing some qualitative features of the physical spectrum, it is unclear to what extent such studies can be trusted. In this section, we therefore focus on exploratory studies including baryon-meson or other five-quark interpolating fields. A first study of low-lying negative parity nucleons has been performed by Lang and Verduci [32,33]. Figure 17 shows their results on a single volume with a pion mass of 266 MeV. Just like the meson studies discussed before, the spectrum using just single hadron interpolators is incomplete. Once meson-baryon interpolators are included, the low-lying energy spectrum can be extracted with reasonable precision. Future lattice QCD studies of negative parity baryon resonances will however require more volumes and/or the use of moving frames as well as additional scattering channels.
A more recent study by the Adelaide group [34] explored the use of local 5-quark interpolators for extracting the positive parity nucleon spectrum. Figure 18 shows the dependence of the energy levels observed when varying the interpolator basis. The complete basis consisted of 3-quark and local 5-quark interpolators with structures resembling Nπ, Nσ, and Na 0 states. Unlike [33] this study does not result in energy levels close to the relevant thresholds and the spectrum should probably considered to be incomplete. The authors suggest that their results favor a scenario where the Roper resonance is a dynamically generated resonance.
All of these calculations are currently of an exploratory nature, and future lattice studies will have to further address the systematic uncertainties, for example arising from discretization effects, exponential volume corrections, the pion mass dependence of observables, and, in some cases, neglected three-particle channels. In light of the many challenges one should recall that, unlike approaches based on models, the results of this programme are parameter free predictions of QCD.