HHadron Structure and Spectrum from the Lattice
C. B. Lang
Inst. of Physics, University of Graz, A-8010 Graz, Austria. [email protected]
Abstract.
Lattice calculations for hadrons are now entering the domain of resonances and scattering, necessitating a better under-standing of the observed discrete energy spectrum. This is a reviewing survey about recent lattice QCD results, with some emphasison spectrum and scattering.
INTRODUCTION
Quantum field theories in four dimensions are not well defined without some regularization. Wilson’s [1] formulationon a Euclidean space-time lattice is such a regularization with the advantage of maintaining gauge invariance andstraightforward accessibility by computer. The path integrals become finite dimensional integrals, however of veryhigh dimensions. The continuum limit is obtained by keeping the physical volume fixed while letting the latticespacing a approaching zero. The scale parameter is determined by comparing a physical observable (e.g., a mass m )with the measured dimensionless lattice observable (e.g., the product a m ). Once a is determined, all further latticeobservables can then be translated to physical values. Since one has to fix also the quark mass parameters one hasto trade n f + a → ff ect is due to the lightest hadron, the pion, thus one wants lattice sizes L where Lm π is large; typical values are largerthan 4. Below that value boundary e ff ects may be sizeable. This gives L > sites. Bringing the quark masses and, equivalently the pion mass, down to physical values necessitates largevolumes or good control on the finite size dependence and of the scaling behaviour in a .The main object of lattice simulations are correlation functions, the Euclidean equivalent of n-point functions.Masses or better: energies are obtained from the exponential decay of hadron propagators. However, in a quantumchannels there will be contribution of (formally: infinitely) many states: C i j ( t ) ≡ (cid:104) X i ( t ) X j (0) (cid:105) = (cid:80) n (cid:104) X i | n (cid:105) e − t E n (cid:104) n | X j (cid:105) .Due to the finiteness of the lattice volume the energies are discrete even in the situation of open scattering channels.Asymptotically, for large t the ground state dominates. However, the statistical errors increasingly obscure the signalwith increasing t and most often one has to work at not so large t .Depending on the type of calculation the excited states may be a nuisance or an advantage. In case one isinterested in hadronic ground state parameters (there are not many such hadrons: the pseudoscalars and the nucleonwith its strange, charmed, beautiful and maybe topped cousins) or in 3-point functions (like form factors or othermatrix elements) the excited states are a “contamination” and one fights to get rid of their influence. If, on the otherhand, one is interested in excited hadrons and decay properties one needs the excitation energies as precise as possible. HADRON STRUCTURE
Obviously I have to concentrate here on a few highlights and the choice is subjective. Recent more complete reviewson the topic are [2, 3]. Also I am considering only information available at the time of this conference (Sept. 2015)with emphasis on results in the recent two years. a r X i v : . [ nu c l - t h ] D ec .05 0.1 0.15 0.2 m π2 g A MainzCLS(2)2013QCDSF(2)2014ETMC(2)2010LHPC(2+1)2014RBC/UKQCD(2+1)2013ETMC(2+1+1)2013ETMC(2+1+1)2015PNDME(2+1+1)2014RQCD(2+1+1)2015Exp.RQCD, extrapol.QCDSF, extrapol.
FIGURE 1.
Left: Matrix element calculation. Right: Axial charge of the nucleon, comparison of values obtained by several groups.The color indicates groups using the same number of dynamical fermions ( N u , d + N s + N c ); no significant di ff erences are seen. Hadron structure calculations are based on studying 3-point functions. A current is inserted between an incomingand an outgoing hadron (cid:104) H | Γ | H (cid:48) (cid:105) . In terms of lattice operators O (also called interpolators) we have (cid:104) O ( t , (cid:126) p ) | Γ ( τ ) | O (0 ,(cid:126) r ) (cid:105) . (1)Fig. 1 shows the situation (and the concerns). Assuming that the temporal distances | τ | and | t − τ | are large enough thatthe ground state hadron dominates the intermediate state, G ≡ (cid:104) O ( t , (cid:126) p ) | H ( (cid:126) p ) (cid:105) e − E p ( t − τ ) E p (cid:104) H ( (cid:126) p ) | Γ ( τ ) | H ( (cid:126) r ) (cid:105) e − E r τ E r (cid:104) H ( (cid:126) r ) | O (0 ,(cid:126) r ) (cid:105) (2)the components factorize. One also determines the 2-point function G = (cid:104) O ( t , (cid:126) q ) O (0 , (cid:126) p ) (cid:105) = (cid:104) O ( t , (cid:126) p ) | H ( (cid:126) p ) (cid:105) e − E p t E p (cid:104) H ( (cid:126) p ) | O (0 ,(cid:126) r ) (cid:105) . (3)and retrieves the wanted matrix element (cid:104) H | Γ | H (cid:105) from a plateau behavior of suitable ratios of G and G .In lattice calculations of matrix elements there are several concerns: volume (finite size e ff ects), lattice hadronoperators, contamination from excited states, disconnected contributions (depending on the insertion type), renormal-ization factors (and possible mixing with other operators), dependence on the pion (quark) mass and lattice spacing.Most studies are at higher pion mass and have to be extrapolated to the physical value. All these aspects have to becarefully considered for a reliable result. Axial Charge of the Nucleon
That axial isovector coupling g A can be obtained from (cid:104) p | ¯ u γ µ γ d | n (cid:105) . Fig. 1, right show results of recent years. All areslightly below the experimental value g A / g V = . ff erent numbers of dynamical quarks do not explainthis. The dependence on lattice spacing or volume (recent results cover a range 3 < m π L <
6) also shows no trend (cf.plots in [2]).The most likely suspect is the influence of excited states that may be still significant in the region of the insertion.Fitting the mentioned ratio at several values of τ to a plateau may lead to an underestimation of the value. Analysisvariants are a summation method [4] or adding excited states to the fit function. Recent studies carefully analyse thoseapproaches [5, 6]. From Fig. 2 of Ref. [6] one clearly sees that the nucleon has admixture from excitations up to about0.6 fm; the source and sink are at t = t = t f ≈ g A / F π ( F π is the pion decay constant) some of the finite volume influence seemsto cancel [7] and extrapolation to physical pion masses gives a value close to experiment [6].Further recent results include a study of the disconnected contribution to the isoscalar (S and A) matrix elements,which are O(7%) [8]. In [6, 5] also isovector couplings g S and g T have been determined and in a ChPT study thenucleon-pion-state contributions in the determination of the nucleon axial charge have been estimated to be a fewpercent [9]. Thanks to Martha Constantinou and Sara Collins for help. π [MeV] ( r ) v [fm ] LHPC [10] 149 0.498(55)Mainz CLS [11] 193 0.501(42)ETMC [12] 135 0.398(126)Exp. ep [17] 0.640(9)Exp. µ p [18] 0.578(2) FIGURE 2.
Dirac charge radius r from lattice data (Figure from [2] with superimposed new points due to [10, 11, 12]). The Tablecompares the results with the numbers from e p scattering and muonic hydrogen experiments (the di ff erence in the numbers fromthe experiments is not yet understood). Nucleon Electromagnetic Form Factors
The so-called Dirac ( F ) and Pauli ( F ) form factors are determined from the matrix element (cid:104) N ( p ) | V µ | N ( r ) (cid:105) ∼ ¯ u N ( p ) (cid:34) F ( q ) γ µ + F ( q ) i σ µν q ν m N (cid:35) u N ( r ) with q ≡ ( p − r ) ≡ − Q . (4)Here recent work has been already at close to physical pion masses [10, 11, 12, 13]. The results are generally consis-tent. They cover, however, only a very small range of Q as compared to experiments. The reason is indigenous to thelattice approach. Due to the finite box size the momenta are quantized. (E.g., q = (cid:126) k (2 π/ L ) for the non-interactingcase, where (cid:126) k is a vector with integer components.) This constrains both, the lowest and the highest achievable values.For small q one need large volumes, for large q the statistical noise increases. Already (cid:126) k =
6, which corresponds to Q ≈ for L = (cid:104) h |O| h (cid:105) and the derivative of a 2-point function may help in going to larger Q [14, 15, 16]Lower values of Q are important for the charge radii which are obtained from extrapolating fits to the latticedata. As can be seen from Fig. 2 the bulk of the values is significantly smaller than both experimental values. The Proton Spin
The proton spin has contributions from the quarks and the gluons: = (cid:80) q J q + J G . where one may distinguish thequark orbital and spinor contributions J q = L q + ∆Σ q , suggested in [19]. The quark contributions need computationof matrix elements like (cid:104) x (cid:105) q and (cid:104) p (cid:48) | T µν | p (cid:105) , involving derivative operators. To determine individual ∆Σ q one needs toconsider disconnected contributions which require high statistics and special methods (stochastic source methods).All lattice calculations need to be extrapolated down to the physical pion mass. In [20] Heavy Baryon ChiralPerturbation Theory was used and the results are quite sensitive on the extrapolation leading to large systematicerrors. For the light quarks the values have stabilized at ∆ u + ∆ d = . m π giving a value ∆ s = − . ∆Σ = ∆ u + ∆ d + ∆ s = . Low Energy Parameters
Low energy parameters like leptonic and semileptonic decay constants, CKM matrix elements, quark masses, thequark condensate, α s and others are collected in the compilation of the Flavor Lattice Averaging Group - FLAG( http://itpwiki.unibe.ch/flag ) [24]. Heavy meson decay constants can be found in recent work by [25]. adiative Decays On the lattice on-shell decays are forbidden due to the Maiani-Testa theorem; however Lellouch and L¨uscher [26]found a method to circumvent that problem. Recently Brice˜no [27] formulated a technique to address readiativedecays like ρ → πγ ∗ (For alternative approaches see [28, 29]). In [30] the ρ was assumed to be stable and basictools for the analysis were formulated. The pion mass there is quite high of O (700 MeV). The CSSM collaborationpresented results at almost physical pion mass (157 MeV)[31].In the real process, however, the ρ is a resonance: πγ ∗ → ρ → ππ . This now has been studied in a lattice sim-ulation [32]. The transition matrix element was computed and a parametrisation of the amplitude allows the analyticcontinuation to the ρ -pole in the unphysical sheet and extraction of the form factor F πρ ( E ∗ ππ , Q ) from the residue. Thecalculation still is for large pion mass of 400 MeV but compares favorable with phenomenological model calculations.For more information see Brice˜no’s contribution to this conference. HADRON SPECTROSCOPYSingle Hadron Approximation
A recent highlight is the determination of the electromagnetic mass di ff erences for p , Σ , Ξ and others [33]. Fourquark species u , d , s , and c were taken into account and QED in its non-compact version was added to QCD, bothnon-perturbatively. QED needs special care: gauge fixing, finite volume corrections O (1 / L ) and regularisation schememake life hard (see also [34, 35]). The results obtained for 197 MeV ≤ m π ≤
440 MeV were extrapolated to thephysical point leading to high precision values in good agreement with experiment, in some cases predictions like for ∆Σ . Milestones in the determination of the hadronic states were [36, 37, 38]. In [36] prominent members of the ( u , d , d ) family of hadrons were obtained, in [37, 38] meson and baryon excitations were determined for several spin-paritychannels. This year has brought results on singly- and doubly charmed baryons with and without strangeness[39]for ground states and first excitations. The pion masses were between 260 and 460 MeV and the results could beextrapolated to the physical point. Ground state energies for baryons with up to three heavy quarks ( c and b ) werecomputed by [40] at several pion masses and extrapolated to the physical point.A challenging problem is the identification of spin, since di ff erent continuum spins couple to the same latticeoperator. Comparing the overlap patterns the Hadron Spectrum Collaboration resolved spins up to 4 in the (excited)charmonium spectrum [41] and for charmed mesons [42]. Based on that experience, in [43] doubly charmed baryonswere studied (for a pion mass of 400 MeV) and spin identification up to 7 / S U (6) × O (3).The bulk of results was extracted from correlation functions of single hadrons, i.e., either baryonic three-quarkoperators or mesonic quark-antiquark operators. Although we know that in quantum field theory all possible multi-quark intermediate states can contribute due to the dynamical vacuum with fermion loops, in practical lattice calcu-lations using single hadron correlations these contributions are suppressed. This explains why one finds signals forresonances although they are not asymptotic states. The influence of coupled channels and associated thresholds ise ff ectively neglected. It also means that an observed excited level do not necessarily give the position of the resonancepeak. One has to allow for multi-hadron operators in the set of lattice interpolators. Multi-Hadron Approach
This led to a changed point of view: One does not study the resonance correlators but the scattering process whereresonances may appear. Due to the finite volume the energy spectrum of the scattering process is discrete. L¨uscherderived a relation [44, 45] between the spectrum in finite volume to the phase shift in the continuum for elastic meson-meson scattering. This has been extended to moving frames and hadron-hadron scattering in general. In recent yearsthere has been an explosion of contributions in that direction.What are the challenges in that approach? First one needs to consider a larger set of operators - single hadron aswell as hadron-hadron operators - and cross-correlations C i j ( t ) between them. In that correlation elements one has (forbaryon-baryon scattering) up to six valence quark propagators. Secondly, there will be quark-antiquark annihilations“backtracking quarks”) in disconnected or partially disconnected terms. This is a notorious problem in such simula-tions and needs high statistics and e ffi cient new tools like stochastic sources or distillation. In the distillation method[46, 47] the hadron operators are constructed from quark sources that are eigenvectors of the spatial Laplacian. Oncethe quark propagators between these sources, the so-called perambulators, have been constructed, it is possible toe ffi ciently compute correlations between di ff erent operators. Changing the operators and projection to momenta canbe done independent of the perambulators and so the method is very versatile.L¨uscher’s original method was valid in the elastic region but meanwhile there are extensions to several coupledchannels [48] including nucleon-nucleon scattering, moving frames and arbitrary spin [49, 50, 51] and generalizationsof the Lellouch-L¨uscher 1 → a → ρπ → πππ ) to a three-hadron state and also there theoretical results werepresented recently [53, 54, 55]. No actual lattice simulation exists yet.Following several studies of elastic ππ and π K scattering as well as coupled channels model calculations [56, 57]the last year has finally brought the first coupled channel simulation. Dudek et al. [58] investigated s -, p - and d -wavesof the coupled π K − η K system. Three lattice sizes with up to 70 identified energy levels and an interpolating modelallows the determination of phase shifts and inelasticity up to 1600 MeV. Due to the large pion mass of 391 MeVthe K ∗ comes as a bound state but in particular in s and d wave the main features are successfully reproduced. Thispromising result was followed by a ππ , KK coupled channel study [59]. These results for a pion mass of 236 MeVhave then been extrapolated to the physical point [60]. More details can be found in the contributions of J. Dudek, D.Wilson and D. Bolton to this conference.A first application of π N scattering was presented already two years ago [61, 62] demonstrating the importance ofscattering states. Earlier results for the − channel did show two energy levels tentatively attributed to the N (1535) and N (1650), but the splitting was too large. In the new study the lowest π N s -wave level was correctly identified closelybelow threshold and the next two level had the right splitting and position of N (1535) and N (1650). Meanwhile furtherresults with multi-hadron interpolators have appeared [63].Nucleon-nucleon scattering needs six valence quark propagators but none of the quarks is backtracking. Such astudy needs large spatial lattice size; results for s , p , d , and f partial waves and spatial extent 4.6 fm was presentedrecently [64]. The pion mass there is quite high (800 MeV) but further studies closer to the physical values are to beexpected. Heavy Quarks
Recent reviews on lattice results in the heavy flavor sector are [65, 66, 67]. At present it is hopeless to perform afull coupled channel phase shift lattice calculation in the charmonium sector - there are too many coupled channelsin the interesting energy regions. On the way towards that far-lying goal we can, however, learn something from themeasured energy levels. An example for this “level hunting” is the search for a signal of the Z + c (3900) state. In [68]18 interpolators of meson-meson type with ccud quark content as well as four tetraquark operators were included inthe cross-correlation matrix. All observed levels (covering the energy range up to 4.1 GeV) could be identifed with(expected) meson-meson states and no extra state (which then could be associated to the Z + c (3900)) was found in this I G ( J PC ) = + (1 + − ) channel. This agrees with other lattice studies [69, 70]. There is an ongoing discussion whether the Z c (3900) might be a threshold e ff ect. This has been also discussed in the so-called HALQCD approach [71]. There apotential related to the Nambu-Bethe-Salpeter equation is determined in a coupled channel formalism [72].Charmonium levels in the single hadron approximation are in good agreement with experiment only below the DD threshold. In [73] charmonium ψ (3770) was studied in a system of 15 operators of cc type as well as two DD interpolators for two pion masses (266 MeV and 157 MeV). Below threshold ψ (2 S ) and above threshold ψ (3770)were identified, both in good agreement with experiment.Of particular interest are resonances or bound states close to thresholds. The reciprocal partial wave scatteringamplitude (in the elastic regime) may be parametrized byRe[ f − (cid:96) ( s )] = ρ ( s ) cot δ (cid:96) ( s ) − i ρ ( s ) ≡ k − ( s ) − i ρ ( s ) with the phase space factor ρ ( s ) = p (cid:96) + / √ s (5)and in the L¨uscher-type analysis each energy level gives a value of Re( f − ) = c Z (cid:18) (cid:16) pL π (cid:17) (cid:19) (cf. Fig. 3). Abovethreshold each point gives a value of the phase shift. Interpolation and continuation below threshold allows to retrievethreshold parameters as well bound state energies or resonance position and coupling. IGURE 3.
Left: Schematic description of the L¨uscher analysis: Red curves are the theoretically possible values, the measuredenergy values then lead to the values of k − ( s ) lying on that curves. Right: Example for this scenario for B s (0 + ) BK scattering (Fig.from [74]); note that below threshold the analytic continuation of the phase space factor contributes to the real part leading to thebound state position. E n [ G e V ] Exp. Lat. Lat. − O D(0) - D*(0)J/ Ψ (0) ω (0)D(1) - D* (-1)J/ Ψ (1) ω (-1) η c (1) σ (-1) χ c1 (0) σ (0) FIGURE 4.
Left: Fig. from [75] for the I = cc and cc ( uu + dd ) interpolators. The tetraquark operators appear tohave little e ff ect on the spectrum. The red squares are dominated by cc operators and are attributed to χ c and X (3872). Right: Fig.from [76]; results for D s states derived for a gauge field ensemble “Ens.2” with pion mass of 157 NeV [77]. In the 0 + (1 ++ ) channel lies the X (3872); this state was postdicted in a lattice study (for the first time) [78]. Thiswas confirmed in [69]. A recent study [75] extended the set of coupled channel operators significantly (22 interpolatorsincluding DD ∗ , J /ψω , η c σ , χ c σ , as well as four tetraquark operators). The X (3872) closely below DD ∗ threshold wasreconfirmed with a strong cc Fock component. It is not seen, if the cc interpolators are not included.Phenomenological models as well as lattice calculations gave controversial results for the D s in 0 ++ and 1 ++ . Inboth cases there is a nearby threshold: KD and KD ∗ , respectively, and it was suggested that these channels may beimportant components of the states [79]. Indeed a lattice study [80, 76] including these channels reproduced the patternfrom experiment and identified bound states D s (2315) and D s (2460) and, above the KD ∗ threshold D s (2536) (Fig.4). The levels were consistently higher than experiment due to larger than physical pion mass of 157 MeV and quarkmass tuning e ff ects but the splitting and distance to threshold agreed with experiment.Motivated by these results a similar study was then done for B s in 0 + , 1 + and 2 + with BK and B ∗ K contri-butions [74]. In 0 + a bound state B s with a mass of 5.711(13)(19) GeV and in 1 + a bound state B s with a massof 5.750(17)(19) GeV was predicted. Close to threshold a weakly coupled B s at a mass of 5.831(9)(6) GeV wasidentified close to the experimental state at 5.8288(4) GeV. Summary
The lattice formulation of QCD is mathematically well defined and provides a controlled continuum limit. Withincreasing compute power and algorithmic improvements we have come close to that ambition. Lattice structureresults approach the quality needed for an input to experiment analysis, although they are not yet precise enough andone still has to understand the origin of deviations. E ffi cient methods for disconnected graph contributions are needed.Our understanding of lattice scattering has improved considerably and hadron spectroscopy has entered a new era.Processes involving several coupled channels are still a challenge. CKNOWLEDGMENTS
Many thanks go to my collaborators of recent years Sasa Prelovsek, Daniel Mohler, Luka Leskovec and PadmanathMadanagopalan. I thank Sara Collins and Martha Constantinou for their help with the data collection and Constan-tia Alexandrou, Raul Brice˜no, Christine Davies, Michael Engelhardt, Gian-Carlo Rossi and Andr´e Walker-Loud forinformation. Support by the Austrian Science Fund FWF: I1313-N27 is gratefully acknowledged,
REFERENCES [1] K. G. Wilson, Phys. Rev. D , 2445 (1974).[2] M. Constantinou, PoS LATTICE2014 , 001 (2015), [arXiv:1411.0078].[3] J. M. Zanotti, Review on hadron structure at lattice 2015, 2015.[4] S. Capitani, B. Knippschild, M. Della Morte and H. Wittig, PoS
LATTICE2010 , 147 (2010),[arXiv:1011.1358].[5] A. Abdel-Rehim et al. , Nucleon and pion structure with lattice QCD simulations at physical value of thepion mass, 2015, [arXiv:1507.04936].[6] G. S. Bali et al. , Phys. Rev. D , 054501 (2015), [arXiv:1412.7336].[7] R. Horsley et al. , Phys. Lett. B , 41 (2014), [arXiv:1302.2233].[8] A. Abdel-Rehim et al. , Phys. Rev. D , 034501 (2014), [arXiv:1310.6339].[9] O. B¨ar, Nucleon-pion-state contributions in the determination of the nucleon axial charge, 2015,[arXiv:1508.01021].[10] J. R. Green et al. , Phys. Rev. D , 074507 (2014), [arXiv:1404.4029].[11] S. Capitani et al. , Phys. Rev. D , 054511 (2015), [arXiv:1504.04628].[12] A. Abdel-Rehim et al. , PoS LATTICE2014 , 148 (2015), [arXiv:1501.01480].[13] PACS, T. Yamazaki, Light nuclei and nucleon form factors in N f = + / UKQCD, CSSM, A. J. Chambers et al. , Phys. Rev. D , 014510 (2014), [arXiv:1405.3019].[15] A. J. Chambers et al. , Applications of the Feynman-Hellmann theorem in hadron structure, 2015,[arXiv:1511.07090].[16] P. E. Shanahan et al. , Phys. Rev. Lett. , 091802 (2015), [arXiv:1403.6537].[17] P. J. Mohr, B. N. Taylor and D. B. Newell, Rev. Mod. Phys. , 1527 (2012), [arXiv:1203.5425].[18] A. Antognini et al. , Science , 417 (2013).[19] X.-D. Ji, Phys. Rev. Lett. , 610 (1997), [arXiv:hep-ph / et al. , Phys. Rev. D , 014509 (2013), [arXiv:1303.5979].[21] Y.-B. Yang, M. Gong, K.-F. Liu and M. Sun, PoS LATTICE2014 , 138 (2014), [arXiv:1504.04052].[22] A. J. Chambers et al. , Disconnected contributions to the spin of the nucleon, 2015, [arXiv:1508.06856].[23] K.-F. Liu, Quark and Glue Components of the Proton Spin from Lattice Calculation, 2015,[arXiv:1504.06601].[24] S. Aoki et al. , Eur. Phys. J.
C74 , 2890 (2014), [arXiv:1310.8555].[25] HPQCD, B. Colquhoun et al. , Phys. Rev. D , 114509 (2015), [arXiv:1503.05762].[26] L. Lellouch and M. L¨uscher, Commun. Math. Phys. , 31 (2001), [arXiv:hep-lat / , 034501 (2015), [arXiv:1406.5965].[28] V. Bernard, D. Hoja, U.-G. Meißner and A. Rusetsky, JHEP , 023 (2012), [arXiv:1205.4642].[29] A. Agadjanov, V. Bernard, U.-G. Meißner and A. Rusetsky, Nucl. Phys. B , 1199 (2014),[arXiv:1405.3476].[30] C. J. Shultz, J. J. Dudek and R. G. Edwards, Phys. Rev. D , 114501 (2015), [arXiv:1501.07457].[31] B. J. Owen, W. Kamleh, D. B. Leinweber, M. S. Mahbub and B. J. Menadue, Phys. Rev. D , 034513(2015), [arXiv:1505.02876].[32] R. A. Brice˜no et al. , The resonant π + γ → π + π amplitude from Quantum Chromodynamics, 2015,[arXiv:1507.06622].[33] S. Borsanyi et al. , Science , 1452 (2015), [arXiv:1406.4088].[34] Z. Davoudi and M. J. Savage, Phys. Rev. D , 054503 (2014), [arXiv:1402.6741].[35] M. G. Endres, A. Shindler, B. C. Tiburzi and A. Walker-Loud, Massive photons: an infrared regularizationscheme for lattice QCD + QED, 2015, [arXiv:1507.08916].36] S. D¨urr et al. , Science , 1224 (2008), [arXiv:0906.3599].[37] R. G. Edwards, J. J. Dudek, D. G. Richards and S. J. Wallace, Phys. Rev. D , 074508 (2011),[arXiv:1104.5152].[38] R. G. Edwards, N. Mathur, D. G. Richards and S. J. Wallace, Phys. Rev. D , 054506 (2013),[arXiv:1212.5236].[39] P. P´erez-Rubio, S. Collins and G. S. Bali, Phys. Rev. D , 034504 (2015), [arXiv:1503.08440].[40] Z. S. Brown, W. Detmold, S. Meinel and K. Orginos, Phys. Rev. D , 094507 (2014), [arXiv:1409.0497].[41] Hadron Spectrum Collaboration, L. Liu et al. , JHEP , 126 (2012), [arXiv:1204.5425].[42] G. Moir, M. Peardon, S. M. Ryan, C. E. Thomas and L. Liu, JHEP , 021 (2013), [arXiv:1301.7670].[43] M. Padmanath, R. G. Edwards, N. Mathur and M. Peardon, Phys. Rev. D , 094502 (2015),[arXiv:1502.01845].[44] M. L¨uscher, Commun. Math. Phys. , 153 (1986).[45] M. L¨uscher, Nucl. Phys. B , 531 (1991).[46] Hadron Spectrum Collaboration, M. Peardon et al. , Phys. Rev. D , 054506 (2009), [arXiv:0905.2160].[47] C. Morningstar et al. , Phys. Rev. D , 114505 (2011), [arXiv:1104.3870].[48] V. Bernard, M. Lage, U. G. Meißner and A. Rusetsky, JHEP , 019 (2011), [arXiv:1010.6018].[49] R. A. Brice˜no and Z. Davoudi, Phys. Rev. D , 094507 (2013), [arXiv:1204.1110].[50] R. A. Brice˜no, Z. Davoudi and T. C. Luu, Phys. Rev. D , 034502 (2013), [arXiv:1305.4903].[51] R. A. Briceno, Phys. Rev. D , 074507 (2014), [arXiv:1401.3312].[52] M. T. Hansen and S. R. Sharpe, Phys. Rev. D , 016007 (2012), [arXiv:1204.0826].[53] M. T. Hansen and S. R. Sharpe, Phys. Rev. D , 116003 (2014), [arXiv:1408.5933].[54] U.-G. Meißner, G. R´ıos and A. Rusetsky, Phys. Rev. Lett. , 091602 (2015), [arXiv:1412.4969].[55] M. T. Hansen and S. R. Sharpe, Expressing the three-particle finite-volume spectrum in terms of the three-to-three scattering amplitude, 2015, [arXiv:1504.04248].[56] V. Bernard, M. Lage, U.-G. Meißner and A. Rusetsky, JHEP , 024 (2008), [arXiv:0806.4495].[57] P. Guo, J. Dudek, R. Edwards and A. P. Szczepaniak, Phys. Rev. D , 014501 (2013), [arXiv:1211.0929].[58] J. J. Dudek, R. G. Edwards, C. E. Thomas and D. J. Wilson, Phys. Rev. Lett. , 182001 (2014),[arXiv:1406.4158].[59] D. J. Wilson, R. A. Briceno, J. J. Dudek, R. G. Edwards and C. E. Thomas, Phys. Rev. D , 094502 (2015),[arXiv:1507.02599].[60] D. R. Bolton, R. A. Briceno and D. J. Wilson, Connecting physical resonant amplitudes and lattice QCD,2015, [arXiv:1507.07928].[61] C. B. Lang and V. Verduci, Phys. Rev. D , 054502 (2013), [arXiv:1212.5055].[62] V. Verduci and C. B. Lang, PoS LATTICE2014 , 121 (2014), [arXiv:1412.0701].[63] A. L. Kiratidis, W. Kamleh, D. B. Leinweber and B. J. Owen, Phys. Rev. D , 094509 (2015),[arXiv:1501.07667].[64] E. Berkowitz et al. , Two-Nucleon Higher Partial-Wave Scattering from Lattice QCD, 2015,[arXiv:1508.00886].[65] S. Prelovsek, Lattice studies of charmonia and exotics, 2015, [arXiv:1508.07322].[66] D. Mohler, Recent Progress in Lattice Calculations of Properties of Open-Charm Mesons, 2015,[arXiv:1508.02753].[67] M. Padmanath and N. Mathur, Charmed baryons on the lattice, 2015, [arXiv:1508.07168].[68] S. Prelovsek, C. B. Lang, L. Leskovec and D. Mohler, Phys. Rev. D , 014504 (2015), [arXiv:1405.7623].[69] S.-H. Lee, C. DeTar, D. Mohler and H. Na, Searching for the X (3872) and Z + c (3900) on HISQ Lattices, 2014,[arXiv:1411.1389].[70] Y. Chen et al. , Phys. Rev. D , 094506 (2014), [arXiv:1403.1318].[71] Y. Ikeda, Zc(3900) from coupled-channel hal qcd approach on the lattice; presented at lattice 2015, 2015.[72] S. Aoki et al. , Phys. Rev. D , 034512 (2013), [arXiv:1212.4896].[73] C. B. Lang, L. Leskovec, D. Mohler and S. Prelovsek, JHEP , 089 (2015), [arXiv:1503.05363].[74] C. B. Lang, D. Mohler, S. Prelovsek and R. M. Woloshyn, Phys. Lett. B , 17 (2015), [arXiv:1501.01646].[75] M. Padmanath, C. B. Lang and S. Prelovsek, Phys. Rev. D , 034501 (2015), [arXiv:1503.03257].[76] C. B. Lang, L. Leskovec, D. Mohler, S. Prelovsek and R. M. Woloshyn, Phys. Rev. D , 034510 (2014),[arXiv:1403.8103].[77] S. Aoki et al. , Phys. Rev. D , 034503 (2009), [arXiv:0807.1661].[78] S. Prelovsek and L. Leskovec, Phys. Rev. Lett. , 192001 (2013), [arXiv:1307.5172].[79] E. van Beveren and G. Rupp, Phys. Rev. Lett. , 012003 (2003), [arXiv:hep-ph /111