Abstract
The extraction of two- and three-body hadronic scattering amplitudes and the properties of the low-lying hadronic resonances from the finite-volume energy levels in lattice QCD represents a rapidly developing field of research. The use of various modifications of the Lüscher finite-volume method has opened a path to calculate infinite-volume scattering amplitudes on the lattice. Many new results have been obtained recently for different two- and three-body scattering processes, including the extraction of resonance poles and their properties from lattice data. Such studies, however, require robust parametrizations of the infinite-volume scattering amplitudes, which rely on basic properties of S-matrix theory and—preferably—encompass systems with quark masses at and away from the physical point. Parametrizations of this kind, provided by unitarized Chiral Perturbation Theory, are discussed in this review. Special attention is paid to three-body systems on the lattice, owing to the rapidly growing interest in the field. Here, we briefly survey the formalism, chiral extrapolation, as well as finite-volume analyses of lattice data.
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Notes
The Euclidean time dimension does not play any role in this review and will be always assumed to be infinite.
References
L. Maiani, M. Testa, Final state interactions from Euclidean correlation functions. Phys. Lett. B 245, 585 (1990)
M. Lüscher, Volume dependence of the energy spectrum in massive quantum field theories. 1. Stable particle states. Commun. Math. Phys. 104, 177 (1986)
M. Lüscher, Volume dependence of the energy spectrum in massive quantum field theories. 2. Scattering states. Commun. Math. Phys. 105, 153 (1986)
M. Lüscher, Two particle states on a torus and their relation to the scattering matrix. Nucl. Phys. B 354, 531 (1991)
M. Lüscher, P. Weisz, On-shell improved lattice gauge theories. Commun. Math. Phys. 97, 59 (1985)
D. Guo, A. Alexandru, R. Molina, M. Döring, Rho resonance parameters from lattice QCD. Phys. Rev. D 94, 034501 (2016). arXiv:1605.03993
M. Mai, C. Culver, A. Alexandru, M. Döring, F.X. Lee, Cross-channel study of pion scattering from lattice QCD. Phys. Rev. D 100, 114514 (2019). arXiv:1908.01847
K. Rummukainen, S.A. Gottlieb, Resonance scattering phase shifts on a nonrest frame lattice. Nucl. Phys. B 450, 397 (1995). arXiv:hep-lat/9503028
X. Li, C. Liu, Two particle states in an asymmetric box. Phys. Lett. B 587, 100 (2004). arXiv:hep-lat/0311035
X. Feng, X. Li, C. Liu, Two particle states in an asymmetric box and the elastic scattering phases. Phys. Rev. D 70, 014505 (2004). arXiv:hep-lat/0404001
C.H. Kim, C.T. Sachrajda, S.R. Sharpe, Finite-volume effects for two-hadron states in moving frames. Nucl. Phys. B 727, 218 (2005). arXiv:hep-lat/0507006
M. Lage, U.-G. Meißner, A. Rusetsky, A method to measure the antikaon-nucleon scattering length in lattice QCD. Phys. Lett. B 681, 439 (2009). arXiv:0905.0069
Z. Fu, Rummukainen–Gottlieb’s formula on two-particle system with different mass. Phys. Rev. D 85, 014506 (2012). arXiv:1110.0319
Z. Davoudi, M.J. Savage, Improving the volume dependence of two-body binding energies calculated with lattice QCD. Phys. Rev. D 84, 114502 (2011). arXiv:1108.5371
M. Döring, U.-G. Meißner, E. Oset, A. Rusetsky, Scalar mesons moving in a finite volume and the role of partial wave mixing. Eur. Phys. J. A 48, 114 (2012). arXiv:1205.4838
L. Leskovec, S. Prelovsek, Scattering phase shifts for two particles of different mass and non-zero total momentum in lattice QCD. Phys. Rev. D 85, 114507 (2012). arXiv:1202.2145
R.A. Briceno, Z. Davoudi, Moving multichannel systems in a finite volume with application to proton–proton fusion. Phys. Rev. D 88, 094507 (2013). arXiv:1204.1110
M. Gockeler, R. Horsley, M. Lage, U.G. Meißner, P.E.L. Rakow, A. Rusetsky et al., Scattering phases for meson and baryon resonances on general moving-frame lattices. Phys. Rev. D 86, 094513 (2012). arXiv:1206.4141
P. Guo, J. Dudek, R. Edwards, A.P. Szczepaniak, Coupled-channel scattering on a torus. Phys. Rev. D 88, 014501 (2013). arXiv:1211.0929
N. Li, C. Liu, Generalized Lüscher formula in multichannel baryon-meson scattering. Phys. Rev. D 87, 014502 (2013). arXiv:1209.2201
R.A. Briceño, Z. Davoudi, T.C. Luu, M.J. Savage, Two-baryon systems with twisted boundary conditions. Phys. Rev. D 89, 074509 (2014). arXiv:1311.7686
F.X. Lee, A. Alexandru, Scattering phase-shift formulas for mesons and baryons in elongated boxes. Phys. Rev. D 96, 054508 (2017). arXiv:1706.00262
C. Morningstar, J. Bulava, B. Singha, R. Brett, J. Fallica, A. Hanlon et al., Estimating the two-particle \(K\)-matrix for multiple partial waves and decay channels from finite-volume energies. Nucl. Phys. B 924, 477 (2017). arXiv:1707.05817
Y. Li, J.-J. Wu, C.D. Abell, D.B. Leinweber, A.W. Thomas, Partial wave mixing in Hamiltonian effective field theory. Phys. Rev. D 101, 114501 (2020). arXiv:1910.04973
S.R. Sharpe, R. Gupta, G.W. Kilcup, Lattice calculation of \(I = 2\) pion scattering length. Nucl. Phys. B 383, 309 (1992)
Y. Kuramashi, M. Fukugita, H. Mino, M. Okawa, A. Ukawa, Lattice QCD calculation of full pion scattering lengths. Phys. Rev. Lett. 71, 2387 (1993)
R. Gupta, A. Patel, S.R. Sharpe, \(I = 2\) pion scattering amplitude with Wilson fermions. Phys. Rev. D 48, 388 (1993). arXiv:hep-lat/9301016
Shape CP-PACS Collaboration, \(I = 2\) pi pi scattering phase shift with two flavors of \(O(a)\) improved dynamical quarks. Phys. Rev. D 70, 074513 (2004). arXiv:hep-lat/0402025
Shape CP-PACS Collaboration, \(I=2\) pion scattering length from two-pion wave functions. Phys. Rev. D 71, 094504 (2005). arXiv:hep-lat/0503025
Shape NPLQCD Collaboration, \(I = 2\) pi–pi scattering from fully-dynamical mixed-action lattice QCD. Phys. Rev. D 73, 054503 (2006). arXiv:hep-lat/0506013
S.R. Beane, T.C. Luu, K. Orginos, A. Parreño, M.J. Savage, A. Torok et al., Precise determination of the \(I=2\) pi pi scattering length from mixed-action lattice QCD. Phys. Rev. D 77, 014505 (2008). arXiv:0706.3026
X. Feng, K. Jansen, D.B. Renner, The \(\pi ^+ \pi ^+\) scattering length from maximally twisted mass lattice QCD. Phys. Lett. B 684, 268 (2010). arXiv:0909.3255
T. Yagi, S. Hashimoto, O. Morimatsu, M. Ohtani, \(I=2\)\(\pi \)–\(\pi \) scattering length with dynamical overlap fermion. arXiv:1108.2970
Z. Fu, Lattice QCD study of the s-wave \(\pi \pi \) scattering lengths in the \(I=0\) and 2 channels. Phys. Rev. D 87, 074501 (2013). arXiv:1303.0517
Shape PACS-CS Collaboration, Scattering lengths for two pseudoscalar meson systems. Phys. Rev. D 89, 054502 (2014). arXiv:1311.7226
HAL QCD Collaboration, \(I=2\)\(\pi \pi \) scattering phase shift from the HAL QCD method with the LapH smearing. PTEP 2018, 043B04 (2018). arXiv: 1711.01883
J.J. Dudek, R.G. Edwards, C.E. Thomas, S and D-wave phase shifts in isospin-2 pi pi scattering from lattice QCD. Phys. Rev. D 86, 034031 (2012). arXiv:1203.6041
J. Bulava, B. Fahy, B. Hörz, K.J. Juge, C. Morningstar, C.H. Wong, \(I=1\) and \(I=2\)\(\pi -\pi \) scattering phase shifts from \(N_{\rm f} = 2+1\) lattice QCD. Nucl. Phys. B 910, 842 (2016). arXiv:1604.05593
Y. Akahoshi, S. Aoki, T. Aoyama, T. Doi, T. Miyamoto, K. Sasaki, \(I=2\)\(\pi \pi \) potential in the HAL QCD method with all-to-all propagators. arXiv:1904.09549
Shape ETM Collaboration, Hadron–hadron interactions from \(N_{f}\) = 2 + 1 + 1 lattice QCD: isospin-2 \(\pi \pi \) scattering length. JHEP 09, 109 (2015). arXiv:1506.00408
C. Helmes, C. Jost, B. Knippschild, B. Kostrzewa, L. Liu, F. Pittler et al., Meson–meson scattering lengths at maximum isospin from lattice QCD. in 9th International Workshop on Chiral Dynamics (CD18) Durham, NC, USA, September 17–21, 2018, 2019. arXiv:1904.00191
B. Hörz, A. Hanlon, Two- and three-pion finite-volume spectra at maximal isospin from lattice QCD. Phys. Rev. Lett. 123, 142002 (2019). arXiv:1905.04277
Shape CP-PACS Collaboration, Lattice QCD calculation of the rho meson decay width. Phys. Rev. D 76, 094506 (2007). arXiv:0708.3705
X. Feng, K. Jansen, D.B. Renner, Resonance parameters of the rho-meson from lattice QCD. Phys. Rev. D 83, 094505 (2011). arXiv:1011.5288
QCDSF Collaboration, Extracting the rho resonance from lattice QCD simulations at small quark masses. PoS LATTICE2008, 136 (2008). arXiv:0810.5337
C.B. Lang, D. Mohler, S. Prelovsek, M. Vidmar, Coupled channel analysis of the rho meson decay in lattice QCD. Phys. Rev. D 84, 054503 (2011). arXiv:1105.5636
Shape RQCD Collaboration, \(\rho \) and \(K^*\) resonances on the lattice at nearly physical quark masses and \(N_f=2\). Phys. Rev. D 93, 054509 (2016). arXiv:1512.08678
C. Pelissier, A. Alexandru, Resonance parameters of the rho-meson from asymmetrical lattices. Phys. Rev. D 87, 014503 (2013). arXiv:1211.0092
Shape CS Collaboration, \(\rho \) meson decay in 2+1 flavor lattice QCD. Phys. Rev. D 84, 094505 (2011). arXiv:1106.5365
Shape Hadron Spectrum Collaboration, Energy dependence of the \(\rho \) resonance in \(\pi \pi \) elastic scattering from lattice QCD. Phys. Rev. D 87, 034505 (2013). arXiv:1212.0830
X. Feng, S. Aoki, S. Hashimoto, T. Kaneko, Timelike pion form factor in lattice QCD. Phys. Rev. D 91, 054504 (2015). arXiv:1412.6319
Budapest-Marseille-Wuppertal Collaboration, Lattice study of \(\pi \pi \) scattering using \(N_f=2+1\) Wilson improved quarks with masses down to their physical values. PoS LATTICE2014, 079 (2015). arXiv:1410.8447
D.J. Wilson, R.A. Briceno, J.J. Dudek, R.G. Edwards, C.E. Thomas, Coupled \(\pi \pi, K\bar{K}\) scattering in \(P\)-wave and the \(\rho \) resonance from lattice QCD. Phys. Rev. D 92, 094502 (2015). arXiv:1507.02599
C. Alexandrou, L. Leskovec, S. Meinel, J. Negele, S. Paul, M. Petschlies et al., \(P\)-wave \(\pi \pi \) scattering and the \(\rho \) resonance from lattice QCD. Phys. Rev. D 96, 034525 (2017). arXiv:1704.05439
C. Andersen, J. Bulava, B. Hörz, C. Morningstar, The \(I=1\) pion-pion scattering amplitude and timelike pion form factor from \(N_{\rm f} = 2+1\) lattice QCD. Nucl. Phys. B 939, 145 (2019). arXiv:1808.05007
Z. Fu, L. Wang, Studying the \(\rho \) resonance parameters with staggered fermions. Phys. Rev. D 94, 034505 (2016). arXiv:1608.07478
M. Werner et al., Hadron–hadron interactions from \(N_f=2+1+1\) lattice QCD: the \(\rho \)-resonance. arXiv:1907.01237
R.A. Briceno, J.J. Dudek, R.G. Edwards, D.J. Wilson, Isoscalar \(\pi \pi \) scattering and the meson resonance from QCD. Phys. Rev. Lett. 118, 022002 (2017). arXiv:1607.05900
L. Liu et al., Isospin-0 \(\pi \pi \) s-wave scattering length from twisted mass lattice QCD. Phys. Rev. D 96, 054516 (2017). arXiv:1612.02061
R.A. Briceno, J.J. Dudek, R.G. Edwards, D.J. Wilson, Isoscalar \(\pi \pi, K\overline{K}, \eta \eta \) scattering and the \( , f_0, f_2\) mesons from QCD. Phys. Rev. D 97, 054513 (2018). arXiv:1708.06667
Z. Fu, X. Chen, \(I=0\)\(\pi \pi \)\(s\)-wave scattering length from lattice QCD. Phys. Rev. D 98, 014514 (2018). arXiv:1712.02219
D. Guo, A. Alexandru, R. Molina, M. Mai, M. Döring, Extraction of isoscalar \(\pi \pi \) phase-shifts from lattice QCD. Phys. Rev. D 98, 014507 (2018). arXiv:1803.02897
S.R. Beane, P.F. Bedaque, T.C. Luu, K. Orginos, E. Pallante, A. Parreño et al., \(\pi K\) scattering in full QCD with domain-wall valence quarks. Phys. Rev. D 74, 114503 (2006). arXiv:hep-lat/0607036
C.B. Lang, L. Leskovec, D. Mohler, S. Prelovsek, K pi scattering for isospin 1/2 and 3/2 in lattice QCD. Phys. Rev. D 86, 054508 (2012). arXiv:1207.3204
Z. Fu, The preliminary lattice QCD calculation of \(\kappa \) meson decay width. JHEP 01, 017 (2012). arXiv:1110.5975
S. Prelovsek, L. Leskovec, C.B. Lang, D. Mohler, K \(\pi \) scattering and the K* decay width from lattice QCD. Phys. Rev. D 88, 054508 (2013). arXiv:1307.0736
T. Janowski, P.A. Boyle, A. Jüttner, C. Sachrajda, K-pi scattering lengths at physical kinematics. PoS LATTICE2014, 080 (2014)
C. Helmes, C. Jost, B. Knippschild, B. Kostrzewa, L. Liu, C. Urbach et al., Hadron–hadron interactions from \(N_f=2+1+1\) lattice QCD: isospin-1 \(KK\) scattering length. Phys. Rev. D 96, 034510 (2017). arXiv:1703.04737
R. Brett, J. Bulava, J. Fallica, A. Hanlon, B. Hörz, C. Morningstar, Determination of \(s\)- and \(p\)-wave \(I=1/2\)\(K\pi \) scattering amplitudes in \(N_{\rm f}=2+1\) lattice QCD. Nucl. Phys. B 932, 29 (2018). arXiv:1802.03100
Shape ETM Collaboration, Hadron–hadron interactions from \(N_f=2+1+1\) lattice QCD: \(I=3/2\)\(\pi K\) scattering length. Phys. Rev. D 98, 114511 (2018). arXiv:1809.08886
D.J. Wilson, R.A. Briceño, J.J. Dudek, R.G. Edwards, C.E. Thomas, The quark-mass dependence of elastic \(\pi K\) scattering from QCD. Phys. Rev. Lett. 123, 042002 (2019). arXiv:1904.03188
A.J. Woss, C.E. Thomas, J.J. Dudek, R.G. Edwards, D.J. Wilson, \(b_1\) resonance in coupled \(\pi \omega \), \(\pi \phi \) scattering from lattice QCD. Phys. Rev. D 100, 054506 (2019). arXiv:1904.04136
A. Woss, C.E. Thomas, J.J. Dudek, R.G. Edwards, D.J. Wilson, Dynamically-coupled partial-waves in \(\rho \pi \) isospin-2 scattering from lattice QCD. arXiv:1802.05580
C.B. Lang, L. Leskovec, D. Mohler, S. Prelovsek, Axial resonances a\(_{1}\)(1260), b\(_{1}\)(1235) and their decays from the lattice. JHEP 04, 162 (2014). arXiv:1401.2088
L. Gayer, N. Lang, S.M. Ryan, D. Tims, C.E. Thomas, D.J. Wilson, Isospin-1/2 \(D\pi \) scattering and the lightest \(D_0^\ast \) resonance from lattice QCD. arXiv:2102.04973
G.K.C. Cheung, C.E. Thomas, D.J. Wilson, G. Moir, M. Peardon, S.M. Ryan, \(DK\)\(I=0,\)\(D\bar{K}\,I=0,1\) scattering and the \(D_{s0}^\ast (2317)\) from lattice QCD. arXiv:2008.06432
S. Prelovsek, S. Collins, D. Mohler, M. Padmanath, S. Piemonte, Charmonium-like resonances with \(J^{PC}=0^{++},2^{++}\) in coupled \(D\bar{D}\), \(D_s\bar{D}_s\) scattering on the lattice. arXiv:2011.02542
S. Piemonte, S. Collins, D. Mohler, M. Padmanath, S. Prelovsek, Charmonium resonances with \(J^{PC}=1^{-}\) and \(3^{-}\) from \(\bar{D}D\) scattering on the lattice. Phys. Rev. D 100, 074505 (2019). arXiv:1905.03506
G.S. Bali, S. Collins, A. Cox, A. Schäfer, Masses and decay constants of the \(D_{s0}^*(2317)\) and \(D_{s1}(2460)\) from \(N_f=2\) lattice QCD close to the physical point. Phys. Rev. D 96, 074501 (2017). arXiv:1706.01247
C.B. Lang, D. Mohler, S. Prelovsek, \(B_s\pi ^+\) scattering and search for X(5568) with lattice QCD. Phys. Rev. D 94, 074509 (2016). arXiv:1607.03185
M. Albaladejo, P. Fernandez-Soler, F.-K. Guo, J. Nieves, Two-pole structure of the \(D^\ast _0(2400)\). Phys. Lett. B 767, 465 (2017). arXiv:1610.06727
C.B. Lang, L. Leskovec, D. Mohler, S. Prelovsek, Vector and scalar charmonium resonances with lattice QCD. JHEP 09, 089 (2015). arXiv: 1503.05363
C.B. Lang, L. Leskovec, D. Mohler, S. Prelovsek, R.M. Woloshyn, Ds mesons with DK and D*K scattering near threshold. Phys. Rev. D 90, 034510 (2014). arXiv:1403.8103
A. Martínez Torres, E. Oset, S. Prelovsek, A. Ramos, Reanalysis of lattice QCD spectra leading to the \(D_{s0}^*(2317)\) and \(D_{s1}^*(2460)\). JHEP 05, 153 (2015). arXiv:1412.1706
D. Mohler, C.B. Lang, L. Leskovec, S. Prelovsek, R.M. Woloshyn, \(D_{s0}^*(2317)\) meson and \(D\)-meson–kaon scattering from lattice QCD. Phys. Rev. Lett. 111, 222001 (2013). arXiv:1308.3175
Z.-H. Guo, L. Liu, U.-G. Meißner, J.A. Oller, A. Rusetsky, Towards a precise determination of the scattering amplitudes of the charmed and light-flavor pseudoscalar mesons. Eur. Phys. J. C 79, 13 (2019). arXiv:1811.05585
C. Alexandrou, J.W. Negele, M. Petschlies, A.V. Pochinsky, S.N. Syritsyn, Study of decuplet baryon resonances from lattice QCD. Phys. Rev. D 93, 114515 (2016). arXiv:1507.02724
C. Alexandrou, J. Negele, M. Petschlies, A. Strelchenko, A. Tsapalis, Determination of \(\Delta \) resonance parameters from lattice QCD. Phys. Rev. D 88, 031501 (2013). arXiv:1305.6081
Shape BGR Collaboration, QCD with two light dynamical chirally improved quarks: baryons. Phys. Rev. D 87, 074504 (2013). arXiv:1301.4318
J.J. Dudek, R.G. Edwards, Hybrid baryons in QCD. Phys. Rev. D 85, 054016 (2012). arXiv:1201.2349
Shape Hadron Spectrum Collaboration, Flavor structure of the excited baryon spectra from lattice QCD. Phys. Rev. D 87, 054506 (2013). arXiv:1212.5236
R.A. Briceño, H.-W. Lin, D.R. Bolton, Charmed-baryon spectroscopy from lattice QCD with \(N_f\) = 2+1+1 flavors. Phys. Rev. D 86, 094504 (2012). arXiv:1207.3536
R.G. Edwards, J.J. Dudek, D.G. Richards, S.J. Wallace, Excited state baryon spectroscopy from lattice QCD. Phys. Rev. D 84, 074508 (2011). arXiv:1104.5152
J. Bulava, R. Edwards, E. Engelson, B. Joo, H.-W. Lin, C. Morningstar et al., Nucleon, \(\Delta \) and \(\Omega \) excited states in \(N_f=2+1\) lattice QCD. Phys. Rev. D 82, 014507 (2010). arXiv:1004.5072
S. Durr et al., Ab-initio determination of light hadron masses. Science 322, 1224 (2008). arXiv:0906.3599
T. Burch, C. Gattringer, L.Y. Glozman, C. Hagen, D. Hierl, C. Lang et al., Excited hadrons on the lattice: baryons. Phys. Rev. D 74, 014504 (2006). arXiv:hep-lat/0604019
Shape European Twisted Mass Collaboration, Light baryon masses with dynamical twisted mass fermions. Phys. Rev. D 78, 014509 (2008). arXiv:0803.3190
B.J. Menadue, W. Kamleh, D.B. Leinweber, M. Mahbub, Isolating the \(\Lambda (1405)\) in lattice QCD. Phys. Rev. Lett. 108, 112001 (2012). arXiv:1109.6716
W. Melnitchouk, S.O. Bilson-Thompson, F. Bonnet, J. Hedditch, F. Lee, D. Leinweber et al., Excited baryons in lattice QCD. Phys. Rev. D 67, 114506 (2003). arXiv:hep-lat/0202022
G. Silvi et al., P-wave nucleon–pion scattering amplitude in the \(\Delta (1232)\) channel from lattice QCD. arXiv:2101.00689
F.M. Stokes, W. Kamleh, D.B. Leinweber, Elastic form factors of nucleon excitations in lattice QCD. Phys. Rev. D 102, 014507 (2020). arXiv:1907.00177
C.W. Andersen, J. Bulava, B. Hörz, C. Morningstar, Elastic \(I=3/2 p\)-wave nucleon-pion scattering amplitude and the \(\Delta \)(1232) resonance from N\(_f\)=2+1 lattice QCD. Phys. Rev. D 97, 014506 (2018). arXiv:1710.01557
C.B. Lang, L. Leskovec, M. Padmanath, S. Prelovsek, Pion–nucleon scattering in the Roper channel from lattice QCD. Phys. Rev. D 95, 014510 (2017). arXiv:1610.01422
C. Lang, V. Verduci, Scattering in the \(\pi \)N negative parity channel in lattice QCD. Phys. Rev. D 87, 054502 (2013). arXiv:1212.5055
M. Döring, M. Mai, U.-G. Meißner, Finite volume effects and quark mass dependence of the \(N\)(1535) and \(N\)(1650). Phys. Lett. B 722, 185 (2013). arXiv:1302.4065
J.M.M. Hall, A.C.P. Hsu, D.B. Leinweber, A.W. Thomas, R.D. Young, Finite-volume matrix Hamiltonian model for a \(\Delta \rightarrow N\pi \) system. Phys. Rev. D 87, 094510 (2013). arXiv:1303.4157
J.-J. Wu, H. Kamano, T.-S.H. Lee, D.B. Leinweber, A.W. Thomas, Nucleon resonance structure in the finite volume of lattice QCD. Phys. Rev. D 95, 114507 (2017). arXiv:1611.05970
Z.-W. Liu, W. Kamleh, D.B. Leinweber, F.M. Stokes, A.W. Thomas, J.-J. Wu, Hamiltonian effective field theory study of the \(\mathbf{N^*(1535)}\) resonance in lattice QCD. Phys. Rev. Lett. 116, 082004 (2016). arXiv:1512.00140
Z.-W. Liu, J.M.M. Hall, D.B. Leinweber, A.W. Thomas, J.-J. Wu, Structure of the \(\mathbf{\Lambda (1405)}\) from Hamiltonian effective field theory. Phys. Rev. D 95, 014506 (2017). arXiv:1607.05856
C. Liu, X. Feng, S. He, Two particle states in a box and the S-matrix in multi-channel scattering. Int. J. Mod. Phys. A 21, 847 (2006). arXiv:hep-lat/0508022
V. Bernard, M. Lage, U.G. Meißner, A. Rusetsky, Scalar mesons in a finite volume. JHEP 01, 019 (2011). arXiv: 1010.6018
M. Döring, U.-G. Meißner, E. Oset, A. Rusetsky, Unitarized chiral perturbation theory in a finite volume: scalar meson sector. Eur. Phys. J. A 47, 139 (2011). arXiv:1107.3988
M. Döring, J. Haidenbauer, U.-G. Meißner, A. Rusetsky, Dynamical coupled-channel approaches on a momentum lattice. Eur. Phys. J. A 47, 163 (2011). arXiv:1108.0676
M. Döring, U.-G. Meißner, Finite volume effects in pion-kaon scattering and reconstruction of the \(\kappa \)(800) resonance. JHEP 01, 009 (2012). arXiv:1111.0616
R.A. Briceño, Two-particle multichannel systems in a finite volume with arbitrary spin. Phys. Rev. D 89, 074507 (2014). arXiv:1401.3312
C.T. Johnson, J.J. Dudek, Excited \(J^{-}\) meson resonances at the SU(3) flavor point from lattice QCD. arXiv:2012.00518
G. Moir, M. Peardon, S.M. Ryan, C.E. Thomas, D.J. Wilson, Coupled-channel \(D\pi \), \(D\eta \) and \(D_{s}\bar{K}\) scattering from lattice QCD. JHEP 10, 011 (2016). arXiv:1607.07093
Shape Hadron Spectrum Collaboration, An \(a_0\) resonance in strongly coupled \(\pi \eta \), \(K\overline{K}\) scattering from lattice QCD. Phys. Rev. D 93, 094506 (2016). arXiv:1602.05122
D.J. Wilson, J.J. Dudek, R.G. Edwards, C.E. Thomas, Resonances in coupled \(\pi K, \eta K\) scattering from lattice QCD. Phys. Rev. D 91, 054008 (2015). arXiv:1411.2004
Shape Hadron Spectrum Collaboration, Resonances in coupled \(\pi K -\eta K\) scattering from quantum chromodynamics. Phys. Rev. Lett. 113, 182001 (2014). arXiv:1406.4158
A.J. Woss, J.J. Dudek, R.G. Edwards, C.E. Thomas, D.J. Wilson, Decays of an exotic \(1^{-+}\) hybrid meson resonance in QCD. arXiv:2009.10034
R.A. Briceno, J.J. Dudek, R.D. Young, Scattering processes and resonances from lattice QCD. Rev. Mod. Phys. 90, 025001 (2018). arXiv:1706.06223
Shape USQCD Collaboration, Hadrons and nuclei. Eur. Phys. J. A 55, 193 (2019). arXiv:1904.09512
C.B. Lang, The hadron spectrum from lattice QCD. Prog. Part. Nucl. Phys. 61, 35 (2008). arXiv:0711.3091
M. Döring, Resonances and multi-particle states. PoS LATTICE2013, 006 (2014)
R.A. Briceño, Z. Davoudi, T.C. Luu, Nuclear reactions from lattice QCD. J. Phys. G42, 023101 (2015). arXiv:1406.5673
T.D. Lee, K. Huang, C.N. Yang, Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties. Phys. Rev. 106, 1135 (1957)
K. Huang, C.N. Yang, Quantum-mechanical many-body problem with hard-sphere interaction. Phys. Rev. 105, 767 (1957)
T.T. Wu, Ground state of a Bose system of hard spheres. Phys. Rev. 115, 1390 (1959)
S. Tan, Three-boson problem at low energy and implications for dilute Bose–Einstein condensates. Phys. Rev. A 78, 013636 (2008). arXiv:0709.2530
S.R. Beane, W. Detmold, M.J. Savage, \(n\)-Boson energies at finite volume and three-boson interactions. Phys. Rev. D 76, 074507 (2007). arXiv:0707.1670
W. Detmold, M.J. Savage, The energy of \(n\) identical bosons in a finite volume at \(O(L^{-7})\). Phys. Rev. D 77, 057502 (2008). arXiv:0801.0763
S.R. Beane et al., Charged multi-hadron systems in lattice QCD + QED. arXiv:2003.12130
J.-Y. Pang, J.-J. Wu, H.W. Hammer, U.-G. Meißner, A. Rusetsky, Energy shift of the three-particle system in a finite volume. Phys. Rev. D 99, 074513 (2019). arXiv:1902.01111
M.T. Hansen, S.R. Sharpe, Threshold expansion of the three-particle quantization condition. Phys. Rev. D 93, 096006 (2016). arXiv:1602.00324
F. Romero-López, A. Rusetsky, N. Schlage, C. Urbach, Relativistic \(N\)-particle energy shift in finite volume. arXiv:2010.11715
B.S. DeWitt, Transition from discrete to continuous spectra. Phys. Rev. 103, 1565 (1956)
D. Agadjanov, M. Döring, M. Mai, U.-G. Meißner, A. Rusetsky, The optical potential on the lattice. JHEP 06, 043 (2016). arXiv: 1603.07205
M.T. Hansen, H.B. Meyer, D. Robaina, From deep inelastic scattering to heavy-flavor semileptonic decays: total rates into multihadron final states from lattice QCD. Phys. Rev. D 96, 094513 (2017). arXiv:1704.08993
P. Guo, B. Long, Visualizing resonances in finite volume. Phys. Rev. D 102, 074508 (2020). arXiv:2007.10895
R.A. Briceño, J.V. Guerrero, M.T. Hansen, A. Sturzu, The role of boundary conditions in quantum computations of scattering observables. arXiv:2007.01155
F. Müller, A. Rusetsky, On the three-particle analog of the Lellouch–Lüscher formula. arXiv:2012.13957
M.T. Hansen, F. Romero-López, S.R. Sharpe, Decay amplitudes to three hadrons from finite-volume matrix elements. arXiv:2101.10246
M. Fischer, B. Kostrzewa, L. Liu, F. Romero-López, M. Ueding, C. Urbach, Scattering of two and three physical pions at maximal isospin from lattice QCD. arXiv:2008.03035
ETM collaboration, The \(\rho \)-resonance with physical pion mass from \(N_f=2\) lattice QCD. arXiv:2006.13805
A. Alexandru, R. Brett, C. Culver, M. Döring, D. Guo, F.X. Lee et al., Finite-volume energy spectrum of the \(K^-K^-K^-\) system. Phys. Rev. D 102, 114523 (2020). arXiv:2009.12358
J. Gasser, H. Leutwyler, Chiral perturbation theory to one loop. Ann. Phys. 158, 142 (1984)
S. Weinberg, Phenomenological Lagrangians. Physica A 96, 327 (1979)
J. Gasser, H. Leutwyler, Chiral perturbation theory: expansions in the mass of the strange quark. Nucl. Phys. B 250, 465 (1985)
J. Gasser, M.E. Sainio, A. Svarc, Nucleons with chiral loops. Nucl. Phys. B 307, 779 (1988)
V. Bernard, N. Kaiser, J. Kambor, U.G. Meißner, Chiral structure of the nucleon. Nucl. Phys. B 388, 315 (1992)
H.-B. Tang, A new approach to chiral perturbation theory for matter fields. arXiv:hep-ph/9607436
T. Becher, H. Leutwyler, Baryon chiral perturbation theory in manifestly Lorentz invariant form. Eur. Phys. J. C 9, 643 (1999). arXiv: hep-ph/9901384
P.J. Ellis, H.-B. Tang, Pion nucleon scattering in a new approach to chiral perturbation theory. Phys. Rev. C 57, 3356 (1998). arXiv:hep-ph/9709354
V. Bernard, U.-G. Meißner, Chiral perturbation theory. Ann. Rev. Nucl. Part. Sci. 57, 33 (2007). arXiv:hep-ph/0611231
V. Bernard, Chiral perturbation theory and baryon properties. Prog. Part. Nucl. Phys. 60, 82 (2008). arXiv:0706.0312
S. Scherer, Introduction to chiral perturbation theory. Adv. Nucl. Phys. 27, 277 (2003). arXiv:hep-ph/0210398
U.G. Meißner, Recent developments in chiral perturbation theory. Rept. Prog. Phys. 56, 903 (1993). arXiv:hep-ph/9302247
B. Kubis, An introduction to chiral perturbation theory. in Workshop on Physics and Astrophysics of Hadrons and Hadronic Matter, vol. 3 (2007). arXiv:hep-ph/0703274
V. Bernard, N. Kaiser, U.-G. Meißner, Chiral dynamics in nucleons and nuclei. Int. J. Mod. Phys. E 4, 193 (1995). arXiv:hep-ph/9501384
V. Bernard, N. Kaiser, U.G. Meißner, Chiral corrections to the S wave pion–nucleon scattering lengths. Phys. Lett. B 309, 421 (1993). arXiv:hep-ph/9304275
Shape Flavour Lattice Averaging Group Collaboration, FLAG review 2019: Flavour Lattice Averaging Group (FLAG). Eur. Phys. J. C 80, 113 (2020). arXiv:1902.08191
J.R. Pelaez, From controversy to precision on the sigma meson: a review on the status of the non-ordinary \(f_0(500)\) resonance. Phys. Rept. 658, 1 (2016). arXiv:1510.00653
M. Mai, Review of the \({\mathbf{\Lambda }}\)(1405): a curious case of a strange-ness resonance. arXiv:2010.00056
C. Hanhart, J.R. Pelaez, G. Rios, Quark mass dependence of the rho and sigma from dispersion relations and chiral perturbation theory. Phys. Rev. Lett. 100, 152001 (2008). arXiv:0801.2871
J. Nebreda, J.R. Peláez, Strange and non-strange quark mass dependence of elastic light resonances from SU(3) unitarized chiral perturbation theory to one loop. Phys. Rev. D 81, 054035 (2010). arXiv:1001.5237
D.R. Bolton, R.A. Briceño, D.J. Wilson, Connecting physical resonant amplitudes and lattice QCD. Phys. Lett. B 757, 50 (2016). arXiv:1507.07928
M. Döring, B. Hu, M. Mai, Chiral extrapolation of the sigma resonance. Phys. Lett. B 782, 785 (2018). arXiv:1610.10070
M. Niehus, M. Hoferichter, B. Kubis, Quark mass dependence of \(\gamma ^{*}\pi \rightarrow \pi \pi \). in 9th International Workshop on Chiral Dynamics (CD18) Durham, NC, USA, September 17-21, 2018 (2019). arXiv:1902.10150
M. Dax, T. Isken, B. Kubis, Quark-mass dependence in \(\omega \rightarrow 3\pi \) decays. Eur. Phys. J. C 78, 859 (2018). arXiv:1808.08957
C. Culver, M. Mai, A. Alexandru, M. Döring, F. Lee, Pion scattering in the isospin \(I=2\) channel from elongated lattices. Phys. Rev. D 100, 034509 (2019). arXiv:1905.10202
NA48-2 Collaboration, Precise tests of low energy QCD from K(e4)decay properties. Eur. Phys. J. C 70, 635 (2010)
C.D. Froggatt, J.L. Petersen, Phase shift analysis of \(\pi ^+ \pi ^-\) scattering between 1.0-GeV and 1.8-GeV based on fixed momentum transfer analyticity. 2. Nucl. Phys. B 129, 89 (1977)
B. Hyams et al., \(\pi \pi \) phase shift analysis from 600-MeV to 1900-MeV. Nucl. Phys. B 64, 134 (1973)
S.D. Protopopescu, M. Alston-Garnjost, A. Barbaro-Galtieri, S.M. Flatte, J.H. Friedman, T.A. Lasinski et al., \(\pi \pi \) partial wave analysis from reactions \(\pi ^+p\rightarrow \pi ^+\pi ^-\Delta ^{++}\) and \(\pi ^+ p \rightarrow K^+ K^- \Delta ^{++}\) at 7.1-GeV/c. Phys. Rev. D 7, 1279 (1973)
G. Grayer et al., High statistics study of the reaction \(\pi ^-p\rightarrow \pi ^-\pi ^+n\): apparatus, method of analysis, and general features of results at 17-GeV/c. Nucl. Phys. B 75, 189 (1974)
L. Rosselet et al., Experimental study of 30,000 K(e4) decays. Phys. Rev. D 15, 574 (1977)
G. Janssen, B.C. Pearce, K. Holinde, J. Speth, On the structure of the scalar mesons \(f0\) (975) and \(a0\) (980). Phys. Rev. D 52, 2690 (1995). arXiv:nucl-th/9411021
P. Estabrooks, A.D. Martin, pi pi Phase shift analysis below the K anti-K threshold. Nucl. Phys. B 79, 301 (1974)
J.A. Oller, E. Oset, J.R. Pelaez, Meson meson interaction in a nonperturbative chiral approach. Phys. Rev. D 59, 074001 (1999). arXiv:hep-ph/9804209
B. Hu, R. Molina, M. Döring, A. Alexandru, Two-flavor simulations of the \(\rho (770)\) and the role of the \(K\bar{K}\) channel. Phys. Rev. Lett. 117, 122001 (2016). arXiv:1605.04823
B. Hu, R. Molina, M. Döring, M. Mai, A. Alexandru, Chiral extrapolations of the \(\varvec {\rho (770)}\) meson in \(\mathbf{N_f=2+1}\) lattice QCD simulations. Phys. Rev. D 96, 034520 (2017). arXiv:1704.06248
J.A. Oller, E. Oset, Chiral symmetry amplitudes in the S wave isoscalar and isovector channels and the , f\(_0\)(980), a\(_0\)(980) scalar mesons. Nucl. Phys. A 620, 438 (1997). arXiv:hep-ph/9702314
M. Albaladejo, J.A. Oller, Identification of a scalar glueball. Phys. Rev. Lett. 101, 252002 (2008). arXiv:0801.4929
Z.-H. Guo, J.A. Oller, Resonances from meson–meson scattering in U(3) CHPT. Phys. Rev. D 84, 034005 (2011). arXiv:1104.2849
Z.-H. Guo, J.A. Oller, J. Ruiz de Elvira, Chiral dynamics in U(3) unitary chiral perturbation theory. Phys. Lett. B 712, 407 (2012). arXiv:1203.4381
X.-K. Guo, Z.-H. Guo, J.A. Oller, J.J. Sanz-Cillero, Scrutinizing the \(\eta \)-\(\eta ^{\prime }\) mixing, masses and pseudoscalar decay constants in the framework of U(3) chiral effective field theory. JHEP 06, 175 (2015). arXiv:1503.02248
Z.-H. Guo, L. Liu, U.-G. Meißner, J.A. Oller, A. Rusetsky, Chiral study of the \(a_0(980)\) resonance and \(\pi \eta \) scattering phase shifts in light of a recent lattice simulation. Phys. Rev. D 95, 054004 (2017). arXiv:1609.08096
T.N. Truong, Chiral perturbation theory and final state theorem. Phys. Rev. Lett. 61, 2526 (1988)
A. Dobado, J.R. Pelaez, The inverse amplitude method in chiral perturbation theory. Phys. Rev. D 56, 3057 (1997). arXiv: hep-ph/9604416
A. Gómez Nicola, J.R. Peláez, G. Rios, The inverse amplitude method and adler zeros. Phys. Rev. D 77, 056006 (2008). arXiv:0712.2763
P.C. Bruns, M. Mai, Chiral symmetry constraints on resonant amplitudes. Phys. Lett. B 778, 43 (2018). arXiv:1707.08983
J.R. Pelaez, G. Rios, Nature of the \(f0\)(600) from its \(N(c)\) dependence at two loops in unitarized chiral perturbation theory. Phys. Rev. Lett. 97, 242002 (2006). arXiv:hep-ph/0610397
M. Niehus, M. Hoferichter, B. Kubis, J. Ruiz de Elvira, Two-loop analysis of the pion-mass dependence of the \(\rho \) meson. arXiv:2009.04479
J.R. Peláez, A. Rodas, J.R. de Elvira, Precision dispersive approaches versus unitarized chiral perturbation theory for the lightest scalar resonances \( /f_0(980)\) and \(\kappa /K_0^*(700)\). arXiv:2101.06506
D. Fernandez-Fraile, A. Gomez Nicola, E.T. Herruzo, Pion scattering poles and chiral symmetry restoration. Phys. Rev. D 76, 085020 (2007). arXiv:0707.1424
R. Molina, J. Ruiz de Elvira, Light- and strange-quark mass dependence of the \(\rho (770)\) meson revisited. arXiv:2005.13584
R. Brett, C. Culver, M. Mai, A. Alexandru, M. Döring, F.X. Lee, Three-body interactions from the finite-volume QCD spectrum. arXiv:2101.06144]
M. Lüscher, U. Wolff, How to calculate the elastic scattering matrix in two-dimensional quantum field theories by numerical simulation. Nucl. Phys. B 339, 222 (1990)
N. Miller et al., \(F_K / F_\pi \) from Möbius domain-wall fermions solved on gradient-flowed HISQ ensembles. Phys. Rev. D 102, 034507 (2020). arXiv:2005.04795
S.R. Beane, W. Detmold, T.C. Luu, K. Orginos, M.J. Savage, A. Torok, Multi-pion systems in lattice QCD and the three-pion interaction. Phys. Rev. Lett. 100, 082004 (2008). arXiv:0710.1827
W. Detmold, M.J. Savage, A. Torok, S.R. Beane, T.C. Luu, K. Orginos et al., Multi-pion states in lattice QCD and the charged-pion condensate. Phys. Rev. D 78, 014507 (2008). arXiv:0803.2728
W. Detmold, K. Orginos, M.J. Savage, A. Walker-Loud, Kaon condensation with lattice QCD. Phys. Rev. D 78, 054514 (2008). arXiv:0807.1856
T.D. Blanton, F. Romero-López, S.R. Sharpe, \(I=3\) three-pion scattering amplitude from lattice QCD. Phys. Rev. Lett. 124, 032001 (2020). arXiv:1909.02973
C. Culver, M. Mai, R. Brett, A. Alexandru, M. Döring, Three pion spectrum in the \(I=3\) channel from lattice QCD. Phys. Rev. D 101, 114507 (2020). arXiv:1911.09047
M.T. Hansen, R.A. Briceño, R.G. Edwards, C.E. Thomas, D.J. Wilson, The energy-dependent \(\pi ^+ \pi ^+ \pi ^+\) scattering amplitude from QCD. Phys. Rev. Lett. 126, 012001 (2021). arXiv:2009.04931
F. Romero-López, A. Rusetsky, C. Urbach, Two- and three-body interactions in \(\varphi ^4\) theory from lattice simulations. Eur. Phys. J. C 78, 846 (2018). arXiv:1806.02367
K. Polejaeva, A. Rusetsky, Three particles in a finite volume. Eur. Phys. J. A 48, 67 (2012). arXiv:1203.1241
M.T. Hansen, S.R. Sharpe, Relativistic, model-independent, three-particle quantization condition. Phys. Rev. D 90, 116003 (2014). arXiv:1408.5933
M.T. Hansen, S.R. Sharpe, Expressing the three-particle finite-volume spectrum in terms of the three-to-three scattering amplitude. Phys. Rev. D 92, 114509 (2015). arXiv:1504.04248
H.-W. Hammer, J.-Y. Pang, A. Rusetsky, Three-particle quantization condition in a finite volume: 1. The role of the three-particle force. JHEP 09, 109 (2017). arXiv:1706.07700
H.W. Hammer, J.Y. Pang, A. Rusetsky, Three particle quantization condition in a finite volume: 2. General formalism and the analysis of data. JHEP 10, 115 (2017). arXiv:1707.02176
M. Mai, M. Döring, Three-body unitarity in the finite volume. Eur. Phys. J. A 53, 240 (2017). arXiv:1709.08222
M. Mai, M. Döring, Finite-volume spectrum of \(\pi ^+\pi ^+\) and \(\pi ^+\pi ^+\pi ^+\) systems. Phys. Rev. Lett. 122, 062503 (2019). arXiv:1807.04746
M.T. Hansen, S.R. Sharpe, Lattice QCD and three-particle decays of resonances. Ann. Rev. Nucl. Part. Sci. 69, 65 (2019). arXiv:1901.00483
A. Rusetsky, Three particles on the lattice. PoS LATTICE2019, 281 (2019). arXiv:1911.01253
T.D. Blanton, S.R. Sharpe, Alternative derivation of the relativistic three-particle quantization condition. Phys. Rev. D 102, 054520 (2020). arXiv:2007.16188
T.D. Blanton, S.R. Sharpe, Equivalence of relativistic three-particle quantization conditions. Phys. Rev. D 102, 054515 (2020). arXiv:2007.16190
R.A. Briceño, Z. Davoudi, Three-particle scattering amplitudes from a finite volume formalism. Phys. Rev. D 87, 094507 (2013). arXiv:1212.3398
L. Roca, E. Oset, Scattering of unstable particles in a finite volume: the case of \(\pi \rho \) scattering and the \(a_1(1260)\) resonance. Phys. Rev. D 85, 054507 (2012). arXiv:1201.0438
S. Bour, H.-W. Hammer, D. Lee, U.-G. Meißner, Benchmark calculations for elastic fermion-dimer scattering. Phys. Rev. C 86, 034003 (2012). arXiv:1206.1765
U.-G. Meißner, G. Ríos, A. Rusetsky, Spectrum of three-body bound states in a finite volume. Phys. Rev. Lett. 114, 091602 (2015). arXiv:1412.4969
M. Jansen, H.W. Hammer, Y. Jia, Finite volume corrections to the binding energy of the \(X(3872)\). Phys. Rev. D 92, 114031 (2015). arXiv:1505.04099
M.T. Hansen, S.R. Sharpe, Perturbative results for two and three particle threshold energies in finite volume. Phys. Rev. D 93, 014506 (2016). arXiv:1509.07929
P. Guo, One spatial dimensional finite volume three-body interaction for a short-range potential. Phys. Rev. D 95, 054508 (2017). arXiv:1607.03184
S. König, D. Lee, Volume dependence of \(N\)-body bound states. Phys. Lett. B 779, 9 (2018). arXiv:1701.00279
R.A. Briceño, M.T. Hansen, S.R. Sharpe, Relating the finite-volume spectrum and the two-and-three-particle \(S\) matrix for relativistic systems of identical scalar particles. Phys. Rev. D 95, 074510 (2017). arXiv:1701.07465
S.R. Sharpe, Testing the threshold expansion for three-particle energies at fourth order in \(\phi ^4\) theory. arXiv:1707.04279
P. Guo, V. Gasparian, Numerical approach for finite volume three-body interaction. Phys. Rev. D 97, 014504 (2018). arXiv:1709.08255
P. Guo, V. Gasparian, An solvable three-body model in finite volume. Phys. Lett. B 774, 441 (2017). arXiv:1701.00438
Y. Meng, C. Liu, U.-G. Meißner, A. Rusetsky, Three-particle bound states in a finite volume: unequal masses and higher partial waves. Phys. Rev. D 98, 014508 (2018). arXiv:1712.08464
P. Guo, M. Döring, A.P. Szczepaniak, Variational approach to \(N\)-body interactions in finite volume. Phys. Rev. D 98, 094502 (2018). arXiv:1810.01261
P. Guo, T. Morris, Multiple-particle interaction in (1+1)-dimensional lattice model. Phys. Rev. D 99, 014501 (2019). arXiv:1808.07397
P. Klos, S. König, H.W. Hammer, J.E. Lynn, A. Schwenk, Signatures of few-body resonances in finite volume. Phys. Rev. C 98, 034004 (2018). arXiv:1805.02029
R.A. Briceño, M.T. Hansen, S.R. Sharpe, Numerical study of the relativistic three-body quantization condition in the isotropic approximation. Phys. Rev. D 98, 014506 (2018). arXiv:1803.04169
R.A. Briceño, M.T. Hansen, S.R. Sharpe, Three-particle systems with resonant subprocesses in a finite volume. Phys. Rev. D 99, 014516 (2019). arXiv:1810.01429
M. Döring, H.W. Hammer, M. Mai, J.Y. Pang, A. Rusetsky, J. Wu, Three-body spectrum in a finite volume: the role of cubic symmetry. Phys. Rev. D 97, 114508 (2018). arXiv:1802.03362
A. Jackura, S. Dawid, C. Fernández-Ramírez, V. Mathieu, M. Mikhasenko, A. Pilloni et al., Equivalence of three-particle scattering formalisms. Phys. Rev. D 100, 034508 (2019). arXiv:1905.12007
M. Mai, M. Döring, C. Culver, A. Alexandru, Three-body unitarity versus finite-volume \(\pi ^+\pi ^+\pi ^+\) spectrum from lattice QCD. Phys. Rev. D 101, 054510 (2020). arXiv:1909.05749
P. Guo, Propagation of particles on a torus. Phys. Lett. B 804, 135370 (2020). arXiv:1908.08081
T.D. Blanton, F. Romero-López, S.R. Sharpe, Implementing the three-particle quantization condition including higher partial waves. JHEP 03, 106 (2019). arXiv: 1901.07095
R.A. Briceño, M.T. Hansen, S.R. Sharpe, A.P. Szczepaniak, Unitarity of the infinite-volume three-particle scattering amplitude arising from a finite-volume formalism. Phys. Rev. D 100, 054508 (2019). arXiv:1905.11188
F. Romero-López, S.R. Sharpe, T.D. Blanton, R.A. Briceño, M.T. Hansen, Numerical exploration of three relativistic particles in a finite volume including two-particle resonances and bound states. JHEP 10, 007 (2019). arXiv:1908.02411
P. Guo, M. Döring, Lattice model of heavy-light three-body system. Phys. Rev. D 101, 034501 (2020). arXiv:1910.08624
S. Zhu, S. Tan, \(d\)-dimensional Lüscher’s formula and the near-threshold three-body states in a finite volume. arXiv:1905.05117
J.-Y. Pang, J.-J. Wu, L.-S. Geng, \(DDK\) system in finite volume. Phys. Rev. D 102, 114515 (2020). arXiv:2008.13014
M.T. Hansen, F. Romero-López, S.R. Sharpe, Generalizing the relativistic quantization condition to include all three-pion isospin channels. JHEP 20, 047 (2020). arXiv:2003.10974
P. Guo, Modeling few-body resonances in finite volume. Phys. Rev. D 102, 054514 (2020). arXiv:2007.12790
P. Guo, Threshold expansion formula of \(N\) bosons in a finite volume from a variational approach. Phys. Rev. D 101, 054512 (2020). arXiv:2002.04111
S. König, Few-body bound states and resonances in finite volume. Few Body Syst. 61, 20 (2020). arXiv:2005.01478
T.D. Blanton, S.R. Sharpe, Relativistic three-particle quantization condition for nondegenerate scalars. arXiv:2011.05520
F. Müller, A. Rusetsky, T. Yu, Finite-volume energy shift of the three-pion ground state. arXiv:2011.14178
S. Kreuzer, H.W. Hammer, The triton in a finite volume. Phys. Lett. B 694, 424 (2011). arXiv:1008.4499
S. Kreuzer, H.W. Hammer, On the modification of the Efimov spectrum in a finite cubic box. Eur. Phys. J. A 43, 229 (2010). arXiv:0910.2191
S. Kreuzer, H.W. Hammer, Efimov physics in a finite volume. Phys. Lett. B 673, 260 (2009). arXiv:0811.0159
S. Kreuzer, H.W. Grießhammer, Three particles in a finite volume: the breakdown of spherical symmetry. Eur. Phys. J. A 48, 93 (2012). arXiv:1205.0277
G. Colangelo, J. Gasser, B. Kubis, A. Rusetsky, Cusps in \(K\rightarrow 3\pi \) decays. Phys. Lett. B 638, 187 (2006). arXiv:hep-ph/0604084
J. Gasser, B. Kubis, A. Rusetsky, Cusps in \(K\rightarrow 3\pi \) decays: a theoretical framework. Nucl. Phys. B 850, 96 (2011). arXiv:1103.4273
R. Aaron, R.D. Amado, J.E. Young, Relativistic three-body theory with applications to pi-minus n scattering. Phys. Rev. 174, 2022 (1968)
M. Mai, B. Hu, M. Döring, A. Pilloni, A. Szczepaniak, Three-body unitarity with isobars revisited. Eur. Phys. J. A 53, 177 (2017). arXiv:1706.06118
Shape JPAC Collaboration, Phenomenology of Relativistic \(\mathbf{3} \rightarrow \mathbf{3}\) reaction amplitudes within the isobar approximation. Eur. Phys. J. C 79, 56 (2019). arXiv:1809.10523
A.W. Jackura, R.A. Briceño, S.M. Dawid, M.H.E. Islam, C. McCarty, Solving relativistic three-body integral equations in the presence of bound states. arXiv:2010.09820
S.M. Dawid, A.P. Szczepaniak, Bound states in the B-matrix formalism for the three-body scattering. Phys. Rev. D 103, 014009 (2021). arXiv:2010.08084
D. Sadasivan, M. Mai, H. Akdag, M. Döring, Dalitz plots and lineshape of \(a_1(1260)\) from a relativistic three-body unitary approach. Phys. Rev. D 101, 094018 (2020). arXiv:2002.12431
M.T. Hansen, S.R. Sharpe, Applying the relativistic quantization condition to a three-particle bound state in a periodic box. Phys. Rev. D 95, 034501 (2017). arXiv:1609.04317
P. Guo, B. Long, Multi- \(\pi ^+\) systems in a finite volume. Phys. Rev. D 101, 094510 (2020). arXiv:2002.09266
H.-W. Hammer, A. Nogga, A. Schwenk, Three-body forces: from cold atoms to nuclei. Rev. Mod. Phys. 85, 197 (2013). arXiv:1210.4273
Shape NPLQCD Collaboration, The \(K^+ K^+\) scattering length from lattice QCD. Phys. Rev. D 77, 094507 (2008). arXiv:0709.1169
S. Beane, P. Bedaque, K. Orginos, M. Savage, \(f_K/f_\pi \) in Full QCD with domain wall valence quarks. Phys. Rev. D 75, 094501 (2007). arXiv:hep-lat/0606023
G. Rendon, L. Leskovec, S. Meinel, J. Negele, S. Paul, M. Petschlies et al., \(I=1/2\)\(S\)-wave and \(P\)-wave \(K\pi \) scattering and the \(\kappa \) and \(K^*\) resonances from lattice QCD. arXiv:2006.14035
J.R. Pelaez, G. Rios, Chiral extrapolation of light resonances from one and two-loop unitarized chiral perturbation theory versus lattice results. Phys. Rev. D 82, 114002 (2010). arXiv:1010.6008
J. Nebreda, J. Pelaez, G. Rios, Chiral extrapolation of pion–pion scattering phase shifts within standard and unitarized chiral perturbation theory. Phys. Rev. D 83, 094011 (2011). arXiv:1101.2171
A. GomezNicola, J.R. Pelaez, Meson meson scattering within one loop chiral perturbation theory and its unitarization. Phys. Rev. D 65, 054009 (2002). arXiv:hep-ph/0109056
L. Lellouch, M. Lüscher, Weak transition matrix elements from finite volume correlation functions. Commun. Math. Phys. 219, 31 (2001). arXiv:hep-lat/0003023
RBC, UKQCD Collaboration, Direct CP violation and the \(\Delta I=1/2\) rule in \(K\rightarrow \pi \pi \) decay from the standard model. Phys. Rev. D 102, 054509 (2020). arXiv:2004.09440
N.H. Christ, C. Kim, T. Yamazaki, Finite volume corrections to the two-particle decay of states with non-zero momentum. Phys. Rev. D 72, 114506 (2005). arXiv:hep-lat/0507009
M.T. Hansen, S.R. Sharpe, Multiple-channel generalization of Lellouch–Lüscher formula. Phys. Rev. D 86, 016007 (2012). arXiv:1204.0826
V. Bernard, D. Hoja, U.G. Meißner, A. Rusetsky, Matrix elements of unstable states. JHEP 09, 023 (2012). arXiv:1205.4642
A. Agadjanov, V. Bernard, U.G. Meißner, A. Rusetsky, A framework for the calculation of the \(\Delta N\gamma ^*\) transition form factors on the lattice. Nucl. Phys. B 886, 1199 (2014). arXiv:1405.3476
A. Agadjanov, V. Bernard, U.-G. Meißner, A. Rusetsky, The \(B\rightarrow K^*\) form factors on the lattice. Nucl. Phys. B 910, 387 (2016). arXiv:1605.03386
R.A. Briceño, M.T. Hansen, Multichannel 0 \(\rightarrow \) 2 and 1 \(\rightarrow \) 2 transition amplitudes for arbitrary spin particles in a finite volume. Phys. Rev. D 92, 074509 (2015). arXiv:1502.04314
R.A. Briceño, M.T. Hansen, A. Walker-Loud, Multichannel 1 \(\rightarrow \) 2 transition amplitudes in a finite volume. Phys. Rev. D 91, 034501 (2015). arXiv:1406.5965
H.B. Meyer, Lattice QCD and the timelike pion form factor. Phys. Rev. Lett. 107, 072002 (2011). arXiv:1105.1892
M. Padmanath, C.B. Lang, S. Prelovsek, X(3872) and Y(4140) using diquark–antidiquark operators with lattice QCD. Phys. Rev. D 92, 034501 (2015). arXiv:1503.03257
V. Baru, E. Epelbaum, A.A. Filin, C. Hanhart, U.G. Meißner, A.V. Nefediev, Quark mass dependence of the \(X(3872)\) binding energy. Phys. Lett. B 726, 537 (2013). arXiv:1306.4108
E.J. Garzon, R. Molina, A. Hosaka, E. Oset, Strategies for an accurate determination of the \(X(3872)\) energy from QCD lattice simulations. Phys. Rev. D 89, 014504 (2014). arXiv:1310.0972
ALICE Collaboration, One-dimensional pion, kaon, and proton femtoscopy in Pb–Pb collisions at \(\sqrt{s_{\rm {NN}}}\) =2.76 TeV. Phys. Rev. C 92, 054908 (2015). arXiv:1506.07884
D. Kaplan, A. Nelson, Strange goings on in dense nucleonic matter. Phys. Lett. B 175, 57 (1986)
G.-Q. Li, C. Lee, G. Brown, Kaons in dense matter, kaon production in heavy ion collisions, and kaon condensation in neutron stars. Nucl. Phys. A 625, 372 (1997). arXiv:nucl-th/9706057
S. Pal, D. Bandyopadhyay, W. Greiner, Anti-K**0 condensation in neutron stars. Nucl. Phys. A 674, 553 (2000). arXiv:astro-ph/0001039
C. Lee, Kaon condensation in dense stellar matter. Phys. Rept. 275, 255 (1996)
D. Lonardoni, A. Lovato, S. Gandolfi, F. Pederiva, Hyperon puzzle: hints from quantum Monte Carlo calculations. Phys. Rev. Lett. 114, 092301 (2015). arXiv:1407.4448
T. Hell, W. Weise, Dense baryonic matter: constraints from recent neutron star observations. Phys. Rev. C 90, 045801 (2014). arXiv:1402.4098
A. Gal, E. Hungerford, D. Millener, Strangeness in nuclear physics. Rev. Mod. Phys. 88, 035004 (2016). arXiv:1605.00557
M.J. Savage, Nuclear physics. PoS LATTICE2016, 021 (2016). arXiv:1611.02078
C. Drischler, W. Haxton, K. McElvain, E. Mereghetti, A. Nicholson, P. Vranas et al., Towards grounding nuclear physics in QCD. 10 (2019). arXiv:1910.07961
B. Hörz et al., Two-nucleon S-wave interactions at the \(SU(3)\) flavor-symmetric point with \(m_{ud}\simeq m_s^{\rm phys}\): a first lattice QCD calculation with the stochastic Laplacian heaviside method. Phys. Rev. C 103, 014003 (2021). arXiv:2009.11825
K. Orginos, A. Parreño, M.J. Savage, S.R. Beane, E. Chang, W. Detmold, Two nucleon systems at \(m_\pi \sim 450~{\rm MeV}\) from lattice QCD. Phys. Rev. D 92, 114512 (2015). arXiv:1508.07583
A. Gade, B.M. Sherrill, NSCL and FRIB at Michigan State University: nuclear science at the limits of stability. Phys. Scripta 91, 053003 (2016)
Acknowledgements
We thank R. Brett for a careful reading of the manuscript. The work of MD and MM is supported by the National Science Foundation under Grant no. PHY-2012289 and by the US Department of Energy under Award No. DE-SC0016582. MD is also supported by the US Department of Energy, Office of Science, Office of Nuclear Physics under contract DE-AC05-06OR23177. The work of AR is funded in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project-ID 196253076—TRR 110, Volkswagenstiftung (Grant no. 93562) and the Chinese Academy of Sciences (CAS) President’s International Fellowship Initiative (PIFI) (Grant no. 2021VMB0007).
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Mai, M., Döring, M. & Rusetsky, A. Multi-particle systems on the lattice and chiral extrapolations: a brief review. Eur. Phys. J. Spec. Top. 230, 1623–1643 (2021). https://doi.org/10.1140/epjs/s11734-021-00146-5
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DOI: https://doi.org/10.1140/epjs/s11734-021-00146-5