Abstract
We derive the three-body quantization condition in a finite volume using an effective field theory in the particle-dimer picture. Moreover, we consider the extraction of physical observables from the lattice spectrum using the quantization condition. To illustrate the general framework, we calculate the volume-dependent three-particle spectrum in a simple model both below and above the three-particle threshold. The relation to existing approaches is discussed in detail.
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M. Lüscher, Two particle states on a torus and their relation to the scattering matrix, Nucl. Phys. B 354 (1991) 531 [INSPIRE].
M. Lage, U.-G. Meissner and A. Rusetsky, A method to measure the antikaon-nucleon scattering length in lattice QCD, Phys. Lett. B 681 (2009) 439 [arXiv:0905.0069] [INSPIRE].
V. Bernard, M. Lage, U.G. Meissner and A. Rusetsky, Scalar mesons in a finite volume, JHEP 01 (2011) 019 [arXiv:1010.6018] [INSPIRE].
S. He, X. Feng and C. Liu, Two particle states and the S-matrix elements in multi-channel scattering, JHEP 07 (2005) 011 [hep-lat/0504019] [INSPIRE].
C. Liu, X. Feng and S. He, Two particle states in a box and the S-matrix in multi-channel scattering, Int. J. Mod. Phys. A 21 (2006) 847 [hep-lat/0508022] [INSPIRE].
M.T. Hansen and S.R. Sharpe, Multiple-channel generalization of Lellouch-Lüscher formula, Phys. Rev. D 86 (2012) 016007 [arXiv:1204.0826] [INSPIRE].
R.A. Briceño and Z. Davoudi, Moving multichannel systems in a finite volume with application to proton-proton fusion, Phys. Rev. D 88 (2013) 094507 [arXiv:1204.1110] [INSPIRE].
N. Li and C. Liu, Generalized Lüscher formula in multichannel baryon-meson scattering, Phys. Rev. D 87 (2013) 014502 [arXiv:1209.2201] [INSPIRE].
P. Guo, J. Dudek, R. Edwards and A.P. Szczepaniak, Coupled-channel scattering on a torus, Phys. Rev. D 88 (2013) 014501 [arXiv:1211.0929] [INSPIRE].
Hadron Spectrum collaboration, J.J. Dudek, R.G. Edwards, C.E. Thomas and D.J. Wilson, Resonances in coupled πK − ηK scattering from quantum chromodynamics, Phys. Rev. Lett. 113 (2014) 182001 [arXiv:1406.4158] [INSPIRE].
D.J. Wilson, J.J. Dudek, R.G. Edwards and C.E. Thomas, Resonances in coupled πK, ηK scattering from lattice QCD, Phys. Rev. D 91 (2015) 054008 [arXiv:1411.2004] [INSPIRE].
Hadron Spectrum collaboration, J.J. Dudek, R.G. Edwards and D.J. Wilson, An a 0 resonance in strongly coupled πη, \( K\overline{K} \) scattering from lattice QCD, Phys. Rev. D 93 (2016) 094506 [arXiv:1602.05122] [INSPIRE].
M. Döring, U.-G. Meissner, E. Oset and A. Rusetsky, Unitarized Chiral Perturbation Theory in a finite volume: Scalar meson sector, Eur. Phys. J. A 47 (2011) 139 [arXiv:1107.3988] [INSPIRE].
M. Döring, U.G. Meissner, E. Oset and A. Rusetsky, Scalar mesons moving in a finite volume and the role of partial wave mixing, Eur. Phys. J. A 48 (2012) 114 [arXiv:1205.4838] [INSPIRE].
A. Martinez Torres, L.R. Dai, C. Koren, D. Jido and E. Oset, The KD, ηD s interaction in finite volume and the nature of the D s*0(2317) resonance, Phys. Rev. D 85 (2012) 014027 [arXiv:1109.0396] [INSPIRE].
M. Döring and U.G. Meissner, Finite volume effects in pion-kaon scattering and reconstruction of the κ(800) resonance, JHEP 01 (2012) 009 [arXiv:1111.0616] [INSPIRE].
M. Döring, M. Mai and U.-G. Meissner, Finite volume effects and quark mass dependence of the N(1535) and N(1650), Phys. Lett. B 722 (2013) 185 [arXiv:1302.4065] [INSPIRE].
D.R. Bolton, R.A. Briceño and D.J. Wilson, Connecting physical resonant amplitudes and lattice QCD, Phys. Lett. B 757 (2016) 50 [arXiv:1507.07928] [INSPIRE].
D. Agadjanov, M. Döring, M. Mai, U.-G. Meißner and A. Rusetsky, The Optical Potential on the Lattice, JHEP 06 (2016) 043 [arXiv:1603.07205] [INSPIRE].
M.T. Hansen, H.B. Meyer and D. Robaina, From deep inelastic scattering to heavy-flavor semi-leptonic decays: Total rates into multi-hadron final states from lattice QCD, arXiv:1704.08993 [INSPIRE].
HAL QCD collaboration, N. Ishii et al., Hadron-hadron interactions from imaginary-time Nambu-Bethe-Salpeter wave function on the lattice, Phys. Lett. B 712 (2012) 437 [arXiv:1203.3642] [INSPIRE].
S. Aoki, Nucleon-nucleon interactions via Lattice QCD: Methodology, Eur. Phys. J. A 49 (2013) 81 [arXiv:1309.4150] [INSPIRE].
S. Aoki, Hadron interactions from lattice QCD, PoS(LATTICE 2007)002 [arXiv:0711.2151] [INSPIRE].
S. Aoki, B. Charron, T. Doi, T. Hatsuda, T. Inoue and N. Ishii, Construction of energy-independent potentials above inelastic thresholds in quantum field theories, Phys. Rev. D 87 (2013) 034512 [arXiv:1212.4896] [INSPIRE].
HAL QCD collaboration, K. Sasaki et al., Coupled-channel approach to strangeness S = −2 baryon-bayron interactions in lattice QCD, PTEP 2015 (2015) 113B01 [arXiv:1504.01717] [INSPIRE].
K. Polejaeva and A. Rusetsky, Three particles in a finite volume, Eur. Phys. J. A 48 (2012) 67 [arXiv:1203.1241] [INSPIRE].
U.-G. Meißner, G. Ríos and A. Rusetsky, Spectrum of three-body bound states in a finite volume, Phys. Rev. Lett. 114 (2015) 091602 [Erratum ibid. 117 (2016) 069902] [arXiv:1412.4969] [INSPIRE].
P. Guo, One spatial dimensional finite volume three-body interaction for a short-range potential, Phys. Rev. D 95 (2017) 054508 [arXiv:1607.03184] [INSPIRE].
P. Guo and V. Gasparian, An solvable three-body model in finite volume, arXiv:1701.00438 [INSPIRE].
R.A. Briceño and Z. Davoudi, Three-particle scattering amplitudes from a finite volume formalism, Phys. Rev. D 87 (2013) 094507 [arXiv:1212.3398] [INSPIRE].
M.T. Hansen and S.R. Sharpe, Relativistic, model-independent, three-particle quantization condition, Phys. Rev. D 90 (2014) 116003 [arXiv:1408.5933] [INSPIRE].
M.T. Hansen and S.R. Sharpe, Expressing the three-particle finite-volume spectrum in terms of the three-to-three scattering amplitude, Phys. Rev. D 92 (2015) 114509 [arXiv:1504.04248] [INSPIRE].
M.T. Hansen and S.R. Sharpe, Perturbative results for two and three particle threshold energies in finite volume, Phys. Rev. D 93 (2016) 014506 [arXiv:1509.07929] [INSPIRE].
M.T. Hansen and S.R. Sharpe, Threshold expansion of the three-particle quantization condition, Phys. Rev. D 93 (2016) 096006 [arXiv:1602.00324] [INSPIRE].
M.T. Hansen and S.R. Sharpe, Applying the relativistic quantization condition to a three-particle bound state in a periodic box, Phys. Rev. D 95 (2017) 034501 [arXiv:1609.04317] [INSPIRE].
R.A. Briceño, M.T. Hansen and 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 (2017) 074510 [arXiv:1701.07465] [INSPIRE].
S. Kreuzer and H.W. Hammer, The triton in a finite volume, Phys. Lett. B 694 (2011) 424 [arXiv:1008.4499] [INSPIRE].
S. Kreuzer and H.W. Hammer, On the modification of the Efimov spectrum in a finite cubic box, Eur. Phys. J. A 43 (2010) 229 [arXiv:0910.2191] [INSPIRE].
S. Kreuzer and H.W. Hammer, Efimov physics in a finite volume, Phys. Lett. B 673 (2009) 260 [arXiv:0811.0159] [INSPIRE].
S. Kreuzer and H.W. Grießhammer, Three particles in a finite volume: The breakdown of spherical symmetry, Eur. Phys. J. A 48 (2012) 93 [arXiv:1205.0277] [INSPIRE].
N. Mathur et al., Roper resonance and S(11)(1535) from lattice QCD, Phys. Lett. B 605 (2005) 137 [hep-ph/0306199] [INSPIRE].
D. Guadagnoli, M. Papinutto and S. Simula, Extracting excited states from lattice QCD: The Roper resonance, Phys. Lett. B 604 (2004) 74 [hep-lat/0409011] [INSPIRE].
D.B. Leinweber, W. Melnitchouk, D.G. Richards, A.G. Williams and J.M. Zanotti, Baryon spectroscopy in lattice QCD, Lect. Notes Phys. 663 (2005) 71 [nucl-th/0406032] [INSPIRE].
K. Sasaki, S. Sasaki and T. Hatsuda, Spectral analysis of excited nucleons in lattice QCD with maximum entropy method, Phys. Lett. B 623 (2005) 208 [hep-lat/0504020] [INSPIRE].
K. Sasaki and S. Sasaki, Excited baryon spectroscopy from lattice QCD: Finite size effect and hyperfine mass splitting, Phys. Rev. D 72 (2005) 034502 [hep-lat/0503026] [INSPIRE].
T. Burch et al., Excited hadrons on the lattice: Baryons, Phys. Rev. D 74 (2006) 014504 [hep-lat/0604019] [INSPIRE].
K.-F. Liu, Y. Chen, M. Gong, R. Sufian, M. Sun and A. Li, The Roper Puzzle, PoS(LATTICE 2013)507 [arXiv:1403.6847] [INSPIRE].
M.S. Mahbub, A.O. Cais, W. Kamleh, D.B. Leinweber and A.G. Williams, Positive-parity Excited-states of the Nucleon in Quenched Lattice QCD, Phys. Rev. D 82 (2010) 094504 [arXiv:1004.5455] [INSPIRE].
B.G. Lasscock et al., Even parity excitations of the nucleon in lattice QCD, Phys. Rev. D 76 (2007) 054510 [arXiv:0705.0861] [INSPIRE].
C.B. Lang, L. Leskovec, M. Padmanath and S. Prelovsek, Pion-nucleon scattering in the Roper channel from lattice QCD, Phys. Rev. D 95 (2017) 014510 [arXiv:1610.01422] [INSPIRE].
S.R. Beane, W. Detmold, K. Orginos and M.J. Savage, Nuclear Physics from Lattice QCD, Prog. Part. Nucl. Phys. 66 (2011) 1 [arXiv:1004.2935] [INSPIRE].
NPLQCD collaboration, S.R. Beane et al., Light Nuclei and Hypernuclei from Quantum Chromodynamics in the Limit of SU(3) Flavor Symmetry, Phys. Rev. D 87 (2013) 034506 [arXiv:1206.5219] [INSPIRE].
NPLQCD collaboration, E. Chang et al., Magnetic structure of light nuclei from lattice QCD, Phys. Rev. D 92 (2015) 114502 [arXiv:1506.05518] [INSPIRE].
E. Epelbaum, H. Krebs, D. Lee and U.-G. Meissner, Lattice effective field theory calculations for A = 3,4,6,12 nuclei, Phys. Rev. Lett. 104 (2010) 142501 [arXiv:0912.4195] [INSPIRE].
E. Epelbaum, H. Krebs, D. Lee and U.-G. Meissner, Ab initio calculation of the Hoyle state, Phys. Rev. Lett. 106 (2011) 192501 [arXiv:1101.2547] [INSPIRE].
A. Rokash, E. Epelbaum, H. Krebs, D. Lee and U.-G. Meißner, Finite volume effects in low-energy neutron-deuteron scattering, J. Phys. G 41 (2014) 015105 [arXiv:1308.3386] [INSPIRE].
S. Elhatisari et al., Ab initio alpha-alpha scattering, Nature 528 (2015) 111 [arXiv:1506.03513] [INSPIRE].
J.R. Taylor, Scattering Theory, Dover, (2006).
P.F. Bedaque and H.W. Griesshammer, Quartet S wave neutron deuteron scattering in effective field theory, Nucl. Phys. A 671 (2000) 357 [nucl-th/9907077] [INSPIRE].
P.F. Bedaque, H.W. Hammer and U. van Kolck, Renormalization of the three-body system with short range interactions, Phys. Rev. Lett. 82 (1999) 463 [nucl-th/9809025] [INSPIRE].
P.F. Bedaque, H.W. Hammer and U. van Kolck, The three boson system with short range interactions, Nucl. Phys. A 646 (1999) 444 [nucl-th/9811046] [INSPIRE].
H.W. Hammer and T. Mehen, Range corrections to doublet S wave neutron deuteron scattering, Phys. Lett. B 516 (2001) 353 [nucl-th/0105072] [INSPIRE].
C. Ji, D.R. Phillips and L. Platter, The three-boson system at next-to-leading order in an effective field theory for systems with a large scattering length, Annals Phys. 327 (2012) 1803 [arXiv:1106.3837] [INSPIRE].
C. Ji and D.R. Phillips, Effective Field Theory Analysis of Three-Boson Systems at Next-To-Next-To-Leading Order, Few Body Syst. 54 (2013) 2317 [arXiv:1212.1845] [INSPIRE].
V. Bernard, M. Lage, U.-G. Meissner and A. Rusetsky, Resonance properties from the finite-volume energy spectrum, JHEP 08 (2008) 024 [arXiv:0806.4495] [INSPIRE].
M. Gockeler et al., Scattering phases for meson and baryon resonances on general moving-frame lattices, Phys. Rev. D 86 (2012) 094513 [arXiv:1206.4141] [INSPIRE].
G. Colangelo, J. Gasser, B. Kubis and A. Rusetsky, Cusps in K → 3π decays, Phys. Lett. B 638 (2006) 187 [hep-ph/0604084] [INSPIRE].
J. Gasser, B. Kubis and A. Rusetsky, Cusps in K → 3π decays: a theoretical framework, Nucl. Phys. B 850 (2011) 96 [arXiv:1103.4273] [INSPIRE].
H.-W. Hammer, J.-Y. Pang and A. Rusetsky, Three-particle quantization condition in a finite volume: 1. The role of the three-particle force, JHEP 09 (2017) 109 [arXiv:1706.07700] [INSPIRE].
M. Mai, B. Hu, M. Döring, A. Pilloni and A. Szczepaniak, Three-body Unitarity with Isobars Revisited, Eur. Phys. J. A 53 (2017) 177 [arXiv:1706.06118] [INSPIRE].
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Hammer, HW., Pang, JY. & Rusetsky, A. Three-particle quantization condition in a finite volume: 2. General formalism and the analysis of data. J. High Energ. Phys. 2017, 115 (2017). https://doi.org/10.1007/JHEP10(2017)115
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DOI: https://doi.org/10.1007/JHEP10(2017)115