Three-particle scattering amplitudes from lattice QCD

Authors

  • Fernando Romero-Lopez MIT

DOI:

https://doi.org/10.31349/SuplRevMexFis.3.0308003

Keywords:

Lattice QCD, finite volume, three-particle scattering

Abstract

Lattice QCD already offers the possibility of extracting three-hadron scattering quantities from first principles. In the last few years, significant progress has been achieved in developing and applying the finite-volume three-body formalism. The formalism is now able to treat physically relevant systems of three mesons, including those with resonances, as well as three-body decays. In this talk, I will review the state of the art, and comment on recent applications to lattice QCD data for systems of three pions and kaons.

References

M. Luscher, Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. 2. Scattering States, Commun. Math. Phys. 105, (1986) 153, https://10.1007/BF01211097.

R. A. Briceno, J. J. Dudek and R. D. Young, Scattering processes and resonances from lattice QCD, Rev. Mod. Phys. 90 (2018) 025001, https://10.1103/RevModPhys.90.025001.

L. D. Roper, Evidence for a P-11 Pion-Nucleon Resonance at 556 MeV, Phys. Rev. Lett. 12 (1964) 340, https://10.1103/PhysRevLett.12.340.

P. A. Zyla et al., Review of Particle Physics, PTEP 2020 (2020) 083C01, https://10.1093/ptep/ptaa104.

M. Luscher, Signatures of unstable particles in finite volume, Nucl. Phys. B 364 (1991) 237, https://10.1016/0550-3213(91)90584-K.

K. Rummukainen and S. A. Gottlieb, Resonance scattering phase shifts on a nonrest frame lattice, Nucl. Phys. B 450 (1995) 397, https://10.1016/0550-3213(95)00313-H.

L. Lellouch and M. Luscher, Weak transition matrix elements from finite volume correlation functions, Commun. Math. Phys. 219 (2001) 31, https://10.1007/s002200100410.

C. h. Kim, C. T. Sachrajda and S. R. Sharpe, Finite-volume effects for two-hadron states in moving frames, Nucl. Phys. B 727 (2005) 218, https://10.1016/j.nuclphysb.2005.08.029.

S. He, X. Feng and C. Liu, Two particle states and the Smatrix elements in multi-channel scattering, JHEP 07 (2005) 011, https://10.1088/1126-6708/2005/07/011,

V. Bernard, M. Lage, U. G. Meissner and A. Rusetsky, Scalar mesons in a finite volume, JHEP 01 (2011) 019, https://10.1007/JHEP01(2011)019.

M. T. Hansen and S. R. Sharpe, Multiple-channel generalization of Lellouch-Luscher formula, Phys. Rev. D 86 (2012) 016007, https://10.1103/PhysRevD.86.016007.

R. A. Briceno and Z. Davoudi, Moving multichannel systems in a finite volume with application to proton-proton fusion, Phys. Rev. D 88 (2013) 094507, https://10.1103/PhysRevD.88.094507.

R. A. Briceno, Two-particle multichannel systems in a finite volume with arbitrary spin, Phys. Rev. D 89 (2014) 074507, https://10.1103/PhysRevD.89.074507.

F. Romero-Lopez, A. Rusetsky and C. Urbach, Vector particle scattering on the lattice, Phys. Rev. D 98 (2018) 014503, https://10.1103/PhysRevD.98.014503.

A. J. Woss et al., Efficient solution of the multichannel Luscher determinant condition through eigenvalue decomposition, Phys. Rev. D 101 (2020) 114505, https://10.1103/PhysRevD.101.114505.

R. A. Briceno, A. W. Jackura, F. G. Ortega-Gama and K. H. Sherman, On-shell representations of two-body transition amplitudes: Single external current, Phys. Rev. D 103 (2021) 114512, https://10.1103/PhysRevD.103.114512.

D. M. Grabowska and M. T. Hansen, Analytic expansions of multi-hadron finite-volume energies: I. Two-particle states, [arXiv:2110.06878 [hep-lat]].

M. T. Hansen and S. R. Sharpe, Lattice QCD and Three-particle Decays of Resonances, Ann. Rev. Nucl. Part. Sci. 69 (2019) 65, https://10.1146/annurev-nucl-101918-023723.

A. Rusetsky, Three particles on the lattice, PoS LATTICE2019 (2019) 281, https://10.22323/1.363.0281.

M. Mai, M. Doring and A. Rusetsky, Multi-particle systems on the lattice and chiral extrapolations: a brief review, Eur. Phys. J. ST 230 (2021) 1623, https://10.1140/epjs/s11734-021-00146-5.

K. Polejaeva and A. Rusetsky, Three particles in a finite volume, Eur. Phys. J. A 48 (2012) 67, https://10.1140/epja/i2012-12067-8.

M. T. Hansen and S. R. Sharpe, Relativistic, modelindependent, three-particle quantization condition, Phys. Rev. D 90 (2014) 116003, https://10.1103/PhysRevD.90.116003.

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 https://10.1103/PhysRevD.92.114509.

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, https://10.1103/PhysRevD.93.014506.

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, https://10.1103/PhysRevD.95.034501.

R. A. Briceno, 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, https://10.1103/PhysRevD.95.074510.

R. A. Briceño, M. T. Hansen and S. R. Sharpe, Numerical study of the relativistic three-body quantization condition in the isotropic approximation, Phys. Rev. D 98 (2018) 014506, https://10.1103/PhysRevD.98.014506.

R. A. Briceño, M. T. Hansen and S. R. Sharpe, Three-particle systems with resonant subprocesses in a finite volume, Phys. Rev. D 99 (2019) 014516, https://10.1103/PhysRevD.99.014516.

T. D. Blanton, F. Romero-Lopez and S. R. Sharpe, Implementing the three-particle quantization condition including higher partial waves, JHEP 03 (2019) 106, https://10.1007/JHEP03(2019)106.

R. A. Briceño, M. T. Hansen, S. R. Sharpe and A. P. Szczepaniak, Unitarity of the infinite-volume three-particle scattering amplitude arising from a finite-volume formalism, Phys. Rev. D 100 (2019) 054508, https://10.1103/PhysRevD.100.054508.

A. W. Jackura et al., Equivalence of three-particle scattering formalisms, Phys. Rev. D 100 (2019) 034508, https://10.1103/PhysRevD.100.034508.

F. Romero-Lopez, S. R. Sharpe, T. D. Blanton, R. A. Briceño and M. T. Hansen, Numerical exploration of three relativistic particles in a finite volume including two-particle resonances and bound states, JHEP 10 (2019) 007, https://10.1007/JHEP10(2019)007.

M. T. Hansen, F. Romero-Lopez and S. R. Sharpe, Generalizing the relativistic quantization condition to include all threepion isospin channels, JHEP 07 (2020) 047, [erratum: JHEP 02 (2021) 014] https://10.1007/JHEP07(2020)047.

T. D. Blanton and S. R. Sharpe, Equivalence of relativistic three-particle quantization conditions, Phys. Rev. D 102 (2020) 054515, https://10.1103/PhysRevD.102.054515.

T. D. Blanton and S. R. Sharpe, Alternative derivation of the relativistic three-particle quantization condition, Phys. Rev. D 102 (2020) 054520, https://10.1103/PhysRevD.102.054520.

T. D. Blanton and S. R. Sharpe, Relativistic three-particle quantization condition for nondegenerate scalars, Phys. Rev. D 103 (2021) 054503, https://10.1103/PhysRevD.103.054503.

M. T. Hansen, F. Romero-Lopez and S. R. Sharpe, Decay amplitudes to three hadrons from finite-volume matrix elements, JHEP 04 (2021) 113, https://10.1007/JHEP04(2021)113.

T. D. Blanton and S. R. Sharpe, Three-particle finite-volume formalism for π+π+K+ and related systems, Phys. Rev. D 104 (2021) 034509, https://10.1103/PhysRevD.104.034509.

T. D. Blanton, F. Romero-Lopez and S. R. Sharpe, Implementing the three-particle quantization condition for π +π +K+ and related systems, [arXiv:2111.12734 [hep-lat]].

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, https://10.1007/JHEP09(2017)109.

H. W. Hammer, J. Y. Pang and A. Rusetsky, Three particle quantization condition in a finite volume: 2. general formalism and the analysis of data, JHEP 10 (2017) 115, https: //10.1007/JHEP10(2017)115.

M. Doring, H. W. Hammer, M. Mai, J. Y. Pang, ¨ §. A. Rusetsky and J. Wu, Three-body spectrum in a finite volume: the role of cubic symmetry, Phys. Rev. D 97 (2018) 114508, https://10.1103/PhysRevD.97.114508.

J. Y. Pang, J. J. Wu, H. W. Hammer, U. G. Meißner and A. Rusetsky, Energy shift of the three-particle system in a finite volume, Phys. Rev. D 99 (2019) 074513, https://10.1103/PhysRevD.99.074513.

F. Romero-Lopez, A. Rusetsky, N. Schlage and C. Urbach, Relativistic N-particle energy shift in finite volume, JHEP 02 (2021) 060, https://10.1007/JHEP02(2021)060.

F. Müller and A. Rusetsky, On the three-particle analog of ¨the Lellouch-Luscher formula, ¨ JHEP 03 (2021) 152, https://10.1007/JHEP03(2021)152.

F. Müller, T. Yu and A. Rusetsky, Finite-volume energy shift of the three-pion ground state, Phys. Rev. D 103 (2021) 054506, https://10.1103/PhysRevD.103.054506.

M. Mai and M. Doring, Three-body Unitarity in the Finite Volume, Eur. Phys. J. A 53 (2017) 240, https://10.1140/epja/i2017-12440-1.

P. Guo and V. Gasparian, A solvable three-body model in finite volume, Phys. Lett. B 774 (2017) 441, https://10.1016/j.physletb.2017.10.009.

P. Klos, S. Konig, H. W. Hammer, J. E. Lynn and A. Schwenk, Signatures of few-body resonances in finite volume, Phys. Rev. C 98 (2018) 034004, https://10.1103/PhysRevC.98.034004.

P. Guo, M. Doring and A. P. Szczepaniak, Variational approach ¨ to N-body interactions in finite volume, Phys. Rev. D 98 (2018) 094502, https://10.1103/PhysRevD.98.094502.

P. Guo, Propagation of particles on a torus, Phys. Lett. B 804 (2020) 135370, https://10.1016/j.physletb.2020.135370.

J. Y. Pang, J. J. Wu and L. S. Geng, DDK system in finite volume, Phys. Rev. D 102 (2020) 114515, https://10.1103/PhysRevD.102.114515.

A. W. Jackura, R. A. Briceño, S. M. Dawid, M. H. E. Islam and C. McCarty, Solving relativistic three-body integral equations in the presence of bound states, Phys. Rev. D 104 (2021) 014507, https://10.1103/PhysRevD.104.014507.

F. Müller, J. Y. Pang, A. Rusetsky and J. J. Wu, Relativistic invariant formulation of the three-particle quantization condition,

F. Romero-Lopez, A. Rusetsky and C. Urbach, Two- and three-body interactions in ϕ 4 theory from lattice simulations, Eur. Phys. J. C 78 (2018) 846, https://10.1140/epjc/s10052-018-6325-8.

M. Mai and M. Doring, Finite-Volume Spectrum of π +π + and π +π +π + Systems, Phys. Rev. Lett. 122 (2019) 062503, https://10.1103/PhysRevLett.122.062503,

T. D. Blanton, F. Romero-Lopez and S. R. Sharpe, I = 3 ThreePion Scattering Amplitude from Lattice QCD, Phys. Rev. Lett. 124 (2020) 032001, https://10.1103/PhysRevLett.124.032001.

M. Mai, M. Doring, C. Culver and A. Alexandru, Three-body unitarity versus finite-volume π +π +π + spectrum from lattice QCD, Phys. Rev. D 101 (2020) 054510, https://10.1103/PhysRevD.101.054510.

C. Culver, M. Mai, R. Brett, A. Alexandru and M. Doring, ¨ Three pion spectrum in the I = 3 channel from lattice QCD, Phys. Rev. D 101 (2020) 114507, https://10.1103/PhysRevD.101.114507.

P. Guo and B. Long, Multi- π + systems in a finite volume, Phys. Rev. D 101 (2020) 094510, https://10.1103/PhysRevD.101.094510.

M. Fischer et al., Scattering of two and three physical pions at maximal isospin from lattice QCD, Eur. Phys. J. C 81 (2021) 436, https://10.1140/epjc/s10052-021-09206-5.

A. Alexandru, R. Brett, C. Culver, M. Doring, D. Guo, F. X. Lee and M. Mai, Finite-volume energy spectrum of the K−K−K− system, Phys. Rev. D 102 (2020) 114523, https://10.1103/PhysRevD.102.114523.

M. T. Hansen et al. [Hadron Spectrum], Energy-Dependent π +π +π + Scattering Amplitude from QCD, Phys. Rev. Lett. 126 (2021) 012001, https://10.1103/PhysRevLett.126.012001,

R. Brett, C. Culver, M. Mai, A. Alexandru, M. Doring and F. X. Lee, Three-body interactions from the finite-volume QCD spectrum, Phys. Rev. D 104 (2021) 014501, https://10.1103/PhysRevD.104.014501.

M. Mai et al., Three-body dynamics of the a1(1260) resonance from lattice QCD, 66. T. D. Blanton et al., Interactions of two and three mesons including higher partial waves from lattice QCD, JHEP 10 (2021) 023, https://10.1007/JHEP10(2021)023.

M. Peardon et al. [Hadron Spectrum], A Novel quark-field creation operator construction for hadronic physics in lattice QCD, Phys. Rev. D 80 (2009) 054506, https://10.1103/PhysRevD.80.054506.

C. Morningstar et al., Improved stochastic estimation of quark propagation with Laplacian Heaviside smearing in lattice QCD, Phys. Rev. D 83 (2011) 114505, https://10.1103/PhysRevD.83.114505.

B. Horz and A. Hanlon, Two- and three-pion finite-volume spectra at maximal isospin from lattice QCD, Phys. Rev. Lett. 123 (2019) 142002, https://10.1103/PhysRevLett.123.142002.

M. Luscher and U. Wolff, How to Calculate the Elastic Scattering Matrix in Two-dimensional Quantum Field Theories by Numerical Simulation, Nucl. Phys. B 339 (1990) 222, https://10.1016/0550-3213(90)90540-T.

M. Luscher, Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. 1. Stable Particle States, Commun. Math. Phys. 104 (1986) 177, https://10.1007/BF01211589.

S. Konig and D. Lee, Volume Dependence of N-Body Bound ¨ States, Phys. Lett. B 779 (2018) 9, https://10.1016/j.physletb.2018.01.060.

S. Konig, Few-body bound states and resonances in finite volume, Few Body Syst. 61 (2020) 20, https://10.1007/s00601-020-01550-8.

L. Maiani and M. Testa, Final state interactions from Euclidean correlation functions, Phys. Lett. B 245 (1990) 585, https://10.1016/0370-2693(90)90695-3.

K. Huang and C. N. Yang, Quantum-mechanical many-body problem with hard-sphere interaction, Phys. Rev. 105 (1957) 767, https://10.1103/PhysRev.105.767.

W. Detmold, M. J. Savage, A. Torok, S. R. Beane, T. C. Luu, K. Orginos and A. Parreno, Multi-Pion States in Lattice QCD and the Charged-Pion Condensate, Phys. Rev. D 78 (2008) 014507, https://10.1103/PhysRevD.78.014507.

B. Smigielski and J. Wasem, Ground-state energy shift of n pions and m kaons in a finite volume, Phys. Rev. D 79 (2009) 054506, https://10.1103/PhysRevD.79.054506.

M. T. Hansen and S. R. Sharpe, Threshold expansion of the three-particle quantization condition, Phys. Rev. D 93 (2016) 096006, [erratum: Phys. Rev. D 96 (2017) 039901.] https://10.1103/PhysRevD.93.096006.

S. R. Beane, W. Detmold, T. C. Luu, K. Orginos, M. J. Savage and A. Torok, Multi-Pion Systems in Lattice QCD and the Three-Pion Interaction, Phys. Rev. Lett. 100 (2008) 082004, https://10.1103/PhysRevLett.100.082004.

W. Detmold and B. Smigielski, Lattice QCD study of mixed systems of pions and kaons, Phys. Rev. D 84 (2011) 014508, https://10.1103/PhysRevD.84.014508.

S. R. Beane et al. [NPLQCD and QCDSF], Charged multihadron systems in lattice QCD+QED, Phys. Rev. D 103 (2021) 054504, https://10.1103/PhysRevD.103.054504.

C. J. D. Lin, G. Martinelli, C. T. Sachrajda and M. Testa, K → pipi decays in a finite volume, Nucl. Phys. B 619 (2001) 467, https://10.1016/S0550-3213(01)00495-3.

W. Detmold and M. J. Savage, Electroweak matrix elements in the two nucleon sector from lattice QCD, Nucl. Phys. A 743 (2004) 170, https://10.1016/j.nuclphysa.2004.07.007.

N. H. Christ, C. Kim and T. Yamazaki, Finite volume corrections to the two-particle decay of states with non-zero momentum, Phys. Rev. D 72 (2005) 114506, https://10.1103/PhysRevD.72.114506.

H. B. Meyer, Lattice QCD and the Timelike Pion Form Factor, Phys. Rev. Lett. 107 (2011) 072002, https://10.1103/PhysRevLett.107.072002.

V. Bernard, D. Hoja, U. G. Meissner and A. Rusetsky, Matrix elements of unstable states, JHEP 09 (2012) 023, https://10.1007/JHEP09(2012)023.

A. Agadjanov, V. Bernard, U. G. Meißner and A. Rusetsky, A framework for the calculation of the transition form factors on the lattice, Nucl. Phys. B 886 (2014) 1199, https://10.1016/j.nuclphysb.2014.07.023.

R. A. Briceño, M. T. Hansen and A. Walker-Loud, Multichannel 1 → 2 transition amplitudes in a finite volume, Phys. Rev. D 91 (2015) 034501, https://10.1103/PhysRevD.91.034501.

X. Feng, S. Aoki, S. Hashimoto and T. Kaneko, Timelike pion form factor in lattice QCD, Phys. Rev. D 91 (2015) 054504, https://10.1103/PhysRevD.91.054504.

R. A. Briceño and M. T. Hansen, Multichannel 0 → 2 and 1 → 2 transition amplitudes for arbitrary spin particles in a finite volume, Phys. Rev. D 92 (2015) 074509, https://10.1103/PhysRevD.92.074509.

R. A. Briceño and M. T. Hansen, Relativistic, model independent, multichannel 2 → 2 transition amplitudes in a finite volume, Phys. Rev. D 94 (2016) 013008, https://10.1103/PhysRevD.94.013008.

A. Baroni, R. A. Bricenñ, M. T. Hansen and F. G. Ortega- Gama, Form factors of two-hadron states from a covariant finite-volume formalism, Phys. Rev. D 100 (2019) 034511, https://10.1103/PhysRevD.100.034511.

R. A. Briceño, M. T. Hansen and A. W. Jackura, Consistency checks for two-body finite-volume matrix elements: I. Conserved currents and bound states, Phys. Rev. D 100 (2019) 114505, https://10.1103/PhysRevD.100.114505.

R. A. Briceño, M. T. Hansen and A. W. Jackura, Consistency checks for two-body finite-volume matrix elements: II. Perturbative systems, Phys. Rev. D 101 (2020) 094508, https://10.1103/PhysRevD.101.094508.

X. Feng, L. C. Jin, Z. Y. Wang and Z. Zhang, Finite-volume formalism in the 2 HI+HI −−−−−→ 2 transition: An application to the lattice QCD calculation of double beta decays, Phys. Rev. D 103 (2021) 034508, https://10.1103/PhysRevD.103.034508.

M. T. Hansen and T. Peterken, Higher partial wave contamination in finite-volume formulae for 1-to-2 transitions, [arXiv:2111.02953 [hep-lat]].

R. Abbott et al. [RBC and UKQCD], Direct CP violation and the ∆I = 1/2 rule in K → ππ decay from the standard model, Phys. Rev. D 102 (2020) 054509, https://10.1103/PhysRevD.102.054509.

R. Aaij et al. [LHCb], Observation of CP Violation in Charm Decays, Phys. Rev. Lett. 122 (2019) 211803, https://10.1103/PhysRevLett.122.211803.

J. Bijnens and T. Husek, Six-pion amplitude, Phys. Rev. D 104 (2021) 054046, https://10.1103/PhysRevD.104.054046.

J. Bulava and M. T. Hansen, Scattering amplitudes from finitevolume spectral functions, Phys. Rev. D 100 (2019) 034521, https://10.1103/PhysRevD.100.034521.

M. Bruno and M. T. Hansen, Variations on the Maiani-Testa approach and the inverse problem, https://10.1007/JHEP06(2021)043. 102. M. Garofalo, F. Romero-Lopez, A. Rusetsky and C. Urbach, Testing a new method for scattering in finite volume in the φ 4 theory, Eur. Phys. J. C 81 (2021) 1034, https://10.1140/epjc/s10052-021-09830-1.

J. Bulava, M. T. Hansen, M. W. Hansen, A. Patella and N. Tantalo, Inclusive rates from smeared spectral densities in the two-dimensional O(3) non-linear σ-model, [arXiv:2111.12774 [hep-lat]].

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Published

2022-05-19

How to Cite

1.
Romero-Lopez F. Three-particle scattering amplitudes from lattice QCD. Supl. Rev. Mex. Fis. [Internet]. 2022 May 19 [cited 2022 Dec. 7];3(3):0308003 1-10. Available from: https://rmf.smf.mx/ojs/index.php/rmf-s/article/view/6116