Abstract
We derive relations between finite-volume matrix elements and infinite-volume decay amplitudes, for processes with three spinless, degenerate and either identical or non-identical particles in the final state. This generalizes the Lellouch-Lüscher relation for two-particle decays and provides a strategy for extracting three-hadron decay amplitudes using lattice QCD. Unlike for two particles, even in the simplest approximation, one must solve integral equations to obtain the physical decay amplitude, a consequence of the nontrivial finite-state interactions. We first derive the result in a simplified theory with three identical particles, and then present the generalizations needed to study phenomenologically relevant three-pion decays. The specific processes we discuss are the CP-violating K → 3π weak decay, the isospin-breaking η → 3π QCD transition, and the electromagnetic γ* → 3π amplitudes that enter the calculation of the hadronic vacuum polarization contribution to muonic g − 2.
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References
R.A. Briceno and Z. Davoudi, Three-particle scattering amplitudes from a finite volume formalism, Phys. Rev. D 87 (2013) 094507 [arXiv:1212.3398] [INSPIRE].
K. Polejaeva and A. Rusetsky, Three particles in a finite volume, Eur. Phys. J. A 48 (2012) 67 [arXiv:1203.1241] [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].
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].
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].
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 [arXiv:1707.02176] [INSPIRE].
M. Mai and M. Döring, Three-body Unitarity in the Finite Volume, Eur. Phys. J. A 53 (2017) 240 [arXiv:1709.08222] [INSPIRE].
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 [arXiv:1810.01429] [INSPIRE].
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 [arXiv:1803.04169] [INSPIRE].
A.W. Jackura et al., Equivalence of three-particle scattering formalisms, Phys. Rev. D 100 (2019) 034508 [arXiv:1905.12007] [INSPIRE].
T.D. Blanton, F. Romero-López and S.R. Sharpe, Implementing the three-particle quantization condition including higher partial waves, JHEP 03 (2019) 106 [arXiv:1901.07095] [INSPIRE].
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 [arXiv:1905.11188] [INSPIRE].
M.T. Hansen and S.R. Sharpe, Lattice QCD and Three-particle Decays of Resonances, Ann. Rev. Nucl. Part. Sci. 69 (2019) 65 [arXiv:1901.00483] [INSPIRE].
F. Romero-López, 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 [arXiv:1908.02411] [INSPIRE].
T.D. Blanton and S.R. Sharpe, Alternative derivation of the relativistic three-particle quantization condition, Phys. Rev. D 102 (2020) 054520 [arXiv:2007.16188] [INSPIRE].
T.D. Blanton and S.R. Sharpe, Equivalence of relativistic three-particle quantization conditions, Phys. Rev. D 102 (2020) 054515 [arXiv:2007.16190] [INSPIRE].
M.T. Hansen, F. Romero-López and S.R. Sharpe, Generalizing the relativistic quantization condition to include all three-pion isospin channels, JHEP 07 (2020) 047 [Erratum ibid. 02 (2021) 014] [arXiv:2003.10974] [INSPIRE].
T.D. Blanton and S.R. Sharpe, Relativistic three-particle quantization condition for nondegenerate scalars, Phys. Rev. D 103 (2021) 054503 [arXiv:2011.05520] [INSPIRE].
F. Müller, A. Rusetsky and T. Yu, Finite-volume energy shift of the three-pion ground state, Phys. Rev. D 103 (2021) 054506 [arXiv:2011.14178] [INSPIRE].
M. Mai and M. Döring, Finite-Volume Spectrum of π+π+ and π+π+π+ Systems, Phys. Rev. Lett. 122 (2019) 062503 [arXiv:1807.04746] [INSPIRE].
B. Hörz and A. Hanlon, Two- and three-pion finite-volume spectra at maximal isospin from lattice QCD, Phys. Rev. Lett. 123 (2019) 142002 [arXiv:1905.04277] [INSPIRE].
T.D. Blanton, F. Romero-López and S.R. Sharpe, I = 3 Three-Pion Scattering Amplitude from Lattice QCD, Phys. Rev. Lett. 124 (2020) 032001 [arXiv:1909.02973] [INSPIRE].
C. Culver, M. Mai, R. Brett, A. Alexandru and M. Döring, Three pion spectrum in the I = 3 channel from lattice QCD, Phys. Rev. D 101 (2020) 114507 [arXiv:1911.09047] [INSPIRE].
M. Mai, M. Döring, C. Culver and A. Alexandru, Three-body unitarity versus finite-volume π+π+π+ spectrum from lattice QCD, Phys. Rev. D 101 (2020) 054510 [arXiv:1909.05749] [INSPIRE].
M. Fischer, B. Kostrzewa, L. Liu, F. Romero-López, M. Ueding and C. Urbach, Scattering of two and three physical pions at maximal isospin from lattice QCD, arXiv:2008.03035 [INSPIRE].
Hadron Spectrum collaboration, Energy-Dependent π+π+π+ Scattering Amplitude from QCD, Phys. Rev. Lett. 126 (2021) 012001 [arXiv:2009.04931] [INSPIRE].
A. Alexandru et al., Finite-volume energy spectrum of the K−K−K− system, Phys. Rev. D 102 (2020) 114523 [arXiv:2009.12358] [INSPIRE].
R. Brett, C. Culver, M. Mai, A. Alexandru, M. Döring and F.X. Lee, Three-body interactions from the finite-volume QCD spectrum, arXiv:2101.06144 [INSPIRE].
F. Romero-López, A. Rusetsky and C. Urbach, Two- and three-body interactions in φ4 theory from lattice simulations, Eur. Phys. J. C 78 (2018) 846 [arXiv:1806.02367] [INSPIRE].
F. Romero-López, A. Rusetsky, N. Schlage and C. Urbach, Relativistic N-particle energy shift in finite volume, JHEP 02 (2021) 060 [arXiv:2010.11715] [INSPIRE].
L. Lellouch and M. Lüscher, Weak transition matrix elements from finite volume correlation functions, Commun. Math. Phys. 219 (2001) 31 [hep-lat/0003023] [INSPIRE].
C.J.D. Lin, G. Martinelli, C.T. Sachrajda and M. Testa, K → ππ decays in a finite volume, Nucl. Phys. B 619 (2001) 467 [hep-lat/0104006] [INSPIRE].
W. Detmold and M.J. Savage, Electroweak matrix elements in the two nucleon sector from lattice QCD, Nucl. Phys. A 743 (2004) 170 [hep-lat/0403005] [INSPIRE].
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 [hep-lat/0507006] [INSPIRE].
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 [hep-lat/0507009] [INSPIRE].
H.B. Meyer, Lattice QCD and the Timelike Pion Form Factor, Phys. Rev. Lett. 107 (2011) 072002 [arXiv:1105.1892] [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. Briceno 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].
V. Bernard, D. Hoja, U.G. Meissner and A. Rusetsky, Matrix elements of unstable states, JHEP 09 (2012) 023 [arXiv:1205.4642] [INSPIRE].
A. Agadjanov, V. Bernard, U.G. Meißner and A. Rusetsky, A framework for the calculation of the ∆Nγ* transition form factors on the lattice, Nucl. Phys. B 886 (2014) 1199 [arXiv:1405.3476] [INSPIRE].
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 [arXiv:1406.5965] [INSPIRE].
X. Feng, S. Aoki, S. Hashimoto and T. Kaneko, Timelike pion form factor in lattice QCD, Phys. Rev. D 91 (2015) 054504 [arXiv:1412.6319] [INSPIRE].
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 [arXiv:1502.04314] [INSPIRE].
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 [arXiv:1509.08507] [INSPIRE].
A. Baroni, R.A. Briceño, 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 [arXiv:1812.10504] [INSPIRE].
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 [arXiv:1909.10357] [INSPIRE].
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 [arXiv:2002.00023] [INSPIRE].
X. Feng, L.-C. Jin, Z.-Y. Wang and Z. Zhang, Finite-volume formalism in the \( 2\overset{H_I+{H}_I}{\to }2 \) transition: An application to the lattice QCD calculation of double beta decays, Phys. Rev. D 103 (2021) 034508 [arXiv:2005.01956] [INSPIRE].
D. Bernecker and H.B. Meyer, Vector Correlators in Lattice QCD: Methods and applications, Eur. Phys. J. A 47 (2011) 148 [arXiv:1107.4388] [INSPIRE].
H.B. Meyer and H. Wittig, Lattice QCD and the anomalous magnetic moment of the muon, Prog. Part. Nucl. Phys. 104 (2019) 46 [arXiv:1807.09370] [INSPIRE].
F. Müller and A. Rusetsky, On the three-particle analog of the Lellouch-Lüscher formula, JHEP 21 (2020) 152 [arXiv:2012.13957] [INSPIRE].
N.N. Khuri and S.B. Treiman, Pion-Pion Scattering and K± → 3π Decay, Phys. Rev. 119 (1960) 1115 [INSPIRE].
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, arXiv:2010.09820 [INSPIRE].
M. Hoferichter, B. Kubis, S. Leupold, F. Niecknig and S.P. Schneider, Dispersive analysis of the pion transition form factor, Eur. Phys. J. C 74 (2014) 3180 [arXiv:1410.4691] [INSPIRE].
M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold and S.P. Schneider, Pion-pole contribution to hadronic light-by-light scattering in the anomalous magnetic moment of the muon, Phys. Rev. Lett. 121 (2018) 112002 [arXiv:1805.01471] [INSPIRE].
M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold and S.P. Schneider, Dispersion relation for hadronic light-by-light scattering: pion pole, JHEP 10 (2018) 141 [arXiv:1808.04823] [INSPIRE].
M. Hoferichter, B.-L. Hoid and B. Kubis, Three-pion contribution to hadronic vacuum polarization, JHEP 08 (2019) 137 [arXiv:1907.01556] [INSPIRE].
B.-L. Hoid, M. Hoferichter and B. Kubis, Hadronic vacuum polarization and vector-meson resonance parameters from e+e− → π0γ, Eur. Phys. J. C 80 (2020) 988 [arXiv:2007.12696] [INSPIRE].
Particle Data Group collaboration, Review of Particle Physics, PTEP 2020 (2020) 083C01 [INSPIRE].
L. Gan, B. Kubis, E. Passemar and S. Tulin, Precision tests of fundamental physics with η and η′ mesons, arXiv:2007.00664 [INSPIRE].
G.M. de Divitiis et al., Isospin breaking effects due to the up-down mass difference in Lattice QCD, JHEP 04 (2012) 124 [arXiv:1110.6294] [INSPIRE].
RM123 collaboration, Leading isospin breaking effects on the lattice, Phys. Rev. D 87 (2013) 114505 [arXiv:1303.4896] [INSPIRE].
G. Colangelo, S. Lanz, H. Leutwyler and E. Passemar, Dispersive analysis of η → 3π, Eur. Phys. J. C 78 (2018) 947 [arXiv:1807.11937] [INSPIRE].
K. Kampf, M. Knecht, J. Novotný and M. Zdráhal, Dispersive construction of two-loop P → πππ (P = K, η) amplitudes, Phys. Rev. D 101 (2020) 074043 [arXiv:1911.11762] [INSPIRE].
RBC and UKQCD collaborations, Standard Model Prediction for Direct CP Violation in K → ππ Decay, Phys. Rev. Lett. 115 (2015) 212001 [arXiv:1505.07863] [INSPIRE].
T. Blum et al., K → ππ ∆I = 3/2 decay amplitude in the continuum limit, Phys. Rev. D 91 (2015) 074502 [arXiv:1502.00263] [INSPIRE].
RBC and UKQCD collaborations, Direct CP-violation and the ∆I = 1/2 rule in K → ππ decay from the standard model, Phys. Rev. D 102 (2020) 054509 [arXiv:2004.09440] [INSPIRE].
V. Cirigliano, G. Ecker, H. Neufeld, A. Pich and J. Portoles, Kaon Decays in the Standard Model, Rev. Mod. Phys. 84 (2012) 399 [arXiv:1107.6001] [INSPIRE].
NA48/2 collaboration, Search for direct CP violating charge asymmetries in K± → π±π+π− and K± → π±π0π0 decays, Eur. Phys. J. C 52 (2007) 875 [arXiv:0707.0697] [INSPIRE].
NA48/2 collaboration, Empirical parameterization of the K± → π±π0π0 decay Dalitz plot, Phys. Lett. B 686 (2010) 101 [arXiv:1004.1005] [INSPIRE].
E. Gamiz, J. Prades and I. Scimemi, Charged kaon K → 3π CP-violating asymmetries at NLO in ChPT, JHEP 10 (2003) 042 [hep-ph/0309172] [INSPIRE].
J. Prades, ChPT Progress on Non-Leptonic and Radiative Kaon Decays, PoS KAON (2008) 022 [arXiv:0707.1789] [INSPIRE].
G. Buchalla, A.J. Buras and M.E. Lautenbacher, Weak decays beyond leading logarithms, Rev. Mod. Phys. 68 (1996) 1125 [hep-ph/9512380] [INSPIRE].
RBC collaboration, Kaon matrix elements and CP-violation from quenched lattice QCD: 1. The three flavor case, Phys. Rev. D 68 (2003) 114506 [hep-lat/0110075] [INSPIRE].
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Hansen, M.T., Romero-López, F. & Sharpe, S.R. Decay amplitudes to three hadrons from finite-volume matrix elements. J. High Energ. Phys. 2021, 113 (2021). https://doi.org/10.1007/JHEP04(2021)113
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DOI: https://doi.org/10.1007/JHEP04(2021)113