Abstract
Recently, formalism has been derived for studying electroweak transition amplitudes for three-body systems both in infinite and finite volumes. The formalism provides exact relations that the infinite-volume amplitudes must satisfy, as well as a relationship between physical amplitudes and finite-volume matrix elements, which can be constrained from lattice QCD calculations. This formalism poses additional challenges when compared with the analogous well-studied two-body equivalent one, including the necessary step of solving integral equations of singular functions. In this work, we provide some non-trivial analytical and numerical tests on the aforementioned formalism. In particular, we consider a case where the three-particle system can have three-body bound states as well as bound states in the two-body subsystem. For kinematics below the three-body threshold, we demonstrate that the scattering amplitudes satisfy unitarity. We also check that for these kinematics the finite-volume matrix elements are accurately described by the formalism for two-body systems up to exponentially suppressed corrections. Finally, we verify that in the case of the three-body bound state, the finite-volume matrix element is equal to the infinite-volume coupling of the bound state, up to exponentially suppressed errors.
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References
T. Aoyama et al., The anomalous magnetic moment of the muon in the Standard Model, Phys. Rep. 887 (2020) 1 [arXiv:2006.04822] [INSPIRE].
R. Abdul Khalek et al., Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report, Nucl. Phys. A 1026 (2022) 122447 [arXiv:2103.05419] [INSPIRE].
L. Leskovec, Electroweak transitions involving resonances, PoS LATTICE2023 (2024) 119 [arXiv:2401.02495] [INSPIRE].
S. Meinel, Quark flavor physics with lattice QCD, PoS LATTICE2023 (2024) 126 [arXiv:2401.08006] [INSPIRE].
M. Luscher, Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. Part 2. Scattering States, Commun. Math. Phys. 105 (1986) 153 [INSPIRE].
M. Luscher, Two particle states on a torus and their relation to the scattering matrix, Nucl. Phys. B 354 (1991) 531 [INSPIRE].
L. Lellouch and M. Luscher, 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. 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-Luscher 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].
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. Part 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. Part 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].
R.A. Briceño, A.W. Jackura, A. Rodas and J.V. Guerrero, Prospects for γ⋆γ⋆ → ππ via lattice QCD, Phys. Rev. D 107 (2023) 034504 [arXiv:2210.08051] [INSPIRE].
R.A. Briceño, J.J. Dudek and L. Leskovec, Constraining 1 + \( \mathcal{J} \) → 2 coupled-channel amplitudes in finite-volume, Phys. Rev. D 104 (2021) 054509 [arXiv:2105.02017] [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].
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, J.J. Dudek, R.G. Edwards, C.J. Shultz, C.E. Thomas and D.J. Wilson, The ππ → πγ⋆ amplitude and the resonant ρ → πγ⋆ transition from lattice QCD, Phys. Rev. D 93 (2016) 114508 [Erratum ibid. 105 (2022) 079902] [arXiv:1604.03530] [INSPIRE].
C. Andersen, J. Bulava, B. Hörz and C. Morningstar, The I = 1 pion-pion scattering amplitude and timelike pion form factor from Nf = 2 + 1 lattice QCD, Nucl. Phys. B 939 (2019) 145 [arXiv:1808.05007] [INSPIRE].
Hadron Spectrum collaboration, Radiative decay of the resonant K* and the γK → Kπ amplitude from lattice QCD, Phys. Rev. D 106 (2022) 114513 [arXiv:2208.13755] [INSPIRE].
C. Alexandrou et al., πγ → ππ transition and the ρ radiative decay width from lattice QCD, Phys. Rev. D 98 (2018) 074502 [Erratum ibid. 105 (2022) 019902] [arXiv:1807.08357] [INSPIRE].
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. Part 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. Part 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, Numerical study of the relativistic three-body quantization condition in the isotropic approximation, Phys. Rev. D 98 (2018) 014506 [arXiv:1803.04169] [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].
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].
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 [arXiv:1902.01111] [INSPIRE].
A.W. Jackura et al., Equivalence of three-particle scattering formalisms, Phys. Rev. D 100 (2019) 034508 [arXiv:1905.12007] [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].
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, 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].
J.-Y. Pang, J.-J. Wu and L.-S. Geng, DDK system in finite volume, Phys. Rev. D 102 (2020) 114515 [arXiv:2008.13014] [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].
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, T. Yu and A. Rusetsky, Finite-volume energy shift of the three-pion ground state, Phys. Rev. D 103 (2021) 054506 [arXiv:2011.14178] [INSPIRE].
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 [arXiv:2105.12094] [INSPIRE].
F. Müller, J.-Y. Pang, A. Rusetsky and J.-J. Wu, Relativistic-invariant formulation of the NREFT three-particle quantization condition, JHEP 02 (2022) 158 [arXiv:2110.09351] [INSPIRE].
T.D. Blanton, F. Romero-López and S.R. Sharpe, Implementing the three-particle quantization condition for π+π+K+ and related systems, JHEP 02 (2022) 098 [arXiv:2111.12734] [INSPIRE].
A.W. Jackura, Three-body scattering and quantization conditions from S-matrix unitarity, Phys. Rev. D 108 (2023) 034505 [arXiv:2208.10587] [INSPIRE].
M. Garofalo, M. Mai, F. Romero-López, A. Rusetsky and C. Urbach, Three-body resonances in the φ4 theory, JHEP 02 (2023) 252 [arXiv:2211.05605] [INSPIRE].
M.T. Hansen, F. Romero-López and S.R. Sharpe, Incorporating DDπ effects and left-hand cuts in lattice QCD studies of the Tcc(3875)+, arXiv:2401.06609 [INSPIRE].
JPAC collaboration, Phenomenology of Relativistic 3 → 3 Reaction Amplitudes within the Isobar Approximation, Eur. Phys. J. C 79 (2019) 56 [arXiv:1809.10523] [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].
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 [arXiv:2010.09820] [INSPIRE].
D. Sadasivan et al., Pole position of the a1(1260) resonance in a three-body unitary framework, Phys. Rev. D 105 (2022) 054020 [arXiv:2112.03355] [INSPIRE].
S.M. Dawid, M.H.E. Islam and R.A. Briceño, Analytic continuation of the relativistic three-particle scattering amplitudes, Phys. Rev. D 108 (2023) 034016 [arXiv:2303.04394] [INSPIRE].
S.M. Dawid, M.H.E. Islam, R.A. Briceno and A.W. Jackura, Evolution of Efimov states, Phys. Rev. A 109 (2024) 043325 [arXiv:2309.01732] [INSPIRE].
M. Mai and M. Doring, Finite-Volume Spectrum of π+π+ and π+π+π+ Systems, Phys. Rev. Lett. 122 (2019) 062503 [arXiv:1807.04746] [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].
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, Eur. Phys. J. C 81 (2021) 436 [arXiv:2008.03035] [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, Phys. Rev. D 104 (2021) 014501 [arXiv:2101.06144] [INSPIRE].
GWQCD collaboration, Three-Body Dynamics of the a1(1260) Resonance from Lattice QCD, Phys. Rev. Lett. 127 (2021) 222001 [arXiv:2107.03973] [INSPIRE].
T.D. Blanton, A.D. Hanlon, B. Hörz, C. Morningstar, F. Romero-López and S.R. Sharpe, Interactions of two and three mesons including higher partial waves from lattice QCD, JHEP 10 (2021) 023 [arXiv:2106.05590] [INSPIRE].
Z.T. Draper, A.D. Hanlon, B. Hörz, C. Morningstar, F. Romero-López and S.R. Sharpe, Interactions of πK, ππK and KKπ systems at maximal isospin from lattice QCD, JHEP 05 (2023) 137 [arXiv:2302.13587] [INSPIRE].
Hadron Spectrum collaboration, Energy-Dependent π+π+π+ Scattering Amplitude from QCD, Phys. Rev. Lett. 126 (2021) 012001 [arXiv:2009.04931] [INSPIRE].
A.W. Jackura and R.A. Briceño, Partial-wave projection of the one-particle exchange in three-body scattering amplitudes, arXiv:2312.00625 [INSPIRE].
R.A. Briceno, J.J. Dudek and R.D. Young, Scattering processes and resonances from lattice QCD, Rev. Mod. Phys. 90 (2018) 025001 [arXiv:1706.06223] [INSPIRE].
M.T. Hansen and S.R. Sharpe, Lattice QCD and Three-particle Decays of Resonances, Annu. Rev. Nucl. Part. Sci. 69 (2019) 65 [arXiv:1901.00483] [INSPIRE].
A.D. Hanlon, Hadron spectroscopy and few-body dynamics from Lattice QCD, in the proceedings of the 40th International Symposium on Lattice Field Theory (LATTICE 2023), 31 July–4 August 2023, Batavia, IL, U.S.A., PoS LATTICE2023 (2024) 106 [arXiv:2402.05185] [INSPIRE].
F. Romero-López, Multi-hadron interactions from lattice QCD, PoS LATTICE2022 (2023) 235 [arXiv:2212.13793] [INSPIRE].
F. Müller and A. Rusetsky, On the three-particle analog of the Lellouch-Lüscher formula, JHEP 03 (2021) 152 [arXiv:2012.13957] [INSPIRE].
F. Müller, J.-Y. Pang, A. Rusetsky and J.-J. Wu, Three-particle Lellouch-Lüscher formalism in moving frames, JHEP 02 (2023) 214 [arXiv:2211.10126] [INSPIRE].
M.T. Hansen, F. Romero-López and S.R. Sharpe, Decay amplitudes to three hadrons from finite-volume matrix elements, JHEP 04 (2021) 113 [arXiv:2101.10246] [INSPIRE].
J.-Y. Pang, R. Bubna, F. Müller, A. Rusetsky and J.-J. Wu, Lellouch-Lüscher factor for the K → 3π decays, arXiv:2312.04391 [INSPIRE].
R.A. Briceño, 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 [arXiv:2012.13338] [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].
L. Leskovec and S. Prelovsek, Scattering phase shifts for two particles of different mass and non-zero total momentum in lattice QCD, Phys. Rev. D 85 (2012) 114507 [arXiv:1202.2145] [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].
Acknowledgments
We thank Max Hansen for useful discussions. DAP and FRL have been supported in part by the U.S. Department of Energy (DOE), Office of Science, Office of Nuclear Physics, under grant Contract Numbers DE-SC000465 (DAP), DE-SC0011090 (FRL) and DE-SC0021006 (FRL). FRL acknowledges support by the Mauricio and Carlota Botton Fellowship. RAB acknowledges the support of the USDOE Early Career award, contract DE-SC0019229. RAB was supported in part by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Awards No. DE-AC02-05CH11231. AWJ acknowledges the support of the USDOE ExoHad Topical collaboration, contract DE-SC0023598.
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Briceño, R.A., Jackura, A.W., Pefkou, D.A. et al. Electroweak three-body decays in the presence of two- and three-body bound states. J. High Energ. Phys. 2024, 279 (2024). https://doi.org/10.1007/JHEP05(2024)279
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DOI: https://doi.org/10.1007/JHEP05(2024)279