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Pion transitions in the Born-Oppenheimer effective field theory: A long distance approach

Joan Soto ^1,2,* and Sandra Tomàs Valls ^1,†

¹Departament de Física Quàntica i Astrofísica and Institut de Ciències del Cosmos, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Catalonia, Spain
²Institut d’Estudis Espacials de Catalunya (IEEC), 08860 Castelldefels, Catalonia, Spain

^*Contact author: joan.soto@ub.edu
^†Contact author: sandratomas@icc.ub.edu

Phys. Rev. D 114, 074010 – Published 5 October, 2026

DOI: https://doi.org/10.1103/lxnt-xpns

Abstract

We address pion transitions involving heavy quarkonium and heavy exotic states in the Born-Oppenheimer effective field theory. Many of these states have a size similar or larger than the typical hadronic scale, and hence the usual QCD multipole expansion breaks down. Unknown low energy functions must be introduced, which at short distances must reproduce the known results of the multipole expansion. In order to determine the long distance behavior of these functions, we propose an interaction Lagrangian for pions and the QCD string. By matching the amplitudes obtained with this Lagrangian to the ones of the Born-Oppenheimer effective field theory in the static limit, we obtain the low energy functions in terms of three universal parameters. As a by-product, we also obtain the light quark mass dependence of the string tension. Assuming that the low energy functions are long-distance dominated, we calculate several quarkonium-to-quarkonium and hybrid-to-quarkonium transitions. We estimate the universal constants and provide a phenomenological analysis of the most relevant transitions.

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References (66)

R. L. Jaffe and K. Johnson, Unconventional states of confined quarks and gluons, Phys. Lett. 60B, 201 (1976).
R. F. Lebed, R. E. Mitchell, and E. S. Swanson, Heavy-quark QCD exotica, Prog. Part. Nucl. Phys. 93, 143 (2017).
N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C.-P. Shen, C. E. Thomas, A. Vairo, and C.-Z. Yuan, The $X Y Z$ states: Experimental and theoretical status and perspectives, Phys. Rep. 873, 1 (2020).
H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu, and S.-L. Zhu, An updated review of the new hadron states, Rep. Prog. Phys. 86, 026201 (2023).
M. Ablikim et al., Observation of a charged charmoniumlike structure in $e^{+} e^{-} \to π^{+} π^{-} J / ψ$ at $\sqrt{s} = 4.26 GeV$ , Phys. Rev. Lett. 110, 252001 (2013).
Z. Q. Liu et al., Study of $e^{+} e^{-} \to π^{+} π^{-} J / ψ$ and observation of a charged charmoniumlike state at Belle, Phys. Rev. Lett. 110, 252002 (2013); 111, 019901(E) (2013).
R. Aaij et al., Observation of $J / ψ p$ resonances consistent with pentaquark states in $Λ_{b}^{0} \to J / ψ K^{-} p$ decays, Phys. Rev. Lett. 115, 072001 (2015).
S. K. Choi et al., Observation of a narrow charmonium-like state in exclusive $B^{\pm} \to K^{\pm} π^{+} π^{-} J / ψ$ decays, Phys. Rev. Lett. 91, 262001 (2003).
D. Acosta et al., Observation of the narrow state $X (3872) \to J / ψ π^{+} π^{-}$ in $\bar{p} p$ collisions at $\sqrt{s} = 1.96 TeV$ , Phys. Rev. Lett. 93, 072001 (2004).
E. Braaten, C. Langmack, and D. H. Smith, Born-Oppenheimer approximation for the XYZ mesons, Phys. Rev. D 90, 014044 (2014).
M. Berwein, N. Brambilla, J. Tarrús Castellà, and A. Vairo, Quarkonium hybrids with nonrelativistic effective field theories, Phys. Rev. D 92, 114019 (2015).
R. Oncala and J. Soto, Heavy quarkonium hybrids: Spectrum, decay and mixing, Phys. Rev. D 96, 014004 (2017).
J. Soto, Heavy quarkonium hybrids, Nucl. Part. Phys. Proc. 294–296, 87 (2018).
N. Brambilla, W. K. Lai, J. Segovia, J. Tarrús Castellà, and A. Vairo, Spin structure of heavy-quark hybrids, Phys. Rev. D 99, 014017 (2019); 101, 099902(E) (2020).
J. Soto and J. Tarrús Castellà, Nonrelativistic effective field theory for heavy exotic hadrons, Phys. Rev. D 102, 014012 (2020).
K. J. Juge, J. Kuti, and C. J. Morningstar, Ab initio study of hybrid anti-b g b mesons, Phys. Rev. Lett. 82, 4400 (1999).
E. Braaten, C. Langmack, and D. H. Smith, Selection rules for hadronic transitions of XYZ mesons, Phys. Rev. Lett. 112, 222001 (2014).
J. Tarrús Castellà and E. Passemar, Exotic to standard bottomonium transitions, Phys. Rev. D 104, 034019 (2021).
N. Brambilla, W. K. Lai, A. Mohapatra, and A. Vairo, Heavy hybrid decays to quarkonia, Phys. Rev. D 107, 054034 (2023).
A. Pineda and J. Tarrús Castellà, Novel implementation of the multipole expansion to quarkonium hadronic transitions, Phys. Rev. D 100, 054021 (2019).
N. Brambilla, W. K. Lai, J. Segovia, J. Tarrús Castellà, and A. Vairo, Spin structure of heavy-quark hybrids, Phys. Rev. D 99, 014017 (2019); 101, 099902(E) (2020).
J. Soto and S. T. Valls, Hyperfine splittings of heavy quarkonium hybrids, Phys. Rev. D 108, 014025 (2023).
K. Gottfried, Hadronic transitions between quark—anti-quark bound states, Phys. Rev. Lett. 40, 598 (1978).
M. B. Voloshin, On dynamics of heavy quarks in nonperturbative QCD vacuum, Nucl. Phys. B154, 365 (1979).
H. Fujii and D. Kharzeev, Long range forces of QCD, Phys. Rev. D 60, 114039 (1999).
N. Brambilla, G. Krein, J. Tarrús Castellà, and A. Vairo, Long-range properties of $1 S$ bottomonium states, Phys. Rev. D 93, 054002 (2016).
M. A. Luty and R. Sundrum, Quarkonium decays and light quark masses, Phys. Lett. B 312, 205 (1993).
M. Luscher and P. Weisz, String excitation energies in SU(N) gauge theories beyond the free-string approximation, J. High Energy Phys. 07 (2004) 014.
N. Brambilla, G. Krein, J. Tarrús Castellà, and A. Vairo, Born-Oppenheimer approximation in an effective field theory language, Phys. Rev. D 97, 016016 (2018).
M. Berwein, N. Brambilla, A. Mohapatra, and A. Vairo, Hybrids, tetraquarks, pentaquarks, doubly heavy baryons, and quarkonia in Born-Oppenheimer effective theory, Phys. Rev. D 110, 094040 (2024).
N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Potential NRQCD: An effective theory for heavy quarkonium, Nucl. Phys. B566, 275 (2000).
M. Luscher and P. Weisz, Quark confinement and the bosonic string, J. High Energy Phys. 07 (2002) 049.
O. Aharony and Z. Komargodski, The effective theory of long strings, J. High Energy Phys. 05 (2013) 118.
M. Billo, M. Caselle, F. Gliozzi, M. Meineri, and R. Pellegrini, The Lorentz-invariant boundary action of the confining string and its universal contribution to the inter-quark potential, J. High Energy Phys. 05 (2012) 130.
B. B. Brandt, Spectrum of the open QCD flux tube and its effective string description I: 3d static potential in $SU (N = 2, 3)$ , J. High Energy Phys. 07 (2017) 008.
C. J. Isham, A. Salam, and J. A. Strathdee, Nonlinear realizations of space-time symmetries. Scalar and tensor gravity, Ann. Phys. (N.Y.) 62, 98 (1971).
D. V. Volkov, Phenomenological Lagrangians, Fiz. Elem. Chastits At. Yadra 4, 3 (1973).
S. Dubovsky, R. Flauger, and V. Gorbenko, Effective string theory revisited, J. High Energy Phys. 09 (2012) 044.
J. Gasser and H. Leutwyler, Chiral perturbation theory to one loop, Ann. Phys. (N.Y.) 158, 142 (1984).
L. Liu, G. Moir, M. Peardon, S. M. Ryan, C. E. Thomas, P. Vilaseca, J. J. Dudek, R. G. Edwards, B. Joo, and D. G. Richards, Excited and exotic charmonium spectroscopy from lattice QCD, J. High Energy Phys. 07 (2012) 126.
G. K. C. Cheung, C. O’Hara, G. Moir, M. Peardon, S. M. Ryan, C. E. Thomas, and D. Tims, Excited and exotic charmonium, $D_{s}$ and $D$ meson spectra for two light quark masses from lattice QCD, J. High Energy Phys. 12 (2016) 089.
J. Bulava, F. Knechtli, V. Koch, C. Morningstar, and M. Peardon, The quark-mass dependence of the potential energy between static colour sources in the QCD vacuum with light and strange quarks, Phys. Lett. B 854, 138754 (2024).
S. Navas et al., Review of particle physics, Phys. Rev. D 110, 030001 (2024).
A. Garmash et al., Amplitude analysis of $e^{+} e^{-} \to ϒ (n S) π^{+} π^{-}$ at $\sqrt{s} = {10.865}^{~} GeV$ , Phys. Rev. D 91, 072003 (2015).
K. J. Juge, J. Kuti, and C. Morningstar, Fine structure of the QCD string spectrum, Phys. Rev. Lett. 90, 161601 (2003).
J. Soto and J. Tarrús Castellà, Effective field theory for double heavy baryons at strong coupling, Phys. Rev. D 102, 014013 (2020); 104, 059901(E) (2021).
R. Oncala and J. Soto, Hybrid to quarkonia transitions, Phys. Rev. D 113, 116004 (2026).
S. Tomàs Valls and J. Soto, Simulation data of decay widths for pion transitions in heavy quarkonium and hybrids, CORA.Repositori de Dades de Recerca, V1, 10.34810/DATA3420 (2026).
J. Soto and J. Tarrús Castellà, Effective QCD string and doubly heavy baryons, Phys. Rev. D 104, 074027 (2021).
N. Brambilla, A. Mohapatra, T. Scirpa, and A. Vairo, Nature of $χ$ c1(3872) and Tcc+(3875), Phys. Rev. Lett. 135, 131902 (2025).
I. Adachi et al., Study of $ϒ (10753)$ decays to $π^{+} π^{-} ϒ (nS)$ final states at Belle II, J. High Energy Phys. 07 (2024) 116.
A. Bondar, A. Garmash, and A. Kuzmin, Amplitude analysis of $e^{+} e^{-} \to ϒ (n S) π^{+} π^{-}$ transitions at $\sqrt{s} = 10.865 GeV$ , Budker Institute of Nuclear Physics Report No. BELLE-NOTE-1292, 2013, p. 033.
Y. S. Surovtsev, P. Bydzovsky, T. Gutsche, R. Kaminski, V. E. Lyubovitskij, and M. Nagy, Unified description of BABAR and Belle data on the bottomonia decays $ϒ (m S) \to ϒ (n S) π^{+} π^{-}$ , Phys. Rev. D 92, 036002 (2015).
D. Cronin-Hennessy et al., Study of di-pion transitions among Upsilon(3S), Upsilon(2S), and Upsilon(1S) states, Phys. Rev. D 76, 072001 (2007).
Y.-H. Chen, M. Cleven, J. T. Daub, F.-K. Guo, C. Hanhart, B. Kubis, U.-G. Meißner, and B.-S. Zou, Effects of $Z_{b}$ states and bottom meson loops on $ϒ (4 S) \to ϒ (1 S, 2 S) π^{+} π^{-}$ transitions, Phys. Rev. D 95, 034022 (2017).
V. Baru, E. Epelbaum, A. A. Filin, C. Hanhart, R. V. Mizuk, A. V. Nefediev, and S. Ropertz, Insights into $Z_{b} (10610)$ and $Z_{b} (10650)$ from dipion transitions from $ϒ (10860)$ , Phys. Rev. D 103, 034016 (2021).
Y.-H. Chen, X.-K. Dong, F.-K. Guo, C. Hanhart, and B. Kubis, Decoding the structure near the $π^{+} π^{-}$ mass threshold in $ψ (3686) \to J / ψ π^{+} π^{-}$ decays, Phys. Rev. D 113, 054018 (2026).
G. Ecker, J. Gasser, A. Pich, and E. de Rafael, The role of resonances in chiral perturbation theory, Nucl. Phys. B321, 311 (1989).
P. Chatzis, A. Faessler, T. Gutsche, and V. E. Lyubovitskij, Hadronic and radiative three-body decays of J/psi involving the scalars f0(1370), f0(1500) and f0(1710), Phys. Rev. D 84, 034027 (2011).
J. A. Oller, Unitarization technics in hadron physics with historical remarks, Symmetry 12, 1114 (2020).
N. Brambilla, M. Groher, H. E. Martinez, and A. Vairo, Effective string theory and the long-range relativistic corrections to the quark-antiquark potential, Phys. Rev. D 90, 114032 (2014).
K. F. Chen et al., Observation of anomalous Upsilon(1S) pi+ pi- and Upsilon(2S) pi+ pi- production near the Upsilon(5S) resonance, Phys. Rev. Lett. 100, 112001 (2008).
A. Bondar et al., Observation of two charged bottomoniumlike resonances in $ϒ (5 s)$ decays, Phys. Rev. Lett. 108, 122001 (2012).
M. B. Voloshin, Charmonium, Prog. Part. Nucl. Phys. 61, 455 (2008).
M. B. Voloshin, Two-pion transitions in quarkonium revisited, Phys. Rev. D 74, 054022 (2006).
R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio, and G. Nardulli, Phenomenology of heavy meson chiral Lagrangians, Phys. Rep. 281, 145 (1997).

Phys. Rev. D 114, 074010– Published 5 October, 2026

Vol. 114, Iss. 7 — 1 October 2026

Received 16 June 2026
Accepted 27 August 2026

DOI: https://doi.org/10.1103/lxnt-xpns

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

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Pion transitions in the Born-Oppenheimer effective field theory: A long distance approach

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