Open Access

Role of electromagnetic corrections in the $π π$ distributions of $ψ^{'} \to J / ψ π π$

Zhao-Sai Jia ^1,2, Gang Li ^1,*, and Zhen-Hua Zhang ^3,†

¹College of Physics and Engineering, Qufu Normal University, Qufu 273165, China
²Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
³Center for High Energy Physics, Peking University, Beijing 100871, China

^*Contact author: gli@qfnu.edu.cn
^†Contact author: zhhzhang@pku.edu.cn

Phys. Rev. D 114, 014013 – Published 6 July, 2026

DOI: https://doi.org/10.1103/t1z7-1c5j

Abstract

The cusp structure at the $π^{+} π^{-}$ threshold in the $π^{0} π^{0}$ invariant mass spectrum serves as a sensitive probe for extracting the $S$ -wave $π π$ scattering lengths in processes where an $S$ -wave $π^{0} π^{0}$ pair is produced in the final states. Within the framework of nonrelativistic effective field theory with coupled channels $π^{0} π^{0}$ and $π^{+} π^{-}$ , we revisit the near-threshold structures in the $π^{0} π^{0}$ spectrum of $ψ^{'} \to J / ψ π π$ . Our analysis incorporates the $π π$ final-state rescattering, including both strong and Coulomb interactions. It turns out that the cusp near the $π^{+} π^{-}$ threshold becomes more prominent when Coulomb interactions are included. The electromagnetic corrections are found to alter the magnitude of the threshold cusp by about 2%–3%, underscoring the necessity of including these effects in precision determinations of the $π π$ scattering lengths. The coupled-channel amplitude constructed in this work provides a ready-to-use theoretical framework for experimental analyses of fine structures near $π π$ thresholds.

Physics Subject Headings (PhySH)

Article Text

References (52)

G. Colangelo, J. Gasser, and H. Leutwyler, The $π π S$ wave scattering lengths, Phys. Lett. B 488, 261 (2000).
G. Colangelo, J. Gasser, and H. Leutwyler, $π π$ scattering, Nucl. Phys. B603, 125 (2001).
S. Weinberg, Phenomenological Lagrangians, Physica A (Amsterdam) 96, 327 (1979).
J. Gasser and H. Leutwyler, Chiral perturbation theory to one loop, Ann. Phys. (N.Y.) 158, 142 (1984).
S. M. Roy, Exact integral equation for pion pion scattering involving only physical region partial waves, Phys. Lett. 36B, 353 (1971).
R. Garcia-Martin, R. Kaminski, J. R. Pelaez, J. Ruiz de Elvira, and F. J. Yndurain, The Pion-pion scattering amplitude. IV: Improved analysis with once subtracted Roy-like equations up to 1100 MeV, Phys. Rev. D 83, 074004 (2011).
B. Ananthanarayan, G. Colangelo, J. Gasser, and H. Leutwyler, Roy equation analysis of $π π$ scattering, Phys. Rep. 353, 207 (2001).
S. Descotes-Genon, N. H. Fuchs, L. Girlanda, and J. Stern, Analysis and interpretation of new low-energy $π π$ scattering data, Eur. Phys. J. C 24, 469 (2002).
L. Rosselet et al., Experimental study of 30,000 $K_{e 4}$ decays, Phys. Rev. D 15, 574 (1977).
S. Pislak et al., High statistics measurement of $K_{e 4}$ decay properties, Phys. Rev. D 67, 072004 (2003); 81, 119903(E) (2010).
J. R. Batley et al. (NA48/2 Collaboration), New high statistics measurement of $K_{e 4}$ decay form factors and $π π$ scattering phase shifts, Eur. Phys. J. C 54, 411 (2008).
J. R. Batley et al. (NA48/2 Collaboration), Precise tests of low energy QCD from $K_{e 4}$ decay properties, Eur. Phys. J. C 70, 635 (2010).
T. R. Palfrey and J. L. Uretsky, Photoproduction and detection of the two meson bound state, Phys. Rev. 121, 1798 (1961).
B. Adeva et al. (DIRAC Collaboration), First measurement of the $π^{+} π^{-}$ atom lifetime, Phys. Lett. B 619, 50 (2005).
P. Budini and L. Fonda, Pion-Pion Interaction from Threshold Anomalies in $K^{+}$ Decay, Phys. Rev. Lett. 6, 419 (1961).
N. Cabibbo, Determination of the $a_{0} - a_{2}$ pion scattering length from $K^{+} \to π^{+} π^{0} π^{0}$ decay, Phys. Rev. Lett. 93, 121801 (2004).
N. Cabibbo and G. Isidori, Pion-pion scattering and the $K \to 3 π$ decay amplitudes, J. High Energy Phys. 03 (2005) 021.
G. Colangelo, J. Gasser, B. Kubis, and A. Rusetsky, Cusps in $K \to 3 π$ decays, Phys. Lett. B 638, 187 (2006).
J. R. Batley et al. (NA48/2 Collaboration), Observation of a cusp-like structure in the $π^{0} π^{0}$ invariant mass distribution from $K^{\pm} \to π^{\pm} π^{0} π^{0}$ decay and determination of the $π π$ scattering lengths, Phys. Lett. B 633, 173 (2006).
J. R. Batley et al., Determination of the S-wave $π π$ scattering lengths from a study of $K^{\pm} \to π^{\pm} π^{0} π^{0}$ decays, Eur. Phys. J. C 64, 589 (2009).
M. Bissegger, A. Fuhrer, J. Gasser, B. Kubis, and A. Rusetsky, Radiative corrections in K— $> 3$ pi decays, Nucl. Phys. B806, 178 (2009).
J. Gasser, B. Kubis, and A. Rusetsky, Cusps in $K \to 3 π$ decays: A theoretical framework, Nucl. Phys. B850, 96 (2011).
S. Gevorkyan, Pion-pion scattering lengths determination from kaon decays, Proc. Sci. CD12 (2013) 066.
M. Bissegger, A. Fuhrer, J. Gasser, B. Kubis, and A. Rusetsky, Cusps in $K_{L} \to 3 π$ decays, Phys. Lett. B 659, 576 (2008).
C. Ditsche, B. Kubis, and U.-G. Meißner, Electromagnetic corrections in $η \to 3 π$ decays, Eur. Phys. J. C 60, 83 (2009).
B. Kubis and S. P. Schneider, The Cusp effect in $η^{'} \to η π π$ decays, Eur. Phys. J. C 62, 511 (2009).
U.-G. Meißner, G. Muller, and S. Steininger, Virtual photons in SU(2) chiral perturbation theory and electromagnetic corrections to $π π$ scattering, Phys. Lett. B 406, 154 (1997); 407, 454(E) (1997).
S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
F.-K. Guo, P.-N. Shen, and H.-Q. Jiang, The $π π$ phase shifts from $ψ^{'} \to J / ψ π^{+} π^{-}$ decays, HEPNP 29, 852 (2005).
Z. X. Zhang, J. J. Sanz-Cillero, X. Y. Shen, N. Wu, L. Y. Xiao, and H. Q. Zheng, Studies on $π^{+} π^{-}$ phase motion in the $ψ^{'} \to J / ψ π^{+} π^{-}$ process, Phys. Rev. D 78, 076010 (2008).
H. Chen, Study of $π π$ phase shifts in $ψ (2 S) \to π^{+} π^{-} J / ψ$ , Phys. Rev. D 80, 034029 (2009).
X.-H. Liu, F.-K. Guo, and E. Epelbaum, Extracting $π π S$ -wave scattering lengths from cusp effect in heavy quarkonium dipion transitions, Eur. Phys. J. C 73, 2284 (2013).
L. Liu, H.-W. Lin, and K. Orginos, Charmed hadron interactions, Proc. Sci. LATTICE2008 (2008) 112 [arXiv:0810.5412].
S. R. Gevorkian, A. V. Tarasov, and O. O. Voskresenskaya, Electromagnetic corrections to final state interactions in $K \to 3 π$ decays, Phys. Lett. B 649, 159 (2007).
M. Ablikim et al. (BESIII Collaboration), Determination of the number of $ψ (3686)$ events taken at BESIII, Chin. Phys. C 48, 093001 (2024).
M. Achasov et al., STCF conceptual design report: Volume 1—Physics & Detector, Front. Phys. (Beijing) 19, 14701 (2023).
M. Ablikim et al. (BES Collaboration), Production of $σ$ in $ψ (2 S) \to π^{+} π^{-} J / ψ$ , Phys. Lett. B 645, 19 (2007).
C. Hati, G. Kumar, and N. Mahajan (ATLAS Collaboration), Measurements of $ψ (2 S)$ and $X (3872) \to J / ψ π^{+} π^{-}$ production in $p p$ collisions at $\sqrt{s} = 8 TeV$ with the ATLAS detector, J. High Energy Phys. 01 (2016) 117.
E. Braaten, E. Johnson, and H. Zhang, Zero-range effective field theory for resonant wino dark matter. Part II. Coulomb resummation, J. High Energy Phys. 02 (2017) 150.
P.-P. Shi, Z.-H. Zhang, F.-K. Guo, and Z. Yang, $D^{+} D^{-}$ hadronic atom and its production in $p p$ and $p \bar{p}$ collisions, Phys. Rev. D 105, 034024 (2022).
Z.-S. Jia, Z.-H. Zhang, F.-K. Guo, and G. Li, Coupled-channel analysis of the near-threshold $e^{+} e^{-} \to N \bar{N}$ cross sections, Phys. Rev. D 111, 054014 (2025).
X. Kong and F. Ravndal, Coulomb effects in low-energy proton proton scattering, Nucl. Phys. A665, 137 (2000).
S. König, H. W. Grießhammer, H. W. Hammer, and U. van Kolck, Effective theory of ${}^{3}H$ and ${}^{3}{He}$ , J. Phys. G 43, 055106 (2016).
S. Sakai, F.-K. Guo, and B. Kubis, Extraction of $N D$ scattering lengths from the $Λ_{b} \to π^{-} p D^{0}$ decay and properties of the $Σ_{c} (2800)^{+}$ , Phys. Lett. B 808, 135623 (2020).
J. Gasser, On the history of pion-pion scattering, Proc. Sci. EFT09, 029 (2009).
E. Braaten and M. Kusunoki, Factorization in the production and decay of the $X (3872)$ , Phys. Rev. D 72, 014012 (2005).
X.-K. Dong, V. Baru, F.-K. Guo, C. Hanhart, A. Nefediev, and B.-S. Zou, Is the existence of a $J / ψ J / ψ$ bound state plausible?, Sci. Bull. 66, 2462 (2021).
B. Wu, X.-K. Dong, M.-L. Du, F.-K. Guo, and B.-S. Zou, Deciphering the mechanism of $J / ψ$ -nucleon scattering, Fund. Res. 5, 2530 (2025).
T. Mannel and R. Urech, Hadronic decays of excited heavy quarkonia, Z. Phys. C 73, 541 (1997).
N. Brambilla, G. a. Krein, J. Tarrús Castellà, and A. Vairo, Long-range properties of $1 S$ bottomonium states, Phys. Rev. D 93, 054002 (2016).
M. B. Voloshin, Charmonium, Prog. Part. Nucl. Phys. 61, 455 (2008).
F.-K. Guo, P.-N. Shen, H.-C. Chiang, and R.-G. Ping, Heavy quarkonium $π^{+} π^{-}$ transitions and a possible $b \bar{b} q \bar{q}$ state, Nucl. Phys. A761, 269 (2005).

Phys. Rev. D 114, 014013– Published 6 July, 2026

Vol. 114, Iss. 1 — 1 July 2026

Received 11 April 2026
Accepted 10 June 2026

DOI: https://doi.org/10.1103/t1z7-1c5j

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.

Published by the American Physical Society

Physical Review D

Reuse & Permissions

Export citation

Export citation

Role of electromagnetic corrections in the $π π$ distributions of $ψ^{'} \to J / ψ π π$

Abstract

Physics Subject Headings (PhySH)

Article Text

References (52)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation

Role of electromagnetic corrections in the ππ distributions of ψ′→J/ψππ

Abstract

Physics Subject Headings (PhySH)

Article Text

References (52)

Role of electromagnetic corrections in the $π π$ distributions of $ψ^{'} \to J / ψ π π$