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REVIEW 5 minor 65 references

Hadronic atoms

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Pith's one-line read The paper establishes a systematic expansion of hadronic-atom energy levels and widths in powers of the fine-structure constant, worked out to next-to-leading order for pionic hydrogen.

desk verdict Solid expert review of the NREFT approach to hadronic atoms; no new results, honest about delegating the central calculation. read the letter →

arxiv 2412.13027 v1 pith:3EFNGZUF submitted 2024-12-17 hep-ph hep-ex

Akaki Rusetsky (Bonn U. , Tbilisi State U.) This is my paper · ORCID
classification hep-phhep-ex
keywords hadronicatomspionichydrogennon-relativisticeffectivefieldtheorypion-nucleonscatteringlengthsisospinbreakingDeser-Goldberger-Baumann-Thirringformulachiralperturbationmatching
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Hadronic atoms—systems like a negative pion bound electromagnetically to a proton—are sensitive probes of the strong interaction at very low energy, but their measured energy shifts and widths can only be used if the known Coulomb binding is cleanly separated from short-range strong and isospin-breaking effects. The paper argues that non-relativistic effective field theory achieves this separation in a systematic way: because the Bohr momentum of the atom is much smaller than the hadron masses, the observables form a power series in the fine-structure constant $\alpha$ and the quark-mass difference, order by order. For pionic hydrogen this yields next-to-leading-order formulas, Eqs. (51)–(52), that express the strong shift and width directly through the threshold pion–nucleon amplitudes, with the effective-theory couplings matched to those amplitudes to all orders. If correct, these formulas let precise atom experiments determine the pion–nucleon scattering lengths, and the same machinery extends to other hadronic atoms and to analogous trapped systems.

What carries the argument

The load-bearing object is the non-relativistic effective field theory (NREFT): a Lagrangian built from single-particle fields with a conserved particle number, local strong couplings $c_0, c_2,\ldots$, and Coulomb photons, valid for momenta far below the hadron masses. Its essential property is that matching to the underlying relativistic theory can be performed perturbatively in $\alpha$, and for the leading coupling the relation $c_0=-2\pi a/m$ holds to all orders, so the atom observables are parameterized directly by physical scattering amplitudes. The bound-state side of the machinery is the Coulomb Green function with its ground-state pole removed; iterating the interaction in Rayleigh–Schr\"odinger perturbation theory gives an expansion in $\delta$, with the otherwise-suppressed effective-range coupling $c_2$ enhanced to $O(\delta^{-1/2})$ by the small $\pi^-p\to\pi^0n$ mass gap, which is what produces the unitary-cusp term in the width.

What would settle it

Search the $\pi^- p$ system for a bound state or narrow resonance with a binding momentum of order $\gamma = \alpha \mu_c \simeq 1$ MeV; the existence of such a state would violate the natural-size assumption on the strong couplings and invalidate the use of local contact terms, making Eqs. (51)–(52) the wrong correction to the Coulomb spectrum.

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Extended reading notes

Core claim

The central claim is that the spectrum of a hadronic atom can be computed as a controlled expansion in the isospin-breaking parameter $\delta \sim \alpha \sim (m_d-m_u)$ by matching a non-relativistic effective Lagrangian to the threshold $S$-matrix elements of QCD+QED. The matching is exact to all orders for the leading strong coupling—the bubble-chain amplitude gives $c_0=-2\pi a/m$—and the infrared-singular Coulomb pieces are factorized so that a finite threshold amplitude can be defined. With the matched Lagrangian, the bound-state problem reduces to Rayleigh–Schr\"odinger perturbation theory around the Coulomb ground state, using the pole-removed Coulomb Green function. The result for pionic hydrogen is $\Delta E_{\rm str} = -2\alpha^3\mu_c^2\,{\rm Re}A_c\,(1-2\alpha\mu_c(\ln\alpha-1){\rm Re}A_c+\delta_{\rm vac})$ and $\Gamma = 8\alpha^3\mu_c^2 p^*(-\gamma^2)({\rm Re}A_x)^2\,[1-4\alpha\mu_c(\ln\alpha-1){\rm Re}A_c+(p^*(0){\rm Re}A_0)^2+\delta_{\rm vac}](1+1/P)$. These formulas contain no remnant of the effective theory; the strong shift and width are fixed by the physical threshold amplitudes $A_c$, $A_x$, $A_0$ and the known vacuum-polarization corrections.

Load-bearing premise

The argument assumes the purely strong pion–nucleon interaction has no shallow bound state at the scale of the atom's Bohr momentum, so the strong couplings can be treated as local contact terms with natural size.

Editorial extensions

If this is right

  • Precise measurements of the pionic hydrogen ground-state shift and width determine the S-wave pion–nucleon scattering lengths $a^+_{0+}$ and $a^-_{0+}$ after known isospin-breaking corrections, without the model dependence of potential theory.
  • The expansion in $\delta$ is systematic: next-to-next-to-leading-order corrections are stated to be very small, so the quoted next-to-leading-order formulas set the practical accuracy of the extraction.
  • The same NREFT machinery applies to other hadronic atoms; for kaonic hydrogen, where the neutral threshold lies above the charged one, the leading isospin-breaking correction becomes $O(\sqrt{\delta})$ and is purely kinematic, expressible in terms of scattering lengths.
  • The method transfers to non-hadronic trapped systems with a known long-range force—cold atoms in harmonic traps and lattice boxes—so the same matching logic reads off short-range scattering parameters from level shifts.
  • Because matching to the 'pure QCD' world is convention-dependent, any extraction must state its convention; the atom observables themselves, however, are convention-independent.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's convention discussion implies that published scattering lengths from hadronic atoms carry an $O(\alpha)$ scheme dependence; comparisons between analyses are only meaningful if they use the same definition of the pure-QCD world, which is a practical caveat the paper mentions but does not make central.
  • The same framework should transfer to proposed electromagnetic bound states of charm mesons, with the caveat that the strong sector there may contain near-threshold states; if a shallow $DD^*$ bound state exists at the Bohr scale, the local contact-term expansion would need modification.
  • The stated smallness of NNLO corrections implies that future precision will be limited by the chiral low-energy constants entering $\delta_c$ and $\delta_x$; reducing the $\pm 2.9\times 10^{-2}$ uncertainty in Eq. (54) becomes the bottleneck for scattering-length extraction.
  • The close analogy with harmonic traps and lattice boxes suggests a unified read-off of scattering lengths from level shifts in any known long-range trap; a cross-check across such systems would test the universality of the matching machinery.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The manuscript is a review chapter that introduces non-relativistic effective field theory (NREFT) as the framework for describing hadronic atoms, with pionic hydrogen as the central worked example. It motivates the scale separation between the Bohr radius and the range of strong interactions, constructs the non-relativistic Lagrangian, explains power counting and matching in the strong sector, and then extends the construction to include photons and Coulomb-gauge exchange. The final sections present the NLO formulas for the strong energy shift and decay width of the ground state of pionic hydrogen, Eqs. (51)-(52), expressed in terms of the threshold amplitudes A_c, A_x, A_0 and isospin-breaking corrections, and discuss how these results are used to extract pion-nucleon scattering lengths. The paper also contains a brief guide to the literature and a comparison with potential-model approaches.

Significance. As a review chapter, the paper's value is pedagogical and consolidating rather than original. The framework it summarizes is mature and important: NREFT provides a systematic expansion of hadronic-atom observables in the isospin-breaking parameter delta~alpha, and the final spectrum formulas quoted in Eqs. (51)-(52) are the basis for precision extractions of pion-nucleon scattering lengths. The paper is transparent about its scope: the bound-state calculation behind Eq. (49) is explicitly delegated to the literature in Section 5.2 ('will not be repeated here'), and the final formulas are imported from Refs. [14,32,33]. This is acceptable for a review, but the abstract's wording 'it will be namely demonstrated' is somewhat stronger than what the chapter itself actually shows. The explicit statement of the naturalness assumption in Section 4.2 and the discussion of the unitary cusp for kaonic hydrogen in Section 5.3 are useful caveats that correctly identify the limits of the power counting. I found no internal inconsistency in the equations shown, and the quoted final results are consistent with the preceding matching and bound-state machinery.

minor comments (5)
  1. [Section 5.2, after Eq. (48)] The text states that the bound-state perturbation theory calculations are standard and 'will not be repeated here'; since Eqs. (51)-(52) are the central quantitative result of the review, the reader should be told explicitly at that point that these formulas are quoted from Refs. [14,32,33] rather than derived in this chapter, and the abstract's claim that the expansion is 'demonstrated' should be softened accordingly.
  2. [Eq. (44)] In the definition of lambda, the expression '1 + 2 mu_p' should read '1 + 2 kappa_p' with kappa_p the anomalous magnetic moment of the proton; as written it is inconsistent with Eq. (31), where the same combination appears as 1 + 2 kappa_p, and the notation mu_p for this quantity is dimensionally confusing.
  3. [Eq. (38)] The notation '(p*(0))' as used in the matching condition for Im c2 is ambiguous; it should be defined explicitly as the derivative of p*(p^2) with respect to p^2 evaluated at p^2 = 0 (or equivalently with respect to s at the charged threshold).
  4. [Section 4.2, third bullet] The naturalness assumption (no shallow bound states in the purely strong sector) is stated in a single sentence, although it is the main validity condition for the delta-counting; a brief remark on why the pion-nucleon and pion-pion systems considered in the review satisfy this condition would strengthen the pedagogical discussion, especially since Section 5.3 shows a case where the counting changes.
  5. [General] The manuscript contains numerous typographical artifacts, including 'a di fference' in Section 1, 'the hadrons are located very far of each other' in the Conclusions, 'Panofski ratio' in Eq. (36), and inconsistent spacing in several displayed equations; a careful proofread is recommended before publication.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the NREFT framework and matching are derived in the text; final pionic-hydrogen formulas are quoted from prior work but are not fitted to the same observables.

full rationale

I walked the derivation chain. Sections 4.1-4.5 construct the non-relativistic effective theory from scale separation, symmetries and threshold expansion, and derive the matching relation c0 = -2 pi a / m (Eq. 21) and the photon-inclusive matching conditions (Eqs. 27-29, 35) without assuming the hadronic-atom spectrum. The matching inputs are threshold scattering amplitudes Ac, Ax, A0 in an underlying relativistic theory; these are not defined in terms of the pionic-hydrogen energy shift or width. The final spectrum formulas (51)-(52) express the shift and width in terms of those amplitudes plus the independently measured Panofsky ratio P, and P is explicitly treated as an input. No parameter is fitted to the pionic-hydrogen shift and then renamed as a prediction. The main caveat is presentational: Section 5.2 says the bound-state calculation 'will not be repeated here' and cites the same author's review [14] for Eq. (49). This is a legitimate review practice rather than a circular reduction, because Eq. (49) is not equivalent to Eq. (51) by definition; it requires a separate Rayleigh-Schrodinger perturbation calculation, and the cited result is parameter-free, published, and externally testable against pionic-hydrogen data and independent ChPT calculations. The naturalness assumption about no shallow strong-sector bound states is explicitly stated as a premise, not smuggled in. I find no step where a claimed prediction equals its input by construction.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The paper is a review, so the ledger reflects assumptions the reviewed framework depends on. No invented entities are introduced. The free parameters are the NREFT couplings, all matched to the underlying theory rather than fitted to hadronic atom data.

free parameters (4)
  • c0 (NREFT contact coupling) = matched, not fitted: c0 = -2 pi a/m in the strong case; Re c0 related to Re A_c in Eq. (35)
    Central low-energy constant of the NREFT Lagrangian; its value is fixed by matching to the pi- p threshold amplitude, not by a fit to hadronic atom data.
  • c2 (derivative coupling) = matched to effective range and unitary cusp; no numerical value quoted
    Enters Eq. (30) and Eq. (32); it is enhanced to order delta^{-1/2} by the integrated-out n pi0 channel, as shown in Eq. (38).
  • h1 (pion charge-radius coupling) = h1 = M_pi^2 <r_pi^2>
    Determined by the pion charge radius through form-factor matching, Eq. (31).
  • Proton electromagnetic couplings cFp, cDp, cSp = cFp = 1 + kappa_p; cDp = 1 + 2 kappa_p + (4/3) m_p^2 <r_p^2>; cSp = 1 + 2 kappa_p
    Matched to the proton magnetic moment and charge radius in Eq. (31); not fitted in this paper.
assumptions (6)
  • domain assumption Typical momenta in the atom are much smaller than hadron masses: p ~ alpha mu_c << m.
    Invoked in Section 4.1 to justify the non-relativistic expansion and the power counting in alpha.
  • domain assumption The strong-sector couplings have natural size: no shallow bound states in the purely strong sector.
    Section 4.2 states this assumption explicitly; if false, the local contact-term expansion and the delta-counting break down.
  • domain assumption The isospin-breaking mass gap Delta = m_p + M_pi - m_n - M_pi0, about 3.3 MeV, counts as O(delta), same order as alpha and m_d - m_u.
    Used in Section 5.1 to set the one-channel NREFT counting and to make c2 of order delta^{-1/2}.
  • standard math Dimensional regularization with minimal subtraction preserves the naive power counting.
    Section 4.3 relies on this to argue that loops do not upset the delta-counting; a cutoff regularization would violate the naive counting.
  • standard math The hadronic atom bound state is governed by the Schrodinger equation, so Rayleigh-Schrodinger perturbation theory applies.
    Section 4.6 and Section 5.2 use the equivalence of NREFT to quantum mechanics for shallow bound states.
  • domain assumption Pure QCD parameters are defined by convention, e.g., by matching QCD+QED parameters at a scale mu1 and setting m_u = m_d = (m_u + m_d)/2.
    Section 3 explains the convention dependence of the 'purely hadronic' scattering lengths that the hadronic atom program extracts.

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Cite this review

Pith. "Pith review of Hadronic atoms." pith.science (2026). https://pith.science/paper/3EFNGZUF

@misc{pith2026241213027,
  author       = {Pith},
  title        = {Pith review of: Hadronic atoms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3EFNGZUF}},
  note         = {Machine review of arXiv:2412.13027}
}
read the original abstract

We give a brief survey of the theory of hadronic atoms, which represent important sources of information for studying hadron interactions at very low energy. It will be namely demonstrated that a systematic expansion of the observables of hadronic atoms (the energy levels and the decay width) in terms of the fine-structure constant can be obtained, using the framework of non-relativistic effective Lagrangians. We also present a pedagogical introduction to the non-relativistic effective theories that includes a review of the main concepts, such as the scale separation, construction of the Lagrangian, power counting and matching.

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Works this paper leans on

65 extracted references · 36 canonical work pages

  1. [1]

    Hadronic shift in pionic hydrogen

    M. Hennebach, et al., Hadronic shift in pionic hydrogen, Eur. Phys. J. A 50 (12) (2014) 190, doi: 10.1140/epja/i2014-14190-x, [Erratum: Eur.Phys.J.A 55, 24 (2019)], 1406.6525

    work page Pith review arXiv 2014

  2. [2]

    Hirtl, et al., Redetermination of the strong-interaction width in pionic hydrogen, Eur

    A. Hirtl, et al., Redetermination of the strong-interaction width in pionic hydrogen, Eur. Phys. J. A 57 (2) (2021) 70, doi: 10.1140/epja/ s10050-021-00387-x

    doi:10.1140/epja/ 2021

  3. [3]

    Pionic deuterium

    Th. Strauch, et al., Pionic deuterium, Eur. Phys. J. A 47 (2011) 88, doi: 10.1140/epja/i2011-11088-1, 1011.2415

    work page Pith review arXiv 2011

  4. [4]

    Determination of $\pi\pi$ scattering lengths from measurement of $\pi^+\pi^-$ atom lifetime

    B. Adeva, et al., Determination of ππ scattering lengths from measurement of π+π− atom lifetime, Phys. Lett. B 704 (2011) 24–29, doi: 10.1016/j.physletb.2011.08.074, 1109.0569

    work page Pith review arXiv 2011

  5. [5]

    First measurement of a long-lived $\pi^+ \pi^-$ atom lifetime

    B. Adeva, et al. (DIRAC), First measurement of a long-lived π+π− atom lifetime, Phys. Rev. Lett. 122 (8) (2019) 082003, doi: 10.1103/ PhysRevLett.122.082003, 1811.08659

    work page Pith review arXiv 2019

  6. [6]

    Measurement of the $\pi K$ atom lifetime and the $\pi K$ scattering length

    B. Adeva, et al. (DIRAC), Measurement of the πK atom lifetime and the πK scattering length, Phys. Rev. D 96 (2017) 052002, doi: 10.1103/PhysRevD.96.052002, 1707.02184

    work page Pith review arXiv 2017

  7. [7]

    Beer, et al

    G. Beer, et al. (DEAR), Measurement of the kaonic hydrogen X-ray spectrum, Phys. Rev. Lett. 94 (2005) 212302, doi:10.1103/PhysRevLett. 94.212302

    doi:10.1103/physrevlett 2005

  8. [8]

    Bazzi, et al

    M. Bazzi, et al. (SIDDHARTA), A New Measurement of Kaonic Hydrogen X-rays, Phys. Lett. B 704 (2011) 113–117, doi:10.1016/j.physletb. 2011.09.011, 1105.3090

    arXiv 2011

Show all 65 references
  1. [9]

    Curceanu, et al., Towards the first kaonic deuterium measurement with the SIDDHARTA-2 experiment, PoS CORFU2022 (2023) 039, doi:10.22323/1.436.0039

    C. Curceanu, et al., Towards the first kaonic deuterium measurement with the SIDDHARTA-2 experiment, PoS CORFU2022 (2023) 039, doi:10.22323/1.436.0039

    2023 doi

  2. [10]

    Doser, Antiprotonic bound systems, Prog

    M. Doser, Antiprotonic bound systems, Prog. Part. Nucl. Phys. 125 (2022) 103964, doi: 10.1016/j.ppnp.2022.103964

    2022

  3. [11]

    Zhen-Hua Zhang, Feng-Kun Guo, D±D∗± Hadronic Atom as a Key to Revealing theX(3872) Mystery, Phys. Rev. Lett. 127 (1) (2021) 012002, doi:10.1103/PhysRevLett.127.012002, 2012.08281

    2021 arXiv

  4. [12]

    Pan-Pan Shi, Zhen-Hua Zhang, Feng-Kun Guo, Zhi Y ang,D+D− hadronic atom and its production in pp and p ¯p collisions, Phys. Rev. D 105 (3) (2022) 034024, doi:10.1103/PhysRevD.105.034024, 2111.13496

    2022 arXiv

  5. [13]

    Stanley Deser, M. L. Goldberger, K. Baumann, Walter E. Thirring, Energy level displacements in pi mesonic atoms, Phys. Rev. 96 (1954) 774–776, doi:10.1103/PhysRev.96.774

    1954 doi

  6. [14]

    Gasser, V

    J. Gasser, V. E. Lyubovitskij, A. Rusetsky, Hadronic atoms in QCD + QED, Phys. Rept. 456 (2008) 167–251, doi: 10.1016/j.physrep.2007.09. 006, 0711.3522

    2008 arXiv

  7. [15]

    A. Gall, J. Gasser, Valery E. Lyubovitskij, A. Rusetsky, On the lifetime of the π+π− atom, Phys. Lett. B 462 (1999) 335–340, doi: 10.1016/ S0370-2693(99)00918-1, hep-ph/9905309

    1999 arXiv

  8. [16]

    Gasser, Valery E

    J. Gasser, Valery E. Lyubovitskij, A. Rusetsky, A. Gall, Decays of the π+π− atom, Phys. Rev. D 64 (2001) 016008, doi:10.1103/PhysRevD.64. 016008, hep-ph/0103157

    2001 arXiv

  9. [17]

    Thomas Busch, Berthold-Georg Englert, Kazimierz Rza ˙zewski, Martin Wilkens, Two Cold Atoms in a Harmonic Trap, Found. Phys. 28 (4) (1998) 549–559, doi:10.1023/a:1018705520999

    1998 doi

  10. [18]

    Savage, Achim Schwenk, James P

    Thomas Luu, Martin J. Savage, Achim Schwenk, James P . Vary, Nucleon-Nucleon Scattering in a Harmonic Potential, Phys. Rev. C 82 (2010) 034003, doi:10.1103/PhysRevC.82.034003, 1006.0427

    2010 arXiv

  11. [19]

    L ¨uscher, Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories

    M. L ¨uscher, Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. 2. Scattering States, Commun. Math. Phys. 105 (1986) 153–188, doi:10.1007/BF01211097

    1986 doi

  12. [20]

    Martin L ¨uscher, Two particle states on a torus and their relation to the scattering matrix, Nucl. Phys. B 354 (1991) 531–578, doi: 10.1016/ 0550-3213(91)90366-6

    1991

  13. [21]

    Aoki, et al

    Y . Aoki, et al. (Flavour Lattice Averaging Group (FLAG)), FLAG Review 2021, Eur. Phys. J. C 82 (10) (2022) 869, doi: 10.1140/epjc/ s10052-022-10536-1 , 2111.09849

    2022 arXiv

  14. [22]

    Gasser, A

    J. Gasser, A. Rusetsky, I. Scimemi, Electromagnetic corrections in hadronic processes, Eur. Phys. J. C 32 (2003) 97–114, doi:10.1140/epjc/ s2003-01383-1, hep-ph/0305260

    2003 arXiv

  15. [23]

    W. E. Caswell, G. P . Lepage, Effective Lagrangians for Bound State Problems in QED, QCD, and Other Field Theories, Phys. Lett. B 167 (1986) 437–442, doi:10.1016/0370-2693(86)91297-9

    1986 doi

  16. [24]

    Nora Brambilla, Antonio Pineda, Joan Soto, Antonio Vairo, Effective Field Theories for Heavy Quarkonium, Rev. Mod. Phys. 77 (2005) 1423, doi:10.1103/RevModPhys.77.1423, hep-ph/0410047

    2005 arXiv

  17. [25]

    Jurg Gasser, Bastian Kubis, Akaki Rusetsky, Cusps in K→ 3π decays: a theoretical framework, Nucl. Phys. B 850 (2011) 96–147, doi: 10.1016/j.nuclphysb.2011.04.013, 1103.4273

    2011 arXiv

  18. [26]

    H. W. Hammer, J. Y . Pang, A. Rusetsky, Three particle quantization condition in a finite volume: 2. general formalism and the analysis of data, JHEP 10 (2017) 115, doi:10.1007/JHEP10(2017)115, 1707.02176

    2017 arXiv

  19. [27]

    Meißner, Akaki Rusetsky, Effective Field Theories, Cambridge University Press 2022, ISBN 978-1-108-68903-8, doi: 10.1017/ 9781108689038

    Ulf-G. Meißner, Akaki Rusetsky, Effective Field Theories, Cambridge University Press 2022, ISBN 978-1-108-68903-8, doi: 10.1017/ 9781108689038

    2022

  20. [28]

    Beneke, Vladimir A

    M. Beneke, Vladimir A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321–344, doi: 10.1016/S0550-3213(98)00138-2, hep-ph/9711391

    1998 arXiv

  21. [29]

    R. F . Mohr, R. J. Furnstahl, R. J. Perry, K. G. Wilson, H. W. Hammer, Precise numerical results for limit cycles in the quantum three-body problem, Annals Phys. 321 (2006) 225–259, doi:10.1016/j.aop.2005.10.002, nucl-th/0509076

    2006 arXiv

  22. [30]

    Dolors Eiras, Joan Soto, Light fermion finite mass effects in non-relativistic bound states, Phys. Lett. B 491 (2000) 101–110, doi: 10.1016/ S0370-2693(00)01004-2, hep-ph/0005066

    2000 arXiv

  23. [31]

    G. H ¨ohler, Methods and Results of Phenomenological Analyses / Methoden und Ergebnisse ph ¨anomenologischer Analysen, Landolt- Boernstein - Group I Elementary Particles, Nuclei and Atoms, vol. 9b2, Springer 1983, ISBN 978-3-540-11282-2, 978-3-540-39059-6, doi:10.1007/b19946

    1983 doi

  24. [32]

    Gasser, M

    J. Gasser, M. A. Ivanov, E. Lipartia, M. Mojzis, A. Rusetsky, Ground state energy of pionic hydrogen to one loop, Eur. Phys. J. C 26 (2002) 13–34, doi:10.1007/s10052-002-1013-z , hep-ph/0206068

    2002 arXiv

  25. [33]

    Peter Zemp, Pionic hydrogen in QCD + QED: Decay width at NNLO, Phd thesis 2004

    2004

  26. [34]

    Meißner, Isospin Violation in Low-Energy Pion-Nucleon Scattering Revisited, Nucl

    Martin Hoferichter, Bastian Kubis, Ulf G. Meißner, Isospin Violation in Low-Energy Pion-Nucleon Scattering Revisited, Nucl. Phys. A 833 (2010) 18–103, doi:10.1016/j.nuclphysa.2009.11.012, 0909.4390. 16 Hadronic atoms

    2010 arXiv

  27. [35]

    Meißner, Isospin breaking in the pion-nucleon scattering lengths, Phys

    Martin Hoferichter, Bastian Kubis, Ulf-G. Meißner, Isospin breaking in the pion-nucleon scattering lengths, Phys. Lett. B 678 (2009) 65–71, doi:10.1016/j.physletb.2009.05.068, 0903.3890

    2009 arXiv

  28. [36]

    V. Baru, C. Hanhart, M. Hoferichter, B. Kubis, A. Nogga, D. R. Phillips, Precision calculation of threshold π−d scattering, πN scattering lengths, and the GMO sum rule, Nucl. Phys. A 872 (2011) 69–116, doi:10.1016/j.nuclphysa.2011.09.015, 1107.5509

    2011 arXiv

  29. [37]

    Meißner, Extracting the σ-term from low-energy pion-nucleon scattering, J

    Jacobo Ruiz de Elvira, Martin Hoferichter, Bastian Kubis, Ulf-G. Meißner, Extracting the σ-term from low-energy pion-nucleon scattering, J. Phys. G 45 (2) (2018) 024001, doi:10.1088/1361-6471/aa9422, 1706.01465

    2018 arXiv

  30. [38]

    Meißner, Roy-Steiner-equation analysis of pion-nucleon scattering, Phys

    Martin Hoferichter, Jacobo Ruiz de Elvira, Bastian Kubis, Ulf-G. Meißner, Roy-Steiner-equation analysis of pion-nucleon scattering, Phys. Rept. 625 (2016) 1–88, doi:10.1016/j.physrep.2016.02.002, 1510.06039

    2016 arXiv

  31. [39]

    Meißner, Remarks on the pion-nucleon σ-term, Phys

    Martin Hoferichter, Jacobo Ruiz de Elvira, Bastian Kubis, Ulf-G. Meißner, Remarks on the pion-nucleon σ-term, Phys. Lett. B 760 (2016) 74–78, doi:10.1016/j.physletb.2016.06.038, 1602.07688

    2016 arXiv

  32. [40]

    Meißner, On the role of isospin violation in the pion-nucleon σ-term, Phys

    Martin Hoferichter, Jacobo Ruiz de Elvira, Bastian Kubis, Ulf-G. Meißner, On the role of isospin violation in the pion-nucleon σ-term, Phys. Lett. B 843 (2023) 138001, doi:10.1016/j.physletb.2023.138001, 2305.07045

    2023

  33. [41]

    Meißner, Udit Raha, Akaki Rusetsky, Spectrum and decays of kaonic hydrogen, Eur

    Ulf G. Meißner, Udit Raha, Akaki Rusetsky, Spectrum and decays of kaonic hydrogen, Eur. Phys. J. C 35 (2004) 349–357, doi:10.1140/epjc/ s2004-01859-4, hep-ph/0402261

    2004 arXiv

  34. [42]

    Lipartia, Valery E

    E. Lipartia, Valery E. Lyubovitskij, A. Rusetsky, Hadronic potentials from effective field theories, Phys. Lett. B 533 (2002) 285–293, doi: 10.1016/S0370-2693(02)01660-X, hep-ph/0110186

    2002 arXiv

  35. [43]

    T. L. Trueman, Energy level shifts in atomic states of strongly-interacting particles, Nucl. Phys. 26 (1961) 57–67, doi: 10.1016/0029-5582(61) 90115-8

    1961 doi

  36. [44]

    Deloff, Fundamentals in hadronic atom theory 2003

    A. Deloff, Fundamentals in hadronic atom theory 2003

    2003

  37. [45]

    C. J. Batty, E. Friedman, A. Gal, Strong interaction physics from hadronic atoms, Phys. Rept. 287 (1997) 385–445, doi: 10.1016/ S0370-1573(97)00011-2

    1997

  38. [46]

    G. C. Oades, G. Rasche, William S. Woolcock, E. Matsinos, A. Gashi, Determination of the s-wave pion-nucleon threshold scattering parameters from the results of experiments on pionic hydrogen, Nucl. Phys. A 794 (2007) 73–86, doi: 10.1016/j.nuclphysa.2007.07.007, hep-ph/0702258

    2007 arXiv

  39. [47]

    Gashi, G

    A. Gashi, G. C. Oades, G. Rasche, William S. Woolcock, New light on electromagnetic corrections to the scattering parameters obtained from experiments on pionium, Nucl. Phys. A 699 (2002) 732–742, doi:10.1016/S0375-9474(01)01292-1, hep-ph/0108116

    2002 arXiv

  40. [48]

    Gashi, G

    A. Gashi, G. Rasche, G. C. Oades, William S. Woolcock, The Extraction of hadronic parameters from experiments on pionium, Nucl. Phys. A 628 (1998) 101–111, doi:10.1016/S0375-9474(97)00645-3, nucl-th/9704017

    1998 arXiv

  41. [49]

    Loiseau, S

    Torleif Erik Oskar Ericson, B. Loiseau, S. Wycech, A phenomenological pi- p scattering length from pionic hydrogen, Phys. Lett. B 594 (2004) 76–86, doi:10.1016/j.physletb.2004.05.009

    2004 doi

  42. [50]

    Krewald, Richard H

    S. Krewald, Richard H. Lemmer, F . P . Sassen, Lifetime of kaonium, Phys. Rev. D 69 (2004) 016003, doi: 10.1103/PhysRevD.69.016003, hep-ph/0307288

    2004 arXiv

  43. [51]

    Jallouli, H

    H. Jallouli, H. Sazdjian, Energy and decay width of the πK atom, Eur. Phys. J. C 48 (2006) 561–578, doi: 10.1140/epjc/s10052-006-0076-7 , hep-ph/0605253

    2006 arXiv

  44. [52]

    Jallouli, H

    H. Jallouli, H. Sazdjian, Relativistic effects in the pionium lifetime, Phys. Rev. D 58 (1998) 014011, doi: 10.1103/PhysRevD.58.099901, [Erra- tum: Phys.Rev.D 58, 099901 (1998)], hep-ph/9706450

    1998 arXiv

  45. [53]

    Ivanov, Valery E

    Mikhail A. Ivanov, Valery E. Lyubovitskij, E. Z. Lipartia, A. G. Rusetsky, pi+ pi- atom in chiral perturbation theory, Phys. Rev. D 58 (1998) 094024, doi:10.1103/PhysRevD.58.094024, hep-ph/9805356

    1998 arXiv

  46. [54]

    Lyubovitskij, Akaki Rusetsky, Lifetime of π+π− atom: Analysis of the role of strong interactions, Phys

    Valery E. Lyubovitskij, Akaki Rusetsky, Lifetime of π+π− atom: Analysis of the role of strong interactions, Phys. Lett. B 389 (1996) 181, doi:10.1016/S0370-2693(96)01431-1, hep-ph/9610217

    1996 arXiv

  47. [55]

    H. W. Hammer, J. N. Ng, Rare pionium decays and pion polarizability, Eur. Phys. J. A 6 (1999) 115–118, doi: 10.1007/s100500050324, hep-ph/9902284

    1999 arXiv

  48. [56]

    Xinwei Kong, Finn Ravndal, Nonrelativistic pion interactions and the pionium lifetime, Phys. Rev. D 59 (1999) 014031, doi:10.1103/PhysRevD. 59.014031

    1999 doi

  49. [57]

    Xin-wei Kong, Finn Ravndal, Relativistic corrections to the pionium lifetime, Phys. Rev. D 61 (2000) 077506, doi: 10.1103/PhysRevD.61. 077506, hep-ph/9905539

    2000 arXiv

  50. [58]

    Holstein, Hadronic atoms and effective interactions, Phys

    Barry R. Holstein, Hadronic atoms and effective interactions, Phys. Rev. D 60 (1999) 114030, doi: 10.1103/PhysRevD.60.114030, nucl-th/ 9901041

    1999 doi

  51. [59]

    Gasser, Valery E

    J. Gasser, Valery E. Lyubovitskij, A. Rusetsky, Numerical analysis of the π+π− atom lifetime in ChPT, Phys. Lett. B 471 (1999) 244–250, doi:10.1016/S0370-2693(99)01334-9, hep-ph/9910438

    1999 arXiv

  52. [60]

    Patrick Labelle, Kirk Buckley, A New order α correction to the decay rate of pionium (1998), hep-ph/9804201

    1998 arXiv

  53. [61]

    Eiras, J

    D. Eiras, J. Soto, Effective field theory approach to pionium, Phys. Rev. D 61 (2000) 114027, doi: 10.1103/PhysRevD.61.114027, hep-ph/ 9905543

    2000 doi

  54. [62]

    Meißner, Udit Raha, Akaki Rusetsky, Kaon-nucleon scattering lengths from kaonic deuterium experiments, Eur

    Ulf-G. Meißner, Udit Raha, Akaki Rusetsky, Kaon-nucleon scattering lengths from kaonic deuterium experiments, Eur. Phys. J. C 47 (2006) 473–480, doi:10.1140/epjc/s2006-02578-6, nucl-th/0603029

    2006 arXiv

  55. [63]

    Meißner, Udit Raha, Akaki Rusetsky, The Pion-nucleon scattering lengths from pionic deuterium, Eur

    Ulf-G. Meißner, Udit Raha, Akaki Rusetsky, The Pion-nucleon scattering lengths from pionic deuterium, Eur. Phys. J. C 41 (2005) 213–232, doi:10.1140/epjc/s2005-02182-4, [Erratum: Eur.Phys.J.C 45, 545 (2006)], nucl-th/0501073

    2005 arXiv

  56. [64]

    Schweizer, Spectra and decays of ππ and πK atoms, Eur

    J. Schweizer, Spectra and decays of ππ and πK atoms, Eur. Phys. J. C 36 (4) (2004) 483–501, doi: 10.1140/epjc/s2004-01959-1, hep-ph/ 0405034

    2004 doi

  57. [65]

    Gasser, V

    J. Gasser, V. E. Lyubovitskij, A. Rusetsky, Hadronic Atoms, Ann. Rev. Nucl. Part. Sci. 59 (2009) 169–190, doi: 10.1146/annurev.nucl.010909. 083806, 0903.0257

    2009 arXiv

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