nLab quantum hadrodynamics

Idea

Quantum hadrodynamics is (or should be) an effective field theory that describes confined hadron bound states in nuclear physics, in particular the interaction between mesons and baryons and specifically with nucleons.

In this sense chiral perturbation theory is the archetype of quantum hadrodynamics, but the latter term is commonly used more specifically for the Walecka model and its generalizations (QHD-I Walecka 74 and QHD-II Serot 79, Serot-Walecka 92, Serot-Walecka 97) to baryon chiral perturbation theory, where the baryons and specifically the nucleons appear as explicit effective fields (as opposed to as emergent Skyrmion fields).

Related concepts

• Walecka model

• hadron supersymmetry, hadron Kaluza-Klein theory

  • flavor branes

• leptonic decay

effective field theories of nuclear physics, hence for confined-phase quantum chromodynamics:

• with effective ligh meson fields:

  • chiral perturbation theory

    • vector meson dominance

    • hidden local symmetry

  • with emergent (solitonic) baryon fields:

    • quark bag model

    • Skyrme model

    • Witten-Sakai-Sugimoto model

  • with explicit effective baryon fields:

    • baryon chiral perturbation theory

    • quantum hadrodynamics

      • Walecka model

  • with explicit quark fields:

    • quark-meson coupling model

References

See also the references at chiral perturbation theory

Walecka hadrodynamics with nucleon fields

On quantum hadrodynamics (relativivist effective field theory of nuclear physics, coupling mesons and nucleons) in the sense of the Walecka model, hence with nucleons appearing as explicit fields (as opposed to being solitonic Skyrmions in the pion field as in chiral perturbation theory).

Precursor:

• Hans-Peter Duerr, Relativistic Effects in Nuclear Forces, Phys. Rev. 103, 469 (1956) (doi:10.1103/PhysRev.103.469)

The original Walecka model (QHD-I model), with nucleons coupled to sigma-mesons and omega-mesons:

• John Dirk Walecka, A Theory of highly condensed matter, Annals Phys. 83 (1974) 491 (spire:91609, dpi:10.1016/0003-4916(74)90208-5)

Inclusion into the Walecka model also of the pion and the rho-meson (the QHD-II model):

• Brian Serot, A relativistic nuclear field theory with $\pi$ and $\rho$ mesons, Physics Letters B Volume 86, Issue 2, 24 (1979), Pages 146-150 (doi:10.1016/0370-2693(79)90804-9)

• T Matsui, Brian Serot, The pion propagator in relativistic quantum field theories of the nuclear many-body problem, Annals of Physics Volume 144, Issue 1, November 1982, Pages 107-167 (doi:10.1016/0003-4916(82)90106-3)

Further discussion of these models:

• S. A. Chin, A relativistic many-body theory of high density matter, Annals of Physics Volume 108, Issue 2, October 1977, Pages 301-367 (doi:10.1016/0003-4916(77)90016-1)

• Brian Serot, John Dirk Walecka, The Relativistic Nuclear Many Body Problem, Adv. Nucl. Phys. 16 (1986) 1-327 (spire:207866)

• Brian Serot, Quantum hadrodynamics, Reports on Progress in Physics, Volume 55, Number 11 (1992) (doi:10.1088/0034-4885/55/11/001)

• Brian Serot, John Dirk Walecka, Chiral QHD with vector mesons, Acta Phys. Polon. B 23 (1992) 655-679 (spire:343513)

• Maciej Nowak, Mannque Rho, Ismail Zahed, Chiral Nuclear Dynamics, World Scientific 1996 (doi:10.1142/1681)

• Brian Serot, John Dirk Walecka, Recent Progress in Quantum Hadrodynamics, Int. J. Mod. Phys. E6:515-631, 1997 (arXiv:nucl-th/9701058)

• R. V. Poberezhnyuk, V. Vovchenko, D. V. Anchishkin, M. I. Gorenstein, Quantum van der Waals and Walecka models of nuclear matter (arXiv:1708.05605)

Further inclusion of electromagnetism (photon field):

• A. Yu. Korchin, D. Van Neck, M. Waroquier, Electromagnetic interaction in chiral quantum hadrodynamics and decay of vector and axial-vector mesons, Phys. Rev. C67 (2003) 015207 (arXiv:nucl-th/0302042)

Relation to quark-meson coupling model:

• Koichi Saito, Relationship between Quark-Meson Coupling Model and Quantum Hadrodynamics, Prog. Theor. Phys. 108 (2002) 609-614 (arXiv:nucl-th/0207053)

Hadrodynamics via Light-front QCD

On light-front QCD for quantum hadrodynamics:

• International Light Cone Advisory Committee, Light-Front Quantum Chromodynamics: A framework for the analysis of hadron physics, Nuclear Physics B - Proceedings Supplements Volumes 251–252, June–July 2014, Pages 165-174 (arXiv:1309.6333, doi:10.1016/j.nuclphysbps.2014.05.004)

• Edward Shuryak, Ismail Zahed, Hadronic structure on the light-front I: Instanton effects and quark-antiquark effective potentials (arXiv:2110.15927)

• Edward Shuryak, Ismail Zahed, Hadronic structure on the light-front II: QCD strings, Wilson lines and potentials (arXiv:2111.01775)

Skyrme hadrodynamics with vector mesons ($\pi$-$\omega$-$\rho$-model)

Inclusion of vector mesons (omega-meson and rho-meson/A1-meson) into the Skyrmion model of quantum hadrodynamics, in addition to the pion:

First, on the equivalence between hidden local symmetry- and massive Yang-Mills theory-description of Skyrmion quantum hadrodynamics:

• Atsushi Hosaka, H. Toki, Wolfram Weise, Skyrme Solitons With Vector Mesons: Equivalence of the Massive Yang-Mills and Hidden Local Symmetry Scheme, 1988, Z. Phys. A332 (1989) 97-102 (spire:24079)

See also

• Marcelo Ipinza, Patricio Salgado-Rebolledo, Meron-like topological solitons in massive Yang-Mills theory and the Skyrme model (arXiv:2005.04920)

Inclusion of the $\omega$-meson

Original proposal for inclusion of the ω-meson in the Skyrme model:

• Gregory Adkins, Chiara Nappi, Stabilization of Chiral Solitons via Vector Mesons, Phys. Lett. 137B (1984) 251-256 (spire:194727, doi:10.1016/0370-2693(84)90239-9)

Relating to nucleon-scattering:

• J. M. Eisenberg, A. Erell, R. R. Silbar, Nucleon-nucleon force in a skyrmion model stabilized by omega exchange, Phys. Rev. C 33, 1531 (1986) (doi:10.1103/PhysRevC.33.1531)

Combination of the omega-meson-stabilized Skyrme model with the bag model for nucleons:

• Atsushi Hosaka, Omega stabilized chiral bag model with a surface $\omega q q$ coupling, Nuclear Physics A Volume 546, Issue 3, 31 (1992) Pages 493-508 (doi:10.1016/0375-9474(92)90544-T)

Discussion of nucleon phenomenology for the $\omega$-stabilized Skyrme model:

• Sven Bjarke Gudnason, James Martin Speight, Realistic classical binding energies in the $\omega$-Skyrme model (arXiv:2004.12862)

• Derek Harland, Paul Leask, Martin Speight, Skyrmion crystals stabilized by $\omega$-mesons [arXiv:2404.11287]

Inclusion of the $\rho$-meson

Original proposal for inclusion of the ρ-meson:

• Y. Igarashi, M. Johmura, A. Kobayashi, H. Otsu, T. Sato, S. Sawada, Stabilization of Skyrmions via $\rho$-Mesons, Nucl.Phys. B259 (1985) 721-729 (spire:213451, doi:10.1016/0550-3213(85)90010-0)

• Gregory Adkins, Rho mesons in the Skyrme model, Phys. Rev. D 33, 193 (1986) (spire:16895, doi:10.1103/PhysRevD.33.193)

Discussion for phenomenology of light atomic nuclei:

• Carlos Naya, Paul Sutcliffe, Skyrmions and clustering in light nuclei, Phys. Rev. Lett. 121, 232002 (2018) (arXiv:1811.02064)

• Carlos Naya, Paul Sutcliffe, Skyrmions in models with pions and rho, JHEP 05 (2018) 174 (arXiv:1803.06098)

  APS Synopsis: Revamping the Skyrmion Model, 2018

See also:

• Miguel Huidobro, Paul Leask, Carlos Naya, Andrzej Wereszczynski, Compressibility of dense nuclear matter in the $\rho$-meson variant of the Skyrme model [arXiv:2405.20757]

Inclusion of the $\omega$- and $\rho$-meson

The resulting $\pi$-$\rho$-$\omega$ model:

• Ulf-G. Meissner, Ismail Zahed, Skyrmions in the Presence of Vector Mesons, Phys. Rev. Lett. 56, 1035 (1986) (doi:10.1103/PhysRevLett.56.1035)

  (includes also the A1-meson)

• Ulf-G. Meissner, Norbert Kaiser, Wolfram Weise, Nucleons as skyrme solitons with vector mesons: Electromagnetic and axial properties, Nuclear Physics A Volume 466, Issues 3–4, 11–18 May 1987, Pages 685-723 (doi:10.1016/0375-9474(87)90463-5)

• Ulf-G. Meissner, Norbert Kaiser, Andreas Wirzba, Wolfram Weise, Skyrmions with $\rho$ and $\omega$ Mesons as Dynamical Gauge Bosons, Phys. Rev. Lett. 57, 1676 (1986) (doi:10.1103/PhysRevLett.57.1676)

• Ulf-G. Meissner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons, Physics Reports Volume 161, Issues 5–6, May 1988, Pages 213-361 (doi:10.1016/0370-1573(88)90090-7)

• L. Zhang, Nimai C. Mukhopadhyay, Baryon physics from mesons: Leading order properties of the nucleon and $\Delta(1232)$ in the $\pi \rho\omega a_1(f_1)$ chiral soliton model, Phys. Rev. D 50, 4668 (1994) (doi:10.1103/PhysRevD.50.4668, spire:384906)

• Yong-Liang Ma, Ghil-Seok Yang, Yongseok Oh, Masayasu Harada, Skyrmions with vector mesons in the hidden local symmetry approach, Phys. Rev. D87:034023, 2013 (arXiv:1209.3554)

  (hidden local symmetry)

• Ju-Hyun Jung, Ulugbek T. Yakhshiev, Hyun-Chul Kim, In-medium modified $\pi$-$\rho$-$\omega$ mesonic Lagrangian and properties of nuclear matter, Physics Letters B Volume 723, Issues 4–5, 25 June 2013, Pages 442-447 (arXiv:1212.4616, doi:10.1016/j.physletb.2013.05.042)

• Ju-Hyun Jung, Ulugbek Yakhshiev, Hyun-Chul Kim, Peter Schweitzerm, In-medium modified energy-momentum tensor form factors of the nucleon within the framework of a $\pi$-$\rho$-$\omgea$ soliton model, Phys. Rev. D 89, 114021 (2014) (arXiv:1402.0161)

• Yongseok Oh, Skyrmions with vector mesons revisited (arXiv:1402.2821)

See also

• Ki-Hoon Hong, Ulugbek Yakhshiev, Hyun-Chul Kim, Modification of hyperon masses in nuclear matter, Phys. Rev. C 99, 035212 (2019) (arXiv:1806.06504)

Review:

• Roland Kaiser, Anomalies and WZW-term of two-flavour QCD, Phys. Rev. D63:076010, 2001 (arXiv:hep-ph/0011377, spire:537600)

• Gottfried Holzwarth, Section 2.3 of: Electromagnetic Form Factors of the Nucleon in Chiral Soliton Models (arXiv:hep-ph/0511194), Chapter 2 in: The Multifaceted Skyrmion, World Scientific 2016 (doi:10.1142/9710)

• Yongseok Oh, Skyrmions with vector mesons: Single Skyrmion and baryonic matter, 2013 (pdf)

Combination of the omega-rho-Skyrme model with the bag model of quark confinement:

• H. Takashita, S. Yoro, H. Toki, Chiral bag plus skyrmion hybrid model with vector mesons for nucleon, Nuclear Physics A Volume 485, Issues 3–4, August 1988, Pages 589-605 (doi:10.1016/0375-9474(88)90555-6)

Inclusion of the $\sigma$-meson

Inclusion of the sigma-meson:

• Thomas D. Cohen, Explicit $\sigma$ meson, topology, and the large-$N$ limit of the Skyrmion, Phys. Rev. D 37 (1988) (doi:10.1103/PhysRevD.37.3344)

For analysis of neutron star equation of state:

• David Alvarez-Castillo, Alexander Ayriyan, Gergely Gábor Barnaföldi, Hovik Grigorian, Péter Pósfay, Studying the parameters of the extended $\sigma$-$\omega$ model for neutron star matter (arXiv:2006.03676)

Skyrme hadrodynamics with heavy quarks/mesons

Inclusion of heavy flavors into the Skyrme model for quantum hadrodynamics:

Inclusion of strange quarks/kaons

Inclusion of strange quarks/kaons into the Skyrme model:

• Curtis Callan, Igor Klebanov, Bound-state approach to strangeness in the Skyrme model, Nuclear Physics B Volume 262, Issue 2, 16 December 1985, Pages 365-382 (doi10.1016/0550-3213(85)90292-5)

• Curtis Callan, K. Hornbostel, Igor Klebanov, Baryon masses in the bound state approach to strangeness in the skyrme model, Physics Letters B Volume 202, Issue 2, 3 March 1988, Pages 269-275 (doi10.1016/0370-2693(88)90022-6)

• Norberto Scoccola, D. P. Min, H. Nadeau, Mannque Rho, The strangeness problem: An $SU(3)$ skyrmion with vector mesons, Nuclear Physics A Volume 505, Issues 3–4, 25 December 1989, Pages 497-524 (doi:10.1016/0375-9474(89)90029-8)

Review:

• Igor Klebanov, section 6 of: Strangeness in the Skyrme model, in: D. Vauthrin, F. Lenz, J. W. Negele, Hadrons and Hadronic Matter, Plenum Press 1989 (doi:10.1007/978-1-4684-1336-6)

• Mannque Rho, Section 2.2 of: Cheshire Cat Hadrons, Phys. Rept. 240 (1994) 1-142 (arXiv:hep-ph/9310300, doi:10.1016/0370-1573(94)90002-7)

Inclusion of charm quarks/D-mesons

Inclusion of charm quarks/D-mesons into the Skyrme model:

• Mannque Rho, D. O. Riska, Norberto Scoccola, Charmed baryons as soliton - D meson bound states, Phys. Lett.B 251 (1990) 597-602 (spire:297771, doi:10.1016/0370-2693(90)90802-D)

• Yongseok Oh, Dong-Pil Min, Mannque Rho, Norberto Scoccola, Massive-quark baryons as skyrmions: Magnetic moments, Nuclear Physics A Volume 534, Issues 3–4 (1991) Pages 493-512 (doi:10.1016/0375-9474(91)90458-I)

Inclusion of bottom quarks/B-mesons

Inclusion of further heavy flavors beyond strange quark/kaons, namely charm quarks/D-mesons and bottom quarks/B-mesons, into the Skyrme model:

• Mannque Rho, D. O. Riska, Norberto Scoccola, The energy levels of the heavy flavour baryons in the topological soliton model, Zeitschrift für Physik A Hadrons and Nuclei volume 341, pages 343–352 (1992) (doi:10.1007/BF01283544)

• Arshad Momen, Joseph Schechter, Anand Subbaraman, Heavy Quark Solitons: Strangeness and Symmetry Breaking, Phys. Rev. D49:5970-5978, 1994 (arXiv:hep-ph/9401209)

• Yongseok Oh, Byung-Yoon Park, Dong-Pil Min, Heavy Baryons as Skyrmion with $1/m_Q$ Corrections, Phys. Rev. D49 (1994) 4649-4658 (arXiv:hep-ph/9402205)

Review:

• Mannque Rho, Massive-quark baryons as Skyrmions, Modern Physics Letters A, Vol. 06, No. 23 (1991) (doi:10.1142/S0217732391002268)

• Norberto Scoccola, Heavy quark skyrmions, (arXiv:0905.2722, doi:10.1142/9789814280709_0004), Chapter 4 in: The Multifaceted Skyrmion, World Scientific 2016 (doi:10.1142/9710)

The WZW term of QCD chiral perturbation theory

The gauged WZW term of chiral perturbation theory/quantum hadrodynamics which reproduces the chiral anomaly of QCD in the effective field theory of mesons and Skyrmions:

General

The original articles:

• Julius Wess, Bruno Zumino, Consequences of anomalous Ward identities, Phys. Lett. B 37 (1971) 95-97 (spire:67330, doi:10.1016/0370-2693(71)90582-X)

• Edward Witten, Global aspects of current algebra, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 422-432 (doi:10.1016/0550-3213(83)90063-9)

See also:

• O. Kaymakcalan, S. Rajeev, J. Schechter, Nonabelian Anomaly and Vector Meson Decays, Phys. Rev. D 30 (1984) 594 (spire:194756)

Corrections and streamlining of the computations:

• Chou Kuang-chao, Guo Han-ying, Wu Ke, Song Xing-kang, On the gauge invariance and anomaly-free condition of the Wess-Zumino-Witten effective action, Physics Letters B Volume 134, Issues 1–2, 5 January 1984, Pages 67-69 (doi:10.1016/0370-2693(84)90986-9))

• H. Kawai, S.-H. H. Tye, Chiral anomalies, effective lagrangians and differential geometry, Physics Letters B Volume 140, Issues 5–6, 14 June 1984, Pages 403-407 (doi:10.1016/0370-2693(84)90780-9)

• J. L. Mañes, Differential geometric construction of the gauged Wess-Zumino action, Nuclear Physics B Volume 250, Issues 1–4, 1985, Pages 369-384 (doi:10.1016/0550-3213(85)90487-0)

• Tomáš Brauner, Helena Kolešová, Gauged Wess-Zumino terms for a general coset space, Nuclear Physics B Volume 945, August 2019, 114676 (doi:10.1016/j.nuclphysb.2019.114676)

See also

• Yasunori Lee, Kantaro Ohmori, Yuji Tachikawa, Revisiting Wess-Zumino-Witten terms (arXiv:2009.00033)

Interpretation as Skyrmion/baryon current:

• Jeffrey Goldstone, Frank Wilczek, Fractional Quantum Numbers on Solitons, Phys. Rev. Lett. 47, 986 (1981) (doi:10.1103/PhysRevLett.47.986)

• Edward Witten, Current algebra, baryons, and quark confinement, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 433-444 (doi:10.1016/0550-3213(83)90064-0)

• Gregory Adkins, Chiara Nappi, Stabilization of Chiral Solitons via Vector Mesons, Phys. Lett. 137B (1984) 251-256 (spire:194727, doi:10.1016/0370-2693(84)90239-9)

  (beware that the two copies of the text at these two sources differ!)

• Mannque Rho et al., Introduction, In: Mannque Rho et al. (eds.) The Multifaceted Skyrmion, World Scientific 2016 (doi:10.1142/9710)

Concrete form for $N$-flavor quantum hadrodynamics in 2d:

• C. R. Lee, H. C. Yen, A Derivation of The Wess-Zumino-Witten Action from Chiral Anomaly Using Homotopy Operators, Chinese Journal of Physics, Vol 23 No. 1 (1985) (spire:16389, pdf)

Concrete form for 2 flavors in 4d:

• Masashi Wakamatsu, On the electromagnetic hadron current derived from the gauged Wess-Zumino-Witten action, (arXiv:1108.1236, spire:922302)

Including light vector mesons

Concrete form for 2-flavor quantum hadrodynamics in 4d with light vector mesons included (omega-meson and rho-meson):

• Ulf-G. Meissner, Ismail Zahed, equation (6) in: Skyrmions in the Presence of Vector Mesons, Phys. Rev. Lett. 56, 1035 (1986) (doi:10.1103/PhysRevLett.56.1035)

• Ulf-G. Meissner, Norbert Kaiser, Wolfram Weise, equation (2.18) in: Nucleons as skyrme solitons with vector mesons: Electromagnetic and axial properties, Nuclear Physics A Volume 466, Issues 3–4, 11–18 May 1987, Pages 685-723 (doi:10.1016/0375-9474(87)90463-5)

• Ulf-G. Meissner, equation (2.45) in: Low-energy hadron physics from effective chiral Lagrangians with vector mesons, Physics Reports Volume 161, Issues 5–6, May 1988, Pages 213-361 (doi:10.1016/0370-1573(88)90090-7)

• Roland Kaiser, equation (12) in: Anomalies and WZW-term of two-flavour QCD, Phys. Rev. D63:076010, 2001 (arXiv:hep-ph/0011377, spire:537600)

Including heavy scalar mesons

Including heavy scalar mesons:

specifically kaons:

• Curtis Callan, Igor Klebanov, equation (4.1) in: Bound-state approach to strangeness in the Skyrme model, Nuclear Physics B Volume 262, Issue 2, 16 December 1985, Pages 365-382 (doi10.1016/0550-3213(85)90292-5)

• Igor Klebanov, equation (99) of: Strangeness in the Skyrme model, in: D. Vauthrin, F. Lenz, J. W. Negele, Hadrons and Hadronic Matter, Plenum Press 1989 (doi:10.1007/978-1-4684-1336-6)

• N. N. Scoccola, D. P. Min, H. Nadeau, Mannque Rho, equation (2.20) in: The strangeness problem: An $SU(3)$ skyrmion with vector mesons, Nuclear Physics A Volume 505, Issues 3–4, 25 December 1989, Pages 497-524 (doi:10.1016/0375-9474(89)90029-8)

specifically D-mesons:

(…)

specifically B-mesons:

• Mannque Rho, D. O. Riska, N. N. Scoccola, above (2.1) in: The energy levels of the heavy flavour baryons in the topological soliton model, Zeitschrift für Physik A Hadrons and Nuclei volume 341, pages343–352 (1992) (doi:10.1007/BF01283544)

Including heavy vector mesons

Inclusion of heavy vector mesons:

specifically K*-mesons:

• S. Ozaki, H. Nagahiro, Atsushi Hosaka, Equations (3) and (9) in: Magnetic interaction induced by the anomaly in kaon-photoproductions, Physics Letters B Volume 665, Issue 4, 24 July 2008, Pages 178-181 (arXiv:0710.5581, doi:10.1016/j.physletb.2008.06.020)

Including electroweak interactions

Including electroweak fields:

• J. Bijnens, G. Ecker, A. Picha, The chiral anomaly in non-leptonic weak interactions, Physics Letters B Volume 286, Issues 3–4, 30 July 1992, Pages 341-347 (doi:10.1016/0370-2693(92)91785-8)

• Gerhard Ecker, Helmut Neufeld, Antonio Pich, Non-leptonic kaon decays and the chiral anomaly, Nuclear Physics B Volume 413, Issues 1–2, 31 January 1994, Pages 321-352 (doi:10.1016/0550-3213(94)90623-8)

Discussion for the full standard model of particle physics:

• Jeffrey Harvey, Christopher T. Hill, Richard J. Hill, Standard Model Gauging of the WZW Term: Anomalies, Global Currents and pseudo-Chern-Simons Interactions, Phys. Rev. D77:085017, 2008 (arXiv:0712.1230)

Baryon chiral perturbation theory

Discussion of baryon chiral perturbation theory, i.e of chiral perturbation theory with explicit effective (as opposed to or in addition to implicit skyrmionic) baryon fields included (see also Walecka model and quantum hadrodynamics):

Review:

• Ulf-G. Meissner, Chiral QCD: Baryon dynamics, in: At The Frontier of Particle Physics, pp. 417-505 (2001) (arxiv:hep-ph/0007092)

• Véronique Bernard, Chiral Perturbation Theory and Baryon Properties, Prog. Part. Nucl. Phys. 60:82-160, 2008 (arXiv:0706.0312)

• Stefan Scherer, Baryon chiral perturbation theory, PoS CD09:075, 2009 (arXiv:0910.0331)

Original articles:

• Elizabeth Jenkins, Aneesh V. Manohar, Baryon chiral perturbation theory using a heavy fermion lagrangian, Physics Letters B Volume 255, Issue 4, 21 February 1991, Pages 558-562 (doi:10.1016/0370-2693(91)90266-S)

• Robert Baur, Joachim Kambor, Generalized Heavy Baryon Chiral Perturbation Theory, Eur. Phys. J. C7:507-524, 1999 (arXiv:hep-ph/9803311)

Higher order terms:

• José Antonio Oller, Michela Verbeni, Joaquim Prades, Meson-baryon effective chiral Lagrangians to $\mathcal{O}(q^3)$, Journal of High Energy Physics, Volume 2006, JHEP09(2006) (arXiv:hep-ph/0608204, doi:10.1088/1126-6708/2006/09/079)

• Matthias Frink, Ulf-G. Meissner, On the chiral effective meson-baryon Lagrangian at third order, Eur. Phys. J. A29:255-260, 2006 (arXiv:hep-ph/0609256)

• Jose Antonio Oller, Joaquim Prades, Michela Verbeni, Meson-Baryon Effective Chiral Lagrangians at $\mathcal{O}(q^3)$ Revisited (arXiv:hep-ph/0701096, spire:742291)

See also:

• Lisheng Geng, Recent developments in $SU(3)$ covariant baryon chiral perturbation theory, Front. Phys., 2013, 8(3): 328-348 (arXiv:1301.6815)