nLab chiral perturbation theory



Fields and quanta

fields and particles in particle physics

and in the standard model of particle physics:

force field gauge bosons

scalar bosons

matter field fermions (spinors, Dirac fields)

flavors of fundamental fermions in the
standard model of particle physics:
generation of fermions1st generation2nd generation3d generation
quarks (qq)
up-typeup quark (uu)charm quark (cc)top quark (tt)
down-typedown quark (dd)strange quark (ss)bottom quark (bb)
neutralelectron neutrinomuon neutrinotau neutrino
bound states:
mesonslight mesons:
pion (udu d)
ρ-meson (udu d)
ω-meson (udu d)
ϕ-meson (ss¯s \bar s),
kaon, K*-meson (usu s, dsd s)
eta-meson (uu+dd+ssu u + d d + s s)

charmed heavy mesons:
D-meson (uc u c, dcd c, scs c)
J/ψ-meson (cc¯c \bar c)
bottom heavy mesons:
B-meson (qbq b)
ϒ-meson (bb¯b \bar b)
proton (uud)(u u d)
neutron (udd)(u d d)

(also: antiparticles)

effective particles

hadrons (bound states of the above quarks)


in grand unified theory

minimally extended supersymmetric standard model




dark matter candidates


auxiliary fields

Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT



What is called chiral perturbation theory in quantum field theory of nuclear physics is the effective field theory of quantum chromodynamics in the confined sector, where the effective fields are hadrons:

Hence some authors also speak of quantum hadrodynamics.

More concretely, chiral perturbation theory is perturbation theory not in the coupling constant of QCD, but in the masses of the light quarks, which in practice means: of the up quark and the down quark and possibly also of the strange quark, with masses m um_u, m dm_d and m sm_s, respectively.

As such, the point about which perturbations are considered in chiral perturbation theory of QCD is that of vanishing quark masses, hence the limit m u,m s,m d0 m_u, m_s, m_d \rightarrow 0 .

Notice that the proton mass (the characteristic scale of QCD witnessing its confinement/mass gap-property, see e.g. Roberts 21) is much larger than the sum of its constituent quarks

m p2m u+m d, m_p \gg 2 m_u + m_d \,,

suggesting that m u,m d0m_u, m_d \to 0 is a good limit around which to inspect the hadronic bound states of QCD.

But since only the quark mass term in the Lagrangian density of QCD mixes the two chiralities of quarks, its Lagrangian density in this massless limit is the sum of a term that only contains the left-chiral quark spinors, and an analogous summand that only contains the right-chiral quark spinors (e.g. Ecker 95, (2.1)):

QCD 0= l=u,d,s(q¯ L li(D+A)q L l+q¯ R li(D+A)q R l)14G aμvG a μv \mathcal{L}^0_QCD \;=\; \sum_{l=u,d,s} \big( \bar{q}^l_L i (\slash{D} + \slash{A}) q^l_L + \bar{q}^l_R i (\slash{D} + \slash{A}) q^l_R \big) - \frac{1}{4}G_{a \mu v}G^{\mu v}_a

(here qq denotes the quark Dirac field with left and right chiral Weyl spinor components q Lq_L and q Rq_R, D\slash{D} denotes the Dirac operator, AA the gluon field (in Feynman slash notation), and GG the field strength of the gluon field AA – see at Yang-Mills theory for more).

Moreover, all two (up and down) or even all three (if including strange) flavours of quarks enter each of these two spinor-chiral summands symmetrically, such that each of them has SU(2) or respectively SU(3) global symmetry, acting by mixing the quark flavours. This yields a total direct product symmetry group

G=SU(N f) L×SU(N f) R, G = SU(N_f)_L \times SU(N_f)_R \,,

where N f{2,3}N_f \in \{ 2,3 \} is the number of flavours of light quarks that are being considered, and where the subscripts indicate the left/right-handed chiral spinor-components of quarks whose flavours are being mixed by these group actions. This symmetry group is hence also called chiral symmetry. Accordingly, its symmetry breaking to the diagonal subgroup

SU(N f) VSU(N f) L×SU(N f) R SU(N_f)_V \;\subset\; SU(N_f)_L \times SU(N_f)_R

(the subscript “VV” is for “vector”) either explicitly by positive quark masses or spontaneously by a vacuum expectation value q¯q=q¯ Lq R+q¯ Rq L\langle \bar q q\rangle = \langle \bar q_L q_R\rangle + \langle \bar q_R q_L\rangle, is called chiral symmetry breaking. This is the origin of the termchiral perturbation theory_.

Additionally, the application of noether's theorem? to the classical global symmetry of the lagrangian density in the chiral limit yields 2×(8+1)=18 2 \times (8+1) = 18 conserved currents, via the method of Gell-Mann? and Levy?. Namely, the promotion of the global symmetry inherent in the lagrangian density to a local symmetry allows us to identify these conserved noether currents?

J a μ=δ μϵ a μJ a μ=δϵ a J^\mu_a = \frac{\partial \delta \mathcal{L}}{\partial \partial_\mu \epsilon_a} \partial_{\mu} J^{\mu}_a = \frac{\partial \delta \mathcal{L}}{\partial \epsilon_a}



Relation to the Skyrme model

It is astounding that Skyrme had suggested his model as early as in 1961 before it has been generally accepted that pions are (pseudo) Goldstone bosons associated with the spontaneous breaking of chiral symmetry, and of course long before Quantum Chromodynamics (QCD) has been put forward as the microscopic theory of strong interactions.

The revival of the Skyrme idea in 1983 is due to Witten who explained the raison d’ˆetre of the Skyrme model from the viewpoint of QCD. In the chiral limit when the light quark masses m um_u, m dm_d, m sm_s tend to zero, such that the octet of the pseudoscalar mesons π, K , η become nearly massless (pseudo) Goldstone bosons, they are the lightest degrees of freedom of QCD. The effective chiral Lagrangian (EχL) for pseudoscalar mesons, understood as an infinite expansion in the derivatives of the pseudoscalar (or chiral) fields, encodes, in principle, full information about QCD. The famous two-term Skyrme Lagrangian can be understood as a low-energy truncation of this infinite series. Witten has added an important four-derivative Wess–Zumino term to the original Skyrme Lagrangian and pointed out that the overall coefficient in front of the EχL is proportional to the number of quark colours N cN_c.


Soon after Witten’s work it has been realized that it is possible to bring the Skyrme model and the Skyrmion even closer to QCD and to the more customary language of constituent quarks. It has been first noticed [[6, 7a, 7b, 8]] that a simple chiral invariant Lagrangian for massive (constituent) quarks QQ interacting with the octet chiral field π A\pi_A (A=1,...,8)(A = 1, ..., 8),

=Q¯(/Me iπ Aλ Aγ 5F π)Q\mathcal{L} = \overline{Q} \left( \partial\!\!\!\!/ - M e^{ \tfrac{i \pi^A \lambda^A \gamma_5}{F_\pi} } \right) Q

induces, via a quark loop in the external pseudoscalar fields (see Fig. 3.1), the EχL whose lowest-derivative terms coincide with the Skyrme Lagrangian, including automatically the Wess–Zumino term, with the correct coefficient!


The condition that the winding number of the trial field is unity needs to be imposed to get a deeply bound state, that is to guarantee that the baryon number is unity. [[9]] The Skyrmion is, thus, nothing but the mean chiral field binding quarks in a baryon.

Baryon Chiral Perturbation Theory

The QCD vacuum breaks chiral symmetry down to the diagonal subgroup of SU(3), U(3) vU(3)_v, whereby 8 massless goldstone bosons appear, each of which are coupled via F 0F_0 to the conserved axial-vector current. The physics of these goldstone bosons, which are themselves the pion fields, describe the low energry effective field theory known as chiral perturbation theory. In this low energy structure of QCD, heavy quarks do not play a role since their degrees of freedom are frozen at low energies. However, when the baryon fields are treated as heavy static fermions in the expansion of the physical vacuum state of QCD, one can write the effective theory in terms of baryon fields with a definite velocity, B vB_v. In the heavy baryon limit, the most general Lagrangian at leading-order is

L v 0=iTrB¯ v(v*𝒟)B v+2DTrB¯ vS v μ{A μ,B v}+2FTrB¯ vS v μ{A μ,B v}+14f 2Tr μΣ μΣ +aTrM(Σ+Σ ) L_v^0 = iTr\overline{B}_v(v*\mathscr{D})B_v + 2DTr\overline{B}_v S_v^\mu \{A_\mu,B_v\} + 2FTr\overline{B}_v S_v^\mu \{A_\mu,B_v\} + \frac{1}{4}f^2Tr\partial_{\mu}\Sigma\partial^\mu \Sigma^\dagger + aTrM(\Sigma + \Sigma^\dagger)

Partially Quenched Chiral Perturbation Theory

In Lattice QCD?, it is notoriously difficult to include the loops of light quarks. One remedy to this problem is to approximate the fermion determinant of a Dirac operator DD, which is proportional to the summation of external sources interacting with one internal fermion loop?. The partially quenched prescription includes the determinant but with sea quark? masses \ggg those of the valence quarks. The simulation of partially quenched QCD (PQQCD) on the lattice can yield information about QCD itself by utilizing chiral perturbation theory. The formulation of XPT in PQQCD is difficult compared to the unquenched case (standard formulation) due to the lack of a physical hilbert space?.

  • S.R.Sharpe and N.Shoresh, Partially quenched chiral perturbation theory without Phi0, Phys. Rev. D 64, 114510 (2001), doi:10.1103/PhysRevD.64.114510, (arXiv:hep-lat/0108003)

  • Claude W. Bernard and Maarten F. L. Golterman, Partially quenched gauge theories and an application to staggered fermions, Phys. Rev. D 49, 486 – Published 1 January 1994, (doi:10.1103/486)

Random Matrix Theory and Chiral Symmetry

It has been posited that the statistics of the low-lying eigenvalues of the QCD Dirac operator can be described by a random matrix theory with the global symmetries of the QCD partition function.

  • J.J.M. Verbaarschot and T.Wettig, Random matrix theory and chiral symmetry in QCD, Ann. Rev. Nucl. Part. Sci. 50, 343-410 (2000), doi:10.1146/annurev.nucl.50.1.343, (arXiv:hep-ph/0003017 [hep-ph]).

  • J.C. Osborn, D.Toublan and J.J.M.Verbaarschot, From chiral random matrix theory to chiral perturbation theory, Nucl. Phys. B 540, 317-344 (1999), doi:10.1016/S0550-3213(98)00716-0 (arXiv:hep-th/9806110).

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


On chiral symmetry breaking:

  • Luca Ciambriello, Roberto Contino, Andrea Luzio, Marcello Romano, Ling-Xiao Xu, A novel strategy to prove chiral symmetry breaking in QCD-like theories [arXiv:2404.02967]

  • Luca Ciambriello, Roberto Contino, Ling-Xiao Xu, On the Proof of Chiral Symmetry Breaking through Anomaly Matching in QCD-like Theories: An Exemplification [arXiv:2404.02971]

Introduction and review

See also:

As quantum hadrodynamics:

Via the S-matrix/scattering amplitudes of mesons:

  • Andrea Guerrieri, Joao Penedones, Pedro Vieira, S-matrix Bootstrap for Effective Field Theories: Massless Pions (arXiv:2011.02802)

  • Eef van Beveren, George Rupp, Modern meson spectroscopy: the fundamental role of unitarity (arXiv:2012.03693)

Original articles

At higher order:

  • Nils Hermansson-Truedsson, Chiral Perturbation Theory at NNNLO (arXiv:2006.01430)

In relation to the theta vacuum (Yang-Mills instanton vacuum):

Specifically for kaon decay:

Further discussion of phenomenology:

  • Prabal Adhikari, Jens O. Andersen, Quark and pion condensates at finite isospin density in chiral perturbation theory (arXiv:2003.12567)

  • Bryan W. Lynn, Brian J. Coffey, Kellen E. McGee, Glenn D. Starkman, Nuclear matter as a liquid phase of spontaneously broken semi-classical SU(2) L×SU(2) RSU(2)_L \times SU(2)_R chiral perturbation theory: Static chiral nucleon liquids (arXiv:2004.01706)

See also

  • Yu-Jia Wang, Feng-Kun Guo, Cen Zhang, Shuang-Yong Zhou, Generalized positivity bounds on chiral perturbation theory (arXiv:2004.03992)

  • Lukas Graf, Brian Henning, Xiaochuan Lu, Tom Melia, Hitoshi Murayama, 2, 12, 117, 1959, 45171, 1170086, …: A Hilbert series for the QCD chiral Lagrangian (arXiv:2009.01239)

Skyrme hadrodynamics with vector mesons (π\pi-ω\omega-ρ\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 ω\omega-meson

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

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:

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

Inclusion of the ρ\rho-meson

Original proposal for inclusion of the ρ-meson:

Discussion for phenomenology of light atomic nuclei:

Inclusion of the ω\omega- and ρ\rho-meson

The resulting π\pi-ρ\rho-ω\omega model:

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)


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 σ\sigma-meson

Inclusion of the sigma-meson:

  • Thomas D. Cohen, Explicit σ\sigma meson, topology, and the large-NN 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 σ\sigma-ω\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:


Inclusion of charm quarks/D-mesons

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

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 Q1/m_Q Corrections, Phys. Rev. D49 (1994) 4649-4658 (arXiv:hep-ph/9402205)


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:


The original articles:

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

Interpretation as Skyrmion/baryon current:

Concrete form for NN-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):

Including heavy scalar mesons

Including heavy scalar mesons:

specifically kaons:

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:

Including electroweak interactions

Including electroweak fields:

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):


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:

See also:

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

Applications to lattice QCD

Application to lattice QCD:

  • S.R.Sharpe, Applications of Chiral Perturbation theory to lattice QCD, ILFTN Workshop on “Perspectives in Lattice QCD”, Nara, Japan, Oct 31-Nov 11 2005, 2006 (arXiv:hep-lat/0607016)

Inclusion of leptons

On chiral perturbation theory including leptons:

Specifically in relation to the Skyrme model:

Antisymmetric tensor representation for vector mesons

The formulation of vector mesons in “antisymmetric tensor representation”:

Original articles:

Relation between the vector- and the antisymmetric tensor-representation of vector mesons:

  • Karol Kampf, Jiri Novotny, Jaroslav Trnka, On different lagrangian formalisms for vector resonances within chiral perturbation theory, Eur. Phys. J. C50:385-403, 2007 (arXiv:hep-ph/0608051)

See also

  • V. Dmitrašinović, around (11) in: Axial baryon number nonconserving antisymmetric tensor four-quark effective interaction, Phys. Rev. D 62, 096010 (2000) (doi:10.1103/PhysRevD.62.096010)

Further application of the antisymmetric tensor representation of vector mesons to quantum hadrodynamics:

  • Stefan Leupold, Carla Terschlusen, Towards an effective field theory for vector mesons, Talk presented at the 50th International Winter Meeting on Nuclear Physics, 23-27 January 2012, Bormio (arXiv:1206.2253)

  • Carla Terschlusen, Stefan Leupold, M. F. M. Lutz, Electromagnetic transitions in an effective chiral Lagrangian with the eta-prime and light vector mesons, Eur. Phys. J. A 48, 190 (2012) (arXiv:1204.4125)

Last revised on April 5, 2024 at 06:35:59. See the history of this page for a list of all contributions to it.