nLab
pion

Contents

Context

Fields and quanta

field (physics)

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)
leptons
chargedelectronmuontauon
neutralelectron neutrinomuon neutrinotau neutrino
bound states:
mesonslight mesons:
pion (udu d)
ρ-meson (udu d)
ω-meson (udu d)
f1-meson
a1-meson
strange-mesons:
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)
baryonsnucleons:
proton (uud)(u u d)
neutron (udd)(u d d)

(also: antiparticles)

effective particles

hadron (bound states of the above quarks)

solitons

minimally extended supersymmetric standard model

superpartners

bosinos:

sfermions:

dark matter candidates

Exotica

auxiliary fields

Contents

Idea

In nuclear physics, specifically in the chiral perturbation theory of quantum chromodynamics, the pion or pi-meson (π\pi-meson) is the isospin-triplet scalar-meson field in the first-generation of fermions, i.e. a bound state of an up quark and a down quark (a light meson):

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)
leptons
chargedelectronmuontauon
neutralelectron neutrinomuon neutrinotau neutrino
bound states:
mesonslight mesons:
pion (udu d)
ρ-meson (udu d)
ω-meson (udu d)
f1-meson
a1-meson
strange-mesons:
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)
baryonsnucleons:
proton (uud)(u u d)
neutron (udd)(u d d)


Details

As the Goldstone boson of chiral symmetry breaking

From the point of view of the quark-model of nuclear physics, the pion is the Goldstone boson corresponding to the spontaneous symmetry breaking of the “chiral”-symmetry group SU(2) R×SU(2) LSU(2)_R \times SU(2)_L to the diagonal subgroup SU(2) VSU(2)_V, by the vacuum expectation value qq¯0\langle q \bar q\rangle \neq 0 of the quark-condensate.

(e.g. Machleidt-Entem 11, 2.1.3)

Plain pion field

Hence, in the sense of the Wigner classification, the pion field transforms in the sign representation of the Lorentz group/Pin group (is a spacetime pseudoscalar) and in the adjoint representation of the isospin group SU(2)

As such, a pion field history is a smooth function from spacetime to the Lie algebra su(2)

(1)iπ: 3,1𝔰𝔲(2), i \vec \pi \;\colon\; \mathbb{R}^{3,1} \longrightarrow \mathfrak{su}(2) \,,

where the vecotr notation on the left is to indicate that this is, at each spacetime point (event) x 3,1x \in \mathbb{R}^{3,1}, an element in a real 3-dimensional vector space

iπ(x)𝔰𝔲(2) 3. i \vec \pi(x) \;\in\; \mathfrak{su}(2) \;\simeq_{\mathbb{R}}\; \mathbb{R}^3 \,.

This means that for any choice of linear basis of 𝔰𝔲(2)\mathfrak{su}(2), the pion field decomposes as three real-valued fields.

In the nuclear physics-literature the common choice is that corresponding to the Cartan-Weyl basis

Span({t +,t ,t 0}) 𝔰𝔲(2) \mathrm{Span} \big( \{t_+, t_-, t_0\} \big) \;\simeq_{\mathbb{R}}\; \mathfrak{su}(2)

in terms of which the components of the pion field are hence denoted as follows

pion field componentquark bound state
π 0\pi^0uu¯u \bar u or dd¯d \bar d
π +\pi^+ud¯u \bar d
π \pi^-du¯d \bar u


Exponentiated pion field

Especially in chiral perturbation theory, the pion field is typically represented as the exponentiation of (1) to an SU(2)-valued field

(2)e iπ/f π: 3,1SU(2), e^{i \vec \pi/f_\pi} \;\colon\; \mathbb{R}^{3,1} \longrightarrow SU(2) \,,

(Witten 83, (2), Adkins-Nappi 84, (1)-(3)) nowadays called the exponentiated pion field or often just the chiral field, for review see Machleidt-Entem 11, (2.29), Rho et al. 16, around (1).

Here the unit f πf_\pi is called the pion decay constant.

Assuming that the pion field vanishes at spatial infinity hence means that the exponentiated pion field is a map

e iπ/f π: 0,1×( 3) cpt= 0,1×S 3S 3SU(2) e^{i \vec \pi/f_\pi} \;\colon\; \mathbb{R}^{0,1} \times (\mathbb{R}^3)^{cpt} \;=\; \mathbb{R}^{0,1} \times S^3 \longrightarrow S^3 \simeq SU(2)

from (the time-axis times) the 3-sphere to SU(2). The homotopy class of this continuous function, an element of the (co-)homotopy group of spheres π 3(S 3)π 3(S 3)\pi_3(S^3) \simeq \pi^3(S^3) \simeq \mathbb{Z}, is the Skyrmion-number, or, in fact, the baryon-number, encoded in the knotted stucture of the pion field.

Relation to baryon current

Explicitly, the baryon current is the Wess-Zumino-Witten term in the exponentiated pion field (Witten 83a, Witten 83b):

B top Tr((e iπ/f πde iπ/f π)(e iπ/f πde iπ/f π)(e iπ/f πde iπ/f π)) =(e iπ/f π) *(θ)(e iπ/f π) *(θ)(e iπ/f π) *(θ)Ω 3( 3,1) \begin{aligned} B_{top} & \coloneqq \; Tr \big( ( e^{-i \vec \pi/f_\pi} d e^{i \vec \pi/f_\pi} ) \wedge ( e^{-i \vec \pi/f_\pi} d e^{i \vec \pi/f_\pi} ) \wedge ( e^{-i \vec \pi/f_\pi} d e^{i \vec \pi/f_\pi} ) \big) \\ & =\; \big\langle (e^{i \vec \pi/f_\pi})^\ast(\theta) \wedge (e^{i \vec \pi/f_\pi})^\ast(\theta) \wedge (e^{i \vec \pi/f_\pi})^\ast(\theta) \big\rangle \;\;\in\; \Omega^3(\mathbb{R}^{3,1}) \end{aligned}

Here the expression in the first line uses the fact that SU(2) is a matrix group, while the second line exporesses the same via pullback of the Maurer-Cartan form θ\theta from the group manifold.

Relation to Skyrmions

The skyrmion-model (see there) realizes baryons as solitons/instantons in the exponentiated pion field.

References

General

Introduction and survey:

See also

Decays and interactions

On γπ 0+π ++π \gamma \to \pi^0 + \pi^+ + \pi^-:

  • Ruvi Aviv, Anthony Zee, Low-Energy Theorem for γ3π\gamma \to 3 \pi Phys. Rev. D 5, 2372 (1972) (doi:10.1103/PhysRevD.5.2372)

  • 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)

  • M. Benayoun, P. David, L. DelBuono, O. Leitner, A Global Treatment Of VMD Physics Up To The ϕ\phi: I. e +e e^+ e^- Annihilations, Anomalies And Vector Meson Partial Widths, Eur. Phys. J. C65:211-245, 2010 (arXiv:0907.4047)

On pion-nucleon interaction:

  • E. Ruiz Arriola, J. E. Amaro, R. Navarro Perez, Three pion nucleon coupling constants, Modern Physics Letters A Vol. 31, No. 28, 1630027 (2016) (arXiv:1606.02171)

On the Dalitz decay of the pion:

  • Richard Dalitz, On an alternative decay process for the neutral π\pi-meson, Proceedings of the Physical Society. Section A 64 (7), 667, 1951 (doi:10.1088/0370-1298/64/7/115)

  • Karol Kampf, Marc Knecht, Jiri Novotny, Some aspects of Dalitz decay π 0e +e γ\pi^0 \to e^+ e^- \gamma, presented at Int. Conf. Hadron Structure ‘02, September 2002, Slovakia (arXiv:hep-ph/0212243)

  • Karol Kampf, Marc Knecht, Jiri Novotny, The Dalitz decay π 0e +e γ\pi^0 \to e^+ e^- \gamma revisited, Eur. Phys. J. C46:191-217, 2006 (arXiv:hep-ph/0510021)

  • Henning Berghäuser, Investigation of the Dalitz decays and the electromagnetic form factors of the η\eta and π 0\pi^0-meson, 2010 (spire:1358057)

  • M. Kunkel, Dalitz Decays of Pseudo-Scalar Mesons, talk at Light Meson Decays Workshop August 5, 2012 (pdf)

  • Sergi González-Solís, Single and double Dalitz decays of π 0\pi^0, η\eta and η\eta' mesons, Nuclear and Particle Physics Proceedings Volumes 258–259, January–February 2015, Pages 94-97 (doi:10.1016/j.nuclphysbps.2015.01.021)

  • Esther Weil, Gernot Eichmann, Christian S. Fischer, Richard Williams, section III.A of: Electromagnetic decays of the neutral pion, Phys. Rev. D 96, 014021 (2017) (arXiv:1704.06046)

Exponentiated pion field and Skyrmions

Discussion of the exponentiated pion field (“chiral field”) in chiral perturbation theory and the interpretation of its winding number as Skyrmion-number / baryon

Via holographic QCD

Discussion via the AdS/QCD correspondence:

  • Domenec Espriu, Alisa Katanaeva, Effects of bulk symmetry breaking on AdS/QCD predictions (arXiv:2001.08723)

Last revised on October 20, 2020 at 06:42:08. See the history of this page for a list of all contributions to it.