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pion

Contents

Context

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)
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

hadrons (bound states of the above quarks)

solitons

in grand unified theory

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.