nLab pion

Idea

In nuclear physics, specifically in the chiral perturbation theory of quantum chromodynamics, the pion or pi-meson ($\pi$-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 fermions	1st generation	2nd generation	3d generation
quarks ($q$)
up-type	up quark ($u$)	charm quark ($c$)	top quark ($t$)
down-type	down quark ($d$)	strange quark ($s$)	bottom quark ($b$)
leptons
charged	electron	muon	tauon
neutral	electron neutrino	muon neutrino	tau neutrino
bound states:
mesons	light mesons: pion ($u d$) ρ-meson ($u d$) ω-meson ($u d$) f1-meson a1-meson	strange-mesons: ϕ-meson ($s \bar s$), kaon, K*-meson ($u s$, $d s$) eta-meson ($u u + d d + s s$) charmed heavy mesons: D-meson ($u c$, $d c$, $s c$) J/ψ-meson ($c \bar c$)	bottom heavy mesons: B-meson ($q b$) ϒ-meson ($b \bar b$)
baryons	nucleons: proton $(u u d)$ neutron $(u d d)$

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 $SU(2)_R \times SU(2)_L$ to the diagonal subgroup $SU(2)_V$, by the vacuum expectation value $\langle q \bar q\rangle \neq 0$ 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)$$i \vec \pi \;\colon\; \mathbb{R}^{3,1} \longrightarrow \mathfrak{su}(2) \,,$$

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

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

This means that for any choice of linear basis of $\mathfrak{su}(2)$, 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

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

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

pion field component	quark bound state
$\pi^0$	$u \bar u$ or $d \bar d$
$\pi^+$	$u \bar d$
$\pi^-$	$d \bar u$

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

(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 $f_\pi$ 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

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

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 $\pi_3(S^3) \simeq \pi^3(S^3) \simeq \mathbb{Z}$, 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):

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

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 $\theta$ from the group manifold.

Relation to Skyrmions

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

Related concepts

• omega-meson,rho meson

• kaon

• B-meson

• quarkonium

References

General

Introduction and survey:

• Rupert Machleidt, David Rodríguez Entem, Chiral effective field theory and nuclear forces, Phys. Rept. 503:1-75, 2011 (arXiv:1105.2919, doi:10.1016/j.physrep.2011.02.001)

• Brambilla et al., Section 3.2.7 of: QCD and strongly coupled gauge theories - challenges and perspectives, Eur Phys J C Part Fields. 2014; 74(10): 2981 (arXiv:1404.3723, doi:10.1140/epjc/s10052-014-2981-5)

See also

• Wikipedia, Pion

Decays and interactions

On $\gamma \to \pi^0 + \pi^+ + \pi^-$:

• Ruvi Aviv, Anthony Zee, Low-Energy Theorem for $\gamma \to 3 \pi$ 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 $\phi$: I. $e^+ e^-$ 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 $\pi$-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 $\pi^0 \to e^+ e^- \gamma$, presented at Int. Conf. Hadron Structure ‘02, September 2002, Slovakia (arXiv:hep-ph/0212243)

• Karol Kampf, Marc Knecht, Jiri Novotny, The Dalitz decay $\pi^0 \to e^+ e^- \gamma$ 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 $\eta$ and $\pi^0$-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 $\pi^0$, $\eta$ and $\eta'$ 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

• 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)

• 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)

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)

• Xuanmin Cao, Matteo Baggioli, Hui Liu, Danning Li, Pion dynamics in a soft-wall AdS-QCD model [arXiv:2210.09088]