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Dalitz decay

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 quantum hadrodynamics a Dalitz decay is the decay of an electrically neutral pseudoscalar meson into a dilepton pair and a either a photon (this is the original sense of Dalitz 51, a radiative leptonic decay) or a vector meson (a semileptonic decay).

Radiative leptonic Dalitz decay

In quantum hadrodynamics a Dalitz decay in the original sense (Dalitz 51) is the radiative leptonic decay of an electromagnetically neutral pseudoscalar meson PP into a photon and a dilepton:

Pγ+l ++l . P \,\to\, \gamma \,+\, l^+ \,+\, l^- \,.

The dominant contribution to this decay is by a purely radiative decay Pγ+γP \to \gamma+ \gamma followed by an electron-photon interaction γ+e ++e \gamma + e^+ + e^- (“Dalitz pair”) of one of the two resulting photons (which thus participates as a virtual photon γ *\gamma^\ast with spacelike wave vector q 2=m e +e 2q^2 = m^2_{e^+ e^-}):

from Kunkel 12, p. 9

The original and archetypical example is the decay of the neutral pion into an electron/positron-pair and a photon:

π 0e ++e +γ. \pi^0 \,\to\, e^+ \,+\, e^- \,+\, \gamma \,.

The Feynman amplitude of the process is proportional to

(1)ϵ μ 1μ 2νκq μ 1ϵ μ 2photonp νscalarmesonL¯γ κLdilepton1(p 1+p 2) 2virtualphoton \epsilon^{\mu_1 \mu_2 \nu \kappa} \underset{ photon }{ \underbrace{ q_{\mu_1} \epsilon_{\mu_2} } } \underset{ {scalar} \atop {meson} }{ \underbrace{ p_\nu } } \underset{ dilepton }{ \underbrace{ \overline{L} \!\cdot\! \gamma_\kappa \!\!\cdot\! L } } \underset{ {virtual} \atop {photon} }{ \underbrace{ \tfrac{1}{(p_1 + p_2)^2} } }

where

(see Yia-Sang 09, (9) (symbol definition on p. 12), also e.g. KKN 02, p. 2, KKN06, p. 6 ).

Remark

The corresponding interaction Lagrangian density for the Feynman amplitude (1) (hence essentially its Fourier transform times the volume form) is, at fixed (p 1+p 2) 2(p_1 + p_2)^2 (all on 4d Minkowski spacetime):

(2)L raddAdπ 0(L¯γ μL)dx μ, \mathbf{L}_{rad} \;\sim\; d A \wedge d \pi^0 \wedge (\overline{L} \!\cdot\!\gamma_\mu \!\!\cdot\! L) d x^\mu \,,

where A=A μdx μA = A_\mu d x^\mu is the differential 1-form defined by the photon field (the gauge symmetry-connection-form), π 0\pi^0 is the function (scalar field) which is the neutral component of the pion field and dd denotes is the de Rham differential.

Semileptonic Dalitz decay

More generally, one speaks of (generalized) Dalitz decays for the analogous process as above with the photon replaced by some vector meson, hence for semileptonic decays AB+l ++l A \to B + l^+ + l^- for AA and BB a pseudoscalar meson and vector meson, respectively, with dominant decay mode given by this Feynman diagram:

from Kunkel 12, p. 9

The Feynman amplitude of the process is, directly analogous to (1), proportional to

(3)ϵ μ 1μ 2νκq μ 1ϵ μ 2vectormesonp νscalarmesonL¯γ κLdilepton1(p 1+p 2) 2virtualphoton \epsilon^{\mu_1 \mu_2 \nu \kappa} \underset{ {vector} \atop {meson} }{ \underbrace{ q_{\mu_1} \epsilon_{\mu_2} } } \underset{ {scalar} \atop {meson} }{ \underbrace{ p_\nu } } \underset{ dilepton }{ \underbrace{ \overline{L} \!\cdot\! \gamma_\kappa \!\!\cdot\! L } } \underset{ {virtual} \atop {photon} }{ \underbrace{ \tfrac{1}{(p_1 + p_2)^2} } }

where

(e.g. Wachs 00, (2.60), FLQY 12, GLMY 19, (2))

Remark

The corresponding interaction Lagrangian density for the Feynman amplitude (3) (hence essentially its Fourier transform times the volume form) is, at fixed (p 1+p 2) 2(p_1 + p_2)^2:

(4)L semdVdπ(L¯γ μL)dx μ, \mathbf{L}_{sem} \;\sim\; d V \wedge d \pi \wedge (\overline{L} \!\cdot\!\gamma_\mu \!\!\cdot\! L) d x^\mu \,,

where V=V μdx μV = V_\mu d x^\mu is the differential 1-form defined by the vector meson field (the hidden local symmetry-connection-form), PP is the scalar field function which is the neutral component of the scalar meson field and dd denotes is the de Rham differential.

Remark

Up to possibly a global sign, it does not matter whether q μq_\mu in (1) or (3) is taken to be the wave vector of the external photon/vector meson (as stated for instance in Yia-Sang 09, (9)) or of the interval virtual photon, equivalently of the dilepton pair (as stated for instance in FLQY 12, GLMY 19, (2)): Due to momentum conservation at the first interaction vertex their difference is proportional to the momentum p μp_\mu of the scalar meson, and due to anti-symmetry of the Levi-Civita symbol this difference vanishes inside the above expression.

Equivalently, in terms of the Lagrangian density (4), this equivalence is reflected by Lagrangian densities defined only up to total derivative:

dVdπ(L¯γ μL)dx μ Vdπd(L¯γ μL)dx μ \begin{aligned} & d V \wedge d \pi \wedge (\overline{L} \!\cdot\!\gamma_\mu \!\cdot\! L) d x^\mu \\ & \sim\; V \wedge d \pi \wedge d (\overline{L} \!\cdot\!\gamma_\mu \!\cdot\! L) \wedge d x^\mu \end{aligned}

References

The original article:

Survey:

Further discussion of the Dalitz decay of pions:

  • 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)

  • 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)

Dalitz decay of/into omega-mesons:

  • Mirko Wachs, Die Selbstenergie des Omega-Mesons, 2000 (epda:000050)

Of pions and eta-mesons:

  • 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)

Of pions, eta-mesons and omega-mesons:

  • 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)

Discussion of Dalitz decay of quarkonium:

  • Yu Jia, Wen-Long Sang, Observation prospects of leptonic and Dalitz decays of pseudoscalar quarkonia, JHEP 0910:090, 2009 (arXiv:0906.4782)

Of charmonium:

  • Jinlin Fu, Hai-Bo Li, Xiaoshuai Qin, Mao-Zhi Yang, Study of the electromagnetic transitions J/ψPl +l J/\psi \to P l^+ l^- and probe dark photon, Modern Physics Letters A Vol. 27, No. 38, 1250223 (2012) (arXiv:1111.4055 doi:10.1142/S0217732312502239)

  • Study of the Dalitz decay J/ψe +e ηJ/\psi \to e^+e^- \eta, Phys. Rev. D 99, 012006 (2019) (arXiv:1810.03091)

Of upsilon-mesons:

  • Li-Min Gu, Hai-Bo Li, Xin-Xin Ma, Mao-Zhi Yang, Study of the electromagnetic Dalitz decays ψ(ϒ)η c(η b)l +l \psi(\Upsilon) \to \eta_c(\eta_b) l^+ l^-, Phys. Rev. D 100, 016018 (2019) (arXiv:1904.06085)

Last revised on May 2, 2020 at 12:13:06. See the history of this page for a list of all contributions to it.