# nLab Dalitz decay

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

flavors of fundamental fermions in the
standard model of particle physics:
generation of fermions1st generation2nd generation3d generation
quarks ($q$)
up-typeup quark ($u$)charm quark ($c$)top quark ($t$)
down-typedown quark ($d$)strange quark ($s$)bottom quark ($b$)
leptons
chargedelectronmuontauon
neutralelectron neutrinomuon neutrinotau neutrino
bound states:
mesonslight 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 ($b \bar b$)
baryonsnucleons:
proton $(u u d)$
neutron $(u d d)$

(also: antiparticles)

effective particles

hadrons (bound states of the above quarks)

solitons

in grand unified theory

minimally extended supersymmetric standard model

superpartners

bosinos:

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).

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

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

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

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

$\pi^0 \,\to\, e^+ \,+\, e^- \,+\, \gamma \,.$

The Feynman amplitude of the process is proportional to

(1)$\epsilon^{\mu_1 \mu_2 \nu \kappa} \underset{ photon }{ \underbrace{ q_{\mu_1} \epsilon_{\mu_2} } } \underset{ {scalar} \atop {meson} }{ \underbrace{ p_\nu } } \underset{ dilepton }{ \underbrace{ \overline{L} \!\cdot\! \gamma_\kappa \!\!\cdot\! L } } \underset{ {virtual} \atop {photon} }{ \underbrace{ \tfrac{1}{(p_1 + p_2)^2} } }$

where

• $\epsilon^{\mu_1 \mu_2 \nu \kappa}$ is the Levi-Civita symbol,

• $q_{\mu_1}$ is the wave vector and $\epsilon_{\mu_2}$ the polarization of the photon,

• $p_\mu$ is the momentum of the (pseudo-)scalar meson,

• $\overline{L} \!\cdot\! \gamma_\kappa \!\!\cdot\! L$ is the lepton current

• $p_1, p_2$ are the momenta of the two leptons, hence $(p_1 + p_2)$ is that of the virtual photon mediating the decay.

(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 $(p_1 + p_2)^2$ (all on 4d Minkowski spacetime):

(2)$\mathbf{L}_{rad} \;\sim\; d A \wedge d \pi^0 \wedge (\overline{L} \!\cdot\!\gamma_\mu \!\!\cdot\! L) d x^\mu \,,$

where $A = A_\mu d x^\mu$ is the differential 1-form defined by the photon field (the gauge symmetry-connection-form), $\pi^0$ is the function (scalar field) which is the neutral component of the pion field and $d$ 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 $A \to B + l^+ + l^-$ for $A$ and $B$ a pseudoscalar meson and vector meson, respectively, with dominant decay mode given by this Feynman diagram:

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

(3)$\epsilon^{\mu_1 \mu_2 \nu \kappa} \underset{ {vector} \atop {meson} }{ \underbrace{ q_{\mu_1} \epsilon_{\mu_2} } } \underset{ {scalar} \atop {meson} }{ \underbrace{ p_\nu } } \underset{ dilepton }{ \underbrace{ \overline{L} \!\cdot\! \gamma_\kappa \!\!\cdot\! L } } \underset{ {virtual} \atop {photon} }{ \underbrace{ \tfrac{1}{(p_1 + p_2)^2} } }$

where

• $\epsilon^{\mu_1 \mu_2 \nu \kappa}$ is the Levi-Civita symbol,

• $q_{\mu_1}$ is the wave vector and $\epsilon_{\mu_2}$ the polarization of the vector meson,

• $p_\mu$ is the momentum of the (pseudo-)scalar meson,

• $\overline{L} \!\cdot\! \gamma_\kappa \!\!\cdot\! L$ is the lepton current

• $p_1, p_2$ are the momenta of the two leptons, hence $(p_1 + p_2)$ is that of the virtual photon mediating the decay.

###### Remark

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

(4)$\mathbf{L}_{sem} \;\sim\; d V \wedge d \pi \wedge (\overline{L} \!\cdot\!\gamma_\mu \!\!\cdot\! L) d x^\mu \,,$

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

###### Remark

Up to possibly a global sign, it does not matter whether $q_\mu$ 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 $p_\mu$ 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:

\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 $\pi^0 \to e^+ e^- \gamma$, 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)

• 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 $\pi^0$, $\eta$ 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)

Of pions, eta-mesons and omega-mesons:

• Henning Berghäuser, Investigation of the Dalitz decays and the electromagnetic form factors of the $\eta$ and $\pi^0$-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 $J/\psi \to P l^+ l^-$ 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 $J/\psi \to e^+e^- \eta$, Phys. Rev. D 99, 012006 (2019) (arXiv:1810.03091)

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

Of baryons:

• G. Ramalho, A covariant model for the decuplet to octet Dalitz decays (arXiv:2002.07280)

Last revised on September 17, 2020 at 03:34:14. See the history of this page for a list of all contributions to it.