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[[File:Dalitz plot.svg|thumb|Dalitz plot for a three-body decay of a spin-0 particle of the mass <math>M</math> into three spin-0 particles of masses <math>m_1</math>, <math>m_2</math>, <math>m_3</math>. The grey area depicts the allowed kinematic region. The blue line shows a possible position of accumulation of events in case a spin-0 [[Resonance (particle physics)|resonance]] is present as an intermediate state in this three-body decay, which then decays to particles 1 and 2. The orange line shows the position of accumulation of events in case another spin-0 resonance is present, decaying to particles 1 and 3.|306x306px]]
[[File:Dalitz plot of D meson decay to two kaons and a pion.png|thumb|Dalitz plot for the decay <math> D^+ \to K^+ K^- \pi^+ </math> in data of the [[LHCb]] experiment at [[CERN]].<ref>{{cite journal
|author=The LHCb collaboration.
|year=2019
|title=Measurement of the branching fractions of the decays D+ → K−K+K+, D+ → π−π+K+ and D+s → π−K+K+
|journal=[[Journal of High Energy Physics]]
|volume=2019 |issue=176
|doi=10.1007/JHEP03(2019)176
|hdl=2445/162294
|hdl-access=free
}}</ref> Clear [[Resonance (particle physics)|resonances]] <math>K^{*0}</math> (vertical region of enhancement) and <math>\phi </math> (horizontal enhancement) can be seen. Distribution of events around resonant regions is not uniform due to spin 1 of the resonances, and interference between the resonant and non-resonant processes.|306x306px]]
The '''Dalitz plot''' is a two-dimensional plot often used in [[particle physics]] to represent the [[Statistical frequency|relative frequency]] of various (kinematically distinct) manners in which the products of certain (otherwise similar) [[Particle decay|three-body decay]]s may move apart.<ref name="D1">{{cite journal
|author=R. H. Dalitz
|year=1953
|title=On the analysis of τ-meson data and the nature of the τ-meson
|journal=[[Philosophical Magazine]]
|volume=44 |issue=357
|pages=1068–1080
|doi=10.1080/14786441008520365
}}</ref><ref name="D2">{{cite journal
|author=R. H. Dalitz
|year=1954
|title=Decay of τ mesons of known charge
|journal=[[Physical Review]]
|volume=94 |pages=1046–1051
|doi=10.1103/PhysRev.94.1046
|bibcode = 1954PhRv...94.1046D
|issue=4 }}</ref>

The [[phase-space]] of a decay of a [[pseudoscalar]] into three spin-0 particles can be completely described using two variables. In a traditional Dalitz plot, the [[Coordinate axis|axes]] of the plot are the squares of the [[invariant mass]]es of two pairs of the decay products. (For example, if particle A decays to particles 1, 2, and 3, a Dalitz plot for this decay could plot m212 on the x-axis and m223 on the y-axis.) If there are no angular correlations between the decay products then the distribution of these variables is flat. However symmetries may impose certain restrictions on the distribution. Furthermore, three-body decays are often dominated by [[Resonance#Quantum_field_theory|resonant]] processes, in which the particle decays into two decay products, with one of those decay products immediately decaying into two additional decay products. In this case, the Dalitz plot will show a non-uniform distribution, with a peak around the mass of the resonant decay. In this way, the Dalitz plot provides an excellent tool for studying the [[Dynamics (mechanics)|dynamics]] of three-body decays.

Dalitz plots play a central role in the discovery of new particles in current high-energy physics experiments, including [[search for the Higgs boson|Higgs boson research]],<ref name=Close2006>{{cite news | first=Frank | last=Close | url=https://www.theguardian.com/obituaries/story/0,3604,1693226,00.html | title=Richard Dalitz: Physicist who mapped the behaviour of exotic particles and argued for the reality of quarks | publisher=[[The Guardian]] | date=24 January 2006}}</ref> and are tools in exploratory efforts that might open avenues beyond the [[Standard Model]].<ref>P. Pakhlov and T. Uglov, Flavor physics at Super B-factories era, ''J. Phys.: Conf. Ser.'' '''675''', 022009 (2016).</ref>

[[Richard Dalitz|R.H. Dalitz]] introduced this technique in 1953<ref name="D1"/><ref name="D2"/> to study decays of [[Kaon|K mesons]] (which at that time were still referred to as "tau-mesons"<ref>{{cite journal
|author=E. Fabri
|year=1954
|title=A study of τ-meson decay
|journal=[[Nuovo Cimento]]
|volume=11 |pages=479–491
|doi=10.1007/BF02781042
|issue=5
|bibcode=1954NCim...11..479F
|s2cid=120859580
}}</ref>). It can be adapted to the analysis of four-body decays as well.<ref>{{cite journal
|author= M. Schulz
|year=2007
|title=Four-particle Dalitz plots to visualize atomic break-up processes
|journal=[[Journal of Physics B]]
|volume=40 |pages=3091–3099
|doi=10.1088/0953-4075/40/15/009
|bibcode = 2007JPhB...40.3091S
|issue= 15 |display-authors=etal}}</ref> A specific form of a four-particle Dalitz plot (for non-relativistic kinematics), which is based on a tetrahedral coordinate system, was first applied to study the few-body dynamics in atomic four-body fragmentation processes.

== Square Dalitz plot ==

Modeling of the common representation of the Dalitz plot can be complicated due to its nontrivial shape. One can however introduce such kinematic variables so that Dalitz plot gets a rectangular (or squared) shape:<ref>{{Cite news|title=Dalitz plot analysis of B s 0 → D ¯ 0 K − π + decays|first3=M.|first9=M.|last8=Alessio|first8=F.|last7=Albrecht|first7=J.|last6=Akar|first6=S.|last5=Ajaltouni|first5=Z.|last4=Affolder|first4=A.|last3=Adinolfi|last2=Adeva|url=https://link.aps.org/doi/10.1103/PhysRevD.90.072003|first2=B.|last=Aaij|first=R.|language=en|issue=7|volume=90|pages=072003|doi=10.1103/PhysRevD.90.072003|issn=1550-7998|accessdate=2021-02-19|date=2014-10-14|work=Physical Review D|last9=Alexander}}</ref>

<math> m'(1,2) = \frac{1}{\pi} \arccos\left(2 * \frac{m(1,2)-m(1,2)_{min}}{m(1,2)_{max}-m(1,2)_{min}} -1\right) </math>;

<math> \theta'(1,2) = \frac{1}{\pi} \theta(1,2) </math>;

where <math>m(1,2)</math>  is the invariant mass of particles 1 and 2 in a given decay event; <math>m(1,2)_{max}</math> and <math>m(1,2)_{min}</math>  are its maximal and minimal kinematically allowed values, while <math> \theta(1,2) </math>  is the angle between particles 1 and 3 in the rest frame of particles 1 and 2. This technique is commonly called "Square Dalitz plot" (SDP).

==References==
{{Reflist}}

== External links ==
* [http://www.slac.stanford.edu/slac/sass/talks/BrianL.pdf Dalitz Plots: Past and Present (a presentation by Brian Lindquist at SLAC)]

{{DEFAULTSORT:Dalitz Plot}}
[[Category:Plots (graphics)]]
[[Category:Scattering]]
[[Category:Experimental particle physics]]

{{scattering-stub}}

Dalitz plot - Revision history

imported>OAbot: Open access bot: hdl added to citation with #oabot.