# Dalitz plot

The Dalitz plot is a two-dimensional plot often used in particle physics to represent the relative frequency of various (kinematically distinct) manners in which the products of certain (otherwise similar) three-body decays may move apart.[2][3]

Dalitz plot for a three-body decay of a spin-0 particle of the mass ${\displaystyle M}$ into three spin-0 particles of masses ${\displaystyle m_{1}}$, ${\displaystyle m_{2}}$, ${\displaystyle m_{3}}$. 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 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.
Dalitz plot for the decay ${\displaystyle D^{+}\to K^{+}K^{-}\pi ^{+}}$ in data of the LHCb experiment at CERN.[1] Clear resonances ${\displaystyle K^{*0}}$ (vertical region of enhancement) and ${\displaystyle \phi }$ (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.

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 axes of the plot are the squares of the invariant masses 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 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 of three-body decays.

Dalitz plots play a central role in the discovery of new particles in current high-energy physics experiments, including Higgs boson research,[4] and are tools in exploratory efforts that might open avenues beyond the Standard Model.[5]

R.H. Dalitz introduced this technique in 1953[2][3] to study decays of K mesons (which at that time were still referred to as "tau-mesons"[6]). It can be adapted to the analysis of four-body decays as well.[7] 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:[8]

${\displaystyle m'(1,2)={\frac {1}{\pi }}\arccos \left(2*{\frac {m(1,2)-m(1,2)_{min}}{m(1,2)_{max}-m(1,2)_{min}}}-1\right)}$ ;

${\displaystyle \theta '(1,2)={\frac {1}{\pi }}\theta (1,2)}$ ;

where ${\displaystyle m(1,2)}$   is the invariant mass of particles 1 and 2 in a given decay event; ${\displaystyle m(1,2)_{max}}$  and ${\displaystyle m(1,2)_{min}}$   are its maximal and minimal kinematically allowed values, while ${\displaystyle \theta (1,2)}$   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

1. ^ The LHCb collaboration. (2019). "Measurement of the branching fractions of the decays D+ → K−K+K+, D+ → π−π+K+ and D+s → π−K+K+". Journal of High Energy Physics. 2019 (176). doi:10.1007/JHEP03(2019)176.
2. ^ a b R. H. Dalitz (1953). "On the analysis of τ-meson data and the nature of the τ-meson". Philosophical Magazine. 44 (357): 1068–1080. doi:10.1080/14786441008520365.
3. ^ a b R. H. Dalitz (1954). "Decay of τ mesons of known charge". Physical Review. 94 (4): 1046–1051. Bibcode:1954PhRv...94.1046D. doi:10.1103/PhysRev.94.1046.
4. ^ Close, Frank (24 January 2006). "Richard Dalitz: Physicist who mapped the behaviour of exotic particles and argued for the reality of quarks". The Guardian.
5. ^ P. Pakhlov and T. Uglov, Flavor physics at Super B-factories era, J. Phys.: Conf. Ser. 675, 022009 (2016).
6. ^ E. Fabri (1954). "A study of τ-meson decay". Nuovo Cimento. 11 (5): 479–491. Bibcode:1954NCim...11..479F. doi:10.1007/BF02781042. S2CID 120859580.
7. ^ M. Schulz; et al. (2007). "Four-particle Dalitz plots to visualize atomic break-up processes". Journal of Physics B. 40 (15): 3091–3099. Bibcode:2007JPhB...40.3091S. doi:10.1088/0953-4075/40/15/009.
8. ^ Aaij, R.; Adeva, B.; Adinolfi, M.; Affolder, A.; Ajaltouni, Z.; Akar, S.; Albrecht, J.; Alessio, F.; Alexander, M. (2014-10-14). "Dalitz plot analysis of B s 0 → D ¯ 0 K − π + decays". Physical Review D. Vol. 90, no. 7. p. 072003. doi:10.1103/PhysRevD.90.072003. ISSN 1550-7998. Retrieved 2021-02-19.