Testing violations of Bose and $CPT$ symmetries via Dalitz plots and their prismatic generalizations

Bose symmetry and $CPT$ symmetry are two very fundamental symmetries of nature. However, the validity of these symmetries in diverse phenomena must be verified by experiments. We propose new techniques to probe these two fundamental symmetries in the realm of mesons by using the Dalitz plot of a few three-body meson decays. Since these symmetries are very fundamental in nature, their violations, if any, are expected to be extremely small. Hence, observing their violations requires study of a huge data sample. In this context we introduce a new three-dimensional plot which we refer to as the Dalitz “prism.” This provides an innovative means for acquiring the huge statistics required for such studies. Using the Dalitz plots and the Dalitz prisms we chart out the way to probe the violations of Bose and $CPT$ symmetries in a significant manner. Since mesons are unstable and composite particles, testing the validity of Bose symmetry and the $CPT$ symmetry in these cases is of paramount importance for fundamental physics.

Figure 1

A hypothetical Dalitz plot with Mandelstam-like variables $s$, $t$, $u$ for the decay $X\to 1+2+3$. The three sides of the equilateral triangle $▵UVW$ are given by $s=0$, $u=0$ and $t=0$. At the three vertices we have $s=M^2$, $t=M^2$ and $u=M^2$. The three vertices in terms of the barycentric rectangular coordinate $(x,y)$ are given by $U=(0,2)$, $V=(\sqrt{3},-1)$ and $W=(-\sqrt{3},-1)$. For the barycentric polar coordinates $(r,\theta )$, the angle $\theta$ is measured from the vertical axis. The blobs with $\mathbf{1}$, $\mathbf{2}$ and $\mathbf{3}$ serve as a mnemonic to suggest that the exchanges $s\leftrightarrow t\leftrightarrow u$ are equivalent to the particle exchanges $1\leftrightarrow 2\leftrightarrow 3$ respectively. The sextants of the Dalitz plot are also shown.

Figure 2

The schematic figure of the Dalitz prism. To get the Dalitz plot corresponding to a certain value of $m_X$ one simply slices the Dalitz prism parallel to the $xy$ plane at that particular value of $m_X$. The sextants of the Dalitz plots are subsumed into the six analogously labeled identical wedges of the Dalitz prism.