Abstract

Two-particle decays of light unflavored, strange mesons and Δ, N, Λ, Σ, Ξ baryons are mapped to isosceles triangles in hyperbolic velocity space. A Lorentz-invariant function is defined on the isosceles triangles—the hyperbolic sine of the triangle, equal to the product of the hyperbolic sine of the altitude dropped from any vertex and the hyperbolic sine of the side opposite that vertex. Integer values of the hyperbolic sine correspond to triangles named Heron’s base triangles. Based on Heron’s triangles, the decays of scalar, strange mesons and Δ, N, Λ, Σ, Ξ baryons, are classified. This classification provides suggestions to search for new resonances.

Keywords

Lobachevsky-Bolyai Velocity Space, Decay Triangles, Hyperbolic Sine of the Triangle, Base Heron Triangle

Share and Cite:

1. Introduction

The endpoints of the velocity vectors of the two decay particles of the resonance represent material points in the Lobachevsky-Boyai hyperbolic velocity space with curvature k = −1/C² (where C = 1 is the speed of light, and the points are assigned the masses of the decay particles) [1]-[5]. Two particles velocity points can be connected by line segment and an arc line, called the oricycle [6]. An oricycle is a line of rotation of curvature 0, whose center “$C_0$ ” lies among infinitely distant points of velocity space. Archimedes’ laws of levers define a third point “m” on the arc of the oricycle, to which an additive mass (the sum of the decay particle masses) is assigned. The roles of the forces in the levers are played by the rest masses of the decay particles, and the lever arms are equal to the lengths of the arcs of the oricycle. By connecting these 3 points with straight line, we get triangle of resonance decay inscribed in the oricycle. The effective resonance mass is determined by the hyperbolic cosine of the length of the side opposite the point “m”, and by the masses of the decay particles [5]. When the masses of the decay particles are equal, we have an isosceles triangle $\text{Δ}{\pi }_{1}m{\pi }_2$ of the decay of the meson $f_o\left(980\right)\to {\pi }_1+{\pi }_2$ (Figure 1). If the masses of the decay particles are unequal, we have a different-sided decay triangle $\text{Δ}\eta m{\pi }_1$ of the decay of the mezon $a_o\left(1450\right)\to \eta +{\pi }_1$ (Figure 2). The triangle $\text{Δ}\eta m{\pi }_1$ can be mapped to the isosceles triangle $\text{Δ}{\pi }_{1}m{\pi }_2$ by rotating the arc $l_{m{\pi }_1}$ of an oricycle around the straight line ($m–G_{\eta {\pi }_1}$ ) (the point “$G_{\eta {\pi }_1}$ ” represents the center of inertia of the pair particles ${\pi }_1$ and $\eta$ ) (Figure 2). Thus, the two-body decays of scalar, strange mesons and Δ, N, Λ, Σ, Ξ baryons are associated with the different-sided decay triangles $\Delta bm{\pi }_1$ and the isosceles triangles $\text{Δ}{\pi }_{1}m{\pi }_2$ inscribed in the oricycle of hyperbolic velocity space (HVS) (Figure 3). Further, we will refer to the isosceles triangle $\text{Δ}{\pi }_{1}m{\pi }_2$ as the base triangle.

Figure 1. Decays of $\rho \left(770\right)\to {\pi }_1+{\pi }_2$ and $f_o\left(980\right)\to {\pi }_1+{\pi }_2$ in the Beltrami model of the hyperbolic velocity space. The separate ellipses of decay oricycles of $\rho \left(770\right),f_o\left(980\right)$ with centers in the “$C_0$ ” points of the circle $X^2+Y^2=1$ , called the Absolute. Due to the congruence of the oricicles in the HVS, the ellipse with semi-axis (C₀ – 0) represents an oricicle in which the individual oricicles of the ρ(770) and $f_o\left(980\right)$ meson decays are combined. The point-velocity “G” represents the centers of inertia of pairs of particles (${\pi }_1,{\pi }_2$ ).

Figure 2. The different sided triangles $\text{Δ}\eta m{\pi }_1$ of the decays of the $a_2\left(1320\right)\to \eta +{\pi }_1$ and $a_o\left(1450\right)\to \eta +{\pi }_1$ , isosceles base triangles $\Delta {\pi }_{1}m{\pi }_2$ , corresponding to the triangles $\text{Δ}\eta m{\pi }_1$ . The point-velocity “G” represents the centers of inertia of pairs of particles (${\pi }_1$ , ${\pi }_2$ ). The point “$G_{\eta {\pi }_1}$ ” represents the center of inertia of the pair particles π₁ and η. The point “$C_0$ ” is the center of rotation of the oricycle.

A Lorentz-invariant function is defined on the base triangles $\text{Δ}{\pi }_{1}m{\pi }_2$ —the hyperbolic sine of the triangle, equal to the product of the hyperbolic sine of the altitude dropped from any vertex and the hyperbolic sine of the side opposite that vertex (this function introduced by Y.P. Panarin [7]). Integer values of the hyperbolic sine (HST) correspond to triangles named Heron’s base triangles.

We attempted to determine how closely the decay triangles of light unflavored, strange mesons and Δ, N, Λ, Σ, Ξ baryons, were related to Heron’s base triangles. For this purpose, data on two-particle decays of all the listed resonances is selected from [8] (Tables A1-A7 in Appendix). Based on the data from Tables A1-A7, we calculated the hyperbolic sines of the $HS{T}_{res}$ triangles of their decays , the integer values of $HS{T}_{her}$ , the masses $m_{her}^{b,{\pi }_1}$ and $m_{her}^{{\pi }_1,{\pi }_2}$ , and the absolute and relative values of the deviations of the mass $m_{her}^{b,{\pi }_1}$ from the effective mass $m_{res}$ (Tables 1-17, Equations (3)-(13), $b={\pi }_2$ , P, n, $\eta$ , $\rho \left(770\right)$ , $\omega \left(782\right)$ , $\Lambda$ , $\Sigma$ , ${\Sigma }_c^0\left(2455\right)$ , ${\Xi }^0$ , $\Delta \left(1232\right)$ , $K\left(493\right)$ , $K^{*}\left(892\right)$ ). Tables 1-17 show that the calculated effective base mass $m_{her}^{b,{\pi }_1}$ for Heron’s triangles agree well with the published values of the effective mass $m_{res}$ (the average deviation over all tables equals 2.2%). In Chapter 2, the correspondence between the decay triangles of the listed resonances and Heron’s base triangles is studied in more details.

Table 18 presents a summary classification of light unflavored, strange mesons and Δ, N, Λ, Σ, Ξ baryons, based on the values of the hyperbolic sines $HS{T}_{her}$ of their Heron base triangles. Table 18 can be used to propose a method for searching for new resonances. Chapter 3 looks more closely at proposals for finding new resonances.

2. Classification of Light Unflavored, Strange Mesons and Δ, N, Λ, Σ, Ξ Baryons Based on Heron’s Triangles of Their Decays in Hyperbolic Velocity Space

Consider a reaction in which the collision of the initial particles B and T produces a pair of particles $b$ and ${\pi }_1$ :

$B+T\to b+{\pi }_1+all$ (1)

Let us assume that the velocities of the particles $B,b$ and ${\pi }_1$ have been measured in reference frame “0”. The ends of the particle velocity vectors represent material velocity points “B,” “b”, “${\pi }_1$ ” in Lobachevsky-Bolyaï hyperbolic velocity space (HVS), located inside a sphere of radius C (hereinafter, the speed of light C = 1; the points are assigned the masses $m_B$ , $m_b$ and $m_{{\pi }_1}$ of the reaction particles) [1]-[5]. Let’s draw a plane in the HVS passing through the points “B”, “${\pi }_1$ ” and “b”. In this plane, we introduce a rectangular coordinate system X0Y with the origin at the point “0” (Figure 3). The orthogonal projections ($X_{{\pi }_1}$ , $Y_{{\pi }_1}$ ) of the velocity vector of the particle ${\pi }_1$ on the axes 0X, 0Y are called the Beltrami coordinates of the point “${\pi }_1$ ” in HVS [6] [9] [10]. The length $S_{b{\pi }_1}$ of the straight line segment ($b –{\pi }_1$ ) with the Beltrami coordinates of its ends “b” ($X_b$ , $Y_b$ ), “${\pi }_1$ ” ($X_{{\pi }_1}$ , $Y_{{\pi }_1}$ ) is represented by the formula [6]:

$\text{ch}\left(S_{b{\pi }_1}\right)=\left(1-X_{{\pi }_1}\ast X_b-Y_{{\pi }_1}\ast Y_b\right)/\left(R_{{\pi }_1}\ast R_b\right)$ (2)

$R_b=\sqrt{1-X_b^2-Y_b^2},R_{{\pi }_1}=\sqrt{1-X_{{\pi }_1}^2-Y_{{\pi }_1}^2}$

The length $S_{b{\pi }_1}$ of the straight line ($b–{\pi }_1$ ) is called the rapidity [11].

Besides the straight line ($b–{\pi }_1$ ), one pair of symmetrical arcs of zero curvature, called oricycles, passes through the velocity points “b” and “${\pi }_1$ ” [6]. An oricycle is a line of rotation with curvature 0, the center “$C_0$ ” of which lies among infinitely distant points of the HVS (infinitely distant points are referred to as the ABSOLUTE of the HVS). All oricycles in HVS are congruent as straight lines of zero curvature in Euclidean space are congruent [6]. Therefore, the ellipse with semi-axis ($C_0-0$ ) represents an oricycle in which the individual oricycles of the $\rho \left(770\right)$ and $f_o\left(980\right)$ meson decays are combined (Figure 1).

Archimedes’ laws of levers define a third point “m” on the arc of the oricycle, to which the additive mass of the decay particles is attributed. Point “m” with additive mass $m_{b{\pi }_1}=m_b+m_{{\pi }_1}$ ($m_b\ge m_{{\pi }_1}$ ) is defined by Equations (3). The role of forces in Archimedes’ levers is played by the masses $m_b$ and $m_{{\pi }_1}$ , and the lever arms are equal to the Euclidean lengths $l_{b{\pi }_1},l_{mb},l_{m{\pi }_1}$ of the arcs of the oricycle [12]-[14]:

$l_{b{\pi }_1}=l_{m{\pi }_1}+l_{mb}$ , $m_{b{\pi }_1}=m_b+m_{{\pi }_1}$

$m_b{l}_{mb}=m_{{\pi }_1}{l}_{m{\pi }_1}=m_{{\pi }_1}\left(l_{b{\pi }_1}-l_{mb}\right)$ (3)

$l_{m{\pi }_1}=m_b{l}_{b{\pi }_1}/\left(m_b+m_{{\pi }_1}\right)$

$l_{mb}=m_{{\pi }_1}{l}_{b{\pi }_1}/\left(m_b+m_{{\pi }_1}\right)$

Equations (3) follow from the fact that Euclidean geometry of zero curvature is valid on the oricycle.

Figure 3. The different sided triangles $\Delta bm{\pi }_1$ of reaction B + T → b + π₁ + all, the base isosceles triangle $\Delta {\pi }_{1}m{\pi }_2$ . The point “$C_0$ ” is the center of rotation of the oricycle. The point “$G_{b{\pi }_1}$ ” represents the center of inertia of the pair particles π₁ and b. The point-velocity “G” represents the centers of inertia of pairs of particles (${\pi }_1$ , ${\pi }_2$ ). A straight line drawn from point “G” through the midpoint “M” of side ($m–{\pi }_2$ ) intersects the oricycle at point “F”. The ratio of the arc lengths $l_{m{\pi }_2}$ to $l_{mF}$ of the oricycle is equal to the golden ratio 1.61803…

Connecting the points “${\pi }_1$ ”, “m”, “b” with each other by straight lines, we obtain a different sided triangle $\Delta bm{\pi }_1$ inscribed in the oricycle (Figure 3). The straight line ($C_0–m$ ) is called the axis of the oricycle [10]. The lengths of the sides of the triangle $\Delta bm{\pi }_1$ are related to the lengths of the arcs of the oricycle by the following relationships [6]:

$l_{b{\pi }_1}=2\text{sh}\left(S_{b{\pi }_1}/2\right)$ , $l_{m{\pi }_1}=2\text{sh}\left(S_{m{\pi }_1}/2\right)$ , $l_{bm}=2\text{sh}\left(S_{bm}/2\right)$ (4)

In Figure 3, in the Beltrami model of HVS, the ellipse tangent to the circle $X_{}^2+Y_{}^2=1$ is a oricycle with center of rotation at the points of tangency “$C_0$ ”. The circle $X_{}^2+Y_{}^2=1$ , called the Absolute, represents, according to formula (2), the infinitely distant points of the HVS.

The straight line ($m-G_{b{\pi }_1}$ ) passes through point “$C_0$ ” - the center of rotation of the oricycle (the point “$G_{b{\pi }_1}$ ” represents the center of inertia of the pair particles ${\pi }_1$ and $b$ ) (Figure 3). The infinitely large radius ($m-C_0$ ) is called the axis of the oricycle. Rotating the arc $l_{m{\pi }_1}$ of the oricycle around the axis ($m-C_0$ ), the point “${\pi }_1$ ” will move to the point “${\pi }_2$ ” on the oricycle. By connecting the points “${\pi }_1$ ” “m,” and “${\pi }_2$ ” with line segments, we obtain an isosceles triangle $\Delta {\pi }_{1}m{\pi }_2$ . Here, the line segment (${\pi }_1-{\pi }_2$ ) of the side of triangle $\Delta {\pi }_{1}m{\pi }_2$ is perpendicular to the axis ($m-C_0$ ) of the oricycle. Accordingly, the line segment ($G-m$ ) is the altitude of $\Delta {\pi }_{1}m{\pi }_2$ (point “G” is the midpoint of the side (${\pi }_1-{\pi }_2$ )). Thus, the pair of particles $b$ and ${\pi }_1$ in reaction (1) corresponds to a different-sided triangle $\Delta bm{\pi }_1$ and the isosceles triangle $\Delta {\pi }_{1}m{\pi }_2$ . And vice versa: every isosceles triangle $\Delta {\pi }_{1}m{\pi }_2$ corresponds to a different-sided triangle $\Delta bm{\pi }_1$ . Further, we will refer to the isosceles triangle $\Delta {\pi }_{1}m{\pi }_2$ as the base triangle. The base triangle $\Delta {\pi }_{1}m{\pi }_2$ , inscribed in an oricycle, has a very interesting property. A straight line drawn from point “G” through the midpoint “M” of side ($m-{\pi }_2$ ) intersects the oricycle at point “F” (Figure 3). In this case, the ratio of the arc lengths $l_{m{\pi }_2}$ to $l_{mF}$ of the oricycle is equal to the golden ratio: 1.61803… [12].

The effective mass $m_{res}$ of the pair of particles $b$ and ${\pi }_1$ is calculated using their rest masses $m_b$ and $m_{{\pi }_1}$ and the length $S_{b{\pi }_1}^{}$ of the side ($b–{\pi }_1$ ) of the triangle $\Delta bm{\pi }_1$ [5]:

$m_{res}^2=m_b^2+m_{{\pi }_1}^2+2m_b^{}{m}_{{\pi }_1}^{}\text{ch}{S}_{b{\pi }_1}^{}$ (5)

The point “${\pi }_2$ ” is assigned the rest mass $m_{{\pi }_1}$ of the ${\pi }_1$ meson ($m_{{\pi }_1}=m_{{\pi }_2}$ ). The base effective mass $m_{res}^{{\pi }_1,{\pi }_2}$ of the particles ${\pi }_1$ and ${\pi }_2$ is calculated using their rest masses and the length $S_{{\pi }_1{\pi }_2}^{}$ of the side (${\pi }_1-{\pi }_2$ ) of the base triangle $\Delta {\pi }_{1}m{\pi }_2$ [5]:

$m_{base}^{{\pi }_1,{\pi }_2}=\sqrt{m_{{\pi }_1}^2+m_{{\pi }_2}^2+2m_{{\pi }_1}^{}{m}_{{\pi }_2}^{}\text{ch}{S}_{{\pi }_1{\pi }_2}^{}}$ (6)

From the correspondence between the triangles $\Delta bm{\pi }_1$ and $\Delta {\pi }_{1}m{\pi }_2$ , it follows that the masses $m_{res}$ and $m_{base}^{{\pi }_1,{\pi }_2}$ are also in correspondence. Each value of the effective mass $m_{res}$ corresponds to a base mass $m_{base}^{{\pi }_1,{\pi }_2}$ . The reverse is also true: each value of the base mass $m_{base}^{{\pi }_1,{\pi }_2}$ corresponds to an effective mass $m_{res}$ .

To classify light unflavored, strange mesons and Δ, N, Λ, Σ, Ξ baryons of the reaction (1) used base triangles $\Delta {\pi }_{1}m{\pi }_2$ , obtained by rotating the arc $l_{m{\pi }_1}$ of the oricycle around the axis ($m-C_0$ ). For the base triangle $\Delta {\pi }_{1}m{\pi }_2$ , the hyperbolic sine $HS{T}_{res}$ (introduced by Y.P. Panarin [7]), defined as the product of the hyperbolic sine of the altitude dropped from any vertex and the hyperbolic sine of the side opposite that vertex (Figures 1-3):

$HS{T}_{res}=\text{sh}{S}_{Gm}\text{sh}{S}_{{\pi }_1{\pi }_2}^{}=\text{sh}{S}_{A{\pi }_1}\text{sh}{S}_{m{\pi }_2}^{}$ (7)

The resonance decay triangle in the HVS provides more information than the line segment whose length is used to calculate its effective mass. Therefore, the hyperbolic sine $HS{T}_{res}$ of the resonance decay triangle provides more information than its effective mass $m_{res}$ .

Using Equations (3)-(7) and standard transformations of hyperbolic functions, we obtain:

$\text{ch}\left(S_{b{\pi }_1}\right)=\left(m_{res}^2-m_{{\pi }_1}^2-m_b^2\right)/2m_{{\pi }_1}^{}{m}_b^{}$ , $\text{sh}\left(S_{b{\pi }_1}/2\right)=\sqrt{\left(\text{ch}\left(S_{b{\pi }_1}/2\right)-1\right)/2}$ (8)

$l_{b{\pi }_1}=2\text{sh}\left(S_{b{\pi }_1}/2\right)$ , $l_{m{\pi }_1}=m_b^{}{l}_{b{\pi }_1}/\left(m_{{\pi }_1}^{}+m_b^{}\right)=\text{sh}\left(S_{{\pi }_1{\pi }_2}/2\right)$ (9)

$\text{ch}\left(S_{{\pi }_1{\pi }_2}/2\right)=\sqrt{{\text{sh}}_{}^2\left(S_{{\pi }_1{\pi }_2}/2\right)+1}$ , $\text{sh}\left(S_{{\pi }_1{\pi }_2}\right)=2\text{sh}\left(S_{{\pi }_1{\pi }_2}/2\right)\text{ch}\left(S_{{\pi }_1{\pi }_2}/2\right)$ (10)

$\text{sh}\left(S_{m{\pi }_1}/2\right)=l_{m{\pi }_1}/2$ , $\text{ch}\left(S_{m{\pi }_1}/2\right)=\sqrt{{\text{sh}}_{}^2\left(S_{m{\pi }_1}/2\right)+1}$ ,(11)

$\text{sh}\left(S_{m{\pi }_1}\right)=2\text{sh}\left(S_{m{\pi }_1}/2\right)\text{ch}\left(S_{m{\pi }_1}/2\right)$ , $\text{ch}\left(S_{m{\pi }_1}\right)=\sqrt{{\text{sh}}_{}^2\left(S_{m{\pi }_1}\right)+1}$ (12)

From the right-angled triangle $\Delta Gm{\pi }_1$ we obtain:

$\text{ch}\left(S_{Gm}\right)=\text{ch}\left(S_{m{\pi }_1}^{}\right)/\text{ch}\left(S_{{\pi }_1{\pi }_2}/2\right)$ , $\text{sh}\left(S_{Gm}\right)=\sqrt{{\text{ch}}_{}^2\left(S_{Gm}\right)-1}$ (13)

The integer values of the hyperbolic sine of a triangle ($HS{T}_{her}$ ) correspond to base triangles $\Delta {\pi }_{1}m{\pi }_2$ that we have named Heron’s base triangles. The integer values of $HS{T}_{her}$ for the Heron’s base triangles $\Delta {\pi }_{1}m{\pi }_2$ was used to classify scalar, strange mesons and Δ, N, Λ, Σ, Ξ baryons, (Tables 1-17).

In the case of the decay of a Light Unflavored Meson into ${\pi }_1$ and ${\pi }_2$ mesons with equal masses, Equations (3)-(13) take the form ($m_b=m_{{\pi }_2}=m_{{\pi }_1}$ ) (Figure 1) [12]-[14]:

$l_{{\pi }_1{\pi }_2}^{}=l_{m{\pi }_1}^{}+l_{m{\pi }_2}^{}$ , $m_{{\pi }_1{\pi }_2}=m_{{\pi }_1}+m_{{\pi }_2}$

$m_{{\pi }_1}{l}_{m{\pi }_1}^{}=m_{{\pi }_2}{l}_{m{\pi }_2}^{}=m_{{\pi }_1}\left(l_{{\pi }_1{\pi }_2}^{}-l_{m{\pi }_1}^{}\right)$ (14)

$l_{m{\pi }_1}^{}=m_{{\pi }_2}{l}_{{\pi }_1{\pi }_2}^{}/\left(m_{{\pi }_2}+m_{{\pi }_1}\right)=l_{m{\pi }_1}^{}=m_{{\pi }_1}{l}_{{\pi }_1{\pi }_2}^{}/\left(m_{{\pi }_1}+m_{{\pi }_2}\right)$

The effective mass $m_{her}^{{\pi }_1,{\pi }_2}$ is calculated using the base length $S_{{\pi }_1{\pi }_2}^{}$ of the Heron’s triangle $\Delta {\pi }_{1}m{\pi }_2$ :

$m_{her}^{{\pi }_1,{\pi }_2}=\sqrt{m_{{\pi }_1}^2+m_{{\pi }_2}^2+2m_{{\pi }_1}^{}{m}_{{\pi }_2}^{}\text{ch}{S}_{{\pi }_1{\pi }_2}^{}}$ (15)

The hyperbolic sine $HS{T}_{res}$ of the base triangle $\Delta {\pi }_{1}m{\pi }_2$ (Figure 2):

$HS{T}_{res}=\text{sh}{S}_{Gm}\text{sh}{S}_{{\pi }_1{\pi }_2}^{}$ (16)

The hyperbolic sine $HS{T}_{her}$ of the Heron’s base triangle $\Delta {\pi }_{1}m{\pi }_2$ rounded to the nearest integer of $HS{T}_{res}$ . Using Equations (3)-(7) and standard transformations of hyperbolic functions, we obtain:

Table 1. Hyperbolic Heron triangles in decay of $\text{Scalar Mezon}\to {\pi }_1+{\pi }_2$ .

$\text{ch}\left(S_{{\pi }_1{\pi }_2}\right)=\left(m_{res}^2-m_{{\pi }_1}^2-m_{{\pi }_2}^2\right)/2m_{{\pi }_1}^{}{m}_{{\pi }_2}^{}$ , $\text{sh}\left(S_{{\pi }_1{\pi }_2}^{}\right)=\sqrt{{\text{ch}}_{}^2\left(S_{{\pi }_1{\pi }_2}^{}\right)-1}$ (17)

$l_{{\pi }_1{\pi }_2}=2\text{sh}\left(S_{{\pi }_1{\pi }_2}/2\right)$ , $l_{m{\pi }_1}=\text{sh}\left(S_{G{\pi }_1}^{}\right)=m_{{\pi }_1}^{}{l}_{{\pi }_1{\pi }_2}/\left(m_{{\pi }_1}^{}+m_{{\pi }_2}^{}\right)$ (18)

$\text{sh}\left(S_{G{\pi }_1}\right)=\text{sh}\left(S_{{\pi }_1{\pi }_2}/2\right)=\sqrt{\left(\text{ch}\left(S_{{\pi }_1{\pi }_2}\right)-1\right)/2}$ ,(19)

$\text{sh}\left(S_{m{\pi }_1}/2\right)=l_{m{\pi }_1}/2=\text{sh}\left(S_{{\pi }_1{\pi }_2}/2\right)/2$ , $\text{ch}\left(S_{m{\pi }_1}/2\right)=\sqrt{{\text{sh}}_{}^2\left(S_{m{\pi }_1}/2\right)+1}$

$\text{sh}\left(S_{m{\pi }_1}\right)=2\text{sh}\left(S_{m{\pi }_1}/2\right)\text{ch}\left(S_{m{\pi }_1}/2\right)$ , $\text{ch}\left(S_{m{\pi }_1}\right)=\sqrt{{\text{sh}}_{}^2\left(S_{m{\pi }_1}\right)+1}$

From the right-angled triangle $\Delta Gm{\pi }_1$ we obtain:

$\text{ch}\left(S_{Gm}\right)=\text{ch}\left(S_{m{\pi }_1}^{}\right)/\text{ch}\left(S_{{\pi }_1{\pi }_2}/2\right)$ , $\text{sh}\left(S_{Gm}\right)=\sqrt{{\text{ch}}_{}^2\left(S_{Gm}\right)-1}$ (20)

In Table 1, based on data from Table A1, the values of $HS{T}_{res}$ for the decay $\text{Scalar Mezon}\to {\pi }_1+{\pi }_2$ in column 4 were calculated using Equations (14)-(20). Columns 2 and 3 of Table 1 list the values of the effective masses $m_{res}$ and widths Γ of scalar mesons from Table A1. Columns 4, 5 list the $HS{T}_{res}$ , $HS{T}_{her}$ values ($HS{T}_{her}$ value for Heron’s triangles is rounded to the nearest integer $HS{T}_{res}$ value). The base mass $m_{her}^{{\pi }_1,{\pi }_2}$ (column 6) is calculated using Equation (14) (the points “${\pi }_1$ ” and “${\pi }_2$ ” on the side (${\pi }_1-{\pi }_2$ ) of the Heron’s base triangle $\Delta {\pi }_{1}m{\pi }_2$ are assigned a mass of $m_{{\pi }_1}=m_{{\pi }_2}$ ). Column 7 lists the absolute deviation $\text{Δ}{m}_{res}^{her}$ of the base mass $m_{her}^{{\pi }_1,{\pi }_2}$ from the published values $m_{res}$ . Column 8 lists the ratio of the absolute deviation $\text{Δ}{m}_{res}^{her}$ to the width Γ, expressed as a percentage. Table 1 shows that the values of the base masses $m_{her}^{{\pi }_1,{\pi }_2}$ for Heron’s triangles agree well with the published values of $m_{res}$ (the average percentage (AP) based on Table 1 equals 3.5%). Note that, overall, the agreement between the mass $m_{her}^{{\pi }_1,{\pi }_2}$ and the measured mass $m_{res}$ lies within the measurement error $\text{Δ}{m}_{res}$ . The main contribution to the calculation error—58%—comes from the $\varphi \left(1020\right)$ meson, whose measured width is very small (4.25 MeV).

In the case of the decay of a Light Unflavored Mezon into daughter mesons $\eta$ and ${\pi }_1$ with different masses ($m_b=m_{\eta }$ , $m_{\eta }>m_{{\pi }_1}$ ), Equation (3) takes the following form (point “$m$ ” with mass $m_{\eta {\pi }_1}=m_{\eta }+m_{{\pi }_1}$ is shifted toward point “$\eta$ ” of the particle with the greater mass) (Figure 2) [12]-[14]:

$l_{\eta {\pi }_1}=l_{m{\pi }_1}+l_{m\eta }$ , $m_{\eta {\pi }_1}=m_{\eta }+m_{{\pi }_1}$

$m_{\eta }{l}_{m\eta }=m_{{\pi }_1}{l}_{m{\pi }_1}=m_{{\pi }_1}\left(l_{\eta {\pi }_1}-l_{m\eta }\right)$ (21)

$l_{m{\pi }_1}=m_{\eta }{l}_{\eta {\pi }_1}/\left(m_{\eta }+m_{{\pi }_1}\right)$

$l_{m\eta }=m_{{\pi }_1}{l}_{\eta {\pi }_1}/\left(m_{\eta }+m_{{\pi }_1}\right)$

By connecting the points “${\pi }_1$ ”, “m” and “$\eta$ ” with straight lines, we obtain a different-sided triangle $\Delta \eta m{\pi }_1$ . The lengths of the sides of the triangle $\Delta \eta m{\pi }_1$ are related to the lengths of the arcs of the oricycle by the following relationships [10]:

$l_{\eta {\pi }_1}=2\text{sh}\left(S_{\eta {\pi }_1}^{}/2\right)$ , $l_{m{\pi }_1}=2\text{sh}\left(S_{m{\pi }_1}/2\right)$ , $l_{m\eta }=2\text{sh}\left(S_{m\eta }/2\right)$ (22)

By rotating the arc $l_{m{\pi }_1}$ of the oricycle around the axis ($m-C_0$ ), the point “${\pi }_1$ ” will move to the point “${\pi }_2$ ” on the oricycle, we obtain an isosceles base triangle $\Delta {\pi }_{1}m{\pi }_2$ , corresponding to the different-sided triangle $\Delta \eta m{\pi }_1$ (Figure 2).

The effective mass $m_{res}$ of the particle pair $\eta$ and ${\pi }_1$ is calculated using their rest masses $m_{\eta }$ and $m_{{\pi }_1}$ and the length $S_{\eta {\pi }_1}^{}$ of the side ($\eta -{\pi }_1$ ) of the triangle $\Delta \eta m{\pi }_1$ :

$m_{res}^2=m_{{\pi }_1}^2+m_{\eta }^2+2m_{{\pi }_1}^{}{m}_{\eta }^{}\text{ch}\left(S_{\eta {\pi }_1}^{}\right)$ (23)

The $HS{T}_{res}$ for the base triangle $\Delta {\pi }_{1}m{\pi }_2$ of the $\text{Scalar Mezon}\to \eta +{\pi }_1$ decay (Figure 2):

$HS{T}_{res}=\text{sh}\left(S_{Gm}\right)\text{sh}\left(S_{{\pi }_1{\pi }_2}^{}\right)$ (24)

Using Equations (3)-(13) and standard transformations of hyperbolic functions, we obtain:

$\text{ch}\left(S_{\eta {\pi }_1}\right)=\left(m_{res}^2-m_{{\pi }_1}^2-m_{\eta }^2\right)/2m_{{\pi }_1}^{}{m}_{\eta }^{}$ , $\text{sh}\left(S_{\eta {\pi }_1}/2\right)=\sqrt{\left(\text{ch}\left(S_{\eta {\pi }_1}/2\right)-1\right)/2}$ (25)

$l_{\eta {\pi }_1}=2\text{sh}\left(S_{\eta {\pi }_1}/2\right)$ , $l_{m{\pi }_1}=m_{\eta }^{}{l}_{\eta {\pi }_1}/\left(m_{{\pi }_1}^{}+m_{\eta }^{}\right)$ (26)

$\text{sh}\left(S_{m{\pi }_1}/2\right)=l_{m{\pi }_1}/2$ , $\text{ch}\left(S_{m{\pi }_1}/2\right)=\sqrt{{\text{sh}}_{}^2\left(S_{m{\pi }_1}/2\right)+1}$ ,

$\text{sh}\left(S_{m{\pi }_1}\right)=2\text{sh}\left(S_{m{\pi }_1}/2\right)\text{ch}\left(S_{m{\pi }_1}/2\right)$ , $\text{ch}\left(S_{m{\pi }_1}\right)=\sqrt{{\text{sh}}_{}^2\left(S_{m{\pi }_1}\right)+1}$ (27)

$\text{sh}\left(S_{{\pi }_1{\pi }_2}^{}\right)=2\text{sh}\left(S_{m{\pi }_1}^{}\right)\text{ch}\left(S_{m{\pi }_1}^{}\right)$ , $\text{ch}\left(S_{{\pi }_1{\pi }_2}^{}\right)=\sqrt{{\text{sh}}_{}^2\left(S_{{\pi }_1{\pi }_2}^{}\right)+1}$ (28)

From the right-angled triangle $\Delta Gm{\pi }_1$ we obtain:

$\text{ch}\left(S_{Gm}\right)=\text{ch}\left(S_{m{\pi }_1}^{}\right)/\text{ch}\left(S_{{\pi }_1{\pi }_2}/2\right)$ , $\text{sh}\left(S_{Gm}\right)=\sqrt{{\text{ch}}_{}^2\left(S_{Gm}\right)-1}$ (29)

Table 2. Hyperbolic Heron triangles in decays of $\text{Scalar Mezon}\to \eta +{\pi }_1$ .

Table 2 shows the results of the calculations for the $\text{Scalar Mezon}\to \eta +{\pi }_1$ decay triangles. Column 4 lists the $HS{T}_{res}$ values calculated using Equations (21)-(29) based on data from Table A1. Columns 2 and 3 of Table 2 list the values of the effective masses $m_{res}$ and widths Γ from Table A1. Columns 4, 5 list the $HS{T}_{res}$ and $HS{T}_{her}$ values ($HS{T}_{her}$ value for Heron’s triangles is rounded to the nearest integer $HS{T}_{res}$ value). Column 6 lists the values of the effective masses $m_{her}^{\eta ,{\pi }_1}$ in triangles $\Delta \eta m{\pi }_1$ . Column 7 lists the absolute deviation $\text{Δ}{m}_{res}^{her}$ of the effective mass $m_{her}^{\eta ,{\pi }_1}$ from the published values $m_{res}$ . Column 8 lists the ratio of the absolute deviation $\text{Δ}{m}_{res}^{her}$ to the width Γ, expressed as a percentage. Column 9 lists the values of the effective masses $m_{her}^{{\pi }_1,{\pi }_2}$ in Heron’s triangles $\Delta {\pi }_{1}m{\pi }_2$ ${\pi }_2$ . Table 2 shows that the values of the effective masses $m_{her}^{\eta ,{\pi }_1}$ agree well with the published values of $m_{res}$ (the average percentage (AP) based on Table 2 equals 1.2%). From Equation (28), we write the following expression:

$\text{ch}\left(S_{{\pi }_1{\pi }_2}\right)=\sqrt{{\text{sh}}_{}^2\left(S_{{\pi }_1{\pi }_2}\right)+1}$

Column 5 lists, separated by a slash, the value of the hyperbolic cosine $\text{ch}\left(S_{{\pi }_1{\pi }_2}\right)$ of the side (${\pi }_1-{\pi }_2$ ) of Heron’s base triangle $\Delta {\pi }_{1}m{\pi }_2$ . For certain values of $HS{T}_{her}$ , the hyperbolic cosines $\text{ch}\left(S_{{\pi }_1{\pi }_2}\right)$ of the side (${\pi }_1-{\pi }_2$ ) of the triangles $\Delta {\pi }_{1}m{\pi }_2$ are also equal to integer values. The integer value of $\text{ch}\left(S_{{\pi }_1{\pi }_2}\right)$ for a Heron triangle corresponds to the integer value of the oricyclic cotangent of the triangle, as introduced in [15].

Table 3. Hyperbolic Heron triangles in decays of $\text{Scalar Mezon}\to \rho \left(770\right)+{\pi }_1$ .

Table 3 shows the results of the calculations for the $\text{Scalar Mezon}\to \rho \left(770\right)+{\pi }_1$ decay triangles. Column 4 lists the $HS{T}_{res}$ values calculated using Equations (3)-(13) based on data from Table A1 ($m_b=m_{\rho \left(770\right)}$ ). Columns 2 and 3 of Table 2 list the values of the effective masses $m_{res}$ and widths Γ from Table A1. Column 5 lists the value of $HS{T}_{her}$ for Heron’s triangles $\Delta {\pi }_{1}m{\pi }_2$ , rounded to the nearest integer of $HS{T}_{res}$ . The nearest integer for values of $HS{T}_{res}$ < 1.1 was determined using inversion. Namely, an inverse value of $HS{T}_{res}^i$ was specified with the coefficient: