Experimental Observation of Hyperbolic Heron Triangles in the Decays of Scalar, Strange Mesons, and Δ, N, Λ, Σ, Ξ Baryons, and the Investigation of New Resonances

Valeriy Pavlovich Khеn, Alexei Valerevich Khen

Limited Liability Partnership “Industry 4.0”, Almaty, Kazakhstan.

DOI: 10.4236/jamp.2026.144085                

Abstract

The endpoints of the velocity vectors of two resonance decay particles represent material velocity points in the hyperbolic Lobachevsky velocity space of curvature k = −1/C² (C = 1 is the speed of light; the points are assigned the masses of the decay particles). Two particle velocity points can be connected by a straight line segment and an arc of constant zero curvature, called the oricycle. Archimedes’ laws of levers define a third point on the oricycle arc, to which an additive mass (the sum of the decay particle masses) is assigned. By connecting these 3 points with straight line segments, we get a triangle of resonance decay inscribed in the oricycle. The effective mass $m_r$ of the resonance is determined by the hyperbolic cosine of the length of one side of its decay triangle and the masses of the decay particles. A Lorentz invariant function called the oricyclic cotangent of the triangle (OCT) is introduced on the triangles of resonance decays (OCT is based on the arc of the oricycle and the angle of the triangle). Using published data on the effective masses $m_r$ of scalar, strange mesons and Δ, N, Λ, Σ, Ξ baryons, the function $OC{T}_r$ and its nearest integer value $OC{T}_M$ were calculated. For triangles with integer values of $OC{T}_M$ , the hyperbolic cosines of the side lengths are also equal to integers. Therefore, triangles with integer values of $OC{T}_M$ are called hyperbolic Heron triangles. The effective masses $m_{her}$ , corresponding to Heron’s triangle, differ from the masses $m_r$ of scalar, strange mesons and Δ, N, Λ, Σ, Ξ baryons within the resonance widths. These results provide grounds for considering the listed resonances as a lattice structure of Heron triangles with integer $OC{T}_M$ . Then, if statistically significant peaks are detected in the distribution for $OC{T}_M<7$ , up to 6 new resonances with masses < 1 GeV may be detected, which would explain the large variance in the measurements of the masses and widths of scalar mesons. In the discrete spectrum at $OC{T}_M>500$ , new resonances with masses > 10 GeV can be detected.

Keywords

Lobachevsky Velocity Space, Resonance Decay Triangles, Oricyclic Cotangent of a Triangle, Hyperbolic Heron Triangle, Lattice Structure of Hyperbolic Heron Triangle, Quantization Resonance Decay

1. Introduction

In inelastic reactions at high energies, the particle velocity vectors are measured in some frame of reference. The ends of the velocity vectors represent material points-velocities in the velocity space located inside a sphere of radius C (C is the speed of light, the points-velocities are assigned rest masses of the particles) [1]-[6]. The Lorentz group defines the Lobachevsky-Bolyai geometry of negative curvature k = −1/C² in the velocity space [1]-[4]. The material points-velocities inside a sphere of radius C represent the Lorentz invariant geometric image of inelastic reaction kinematics in hyperbolic Lobachevsky velocity space (HLVS) (further everywhere the speed of light C = 1) [5]-[8]. Two material points-velocities of resonance decay particles in HLVS can be connected by a straight line segment and an arc of a line of zero curvature, called an oricycle [9]. Archimedes’ lever laws (3), (10) define a third point-velocity on the oricycle arc, to which an additive mass (the sum of the masses of the decay particles) is assigned. By connecting the three points by straight line segments, we obtain triangles of resonance decays inscribed in the oricycle.

Figure 1 in the Beltrami model of HLVS shows the triangles of scalar meson decays inscribed in the oricycle [7]. The circle $X^2+Y^2=1$ , called the Absolute, represents infinitely distant points of HLVS. The effective mass $m_r$ of a resonance is determined by the hyperbolic cosine of the length of one side of its decay triangle and the masses of the decay particles (2). For resonance decay triangles, function (4) is introduced—the product of the arc length of the oricycle and the cotangent of half the angle. This function is called the oricyclic cotangent of a triangle (OCT). Using published data on the masses $m_r$ of scalar and strange mesons and Δ, N, Λ, Σ, Ξ baryons, the function $OC{T}_r$ and its nearest integer value $OC{T}_r=OC{T}_M$ are calculated. Integer values of $OC{T}_M$ correspond to triangles for which the sum of the hyperbolic cosines of the side lengths and the hyperbolic cosines of the base lengths are also integers. Therefore, triangles with integer values of $OC{T}_M$ are called hyperbolic Heron triangles [9]-[12]. The effective masses $m_{her}^{}$ , corresponding to Heron’s triangle, differ from the masses $m_r$ of the decay triangles of the listed resonances within their widths (Tables 1-13) [13]. These results provide grounds for considering the listed resonances as a lattice structure of Heron triangles with integer (Figure 4). In this approach, many two-particle (π, π), (p, π), (n, π), (η, π), (ρ(770), π), (ω(782), π), (Δ(1232), π), (Σ, π), (Λ, π), (Ξ⁰, π), (k493, π), (k892, π) decays of new resonances can be detected experimentally (Table 14, Table 15). These may be resonances with both small masses (<1 GeV) and resonances with large masses (>10 GeV).

Figure 1. Decays of scalar mesons in the Beltrami model of the Lobachevsky velocity space. The separate ellipses of decay oricycles of $\rho \left(770\right)$ , $f_o\left(980\right)$ scalar mesons with centers in the “$C_0$ ” points of the circle $X^2+Y^2=1,$ called the Absolute, and Δπ₁mπ₂ tringles of $\rho \left(770\right)$ , $f_o\left(980\right)$ scalar mesons decays, combined into one oricycle with the center at the point “$C_0$ ” (1, 0) on the Absolute. The point-velocity “G” represents the centers of inertia of pairs of particles $\left({\pi }_1,{\pi }_2\right)$ , (P, π₁).

2. Decay of Scalar Mesons with Equal Masses of Decay Particles

Suppose that the velocities of particles ${\pi }_1$ and ${\pi }_2$ decay of a scalar meson in some reference frame “0” are measured. The ends of the particle velocity vectors represent material points-velocities “${\pi }_1$ ” and “${\pi }_2$ ” in the hyperbolic Lobachevsky velocity space (HLVS) located inside a sphere of radius C (hereinafter the speed of light C = 1, the masses $m_{{\pi }_1}=m_{{\pi }_2}$ of decay particles are attributed to the points “${\pi }_1$ ” and “${\pi }_2$ ”) [5]-[7]. Let’s draw a plane through the points “0”, “${\pi }_1$ ”, “${\pi }_2$ ”. In this plane, we introduce a rectangular coordinate system X0Y with the origin at the point “0” (Figure 1). 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 HLVS. The length $S_{{\pi }_1{\pi }_2}$ of the line segment (${\pi }_1-{\pi }_2$ ) with the Beltrami coordinates of its ends “${\pi }_1$ ” ($X_{{\pi }_1}$ , $Y_{{\pi }_1}$ ), “${\pi }_2$ ” ($X_{{\pi }_2}$ ,$Y_{{\pi }_2}$ ) is represented by the formula [9]:

$\text{ch}\left(S_{{\pi }_1{\pi }_2}\right)=\left(1-X_{{\pi }_1}\ast X_{{\pi }_2}-Y_{{\pi }_1}\ast Y_{{\pi }_2}\right)/\left(R_{{\pi }_1}\ast R_{{\pi }_2}\right)$ (1)

$R_{{\pi }_2}=\sqrt{1-X_{{\pi }_2}^2-Y_{{\pi }_2}^2},R_{{\pi }_1}=\sqrt{1-X_{{\pi }_1}^2-Y_{{\pi }_1}^2}$

The length $S_{{\pi }_1{\pi }_2}$ of the line segment (${\pi }_1-{\pi }_2$ ) is called the rapidity [14]. The effective mass $m_r$ of a scalar meson is calculated using the formula [6]:

$m_r^2=m_{{\pi }_1}^2+m_{{\pi }_2}^2+2m_{{\pi }_1}{m}_{{\pi }_2}\text{ch}\left(S_{{\pi }_1{\pi }_2}\right),\frac{m_r^2-2m_{{\pi }_1}^2}{2m_{{\pi }_1}^2}=\text{ch}\left(S_{{\pi }_1{\pi }_2}\right)$ (2)

Besides the straight line (${\pi }_1-{\pi }_2$ ), one pair of symmetrical arcs of zero curvature, called oricycles, passes through the velocity points “${\pi }_1$ ” and “${\pi }_2$ ”. In Figure 1, in the Beltrami model of HLVS, the ellipses tangent to the circle $X^2+Y^2=1$ are oricycles with centers of rotation at the points of tangency “$C_0$ ”. The straight line connecting the center of rotation “$C_0$ ” with an arbitrary point of the oricycle is called its axis. The circle $X^2+Y^2=1$ , called the Absolute, represents, according to formula (1), the infinitely distant points of the HLVS.

All oricycles in HLVS are congruent as straight lines of zero curvature in Euclidean space are congruent [9]. Thus, the ellipse with axis ($C_0-0$ ) in Figure 1 represents an oricycle, which combines the oricycles of the decays of individual scalar mesons.

The point “m” on the oricycle with additive mass $m_{{\pi }_1{\pi }_2}=m_{{\pi }_1}+m_{{\pi }_2}$ are determined by Archimedes’ laws of levers (3). The roles of forces in the levers are played by the masses $m_{{\pi }_1}$ and $m_{{\pi }_2}$ , and the arms of the levers are equal to the Euclidean lengths $l_{{\pi }_1{\pi }_2},l_{m{\pi }_1},l_{m{\pi }_2}$ of the arcs of the oricycle [6] [10]-[12]. For the case of equal rest masses of particles ${\pi }_1$ , ${\pi }_2$ ($m_{{\pi }_1}=m_{{\pi }_2}$ ) the point “m” lies in the center of the arc (${\pi }_1$ , m, ${\pi }_2$ ) of the oricycle (Figure 1):

$l_{{\pi }_1{\pi }_2}=l_{m{\pi }_1}+l_{m{\pi }_2},m_{{\pi }_1{\pi }_2}=m_{{\pi }_1}+m_{{\pi }_2}=2m_{{\pi }_1}$

$m_{{\pi }_1}{l}_{m{\pi }_1}=m_{{\pi }_2}{l}_{m{\pi }_2}=m_{{\pi }_1}{l}_{{\pi }_1{\pi }_2}/\left(m_{{\pi }_1}+m_{{\pi }_2}\right)=m_{{\pi }_1}{l}_{{\pi }_1{\pi }_2}/2$ (3)

Connecting the points “${\pi }_1$ ”, “m”, “${\pi }_2$ ” with each other by straight line segments, we obtain an isosceles triangle $\Delta {\pi }_{1}m{\pi }_2$ of the $f_o\left(980\right)$ meson decays inscribed in the oricycle (Figure 1). On the triangle $\Delta {\pi }_{1}m{\pi }_2$ we introduce a dimensionless Lorentz invariant function:

$OC{T}_r=l_{{\pi }_1{\pi }_2}\text{ctg}\left(\frac{M}{2}\right)={\text{sh}}^2\left(\frac{S_{{\pi }_1{\pi }_2}}{2}\right)={\text{sh}}^2\left(S_{G{\pi }_1}\right),l_{{\pi }_1{\pi }_2}=2\ast \text{sh}\left(S_{G{\pi }_1}\right)$ (4)

where $l_{{\pi }_1{\pi }_2}$ is the length of the oricycle arc subtending the base (${\pi }_1-{\pi }_2$ ) with a rapidity $S_{{\pi }_1{\pi }_2}$ , M is the angle at the vertex “m”, the “G” point represents the center of inertia of the pairs (${\pi }_1$ , ${\pi }_2$ ) of decay particles. The function $OC{T}_r$ is named oricyclic cotangent of a triangle.

In Table 1, the $OC{T}_r$ values are calculated from published data on the effective masses of Scalar Mezon $\to {\pi }_1+{\pi }_2$ decays [13]. Columns 2 and 3 of Table 1 present the values of the effective masses $m_r$ and the widths Γ of the Scalar Mezon $\to {\pi }_1+{\pi }_2$ decays [13]. According to formula (2), the mass $m_r$ corresponds to the rapidity $S_{{\pi }_1{\pi }_2}$ of the bases (${\pi }_1-{\pi }_2$ ) of the triangles $\Delta {\pi }_{1}m{\pi }_2$ of the decays of Scalar Mesons. Columns 3 and 4 of Table 1 show the values of the function $OC{T}_r$ , calculated using formula (4), and its nearest integer value $OC{T}_M$ . Integer values of $OC{T}_M$ correspond to triangles $\text{Δ}\pi m\pi$ (Figure 1). Calculations have shown that when $OC{T}_M=L$ , where $L$ is an integer, then the lengths $S_{m\pi }$ and $S_{\pi \pi }$ of the lateral side and base of the triangle $\Delta \pi m\pi$ are related by the relations:

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

Name Scalar Mezon	Mass Resonance $m_r$ (Mev)	Width Γ (Mev)	$OC{T}_r$	$OC{T}_M$	Mass in H $\text{Δ}\pi m\pi$ $m_{her}^{\pi ,\pi }$ (Mev)	$\text{Δ}{m}_r^{her}$ = $m_r$ − $m_{her}^{\pi ,\pi }$ (Mev)	$\frac{\Delta m_r^{her}}{\Gamma }$ (%)
1	2	3	4	5	6	7	8
$\text{Scalar Mezon}\to {\pi }_1+{\pi }_2$ , $m_{{\pi }_1}=m_{{\pi }_2}=m_{\pi }=139.57001\text{Mev}$
$f_o\left(500\right)$	500.0 ± 100.0	300.0	${2.21}_{-1.155}^{+1.412}$	2	483.5	16.5	5.5
$\rho \left(770\right)$	766.5 ± 1.1	150.0	${6.54}_{-0.022}^{+0.022}$	7	789.5	23.0	15.4
$\omega \left(782\right)$	782.7 ± 0.13	8.68	${6.86}_{-0.026}^{+0.026}$	7	789.5	6.8	78.6
$f_o\left(980\right)$	980.0 ± 20.0	55.0	${11.58}_{-0.503}^{+0.513}$	12	1006.4	16.4	29.9
$\varphi \left(1020\right)$	1019.46 ± 0.02	4.25	${12.34}_{-0.0003}^{+0.0005}$	12	1006.4	13.0	307.0
$f_2\left(1270\right)$	1275.4 ± 0.80	186.6	${19.88}_{-0.026}^{+0.026}$	20	1279.2	3.8	2.0
$f_o\left(1370\right)$	1350.0 ± 50.0	350.8	${22.39}_{-1.700}^{+1.765}$	22	1338.7	11.3	3.2
$f_o\left(1500\right)$	1522.0 ± 25.0	108.0	${28.73}_{-0.968}^{+0.985}$	29	1528.9	6.9	6.4
${\rho }_3\left(1690\right)$	1686.0 ± 4.0	161.0	${35.48}_{-0.173}^{+0.173}$	35	1674.8	11.2	6.9
$\rho \left(1700\right)$	1720.0 ± 20.0	250.0	${36.97}_{-0.877}^{+0.888}$	37	1720.7	0.7	0.3
$f_o\left(1710\right)$	1733.0 ± 8.0	150.0	${37.54}_{-0.355}^{+0.357}$	38	1743.2	10.2	6.8
$f_o\left(1770\right)$	1804.0 ± 16.0	138.0	${40.77}_{-0.737}^{+0.745}$	41	1809.0	5.0	3.7
$f_2\left(1810\right)$	1815.0 ± 12.0	197.0	${41.28}_{-0.557}^{+0.561}$	41	1809.0	6.0	3.0
$f_2\left(1950\right)$	1936.0 ± 12.0	197.0	${47.10}_{-0.594}^{+0.591}$	47	1933.9	2.1	0.4
$f_0\left(2020\right)$	1982.0 ± 54.0	464.0	${49.42}_{-2.708}^{+2.785}$	49	1973.8	8.2	1.9
$f_4\left(2050\right)$	2018.0 ± 11.0	237.0	${51.26}_{-0.568}^{+0.572}$	52	2032.2	14.2	6.0
${\rho }_2\left(2250\right)$	2248.0 ± 17.0	185.0	${63.86}_{-0.975}^{+0.985}$	64	2250.5	2.5	1.4
${\rho }_5\left(2350\right)$	2330.0 ± 35.0	400.0	${68.67}_{-2.077}^{+2.109}$	69	2335.4	5.5	1.4
$f_6\left(2510\right)$	2470.0 ± 50.0	260.0	${77.30}_{-3.136}^{+3.204}$	78	2481.0	11.0	4.3

$OC{T}_M=l_{\pi \pi }\ast \text{ctg}\left(M/2\right)=L,l_{\pi \pi }=2\text{sh}\left(S_{\pi \pi }/2\right)$

$\text{ch}{S}_{m\pi }=\left(L+2\right)/2,\text{ch}{S}_{\pi \pi }=2L+1$ (5)

Therefore, triangles $\text{Δ}\pi m\pi$ with integer values of $OC{T}_M$ are called hyperbolic Heron triangles [10] [12]. The effective mass $m_{her}^{}=m_{her}^{\pi ,\pi }$ (column 6) is calculated using formula (6) (points “$\pi$ ”, “$\pi$ ” of the base ($\pi -\pi$ ) of triangle $\Delta \pi m\pi$ are assigned the mass $m_{{\pi }_{}}=m_{{\pi }_1}$ ):

$m_{her}=m_{her}^{\pi ,\pi }=\sqrt{2m_{\pi }^2\left(1+\text{ch}\left(S_{\pi \pi }\right)\right)},\text{ch}{S}_{\pi \pi }=\left(m_{her}^2-2m_{\pi }^2\right)/2m_{\pi }^2$ (6)

where $S_{\pi \pi }$ is the rapidity of the base ($\pi -\pi$ ) of triangle $\text{Δ}\pi m\pi$ . Columns 7 and 8 of Table 1 show the absolute and relative deviations of the mass $m_{her}^{\pi ,\pi }$ from the experimental values $m_r$ . From Table 1 it can be seen that the masses $m_{r }$ differ from masses $m_{her}^{\pi ,\pi }$ within the resonance widths. The maximum deviation is 307% of the resonance width, the minimum is 0.3% of the resonance width, and the average deviation is 25.3% of the resonance width. Only the $\varphi \left(1020\right)$ meson mass $m_r$ differs from the $m_{her}^{\pi ,\pi }$ mass by 307% widths. In the experiment, the very small value of the $\varphi \left(1020\right)$ meson width (4.25 MeV) was obtained from the parameterization. Direct calculation of the $OC{T}_M$ values using real data might yield an acceptable result. Or the production of the $\varphi \left(1020\right)$ meson is not associated with Heron’s triangles.

It should be noted that Archimedes’ levers in HLVS were first used by N.A. Chernikov, who used the following expressions for the momenta $P_{G{\pi }_1}$ and $P_{G{\pi }_2}$ and kinetic energies $T_{G{\pi }_1}$ and $T_{G{\pi }_2}$ of particles ${\pi }_1$ and ${\pi }_2$ in the system of their center of mass “G” (Figure 1) [6]:

$P_{G{\pi }_1}=m_{{\pi }_1}\text{sh}\left(S_{G{\pi }_1}\right)=P_{G{\pi }_2}=m_{{\pi }_2}\text{sh}\left(S_{G{\pi }_2}\right)$ (7)

$T_{G{\pi }_1}=m_{{\pi }_1}\left(\text{ch}\left(S_{G{\pi }_1}\right)-1\right),T_{G{\pi }_2}=m_{{\pi }_2}\left(\text{ch}\left(S_{G{\pi }_2}\right)-1\right)$ (8)

$S_{{\pi }_1{\pi }_2}=S_{G{\pi }_1}+S_{G{\pi }_2}$

Since in the reference frame “G” the momenta $P_{G{\pi }_1}=P_{G{\pi }_2}$ are equal, then:

$m_{{\pi }_1}\text{sh}\left(S_{G{\pi }_1}\right)=m_{{\pi }_2}\text{sh}\left(S_{G{\pi }_2}\right)$ , $\frac{m_{{\pi }_1}}{2\pi }2\pi \text{sh}\left(S_{G{\pi }_1}\right)=\frac{m_{{\pi }_2}}{2\pi }2\pi \text{sh}\left(S_{G{\pi }_2}\right)$

The expression $2\pi \text{sh}\left(S_{G{\pi }_1}\right)$ represents the length of a circle of radius $S_{G{\pi }_1}$ in HLVS. Therefore, N.A. Chernikov used the lengths of circles of radii $S_{G{\pi }_1}$ and $S_{G{\pi }_2}$ as the lever arms (point “G” is assigned an effective mass $m_r$ ) (Figure 1). However, the expression $\text{sh}\left(S_{G{\pi }_1}\right)$ represents the length $l_{m{\pi }_1}$ of the oricycle arc and Archimedes’ laws of levers can be represented in the form (3) [10]-[12].

According to (2), (7), formula (4) for $OC{T}_r$ can be represented as:

$OC{T}_r={\text{sh}}^2\left(S_{G{\pi }_1}\right)={\left(P_{G{\pi }_1}/m_{{\pi }_1}\right)}^2=\left(m_r^2-4m_{{\pi }_1}^2\right)/\left(4m_{{\pi }_1}^2\right)$ (9)

The expression ($P_{G{\pi }_1}/m_{{\pi }_1}$ ) will be called the reduced momentum of particles ${\pi }_1$ in the reference frame “G”.

3. Two-Particle Decays of Scalar, Strange Mesons and Δ, N, Λ, Σ, Ξ Baryons into Particles with Different Masses

Figure 1 shows the different sided triangles $\Delta Pm{\pi }_1$ of the decays of $\Delta \left(1232\right)\to P+{\pi }_1$ . The point “m” of the additive mass $m_{P{\pi }_1}=m_{{\pi }_1}+m_P$ is determined by the laws of the levers of Archimedes (10) ($m_P$ is mass of a proton, $m_{{\pi }_1}$ is mass of a pi meson). In the case of different masses of decay particles ($m_P>m_{{\pi }_1}$ ), the point “m” is shifted along the arc of the oricycle to the point “P” of the particle with a higher rest mass:

$l_{P{\pi }_1}=l_{m{\pi }_1}+l_{mP},m_{P{\pi }_1}=m_P+m_{{\pi }_1}$

$m_P{l}_{m{P}_{}}=m_{{\pi }_1}{l}_{m{\pi }_1}=m_{{\pi }_1}\left(l_{P{\pi }_1}-l_{mP}\right)$ (10)

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

$l_{mP}=m_{{\pi }_1}{l}_{P{\pi }_1}/\left(m_P+m_{{\pi }_1}\right)$

The different sided triangle $\Delta Pm{\pi }_1$ of the decay of the Δ(1232) baryon is obtained by connecting the points “${\pi }_1$ ”, “m”, “P” with each other by straight line segments (Figure 1). The effective mass $m_r$ of the decays Δ Barions $\to P+{\pi }_1$ can be calculated using formula (11) (the points “b” and “${\pi }_1$ ” are associated with the rest mass $m_b$ of the particle “b” and the rest mass $m_{{\pi }_1}$ of the pi meson,$m_b\ge m_{{\pi }_1}$ ) [4]:

$m_r^2=m_b^2+m_{{\pi }_1}^2+2m_b{m}_{{\pi }_1}\text{ch}\left(S_{b{\pi }_1}\right)$ (11)

For “b” = “P” and $m_b=m_P$ :

$m_r^2=m_P^2+m_{{\pi }_1}^2+2m_P{m}_{{\pi }_1}\text{ch}\left(S_{P{\pi }_1}\right)$ (12)

The mass $m_r$ is related to the length $S_{P{\pi }_1}$ of the side ($P-{\pi }_1$ ) of the triangle $\Delta Pm{\pi }_1$ (the points “P” and “${\pi }_1$ ” are associated with the rest mass $m_P$ of the proton and the rest mass $m_{{\pi }_1}$ of the pi meson).

Rotate the segment ($m-{\pi }_1$ ) around the axis ($C_0-m$ ) of the oricycle until the point “${\pi }_1$ ” coincides with the point “${\pi }_2$ ” (Figure 1). By connecting the points “${\pi }_1$ ”, “m”, “${\pi }_2$ ” with each other by straight line segments, we obtain isosceles rotary triangle $\Delta {\pi }_{1}m{\pi }_2$ of Δ(1232) baryon decay inscribed in the oricycle (the lengths of the sides $S_{m{\pi }_1}$ and $S_{m{\pi }_2}$ are equal). The triangles $\Delta Pm{\pi }_1$ and $\Delta {\pi }_{1}m{\pi }_2$ are shown in Figure 2 (the point “m” is placed at the origin “0” of coordinates, the different sided triangles $\Delta {\pi }_{1}m\Delta \left(1232\right)$ represent the decay of the $N\left(1520\right)\to \Delta \left(1232\right)+{\pi }_1$ ).

In Table 2, the $OC{T}_r$ values for rotary triangle $\Delta {\pi }_{1}m{\pi }_2$ are calculated from published data on the effective masses of Δ baryon $\to P+{\pi }_1$ decays [13]. Columns 2 and 3 of Table 2 give the values of the effective masses $m_r$ and widths Γ of the decays Δ Barions. The $OC{T}_r$ values for rotary triangle $\Delta {\pi }_{1}m{\pi }_2$ are calculated using the formulas (4) and (12):

$OC{T}_r={\left(P_{G{\pi }_1}/m_{{\pi }_1}\right)}^2=\left(m_b/m_{{\pi }_1}\right)\left(m_r^2-{\left(m_b+m_{{\pi }_1}\right)}^2\right)/{\left(m_b+m_{{\pi }_1}\right)}^2$ (13)

For values $m_b=m_P$ :

$OC{T}_r={\left(P_{G{\pi }_1}/m_{{\pi }_1}\right)}^2=\left(m_P/m_{{\pi }_1}\right)\left(m_r^2-{\left(m_P+m_{{\pi }_1}\right)}^2\right)/{\left(m_P+m_{{\pi }_1}\right)}^2$ (14)

The “G” point represents the center of inertia of the pair (${\pi }_1$ , ${\pi }_2$ ) of the rotary triangle $\Delta {\pi }_{1}m{\pi }_2$ . It is important to note that the “G” points of the (P, ${\pi }_1$ ) and (${\pi }_1$ , ${\pi }_2$ ) pairs are located on the ($C_0-m$ ) axis of the oricycle. The effective mass of the corresponding particle pairs is concentrated at the “G” points (Figure 2).

Table 2. Hyperbolic Heron triangles in decays of $\Delta \text{Barion}\to P+{\pi }_1$ .

Name Δ Barion	Mass Resonance $m_r$ (Mev)	Width Γ (Mev)	$OC{T}_r$	$OC{T}_M$	Heron Mass in $\Delta Pm\pi$ $m_{her}^{P,\pi }$ (Mev)	$\text{Δ}{m}_r^{her}$ = $m_r$ − $m_{her}^{P,\pi }$ (Mev)	$\frac{\text{Δ}{m}_r^{her}}{\text{Γ}}$ (%)	Mass in Rot_H $\text{Δ}\pi m\pi$ $m_{her}^{\pi ,\pi }$ (Mev)
1	2	3	4	5	6	7	8	9
$\Delta \text{Barion}\to P+{\pi }_1$ , $m_P=938.27209\text{Mev}$ , $m_{{\pi }_1}=m_{\pi }=139.57001\text{Mev}$
Δ(1232)	1232.0 ± 0.40	117.0	${2.06}_{-0.006}^{+0.006}$	2	1227.8	4.2	3.63	483.5
Δ(1600)	1570.0 ± 0.25	250.0	${7.54}_{-0.051}^{+0.005}$	8	1595.1	25.1	10.0	837.4
Δ(1620)	1610.0 ± 20.0	130.0	${8.28}_{-0.370}^{+0.375}$	8	1595.1	14.9	11.5	837.4
Δ(1710)	1710.0 ± 10.0	300.0	${10.20}_{-0.197}^{+0.199}$	10	1700.0	10.0	3.4	925.8
Δ(1900)	1860.0± 30.0	300.0	${13.30}_{-0.640}^{+0.651}$	13	1946.2	13.8	4.6	1044.5
Δ(1950)	1930.0 ± 10.0	285.0	${14.83}_{-0.223}^{+0.224}$	15	1937.5	7.5	2.6	1116.6
Δ(2150)	2150.0±100.0	200.0	${20.03}_{-2.430}^{+2.547}$	20	2149.0	1.1	0.5	1279.2
Δ(2300)	2300.0±100.0	350.0	${23.89}_{-2.604}^{+2.720}$	24	2304.2	4.2	1.2	1395.7
Δ(2400)	2450.0±100.0	500.0	${20.01}_{-2.777}^{+2.894}$	28	2449.6	0.4	0.1	1503.2
Δ(2750)	2794.0 ± 80.0	350.0	${38.35}_{-2.550}^{+2.624}$	38	2780.0	14.0	4.0	1743.2
Δ(2950)	2990.0±100.0	330.0	${45.01}_{-3.402}^{+3.520}$	45	2989.7	0.3	0.1	1893.2

Figure 2. The different sided triangles ΔPmπ₁ of baryon decays Δ(1232) ––> P + π₁, N(1520) ––> Δ(1232) + π₁, isosceles triangle Δπ₁ m π₂ of baryon decays, isosceles Heron triangle Δπ m π. The point-velocity “G” represents the centers of inertia of pairs of particles (P, π₁), (Δ(1232), π₁).

Columns 3 and 4 of Table 2 show the values of the function $OC{T}_r$ for rotary triangle $\Delta {\pi }_{1}m{\pi }_2$ of Δ(1232) baryon decay, calculated using formula (14) and its nearest integer value $OC{T}_M$ (Figure 2). The integer values $OC{T}_M$ correspond to rotary Heron triangles $\Delta \pi m\pi$ (Rot_H triangles). In turn, the rotary Heron triangle $\Delta \pi m\pi$ corresponds to a triangle with different sides $\Delta Pm\pi$ (similar to how a triangle with different sides $\Delta Pm{\pi }_1$ corresponds to a rotary triangle $\Delta {\pi }_{1}m{\pi }_2$ ). Column 6 of Table 2 shows the effective mass $m_{her}^{P,\pi }$ of the pair proton and pi meson, calculated using formula (15) through the length $S_{\pi P}$ of the side ($\pi -P$ ) $\Delta Pm\pi$ (the points “P” and “$\pi$ ” are associated with the rest mass $m_P$ of the proton and the rest mass $m_{\pi }$ of the pi meson, the center of inertia “G” of the pair $\left(P,\pi \right)$ lies on the axis $\left(C_0-m\right)$ of the oricycle):

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

For values $m_b=m_P$

$m_{her}=m_{her}^{P,\pi }=\sqrt{m_P^2+m_{\pi }^2+2m_P{m}_{\pi }\text{ch}{S}_{\pi P}}$

Columns 7 and 8 of Table 2 show the absolute and relative deviations of the mass $m_{her}^{P,\pi }$ from the experimental values $m_r$ . From Table 2 it is evident that the masses $m_{her}^{P,\pi }$ differ from the masses $m_r$ of Δ baryon decays within their widths. The maximum deviation is 10% of the resonance width, the minimum is 0.1% of the resonance width, and the average deviation is 3.8% of the resonance width. Column 9 of Table 2 shows the effective mass $m_{her}^{\pi ,\pi }$ of the pair of pi mesons, calculated using formula (6) through the length $S_{\pi \pi }^{}$ of the base of the rotary Heron triangle $\Delta \pi m\pi$ (points “$\pi$ ”, “$\pi$ ” of the base ($\pi - \pi$ ) are associated with the rest mass $m_{\pi }$ ).

In Table 3, the $OC{T}_r$ values for rotary triangle $\Delta {\pi }_{1}m{\pi }_2$ are calculated from published data on the effective masses of 2-particle decays of N Barion$\to \Delta \left(1232\right)+{\pi }_1$ [13]. Columns 2 and 3 of Table 3 give the values of the effective masses $m_r$ and widths Γ of the N Barions decays. The mass $m_r$ is related to the length $S_{\Delta \left(1232\right){\pi }_1}$ of the side ($\Delta \left(1232\right),{\pi }_1$ ) of the triangle $\Delta {\pi }_{1}m\Delta \left(1232\right)$ (the points “Δ(1232)” and “${\pi }_1$ ” are associated with the mass $m_{\Delta \left(1232\right)}$ and the mass $m_{{\pi }_1}$ ) (Figure 2):

$m_r^2=m_{\Delta \left(1232\right)}^2+m_{{\pi }_1}^2+2m_{\Delta \left(1232\right)}{m}_{{\pi }_1}\text{ch}\left(S_{\Delta \left(1232\right){\pi }_1}\right)$ (16)

The “G” points represent the centers of inertia of the pairs (Δ(1232), ${\pi }_1$ ) of decay particles. Columns 3 and 4 of Table 3 show the values of the function $OC{T}_r$ for rotary triangle $\Delta {\pi }_{1}m{\pi }_2$ of N baryon decay, calculated using formula (13) (for values $m_b=m_{\Delta \left(1232\right)}$ ) and its nearest integer value $OC{T}_M$ (Figure 2). Integer values of $OC{T}_M$ correspond to rotary Heron triangles $\text{Δ}\pi m\pi$ . In turn, the rotary Heron triangle $\text{Δ}\pi m\pi$ corresponds to the different sided triangle $\Delta \pi m\Delta \left(1232\right)$ (similarly to the fact that the different sided triangle $\Delta {\pi }_{1}m\Delta \left(1232\right)$ corresponds to the rotary triangle $\Delta {\pi }_{1}m{\pi }_2$ ). Column 6 of Table 3 shows the effective mass $m_{N,her}^{\Delta \left(1232\right),\pi }$ of the pair (Δ(1232), π), calculated using formula (16) through the length $S_{\pi \Delta \left(1232\right)}$ of the side ($\Delta \left(1232\right)-\pi$ ) of the triangle $\Delta \pi m\Delta \left(1232\right)$ (the points “Δ(1232)” and “$\pi$ ” are associated with the mass $m_{\Delta \left(1232\right)}$ and the mass $m_{\pi }$ ):

Table 3. Hyperbolic Heron triangles in decays of $N\text{Barion}\to \Delta \left(1232\right)+{\pi }_1$ .

Name N Barion	Mass Resonance $m_r$ (Mev)	Width Γ (Mev)	$OC{T}_r$	$OC{T}_M$	Heron Mass in $\text{Δ}\pi m\Delta \left(1232\right)$ $m_{N,her}^{\Delta 232,\pi }$ (Mev)	$\text{Δ}{m}_r^{her}$ = $m_r$ − $m_{N,her}^{\Delta 232,\pi }$ (Mev)	$\frac{\text{Δ}{m}_r^{her}}{\text{Γ}}$ (%)	Mass in Rot_H $\text{Δ}\pi m\pi$ $m_{her}^{\pi ,\pi }$ (Mev)
1	2	3	4	5	6	7	8	9
$N\text{Barion}\to \Delta \left(1232\right)+{\pi }_1$ , $m_{\Delta 232}=1232.0\text{Mev}$ , $m_{{\pi }_1}=m_{\pi }=139.57001\text{Mev}$
N(1440)	1440.0 ± 30.0	350.0	${0.90}_{-0.135}^{+0.136}$	1	1447.2	7.2	2.0	394.8
N(1520)	1515.0 ± 10.0	110.0	${1.94}_{-0.142}^{+0.143}$	2	1519.0	4.0	3.7	483.5
N(1535)	1530.0 ± 5.0	150.0	${2.16}_{-0.172}^{+0.072}$	2	1519.0	10.0	7.3	483.5
N(1650)	1650.0 ± 5.0	125.0	${3.95}_{-0.077}^{+0.077}$	4	1653.4	3.4	2.7	624.8
N(1675)	1675.0 ± 5.0	150.0	${4.34}_{-0.078}^{+0.079}$	4	1653.4	21.6	14.4	624.8
N(1680)	1685.0 ± 5.0	120.0	${4.50}_{-0.079}^{+0.079}$	4	1653.4	31.6	26.3	624.8
N(1700)	1720.0 ± 10.0	200.0	${5.05}_{-0.160}^{+0.162}$	5	1716.6	3.4	1.7	683.7
N(1710)	1710.0 ± 20.0	200.0	${4.89}_{-0.319}^{+0.323}$	5	1716.6	6.6	3.3	683.7
N(1720)	1720.0 ± 10.0	250.0	${5.05}_{-0.161}^{+0.162}$	5	1716.6	3.4	1.3	683.7
N(1860)	1860.0 ± 10.0	250.0	${7.41}_{-0.174}^{+0.175}$	7	1836.6	23.4	9.4	789.5
N(1875)	1875.0 ± 10.0	200.0	${7.67}_{-0.176}^{+0.175}$	8	1893.7	18.7	9.3	837.4
N(1880)	1880.0 ± 20.0	300.0	${7.76}_{-0.351}^{+0.355}$	8	1893.7	13.7	4.6	837.4
N(1895)	1985.0 ± 10.0	200.0	${8.02}_{-0.177}^{+0.178}$	8	1893.7	1.3	0.7	837.4
N(1900)	1920.0 ± 10.0	200.0	${8.47}_{-0.180}^{+0.181}$	8	1893.7	26.3	13.1	837.4
N(2000)	2000.0 ± 10.0	300.0	${9.94}_{-0.187}^{+0.188}$	10	2003.1	3.1	1.0	925.8
N(2060)	2100.0 ± 15.0	400.0	${11.87}_{-0.295}^{+0.297}$	12	2106.8	6.8	1.7	1006.5
N(2100)	2100.0 ± 20.0	260.0	${11.87}_{-0.392}^{+0.396}$	12	2106.8	6.8	1.7	1006.5
N(2190)	2180.0 ± 20.0	400.0	${13.47}_{-0.307}^{+0.412}$	13	2156.8	23.2	5.8	1044.5

$m_{her}=m_{her}^{\Delta \left(1232\right),\pi }=\sqrt{m_{\Delta \left(1232\right)}^2+m_{\pi }^2+2m_{\Delta \left(1232\right)}{m}_{\pi }\text{ch}{S}_{\Delta \left(1232\right)\pi }}$ (17)

Columns 7 and 8 of Table 3 show the absolute and relative deviations of the mass $m_{N,her}^{\Delta \left(1232\right),\pi }$ from the experimental values $m_r$ . From Table 3 it is evident that the masses $m_{N,her}^{\Delta \left(1232\right),\pi }$ differ from the masses $m_r$ of N baryon decays within their widths. The maximum deviation is 23.4% of the resonance width, the minimum is 1.3% of the resonance width, and the average deviation is 6.1% of the resonance width. Column 9 of Table 3 shows the effective mass $m_{her}^{\pi ,\pi }$ of the pair of pi mesons, calculated using formula (6) through the length $S_{\pi \pi }^{}$ of the base of the rotary Heron triangle $\text{Δ}\pi m\pi$ (points “$\pi$ ”, “$\pi$ ” of the base ($\pi -\pi$ ) are associated with the mass $m_{\pi }$ ).

Table 4 shows the published effective masses $m_r$ and widths Γ of Λ Barion $\to \Sigma +{\pi }_1$ decays [13]. Columns 4 - 9 contain the calculated $OC{T}_r$ values for the rotary triangles $\Delta {\pi }_{1}m{\pi }_2$ , the $OC{T}_M$ values for the rotary Heron triangles $\text{Δ}\pi m\pi$ and the absolute and relative deviations of the masses $m_{her}^{\Sigma ,\pi }$ from the experimental values $m_r$ . From Table 4 it is evident that the masses $m_{her}^{\Sigma ,\pi }$ differ from the masses $m_r$ of Λ baryon decays within their widths. The average relative deviation is 18.4% of the resonance width. Column 9 shows the effective mass $m_{her}^{\pi ,\pi }$ of the pair of pi mesons, calculated using formula (6) through the length $S_{\pi \pi }^{}$ of the base of the rotary Heron triangle $\text{Δ}\pi m\pi$ .

Table 4. Hyperbolic Heron triangles in decays of $\Lambda Barion\to \Sigma +{\pi }_1$ .

Name Λ Barion	Mass Resonance $m_r$ (Mev)	Width Γ (Mev)	$OC{T}_r$	$OC{T}_M$	Heron Mass in $\Delta \Sigma m\pi$ $m_{her}^{\Sigma ,\pi }$ (Mev)	$\text{Δ}{m}_r^{her}$ = $m_r^{}$ − $m_{her}^{\Sigma ,\pi }$ (Mev)	$\frac{\text{Δ}{m}_r^{her}}{\text{Γ}}$ (%)	Mass in Rot_H $\text{Δ}\pi m\pi$ $m_{her}^{\pi ,\pi }$ (Mev)
1	2	3	4	5	6	7	8	9
$\Lambda Barion\to \Sigma +{\pi }_1$ , $m_{\Sigma }=1189.37\text{Mev}$ , $m_{{\pi }_1}=m_{\pi }=139.57001\text{Mev}$
Λ(1405)	1405.1 ± 1.3	50.0	${1.00}_{-0.018}^{+0.018}$	1	1404.8	0.4	0.7	324.8
Λ(1520)	1519.4 ± 0.2	15.7	${2.62}_{-0.003}^{+0.003}$	3	1545.3	25.8	164.3	558.3
Λ(1600)	1600.0 ± 10.0	200.0	${3.83}_{-0.154}^{+0.155}$	4	1610.9	10.9	5.5	624.2
Λ(1670)	1670.0 ± 5.0	30.0	${4.94}_{-0.080}^{+0.081}$	5	1674.0	4.0	13.4	683.7
Λ(1690)	1690.0 ± 5.0	70.0	${5.26}_{-0.081}^{+0.082}$	5	1674.0	16.0	22.8	683.7
Λ(1710)	1713.0 ± 13.0	180.0	${5.64}_{-0.214}^{+0.216}$	6	1734.8	21.8	12.1	738.5
Λ(1800)	1800.0 ± 10.0	200.0	${7.11}_{-0.173}^{+0.174}$	7	1793.6	6.6	3.2	789.5
Λ(1810)	1790.0 ± 10.0	110.0	${6.94}_{-0.172}^{+0.173}$	7	1793.6	3.6	3.2	789.5
Λ(1820)	1820.0 ± 4.0	80.0	${7.46}_{-0.070}^{+0.070}$	7	1793.6	26.4	33.0	789.5
Λ(1830)	1825.0 ± 10.0	90.0	${7.55}_{-0.176}^{+0.177}$	8	1850.4	25.4	28.2	837.4
Λ(1890)	1890.0 ± 5.0	120.0	${8.71}_{-0.091}^{+0.091}$	9	1905.6	15.6	13.0	882.7
Λ(2050)	2056.0 ± 22.0	493.0	${11.88}_{-0.434}^{+0.439}$	12	2062.3	6.3	1.3	1006.5
Λ(2070)	2070.0 ± 24.0	370.0	${12.15}_{-0.477}^{+0.482}$	12	2062.3	7.7	2.0	1006.5
Λ(2080)	2082.0 ± 13.0	181.0	${12.39}_{-0.260}^{+0.262}$	12	2062.3	7.7	2.0	1006.5
Λ(2100)	2100.0 ± 22.0	200.0	${12.76}_{-0.443}^{+0.448}$	13	2111.9	11.9	6.0	1044.5
Λ(2110)	2090.0 ± 22.0	250.0	${12.56}_{-0.441}^{+0.447}$	13	2111.9	21.9	8.8	1044.5
Λ(2250)	2350.0 ± 22.0	150.0	${18.13}_{-0.497}^{+0.501}$	18	2344.5	5.5	3.7	1216.7

Table 5 shows the published effective masses $m_r$ and widths Γ of N Barion→ n + ${\pi }_1$ decays [13]. Columns 4 - 9 contain the calculated $OC{T}_r$ values for the rotary triangles $\Delta {\pi }_{1}m{\pi }_2$ , the $OC{T}_M$ values for the rotary Heron triangles $\text{Δ}\pi m\pi$ , and the absolute and relative deviations of the masses $m_{her}^{n,\pi }$ from the experimental values $m_r$ . From Table 5 it is evident that the masses $m_{her}^{n,\pi }$ differ from the masses $m_r$ of N Baryon→ n + ${\pi }_1$ decays within their widths. The average relative deviation is 5.7% of the resonance width. Column 9 shows the effective mass $m_{her}^{\pi ,\pi }$ of the pair of pi mesons, calculated using formula (6) through the length $S_{\pi \pi }^{}$ of the base of the rotary Heron triangle $\text{Δ}\pi m\pi$ .

Table 5. Hyperbolic Heron triangles in decays of $N\text{Barion}\to n+{\pi }_1$ .

Name N Barion	Mass Resonance $m_r$ (Mev)	Width Γ (Mev)	$OC{T}_r$	$OC{T}_M$	Heron Mass in $\text{Δ}nm\pi$ $m_{her}^{n,\pi }$ (Mev)	$\text{Δ}{m}_r^{her}$ = $m_r$ − $m_{her}^{n,\pi }$ (Mev)	$\frac{\text{Δ}{m}_r^{her}}{\text{Γ}}$ (%)	Mass in Rot_H $\text{Δ}\pi m\pi$ $m_{her}^{\pi ,\pi }$ (Mev)
1	2	3	4	5	6	7	8	9
$N\text{Barion}\to n+{\pi }_1$ , $m_n=939.56542052\text{Mev}$ , $m_{{\pi }_1}=m_{\pi }=139.57001\text{Mev}$
N(1440)	1440.0 ± 30.0	350.0	${5.28}_{-0.495}^{+0.505}$	5	1423.3	16.7	4.8	683.8
N(1520)	1515.0 ± 10.0	110.0	${6.56}_{-0.175}^{+0.176}$	7	1539.9	24.9	22.7	789.5
N(1535)	1530.0 ± 15.0	150.0	${6.82}_{-0.264}^{+0.267}$	7	1599.4	9.9	6.6	789.5
N(1650)	1650.0 ± 6.0	125.0	${9.03}_{-0.114}^{+0.115}$	9	1648.3	1.6	1.3	882.7
N(1675)	1675.0 ± 5.0	145.0	${9.51}_{-0.097}^{+0.097}$	10	1700.0	25.0	17.2	925.8
N(1680)	1685.0 ± 5.0	120.0	${9.71}_{-0.097}^{+0.977}$	10	1700.0	15.0	12.5	925.8
N(1700)	1720.0 ± 20.0	200.0	${10.40}_{-0.396}^{+0.400}$	10	1700.0	17.4	8.7	925.8
N(1875)	1875.0 ± 20.0	200.0	${13.62}_{-0.432}^{+0.436}$	14	1892.4	12.4	4.1	1081.1
N(1880)	1880.0 ± 20.0	300.0	${13.73}_{-0.433}^{+0.437}$	14	1892.4	0.4	0.1	1081.1
N(1895)	1985.0 ± 20.0	120.0	${14.06}_{-0.436}^{+0.441}$	14	1892.4	2.6	2.2	1081.1
N(1920)	1920.0 ± 20.0	200.0	${14.61}_{-0.442}^{+0.447}$	15	1937.5	17.5	8.8	1116.6
N(1990)	2020.0 ± 40.0	300.0	${16.89}_{-0.926}^{+0.945}$	17	2024.7	4.7	1.6	1184.3
N(2060)	2100.0 ± 15.0	400.0	${18.80}_{-0.363}^{+0.366}$	19	2108.4	8.4	2.1	1248.3
N(2220)	2250.0 ± 15.0	400.0	${22.57}_{-0.389}^{+0.392}$	23	2266.3	16.4	4.1	1367.5
N(2250)	2280.0 ± 15.0	500.0	${23.36}_{-0.394}^{+0.400}$	23	2266.3	13.6	2.7	1367.5
N(2600)	2600.0 ± 100.0	650.0	${32.40}_{-2.951}^{+3.067}$	32	2586.9	13.2	2.0	1603.5
N(2700)	2612.0 ± 45.0	650.0	${32.76}_{-1.348}^{+1.372}$	33	2620.0	8.0	1.2	1627.7
N(3000)	3000.0 ± 200.0	1650.	${45.36}_{-6.712}^{+7.177}$	45	2989.7	10.3	0.6	1893.2

Table 6 shows the published effective masses $m_r$ and widths Γ of Scalar Mezon $\to \eta +{\pi }_1$ decays [13]. Columns 4 - 9 contain the calculated $OC{T}_r$ values for the rotary triangles $\Delta {\pi }_{1}m{\pi }_2$ , the $OC{T}_M$ values for the rotary Heron triangle $\text{Δ}\pi m\pi$ , and the absolute and relative deviations of the masses $m_{her}^{\eta ,\pi }$ from the experimental values $m_r$ . From Table 6 it is evident that the masses $m_{her}^{\eta ,\pi }$ differ from the masses $m_r$ of Scalar Mezon $\to \eta +{\pi }_1$ decays within their widths. The average relative deviation is 7.6% of the resonance width. Column 9 shows the effective mass $m_{her}^{\pi ,\pi }$ of the pair of pi mesons, calculated using formula (6) through the length $S_{\pi \pi }$ of the base of the rotary Heron triangle $\text{Δ}\pi m\pi$ .

Table 6. Hyperbolic Heron triangles in decays of $\text{Scalar Mezon}\to \eta +{\pi }_1$ .

Name Scalar Mezon	Mass Resonance $m_r$ (Mev)	Width Γ (Mev)	$OC{T}_r$	$OC{T}_M$	Heron Mass in $\text{Δ}\eta m\pi$ $m_{her}^{\eta ,\pi }$ (Mev)	$\text{Δ}{m}_r^{her}$ = $m_r$ − $m_{her}^{\eta ,\pi }$ (Mev)	$\frac{\text{Δ}{m}_r^{her}}{\text{Γ}}$ (%)	Mass in Rot_H $\text{Δ}\pi m\pi$ $m_{her}^{\pi ,\pi }$ (Mev)
1	2	3	4	5	6	7	8	9
$\text{Scalar Mezon}\to \eta +{\pi }_1$ , $m_{\eta }=547.862\text{Mev}$ , $m_{{\pi }_1}=m_{\pi }=139.57001\text{Mev}$
$a_o\left(980\right)$	980.0 ± 20.0	75.0	${4.05}_{-0.322}^{+0.330}$	4	976.8	3.2	4.3	624.2
$a_2\left(1320\right)$	1317.7 ± 1.4	107.0	${10.50}_{-0.031}^{+0.031}$	10	1294.8	22.9	21.4	925.8
${\pi }_1\left(1600\right)$	1354.0 ± 25.0	330.0	${11.30}_{-0.557}^{+0.568}$	11	1340.5	13.5	4.1	967.0
$a_o\left(1450\right)$	1439.0 ± 34.0	258.0	${13.28}_{-0.803}^{+0.822}$	13	1427.5	11.6	4.5	1044.5
$a_2\left(1700\right)$	1706.0 ± 14.0	380.0	${20.25}_{-0.395}^{+0.398}$	20	1697.1	8.5	2.3	1279.2
$a_0\left(1710\right)$	1713.0 ± 19.0	107.0	${20.45}_{-0.537}^{+0.544}$	20	1697.1	15.9	14.8	1279.2
$a_4\left(1970\right)$	1967.0± 16.0	324.0	${28.21}_{-0.520}^{+0.525}$	28	1960.5	6.5	2.0	1503.2

Table 7 shows the published masses $m_r$ and widths Γ of Scalar Mezon$\to \rho \left(770\right)+{\pi }_1$ decays [13]. Columns 4 - 9 contain the calculated $OC{T}_r$ values for the rotary triangles $\Delta {\pi }_{1}m{\pi }_2$ , the $OC{T}_M$ values for the rotary Heron triangles $\text{Δ}\pi m\pi$ , and the absolute and relative deviations of the masses $m_{her}^{\rho \left(770\right),\pi }$ from the experimental values $m_r$ . Decays ${\eta }^{\prime }\left(958\right)\to \rho \left(770\right)+{\pi }_1$ with a relative deviation equal to 15,652% from Table 7 are missing from all the calculations presented. This can only be explained by the small values of the resonance widths given (width = 0.23 Mev). Or the production of the ${\eta }^{\prime }\left(958\right)$ meson is not associated with Heron’s triangles. If we exclude the 15,652% deviation, the average relative deviation will be 4.4% of the resonance width. Column 9 shows the effective mass $m_{her}^{\pi ,\pi }$ of the pair of pi mesons, calculated using formula (6) through the length $S_{\pi ,\pi }$ of the base of the rotary Heron triangle $\Delta \pi m\pi$ .

Table 7. Hyperbolic Heron triangles in decays of $\text{Scalar Mezon}\to \rho \left(770\right)+{\pi }_1$ .

Name Scalar Mezon	Mass Resonance $m_r$ (Mev)	Width Γ (Mev)	$OC{T}_r$	$OC{T}_M$	Heron Mass in $\text{Δ}\rho \left(770\right)m\pi$ $m_{her}^{\rho ,\pi }$ (Mev)	$\text{Δ}{m}_r^{her}$ = $m_r$ − $m_{her}^{\rho ,\pi }$ (Mev)	$\frac{\text{Δ}{m}_r^{her}}{\text{Γ}}$ (%)	Mass in Rot_H $\text{Δ}\pi m\pi$ $m_{her}^{\pi ,\pi }$ (Mev)
1	2	3	4	5	6	7	8	9
$\text{Scalar Mezon}\to \rho \left(770\right)+{\pi }_1$ , $m_{\rho }=775.29999\text{Mev}$ , $m_{{\pi }_1}=m_{\pi }=139.57001\text{Mev}$
${\eta }^{\prime }\left(958\right)$	957.8 ± 0.06	0.23	${0.53}_{-0.001}^{+0.001}$	1	993.8	36.0	15,652	394.8
$\pi \left(1300\right)$	1300.0±100.0	400.0	${5.66}_{-1.659}^{+1.792}$	6	1319.5	19.5	4.9	738.5
$a_2\left(1320\right)$	1318.2 ± 0.60	107.0	${5.98}_{-0.011}^{+0.010}$	6	1319.5	1.3	1.2	738.5
$\omega \left(1420\right)$	1410.0 ± 60.0	290.0	${7.64}_{-1.099}^{+1.147}$	8	1429.1	19.2	1.2	837.2
$a_2\left(1700\right)$	1706.0 ± 14.0	380.0	${20.25}_{-0.395}^{+0.398}$	20	1697.1	8.5	2.3	1279.2
$a_0\left(1710\right)$	1713.0 ± 19.0	107.0	${20.45}_{-0.537}^{+0.544}$	20	1697.1	15.9	14.8	1279.2
$a_4\left(1970\right)$	1967.0± 16.0	324.0	${28.21}_{-0.520}^{+0.525}$	28	1960.5	6.5	2.0	1503.2

Table 8 shows the published masses $m_r$ and widths Γ of Scalar Mezon $\to \omega \left(782\right)+{\pi }_1$ decays [13]. Columns 4 - 9 contain the calculated $OC{T}_r$ values for the rotary triangles $\Delta {\pi }_{1}m{\pi }_2$ , the $OC{T}_M$ values for the rotary Heron triangles $\Delta \pi m\pi$ , and the absolute and relative deviations of the masses $m_{her}^{\omega \left(782\right),\pi }$ from the experimental values $m_r$ . From Table 8 it is evident that the masses $m_{her}^{\omega \left(782\right),\pi }$ differ from the masses $m_r$ of Scalar Mezon$\to \omega \left(782\right)+{\pi }_1$ decays within their widths. The average relative deviation is 7.1% of the resonance width. Column 9 shows the effective mass $m_{her}^{\pi ,\pi }$ of the pair of pi mesons, calculated using formula (6) through the length $S_{\pi ,\pi }$ of the base of the rotary Heron triangle $\Delta \pi m\pi$ .

Table 8. Hyperbolic Heron triangles in decays of $\text{Scalar Mezon}\to \omega \left(782\right)+{\pi }_1$ .

Name Scalar Mezon	Mass Resonance $m_r$ (Mev)	Width Γ (Mev)	$OC{T}_r$	$OC{T}_M^{}$	Heron Mass in $\text{Δ}\omega \left(782\right)m\pi$ $m_{her}^{\omega ,\pi }$ (Mev)	$\text{Δ}{m}_r^{her}$ = $m_r$ − $m_{her}^{\omega ,\pi }$ (Mev)		$\frac{\text{Δ}{m}_r^{her}}{\text{Γ}}$ (%)	Mass in Rot_H $\text{Δ}\pi m\pi$ $m_{her}^{\pi ,\pi }$ (Mev)
1	2	3	4	5	6	7		8	9
$\text{Scalar Mezon}\to \omega \left(782\right)+{\pi }_1$ , $m_{\omega }=782.65002\text{Mev}$ , $m_{{\pi }_1}=m_{\pi }=139.57001\text{Mev}$
$b_1\left(1235\right)$	1229.5 ± 3.30	142.9	${4.36}_{-0.052}^{+0.052}$	4	1207.1	22.4	15.6		624.2
$\rho \left(1450\right)$	1465.8 ± 25.0	400.0	${8.56}_{-0.479}^{+0.487}$	9	1488.5	22.7	5.7		882.7
${\rho }_3\left(1690\right)$	1688.8± 2.1	161.0	${13.20}_{-0.047}^{+0.047}$	13	1679.9	8.9	5.5		1044.5
$\rho \left(2250\right)$	2150.0 ± 40.0	300.0	${24.87}_{-1.123}^{+1.145}$	25	2154.6	4.6	1.5		1423.3

Table 9 shows the published masses $m_r$ and widths Γ of Σ Barion$\to \Lambda +{\pi }_1$ decays [13]. Columns 4 - 9 contain the calculated $OC{T}_r$ values for the rotary triangles $\Delta {\pi }_{1}m{\pi }_2$ , the $OC{T}_M$ values for the rotary Heron triangles $\Delta \pi m\pi$ , and the absolute and relative deviations of the masses $m_{her}^{\Lambda ,\pi }$ from the experimental values $m_r$ . From Table 9 it is evident that the masses $m_{her}^{\Lambda ,\pi }$ differ from the masses $m_r$ of Σ Barion decays within their widths. The average relative deviation is 12.0% of the resonance width. Column 9 shows the effective mass $m_{her}^{\pi ,\pi }$ of the pair of pi mesons, calculated using formula (6) through the length $S_{\pi ,\pi }$ of the base of the rotary Heron triangle $\Delta \pi m\pi$ .

Table 9. Hyperbolic Heron triangles in decays of $\Sigma \text{Barion}\to \Lambda +{\pi }_1$ .

Name Σ Barion	Mass Resonance $m_r$ (Mev)	Width Γ (Mev)	$OC{T}_r$	$OC{T}_M$	Heron Mass in $\text{Δ}\Lambda m\pi$ $m_{her}^{\Lambda ,\pi }$ (Mev)	$\text{Δ}{m}_r^{her}$ = $m_r$ − $m_{her}^{\Lambda ,\pi }$ (Mev)	$\frac{\text{Δ}{m}_r^{her}}{\text{Γ}}$ ( % )	Mass in Rot_H $\text{Δ}\pi m\pi$ $m_{her}^{\pi ,\pi }$ (Mev)
1	2	3	4	5	6	7	8	9
$\Sigma \text{Barion}\to \Lambda +{\pi }_1$ , $m_{\Lambda }=1115.68298\text{Mev}$ , $m_{{\pi }_1}=m_{\pi }=139.57001\text{Mev}$
Σ(1385)	1383.7 ± 2.0	39.4	${1.72}_{-0.028}^{+0.028}$	2	1403.5	19.8	50.3	483.5
Σ(1620)	1620.0 ± 2.0	70.0	${5.32}_{-0.033}^{+0.033}$	5	1600.4	19.6	28.0	683.8
Σ(1660)	1660.0 ± 20.0	200.4	${5.99}_{-0.335}^{+0.339}$	6	1660.8	0.8	0.4	738.5
Σ(1670)	1675.0 ± 2.0	70.0	${6.24}_{-0.034}^{+0.034}$	6	1660.8	14.2	20.2	738.5
Σ(1750)	1750.0 ± 2.0	150.0	${7.54}_{-0.035}^{+0.036}$	8	1775.5	25.5	17.0	837.4
Σ(1775)	1775.0 ± 10.0	120.0	${7.99}_{-0.180}^{+0.180}$	8	1775.5	0.6	0.5	837.4
Σ(1780)	1780.0 ± 30.0	200.0	${8.08}_{-0.537}^{+0.546}$	8	1775.5	4.5	2.2	837.4
Σ(1900)	1925.0 ± 20.0	165.0	${10.81}_{-0.389}^{+0.393}$	11	1935.0	10.0	6.0	967.0
Σ(1910)	1910.0 ± 50.0	220.0	${10.51}_{-0.956}^{+0.982}$	11	1935.0	24.9	11.3	967.0
Σ(1915)	1915.0 ± 22.0	120.0	${10.61}_{-0.425}^{+0.430}$	11	1935.0	20.0	16.6	967.0
Σ(1940)	1940.0±100.0	250.0	${11.10}_{-1.917}^{+2.019}$	11	1935.0	5.1	2.0	967.0
Σ(2010)	2005.0 ± 14.0	178.0	${12.40}_{-0.284}^{+0.286}$	12	1985.2	19.8	11.1	1006.5
Σ(2030)	2030.0 ± 10.0	180.0	${12.91}_{-0.205}^{+0.207}$	13	2034.2	4.2	2.4	1044.5
Σ(2070)	2060.0 ± 10.0	200.0	${13.54}_{-0.208}^{+0.210}$	14	2082.1	22.1	11.1	1081.1
Σ(2100)	2100.0 ± 50.0	310.0	${14.38}_{-0.424}^{+0.428}$	14	2082.1	18.9	6.1	1116.6
Σ(2230)	2240.0 ± 27.0	347.0	${17.46}_{-0.610}^{+0.617}$	17	2119.6	20.4	5.9	1367.5
Σ(2250)	2250.0 ± 20.0	140.0	${17.69}_{-0.454}^{+0.459}$	18	2263.6	13.6	13.6	1216.7
Σ(2455)	2455.0 ± 20.0	140.0	${22.58}_{-0.496}^{+0.500}$	23	2471.1	16.7	11.9	1216.7

Table 10 shows the published masses $m_r$ and widths Γ of Ξ Barion $\to {\Xi }^0+{\pi }_1$ decays [13]. Columns 4 - 9 contain the calculated $OC{T}_r$ values for the rotary triangles $\Delta {\pi }_{1}m{\pi }_2$ , the $OC{T}_M$ values for the rotary Heron triangles $\Delta \pi m\pi$ , and the absolute and relative deviations of the masses $m_{her}^{ {\Xi }^0,\pi }$ from the experimental values $m_r$ . From Table 10 it is evident that the masses $m_{her}^{ {\Xi }^0,\pi }$ differ from the masses $m_r$ of Ξ Barion $\to {\Xi }^0+{\pi }_1$ decays within their widths. The average relative deviation is 51.7% of the resonance width. Column 9 shows the effective mass $m_{her}^{\pi ,\pi }$ of the pair of pi mesons, calculated using formula (6) through the length $S_{\pi ,\pi }$ of the base of the rotary Heron triangle $\Delta \pi m\pi$ .

Table 10. Hyperbolic Heron triangles in decays of $\Xi \text{ Barion}\to {\Xi }_{}^0+{\pi }_1$ .

Name Ξ Barion	Mass Resonance $m_r$ (Mev)	Width Γ (Mev)	$OC{T}_r$	$OC{T}_M$	Heron Mass in $\text{Δ}{\Xi }^{0}m\pi$ $m_{her}^{{\text{Ξ}}^0,\pi }$ (Mev)	$\text{Δ}{m}_r^{her}$ = $m_r$ − $m_{her}^{{\text{Ξ}}^0,\pi }$ (Mev)	$\frac{\text{Δ}{m}_r^{her}}{\text{Γ}}$ (%)	Mass in Rot_H $\text{Δ}\pi m\pi$ $m_{her}^{\pi ,\pi }$ (Mev)
1	2	3	4	5	6	7	8	9
$\Xi \text{Barion}\to {\Xi }^0+{\pi }_1$ , $m_{{\text{Ξ}}^0}=1314.85999\text{Mev}$ , $m_{{\pi }_1}=m_{\pi }=139.57001\text{Mev}$
Ξ(1530)	1531.8 ± 0.3	9.1	${1.03}_{-0.004}^{+0.004}$	1	1529.7	2.1	23.3	394.8
Ξ(1620)	1620.8 ± 6.0	32.0	${2.28}_{-0.086}^{+0.087}$	2	1601.4	19.4	60.7	483.5
Ξ(1690)	1690.0 ± 10.0	20.0	${3.30}_{-0.150}^{+0.151}$	3	1670.0	20.0	100.0	558.3
Ξ(1820)	1823.0 ± 5.0	20.0	${5.38}_{-0.081}^{+0.081}$	5	1799.5	23.5	117.7	683.7
Ξ(1950)	1950.0 ± 15.0	60.0	${7.51}_{-0.259}^{+0.261}$	8	1977.8	8.8	14.6	837.4
Ξ(2030)	2025.0 ± 20.0	60.0	${8.84}_{-0.359}^{+0.363}$	9	2033.8	8.8	14.6	882.7
Ξ(2500)	2500.0±150.0	60.0	${18.41}_{-3.240}^{+3.440}$	18	2481.4	18.7	31.1	1216.7

Table 11 shows the published masses $m_r$ and widths Γ of Δ Barion$\to \Delta \left(1232\right)+{\pi }_1$ decays [13]. Columns 4 - 9 contain the calculated $OC{T}_r$ values for the rotary triangles $\Delta {\pi }_{1}m{\pi }_2$ , the $OC{T}_M$ values for the rotary Heron triangles $\Delta \pi m\pi$ , and the absolute and relative deviations of the masses $m_{\Delta ,her}^{\Delta \left(1232\right),\pi }$ from the experimental values $m_r$ . From Table 11 it is evident that the masses $m_{\Delta ,her}^{\Delta \left(1232\right),\pi }$ differ from the masses $m_r$ of Δ Barions $\to \Delta \left(1232\right)+{\pi }_1$ decays within their widths. The average relative deviation is 5.1% of the resonance width. Column 9 shows the effective mass $m_{her}^{\pi ,\pi }$ of the pair of pi mesons, calculated using formula (6) through the length $S_{\pi ,\pi }$ of the base of the rotary Heron triangle $\Delta \pi m\pi$ .

Table 11. Hyperbolic Heron triangles in decays of $\Delta \text{ Barion}\to \Delta \left(1232\right)+{\text{π}}_1$ .

Name Δ Barion	Mass Resonance $m_r$ (Mev)	Width Γ (Mev)	$OC{T}_r$	$OC{T}_M$	Heron Mass in $\text{Δ}\pi m\Delta \left(1232\right)$ $m_{\Delta ,her}^{\Delta 232,\pi }$ (Mev)	$\text{Δ}{m}_r^{her}$ = $m_r$ − $m_{\Delta ,her}^{\Delta 232,\pi }$ (Mev)	$\frac{\text{Δ}{m}_r^{her}}{\text{Γ}}$ (%)	Mass in Rot_H $\text{Δ}\pi m\pi$ $m_{her}^{\pi ,\pi }$ (Mev)
1	2	3	4	5	6	7	8	9
$\Delta \text{Barion}\to \Delta \left(1232\right)+{\pi }_1$ , $m_{\Delta 232}=1232.0\text{Mev}$ , $m_{{\pi }_1}=m_{\pi }=139.57001\text{Mev}$
Δ(1600)	1570.0 ± 20.0	250.0	${2.74}_{-0.293}^{+0.300}$	3	1587.6	17.6	7.0	558.3
Δ(1620)	1610.0 ± 10.0	130.0	${3.34}_{-0.150}^{+0.152}$	3	1587.6	23.4	17.2	558.3
Δ(1700)	1710.0 ± 2.0	300.0	${4.89}_{-0.032}^{+0.032}$	5	1716.6	6.6	2.2	683.8
Δ(1900)	1860.0± 20.0	250.0	${7.41}_{-0.347}^{+0.351}$	7	1836.6	23.4	9.4	789.5
Δ(1905)	1880.0 ± 20.0	330.0	${7.76}_{-0.351}^{+0.355}$	8	1893.7	13.7	4.2	837.4
Δ(1910)	1900.0 ± 20.0	300.0	${8.11}_{-0.355}^{+0.358}$	8	1893.7	6.3	2.1	837.4
Δ(1950)	1930.0± 10.0	300.0	${8.65}_{-0.180}^{+0.182}$	9	1949.2	19.2	6.4	882.7
Δ(1940)	2000.0± 40.0	400.0	${9.94}_{-0.743}^{+0.758}$	10	2003.1	3.1	0.1	925.8
Δ(2000)	2100.0 ± 20.0	450.0	${11.87}_{-0.392}^{+0.396}$	12	2106.8	6.8	1.5	1006.5
Δ(2200)	2200.0± 30.0	300.0	${13.88}_{-0.615}^{+0.624}$	14	2205.6	5.6	1.2	1081.1

Table 12 shows the published masses $m_r$ and widths Γ of Strange Mezon$\to K\left(493\right)+{\pi }_1$ decays [13]. Columns 4 - 9 contain the calculated $OC{T}_r$ values for the rotary triangles $\Delta {\pi }_{1}m{\pi }_2$ , the $OC{T}_M$ values for the rotary Heron triangles $\Delta \pi m\pi$ , and the absolute and relative deviations of the masses $m_{her}^{K\left(493\right),\pi }$ from the experimental values $m_r$ . From Table 12 it is evident that the masses $m_{her}^{K\left(493\right),\pi }$ differ from the masses $m_r$ of Strange Mezon $\to K\left(493\right)+{\pi }_1$ decays within their widths. The average relative deviation is 12.1% of the resonance width. Column 9 shows the effective mass $m_{her}^{\pi ,\pi }$ of the pair of pi mesons, calculated using formula (6) through the length $S_{\pi ,\pi }$ of the base of the rotary Heron triangle $\Delta \pi m\pi$ .

Table 13 shows the published masses $m_r$ and widths Γ of Strange Mezon $\to K^{*}\left(892\right)+{\pi }_1$ decays [13]. Columns 4 - 9 contain the calculated $OC{T}_r$ values for the rotary triangles $\Delta {\pi }_{1}m{\pi }_2$ , the $OC{T}_M$ values for the rotary Heron triangles $\Delta \pi m\pi$ , and the absolute and relative deviations of the masses $m_{her}^{K\left(892\right),\pi }$ from the experimental values $m_r$ . From Table 13 it is evident that the masses $m_{her}^{K\left(892\right),\pi }$ differ from the masses $m_r$ of Strange Mezon $\to K^{*}\left(892\right)+{\pi }_1$ decays within their widths. The average relative deviation is 7.1% of the resonance width. Column 9 shows the effective mass $m_{her}^{\pi ,\pi }$ of the pair of pi mesons, calculated using formula (6) through the length $S_{\pi ,\pi }$ of the base of the rotary Heron triangle $\Delta \pi m\pi$ .

Table 12. Hyperbolic Heron triangles in decays of $\text{Strange Mezon}\to K\left(493\right)+{\pi }_1$ .

Name Strange Mezon	Mass Resonance $m_r$ (Mev)	Width Γ (Mev)	$OC{T}_r$	$OC{T}_M$	Heron Mass in $\Delta k493m\pi$ $m_{her}^{k493,\pi }$ (Mev)	$\text{Δ}{m}_{res}^{her}$ = $m_r$ − $m_{her}^{k493,\pi }$ (Mev)	$\frac{\text{Δ}{m}_{res}^{her}}{\text{Γ}}$ (%)	Mass in Rot_H $\text{Δ}\pi m\pi$ $m_{her}^{\pi ,\pi }$ (Mev)
1	2	3	4	5	6	7	8	9
$\text{Strange Mezon}\to K\left(493\right)+{\pi }_1$ , $m_{k493}=493.677\text{Mev}$ , $m_{{\pi }_1}=m_{\pi }=139.57001\text{Mev}$
$K_{}^{*}\left(700\right)$	838.0 ± 11.0	463.0	${2.66}_{-0.161}^{+0.164}$	3	860.9	22.9	4.9	558.3
$K_{}^{*}\left(892\right)$	891.8± 0.25	50.3	${3.48}_{-0.004}^{+0.004}$	3	860.9	30.9	61.4	558.3
$K_{}^{*}\left(1410\right)$	1421.8 ± 9.0	236.0	${14.27}_{-0.225}^{+0.226}$	14	1410.0	11.0	4.6	1081.8
$K_{}^{*}\left(1430\right)$	1425.6 ± 18.0	270.0	${14.39}_{-0.450}^{+0.457}$	14	1410.0	15.6	5.8	1081.8
$K_{}^{*}\left(1780\right)$	1718.0 ± 18.0	322.5	${22.5}_{-0.542}^{+0.549}$	22	1701.5	16.5	5.1	1338.7
$K_0^{*}\left(1950\right)$	1957.0 ± 14.0	170.5	${30.25}_{-0.481}^{+0.485}$	30	1949.9	7.1	4.2	1554.2
$K_0^{*}\left(2045\right)$	2045.0 ± 9.0	199.0	${33.35}_{-0.324}^{+0.326}$	33	2035.2	9.8	4.9	1627.7
$K_5^{*}\left(2380\right)$	2382.8 ± 14.0	178.0	${46.54}_{-0.586}^{+0.591}$	47	2393.6	10.8	6.1	1933.9

Table 13. Hyperbolic Heron triangles in decays of $\text{Strange Mezon}\to K^{*}\left(892\right)+{\pi }_1$ .

Name Strange Mezon	Mass Resonance $m_r$ (Mev)	Width Γ (Mev)	$OC{T}_r$	$OC{T}_M$	Heron Mass in $\Delta k892m\pi$ $m_{her}^{k892,\pi }$ (Mev)	$\text{Δ}{m}_r^{her}$ = $m_r$ − $m_{her}^{k892,\pi }$ (Mev)	$\frac{\text{Δ}{m}_r^{her}}{\text{Γ}}$ (%)	Mass in Rot_H $\text{Δ}\pi m\pi$ $m_{her}^{\pi ,\pi }$ (Mev)
1	2	3	4	5	6	7	8	9
$\text{Strange Mezon}\to K_{}^{*}\left(892\right)+{\pi }_1$ , $m_{k892}=891.76001\text{Mev}$ , $m_{{\pi }_1}=m_{\pi }=139.57001\text{Mev}$
$K_1\left(1270\right)$	1253.0 ±7.0	10.0	${3.04}_{-0.105}^{+0.107}$	3	1250.2	2.8	27.8	558.3
$K_1\left(1400\right)$	1403.0± 7.0	174.0	${5.43}_{-0.117}^{+0.118}$	5	1378.9	26.1	15.0	683.8
$K_1\left(1410\right)$	1414.0 ± 15.0	232.0	${5.62}_{-0.253}^{+0.256}$	6	1436.1	22.1	9.5	738.5
$K_2^{*}\left(1430\right)$	1427.3 ± 1.5	100.0	${5.85}_{-0.026}^{+0.026}$	6	1436.1	8.8	8.8	738.5
$K\left(1460\right)$	1482.4 ± 15.0	335.6	${6.81}_{-0.266}^{+0.269}$	7	1493.0	10.6	3.2	789.5
$K_2^{*}\left(1580\right)$	1580.8 ± 5.0	110.0	${8.62}_{-0.095}^{+0.095}$	9	1600.6	19.8	18.0	882.7
$K_{}^{*}\left(1680\right)$	1718.8 ± 18.0	320.0	${11.36}_{-0.370}^{+0.374}$	11	1701.4	17.4	5.4	967.0
$K_3^{*}\left(1780\right)$	1779.0 ± 8.0	161.0	${13.62}_{-0.170}^{+0.171}$	13	1796.6	17.6	10.9	1044.5
$K_2\left(1770\right)$	1773.8 ± 8.0	186.0	${12.51}_{-0.170}^{+0.171}$	13	1796.6	22.8	12.3	1044.5
$K_2\left(1820\right)$	1819.0 ± 12.0	264.0	${13.49}_{-0.261}^{+0.363}$	13	1796.6	22.4	8.5	1044.5
$K_2^{*}\left(1980\right)$	1990.0 ± 50.0	348.0	${17.40}_{-1.180}^{+1.210}$	17	1973.2	16.8	4.8	1184.3

4. Physics Interpretation of Heron’s Triangles

As can be seen from Tables 1-13, the rotary triangles $\Delta {\pi }_{1}m{\pi }_2$ of the decays of scalar, strange mesons and Δ, N, Λ, Σ, Ξ baryons coincide with the rotary Heron triangles $\Delta \pi m\pi$ (within the resonance widths). For the rotary Heron triangle $\Delta \pi m\pi$ , formula (12) takes the form (L is an integer):

$OC{T}_M=L={\left(P_{G\pi }/m_{\pi }\right)}^2=\left(m_b/m_{\pi }\right)\left(m_r^2-{\left(m_b+m_{\pi }\right)}^2\right)/{\left(m_b+m_{\pi }\right)}^2$ (18)

Here $m_b\ge m_{\pi }$ and $m_b=m_{\pi }$ , $m_p$ , $m_{\omega \left(782\right)}$ , $m_{\eta }$ , $m_{\rho \left(770\right)}$ , $m_{\Delta \left(1232\right)}$ , $m_{ {\Xi }^0}$ , $m_{\Lambda }$ , $m_{\Sigma }$ , $m_{k\left(493\right)}$ , $m_{k\left(892\right)}$ .

Formula (18) represents the dependence of the integer L on the square of the effective mass $m_r$ of the resonance, similar to the Regge dependence of the square of the resonance mass on its spin.

According to formulas (7) and (18) for a rotary Heron triangle $\Delta \pi m\pi$ , the integer values $L=OC{T}_M$ are equal to the square of the reduced momentum ${\left(P_{G\pi }/m_{\pi }\right)}^2$ of the π meson in the reference frame “G” (“G” is the center of inertia of the pair (π, π) of the π mesons).

According to formulas (5) and (6) for a rotary Heron triangle $\Delta \pi m\pi$ , the value $\text{ch}{S}_{\pi \pi }^{}$ , called the square of the reduced absorbed mass, is equal to an integer:

$\text{ch}{S}_{\pi \pi }=\left(m_{her}^2-2m_{\pi }^2\right)/2m_{\pi }^2$ (19)

According to formula (8), the expression ($\text{ch}{S}_{m\pi }-1$ ) is equal to the kinetic energy $T_{m\pi }^{}$ of the π meson in the reference frame “m” of additive mass. From formulas (5) and (8), it follows that the expression ($2T_{m\pi }/m_{\pi }$ ), called the reduced kinetic energy of the pair (π, π) of π mesons in the reference frame “m”, is equal to an integer.

Point “m” of the additive mass is the equilibrium point of Archimedes’ levers, at the ends of which the gravitational forces of the decaying particles are applied. Therefore, discrete hyperbolic Heron triangles can reflect the self-oscillations that occur during particle decay, caused by the constant action of free-fall acceleration.

5. Looking for New Resonances Based on Heron’s Triangles

For values of $OC{T}_M$ from the intervals (1 - 80), (200 - 301), (398 - 498), Table 14 shows some discrete characteristics of Heron triangles and effective masses of decays of scalar mesons and Δ, N baryons. Columns 2 and 3 show the calculations of the hyperbolic cosines of the lengths of the sides and bases of Heron’s triangles.

Column 4 show the calculations of the values of $Ka{s}_{\pi }^{\pi }$ —the generalized cosine of the angle θ between the tangents at points “π” and “π” of the base of Heron’s triangles Δπmπ. The values of $Ka{s}_{\pi }^{\pi }$ for various cases of the location of point “p” ($x_p^{},y_p^{}$ ) of intersection of these tangents relative to the Absolute (Figure 3):

$Ka{s}_{\pi }^{\pi }=\cos \left(\theta \right)<1,x_p^2+y_p^2<1$ (20)

$Ka{s}_{\pi }^{\pi }=\text{ch}\left(S_{cd}\right)>1,x_p^2+y_p^2>1$ (21)

Formula (20) corresponds to the case where the tangents (m – p) and (π – p) intersect inside the Absolute. Formula (21) corresponds to the case where the tangents (m – p), (π – p) intersect outside the Absolute, then the angle θ between them corresponds to a segment (c – d) of length $S_{cd}$ which the tangents cut off on the line (A₁ – A₂). The straight lines (A₁ – p), (A₂ – p) are tangents to the Absolute, drawn from the point “p”.

As can be seen from Table 14, the values of $Ka{s}_{\pi }^{\pi }=2\left(OC{T}_M-1\right)+1$ . From formulas (5), (6) it follows:

Table 14. Discrete characteristics and masses of Hyperbolic Heron Triangles.

$OC{T}_M^{}$	$ch{S}_{m\pi }^{}$	$ch{S}_{\pi \pi }^{}$	$Ka{s}_{\pi }^{\pi }$	$Ka{s}_m^{\pi }$	$m_{her}^{\pi ,\pi }$ (Mev)	$m_{\Delta ,her}^{P,\pi }$ (Mev)	$m_{N,her}^{P,\pi }$ (Mev)	$m_{her}^{\eta ,\pi }$ (Mev)	$m_{her}^{\rho \left(770\right),\pi }$ (Mev)	$m_{her}^{\text{ω}\left(782\right),\pi }$ (Mev)
1	2	3	4	5	6	7	8	9	10	11
1	1.5	3	1	−0.5	394.8	1155.2	1155.9	770.0	938.8	1001.1
2	2.0	5	3	0.0	485.3	1277.7	1228.4	844.6	1066.9	1074.2
3	2.5	7	5	0.5	558.3	1296.2	1296.9	913.1	1135.3	1142.6
4	3.0	9	7	1.0	624.8	1361.2	1361.9	976.8	1199.9	1207.1
5	3.5	11	9	1.5	683.8	1423.3	1423.3	1036.6	1261.1	1268.4
6	4.0	13	10	2.0	738.5	1482.8	1484.4	1093.1	1319.5	1326.8
7	4.5	15	13	2.5	789.5	1539.9	1599.4	1146.9	1375.4	1382.8
8	5.0	17	14	3.0	837.4	1595.1	1595.7	1198.2	1429.1	1436.6
9	5.5	19	17	3.5	882.7	1648.3	1648.3	1247.4	1480.9	1488.5
10	6.0	21	19	4.0	925.8	1700.0	1700.0	1294.8	1530.9	1538.6
11	6.5	23	21	4.5	967.0	1750.1	1750.7	1340.5	1579.4	1587.1
12	7.0	25	23	5.0	1006.5	1789.7	1799.4	1384.6	1624.4	1634.2
13	7.5	27	25	5.5	1044.4	1846.2	1846.8	1427.4	1672.1	1679.9
14	8.0	29	26	6.0	1081.1	1892.4	1893.1	1469.0	1716.5	1724.5
15	8.5	31	29	6.5	1116.6	1937.5	1938.2	1509.4	1759.9	1767.9
16	9.0	33	31	7.0	1150.9	1981.6	1982.3	1548.8	1802.2	1810.3
17	9.5	35	33	7.5	1184.3	2024.7	2025.4	1587.2	1843.5	1851.7
18	10.0	37	34	8.0	1216.7	2067.0	2067.7	1624.7	1883.9	1892.2
19	10.5	39	37	8.5	1248.4	2108.4	2109.1	1661.3	192359	1931.9
20	11.0	41	39	9.0	1279.2	2148.9	2149.7	1697.1	1962.3	1970.7
21	11.5	43	41	9.5	1309.3	2188.8	2189.5	1732.3	2000.3	2008.9
22	12.0	45	43	10.0	1338.7	2227.9	2228.7	1766.7	2037.6	2046.3
23	12.5	47	44	10.5	1367.5	2266.4	2267.1	1800.4	2074.2	2083.0
24	13.0	49	47	11.0	1395.7	2304.2	2304.9	1833.5	2110.3	2119.1
25	13.5	51	49	11.5	1423.3	2341.4	2342.1	1866.1	2145.7	2154.6
26	14.0	53	51	12.0	1450.5	2378.0	2378.8	1891.1	2180.3	2189.5
27	14.5	55	53	12.5	1477.1	2414.1	2414.8	1929.5	2214.8	2223.9
28	15.0	57	55	13.0	1503.2	2449.6	2450.4	1960.5	2248.5	2257.7
29	15.5	59	57	13.5	1528.9	2484.6	2485.4	1990.9	2281.8	2291.0
30	16.0	61	59	14.0	1554.2	2519.1	2519.9	2020.9	2314.6	2323.9
31	16.5	63	61	14.5	1579.1	2553.2	2554.0	2050.5	2346.9	2356.3
32	17.0	65	63	15.0	1603.5	2586.8	2587.7	2079.7	2378.8	2388.3
33	17.5	67	65	15.5	1627.7	2620.2	2620.9	2108.4	2410.3	2419.8
34	18.0	69	67	16.0	1651.4	2252.8	2653.6	2136.8	2441.3	2451.0
35	18.5	71	69	16.5	1674.8	2685.2	2686.0	2164.7	2472.0	2481.7
36	19.0	73	71	17.0	1698.0	2717.2	2718.0	2192.4	2502.3	2512.1
37	19.5	75	73	17.5	1720.7	2748.8	2749.6	2219.7	2532.2	2542.1
38	20.0	77	74	18.0	1743.2	2780.0	2780.9	2246.6	2561.8	2571.7
39	20.5	79	77	18.5	1765.4	2811.0	2811.8	2272.3	2591.0	2601.1
40	21.0	81	79	19.0	1787.4	2841.5	2842.4	2299.6	2619.9	2630.1
41	21.5	83	81	19.5	1809.0	2871.8	2872.7	2325.6	2648.5	2658.7
42	22.0	85	83	20.0	1830.5	2901.7	2902.6	2351.4	2676.8	2687.1
43	22.5	87	85	20.5	1851.6	2931.3	2932.2	2376.8	2704.8	2715.2
44	23.0	89	87	21.0	1872.5	2960.7	2961.6	2402.6	2372.0	2743.0
45	23.5	91	89	21.5	1893.2	2989.7	2990.6	2426.9	2760.0	2770.5
46	24.0	93	91	22.0	1913.7	3018.5	3019.4	2451.6	2787.1	2797.7
47	24.5	95	93	22.5	1933.9	3047.0	3047.9	2576.0	2814.0	2824.7
48	25.0	97	95	23.0	1954.0	3075.2	3076.1	2500.2	2840.7	2851.4
49	25.5	99	97	23.5	1973.8	3103.2	3140.1	2524.2	2867.1	2877.9
50	26.0	101	99	24.0	1993.5	3130.9	3131.8	2547.9	2893.2	2904.1
51	26.5	103	101	24.5	2012.9	3158.3	3159.3	2571.4	2919.1	2930.1
52	27.0	105	103	25.0	2032.2	3185.6	3186.5	2594.7	2944.8	2955.9
53	27.5	107	105	25.5	2051.3	3212.6	3213.6	2617.9	2970.3	2981.4
54	28.0	109	107	26.0	2070.2	3239.4	3240.3	2640.7	2995.6	3006.8
56	29.0	113	110	27.0	2107.5	3292.3	3293.2	2685.9	3045.4	3056.8
60	31.0	121	119	29.0	2180.2	3395.7	3396.6	2774.1	3142.8	3154.5
63	32.5	127	125	30.5	2233.1	3471.2	3472.2	2838.5	3214.0	3225.8
64	33.0	129	127	31.0	2250.5	3496.0	3497.0	2859.6	3237.3	3249.2
68	35.0	137	135	33.0	2318.7	3593.5	3594.5	2942.6	3329.1	3341.2
69	35.5	139	137	33.5	2335.5	3617.4	3618.5	2963.0	3351.7	3363.9
70	36.0	141	139	34.0	2352.1	3641.2	3642.3	2983.2	3374.1	3383.6
77	39.5	155	153	37.5	2465.3	3803.7	3804.8	3121.3	3526.9	3539.6
78	40.0	157	154	38.0	2481.1	3826.4	3827.4	3140.5	3548.2	3561.0
79	40.5	159	157	38.5	2496.7	3848.9	3850.0	3159.6	3569.4	3582.2
80	41.0	161	159	39.0	2512.3	3871.3	3872.4	3178.6	3590.4	3603.3
200	101.0	401	399	99.0	3957.5	5977.0	5977.0	4954.8	5565.2	5584.3
215	108.5	431	428	106.5	4102.5	6190.0	6190.0	5138.8	5764.7	5784.4
217	109.5	435	433	107.5	4121.5	6217.9	6217.9	5157.2	5790.8	5810.5
221	111.5	443	440	109.5	4159.1	6273.2	6273.2	5203.7	5842.6	5862.5
226	114.0	453	450	112.0	4205.7	6341.7	6341.7	5261.2	5906.7	5926.8
248	125.0	497	494	123.0	4404.8	6634.7	6634.7	5507.1	6181.0	6209.9
254	128.0	509	506	126.0	4457.5	6712.4	6712.4	5572.3	6253.7	6274.9
263	132.5	527	524	130.5	4535.5	6827.2	6827.2	5668.7	6361.2	6382.7
272	137.0	545	543	135.0	4612.1	6940.2	6940.2	5763.5	6466.9	6488.8
301	151.5	603	601	149.5	4850.9	7292.3	7292.3	6058.8	6796.3	6819.3
398	200.0	797	794	198.0	5575.8	8363.1	8363.1	6956.1	7797.8	7824.0
409	205.5	819	817	203.5	5652.2	8476.0	8476.0	7050.6	7903.4	7929.9
436	219.0	873	871	217.0	5835.3	8746.9	8746.9	7277.5	8156.7	8184.9
498	250.0	997	995	248.0	6235.5	9339.3	9339.3	7773.4	8710.5	8739.6

Figure 3. The values of $Ka{s}_{\pi }^{\pi }$ —the generalized cosine of the angle $\theta$ of decay between the tangents at the points “π”, “π” of the base of Heron’ s triangles Δπmπ are calculated. The values of $Ka{s}_m^{\pi }$ —the generalized cosine of the angle α of decay between the tangent at point “m” of the additive mass and the tangent at points “$\pi$ ” of the base of Heron's triangles Δπmπ are calculated.

$\text{ch}\left(S_{\pi \pi }\right)=Ka{s}_{\pi }^{\pi }+2$ (22)

From formula (22) it follows that the expression $\text{ch}\left(S_{\pi \pi }\right)$ linearly depends on the $Ka{s}_{\pi }^{\pi }$ . We have a dependence similar to the Regge dependence of the square of the resonance mass on its spin.

Column 5 shows the values of $Ka{s}_m^{\pi }$ —the generalized cosine of the angle α between the tangent at point “m” of the additive mass and the tangent at points “$\pi$ ” of the of the side of Heron’s triangles Δπmπ. The values of $Ka{s}_m^{\pi }$ for various cases of the location of point “t” ($x_t^{},y_t^{}$ ) of intersection of these tangents relative to the Absolute (Figure 3):

$Ka{s}_m^{\pi }=\cos \left(\alpha \right)<1,x_t^2+y_t^2<1$ (23)

$Ka{s}_m^{\pi }=\text{ch}\left(S_{ab}\right)>1,x_t^2+y_t^2>1$ (24)

Formula (23) corresponds to the case where the tangents (m – t) and (π – t) intersect inside the Absolute. Formula (24) corresponds to the case where the tangents (m – t), (π – t) intersect outside the Absolute, then the angle α between them corresponds to a segment (a – b) of length $S_{ab}$ which the tangents cut off on the line (A₃ – A₂). The straight lines (A₃ – t), (A₂ – t) are tangents to the Absolute, drawn from the point “t”. As can be seen from Table 14, the values of $Ka{s}_m^{\pi }=\left(OC{T}_M-2\right)/2$ . From formula (7) it follows:

$\text{ch}\left(S_{m\pi }\right)-1=T_{m\pi }/m_{\pi }=Ka{s}_m^{\pi }+1$ (25)

From formula (25) it follows that expression $(\text{ch}{S}_{m\pi }-1$ ) is equal to the kinetic energy $T_{m\pi }^{}$ π meson in the reference frame “m” of the additive mass, divided by the mass $m_{\pi }$ of the π meson. From formula (25) it follows that the expression ($T_{m\pi }/m_{\pi }$ ), called the reduced kinetic energy of the $\pi$ meson.

Column 6 shows the masses $m_{her}^{\pi ,\pi }$ in the rotary Heron triangles $\Delta \pi m\pi$ of decays Scalar Mezon $\to {\pi }_1+{\pi }_2$ , calculated using formula (6). Column 7 shows the masses $m_{\Delta ,her}^{P,\pi }$ of Δ Barions $\to P+{\pi }_1$ decays calculated using formula (15) for values $m_b=m_P$ . Column 8 shows the masses $m_{N,her}^{P,\pi }$ of N Barions $\to P+{\pi }_1$ decays calculated using formula (15) for values $m_b=m_P$ . Column 9 shows the masses $m_{her}^{\eta ,\pi }$ of Scalar Mezon $\to \eta +{\pi }_1$ decays calculated using formula (15) for values $m_b=m_{\eta }$ . Columns 9 and 10 show the masses $m_{her}^{\rho \left(770\right),\pi }$ and $m_{her}^{\omega \left(782\right),\pi }$ calculates using formula (15) for values $m_b=m_{\rho \left(770\right)}$ , $m_{\omega \left(782\right)}$ .

The mass values in Table 14 corresponding to the resonances from Tables 1-13 are highlighted in bold. The remaining unhighlighted mass values may correspond to the masses of new resonances in the decays of scalar mesons and Δ, N baryons. To detect these new resonances, it is necessary to investigate the distribution $OC{T}_r$ calculated using formula (13) for $m_b=m_{\pi }$ , $m_p$ , $m_{\omega \left(782\right)}$ , $m_{\eta }$ , $m_{\rho \left(770\right)}$ .

In Table 15, the effective masses of the decays of strange mesons and Δ, N, Λ, Σ, Ξ baryons are given for the values of $OC{T}_M$ from the intervals (1 - 80), (200 - 301), (398 - 498). Columns 2 - 9 show the masses $m_{her}^{\pi ,\pi }$ , $m_{her}^{\Sigma ,\pi }$ , $m_{her}^{\Lambda ,\pi }$ , $m_{her}^{{\text{ Ξ}}^0,\pi }$ , $m_{\Delta ,her}^{\Delta 232,\pi }$ , $m_{N,her}^{\Delta 232,\pi }$ , $m_{\Delta ,her}^{k493,\pi }$ , $m_{\Delta ,her}^{k892,\pi }$ calculated using formula (15) for $m_b=m_{\pi }$ , $m_{\Sigma }$ , $m_{\Lambda }$ , $m_{{\text{ Ξ}}^0}$ , $m_{\Delta 232}$ , $m_{k493}$ , $m_{k892}$ . The mass values in Table 15 corresponding to the resonances from Tables 1-13 are highlighted in bold. The remaining unhighlighted mass values may correspond to the masses of new resonances in the decays of strange mesons and Δ, N, Λ, Σ, Ξ baryons. To detect these new resonances, it is necessary to investigate the distribution $OC{T}_r$ calculated using formula (13) for $m_b=m_{\pi }$ , $m_{\Sigma }$ , $m_{\Lambda }$ , $m_{{\text{Ξ}}^0}$ , $m_{\Delta 232}$ , $m_{k493}$ , $m_{k892}$ .

Note that from Table 2, Table 3, Tables 8-13 it follows that rotary Heron triangles with the values $OC{T}_M=1,2,3,4,5,6,7$ are detected in the decays of strange mesons and Δ, N, Λ, Ξ, Σ baryons. Figure 4 expands on the data from Tables 1-13 for the first $OC{T}_M=1,\cdots ,7$ rotary Heron triangles. The oricycles with inscribed rotary Heron triangles are shifted upward along their axes. Each

$OC{T}_M$ level has Heron triangles Δπmπ with own lattice $\frac{\text{ch}{S}_{m\pi }}{0.5}\times \text{ch}{S}_{\pi \pi }$ (see formulas (5)).

Table 15. $OC{T}_M$ and masses of Hyperbolic Heron Triangle.

$OC{T}_M^{}$	$m_{her}^{\pi ,\pi }$ (Mev)	$m_{her}^{\Sigma ,\pi }$ (Mev)	$m_{her}^{\Lambda ,\pi }$ (Mev)	$m_{her}^{{\text{ Ξ}}^0,\pi }$ (Mev)	$m_{\Delta ,her}^{\Delta 232,\pi }$ (Mev)	$m_{N,her}^{\Delta 232,\pi }$ (Mev)	$m_{her}^{k493,\pi }$ (Mev)	$m_{her}^{k892,\pi }$ (Mev)
1	2	3	4	5	6	7	8	9
1	394.8	1404.8	1331.5	1579.7	1447.2	1447.2	717.2	1109.1
2	483.5	1476.7	1403.5	1601.4	1519.0	1519.0	792.3	1181.8
3	558.3	1545.3	1472.1	1670.0	1587.6	1587.6	860.9	1250.2
4	624.2	1610.9	1537.6	1736.0	1653.4	1653.4	924.4	1315.1
5	683.8	1674.0	1600.4	1799.5	1716.6	1716.6	983.8	1377.0
6	738.5	1734.8	1660.8	1860.8	1776.6	1776.6	1039.8	1436.1
7	789.5	1793.5	1719.1	1920.2	1836.6	1836.6	1093.0	1493.0
8	837.4	1850.4	1775.5	1977.8	1893.7	1893.7	1143.7	1547.7
9	882.7	1905.6	1830.2	2033.8	1949.2	1949.2	1192.2	1600.6
10	925.8	1959.2	1883.3	2088.3	2003.1	2003.1	1238.8	1651.8
11	967.0	2011.4	1934.9	2141.3	2055.6	2055.6	1283.8	1704.4
12	1006.5	2062.3	1985.2	2193.1	2106.8	2106.8	1327.2	1749.7
13	1044.4	2111.9	2034.2	2243.8	2156.8	2156.8	1369.2	1796.6
14	1081.1	2160.4	2082.1	2088.3	2205.6	2205.6	1410.0	1842.3
15	1116.6	2207.9	2128.9	2541.7	2253.4	2253.4	1449.7	1887.0
16	1150.9	2254.3	2174.7	2389.2	2300.2	2300.2	1488.3	1930.6
17	1184.3	2299.8	2219.6	2435.7	2346.1	2346.1	1525.9	1973.2
18	1216.7	2344.5	2263.6	2481.4	2391.1	2391.1	1562.6	2015.0
19	1248.4	2388.3	2306.7	2526.2	2435.2	2435.2	1598.4	2055.9
20	1279.2	2431.3	2349.0	2570.3	2478.6	2478.6	1633.5	2096.0
21	1309.3	2473.5	2390.6	2613.6	2521.2	2521.2	1667.9	2135.3
22	1338.7	2515.1	2431.5	2656.2	2563.2	2563.2	1701.5	2173.9
23	1367.5	2555.9	2471.7	2698.1	2604.4	2604.4	1734.5	2211.9
24	1395.7	2596.1	2511.2	2379.4	2645.0	2645.0	1766.9	2249.2
25	1423.3	2635.8	2550.2	2780.1	2685.0	2685.0	1798.7	2285.9
26	1450.5	2674.8	2588.6	2820.2	2724.4	2724.4	1829.9	2322.1
27	1477.1	2713.3	2626.4	2859.7	2763.2	2763.2	1860.6	2357.6
28	1503.2	2751.2	2663.6	2898.7	2801.5	2801.5	1890.9	2392.9
29	1528.9	2788.6	2700.4	2937.2	2839.3	2839.3	1920.6	2427.2
30	1554.2	2825.2	2736.6	2975.2	2876.6	2876.6	1949.9	2461.3
31	1579.1	2861.9	2772.4	3012.7	2913.4	2913.4	1978.8	2494.8
32	1603.5	2897.9	2807.6	3049.7	2949.7	2949.7	2007.2	2528.0
33	1627.6	2933.5	2842.6	3086.3	2985.7	2985.7	2035.3	2560.7
34	1651.4	2968.6	2877.1	3122.5	3021.1	3021.1	2062.9	2593.0
35	1674.8	3003.3	2911.1	3158.2	3056.2	3056.2	2090.2	2624.9
36	1698.0	3037.6	2944.8	3193.6	3090.9	3090.9	2117.2	2656.4
37	1720.7	3071.5	2978.1	3228.5	3125.1	3125.1	2143.8	2687.6
38	1743.2	3105.1	3011.0	3263.1	3159.1	3159.1	2170.0	2718.4
39	1765.4	3138.3	3043.5	3297.4	3192.6	3192.6	2192.6	2748.8
40	1787.4	3171.1	3075.7	3331.2	3225.8	3225.8	2221.7	2778.9
41	1809.0	3203.6	3107.6	3364.8	3258.7	3258.7	2247.0	2808.7
42	1830.5	3235.8	3139.2	3398.0	3291.2	3291.2	2271.1	2838.2
43	1851.6	3267.7	3170.4	3430.9	3323.4	3323.4	2296.9	2867.4
44	1872.5	3299.2	3201.4	3463.4	3355.4	3355.4	2321.5	2896.3
45	1893.2	3330.5	3232.0	3495.7	3387.0	3387.0	2345.8	2924.9
46	1913.7	3361.5	3262.4	3527.7	3418.3	3418.3	2369.8	2953.2
47	1933.9	3392.2	3292.4	3559.4	3449.3	3449.3	2393.6	2981.3
48	1954.0	3422.6	3322.2	3590.7	3480.1	3480.1	2417.2	3009.0
49	1973.8	3452.7	3351.8	3621.9	3510.6	3510.6	2440.5	3036.6
50	1993.5	3482.6	3381.0	3652.8	3540.8	3540.8	2463.6	3063.9
51	2012.9	3512.2	3410.1	3683.4	3540.8	3540.8	2463.6	3063.9
52	2032.2	3541.6	3438.8	3713.7	3600.5	3600.5	2509.2	3117.7
53	2051.3	3570.8	3467.4	3743.8	3630.0	3630.0	2531.7	3144.3
54	2070.2	3599.6	3495.7	3652.8	3659.2	3659.2	2554.0	3170.7
55	2088.9	3628.3	3523.8	3803.3	3688.2	3688.2	2576.1	3196.8
56	2107.5	3656.8	3551.6	3832.8	3717.0	3717.0	2598.0	3222.8
57	2125.9	3685.0	3579.3	3861.9	3745.5	3745.5	2619.8	3248.5
58	2144.1	3713.0	3606.7	3890.9	3773.9	3773.9	2641.3	3274.0
59	2162.2	3740.3	3634.0	3919.7	3802.0	3802.0	2662.7	3299.4
60	2180.2	3768.4	3660.9	3948.2	3829.9	3829.9	2683.9	3324.4
63	2233.1	3850.0	3740.8	4032.6	3912.5	3912.5	2746.5	3398.7
64	2250.5	3876.1	3767.1	4060.3	3939.6	3939.6	2767.1	3423.1
68	2318.7	3982.3	3870.3	4169.5	4046.4	4046.4	2847.8	3519.1
69	2335.5	4008.2	3895.7	4196.3	4072.6	4072.6	2867.7	3542.6
70	2352.1	4034.1	3921.0	4223.0	4098.8	4098.8	2887.4	3566.1
77	2465.3	4210.0	4093.1	4405.2	4276.9	4276.9	3021.7	3725.9
78	2481.1	4234.5	4117.1	4430.5	4301.7	4301.7	3040.4	3748.1
79	2496.7	4259.0	4141.0	4455.8	4326.4	4326.4	3059.0	3770.3
80	2512.3	4283.2	4164.7	4481.0	4350.9	4350.9	3077.4	3792.3
200	3957.5	6573.8	6403.0	6857.4	6671.2	6671.2	4803.6	5861.6
200	3957.5	6573.8	6403.0	6857.4	6671.2	6671.2	4803.6	5861.6
215	4102.5	6806.2	6629.8	7098.7	6906.6	6906.6	4977.5	6070.8
217	4121.5	6836.6	6659.5	7130.3	6937.4	6937.4	5000.2	6098.2
221	4169.1	6896.9	6718.4	7193.0	6998.6	6998.6	5045.4	6152.5
226	4205.7	6971.6	6791.4	7270.6	7074.3	7074.3	5101.2	6219.8
248	4404.8	7291.3	7103.5	7602.8	7398.3	7398.3	5340.1	6307.6
254	4457.5	7376.1	7186.2	7690.8	7484.2	7484.2	5403.4	6583.9
263	4535.5	7501.4	7308.6	7821.1	7611.2	7611.2	5497.0	6696.7
272	4612.1	7624.8	7429.0	7949.3	7736.2	7736.2	5589.0	6807.6
301	4850.9	8009.2	7804.3	8348.8	8125.8	8125.8	5875.8	7153.4
398	5575.8	9178.8	8945.7	9564.7	9311.4	9311.4	6747.0	8204.8
409	5652.2	9302.1	9066.1	9692.9	9436.4	9436.4	6838.8	8315.7
436	5835.3	9598.2	9355.0	10000.8	9736.5	9736.5	7059.0	8581.7
498	6235.5	10245.7	9986.9	10674.1	10392.9	10392.9	7540.5	9163.3

Figure 4. The data from Table 14, Table 15 for the first $OC{T}_M$ = 1, …, 7 Heron triangles. The oriycles with inscribed Heron triangles are shifted upward along there axes ($C_0-m$ ). Each $OC{T}_M$ level has Heron triangles Δπmπ with own lattice $\frac{ch{S}_{m\pi }^{}}{0.5}$ × $ch{S}_{\pi \pi }^{}$ .

The value $OC{T}_M=2$ from Table 1 for the scalar meson $f_o\left(500\right)$ corresponds to an average mass of 500 Mev and a width of 300 Mev. However, all measurements of the mass and width $f_o\left(500\right)$ meson have a large variance of values (mass 400 - 800 MeV, width 100 - 800 MeV) [13]. Then it is quite possible that in the region $OC{T}_r<7$ (calculated using formula (9)) instead of $f_o\left(500\right)$ meson, 6 or less than 6 new scalar mesons with $OC{T}_M=1,2,3,4,5,6$ and masses of (394.8, 483.5, 558.3, 624.2, 683.7, 738.5) Mev may be detected. These new scalar mesons will appear in (${\pi }^{+},{\pi }^{+}$ ), (${\pi }^{-},{\pi }^{-}$ ), (${\pi }^{+},{\pi }^{-}$ ), (${\pi }^{+},{\pi }^0$ ), (${\pi }^{-},{\pi }^0$ ), (${\pi }^0,{\pi }^0$ ) decays.

Thus, based on actual data on effective masses of pairs (${\pi }_1,{\pi }_2$ ), ($P,{\pi }_1$ ), (${\Xi }^0,{\pi }_1$ ), ($\Sigma ,{\pi }_1$ ), ($\eta ,{\pi }_1$ ), ($\rho \left(770\right),{\pi }_1$ ), ($\omega \left(782\right),{\pi }_1$ ), ($\Lambda ,{\pi }_1$ ), ($\Delta \left(1232\right),{\pi }_1$ ), ($K\left(493\right),{\pi }_1$ ), ($K^{*}\left(892\right),{\pi }_1$ ) it is sufficient to examine the distribution of $OC{T}_r$ , calculated using formula (13). Statistically significant peaks in the distribution at $OC{T}_r<50$ will correspond to new resonances with masses < 2.5 GeV.

Table 14, Table 15 show that at $50<OC{T}_M<200$ , new resonances with masses (3 - 4) GeV can be detected. At $200<OC{T}_M<300$ , new resonances with masses (4 - 6) Gev can be detected. At $300<OC{T}_M<400$ , new resonances with masses (6 - 8) Gev can be detected. At $400<OC{T}_M<500$ , new resonances with masses (8 - 10) Gev can be detected. It would be interesting to detect resonances with masses > 10 Gev, corresponding to $OC{T}_M$ values > 500.

Finally, we present three functions whose integer values define new types of hyperbolic Heron triangles. The first function (HSASB2) is the product of the hyperbolic sine of the altitude and the hyperbolic sine of half the base of the rotary Heron triangles $\Delta {\pi }_{1}m{\pi }_2$ :

$\text{HSASB2}=\text{sh}\left(S_{Gm}\right)\text{sh}\left(S_{\pi }{{}_{{}_1}}_{\pi }{}_{{}_2}/2\right)=0.5*OC{T}_r\sqrt{OC{T}_r/\left(OC{T}_r+1\right)}$ (26)

Here, $S_{Gm}^{}$ is the altitude dropped from vertex “m” to the base, and $S_{\pi \pi }^{}$ is the base length of the triangle. The second function (HSP2) is the hyperbolic sine of the semi-perimeter of the $\Delta {\pi }_{1}m{\pi }_2$ :

$\text{HSP2}=\text{sh}\left(S_m{{}_{\pi }}_{{}_1}+S_{\pi }{{}_{{}_1}}_{\pi }{}_{{}_2}/2\right)=0.5\sqrt{OC{T}_r}\left(\sqrt{\left(OC{T}_r+4\right)\left(OC{T}_r+1\right)}+OC{T}_r+2\right)$ (27)

Here, $S_{m{\text{π}}_1}^{}$ is the length of the lateral side, and $S_{\pi }{{}_{{}_1}}_{\pi }{}_{{}_2}$ is the base length of the triangle. The third function (CotHArea) is the cotangent of the hyperbolic area of triangle $\Delta {\pi }_{1}m{\pi }_2$ :

$\text{CotHArea}=\text{Ctg}\left(\pi -M-2A\right)$ (28)

Here, M is the angle at vertex “m”, and A is the angle at the base of the triangle $\Delta {\pi }_{1}m{\pi }_2$ .

6. Conclusions

The published data show that the decays of scalar, strange mesons, and Δ, N, Λ, Σ, Ξ baryons correspond (within the resonance width) to hyperbolic Heron triangles with integer values of $OC{T}_r$ . Based on Heron’s triangles, the existence of new resonances is predicted. This can be confirmed by actual measurements of effective masses in the decays of scalar, strange mesons, and Δ, N, Λ, Σ, Ξ baryons.

In addition, Heron’s triangles have about 10 other discrete characteristics. Therefore, the detection of Heron triangles in hadron spectra can experimentally relate these discrete characteristics to quantum resonance numbers.

Further development of the described approach will consist of:

• isolation of resonances by the Heron triangle method and analysis of the angular distributions of their decays using parametrization of the dynamic axis of spin quantization by Lobachevsky straight line beams [15];

• analysis of 3 particles decays of resonances based on a 3-dimensional analogue of Heron triangles.

Funding

The work was financed by the LLP “Industry 4.0”, 050020 Almaty, Kazakhstan.

Data Availability Statement

The data used in the article are taken from open sources [13].

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

[1]	Lobachevsky, N.I. (1956) On the Principles of Geometry. In: Collection of Classic Works on Lobachevsky’s Geometry and the Development of Its Ideas, GITTL Publication, 27-49.
[2]	Bolyai, J. (1956) Appendix. Maros-Vasarhely. 1832. In: Collection of Classic Works on Lobachevsky’s Geometry and the Development of Its Ideas, GITTL Publication, 71.
[3]	Beltrami, E. (1956) Essay on the Interpretation of Non-Euclidean Geometry. In: Collection of Classic Works on Lobachevsky’s Geometry and the Development of Its Ideas, GITTL Publication, 182.
[4]	Klein, F. (1924) On the Geometric Bases of the Lorentz Group. In: New Ideas in Mathematics, Vol. 5, Springer, 144-174.
[5]	Kotelnikov, A.P. (1927) Relativity Principle and Lobachevsky’s Geometry. Memorial Nikolaĭ Ivanovich Lobachevskiĭ, 2, 37-36,
[6]	Chernikov, N.A. (1973) Lobachevsky’s Geometry and Relativistic Mechanics. Elementary Particles and Atomic Nuclei. Atomizdat.
[7]	Khen, V.P. (1975) Beltrami Model of the Lobachevsky Space Applied to the Kinematics of Hadron Reactions. https://inis.iaea.org/records/eat3k-r4t34
[8]	Khen, V.P. (1977) Application of the Lobachevsky Velocity Space to the Analysis of Reactions with the Birth of Resonances. PhD Thesis, Joint Institute for Nuclear Research.
[9]	Kagan, V.F. (1949) Foundations of Geometry. Part I. Lobachevsky’s Geometry and Its Prehistory. GITTL Publication.
[10]	Khen, V.P. and Khen, A.V. (2025) Heron’s Triangles, Golden Ratio and Quantization of Decays of Scalar, Strange Mesons and Δ, N Baryons in Hyperbolic Lobachevsky Velocity Space. (In Russian) https://www.litres.ru/book/aleksey-valerevich-h/treugolniki-gerona-zolotoe-sechenie-i-kvantovanie-eff-71529268/
[11]	Khen, V.P. and Khen, A.V. (2026) Heron’s Triangles and Resonance Decays in Lobachevsky Velocity Space. https://www.amazon.com/dp/B0DYJVD2MB/
[12]	Khen, V.P. and Khen, A.V. (2025) Heron’s Triangles, Golden Ratio and Quantization of Decays of Scalar, Strange Mesons and Δ, N Baryons in Hyperbolic Lobachevsky Velocity Space. Journal of Applied Mathematics and Physics, 13, 3337-3351.[CrossRef]
[13]	Navas, S., Amsler, C., Gutsche, T., Hanhart, C., Hernández-Rey, J.J., Lourenço, C., et al. (2024) Review of Particle Physics. Physical Review D, 110, Article 030001.[CrossRef]
[14]	Robb, A.A. (1911) Optical Geometry of Motion, a New View of the Theory Relativity. W. Heffer & Sons Ltd.
[15]	Bubelev, E.G., Khen, V.P. and Yatsyuk, V.G. (1976) Parameterization of the Directions of the Dynamic Axis for Rho, Ksup (ω) and Δsup (++) Resonances by an Arbitrary Pencil of Lobachevsky’s Straight Lines. https://inis.iaea.org/records/ra8x3-5nm97