jamp Journal of Applied Mathematics and Physics 2327-4379 2327-4352 Scientific Research Publishing 10.4236/jamp.2026.1411215 jamp-155255 Article Physics Mathematics 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, and the Proposal to Search for New Resonances Khеn Valeriy Pavlovich 1 Khen Aleksey Valerevich 1 Limited Liability Partnership “Industry 4.0”, Almaty, Kazakhstan

There is no conflict of interest among the authors of this article.

 05 11 2026 11 2026 14 11 4369 4404 20 07 2026 09 11 2026 12 11 2026 © 2026 by the authors and Scientific Research Publishing Inc. 2026 This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( https://creativecommons.org/licenses/by/4.0/ ). https://doi.org/10.4236/jamp.2026.1411215

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.

 Lobachevsky-Bolyai Velocity Space Decay Triangles Hyperbolic Sine of the Triangle Base Heron Triangle 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 Δ π 1 m π 2 of the decay of the meson f o ( 980 ) → π 1 + π 2 (Figure 1). If the masses of the decay particles are unequal, we have a different-sided decay triangle Δ η m π 1 of the decay of the mezon a o ( 1450 ) → η + π 1 (Figure 2). The triangle Δ η m π 1 can be mapped to the isosceles triangle Δ π 1 m π 2 by rotating the arc l m π 1 of an oricycle around the straight line ( m – G η π 1 ) (the point “ G η π 1 ” represents the center of inertia of the pair particles π 1 and η ) (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 Δ b m π 1 and the isosceles triangles Δ π 1 m π 2 inscribed in the oricycle of hyperbolic velocity space (HVS) (Figure 3). Further, we will refer to the isosceles triangle Δ π 1 m π 2 as the base triangle.

Decays of ρ ( 770 ) → π 1 + π 2 and f o ( 980 ) → π 1 + π 2 in the Beltrami model of the hyperbolic velocity space. The separate ellipses of decay oricycles of ρ ( 770 ) , f o ( 980 ) with centers in the “ C 0 ” points of the circle X 2 + Y 2 = 1 , called theAbsolute. 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 ( 980 ) meson decays are combined. The point-velocity “G” represents the centers of inertia of pairs of particles ( π 1 , π 2 ).

The different sided triangles Δ η m π 1 of the decays of the a 2 ( 1320 ) → η + π 1 and a o ( 1450 ) → η + π 1 , isosceles base triangles Δ π 1 m π 2 , corresponding to the triangles Δ η m π 1 . The point-velocity “G” represents the centers of inertia of pairs of particles ( π 1 , π 2 ). The point “ G η π 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 Δ π 1 m π 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 H S T r e s triangles of their decays , the integer values of H S T h e r , the masses m h e r b , π 1 and m h e r π 1 , π 2 , and the absolute and relative values of the deviations of the mass m h e r b , π 1 from the effective mass m r e s (Tables 1-17, Equations (3)-(13), b = π 2 , P, n, η , ρ ( 770 ) , ω ( 782 ) , Λ , Σ , Σ c 0 ( 2455 ) , Ξ 0 , Δ ( 1232 ) , K ( 493 ) , K * ( 892 ) ). Tables 1-17 show that the calculated effective base mass m h e r b , π 1 for Heron’s triangles agree well with the published values of the effective mass m r e s (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 H S T h e r 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 <italic>Δ</italic> , <italic>N</italic> , <italic>Λ</italic> , <italic>Σ</italic> , <italic>Ξ</italic> 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 π 1 :

B + T → b + π 1 + a l l

Let us assume that the velocities of the particles B , b and π 1 have been measured in reference frame “0”. The ends of the particle velocity vectors represent material velocity points “B,” “b”, “ π 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 π 1 of the reaction particles) [1]-[5]. Let’s draw a plane in the HVS passing through the points “B”, “ π 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 π 1 , Y π 1 ) of the velocity vector of the particle π 1 on the axes 0X, 0Y are called the Beltrami coordinates of the point “ π 1 ” in HVS [6] [9] [10]. The length S b π 1 of the straight line segment ( b – π 1 ) with the Beltrami coordinates of its ends “b” ( X b , Y b ), “ π 1 ” ( X π 1 , Y π 1 ) is represented by the formula [6]:

ch ( S b π 1 ) = ( 1 − X π 1 ∗ X b − Y π 1 ∗ Y b ) / ( R π 1 ∗ R b ) R b = 1 − X b 2 − Y b 2 ,     R π 1 = 1 − X π 1 2 − Y π 1 2

The length S b π 1 of the straight line ( b – π 1 ) is called the rapidity [11].

Besides the straight line ( b – π 1 ), one pair of symmetrical arcs of zero curvature, called oricycles, passes through the velocity points “b” and “ π 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 ρ ( 770 ) and f o ( 980 ) 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 π 1 = m b + m π 1 ( m b ≥ m π 1 ) is defined by Equations (3). The role of forces in Archimedes’ levers is played by the masses m b and m π 1 , and the lever arms are equal to the Euclidean lengths l b π 1 , l m b , l m π 1 of the arcs of the oricycle [12]-[14]:

l b π 1 = l m π 1 + l m b , m b π 1 = m b + m π 1

m b l m b = m π 1 l m π 1 = m π 1 ( l b π 1 − l m b ) l m π 1 = m b l b π 1 / ( m b + m π 1 ) l m b = m π 1 l b π 1 / ( m b + m π 1 )

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

The different sided triangles Δ b m π 1 of reaction B+ T→ b+ π1 + all, the base isosceles triangle Δ π 1 m π 2 . The point “ C 0 ” is the center of rotation of the oricycle. The point “ G b π 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 ( π 1 , π 2 ). A straight line drawn from point “G” through the midpoint “M” of side ( m – π 2 ) intersects the oricycle at point “F”. The ratio of the arc lengths l m π 2 to l m F of the oricycle is equal to the golden ratio 1.61803…

Connecting the points “ π 1 ”, “m”, “b” with each other by straight lines, we obtain a different sided triangle Δ b m π 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 Δ b m π 1 are related to the lengths of the arcs of the oricycle by the following relationships [6]:

l b π 1 = 2 sh ( S b π 1 / 2 ) , l m π 1 = 2 sh ( S m π 1 / 2 ) , l b m = 2 sh ( S b m / 2 ) (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 π 1 ) passes through point “ C 0 ” - the center of rotation of the oricycle (the point “ G b π 1 ” represents the center of inertia of the pair particles π 1 and b ) (Figure 3). The infinitely large radius ( m − C 0 ) is called the axis of the oricycle. Rotating the arc l m π 1 of the oricycle around the axis ( m − C 0 ), the point “ π 1 ” will move to the point “ π 2 ” on the oricycle. By connecting the points “ π 1 ” “m,” and “ π 2 ” with line segments, we obtain an isosceles triangle Δ π 1 m π 2 . Here, the line segment ( π 1 − π 2 ) of the side of triangle Δ π 1 m π 2 is perpendicular to the axis ( m − C 0 ) of the oricycle. Accordingly, the line segment ( G − m ) is the altitude of Δ π 1 m π 2 (point “G” is the midpoint of the side ( π 1 − π 2 )). Thus, the pair of particles b and π 1 in reaction (1) corresponds to a different-sided triangle Δ b m π 1 and the isosceles triangle Δ π 1 m π 2 . And vice versa: every isosceles triangle Δ π 1 m π 2 corresponds to a different-sided triangle Δ b m π 1 . Further, we will refer to the isosceles triangle Δ π 1 m π 2 as the base triangle. The base triangle Δ π 1 m π 2 , inscribed in an oricycle, has a very interesting property. A straight line drawn from point “G” through the midpoint “M” of side ( m − π 2 ) intersects the oricycle at point “F” (Figure 3). In this case, the ratio of the arc lengths l m π 2 to l m F of the oricycle is equal to the golden ratio: 1.61803… [12].

The effective mass m r e s of the pair of particles b and π 1 is calculated using their rest masses m b and m π 1 and the length S b π 1 of the side ( b – π 1 ) of the triangle Δ b m π 1 [5]:

m r e s 2 = m b 2 + m π 1 2 + 2 m b m π 1 ch   S b π 1

The point “ π 2 ” is assigned the rest mass m π 1 of the π 1 meson ( m π 1 = m π 2 ). The base effective mass m r e s π 1 , π 2 of the particles π 1 and π 2 is calculated using their rest masses and the length S π 1 π 2 of the side ( π 1 − π 2 ) of the base triangle Δ π 1 m π 2 [5]:

m b a s e π 1 , π 2 = m π 1 2 + m π 2 2 + 2 m π 1 m π 2 ch S π 1 π 2

From the correspondence between the triangles Δ b m π 1 and Δ π 1 m π 2 , it follows that the masses m r e s and m b a s e π 1 , π 2 are also in correspondence. Each value of the effective mass m r e s corresponds to a base mass m b a s e π 1 , π 2 . The reverse is also true: each value of the base mass m b a s e π 1 , π 2 corresponds to an effective mass m r e s .

To classify light unflavored, strange mesons and Δ,N,Λ,Σ,Ξ baryons of the reaction (1) used base triangles Δ π 1 m π 2 , obtained by rotating the arc l m π 1 of the oricycle around the axis ( m − C 0 ). For the base triangle Δ π 1 m π 2 , the hyperbolic sine H S T r e s (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):

H S T r e s = sh   S G m   sh   S π 1 π 2 = sh   S A π 1   sh   S m π 2

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 H S T r e s of the resonance decay triangle provides more information than its effective mass m r e s .

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

ch ( S b π 1 ) = ( m r e s 2 − m π 1 2 − m b 2 ) / 2 m π 1 m b , sh ( S b π 1 / 2 ) = ( ch ( S b π 1 / 2 ) − 1 ) / 2 (8)

l b π 1 = 2 sh ( S b π 1 / 2 ) , l m π 1 = m b l b π 1 / ( m π 1 + m b ) = sh ( S π 1 π 2 / 2 ) (9)

ch ( S π 1 π 2 / 2 ) = sh 2 ( S π 1 π 2 / 2 ) + 1 , sh ( S π 1 π 2 ) = 2 sh ( S π 1 π 2 / 2 ) ch ( S π 1 π 2 / 2 ) (10)

sh ( S m π 1 / 2 ) = l m π 1 / 2 , ch ( S m π 1 / 2 ) = sh 2 ( S m π 1 / 2 ) + 1 ,(11)

sh ( S m π 1 ) = 2 sh ( S m π 1 / 2 ) ch ( S m π 1 / 2 ) , ch ( S m π 1 ) = sh 2 ( S m π 1 ) + 1 (12)

From the right-angled triangle Δ G m π 1 we obtain:

ch ( S G m ) = ch ( S m π 1 ) / ch ( S π 1 π 2 / 2 ) , sh ( S G m ) = ch 2 ( S G m ) − 1 (13)

The integer values of the hyperbolic sine of a triangle ( H S T h e r ) correspond to base triangles Δ π 1 m π 2 that we have named Heron’s base triangles. The integer values of H S T h e r for the Heron’s base triangles Δ π 1 m π 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 π 1 and π 2 mesons with equal masses, Equations (3)-(13) take the form ( m b = m π 2 = m π 1 ) (Figure 1) [12]-[14]:

l π 1 π 2 = l m π 1 + l m π 2 , m π 1 π 2 = m π 1 + m π 2

m π 1 l m π 1 = m π 2 l m π 2 = m π 1 ( l π 1 π 2 − l m π 1 ) l m π 1 = m π 2 l π 1 π 2 / ( m π 2 + m π 1 ) = l m π 1 = m π 1 l π 1 π 2 / ( m π 1 + m π 2 )

The effective mass m h e r π 1 , π 2 is calculated using the base length S π 1 π 2 of the Heron’s triangle Δ π 1 m π 2 :

m h e r π 1 , π 2 = m π 1 2 + m π 2 2 + 2 m π 1 m π 2 ch S π 1 π 2

The hyperbolic sine H S T r e s of the base triangle Δ π 1 m π 2 (Figure 2):

H S T r e s = sh   S G m   sh   S π 1 π 2

The hyperbolic sine H S T h e r of the Heron’s base triangle Δ π 1 m π 2 rounded to the nearest integer of H S T r e s . Using Equations (3)-(7) and standard transformations of hyperbolic functions, we obtain:

Table 1. Hyperbolic Heron triangles in decay of Scalar Mezon → π 1 + π 2 .

NameScalarMezon Effective Mass Resonance m r e s ± Δ m r e s (Mev) WidthΓ(Mev) H S T r e s H S T h e r Base Mass m h e r π 1 , π 2 in Δ π 1 m π 2 (Mev) Δ m r e s h e r = m r e s − m h e r π 1 , π 2 (Mev) Δ m r e s h e r Γ (%)
1 2 3 4 5 6 7 8
Scalar Mezon → π 1 + π 2 , m π 1 = m π 2 139.57001   Mev , AP = 3.5 %
f o ( 500 ) 600.0 ± 50.0 450.0 6.89 − 1.99 + 2.41 7 602.5 2.5 0.6
ρ ( 770 ) 775.3 ± 0.2 147.0 17.39 − 0.02 + 0.02 17 770.2 5.1 3.5
ω ( 782 ) 782.7 ± 0.13 8.68 17.97 − 0.01 + 0.01 18 783.0 0.3 3.9
f o ( 980 ) 980.0 ± 20.0 55.0 39.40 − 2.54 + 2.65 39 986.9 3.1 5.6
ϕ ( 1020 ) 1019.46 ± 0.02 4.25 43.34 − 0.02 + 0.02 43 1017.0 2.5 58.0
f 2 ( 1275 ) 1275.4 ± 0.80 186.6 88.61 − 0.18 + 0.18 89 1277.2 1.8 1.0
f o ( 1370 ) 1350.0 ± 50.0 350.8 105.94 − 11.84 + 12.77 106 1350.2 0.2 0.1
ρ ( 1450 ) 1465.0 ± 25.0 400.0 137.76 − 7.14 + 7.39 138 1465.8 0.8 0.2
f o ( 1500 ) 1522.0 ± 25.0 108.0 153.99 − 7.22 + 7.98 154 1522.0 0.03 0.04
f ′ 2 ( 1525 ) 1517.0 ± 13.0 132.0 152.52 − 7.22 + 7.98 153 1517.3 1.5 2.1
f 2 ( 1565 ) 1571.0 ± 13.0 132.0 169.89 − 0.80 + 0.81 170 1571.3 0.3 0.3
ρ 3 ( 1690 ) 1686.0 ± 4.0 161.0 211.35 − 1.54 + 1.55 211 1685.1 0.9 0.6
ρ ( 1700 ) 1720.0 ± 20.0 250.0 224.77 − 7.96 + 8.15 225 1720.6 0.6 0.2
f o ( 1710 ) 1733.0 ± 8.0 150.0 230.04 − 3.25 + 3.29 230 1732.9 0.1 0.1
f o ( 1770 ) 1804.0 ± 16.0 138.0 260.29 − 7.03 + 7.16 260 1803.3 0.7 0.5
f 2 ( 1810 ) 1815.0 ± 12.0 197.0 265.20 − 5.35 + 5.42 265 1814.6 0.4 0.2
f 2 ( 1950 ) 1936.0 ± 12.0 197.0 323.27 − 6.10 + 6.18 323 1935.5 0.5 0.1
f 0 ( 2020 ) 1982.0 ± 54.0 464.0 347.37 − 28.18 + 29.77 347 1981.3 0.7 0.2
f 4 ( 2050 ) 2018.0 ± 11.0 237.0 367.04 − 6.09 + 6.15 367 2017.9 0.1 0.03
ρ 4 ( 2300 ) 2312.0 ± 15.0 300.0 555.82 − 10.91 + 11.05 556 2312.2 0.2 0.1
f 6 ( 2510 ) 2470.0 ± 50.0 260.0 679.60 − 40.96 + 42.66 680 2470.5 0.5 0.2

ch ( S π 1 π 2 ) = ( m r e s 2 − m π 1 2 − m π 2 2 ) / 2 m π 1 m π 2 , sh ( S π 1 π 2 ) = ch 2 ( S π 1 π 2 ) − 1 (17)

l π 1 π 2 = 2 sh ( S π 1 π 2 / 2 ) , l m π 1 = sh ( S G π 1 ) = m π 1 l π 1 π 2 / ( m π 1 + m π 2 ) (18)

sh ( S G π 1 ) = sh ( S π 1 π 2 / 2 ) = ( ch ( S π 1 π 2 ) − 1 ) / 2

sh ( S m π 1 / 2 ) = l m π 1 / 2 = sh ( S π 1 π 2 / 2 ) / 2 , ch ( S m π 1 / 2 ) = sh 2 ( S m π 1 / 2 ) + 1

sh ( S m π 1 ) = 2 sh ( S m π 1 / 2 ) ch ( S m π 1 / 2 ) , ch ( S m π 1 ) = sh 2 ( S m π 1 ) + 1

From the right-angled triangle Δ G m π 1 we obtain:

ch ( S G m ) = ch ( S m π 1 ) / ch ( S π 1 π 2 / 2 ) , sh ( S G m ) = ch 2 ( S G m ) − 1 (20)

In Table 1, based on data from Table A1, the values of H S T r e s for the decay Scalar Mezon → π 1 + π 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 r e s and widths Γ of scalar mesons from Table A1. Columns 4, 5 list the H S T r e s , H S T h e r values ( H S T h e r value for Heron’s triangles is rounded to the nearest integer H S T r e s value). The base mass m h e r π 1 , π 2 (column 6) is calculated using Equation (14) (the points “ π 1 ” and “ π 2 ” on the side ( π 1 − π 2 ) of the Heron’s base triangle Δ π 1 m π 2 are assigned a mass of m π 1 = m π 2 ). Column 7 lists the absolute deviation Δ m r e s h e r of the base mass m h e r π 1 , π 2 from the published values m r e s . Column 8 lists the ratio of the absolute deviation Δ m r e s h e r to the width Γ, expressed as a percentage. Table 1 shows that the values of the base masses m h e r π 1 , π 2 for Heron’s triangles agree well with the published values of m r e s (the average percentage (AP) based on Table 1 equals 3.5%). Note that, overall, the agreement between the mass m h e r π 1 , π 2 and the measured mass m r e s lies within the measurement error Δ m r e s . The main contribution to the calculation error—58%—comes from the ϕ ( 1020 ) meson, whose measured width is very small (4.25 MeV).

In the case of the decay of a Light Unflavored Mezon into daughter mesons η and π 1 with different masses ( m b = m η , m η > m π 1 ), Equation (3) takes the following form (point “ m ” with mass m η π 1 = m η + m π 1 is shifted toward point “ η ” of the particle with the greater mass) (Figure 2) [12]-[14]:

l η π 1 = l m π 1 + l m η , m η π 1 = m η + m π 1

m η l m η = m π 1 l m π 1 = m π 1 ( l η π 1 − l m η ) l m π 1 = m η l η π 1 / ( m η + m π 1 ) l m η = m π 1 l η π 1 / ( m η + m π 1 )

By connecting the points “ π 1 ”, “m” and “ η ” with straight lines, we obtain a different-sided triangle Δ η m π 1 . The lengths of the sides of the triangle Δ η m π 1 are related to the lengths of the arcs of the oricycle by the following relationships [10]:

l η π 1 = 2 sh ( S η π 1 / 2 ) , l m π 1 = 2 sh ( S m π 1 / 2 ) , l m η = 2 sh ( S m η / 2 ) (22)

By rotating the arc l m π 1 of the oricycle around the axis ( m − C 0 ), the point “ π 1 ” will move to the point “ π 2 ” on the oricycle, we obtain an isosceles base triangle Δ π 1 m π 2 , corresponding to the different-sided triangle Δ η m π 1 (Figure 2).

The effective mass m r e s of the particle pair η and π 1 is calculated using their rest masses m η and m π 1 and the length S η π 1 of the side ( η − π 1 ) of the triangle Δ η m π 1 :

m r e s 2 = m π 1 2 + m η 2 + 2 m π 1 m η ch ( S η π 1 )

The H S T r e s for the base triangle Δ π 1 m π 2 of the Scalar Mezon → η + π 1 decay (Figure 2):

H S T r e s = sh ( S G m ) sh ( S π 1 π 2 )

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

ch ( S η π 1 ) = ( m r e s 2 − m π 1 2 − m η 2 ) / 2 m π 1 m η , sh ( S η π 1 / 2 ) = ( ch ( S η π 1 / 2 ) − 1 ) / 2 (25)

l η π 1 = 2 sh ( S η π 1 / 2 ) , l m π 1 = m η l η π 1 / ( m π 1 + m η ) (26)

sh ( S m π 1 / 2 ) = l m π 1 / 2 , ch ( S m π 1 / 2 ) = sh 2 ( S m π 1 / 2 ) + 1 ,

sh ( S m π 1 ) = 2 sh ( S m π 1 / 2 ) ch ( S m π 1 / 2 ) , ch ( S m π 1 ) = sh 2 ( S m π 1 ) + 1 (27)

sh ( S π 1 π 2 ) = 2 sh ( S m π 1 ) ch ( S m π 1 ) , ch ( S π 1 π 2 ) = sh 2 ( S π 1 π 2 ) + 1 (28)

From the right-angled triangle Δ G m π 1 we obtain:

ch ( S G m ) = ch ( S m π 1 ) / ch ( S π 1 π 2 / 2 ) , sh ( S G m ) = ch 2 ( S G m ) − 1 (29)

Table 2. Hyperbolic Heron triangles in decays of Scalar Mezon → η + π 1 .

NameScalarMezon Effective Mass Resonance m r e s ± Δ m r e s (Mev) WidthΓ(Mev) H S T r e s H S T h e r H S T h e r / ch   S π 1 π 2 Effective Mass m h e r η , π 1 in Δ η m π 1 (Mev) Δ m r e s h e r = m r e s − m h e r η , π 1 (Mev) Δ m r e s h e r Γ (%) Base Mass m h e r π 1 , π 2 in Δ π 1 m π 2 (Mev)
1 2 3 4 5 6 7 8 9
Scalar Mezon → η + π 1 , m η = 547.862   Mev , m π 1 = 139.57   Mev , AP = 1.2 %
a o ( 980 ) 980.0 ± 20.0 75.0 8.16 − 0.94 + 1.01 8/9 976.8 3.2 4.3 624.2
a 2 ( 1320 ) 1317.7 ± 1.4 107.0 34.01 − 0.16 + 0.17 34 1317.6 0.1 0.1 946.4
π 1 ( 1600 ) 1354.0 ± 25.0 330.0 38.00 − 3.08 + 3.25 38 1354.0 0.1 0.0 979.1
a o ( 1450 ) 1439.0 ± 34.0 258.0 48.37 − 4.86 + 5.20 48 1436.2 2.8 1.1 1052.2
a 2 ( 1700 ) 1706.0 ± 14.0 380.0 91.13 − 3.08 + 3.16 91 1705.3 0.7 0.2 1286.2
a 0 ( 1710 ) 1713.0 ± 19.0 107.0 92.47 − 4.21 + 4.34 92 1710.6 2.5 2.3 1290.7
a 4 ( 1970 ) 1967.0 ± 16.0 324.0 149.86 − 4.88 + 4.99 150 1967.5 0.5 0.2 1509.2

Table 2 shows the results of the calculations for the Scalar Mezon → η + π 1 decay triangles. Column 4 lists the H S T r e s 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 r e s and widths Γ from Table A1. Columns 4, 5 list the H S T r e s and H S T h e r values ( H S T h e r value for Heron’s triangles is rounded to the nearest integer H S T r e s value). Column 6 lists the values of the effective masses m h e r η , π 1 in triangles Δ η m π 1 . Column 7 lists the absolute deviation Δ m r e s h e r of the effective mass m h e r η , π 1 from the published values m r e s . Column 8 lists the ratio of the absolute deviation Δ m r e s h e r to the width Γ, expressed as a percentage. Column 9 lists the values of the effective masses m h e r π 1 , π 2 in Heron’s triangles Δ π 1 m π 2 π 2 . Table 2 shows that the values of the effective masses m h e r η , π 1 agree well with the published values of m r e s (the average percentage (AP) based on Table 2 equals 1.2%). From Equation (28), we write the following expression:

ch ( S π 1 π 2 ) = sh 2 ( S π 1 π 2 ) + 1

Column 5 lists, separated by a slash, the value of the hyperbolic cosine ch ( S π 1 π 2 ) of the side ( π 1 − π 2 ) of Heron’s base triangle Δ π 1 m π 2 .For certain values of H S T h e r , the hyperbolic cosines ch ( S π 1 π 2 ) of the side ( π 1 − π 2 ) of the triangles Δ π 1 m π 2 are also equal to integer values. The integer value of ch ( S π 1 π 2 ) 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 Scalar Mezon → ρ ( 770 ) + π 1 .

NameScalarMezon Effective Mass Resonance m r e s ± Δ m r e s (Mev) WidthΓ(Mev) H S T r e s i H S T r e s H S T h e r i H S T h e r Effective Mass m h e r ρ ( 770 ) , π 1 in Δ ρ ( 770 ) m π 1 (Mev) Δ m r e s h e r = m r e s − m h e r ρ ( 770 ) , π 1 (Mev) Δ m r e s h e r Γ (%) Base Mass m h e r π 1 , π 2 in Δ π 1 m π 2 (Mev)
1 2 3 4 5 6 7 8 9
Scalar Mezon → ρ ( 770 ) + π 1 , m ρ = 775.29999   Mev , m π 1 = 139.57   Mev , AP = 1.2 %
η ′ ( 958 ) 957.8 ± 0.06 0.23 − 19.98 − 0.02 + 0.02 -20 957.8 0.0 0.03 345.6
h 1 ( 1170 ) 1166.2 ± 19.0 107.0 6.47 − 0.47 + 0.47 6 1152.2 11.0 2.9 579.0
a 1 ( 1260 ) 1240.0 ± 40.0 422.0 10.03 − 0.03 + 0.03 10 1239.5 0.5 0.1 663.0
π ( 1300 ) 1300.0 ± 100.0 400.0 13.47 − 5.46 + 6.88 13 1292.3 7.7 1.9 713.2
a 2 ( 1320 ) 1317.7 ± 1.4 107.0 14.58 − 0.10 + 0.10 15 1324.2 6.5 6.0 742.9
ω ( 1420 ) 1410.0 ± 60.0 290.0 21.12 − 4.84 + 5.66 21 1408.5 1.5 0.5 819.2
π 1 ( 1600 ) 1645.0 ± 16.0 370.0 43.69 − 2.18 + 2.25 44 1647.7 2.7 0.7 1024.2
ω ( 1650 ) 1670.0 ± 30.0 315.0 46.63 − 4.20 + 4.47 47 1673.1 3.1 1.0 1045.3
ω 3 ( 1670 ) 1667.0 ± 4.0 168.0 46.27 − 0.57 + 0.58 46 1664.8 2.2 1.3 1038.4
π 2 ( 1670 ) 1670.6 ± 2.9 288.0 46.70 − 0.42 + 0.42 47 1673.1 2.5 0.9 1045.3
a 2 ( 1700 ) 1706.0 ± 14.0 380.0 51.05 − 1.75 + 1.78 51 1705.6 0.4 0.1 1072.1
a 0 ( 1710 ) 1713.0 ± 19.0 107.0 52.05 − 0.07 + 0.07 52 1713.5 0.5 0.5 1078.6
a 4 ( 1970 ) 1967.0 ± 16.0 324.0 90.27 − 3.50 + 3.59 90 1965.5 1.6 0.5 1281.7
π 2 ( 2100 ) 2090.0 ± 29.0 620.0 113.45 − 7.30 + 7.62 113 2087.8 2.2 0.4 1378.1

Table 3 shows the results of the calculations for the Scalar Mezon → ρ ( 770 ) + π 1 decay triangles. Column 4 lists the H S T r e s values calculated using Equations (3)-(13) based on data from Table A1 ( m b = m ρ ( 770 ) ). Columns 2 and 3 of Table 2 list the values of the effective masses m r e s and widths Γ from Table A1. Column 5 lists the value of H S T h e r for Heron’s triangles Δ π 1 m π 2 , rounded to the nearest integer of H S T r e s . The nearest integer for values of H S T r e s < 1.1 was determined using inversion. Namely, an inverse value of H S T r e s i was specified with the coefficient:

H S T r e s i = PI / ( sh   S G m   sh   S G π 1 )

Value PI = 3.1415… Column 4 list the H S T r e s and H S T r e s i values calculated using Equations (3)-(13) ( m b = m ρ ( 770 ) ) based on data from Table A1. Values of H S T r e s i are shown with a minus sign. Columns 2 and 3 of Table 3 list the values of the effective masses m r e s and widths Γ from Table A1. Column 5 list the H S T h e r and H S T h e r i values for Heron’s triangles Δ π 1 m π 2 , rounded to the nearest integer. Column 6 lists the values of the effective masses m h e r ρ ( 770 ) , π 1 in triangles Δ ρ ( 770 ) m π 1 . Column 7 lists the absolute deviation Δ m r e s h e r of the effective mass m h e r ρ ( 770 ) , π 1 from the published values m r e s . Column 8 lists the ratio of the absolute deviation Δ m r e s h e r to the width Γ, expressed as a percentage. Column 9 lists the values of the effective masses m h e r π 1 , π 2 in Heron’s triangles Δ π 1 m π 2 . Table 3 shows that the values of the effective masses m h e r ρ ( 770 ) , π 1 agree well with the published values of m r e s (the average percentage (AP) based on Table 3 equals 1.2%).

Table 4 shows the results of the calculations for the Scalar Mezon → ω ( 782 ) + π 1 decay triangles. Column 4 lists the H S T r e s values calculated using Equations (3)-(13) ( m b = m ω ( 782 ) ) based on data from Table A1. The calculations in the remaining columns of Table 4 are similar to those in the columns of Table 2. Table 4 shows that the values of the effective masses m h e r ω ( 782 ) , π 1 agree well with the published values of m r e s (average percentage based on Table 4 equals 0.5%).

Table 4. Hyperbolic Heron triangles in decays of Scalar Mezon → ω ( 782 ) + π 1 .

NameScalarMezon Effective Mass Resonance m r e s ± Δ m r e s (Mev) WidthΓ(Mev) H S T r e s H S T h e r Effective Mass m h e r ω ( 782 ) , π 1 in Δ ω ( 782 ) m π 1 (Mev) Δ m r e s h e r = m r e s − m h e r ω ( 782 ) , π 1 (Mev) Δ m r e s h e r Γ (%) Base Mass m h e r π 1 , π 2 in Δ π 1 m π 2 (Mev)
1 2 3 4 5 6 7 8 9
Scalar Mezon → ω ( 782 ) + π 1 , m ω = 782.65002   Mev , m π 1 = 139.57   Mev , AP = 0.5 %
b 1 ( 1235 ) 1229.5 ± 3.30 142.9 9.10 − 0.16 + 0.17 9 1227.5 2.0 1.4 644.2
ρ ( 1450 ) 1465.8 ± 25.0 400.0 25.04 − 2.35 + 2.50 25 1465.3 0.5 0.1 862.6
ρ 3 ( 1690 ) 1688.8 ± 2.1 161.0 47.94 − 0.31 + 0.31 48 1689.3 0.5 0.3 1052.2
ρ 3 ( 2150 ) 2150.8± 40.0 300.0 124.03 − 0.04 + 0.04 124 2149.9 0.1 0.04 1419.7

Table 5 shows the results of the calculations for the StrangeMezon → K ( 493 ) + π 1 decay triangles. Column 4 lists the H S T r e s values calculated using Equations (3)-(13) ( m b = m k ( 493 ) ) based on data from Table A2. The calculations in the remaining columns of Table 5 are similar to those in the columns of Table 2. Table 5 shows that the values of the effective masses m h e r k ( 493 ) , π 1 agree well with the published values of m r e s (the average percentage (AP) based on Table 5 equals 3.0%).

Table 5. Hyperbolic Heron triangles in decays of Strange Mezon → K ( 493 ) + π 1 .

NameStrangeMezon Effective Mass Resonance m r e s ± Δ m r e s (Mev) WidthΓ(Mev) H S T r e s H S T h e r Effective Mass m h e r k 493 , π 1 in Δ k 493 m π 1 (Mev) Δ m r e s h e r = m r e s − m h e r k 493 , π 1 (Mev) Δ m r e s h e r Γ (%) Base Mass m h e r π 1 , π 2 in Δ π 1 m π 2 (Mev)
1 2 3 4 5 6 7 8 9
Strange Mezon → K ( 493 ) + π 1 , m k 493 = 493.677   Mev , m π 1 = 139.57   Mev , AP = 3.0 %
K * ( 700 ) 838.0 ± 11.0 463.0 4.33 − 0.36 + 0.38 4 828.7 9.3 2.0 523.7
K * ( 892 ) 891.8 ± 0.25 50.3 6.48 − 0.01 + 0.01 6 880.5 11.2 22.3 579.0
K * ( 1410 ) 1421.8 ± 9.0 236.0 53.93 − 1.42 + 1.44 54 1421.5 0.5 0.2 1091.4
K 0 * ( 1430 ) 1425.6 ± 18.0 270.0 54.58 − 2.83 + 2.93 55 1428.5 2.9 1.1 1097.6
K 2 * ( 1430 ) 1427.3 ± 1.5 100.0 54.83 − 0.24 + 0.24 55 1428.5 1.2 1.2 1097.6
K 3 * ( 1780 ) 1779.0 ± 8.0 161.0 120.37 − 2.13 + 2.15 120 1777.4 1.6 1.0 1404.9
K 0 * ( 1950 ) 1957.0 ± 14.0 170.5 166.33 − 4.58 + 4.67 166 1955.8 1.2 0.7 1559.3
K 4 * ( 2045 ) 2045.0 ± 9.0 199.0 192.61 − 3.26 + 3.29 193 2046.3 1.3 0.6 1637.1
K 5 * ( 2380 ) 2382.8 ± 14.0 178.0 317.54 − 7.02 + 7.13 318 2383.9 1.1 0.6 1925.6

Table 6. Hyperbolic Heron triangles in decays of Strange Mezon → K * ( 892 ) + π 1 .

NameStrangeMezon Effective Mass Resonance m r e s ± Δ m r e s (Mev) WidthΓ(Mev) H S T r e s H S T h e r Effective Mass m h e r k 892 , π 1 in Δ k 892 m π 1 (Mev) Δ m r e s h e r = m r e s − m h e r k 892 , π 1 (Mev) Δ m r e s h e r Γ (%) Base Mass m h e r π 1 , π 2 in Δ π 1 m π 2 (Mev)
1 2 3 4 5 6 7 8 9
Strange Mezon → K * ( 892 ) + π 1 , m k 892 = 891.76   Mev , m π 1 = 139.57   Mev , AP = 2.7 %
K 1 ( 1270 ) 1253.0 ± 7.0 10.0 5.42 − 0.23 + 0.23 5 1245.2 10.9 12.1 552.9
K 1 ( 1400 ) 1403.0 ± 7.0 174.0 12.67 − 0.44 + 0.45 13 1408.6 5.6 3.2 713.2
K 1 * ( 1410 ) 1414.0 ± 15.0 232.0 13.33 − 0.96 + 1.00 13 1408.6 5.4 2.4 713.2
K 2 * ( 1430 ) 1427.3 ± 1.5 100.0 14.14 − 0.10 + 0.10 14 1425.0 2.3 2.3 728.4
K ( 1460 ) 1482.4 ± 15.0 335.6 17.78 − 1.16 + 1.20 18 1485.6 3.2 0.9 783.0
K 2 ( 1580 ) 1580.8 ± 5.0 110.0 25.32 − 0.49 + 0.49 25 1577.0 3.8 3.5 862.6
K * ( 1680 ) 1718.8 ± 18.0 320.0 38.19 − 2.25 + 2.34 38 1716.2 1.8 0.6 979.1
K 3 * ( 1780 ) 1779.0 ± 8.0 161.0 44.84 − 1.12 + 1.13 45 1780.4 1.4 0.9 1031.4
K 2 ( 1770 ) 1773.8 ± 8.0 186.0 44.25 − 1.10 + 1.13 44 1771.6 2.2 1.2 1024.2
K 2 ( 1820 ) 1819.0 ± 12.0 264.0 49.53 − 1.78 + 1.82 50 1822.9 3.9 1.5 1065.6
K 2 * ( 1980 ) 1990.0 ± 50.0 348.0 72.58 − 9.21 + 10.03 73 1992.8 2.8 0.8 1199.5

Table 6 shows the results of the calculations for the Strange Mezon → K * ( 892 ) + π 1 decay triangles. Column 4 lists the H S T r e s values calculated using Equations (3)-(13) ( m b = m k ( 892 ) ) based on data from Table A2. The calculations in the remaining columns of Table 6 are similar to those in the columns of Table 2. Table 6 shows that the values of the effective masses m h e r k ( 892 ) , π 1 agree well with the published values of m r e s (the average percentage (AP) based on Table 6 equals 2.7%).

Table 7. Hyperbolic Heron triangles in decays of Δ Barion → P + π 1 .

Name Δ Barion Effective Mass Resonance m r e s ± Δ m r e s (Mev) WidthΓ(Mev) H S T r e s H S T h e r Effective Mass m h e r P , π 1 in Δ P m π 1 (Mev) Δ m r e s h e r = m r e s − m h e r P , π 1 (Mev) Δ m r e s h e r Γ (%) Base Mass m h e r π 1 , π 2 in Δ π 1 m π 2 (Mev)
1 2 3 4 5 6 7 8 9
Δ   Barion → P + π 1 , m P = 938.27209   Mev , m π 1 = 139.57   Mev , AP = 0.5 %
Δ ( 1232 ) 1232.0 ± 0.40 117.0 2.96 − 0.01 + 0.01 3 1233.4 1.4 1.2 489.9
Δ ( 1600 ) 1570.0 ± 0.25 250.0 20.71 − 0.02 + 0.02 21 1573.9 3.9 1.6 819.2
Δ ( 1620 ) 1610.0 ± 20.0 130.0 23.81 − 1.58 + 1.64 24 1612.3 2.3 1.8 852.2
Δ ( 1750 ) 1715.0 ± 10.0 350.0 32.57 − 0.94 + 0.96 33 1714.6 0.5 0.1 937.9
Δ ( 1900 ) 1865.0 ± 30.0 300.0 49.08 − 3.46 + 3.60 49 1864.3 0.6 0.3 1058.9
Δ ( 1905 ) 1880.0 ± 30.0 300.0 50.87 − 3.46 + 3.60 51 1881.3 1.0 0.3 1072.1
Δ ( 1910 ) 1900.0 ± 30.0 300.0 53.32 − 4.59 + 4.86 53 1897.4 2.6 0.9 1085.0
Δ ( 1920 ) 1920.0 ± 30.0 300.0 55.84 − 4.84 + 5.00 56 1921.3 1.3 0.4 1103.8
Δ ( 1930 ) 1950.0 ± 30.0 400.0 59.74 − 4.95 + 5.22 60 1952.0 2.0 0.7 1127.9
Δ ( 1940 ) 2000.0 ± 24.0 500.0 66.56 − 4.27 + 4.46 67 2003.1 3.1 0.8 1167.6
Δ ( 1950 ) 1930.0 ± 10.0 285.0 57.12 − 1.28 + 1.30 57 1929.1 0.9 0.3 1109.9
Δ ( 2000 ) 2015.0 ± 24.0 500.0 68.69 − 5.42 + 5.71 69 2017.2 2.2 0.4 1178.4
Δ ( 2150 ) 2150.0 ± 100.0 200.0 89.62 − 20.33 + 23.87 90 2152.3 2.3 1.1 1281.7
Δ ( 2200 ) 2200.0 ± 30.0 300.0 98.20 − 21.63 + 25.27 98 2198.9 1.1 0.4 1316.9
Δ ( 2300 ) 2300.0 ± 100.0 350.0 116.76 − 24.33 + 28.17 117 2301.2 1.2 0.3 1393.5
Δ ( 2350 ) 2350.0 ± 100.0 400.0 126.76 − 27.15 + 31.20 127 2351.2 1.2 0.3 1430.6
Δ ( 2390 ) 2375.0 ± 100.0 300.0 131.95 − 27.15 + 31.20 132 2375.3 0.3 0.1 1448.4
Δ ( 2400 ) 2450.0 ± 100.0 500.0 148.26 − 28.61 + 32.76 148 2470.9 0.9 0.2 1518.5
Δ ( 2420 ) 2483.0 ± 100.0 500.0 155.80 − 28.61 + 32.76 156 2483.9 0.9 0.2 1528.4
Δ ( 2750 ) 2794.0 ± 80.0 350.0 238.43 − 31.97 + 35.01 238 2792.6 1.4 0.4 1752.3
Δ ( 2950 ) 2990.0 ± 100.0 330.0 301.98 − 46.35 + 51.57 302 2990.1 0.1 0.02 1893.5

Table 7 shows the results of the calculations for the Δ   Barion → P + π 1 decay triangles. Column 4 lists the H S T r e s values calculated using Equations (3)-(13) ( m b = m P ) based on data from Table A3. The calculations in the remaining columns of Table 7 are similar to those in the columns of Table 2. Table 6 shows that the values of the effective masses m h e r P , π 1 agree well with the published values of m r e s (the average percentage (AP) based on Table 7 equals 0.5%).

Table 8 shows the results of the calculations for the Δ   Barion → Δ ( 1232 ) + π 1 decay triangles. Column 4 lists the H S T r e s values calculated using Equations (3)-(13) ( m b = m Δ ( 1232 ) ) based on data from Table A3. The calculations in the remaining columns of Table 8 are similar to those in the columns of Table 2. Table 8 shows that the values of the effective masses m Δ , h e r Δ ( 1232 ) , π 1 agree well with the published values of m r e s (the average percentage based on Table 8 equals 1.4%).

Table 8. Hyperbolic Heron triangles in decays of Δ   Barion → Δ ( 1232 ) + π 1 .

Name Δ Barion Effective Mass Resonance m r e s ± Δ m r e s (Mev) WidthΓ(Mev) H S T r e s H S T h e r H S T h e r / ch   S π 1 π 2 Effective Mass m Δ , h e r Δ ( 1232 ) , π 1 in Δ π 1 m Δ ( 1232 ) (Mev) Δ m r e s h e r = m r e s − m Δ , h e r Δ ( 1232 ) , π 1 (Mev) Δ m r e s h e r Γ (%) Base Mass m h e r π 1 , π 2 in Δ π 1 m π 2 (Mev)
1 2 3 4 5 6 7 8 9
Δ   Barion → Δ ( 1232 ) + π 1 , m Δ ( 1232 ) = 1232.0   Mev , m π 1 = 139.57   Mev , AP = 1.4 %
Δ ( 1600 ) 1570.0 ± 20.0 250.0 4.53 − 0.70 + 0.76 5 1582.5 12.5 5.0 553.0
Δ ( 1620 ) 1610.0 ± 10.0 130.0 6.09 − 0.41 + 0.42 6 1607.8 2.8 1.8 579.0
Δ ( 1700 ) 1710.0 ± 2.0 300.0 10.83 − 0.11 + 0.11 11 1713.3 3.3 1.1 680.7
Δ ( 1900 ) 1860.0 ± 20.0 250.0 20.16 − 0.40 + 0.45 20 1857.8 2.2 0.9 807.5
Δ ( 1905 ) 1880.0 ± 20.0 330.0 21.61 − 1.45 + 1.50 22 1885.3 5.3 1.6 830.5
Δ ( 1910 ) 1900.0 ± 20.0 300.0 23.10 − 1.50 + 1.55 23 1898.5 1.4 0.5 841.5
Δ ( 1920 ) 1920.0 ± 30.0 300.0 24.65 − 2.78 + 3.00 25 1924.4 4.4 1.5 862.6
Δ ( 1930 ) 1950.0 ± 30.0 300.0 27.07 − 2.95 + 3.17 27/19 1949.2 0.8 0.3 882.7
Δ ( 1950 ) 1930.0 ± 10.0 300.0 25.45 − 0.79 + 0.80 25 1924.4 5.6 1.9 862.6
Δ ( 1940 ) 2000.0 ± 40.0 400.0 31.35 − 3.45 + 3.65 31 1996.1 3.9 1.0 920.2
Δ ( 2000 ) 2150.0 ± 24.0 500.0 46.13 − 3.36 + 3.53 46 2148.8 1.2 0.2 1038.4
Δ ( 2200 ) 2200.0 ± 30.0 300.0 51.73 − 3.40 + 3.52 52 2202.3 2.3 0.8 1078.6

Table 9 shows the results of the calculations for the N   Barion → n + π 1 decay triangles. Column 4 lists the H S T r e s values calculated using Equations (3)-(13) ( m b = m n ) based on data from Table A4. The calculations in the remaining columns of Table 9 are similar to those in the columns of Table 2. Table 9 shows that the values of the effective masses m N , h e r n , π 1 agree well with the published values of m r e s (the average percentage (AP) based on Table 9 equals 1.0%).

Table 9. Hyperbolic Heron triangles in decays of N   Barion → n + π 1 .

Name N Barion Effective Mass Resonance m r e s ± Δ m r e s (Mev) WidthΓ(Mev) H S T r e s H S T h e r H S T h e r / ch   S π 1 π 2 Effective Mass m h e r n , π 1 in Δ n m π 1 (Mev) Δ m r e s h e r = m r e s − m h e r n , π 1 (Mev) Δ m r e s h e r Γ (%) Base Mass m h e r π 1 , π 2 in Δ π 1 m π 2 (Mev)
1 2 3 4 5 6 7 8 9
N   Barion → n + π 1 , m n = 939.56542   Mev , m π 1 = 139.57   Mev , AP = 1.0 %
N ( 1440 ) 1440.0 ± 30.0 350.0 12.05 − 1.66 + 1.78 12 1439.2 0.8 0.2 697.4
N ( 1520 ) 1515.0 ± 10.0 110.0 16.71 − 0.66 + 0.68 17 1519.3 4.3 3.9 770.1
N ( 1535 ) 1530.0 ± 15.0 150.0 17.73 − 0.34 + 0.35 18 1533.8 3.8 2.6 783.0
N ( 1650 ) 1650.0 ± 6.0 125.0 27.03 − 0.51 + 0.52 27/19 1649.7 0.3 0.3 882.7
N ( 1675 ) 1675.0 ± 5.0 145.0 29.22 − 0.45 + 0.45 29 1672.5 2.5 1.7 901.9
N ( 1680 ) 1685.0 ± 5.0 120.0 30.12 − 0.45 + 0.46 30 1683.7 1.3 1.1 911.2
N ( 1700 ) 1720.0 ± 20.0 200.0 33.39 − 1.89 + 1.95 33 1715.9 4.1 2.0 937.9
N ( 1875 ) 1875.0 ± 20.0 200.0 50.10 − 2.37 + 2.43 50 1874.1 0.9 0.4 1065.6
N ( 1880 ) 1880.0 ± 20.0 300.0 50.71 − 2.38 + 2.45 51 1822.4 2.4 0.8 1072.1
N ( 1900 ) 1920.0 ± 10.0 200.0 55.66 − 2.43 + 2.49 56 1922.7 2.6 1.3 1103.8
N ( 1895 ) 1985.0 ± 20.0 120.0 52.53 − 2.43 + 2.49 53 1898.8 3.8 3.1 1085.1
N ( 2000 ) 2050.0 ± 20.0 400.0 73.60 − 3.81 + 3.95 74 2052.7 2.7 0.7 1204.7
N ( 1990 ) 2020.0 ± 40.0 300.0 69.20 − 5.62 + 5.89 69 2018.6 1.4 0.5 1178.5
N ( 2060 ) 2100.0 ± 15.0 400.0 81.26 − 2.35 + 2.39 81 2098.3 1.7 0.4 1239.6
N ( 2100 ) 2100.0 ± 20.0 260.0 81.26 − 2.35 + 2.39 81 2098.3 1.7 0.6 1239.6
N ( 2120 ) 2120.0 ± 20.0 400.0 84.45 − 4.17 + 4.31 84 2117.2 2.8 0.9 1254.0
N ( 2190 ) 2180.0 ± 20.0 400.0 94.46 − 4.49 + 4.64 94 2177.3 2.6 0.7 1299.5
N ( 2220 ) 2250.0 ± 15.0 400.0 106.96 − 0.39 + 0.39 107 2250.2 0.2 0.1 1354.3
N ( 2250 ) 2280.0 ± 15.0 500.0 112.60 − 7.49 + 7.79 113 2282.1 2.1 0.4 1378.1
N ( 2600 ) 2600.0 ± 100.0 650.0 183.96 − 36.06 + 40.96 184 2611.8 0.2 0.02 1612.0
N ( 2700 ) 2612.0 ± 45.0 650.0 187.06 − 11.44 + 11.88 187 2611.8 0.2 0.1 1620.5
N ( 3000 ) 3000.0 ± 200.0 1650.0 304.84 − 65.12 + 75.15 305 3000.5 0.5 0.03 1899.6

Table 10 shows the results of the calculations for the N   Barion → Δ ( 1232 ) + π 1 decay triangles. Column 4 lists the H S T r e s values calculated using Equations (3)-(13) ( m b = m Δ ( 1232 ) ) based on data from Table A4. The calculations in the remaining columns of Table 10 are similar to those in the columns of Table 2. Table 10 shows that the values of the effective masses m N , h e r Δ ( 1232 ) , π 1 agree well with the published values of m r e s (the average percentage (AP) based on Table 10 equals 1.7%). From Equation (28), we write the following expression:

ch ( S π 1 π 2 ) = sh 2 ( S π 1 π 2 ) + 1

Column 5 lists, separated by a slash, the value of the hyperbolic cosine ch ( S π 1 π 2 ) of the base ( π 1 − π 2 ) of Heron’s basic triangle Δ π 1 m π 2 .

Table 10. Hyperbolic Heron triangles in decays of N Barion → Δ ( 1232 ) + π 1 .

NameN Barion Effective Mass Resonance m r e s ± Δ m r e s (Mev) WidthΓ(Mev) H S T r e s H S T h e r H S T h e r / ch   S π 1 π 2 Effective Mass m N , h e r Δ ( 1232 ) , π 1 in Δ π 1 m Δ ( 1232 ) (Mev) Δ m r e s h e r = m r e s − m N , h e r Δ ( 1232 ) , π 1 (Mev) Δ m r e s h e r Γ (%) Base Mass m h e r π 1 , π 2 in Δ π 1 m π 2 (Mev)
1 2 3 4 5 6 7 8 9
N   Barion → Δ ( 1232 ) + π 1 , m Δ ( 1232 ) = 1232.0   Mev , m π 1 = 139.57   Mev , AP = 1.7 %
N ( 1440 ) 1440.0 ± 30.0 350.0 − 10.01 − 0.19 + 0.20 −10 1440.5 0.5 0.2 394.8
N ( 1520 ) 1515.0 ± 10.0 110.0 2.71 − 0.29 + 0.30 3 1524.3 9.6 8.8 489.9
N ( 1535 ) 1530.0 ± 5.0 150.0 3.17 − 0.16 + 0.16 3 1524.6 5.4 3.6 489.9
N ( 1650 ) 1650.0 ± 5.0 125.0 7.84 − 0.23 + 0.23 8/9 1653.4 3.4 2.7 624.2
N ( 1675 ) 1675.0 ± 5.0 150.0 9.03 − 0.24 + 0.25 9 1674.3 0.7 0.5 644.3
N ( 1680 ) 1685.0 ± 5.0 120.0 9.53 − 0.25 + 0.25 10 1694.2 9.2 7.7 663.0
N ( 1700 ) 1720.0 ± 10.0 200.0 11.36 − 0.54 + 0.55 11 1713.3 6.7 3.4 680.7
N ( 1710 ) 1710.0 ± 20.0 200.0 10.83 − 1.04 + 1.09 11 1713.3 3.3 1.6 680.7
N ( 1720 ) 1720.0 ± 10.0 250.0 11.36 − 0.54 + 0.55 11 1713.3 6.7 2.7 680.7
N ( 1860 ) 1860.0 ± 10.0 250.0 20.16 − 0.71 + 0.72 20 1857.8 2.2 0.9 807.5
N ( 1875 ) 1875.0 ± 10.0 200.0 21.24 − 0.73 + 0.74 21 1871.7 3.3 1.6 819.1
N ( 1880 ) 1880.0 ± 20.0 300.0 21.61 − 1.45 + 1.50 22 1885.3 5.3 1.8 830.5
N ( 1895 ) 1985.0 ± 10.0 200.0 22.72 − 0.75 + 0.76 23 1898.6 3.6 1.8 841.5
N ( 1900 ) 1920.0 ± 10.0 200.0 24.65 − 0.78 + 0.79 25 1924.4 4.4 2.2 862.6
N ( 2000 ) 2050.0 ± 20.0 400.0 35.95 − 2.40 + 2.50 36 2050.6 0.5 0.1 963.0
N ( 2060 ) 2100.0 ± 15.0 400.0 40.87 − 2.60 + 2.71 41 2101.2 1.2 0.3 1002.2
N ( 2100 ) 2100.0 ± 20.0 260.0 40.87 − 1.51 + 1.54 41 2101.2 1.2 0.5 1002.2
N ( 2120 ) 2120.0 ± 30.0 300.0 42.94 − 3.97 + 4.24 43 2120.6 0.6 0.2 1017.0
N ( 2190 ) 2180.0 ± 20.0 400.0 49.45 − 2.94 + 3.06 49 2176.0 4.0 1.0 1058.9

Table 11. Hyperbolic Heron triangles in decays of Σ   Barion → Λ + π 1 .

Name Σ Barion Effective Mass Resonance m r e s ± Δ m r e s (Mev) WidthΓ(Mev) H S T r e s H S T h e r Effective Mass m h e r Λ , π 1 in Δ Λ m π 1 (Mev) Δ m r e s h e r = m r e s − m h e r Λ , π 1 (Mev) Δ m r e s h e r Γ (%) Base Mass m h e r π 1 , π 2 in Δ π 1 m π 2 (Mev)
1 2 3 4 5 6 7 8 9
Σ   Barion → Λ + π 1 , m Λ = 1115.68298   Mev , m π 1 = 139.57   Mev , AP = 3.9%
Σ ( 1385 ) 1383.7 ± 2.0 39.4 2.26 − 0.06 + 0.06 2 1374.3 9.5 24.0 449.0
Σ ( 1620 ) 1620.0 ± 2.0 70.0 12.27 − 0.11 + 0.11 12 1615.2 4.8 6.9 697.4
Σ ( 1660 ) 1660.0 ± 20.0 200.4 14.65 − 1.21 + 1.26 15 1665.7 5.7 2.8 742.9
Σ ( 1670 ) 1675.0 ± 2.0 70.0 15.59 − 0.13 + 0.13 16 1681.4 6.4 9.2 756.8
Σ ( 1750 ) 1750.0 ± 2.0 150.0 20.72 − 0.15 + 0.15 21 1753.8 3.8 2.6 819.1
Σ ( 1775 ) 1775.0 ± 10.0 120.0 22.59 − 0.76 + 0.77 23 1780.4 5.4 4.5 841.5
Σ ( 1780 ) 1780.0 ± 30.0 200.0 22.97 − 2.25 + 2.37 23 1780.4 0.4 0.2 841.5
Σ ( 1900 ) 1925.0 ± 20.0 165.0 35.52 − 1.90 + 1.95 36 1930.0 4.9 3.0 963.0
Σ ( 1910 ) 1910.0 ± 50.0 220.0 34.09 − 4.54 + 4.88 34 1909.0 1.0 0.4 946.4
Σ ( 1915 ) 1915.0 ± 22.0 120.0 34.56 − 2.06 + 2.12 35 1919.6 4.6 3.8 954.8
Σ ( 1940 ) 1940.0 ± 100.0 250.0 36.98 − 9.16 + 10.54 37 1940.2 0.2 0.1 971.1
Σ ( 2010 ) 2005.0 ± 14.0 178.0 43.67 − 1.49 + 1.52 44 2008.1 3.1 1.7 1024.2
Σ ( 2030 ) 2030.0 ± 10.0 180.0 46.40 − 1.10 + 1.12 46 2026.4 3.6 2.0 1038.4
Σ ( 2100 ) 2100.0 ± 50.0 310.0 54.53 − 2.39 + 2.45 55 2103.9 3.9 1.9 1097.6
Σ ( 2110 ) 2100.0 ± 50.0 310.0 55.75 − 3.19 + 3.30 56 2112.1 2.1 2.1 1103.8
Σ ( 2230 ) 2240.0 ± 27.0 347.0 72.97 − 3.79 + 3.90 73 2240.2 0.2 0.1 1199.6
Σ ( 2250 ) 2250.0 ± 20.0 140.0 74.40 − 2.85 + 2.91 74 2247.2 2.8 0.8 1204.7

Table 12. Hyperbolic Heron triangles in decays of Σ   Barion → Σ + π 1 .

Name Σ Barion Effective Mass Resonance m r e s ± Δ m r e s (Mev) WidthΓ(Mev) H S T r e s H S T h e r i H S T h e r H S T h e r / ch   S π 1 π 2 Effective Mass m h e r Λ , π 1 in Δ Λ m π 1 (Mev) Δ m r e s h e r = m r e s − m h e r Λ , π 1 (Mev) Δ m r e s h e r Γ (%) Base Mass m h e r π 1 , π 2 in Δ π 1 m π 2 (Mev)
1 2 3 4 5 6 7 8 9
Σ   Barion → Σ + π 1 , m Σ = 1189.37   Mev , m π 1 = 139.57   Mev , AP = 3.1 %
Σ ( 1385 ) 1383.7 ± 2.0 39.4 − 13.57 − 0.5 + 0.5 −14 1382.4 1.3 3.3 363.9
Σ ( 1620 ) 1620.0 ± 2.0 70.0 8.43 − 0.10 + 0.10 8/9 1610.9 9.9 13.0 624.2
Σ ( 1660 ) 1660.0 ± 20.0 200.4 10.43 − 1.09 + 1.17 10 1651.7 8.3 4.1 663.0
Σ ( 1670 ) 1675.0 ± 2.0 70.0 11.23 − 0.12 + 0.12 11 1670.7 4.3 6.2 680.7
Σ ( 1750 ) 1750.0 ± 2.0 150.0 15.65 − 0.14 + 0.14 16 1755.6 5.6 3.7 756.7
Σ ( 1775 ) 1775.0 ± 10.0 120.0 17.27 − 0.76 + 0.78 17 1771.0 4.0 3.4 770.1
Σ ( 1780 ) 1780.0 ± 30.0 200.0 17.60 − 2.24 + 2.34 18 1785.9 5.9 3.0 783.0
Σ ( 1900 ) 1925.0 ± 20.0 165.0 28.63 − 2.07 + 2.17 29 1929.3 4.3 2.6 901.9
Σ ( 1910 ) 1910.0 ± 50.0 220.0 27.37 − 4.84 + 5.45 27/19 1905.6 4.4 2.0 882.7
Σ ( 1915 ) 1915.0 ± 22.0 120.0 27.78 − 2.25 + 2.34 28 1917.6 2.6 2.1 892.4
Σ ( 1940 ) 1940.0 ± 100.0 250.0 29.92 − 9.67 + 12.19 30 1940.9 0.9 0.3 911.2
Σ ( 2010 ) 2005.0 ± 14.0 178.0 35.87 − 1.68 + 1.74 36 2006.4 1.4 0.8 963.0
Σ ( 2030 ) 2030.0 ± 10.0 180.0 38.30 − 1.26 + 1.29 38 2027.0 3.0 1.7 979.1
Σ ( 2070 ) 2060.0 ± 10.0 200.0 41.33 − 1.26 + 1.29 41 2056.8 3.2 1.6 1002.2
Σ ( 2100 ) 2100.0 ± 50.0 310.0 45.57 − 1.41 + 1.44 46 2104.0 4.0 1.5 1038.4
Σ ( 2230 ) 2240.0 ± 27.0 347.0 62.14 − 3.42 + 3.54 62 2238.9 1.1 0.3 1139.5
Σ ( 2250 ) 2250.0 ± 20.0 140.0 63.44 − 4.64 + 4.87 63 2246.6 3.4 1.0 1145.2
Σ ( 2110 ) 2110.0 ± 20.0 100.0 46.66 − 2.83 + 2.95 47 2113.1 3.4 3.4 1045.3

Table 11 shows the results of the calculations for the Σ   Barion → Λ + π 1 decay triangles. Column 4 lists the H S T r e s values calculated using Equations (3)-(13) ( m b = m Λ ) based on data from Table A5. The calculations in the remaining columns of Table 11 are similar to those in the columns of Table 2. Table 11 shows that the values of the effective masses m h e r Λ , π 1 agree well with the published values of m r e s (the average percentage (AP) based on Table 11 equals 3.9%).

Table 12 shows the results of the calculations for the Σ   Barion → Σ ( 1189 ) + π 1 decay triangles. Column 4 lists the H S T r e s values calculated using Equations (3)-(13) ( m b = m Σ ) based on data from Table A5. The calculations in the remaining columns of Table 12 are similar to those in the columns of Table 2. Table 12 shows that the values of the effective masses m h e r Σ ( 1189 ) , π 1 agree well with the published values of m r e s (the average percentage (AP) based on Table 12 equals 3.1%).

Table 13 shows the results of the calculations for the Σ c   Barion → Λ c + ( 2286 ) + π 1 decay triangles. Column 4 lists the H S T r e s values calculated using Equations (3)-(13) ( m b = m Λ c + ( 2286 ) ) based on data from Table A5. The calculations in the remaining columns of Table 13 are similar to those in the columns of Table 2. Table 13 shows that the values of the effective masses m h e r Λ c + ( 2286 ) , π 1 agree well with the published values of m r e s (the average percentage (AP) based on Table 13 equals 16.5%).

Table 13. Hyperbolic Heron triangles in decays of Σ c   Barion → Λ c + ( 2286 ) + π 1 .

Name Σ c Barion Effective Mass Resonance m r e s ± Δ m r e s (Mev) WidthΓ(Mev) H S T r e s H S T h e r i H S T h e r Mass m h e r Λ c + ( 2286 ) , π 1 in Δ Λ c + ( 2286 ) m π 1 (Mev) Δ m r e s h e r = m r e s − m h e r Λ c + ( 2286 ) , π 1 (Mev) Δ m r e s h e r Γ (%) Mass m h e r π 1 , π 2 in Δ π 1 m π 2 (Mev)
1 2 3 4 5 6 7 8 9
Σ c   Barion → Λ c + ( 2286 ) + π 1 , m Λ c + = 2286.46   Mev , m π 1 = 139.57   Mev , AP = 16.5 %
Σ c ( 2455 ) 2454.0 ± 0.1 1.9 − 31.56 − 0.1 + 0.1 −32 2453.7 0.3 14.7 327.4
Σ c ( 2520 ) 2518.4 ± 0.2 14.8 − 6.61 − 0.3 + 0.3 −7 2515.0 4.2 28.4 415.2
Σ c ( 2800 ) 2801.0 ± 5.0 75.0 12.74 − 0.32 + 0.32 13 2805.7 4.7 6.3 713.2

Table 14 shows the results of the calculations for the Λ   Barion → Σ + π 1 decay triangles. Column 4 lists the H S T r e s values calculated using Equations (3)-(13) ( m b = m Σ ) based on data from Table A6. The calculations in the remaining columns of Table 14 are similar to those in the columns of Table 2. Table 11 shows that the values of the effective masses m h e r Σ , π 1 agree well with the published values of m r e s (the average percentage (AP) based on Table 14 equals 4.6%).

Table 15 shows the results of the calculations for the Λ c   Barion → Σ c 0 ( 2455 ) + π 1 + decay triangles. Column 4 lists the H S T r e s values calculated using Equations (3)-(13) ( m b = m Σ c 0 ( 2455 ) ) based on data from Table A6. The calculations in the remaining columns of Table 12 are similar to those in the columns of Table 2. Table 15 shows that the values of the effective masses m h e r Σ c 0 , π 1 agree well with the published values of m r e s (the average percentage (AP) based on Table 15 equals 10.2%).

Table 14. Hyperbolic Heron triangles in decays of Λ   Barion → Σ + π 1 .

Name Λ Barion Effective Mass Resonance m r e s ± Δ m r e s (Mev) WidthΓ(Mev) H S T r e s H S T h e r H S T h e r / ch   S π 1 π 2 Effective Mass m h e r Σ , π 1 in Δ Σ m π 1 (Mev) Δ m r e s h e r = m r e s − m h e r Σ , π 1 (Mev) Δ m r e s h e r Γ (%) Base Mass m h e r π 1 , π 2 in Δ π 1 m π 2 (Mev)
1 2 3 4 5 6 7 8 9
Λ   Barion → Σ + π 1 , m Σ = 1189.37   Mev , m π 1 = 139.57   Mev , AP = 4.6 %
Λ ( 1405 ) 1405.1 ± 1.3 50.0 − 6.97 − 0.07 + 0.07 −7/3 1404.8 0.3 0.7 394.8
Λ ( 1520 ) 1519.4 ± 0.2 15.7 4.24 − 0.01 + 0.01 4 1512.7 6.3 39.9 523.7
Λ ( 1600 ) 1600.0 ± 10.0 200.0 7.50 − 0.46 + 0.45 7 1588.9 11.1 5.6 602.4
Λ ( 1670 ) 1670.0 ± 5.0 30.0 10.96 − 0.27 + 0.27 11 1670.7 0.6 2.2 680.7
Λ ( 1690 ) 1690.0 ± 5.0 70.0 12.06 − 0.28 + 0.28 12 1688.9 1.1 1.6 697.4
Λ ( 1710 ) 1713.0 ± 13.0 180.0 13.38 − 0.76 + 0.78 13 1706.4 6.6 3.6 713.2
Λ ( 1800 ) 1800.0 ± 10.0 200.0 18.97 − 0.07 + 0.69 19 1800.5 0.5 0.2 795.4
Λ ( 1810 ) 1790.0 ± 10.0 110.0 18.28 − 0.68 + 0.69 18 1785.9 4.1 3.7 783.0
Λ ( 1820 ) 1820.0 ± 4.0 80.0 20.38 − 0.29 + 0.29 20 1814.7 5.3 6.6 807.5
Λ ( 1830 ) 1825.0 ± 10.0 90.0 20.74 − 0.72 + 0.73 21 1828.5 3.5 3.9 819.1
Λ ( 1890 ) 1890.0 ± 5.0 120.0 25.72 − 0.40 + 0.40 26 1893.4 3.4 2.8 872.8
Λ ( 2050 ) 2056.0 ± 22.0 493.0 40.92 − 2.22 + 2.29 41 2056.8 0.8 0.2 1002.2
Λ ( 2070 ) 2070.0 ± 24.0 370.0 42.37 − 2.47 + 2.55 42 2066.4 3.6 1.0 1009.6
Λ ( 2080 ) 2082.0 ± 13.0 181.0 43.63 − 1.37 + 1.39 44 2085.4 3.4 1.9 1024.2
Λ ( 2100 ) 2100.0 ± 22.0 200.0 45.57 − 2.35 + 2.42 46 2104.0 4.0 2.0 1038.4
Λ ( 2110 ) 2090.0 ± 22.0 250.0 44.49 − 2.32 + 2.39 44 2085.4 4.6 1.8 1024.2
Λ ( 2350 ) 2350.0 ± 22.0 150.0 77.17 − 3.15 + 3.22 77 2348.8 1.2 0.8 1219.9

Table 15. Hyperbolic Heron triangles in decays of Λ c   Barion → Σ c 0 ( 2455 ) + π 1 + .

Name Λ c Barion Effective Mass Resonance m r e s ± Δ m r e s (Mev) WidthΓ(Mev) H S T r e s H S T h e r i H S T h e r H S T h e r / ch   S π 1 π 2 Effective Mass m h e r Σ c 0 ( 2455 ) , π 1 in Δ Σ c 0 ( 2455 ) m π 1 (Mev) Δ m r e s h e r = m r e s − m h e r Σ c 0 ( 2455 ) , π 1 (Mev) Δ m r e s h e r Γ (%) Base Mass m h e r π 1 , π 2 in Δ π 1 m π 2 (Mev)
1 2 3 4 5 6 7 8 9
Λ c   Barion → Σ c 0 ( 2455 ) + π 1 + , m Σ = 2453.75   Mev , m π 1 = 139.57   Mev , AP = 10.2 %
Λ c ( 2625 ) 2628.0 ± 0.2 52.0 − 13.41 − 0.03 + 0.03 −23 2628.5 0.5 0.9 339.6
Λ c ( 2880 ) 2881.6 ± 0.2 51.8 8.38 − 0.01 + 0.01 8/9 2873.2 8.4 16.2 624.2
Λ c ( 2910 ) 2913.8 ± 3.8 51.8 9.91 − 0.21 + 0.21 10 2915.6 1.8 3.5 663.0
Λ c ( 2940 ) 2939.6 ± 1.5 20.0 11.21 − 0.09 + 0.09 11 2935.5 4.0 20.3 680.7

Table 16 shows the results of the calculations for the Ξ   Barion → Ξ 0 + π 1 decay triangles. Column 4 lists the H S T r e s values calculated using Equations (3)-(13) ( m b = m Ξ 0 ) based on data from Table A7. The calculations in the remaining columns of Table 16 are similar to those in the columns of Table 2. Table 16 shows that the values of the effective masses m h e r Ξ 0 , π 1 agree well with the published values of m r e s (the average percentage (AP) based on Table 16 equals 15.5%).

Table 17 shows the results of the calculations for the Ξ   Barion → Ξ ( 1530 ) + π 1 decay triangles. Column 4 lists the H S T r e s values calculated using Equations (3)-(13) ( m b = m Ξ ( 1530 ) ) based on data from Table A7. The calculations in the remaining columns of Table 17 are similar to those in the columns of Table 2. Table 17 shows that the values of the effective masses m h e r Ξ ( 1530 ) , π 1 agree well with the published values of m r e s (the average percentage (AP) based on Table 17 equals 5.4%).

Table 16. Hyperbolic Heron triangles in decays of Ξ   Barion → Ξ 0 + π 1 .

Name Ξ Barion Effective Mass Resonance m r e s ± Δ m r e s (Mev) WidthΓ(Mev) H S T r e s H S T h e r i H S T h e r H S T h e r / ch   S π 1 π 2 Effective Mass m h e r Ξ 0 , π 1 in Δ Ξ 0 m π 1 (Mev) Δ m r h e r = m r e s − m h e r Ξ 0 , π 1 (Mev) Δ m r e s h e r Γ (%) Base Mass m h e r π 1 , π 2 in Δ π 1 m π 2 (Mev)
1 2 3 4 5 6 7 8 9
Ξ   Barion → Ξ 0 + π 1 , m Ξ 0 = 1314.85999   Mev , m π 1 = 139.57   Mev , AP = 15.5 %
Ξ ( 1530 ) 1531.8 ± 0.3 9.1 − 9.02 − 0.04 + 0.04 −9 1532.3 0.1 1.4 393.8
Ξ ( 1620 ) 1620.8 ± 6.0 32.0 3.33 − 0.19 + 0.20 3 1610.4 10.4 32.5 489.9
Ξ ( 1690 ) 1690.0 ± 10.0 20.0 5.85 − 0.40 + 0.41 6 1693.6 3.6 18.0 579.0
Ξ ( 1820 ) 1823.0 ± 5.0 20.0 12.29 − 0.28 + 0.28 12 1817.9 5.1 25.6 697.4
Ξ ( 1950 ) 1950.0 ± 15.0 60.0 20.35 − 1.05 + 1.08 20 1945.1 5.0 24.7 807.5
Ξ ( 2030 ) 2025.0 ± 20.0 60.0 26.01 − 1.58 + 1.62 26 2024.9 0.1 0.2 872.8
Ξ ( 2500 ) 2500.0 ± 150.0 60.0 78.48 − 19.82 + 23.05 78 2496.6 3.6 5.6 1224.9

Table 17. Hyperbolic Heron triangles in decays of Ξ   Barion → Ξ ( 1530 ) + π 1 .

Name Ξ Barion Effective Mass Resonance m r e s ± Δ m r e s (Mev) WidthΓ(Mev) H S T r e s i H S T r e s H S T h e r i H S T h e r H S T h e r / ch   S π 1 π 2 Effective Mass m h e r Ξ ( 1530 ) , π 1 in Δ Ξ ( 1530 ) m π 1 (Mev) Δ m r h e r = m r e s − m h e r Ξ ( 1530 ) , π 1 (Mev) Δ m r e s h e r Γ (%) Base Mass m h e r π 1 , π 2 in Δ π 1 m π 2 (Mev)
1 2 3 4 5 6 7 8 9
Ξ   Barion → Ξ ( 1530 ) + π 1 , m Ξ ( 1530 ) = 1531.8   Mev , m π 1 = 139.57   Mev , AP = 5.4 %
Ξ ( 1690 ) 1690.0 ± 10.0 20.0 − 59.26 − 0.26 + 0.26 −59 1690.1 0.1 0.2 311.1
Ξ ( 1820 ) 1823.0 ± 5.0 20.0 2.99 − 0.26 + 0.28 3 1823.3 0.3 1.4 697.4
Ξ ( 1950 ) 1950.0 ± 15.0 60.0 7.87 − 0.24 + 0.24 8/9 1952.7 2.7 13.6 624.2
Ξ ( 2030 ) 2025.0 ± 20.0 60.0 11.62 − 0.88 + 0.93 12 2032.0 7.0 11.7 697.4
Ξ ( 2500 ) 2500.0 ± 150.0 60.0 50.00 − 2.96 + 3.08 50 2500.0 0.01 0.01 1065.6
3. The Proposal to Search for New Resonances

Table 18 presents a summary classification of two-body decays of light unflavored, strange mesons and Δ,N,Λ,Σ,Ξ baryons, based on the values of the hyperbolic sines H S T h e r of their Heron triangles, using data from Tables 1-17. Column 1 of Table 18 list the H S T h e r i , H S T h e r and H S T h e r / ch ( S π 1 π 2 ) values. Values of H S T h e r i are shown with a minus sign. The values of H S T h e r / ch ( S π 1 π 2 ) are shown in bold. Column 2 lists the base mass m h e r π 1 , π 2 . Column 3 list the decays Scalar Mezon → π 1 + π 2 and Σ c   Barion → Λ c + ( 2286 ) + π 1 . Column 4 list the decays Δ   Barion → P + π 1 and N   Barion → n + π 1 , and Scalar Mezon → η + π 1 . Column 5 list the decays Σ   Barion → Λ + π 1 and Λ   Barion → Σ + π 1 . Column 6 list the decays Ξ   Barion → Ξ 0 + π 1 and Scalar Mezon → ω ( 782 ) + π 1 . Column 7 list the decays Scalar Mezon → ρ ( 770 ) + π 1 and Λ c   Barion → Σ c 0 ( 2455 ) + π 1 , and Σ   Barion → Σ + π 1 . Column 8 list the decays Strange Mezon → K ( 493 ) + π 1 and N   Barion → Δ ( 1232 ) + π 1 . Column 9 list the decays Δ   Barion → Δ ( 1232 ) + π 1 and Strange Mezon → K * ( 892 ) + π 1 , and Ξ   Barion → Ξ ( 1530 ) + π 1 .

Table 18. Hyperbolic Sine H S T h e r , H S T h e r / ch   S π 1 π 2 , mass m h e r π 1 , π 2 of Heron triangles of light unflavored, strange meson and Δ, N, Λ, Σ, Ξ baryons decays.

H S T h e r i H S T h e r H S T h e r / ch   S π 1 π 2 BaseMass m h e r π 1 , π 2 (Mev) Scalar Mez → π 1 + π 2 Σ c   Barion → Λ c + ( 2286 ) + π 1 Δ   Barion → P + π 1 N   Barion → n + π 1 Scalar Mez → η + π 1 Σ   Barion → Λ + π 1 Λ   Barion → Σ + π 1 Ξ   Barion → Ξ 0 + π 1 Scalar   Mez → ω ( 782 ) + π 1 Scalar   Mez → ρ ( 770 ) + π 1 Λ c   Barion → Σ c 0 ( 2455 ) + π 1 + Σ   Barion → Σ + π 1 Strange Mez → K ( 493 ) + π 1 N   Barion → Δ ( 1232 ) + π 1 Δ   Barion → Δ ( 1232 ) + π 1 Strange   Mez → K * ( 892 ) + π 1 Ξ   Barion → Ξ ( 1530 ) + π 1
1 2 3 4 5 6 7 8 9
−59 311.1 Ξ ( 1690 )
−32 327.4 Σ c ( 2455 )
−23 339.6 Λ c ( 2625 )
−20 345.6 η ′ ( 958 )
−14 363.9 Σ ( 1385 )
−10 394.8 N ( 1440 )
−9 393.8 Ξ ( 1530 ) empty
−7 415.2 Σ c ( 2520 )
1 / 3 394.8 Λ ( 1405 ) Ξ ( 1530 ) empty
2 449.0 Σ ( 1385 )
3 489.9 Δ ( 1232 ) Ξ ( 1620 ) N ( 1520 ) N ( 1535 ) Ξ ( 1820 )
4 523.9 Λ ( 1520 ) K * ( 700 )
5 552.9 Δ ( 1600 ) K 1 ( 1270 )
6 579.0 Ξ ( 1690 ) h 1 ( 1170 ) K * ( 892 ) Δ ( 1620 )
7 602.5 f o ( 500 ) Λ ( 1600 )
8 /9 624.2 a o ( 980 ) Σ ( 1620 ) Λ c ( 2880 ) N ( 1650 ) Ξ ( 1950 )
9 644.2 b 1 ( 1235 ) N ( 1675 )
10 663.0 Σ ( 1660 ) Λ c ( 2910 ) a 1 ( 1260 ) N ( 1680 )
11 680.7 Λ ( 1670 ) Σ ( 1670 ) Λ c ( 2910 ) N ( 1700 ) N ( 1710 ) N ( 1720 ) Δ ( 1700 )
12 697.4 N ( 1440 ) Σ ( 1620 ) Λ ( 1690 ) Ξ ( 1820 ) Λ c ( 2940 ) Ξ ( 2030 )
13 713.2 Σ c ( 2800 ) Λ ( 1710 ) π ( 1300 ) K 1 ( 1400 ) K 1 * ( 1410 )
14 728.4 K 2 * ( 1430 )
15 742.9 Σ ( 1660 ) a 2 ( 1320 )
16 756.8 Σ ( 1670 ) Σ ( 1750 )
17 771.6 ρ ( 770 ) N ( 1520 ) Σ ( 1775 )
18 783.0 ω ( 782 ) N ( 1535 ) Λ ( 1810 ) Σ ( 1780 ) K ( 1460 )
19 795.4 Λ ( 1800 )
20 807.5 Λ ( 1820 ) Ξ ( 1950 ) N ( 1860 ) Δ ( 1900 )
21 819.2 Δ ( 1600 ) Σ ( 1750 ) Λ ( 1830 ) ω ( 1420 ) N ( 1875 )
22 830.5 N ( 1880 ) Δ ( 1905 )
23 841.5 Σ ( 1775 ) Σ ( 1780 ) N ( 1895 ) Δ ( 1910 )
24 852.2 Δ ( 1620 )
25 862.6 ρ ( 1450 ) N ( 1900 ) Δ ( 1950 ) Δ ( 1920 ) K 2 ( 1580 )
26 872.8 Λ ( 1890 ) Ξ ( 2030 )
27 /19 882.7 N ( 1650 ) Σ ( 1910 ) Δ ( 1930 )
28 892.4 Σ ( 1915 )
29 901.9 N ( 1675 ) Σ ( 1900 )
30 911.2 N ( 1680 ) Σ ( 1940 )
31 920.2 Δ ( 1940 )
32 929.4
33 937.8 Δ ( 1710 ) N ( 1700 )
34 946.4 a 2 ( 1320 ) Σ ( 1910 )
35 954.8 Σ ( 1915 )
36 963.0 Σ ( 1900 ) Σ ( 2010 ) N ( 2000 )
37 971.1 Σ ( 1940 )
38 979.1 π 1 ( 1600 ) Σ ( 2030 ) K * ( 1680 )
39 986.9 f o ( 980 )
40 994.6
41 1002.2 Λ ( 2050 ) Σ ( 2070 ) N ( 2060 ) N ( 2100 ) Δ ( 2000 )
42 1009.6 Λ ( 2070 )
43 1017.0 ϕ ( 1020 ) N ( 2120 )
44 1024.2 Σ ( 2010 ) Λ ( 2080 ) Λ ( 2110 ) π 1 ( 1600 ) K 2 ( 1770 )
45 1031.4 K 3 * ( 1780 )
46 1038.4 Σ ( 2030 ) Λ ( 2100 ) ω 3 ( 1670 ) Σ ( 2100 ) Δ ( 2000 )
47 1045.3 π 2 ( 1670 ) Σ ( 2110 )
48 1052.2 a o ( 1450 ) ρ 3 ( 1690 )
49 1058.9 Δ ( 1900 ) N ( 2190 )
50 1065.6 N ( 1650 ) Σ ( 2070 ) K 2 ( 1820 ) Ξ ( 2500 )
51 1072.1 N ( 1650 ) Δ ( 1905 ) a 2 ( 1700 )
52 1078.6 a 0 ( 1710 ) Δ ( 2200 )
53 1085.0 N ( 1650 ) Δ ( 1910 )
54 1091.4 K * ( 1410 )
55 1097.6 Σ ( 2100 ) K 2 * ( 1430 ) K 0 * ( 1430 )
56 1103.8 N ( 1900 ) Δ ( 1920 ) Σ ( 2110 )
57 1109.9 Δ ( 1950 )
58 1116.0
59 1122.0
60 1127.9 Δ ( 1905 )
… …
62 1139.5 Σ ( 2230 )
63 1145.2 Σ ( 2250 )
64/33 1150.9
… …
67 1167.6 Δ ( 1940 )
69 1178.6 N ( 1990 ) Δ ( 2000 )
70-72 …
73 1199.5 Σ ( 2230 ) K 2 * ( 1980 )
74 1204.7 N ( 2000 ) Σ ( 2250 )
… …
77 1219.9 Λ ( 2350 )
78 1224.8 Ξ ( 2500 ) Ξ ( 2500 )
… …
81 1239.6 N ( 2060 ) N ( 2100 )
82 3957.5
83 4102.5
84 4121.5 N ( 2120 )
85 4169.1
86 4205.7
87 4404.8
88 4457.5
89 1277.1 f 2 ( 1275 )
90 1281.7 Δ ( 1950 ) a 4 ( 1970 )
91 1286.2 a 2 ( 1700 )
92 1290.0 a 0 ( 1710 )
… …
94 1299.5 N ( 2190 )
98 1316.9 Δ ( 2200 )
106 1350.2 f o ( 1370 )
107 1354.2 N ( 2220 ) Σ ( 2455 ) K * ( 1780 )
… …
113 1378.1 N ( 2190 ) π 2 ( 2100 )
… …
117 1393.5 Δ ( 2300 )
… …
120 1404.9 K 3 * ( 1780 )
… …
124 1419.6 ρ ( 2250 )
125/51 1423.3
127 1430.6 Δ ( 2350 )
132 1448.4 Δ ( 2390 )
138 1465.8 ρ ( 1450 )
… …
148 1502.6 Δ ( 2400 )
… …
150 1509.2 a 4 ( 1970 )
… …
153 1518.8 f ′ 2 ( 1525 )
154 1522.0 f o ( 1500 )
156 1528.4 Δ ( 2420 )
166 1554.2 K 0 * ( 1950 )
… …
170 f 2 ( 1565 )
… …
184 1612.0 N ( 2600 )
… …
187 1620.5 N ( 2700 )
… …
193 1637.1 K 0 * ( 2045 )
… …
211 1685.1 ρ 3 ( 1690 )
… …
216/73 1697.9
… …
225 1720.6 ρ ( 1700 )
… …
230 1732.9 f o ( 1710 )
… …
238 1752.5 Δ ( 2750 )
… …
252 1785.1 f o ( 1810 )
… …
260 1803.3 f o ( 1770 )
… …
265 1814.5 f 2 ( 1810 )
… …
302 1893.5 Δ ( 2950 )
… …
305 1899.6 N ( 3000 )
… …
318 1925.6 K 5 * ( 2380 )
… …
323 1935.5 f 2 ( 1950 )
… …
343 /99 1973.8
… …
347 1981.3 f 0 ( 2020 )
… …
367 2017.9 f 4 ( 2050 )
… …
510 2247.6 ρ 2 ( 2250 )
511 2249.1
512/129 2250.5
… …
556 2312.2 ρ 4 ( 2300 )
… …
680 2470.5 f 6 ( 2510 )
… …
729/163 2527.7
… …
1000/201 2805.3
… …
1331 /243 3083.2
… …
1728/289 3361.3
… …
2197/339 3639.5
… …
2744/390 3917.9
… …
3375/451 4196.4
… …

Table 18 can be used to propose a method for searching for new resonances. Generally speaking, new resonances are possible in every empty cell of Table 18. First, let’s consider the non-empty cell in row 1, column 9, with the value H S T h e r i = − 59 , corresponding to the decay of Ξ ( 1690 ) → Ξ ( 1530 ) + π 1 . The empty cell in row 1, column 3 corresponds to ( π 1 , π 2 ) decays of scalar mesons and ( Λ c + , π 1 ) decays of Σ c baryons. Then, using the effective masses m r e s of the pairs ( π 1 , π 2 ) and ( Λ c + , π 1 ) and the formulas (3)-(13) and (16)-(20), (30), we can construct a distributions of H S T h e r i . A statistically significant peak in the distribution of H S T h e r i at a value equal to −59 can be interpreted as the new scalar meson and new Σ c baryon with a base mass m h e r π 1 , π 2 = 311.0 MeV. Similar assumptions can be made regarding the decays listed in the remaining blank cells in row 1 and columns 4-9.

The cell in row with the value H S T r e s = 7 and column 3 corresponds to the decay f o ( 500 ) → π 1 + π 2 . Therefore, a statistically significant peak at a value of H S T r e s = 7 will necessarily appear. The value H S T h e r = 7 for the scalar meson f o ( 500 ) corresponds to an average mass of 600 Mev and a width of 300 Mev. However, all measurements of the mass and width f o ( 500 ) meson have a large variance of values (mass 400 - 800 MeV, width 100 - 800 MeV) [8]. Therefore, no clear peak is observed in the distribution for m r e s < 600 MeV in ( π 1 , π 2 ) decays. This situation can be resolved using Heron’s base triangles. The hyperbolic sine H S T r e s ( H S T r e s i ) depends nonlinearly on the effective mass m r e s , of the pair ( π 1 , π 2 ) of pi mesons. Therefore, it is quite likely that statistically significant peaks will appear in the distribution of H S T r e s ( H S T r e s i ) at the values equal to −59, −32, −23, −20, −14, −10, −9, −7, 1, 2, 3, 4, 5, 6, and 7. These peaks can be interpreted as ( π 1 , π 2 ) decays of new scalar mesons with masses m h e r π 1 , π 2 < 600.0 MeV. It would be interesting to study the effective masses m r e s of ( π 1 , π 2 ) pairs for values of H S T r e s > 1000, which corresponds to masses m h e r π 1 , π 2 > 2805.3 MeV.

The non-empty cells in row with the value H S T h e r = 8 corresponds to the decays a o ( 980 ) → η + π 1 , Σ ( 1620 ) → Σ + π 1 , Λ c ( 2880 ) → Σ c 0 ( 2455 ) + π 1 , Ξ ( 1950 ) → Ξ ( 1530 ) + π 1 , N ( 1650 ) → Δ ( 1232 ) + π 1 . The hyperbolic cosine of the base length ch ( S π 1 π 2 ) = 9 for Heron’s base triangles Δ π 1 m π 2 of these resonances is also an integer.

Consider the cell column 4 of this in row H S T h e r = 8. Based on the actual values of the effective masses m r e s of the ( η , π 1 ) pairs, in the distribution of H S T r e s a statistically significant peak at a value of H S T r e s = 8 will necessarily appear. Similarly, based on the actual values of the effective masses m r e s of the pairs ( P , π 1 ) from the same column, one can obtain the H S T r e s distribution, calculated using Equations (21)-(29) for m b = m P . A statistically significant peak in the distribution of H S T r e s at a value equal to 8 can be interpreted as a new Δ baryon with a base mass m h e r π 1 , π 2 = 624.2 MeV. The empty cells in this row correspond to the decays of the pairs ( π 1 , π 2 ) and ( Λ c + , π 1 ) (column 3), (Λ, π 1 ) and (Σ, π 1 ) (column 5), ( Ξ 0 , π 1 ) and (ω(782), π 1 ) (column 6). The statistically significant peaks in the distributions of H S T r e s = 8 can be interpreted as the new light unflavored mezon and Λ c Barion of the column 3, the new Ξ Barion and light unflavored mezon of the column 5, the new Ξ Barion and light unflavored mezon of the column 6.

It would be interesting to study the effective masses m r e s of ( η , π 1 ), ( P , π 1 ) pairs for values of H S T r e s = 1000, 1331, 1728, 2197, 2744, 3375 which correspond to base masses m h e r π 1 , π 2 = 2805.3, 3083.2, 3361.3, 3639.5, 3917.9, 4196.4 MeV.

To conclude this chapter, we note that the decay triangles originated from Archimedes’ levers (3), in which the masses of the daughter particles of resonance decay act as forces. Therefore, the experimentally observed (Tables 1-18) close relationship between the resonance decay triangles and Heron’s base triangles indicates that, in some way, the gravitational forces of the daughter particle masses are connected with the formation of resonances.

 4. 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 base Heron triangles with integer values of H S T r e s . Based on Heron’s triangles, the existence of new resonances is predicted. This can be confirmed by direct measurements of the effective masses of the pairs ( π 1 , π 2 ), ( ω , π 1 ), ( η , π 1 ), ( ρ ( 770 ) , π 1 ), (P, π 1 ), (n, π 1 ), (Δ(1232), π 1 ), ( K ( 493 ) , π 1 ), ( K ( 892 ) , π 1 ), ( Ξ 0 , π 1 ), (Λ, π 1 ) of the particles in reaction (1) and the detection of statistically significant peaks in the distribution of H S T r e s .

Further development of the described approach will consist in:

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 [16];analysis of reaction (1) with the formation of more than two particles, based on a 3-dimensional analog of Heron’s triangles.

 Funding of the Work

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 [8].

 Appendix

Table A1.Data on two-particle decays of scalar mesons from [8].

Name Scalar Mezon I G (J PC ) Effective Mass Resonance m r e s (Mev) WidthΓ (Mev) Decay π + π Decay η + π Decay ρ ( 770 ) + π Decay ω ( 782 ) + π
1 2 3 4 5 6 7
f o ( 500 ) 0 + (0 + + ) 600.0 ± 50.0 450.0 seen - - -
ρ ( 770 ) 1 + (1 − − ) 766.5 ± 1.1 150.0 ~100% - - -
ω ( 782 ) 0 − (1 − − ) 782.7 ± 0.13 8.68 (1.53 ± 0.12)% - - -
η ′ ( 958 ) 0 + (0 − + ) 957.8 ± 0.06 0.23 - - (7.4 ± 2.3)10 − 4 -
f o ( 980 ) 0 + (0 + + ) 980.0 ± 20.0 55.0 seen - - -
ϕ ( 1020 ) 0 − (1 − − ) 1019.46 ± 0.02 4.25 (9.5 ± 1.9)10 − 5 43 - (4.7 ± 0.5)10 − 5
h 1 ( 1170 ) 0 − (1 + − ) 1166.4 ± 5.00 375.0 - - seen -
b 1 ( 1235 ) 1 + (1 + − ) 1229.5 ± 3.30 142.9 - - - seen
a 1 ( 1260 ) 1 − (1 + + ) 1230.0 ± 40.0 420.0 - - seen -
f 2 ( 1270 ) 0 + (2 + + ) 1275.4 ± 0.80 186.6 ~84% - - -
π ( 1300 ) 1 − (0 − + ) 1300.0 ± 100.0 400.0 - - seen -
a 2 ( 1320 ) 1 − (2 + + ) 1317.7 ± 1.4 107.0 - (14.5 ± 1.2)% - -
f o ( 1370 ) 0 + (0 + + ) 1350.0 ± 50.0 350.8 seen - - -
ω ( 1420 ) 0 − (1 − − ) 1410.0 ± 60.0 290.0 - - seen -
a o ( 1450 ) 1 − (0 + + ) 1439.0 ± 34.0 258.0 - (0.93 ± 0.02)% - -
ρ ( 1450 ) 1 + (1 − − ) 1465.0 ± 25.0 400.0 seen - - -
f o ( 1500 ) 0 + (0 + + ) 1522.0 ± 25.0 108.0 (34.5 ± 2.2)% - - -
f ′ 2 ( 1525 ) 0 + (2 + + ) 1517.3 ± 2.4 72.0 (8.2 ± 1.3)10 − ³ - - -
f 2 ( 1565 ) 0 + (2 + + ) 1571.0 ± 13.0 132.0 seen - - -
π 1 ( 1600 ) 1 − (1 − + ) 1645.0 ± 16.0 370.0 - seen seen -
a 1 ( 1640 ) 1 − (1 + + ) 1655.0 ± 16.0 250.0 - - seen -
ω ( 1650 ) 0 − (1 − − ) 1670.0 ± 30.0 315.0 - - seen -
ω 3 ( 1670 ) 0 − (3 − − ) 1667.0± 4.0 168.0 - - seen -
π 2 ( 1670 ) 1 − (2 − + ) 1670.6 ± 2.9 288.0 - - (31.0 ± 4.0)% -
ρ 3 ( 1690 ) 1 + (3 − − ) 1686.0 ± 4.0 161.0 - - - (16.0 ± 6.0)%
ρ ( 1700 ) 1 + (1 − − ) 1720.0 ± 20.0 250.0 seen - - -
a 2 ( 1700 ) 1 − (2 − + ) 1706.0 ± 14.0 380.0 - (2.5 ± 0.6)% seen -
a 0 ( 1710 ) 1 − (0 + + ) 1713.0 ± 19.0 107.0 - seen - -
f o ( 1710 ) 0 + (0 + + ) 1733.0 ± 8.0 150.0 seen - - -
f o ( 1770 ) 0 + (0 − + ) 1804.0 ± 16.0 138.0 - - -
f 0 ( 1810 ) 0 + (0 + + ) 1785.0 ± 13.0 137.0 seen - - -
f 2 ( 1810 ) 0 + (0 + + ) 1815.0 ± 12.0 197.0 seen - - -
f 2 ( 1950 ) 0 + (0 − + ) 1936.0 ± 12.0 197.0 seen - - -
a 4 ( 1970 ) 1 − (4 + + ) 1967.0 ± 16.0 324.0 - - seen -
f 0 ( 2020 ) 0 + (0 − + ) 1982.0 ± 54.0 464.0 seen - - -
f 4 ( 2050 ) 0 + (4 + + ) 2018.0 ± 11.0 237.0 (17.0 ± 1.5)% - - -
π 2 ( 2100 ) 0 + (0 − + ) 2090.0 ± 29.0 620.0 - - seen -
f 4 ( 2300 ) 0 + (4 + + ) 2312.0 ± 15.0 200.0 seen - - -
f 6 ( 2510 ) 0 + (6 + + ) 2470.0 ± 50.0 260.0 (6.0 ± 1.0)% - - -

Table A2.Data on two-particle decays of Strange Mezons from [8].

Name StrangeMezon I (J P ) Effective Mass Resonance m r e s (Mev) WidthΓ (Mev) Decay K ( 493 ) + π Decay K * ( 892 ) + π
1 2 3 4 5
K * ( 700 ) 1/2 (0 + ) 838.0 ± 11.0 450.0 ~100% -
K * ( 892 ) 1/2(1 + ) 891.8± 0.25 150.0 ~100% -
K 1 ( 1270 ) 1/2(1 + ) 1253.0 ± 7.0 10.0 - (18.0 ± 4.0)%
K 1 ( 1400 ) 1/2(1 + ) 1403.0± 7.0 174.0 - (94.0 ± 6.0)%
K * ( 1410 ) 1/2 (1 + ) 1421.8 ± 9.0 8.68 > 40% (6.6 ± 1.3)%
K 0 * ( 1430 ) 1/2(0 + ) 1425.0 ± 50.0 270.0 (93.0 ± 10.0)% -
K 2 * ( 1430 ) 1/2(2 + ) 1423.6 ± 1.5 100.0 (49.9 ± 1.2)% (24.7 ± 1.5)%
K ( 1460 ) 1/2(0 − ) 1482.4 ± 15.0 335.6 - seen
K 2 ( 1580 ) 1/2(2 − ) 1580.0 ± 10.0 110.0 - seen
K * ( 1680 ) 1/2(2 − ) 1718.8 ± 18.0 320.0 (38.7 ± 2.5)% (29.9 ± 5.0)%
K 2 ( 1770 ) 1/2(2 − ) 1773.8 ± 8.0 186.0 - seen
K 3 * ( 1780 ) 1/2(3 − ) 1779.0 ± 8.0 161.0 (18.8 ± 1.0)% (20.0 ± 5.0)%
K 2 ( 1820 ) 1/2(2 − ) 1819.0 ± 12.0 264.0 - seen
K 0 * ( 1950 ) 1/2(0 + ) 1957.0 ± 14.0 170.0 (52.0 ± 14.) % -
K 2 * ( 1980 ) 1/2(0 + ) 1990.0 ± 50.0 348.0 - possibly seen
K 4 * ( 2045 ) 1/2(4 + ) 2045.0 ± 9.0 375.0 (9.9 ± 1.2)% -
K 5 * ( 2380 ) 1/2(5 − ) 2382.8 ± 14.0 142.9 (6.1 ± 1.2)% -

Table A3.Data on two-particle decays of Δ Barions from [8].

Name Δ Barions I (J P ) Effective Mass Resonance m r e s (Mev) WidthΓ (Mev) Decay P + π Decay Δ ( 1232 ) + π
1 2 3 4 5
Δ (1232) 3/2 (3/2 + ) 1232.0 ± 0.40 117.0 99.4% -
Δ (1600) 3/2 (3/2 + ) 1570.0 ± 0.25 250.0 (8.0 - 24.0)% (58.0 - 82.0)%
Δ (1620) 3/2 (1/2 − ) 1610.0 ± 20.0 130.0 (25.0 - 35.0)% (44.0 - 72.0)%
Δ (1700) 3/2 (3/2 − ) 1710.0 ± 2.0 300.0 (10.0 - 20.0)% (9.0 - 10.0)%
Δ (1750) 3/2 (1/2 + ) 1715.0 ± 10.0 350.0 seen -
Δ (1900) 3/2 (1/2 − ) 1860.0± 30.0 300.0 (4.0 - 12.0)% (30.0 - 70.0)%
Δ (1905) 3/2 (5/2 + ) 1880.0± 20.0 330.0 (9.0 - 15.0)% >48%
Δ (1910) 3/2 (1/2 + ) 1900.0± 20.0 300.0 (10.0 - 30.0)% (34.0 - 66.0)%
Δ (1920) 3/2 (3/2 + ) 1920.0± 30.0 300.0 (5.0 - 20.0)% >46%
Δ (1930) 3/2 (5/2 + ) 1950.0± 30.0 300.0 (5.0 - 15.0)% (33.0 ± 9.0)%
Δ (1940) 3/2 (3/2 − ) 2000.0± 30.0 400.0 (1.0 - 20.0)% (38.0 ± 10.0)%
Δ (1950) 3/2 (7/2 + ) 1930.0 ± 10.0 285.0 (1.0 - 20.0)% (4.0 ± 3.0)%
Δ (1950) 3/2 (5/2 − ) 1930.0 ± 10.0 285.0 (93.0 ± 10.0)% -
Δ (2000) 3/2 (7/2 − ) 2015.0± 24.0 500.0 (3.0 - 11.0)% <9%
Δ (2150) 3/2 (1/2 − ) 2150.0 ± 100.0 200.0 (6.0 - 10.0)% -
Δ (2200) 3/2 (9/2 + ) 2200.0± 30.0 300.0 (2.0 - 8.0)% >45%
Δ (2300) 3/2 (3/2 + ) 2300.0 ± 100.0 350.0 (1.0 - 8.0)% -
Δ (2350) 3/2 (5/2 − ) 2400.0 ± 100.0 400.0 (3.0 - 30.0)% -
Δ (2390) 3/2 (7/2 + ) 2375.0 ± 100.0 300.0 (3.0 - 12.0)% -
Δ (2400) 3/2 (9/2 − ) 2450.0 ± 100.0 500.0 (3.0 - 9.0)% -
Δ (2420) 3/2 (11/2 + ) 2450.0 ± 100.0 500.0 (5.0 - 10.0)% -
Δ (2750) 3/2 (13/2 − ) 2794.0 ± 80.0 350.0 (2.0 - 6.0)% -
Δ (2950) 3/2 (15/2 + ) 2990.0 ± 100.0 330.0 (2.0 - 6.0)% -

Table A4.Data on two-particle decays of N Barions from [13].

Name N Barions I (J P ) Effective Mass Resonance m r e s (Mev) WidthΓ (Mev) Decay N + π Decay Δ ( 1232 ) + π
1 2 3 4 5
N (1440) 1/2 (1/2 + ) 1440.0 ± 30.0 350.0 (55.0 - 65.0)% (6.0 - 27.0)%
N (1520) 1/2 (3/2 − ) 1515.0 ± 10.0 110.0 (55.0 - 65.0)% (15.0 - 23.0)%
N (1535) 1/2 (1/2 − ) 1530.0 ± 5.0 150.0 (32.0 - 52.0)% (1.0 - 4.0)%
N (1650) 1/2 (1/2 − ) 1650.0 ± 5.0 125.0 (50.0 - 70.0)% (6.0 - 18.0)%
N (1675) 1/2 (5/2 − ) 1675.0 ± 5.0 150.0 (38.0 - 42.0)% (19.0 ± 4.0)%
N (1680) 1/2 (5/2 + ) 1685.0 ± 5.0 120.0 (60.0 - 70.0)% (11.0 - 23.0)%
N (1700) 1/2 (5/2 − ) 1720.0 ± 10.0 200.0 (7.0 - 17.0)% (55.0 - 85.0)%
N (1710) 1/2 (1/2 + ) 1710.0 ± 20.0 200.0 (5.0 - 20.0)% (3.0 - 9.0)%
N (1720) 1/2 (3/2 + ) 1720.0 ± 10.0 250.0 (8.0 - 14.0)% (47.0 - 89.0)%
N (1860) 1/2 (5/2 + ) 1860.0 ± 10.0 250.0 (4.0 - 20.0)% (8.0 - 54.0)%
N (1875) 1/2 (3/2 − ) 1875.0 ± 20.0 200.0 (3.0 - 11.0)% (4.0 - 44.0)%
N (1880) 1/2 (1/2 + ) 1880.0 ± 20.0 300.0 (3.0 - 11.0)% (5.0 - 42.0)%
N (1895) 1/2 (1/2 − ) 1985.0 ± 20.0 120.0 (2.0 - 11.0)% (3.0 - 18.0)%
N (1900) 1/2 (3/2 + ) 1920.0 ± 10.0 200.0 (1.0 - 20.0)% (3.0 - 70.0)%
N (2000) 1/2 (5/2 + ) 1920.0 ± 20.0 400.0 (6.0 - 10.0)% (30.0 - 80.0)%
N (1990) 1/2 (7/2 + ) 2020.0 ± 40.0 300.0 (2.0 - 6.0)% (72.0 ± 15.0)%
N (2060) 1/2 (5/2 − ) 2050.0 ± 40.0 400.0 (7.0 - 12.0)% (12.0 ± 3.0)%
N (2100) 1/2 (1/2 + ) 2100.0 ± 20.0 260.0 (8.0 - 32.0)% (6.0 - 14.0)%
N (2120) 1/2 (3/2 − ) 2120.0 ± 30.0 300.0 (5.0 - 15.0)% >23%
N (2190) 1/2 (7/2 − ) 2180.0 ± 20.0 400.0 (1.0 - 20.0)% (4.0 ± 2.0)%
N (2220) 1/2 (9/2 + ) 2250.0 ± 15.0 400.0 (15.0 - 30.0)% (15.0 ± 10.0)%
N (2250) 1/2 (9/2 − ) 2280.0 ± 15.0 500.0 (5.0 - 15.0)% (10.0 ± 7.0)%
N (2600) 1/2 (11/2 − ) 2600.0 ± 100.0 650.0 (3.0 - 8.0)% -
N (2700) 1/2 (3/2 + ) 2612.0 ± 45.0 350.0 (3.0 - 5.0)% -
N (3000) 1/2 (5/2 − ) 3000.0 ± 200.0 1650.0 (2.0 - 8.0)% -

Table A5.Data on two-particle decays of Σ Barions from [8].

Name Σ Barions I (J P ) Effective Mass Resonance m r e s (Mev) WidthΓ (Mev) Decay Λ + π Decay Σ + π Decay Λ c + ( 2286 ) + π
1 2 3 4 5 6
Σ (1385) 1 (3/2 + ) 1383.7 ± 2.0 39.4 (87.0 ± 1.5)% (11.7 ± 1.5)%
Σ (1620) 1 (3/2 − ) 1620.0 ± 2.0 70.0 (9.0 ± 3.0)% (17.0 ± 5.0)%
Σ (1660) 1 (1/2 + ) 1660.0 ± 20.0 200.4 (35.0 ± 12.0)% (30.0 ± 10.0)%
Σ (1670) 1 (3/2 − ) 1675.0 ± 2.0 70.0 (5.0 - 15.0)% (30.0 - 60.0)%
Σ (1750) 1 (1/2 − ) 1750.0 ± 2.0 150.0 (14.0 ± 5.0)% (16.0 ± 4.0)%
Σ (1775) 1 (5/2 − ) 1775.0 ± 10.0 120.0 (14.0 - 20.0)% (2.0 - 5.0)%
Σ (1780) 1 (3/2 + ) 1780.0 ± 30.0 200.0 (70.0 ± 7.0)% (12.0 ± 6.0)%
Σ (1900) 1 (1/2 − ) 1925.0 ± 20.0 165.0 (6.0 ± 2.0)% 0.25
Σ (1910) 1 (3/2 − ) 1910.0 ± 50.0 220.0 (6.0 ± 4.0)% (86.0 ± 21.0)%
Σ (1915) 1 (5/2 + ) 1915.0 ± 22.0 120.0 (6.0 ± 2.0)% (10.0 ± 2.0)%
Σ (1940) 1 (3/2 + ) 1940.0 ± 100.0 250.0 - (4.0 ± 2.0)%
Σ (2010) 1 (3/2 − ) 2005.0 ± 14.0 178.0 (5.0 ± 2.0)% (3.0 ± 2.0)%
Σ (2030) 1 (7/2 + ) 2030.0 ± 10.0 180.0 (17.0 - 23.0)% (5.0 - 10.0)%
Σ (2070) 1 (5/2 + ) 2060.0 ± 10.0 200.0 seen
Σ (2080) 1 (3/2 + ) 2090.0 ± 10.0 170.0 seen
Σ (2100) 1 (7/2 − ) 2100.0 ± 50.0 310.0 (1.5 ± 1.0)% (2.0 ± 1.0)%
Σ (2110) 1 (1/2 − ) 2110.0 ± 50.0 310.0 (1.5 ± 1.0)% (2.0 ± 1.0)%
Σ (2230) 1 (3/2 + ) 2240.0 ± 27.0 347.0 (12.0 ± 6.0)% (2.0 ± 1.0)%
Σ (2250) 1 (1/2 + ) 2250.0 ± 20.0 140.0 seen seen
Σ (2455) 1 (3/2 − ) 2455.0 ± 20.0 140.0 (8.0 - 32.0)% (6.0 - 14.0)%
Σ c (2455) 1 (1/2 + ) 2454.0 ± 0.1 1.9 100%
Σ c (2520) 1 (3/2 + ) 2518.4 ± 0.2 14.8 100%
Σ c (2800) 1 (1/2 + ) 2801.0 ± 5.0 75.0 seen

Table A6.Data on two-particle decays of Λ Barions from [8].

Name Λ Barions I (J P ) Effective Mass Resonance m r e s (Mev) WidthΓ (Mev) Decay Σ + π Σ c 0 ( 2455 ) + π +
1 2 3 4
Λ (1405) 0 (1/2 − ) 1405.1 ± 1.3 50.0 100%
Λ (1520) 0 (3/2 − ) 1519.4 ± 0.2 15.7 (42.0 ± 1.0)%
Λ (1600) 0 (1/2 + ) 1600.0 ± 10.0 200.0 (10.0 - 60.0)%
Λ (1670) 0 (1/2 − ) 1670.0 ± 5.0 30.0 (25.0 - 55.0)%
Λ (1690) 0 (3/2 − ) 1690.0 ± 5.0 70.0 (20.0 - 40.0)%
Λ (1710) 0 (1/2 + ) 1713.0 ± 13.0 180.0 (21.0 ± 5.0)%
Λ (1800) 0 (1/2 − ) 1800.0 ± 10.0 200.0 (27.0 ± 6.0)%
Λ (1810) 0 (1/2 + ) 1790.0 ± 10.0 110.0 (16.0 ± 5.0)%
Λ (1820) 0 (5/2 + ) 1820.0 ± 4.0 80.0 (8.0 - 14.0)%
Λ (1830) 0 (5/2 − ) 1825.0 ± 10.0 90.0 (35.0 - 75.0)%
Λ (1890) 0 (3/2 + ) 1890.0 ± 5.0 120.0 (3.0 - 10.0)%
Λ (2050) 0 (3/2 − ) 2056.0 ± 22.0 493.0 (6.0 ± 3.0)%
Λ (2070) 0 (3/2 + ) 2070.0 ± 24.0 370.0 (7.0 ± 3.0)%
Λ (2080) 0 (5/2 − ) 2082.0 ± 13.0 181.0 (5.0 ± 2.0)%
Λ (2100) 0 (7/2 − ) 2100.0 ± 22.0 200.0 5.0%
Λ (2110) 0 (5/2 + ) 2090.0 ± 22.0 250.0 (10.0 - 40.0)%
Λ (2350) 0 (9/2 + ) 2350.0 ± 22.0 150.0 10.0%
Λ c (2595)not included 0 (3/2 − ) 2592.0 ± 0.3 2.6 2453.75 + 139.57=2593.32 > 2592.0
Λ c (2625) 0 (3/2 − ) 2628.0 ± 0.2 <52.0 (2.6 ± 0.4)%
Λ c (2880) 0 (5/2 + ) 2881.6 ± 0.2 51.8 seen
Λ c (2910) 0 (5/2 + ) 2913.8 ± 3.8 51.8 seen
Λ c (2940) 0 (3/2 − ) 2939.6 ± 1.5 20.0 seen

Table A7.Data on two-particle decays of Ξ Barions from [8].

Name Ξ Barion I (J P ) Effective Mass Resonance m r e s (Mev) WidthΓ (Mev) Decay Ξ + π Decay Ξ ( 1530 ) + π
1 2 3 4
Ξ (1530) 1/2 (3/2 − ) 1531.8 ± 0.3 9.1 100% -
Ξ (1620) 1/2 (3/2 + ) 1620.8 ± 6.0 32.0 seen -
Ξ (1690) 1/2 (3/2 − ) 1690.0 ± 10.0 20.0 seen possibly seen
Ξ (1820) 1/2 (3/2 − ) 1823.0 ± 5.0 20.0 small small
Ξ (1950) 1/2 (3/2 − ) 1950.0 ± 15.0 60.0 seen seen
Ξ (2030) 1/2 (3/2 + ) 2025.0 ± 20.0 60.0 small small
Ξ (2500) 1/2 (3/2 − ) 2500.0 ± 150.0 60.0 seen seen
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