TITLE:
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
AUTHORS:
Valeriy Pavlovich Khеn, Alexei Valerevich Khen
KEYWORDS:
Lobachevsky Velocity Space, Resonance Decay Triangles, Oricyclic Cotangent of a Triangle, Hyperbolic Heron Triangle, Lattice Structure of Hyperbolic Heron Triangle, Quantization Resonance Decay
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.4,
April
30,
2026
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
OC
T
M
>500
, new resonances with masses > 10 GeV can be detected.