The free neutron decays into a proton, electron, and antineutrino [8] is shown in Figure 1.

n → p + e − + ν ¯ e

The rest energy ( m n − m p − m e ) c 2 = 782   keV is carried away by e and ν

The transition matrix of the decay is [8] [10]

M = ( G V p ¯   γ μ n − G A p ¯   γ μ γ 5 n ) ( e ¯ γ μ ( 1 − γ 5 ) ν ) δ ( E n − E p − E e − E ν )

from the interaction Hamiltonian [10]

H int = G F V u d ( p ¯   γ μ ( 1 − G A G V γ 5 ) n ) ( e ¯ γ μ ( 1 − γ 5 ) ν )

with G A / G V = 1.255 ± 0.005

E ( G V ) = 1 G V = 296.7   GeV

G F = 1.1663787 ( 6 ) × 10 − 5   GeV − 2 is the Fermi weak coupling constant, V u d = 0.97417 ( 21 ) (V is the CKM-matrix),

and the weak V-constant is

G V = G F V u d = 1.135 × 10 − 5   GeV − 2 , G A = G F V u d λ and λ is the hadronic strong interaction correction.

We compute the neutron decay probability per unit time using Fermi’s golden rule [11] :

Γ i → f = 2 π ℏ | 〈 f | H | i 〉 | 2 ρ ( E f ) , where ρ ( E f ) = final state energy density

or in differential form [11]

d Γ = | M ( k 1 , k 2 , k 3 , k 4 ) | 2 2 m 1 d 3 k 2 ( 2 π ) 3 2 E 2 d 3 k 3 ( 2 π ) 3 2 E 3 d 3 k 4 ( 2 π ) 3 2 E 4 ( 2 π ) 4 δ 4 ( k 1 − k 2 − k 3 − k 4 ) (1)

where k 1 = p n , k 2 = p p , k 3 = p e , k 4 = p ν , m 1 = m n with the (dimensionless) transition matrix M ( k 1 , k 2 , k 3 , k 4 ) = 〈 f | H int | i 〉 δ ( E f − E i ) of the interaction Hamiltonian H int .

Here E_e, p_e, E_n, and p_n are the electron and antineutrino total energy and momentum Δ is the neutron-proton mass difference Δ = 1.29333205 ( 51 ) MeV .

Integration over the antineutrino and electron momenta gives the beta electron energy spectrum

d Γ d E = G V 2 + 3 G A 2 2 π 3 E e p e ( Δ − E e ) 2

Additional integration over electron energy yields

Γ = ( G V 2 + 3 G A 2 ) m e 5 2 π 3 f R

Here f_R is the phase-space term, i.e. the value of the integral over the Fermi energy spectrum, including Coulomb, recoil order, and radiative corrections.

The decay width of the decay becomes [8]

Γ ( n → p   e   ν e ) = ( G V 2 + 3 G A 2 ) m e 5 2 π 3 f R = G F 2 V u d 2 ( 1 + 3 λ 2 ) m e 5 2 π 3 f R (2)

where G V = G F V u d , G A = G F V u d λ , λ = 1.255 , V u d = 0.974 , G F = 1.166 × 10 − 5   GeV − 2

with the phase-space term [9] f R ( ξ ) = 1 60 ( 2 ξ 4 − 9 ξ 2 − 8 ) ( ξ 2 − 1 ) 1 / 2 + 1 4 ξ ln ( ξ + ( ξ 2 − 1 ) 1 / 2 ) , ξ = m n − m p m e ,

f R = 1.6332

here the transition probability per unit time is W = Γ ( n → p   e   ν e ) / ℏ .

The neutron lifetime τ_n becomes

τ n = ℏ / Γ ( n → p   e   ν e ) = 2 π 3 ( G V 2 + 3 G A 2 ) m e 5 f R = 881.5   s

The muon decays into an electron, an electron-antineutrino and a muon-neutrino is shown in Figure 2.

μ → e − + ν ¯ e + ν μ

For the muon decay we derive the formula for the decay width Γ [6] [12] .

The interaction Hamiltonian is the current-current interaction

H int = g 2 2 M W 2 u ¯ 2 ( k 2 , s 2 ) ( γ μ 1 − γ 5 2 ) u 1 ( k 1 , s 1 ) u ¯ 4 ( k 4 , s 4 ) ( γ μ 1 − γ 5 2 ) u 3 ( k 3 , s 3 )

From the transition matrix element

M = g 2 2 M W 2 u ¯ 2 ( k 2 , s 2 ) ( γ μ 1 − γ 5 2 ) u 1 ( k 1 , s 1 ) u ¯ 4 ( k 4 , s 4 ) ( γ μ 1 − γ 5 2 ) u 3 ( k 3 , s 3 ) δ ( E 1 − E 2 − E 3 − E 4 )

with G 2 = g 2 8 M W 2 and G = G_F, g is the weak dimensionless interaction constant,

we obtain after some γ-algebra averaging over the spins and trace-manipulation.

| M ( k 1 , k 2 , k 3 , k 4 ) | 2 = 64 G 2 ( k 1 ⋅ k 3 ) ( k 2 ⋅ k 4 ) and for the decay rate we have Fermi’s golden rule

d Γ = | M ( k 1 , k 2 , k 3 , k 4 ) | 2 2 m d 3 k 2 ( 2 π ) 3 2 E 2 d 3 k 3 ( 2 π ) 3 2 E 3 d 3 k 4 ( 2 π ) 3 2 E 4 ( 2 π ) 4 δ 4 ( k 1 − k 2 − k 3 − k 4 )

d Γ = 64 G 2 2 m ( ( k 1 ⋅ k 3 ) ( k 2 ⋅ k 4 ) ) d 3 k 2 ( 2 π ) 3 2 E k 2 d 3 k 3 ( 2 π ) 3 2 E k 3 d 3 k 4 ( 2 π ) 3 2 E k 4 ( 2 π ) 4 δ 4 ( k 1 − k 2 − k 3 − k 4 )

In the muon rest frame k 1 = ( m , 0 , 0 , 0 ) and ( k 1 ⋅ k 3 ) = m E 3 and with k 1 = k 2 + k 3 + k 4

d Γ = G 2 8 m π 5 ( ( k 2 ⋅ k 4 ) m E 3 ) d 3 k 2 | k → 2 | d 3 k 3 | k → 3 | d 3 k 4 | k → 4 | δ ( m − | k → 2 | − | k → 3 | − | k → 4 | ) δ 3 ( k → 2 + k → 3 + k → 4 )

in spherical coordinates

d Γ = m G 2 | k → 3 | 2 8 π 4 ( m − 2 | k → 3 | ) sin θ   d θ   d | k → 3 | d 3 k 4 ( | k → 3 | 2 + | k → 4 | 2 + 2 | k → 3 | | k → 4 | cos θ ) | k → 4 | δ ( m − | k → 3 + k → 4 | − | k → 3 | − | k → 4 | )

with variable u 2 = | k → 3 | 2 + | k → 4 | 2 + 2 | k → 3 | | k → 4 | cos θ

d Γ = m G 2 | k → 3 | 8 π 4 ( m − 2 | k → 3 | ) d | k → 3 | d 3 k 4 | k → 4 | 2 ∫ d u   δ ( m − u 2 − | k → 3 | − | k → 4 | ) ,

and with E = k₄

we obtain

d Γ d E = m G 2 2 π 3 E 2 ( m 2 − 2 E 3 )

Γ = m 2 G 2 4 π 3 ∫ 0 m / 2 E 2 ( 1 − 4 E 3 m ) d E , Γ = m 5 G 2 192 π 3 (3)

and the decay time τ = ℏ Γ − 1 ,

where m = 0.1056584   GeV

τ = ℏ 192 π 3 ( 0.1056584   GeV ) 5 ( 1.17 × 10 − 5   GeV − 2 ) 2 = ℏ   3.30 × 10 18   GeV ,

or in seconds, multiplied by ℏ = 6.58 × 10 − 25   s ⋅ GeV ,

we obtain the lifetime τ = 2.17   μ s .

The decay modes of the tauon are

τ → μ + ν ¯ μ + ν τ

τ → e + ν ¯ e + ν τ

τ → d + u ¯ + ν τ

τ → s + u ¯ + ν τ

The leptonic modes give a factor f l = 2 , the hadronic modes a factor f h = 3 | V u d | 2 + 3 | V u s | 2 = 2.99 ,

and m τ = 16.82   m μ

t l i f e ( τ ) = t l i f e ( μ ) ( f l + f h ) ( m τ / m μ ) 2 = 3.23 × 10 − 13   s

The particle data for the pions are shown in Table 1.

Charged pion decays

The diagram of charged pion decays is shown in Figure 3.

The π^± mesons have a mass of 139.6 MeV/c² and a mean lifetime of 2.6033 × 10⁻⁸ s. They decay due to the weak interaction. The primary decay mode of a pion, with a branching fraction of 0.999877, is a leptonic decay into a muon and a muon neutrino:

π + → μ + + ν μ

π − → μ − + ν ¯ μ

The pion-muon decay width is [14]

Γ = f π 2 G 2 8 π m π 3 m μ 2 ( 1 − m μ 2 m π 2 ) 2

where f π is the (dimensionless) pion decay factor, f π = 130.41   MeV m π = 0.934 .

The second most common decay mode of a pion, with a branching fraction of 0.000123, is also a leptonic decay into an electron and the corresponding electron antineutrino. This “electronic mode” was discovered at CERN in 1958

π + → e + + ν e

π − → e − + ν ¯ e

The suppression of the electronic decay mode with respect to the muonic one is given approximately (up to a few percent effect of the radiative corrections) by the ratio of the half-widths of the pion-electron and the pion-muon decay reactions:

R π = ( m e / m μ ) 2 ( m π 2 − m e 2 m π 2 − m μ 2 ) 2 = 1.283 × 10 − 4

and is a spin effect known as helicity suppression.

Also observed, for charged pions only, is the very rare “pion beta decay” (with branching fraction of about 10⁻⁸) into a neutral pion, an electron and an electron antineutrino (or for positive pions, a neutral pion, a positron, and electron neutrino).

π + → π 0 + e + + ν e

π − → π 0 + e − + ν ¯ e

The rate at which pions decay is a prominent quantity in many sub-fields of particle physics, such as chiral perturbation theory. This rate is parametrized by the pion decay constant (ƒ_π), related to the wave function overlap of the quark and antiquark, which is about 130 MeV.

Neutral pion decays

The π⁰ meson has a mass of 135.0 MeV/c² and a mean lifetime of 8.4 × 10⁻¹⁷ s. It decays via the electromagnetic force, which explains why its mean lifetime is much smaller than that of the charged pion (which can only decay via the weak force).

The neutral pion decay is shown in Figure 4.

The dominant π⁰ decay mode (anomaly-induced neutral pion decay), with a branching ratio of BR = 0.98823, is into two photons:

π 0 → γ + γ

The second largest π⁰ decay mode (BR = 0.01174) is the Dalitz decay (named after Richard Dalitz), which is a two-photon decay with an internal photon conversion resulting a photon and an electron-positron pair in the final state:

π 0 → γ + e − + e +

The third largest established decay mode (BR = 3.34 × 10⁻⁵) is the double Dalitz decay, with both photons undergoing internal conversion which leads to further suppression of the rate:

π 0 → e − + e + + e − + e +

The fourth largest established decay mode is the loop-induced and therefore suppressed (and additionally helicity-suppressed) leptonic decay mode (BR = 6.46 × 10⁻⁸):

π 0 → e − + e +

The Lagrangian is [6] [15]

L = i g   ψ ¯ ( x ) γ 5 σ → ⋅ ϕ ( x ) ψ ( x )

with pion ϕ ( x ) = ( ϕ 1 , ϕ 2 , ϕ 3 ) , nucleon ψ ( x ) = ( ψ p , ψ n ) , and Pauli-matrix-vector σ → , explicitly

L = i g   ( ψ ¯ p , ψ ¯ n ) γ 5 ( ϕ 3 ϕ 1 − i   ϕ 2 ϕ 1 + i   ϕ 2 ϕ 3 ) ( ψ p , ψ n )   ( ϕ 1 , ϕ 2 , ϕ 3 )

with Feynman diagrams shown in Figure 5.

and with the corresponding hadronic transformations

p → π 0 p , n → π 0 n , n → π − p , p → π + n

Kaons exist in charged form ch-kaon = ( K + , K − ) and neutral form n-kaon = ( K 0 , K ¯ 0 ) , where n-kaon appears in nature as a symmetric and antisymmetric mixture of d and s: the long-lived kaon-L and the short-lived kaon-S.

The particle data for the kaons are shown in Table 2, their decay reactions in Table 3, and their quark decay processes in Figure 6.

K0 decay and CP-violation

The full kaon-L and kaon-S contain a CP-violating term:

K L = 1 1 + ε 2 ( K 0 + K ¯ 0 2 + ε K 0 − K ¯ 0 2 )

K S = 1 1 + ε 2 ( K 0 − K ¯ 0 2 + ε K 0 + K ¯ 0 2 )

ε = 2.25 × 10 − 3 CP violation factor.

In [17] a semi-empirical formula for the transition matrix element A + + in Γ ( K + ( k ) → π +   ( p 1 ) π + ( p 2 )   π − ( p 3 ) ) is derived.

First, the kinematic momentum variables s₀, s₁, s₂, s₃ are introduced

s 1 = ( k − p 1 ) 2 , s 2 = ( k − p 2 ) 2 , s 3 = ( k − p 3 ) 2 , s 0 = ( m 2 + m 1 2 + m 2 2 + m 3 2 ) / 3

then, the Dalitz plot variables x = s 2 − s 1 ( m 1 2 + m 2 2 ) / 2 , y = s 3 − s 0 m 3 2 are defined.

We obtain for A + + (x,y) the expression

A + + − ( x , y ) = ( − 2 α 1 + α 3 ) + ( − β 1 + 1 2 β 3 − 3 γ 3 ) y   − ( 2 ζ 1 + 2 ζ 3 ) ( y 2 + 1 3 x 2 ) + ( ξ 1 + ξ 3 − ξ 3 ' ) ( y 2 − 1 3 x 2 )

with the constants:

α 1 = ( 91.71 ± 0.32 ) × 10 − 8 , α 2 = ( − 7.36 ± 0.47 ) × 10 − 8 , β 1 = ( − 25.68 ± 0.27 ) × 10 − 8 , β 2 = ( − 2.43 ± 0.41 ) × 10 − 8 , γ 3 = ( 2.26 ± 0.23 ) × 10 − 8

ζ 1 = ( − 0.47 ± 0.15 ) × 10 − 8 , ζ 3 = ( − 0.21 ± 0.08 ) × 10 − 8 , ξ 1 = ( − 1.51 ± 0.30 ) × 10 − 8 , ξ 3 = ( − 0.12 ± 0.17 ) × 10 − 8 , ξ 3 ' = ( − 0.21 ± 0.51 ) × 10 − 8

and m 2 = m 3 = m 1 , s 3 = ( p 1 + p 2 ) 2 = ( E 1 + E 2 ) 2 − ( p → 1 + p → 2 ) 2 .

We obtain the following expression for the differential transition width from Fermi’s golden rule:

d Γ = | M ( k , p 1 , p 2 , p 3 ) | 2 2 m d 3 p 1 ( 2 π ) 3 2 E 1 d 3 p 2 ( 2 π ) 3 2 E 2 d 3 p 3 ( 2 π ) 3 2 E 3 ( 2 π ) 4 δ 4 ( k − p 1 − p 2 − p 3 )

or, with Dalitz variables

d Γ = | M ( k , p 1 , p 2 , p 3 ) | 2 2 m d 3 p 1 ( 2 π ) 3 2 m 1 2 + p 1 2 d 3 p 2 ( 2 π ) 3 2 m 2 2 + p 2 2 d 3 p 3 ( 2 π ) 3 2 m 3 2 + p 3 2 ( 2 π ) 4 δ 4 ( k − p 1 − p 2 − p 3 )

we choose k → = 0 , k 0 = m , i.e. p → 3 = − ( p → 1 + p → 2 )

s 1 = ( k − p 1 ) 2 = ( m − E 1 ) 2 − p → 1 2 = ( m − E 1 ) 2 − ( E 1 2 − m 1 2 ) = m 2 + m 1 2 − 2 m E 1

s 2 = ( m − E 2 ) 2 − p → 2 2 = m 2 + m 1 2 − 2 m E 2

p → 1 2 = E 1 2 − m 1 2 , p → 2 2 = E 2 2 − m 1 2

p → 2 2 = p → 1 2 + p → 2 2 + 2 p → 1 p → 2 cos θ 12 = E 1 2 + E 2 2 − 2 m 1 2 + 2 E 1 2 − m 1 2 E 2 2 − m 1 2 cos θ 12 , m = E 1 + E 2 + E 3

s 3 = ( p 1 + p 2 ) 2 = ( m − ( E 1 + E 2 ) ) 2 − ( p → 1 + p → 2 ) 2 = ( m − ( E 1 + E 2 ) ) 2 − ( E 1 2 + E 2 2 − 2 m 1 2 + 2 E 1 2 − m 1 2 E 2 2 − m 1 2 cos θ 12 )

s 3 = ( p 1 + p 2 ) 2 = 2 E 1 E 2 − 2 m ( E 1 + E 2 ) + m 2 + 2 m 1 2 − 2 E 1 2 − m 1 2 E 2 2 − m 1 2 cos θ 12

E 1 = m 1 2 + p → 1 2 , E 2 = m 1 2 + p → 2 2 ,

E 3 = m 1 2 + ( p → 1 + p → 2 ) 2 = ( E 1 2 + E 2 2 − m 1 2 + 2 E 1 2 − m 1 2 E 2 2 − m 1 2 cos θ 12 ) 1 / 2

x = s 2 − s 1 m 1 2 = 2 m ( E 1 − E 2 ) m 1 2

y = s 3 − s 0 m 1 2 = 2 E 1 E 2 − 2 m ( E 1 + E 2 ) + 2 m 2 / 3 + m 1 2 − 2 E 1 2 − m 1 2 E 2 2 − m 1 2 cos θ 12 m 1 2

We insert | M ( k , p 1 , p 2 , p 3 ) | 2 = 1 8 π | A + + − ( x , y ) | 2

and calculate Γ as an integral over | p 1 | = | p → 1 | , | p 2 | = | p → 2 | θ₁₂, integration d 3 p 3 δ 3 ( p → 3 + p → 1 + p → 2 ) cancels out

d Γ = | A + + − ( x , y ) | 2 2 m 4 π ( E 1 2 − m 1 2 ) d | p 1 | ( 2 π ) 3 2 E 1 2 π ( E 2 2 − m 1 2 ) d | p 2 | ( 2 π ) 3 2 E 2 1 ( 2 π ) 3 2 E 3 ( 2 π ) 4 δ ( m − E 1 − E 2 − E 3 )

and changing to E₁, E₂, θ₁₂: d | p 1 | = E 1 d E 1 E 1 2 − m 1 2 , d | p 2 | = E 2 d E 2 E 2 2 − m 1 2 ,

| p 1 | = E 1 2 − m 1 2 , | p 2 | = E 2 2 − m 1 2

d Γ = | A + + − ( x , y ) | 2 8 m ( 2 π ) 3 E 1 2 − m 1 2 d E 1 E 2 2 − m 1 2 d E 2   × sin θ 12 d θ 12 ( E 1 2 + E 2 2 − 2 m 1 2 + 2 E 1 2 − m 1 2 E 2 2 − m 1 2 cos θ 12 ) 1 / 2   × δ ( m − E 1 − E 2 − ( E 1 2 + E 2 2 − 2 m 1 2 + 2 E 1 2 − m 1 2 E 2 2 − m 1 2 cos θ 12 ) 1 / 2 )

we solve dΓ for E₂ [2] :

E 2 = − 2 E 1 2 m − m 3 − m m 1 2 + E 1 ( 3 m 2 + m 1 2 ) + cos ( θ 12 ) ( ( E 1 2 − m 1 2 ) ( ( m 2 − m 1 2 ) ( 4 E 1 2 − 4 E 1 m + m 2 − m 1 2 ) + 4 m 1 2 ( E 1 2 − m 1 2 ) cos ( θ 12 ) 2 ) ) 1 / 2 − 2 ( E 1 − m ) 2 + 2 ( E 1 2 − m 1 2 ) cos ( θ 12 ) 2

and m − E 1 − E 2 = E 1 2 + E 2 2 − m 1 2 + 2 E 1 2 − m 1 2 E 2 2 − m 1 2 cos θ 12

( m − ( E 1 + E 2 ) ) 2 − E 1 2 − E 2 2 + m 1 2 2 E 1 2 − m 1 2 cos θ 12 = E 2 2 − m 1 2

now we carry out the integration over E₂ with the delta-function:

Γ = ∫ | A + + − ( x , y ) | 2 8 m ( 2 π ) 3 d E 1 ( m − ( E 1 + E 2 ) ) 2 − E 1 2 − E 2 2 + m 1 2 2 cos θ 12 sin θ 12 d θ 12 m − ( E 1 + E 2 )

and after simplification

Γ = ∫ | A + + − ( x , y ) | 2 8 m ( 2 π ) 3 E 2 2 − m 1 2 E 1 2 − m 1 2 m − ( E 1 + E 2 ) sin θ 12   d θ 12 d E 1 (4)

The integration boundary in E₁ is m 1 ≤ E 1 ≤ ( 1 + f e 1 b 1 ( m , m 1 , θ 12 ) ) m 1 , in θ₁₂ 0 ≤ θ 12 ≤ π

where f e 1 b 1 ( m , m 1 , θ 12 ) = | p 1 | m 1 is the relative momentum, at which E₂ becomes

complex

f e 1 b 1 ( m , m 1 , θ 12 ) = 2 m 3 m 1 − 4 m 2 m 1 2 − 2 m m 1 3 + 4 m 1 4 sin ( θ 12 ) 2 − 2 ( m 2 m 1 4 ( m 2 ( 1 − cos ( 2 θ 12 ) ) + 2 m 1 2 + 6 m 1 4 cos ( 2 θ 12 ) ) + m 1 8 ( cos ( 4 θ 12 ) − cos ( 2 θ 12 ) ) ) 1 / 2 4 ( m 2 m 1 2 − m 1 4 sin ( θ 12 ) 2 )

Numerical integration yields for m 1 = m ( π + ) = 0.139   GeV , m = m ( K + ) = 0.493   GeV [2] .

Γ ( m 1 , m ) = 0.033 × 10 − 16   GeV , the measured decay width is 0.0297 × 10⁻¹⁶ GeV (see below).

We use the momentum notations Γ ( P 0 ( k , m ) → P ( p 1 , m 1 ) P ( p 2 , m 2 ) P ( p 3 , m 3 ) ) [14] [18] .

We start again with Fermi’s golden rule:

d Γ = | M ( k , p 1 , p 2 , p 3 ) | 2 2 m d 3 p 1 ( 2 π ) 3 2 E 1 d 3 p 2 ( 2 π ) 3 2 E 2 d 3 p 3 ( 2 π ) 3 2 E 3 ( 2 π ) 4 δ 4 ( k − p 1 − p 2 − p 3 ) (5)

we choose k → = 0 , k 0 = m , i.e. p → 3 = − ( p → 1 + p → 2 )

p → 1 2 = E 1 2 − m 1 2 , p → 2 2 = E 2 2 − m 2 2

p → 3 2 = p → 1 2 + p → 2 2 + 2 p → 1 p → 2 cos θ 12 = E 1 2 + E 2 2 − m 1 2 − m 2 2 + 2 E 1 2 − m 1 2 E 2 2 − m 2 2 cos θ 12 , m = E 1 + E 2 + E 3

E 1 = m 1 2 + p → 1 2 , E 2 = m 2 2 + p → 2 2 ,

E 3 = m 3 2 + ( p → 1 + p → 2 ) 2 = ( E 1 2 + E 2 2 + m 3 2 − m 1 2 − m 2 2 + 2 E 1 2 − m 1 2 E 2 2 − m 2 2 cos θ 12 ) 1 / 2

now we calculate Γ as an integral over | p 1 | = | p → 1 | , | p 2 | = | p → 2 | θ₁₂, integration d 3 p 3 δ 3 ( p → 3 + p → 1 + p → 2 ) cancels out

d Γ = | M ( k , p 1 , p 2 , p 3 ) | 2 2 m 4 π ( E 1 2 − m 1 2 ) d | p 1 | ( 2 π ) 3 2 E 1 2 π ( E 2 2 − m 2 2 ) d | p 2 | ( 2 π ) 3 2 E 2 1 ( 2 π ) 3 2 E 3 ( 2 π ) 4 δ ( m − E 1 − E 2 − E 3 )

and changing to E₁, E₂, θ₁₂: d | p 1 | = E 1 d E 1 E 1 2 − m 1 2 , d | p 2 | = E 2 d E 2 E 2 2 − m 2 2 ,

| p 1 | = E 1 2 − m 1 2 , | p 2 | = E 2 2 − m 2 2

d Γ = | M ( k , p 1 , p 2 , p 3 ) | 2 8 m ( 2 π ) 3 E 1 2 − m 1 2 d E 1 E 2 2 − m 2 2 d E 2   × sin θ 12 d θ 12 ( E 1 2 + E 2 2 + m 3 2 − m 1 2 − m 2 2 + 2 E 1 2 − m 1 2 E 2 2 − m 2 2 cos θ 12 ) 1 / 2   × δ ( m − E 1 − E 2 − ( E 1 2 + E 2 2 + m 3 2 − m 1 2 − m 2 2 + 2 E 1 2 − m 1 2 E 2 2 − m 2 2 cos θ 12 ) 1 / 2 )

we solve for E₂ [2] :

E 2 = ( m ( − 2 E 1 2 − m 2 − m 1 2 + m 3 2 ) + E 1 ( 3 m 2 + m 1 2 + m 2 2 − m 3 2 ) + cos ( θ 12 ) ( m 4 + 2 m 2 m 1 2       + m 1 4 − 2 m 2 m 2 2 + m 2 4 − 2 m 2 m 3 2 − 2 ( m 1 2 + m 2 2 ) m 3 2 + m 3 4 + 2 E 1 2 ( 2 m 2 − m 2 2 )   − 4 E 1 m ( m 2 + m 1 2 − m 2 2 − m 3 2 ) + 2 m 2 2 ( E 1 2 − m 1 2 ) cos ( 2 θ 12 ) ) 1 / 2 )   / ( − 2 ( E 1 − m ) 2 + 2 ( E 1 2 − m 1 2 ) cos ( θ 12 ) 2 )

and m − E 1 − E 2 = ( E 1 2 + E 2 2 + m 3 2 − m 1 2 − m 2 2 + 2 E 1 2 − m 1 2 E 2 2 − m 2 2 cos θ 12 ) 1 / 2

( m − ( E 1 + E 2 ) ) 2 − E 1 2 − E 2 2 − m 3 2 + m 1 2 + m 2 2 2 E 1 2 − m 1 2 cos θ 12 = E 2 2 − m 2 2

now we carry out the integration over E₂ with the delta-function:

Γ = ∫ | M ( k , p 1 , p 2 , p 3 ) | 2 8 m ( 2 π ) 3 d E 1 ( m − ( E 1 + E 2 ) ) 2 − E 1 2 − E 2 2 − m 3 2 + m 1 2 + m 2 2 2 cos θ 12 sin θ 12 d θ 12 m − ( E 1 + E 2 )

and after simplification

Γ = ∫ | M ( k , p 1 , p 2 , p 3 ) | 2 8 m ( 2 π ) 3 E 2 2 − m 1 2 E 1 2 − m 2 2 m − ( E 1 + E 2 ) sin θ 12   d θ 12 d E 1

setting | M ( k , p 1 , p 2 , p 3 ) | = 1 we obtain the partial kinematic factor for 3-body

decay I Γ 3 ( m 1 m , m 2 m , m 3 m )

Γ = ∫ 1 8 m ( 2 π ) 3 E 2 2 − m 1 2 E 1 2 − m 2 2 m − ( E 1 + E 2 ) sin θ 12   d θ 12 d E 1 = m   I Γ 3 ( m , m 1 , m 2 , m 3 ) (6)

The integration boundary in E₁ is m 1 ≤ E 1 ≤ ( 1 + f e 1 b 1 ( m , m 1 , m 2 , m 3 , θ 12 ) ) m 1 , in θ₁₂ 0 ≤ θ 12 ≤ π

where f e 1 b 1 ( m , m 1 , m 2 , m 3 , θ 12 ) = | p 1 | m 1 is the relative momentum, at which E₂

becomes complex

f e 1 b 1 ( m , m 1 , m 2 , m 3 , θ 12 ) = ( 2 ( m 3 m 1 − 2 m 2 m 1 2 + m m 1 3 − m m 1 ( m 2 2 + m 3 2 ) + m 1 2 m 2 2 − m 1 2 m 2 2 cos ( 2 θ 12 ) )           − ( − ( m 2 ( m 2 − 4 m m 1 + 6 m 1 2 − 2 m 2 2 − 2 m 3 2 ) − 4 m m 1 3 + m 1 4 + 4 m m 1 ( m 2 2 + m 3 2 )    

    − 2 m 1 2 ( m 2 2 + m 3 2 ) + m 2 4 + m 3 4 − 2 m 2 2 m 3 2 ) ( 4 m 2 m 1 2 − 2 m 1 2 m 2 2 + 2 m 1 2 m 2 2 cos ( 2 θ 12 ) )       + 4 ( − m 3 m 1 + 2 m 2 m 1 2 − m m 1 3 + m m 1 ( m 2 2 + m 3 2 ) + m 1 2 m 2 2 ( cos ( 2 θ 12 ) − 1 ) ) 2 ) 1 / 2 )       / ( 2 ( 2 m 2 m 1 2 − m 1 2 m 2 2 + m 1 2 m 2 2 cos ( 2 θ 12 ) ) ) (6a)

The kinematic factor I Γ 3 ( m , m 1 , m 2 , m 3 ) can be calculated numerically.

The total kinematic factor results from I Γ 3 ( m , m 1 , m 2 , m 3 ) by symmetrization over all 6 index permutations

I Γ 3 s ( m , m 1 , m 2 , m 3 ) = ( I Γ 3 ( m , m 1 , m 2 , m 3 ) + I Γ 3 ( m , m 1 , m 3 , m 2 ) + ⋯ ) / 6

Here is the plot of I Γ 3 s ( 1 , m 1 , 0.1 , 0.1 ) 10 6 shown in Figure 7.

Example: Γ ( μ → e   ν ¯ e ν μ ) with kinematic factor: I Γ 3 s ( m ( μ ) , m ( e ) , 0 , 0 ) = 0.3835 .

We use the momentum notations Γ ( P 0 ( k , m ) → P ( p 1 , m 1 ) P ( p 2 , m 2 ) ) [14] [18] .

We start again with Fermi’s golden rule for 2-body decay [11] :

d Γ = m | M ( k , p 1 , p 2 ) | 2 2 d 3 p 1 ( 2 π ) 3 2 E 1 d 3 p 2 ( 2 π ) 3 2 E 2 ( 2 π ) 4 δ 4 ( k − p 1 − p 2 ) (7)

we choose k → = 0 , k 0 = m , i.e. p → 2 = − p → 1

p → 1 2 = E 1 2 − m 1 2 , p → 2 2 = E 2 2 − m 2 2

m = E 1 + E 2

E 1 = m 1 2 + p → 1 2 , E 2 = m 2 2 + p → 1 2 = E 1 2 + m 2 2 − m 1 2 = m − E 1 , δ 4 ( k − p 1 − p 2 ) = δ ( m − E 1 − E 2 ) δ 3 ( p → 1 + p → 2 )

now we calculate Γ as an integral over | p 1 | = | p → 1 | , integration d 3 p 2 δ 3 ( p → 1 + p → 2 ) cancels out

and changing to E 1 : d | p 1 | = E 1 d E 1 E 1 2 − m 1 2 , | p 1 | = E 1 2 − m 1 2

d Γ = m | M ( k , p 1 , p 2 ) | 2 8 ( 2 π ) 2 4 π E 1 2 − m 1 2 d E 1 E 1 2 + m 2 2 − m 1 2 δ ( m − E 1 − E 2 )

from E 1 2 + m 2 2 − m 1 2 = m − E 1 we obtain the solution [2]

E 10 = m 2 + m 1 2 − m 2 2 2 m ,

the integration d E 1 δ ( m − E 1 − E 2 ) cancels out and we obtain, setting | M ( k , p 1 , p 2 , p 3 ) | = 1

Γ = m 4 ( 2 π ) E 10 2 − m 1 2 E 10 2 + m 2 2 − m 1 2 = 1 8 m π ( m 4 + m 1 4 + m 2 4 − 2 m 2 m 1 2 − 2 m 2 m 2 2 − 2 m 1 2 m 2 2 ) 1 / 2 ( m 4 + m 1 4 + m 2 4 − 2 m 2 m 1 2 + 2 m 2 m 2 2 − 2 m 1 2 m 2 2 ) 1 / 2 (8)

we obtain the kinematic factor for 2-body decay I Γ 2 ( m , m 1 )

I Γ 2 ( m , m 1 , m 2 ) = Γ m = 1 8 π ( m 4 + m 1 4 + m 2 4 − 2 m 2 m 1 2 − 2 m 2 m 2 2 − 2 m 1 2 m 2 2 ) 1 / 2 ( m 4 + m 1 4 + m 2 4 − 2 m 2 m 1 2 + 2 m 2 m 2 2 − 2 m 1 2 m 2 2 ) 1 / 2 (8a)

The total kinematic factor for 2-body decay results from the symmetrized I Γ 2 ( m , m 1 )

I Γ 2 s ( m , m 1 , m 2 ) = I Γ 2 ( m , m 1 ) + I Γ 2 ( m , m 2 ) 2

As an example, here is the plot I Γ 2 s ( 1 , m 1 , 0.5 ) shown in Figure 8.

Example: Γ ( π → μ   ν ) , with kinematic factor: I Γ 2 s ( m ( π ) , m ( μ ) , 0 ) = 0.0251 .

The phenomenological decay formula

The phenomenological formula for the decay width is [1]

Γ = G ˜ 2 m i k | P l m ( x ) | 2 = G 2 C 1 m i k | P l m ( x ) | 2 (9)

where P l m ( x ) Legendre polynomial m = l or m = l + 1 , l = isospin I, x = m f m i mass ratio, G ˜ = G C 1 with G = interaction constant, m i is the initial mass.

The constant C₁ is process-dependent, standard value C 1 = 4 π , with exceptions for muon: C 1 = 192 π 3 , and for neutron: C 1 = 2 π 3 .

The G constants are: for kaons G 2 = g 1 2 = 2.06 × 10 − 14 , for pions G 2 = g 0 2 = 2.18 × 10 − 14 , for leptonic decays A → A ′ e     ν ¯ e ( Δ S = 0 )

G = G l = 1.02 × 10 − 5 ( 1 m p 2 ) = 1.16 × 10 − 5   GeV − 2 , for hyperons

G = g h 1 = 5.81 × 10 − 7   GeV or G = g h 2 = 1.4 × 10 − 8   GeV − 3 / 2 .

The interaction constants for hyperons in [1] were given by

G = g h = 6.2 × 10 − 7 ( m p m i ) or G = g h ' = 1.28 × 10 − 8 ( m i m p ) 3 / 2 , they have been corrected, since G must not depend on initial mass m i , also the power coefficient k

and the data tables were corrected accordingly.

The power coefficient is

k = 1 for a dimensionless G ˜ , like in pion decay

Γ ( π → μ ν μ ) = G ˜ 2 m i x 2   ( 1 − x 2 ) 2 = G 2 m i 4   π x 2 ( 1 − x 2 ) 2 , G 2 = g 0 2 = 2.18 × 10 − 14

k = 5 for a dimensional G ˜ , [ G ˜ ] = GeV − 2 , like in muon decay

Γ ( μ → e   ν e   ν μ ) = G ˜ 2 m i 5 ( 1 − x 2 ) 4 = G F 2 m i 5 192 π 3 ( 1 − x 2 ) 4

k = 3 for a dimensional G ˜ , [ G ˜ ] = GeV − 1 , like in π⁰ decay Γ ( π 0 → γ   γ ) = G ˜ 2 m i 3 ( 1 − x 2 ) 3

k = −1 for a dimensional G ˜ , [ G ˜ ] = GeV , for kaon decays with G ˜ ≅ g h 1

k = 4 for a dimensional G ˜ , [ G ˜ ] = GeV − 3 / 2 , for non-kaon hyperon decays with G ˜ ≅ g h 2

so in all cases [ Γ ] = m , i.e. the dimension is energy, as it must be.

The extended isospin I [5] [19] includes higher generation quarks, I ( s ) = I ( c ) = 1 / 2 and I ( l ) = 1 for leptons l as well as I ( γ ) = 1 for the photon.

The extended isospin has the following values shown in Table 4.

The angular momentum in decay width: l = | Δ I | = | I i ± I f | is the difference or sum of the initial and final isospin.

The interaction energy

The interaction energy m_X is the (excitation) energy of the mediating virtual exchange boson (for pure weak decays: W or Z-boson).

We can deduce the interaction energy from the phenomenological formula in

the following way.

We consider the phenomenological formula for k = 5 (e.g. muon decay) in the

form Γ = G ˜ 2 m i 5 ( 1 − x 2 ) 4 , where x = m f m i .

We separate the kinematic factor as in chap. 2.8 and 2.9 in the form Γ = | M | 2 I Γ m i .

So we conclude I Γ = ( 1 − x 2 ) 4 , M 2 = G ˜ 2 m i 4 .

In analogy to the weak interaction mediated by the W-boson (as in muon

decay chap.3.3) G ˜ = G F 2 = g 2 8 M W 2 (g weak interaction constant), we make the

ansatz for the matrix element M and the interaction energy m_X

M = m i 2 8 m X 2

The decay width with the matrix element M and the kinematic I Γ factor becomes then

Γ = | M | 2 I Γ m i = ( m i 2 8 m X 2 ) 2 I Γ m i or in general m X m i = f I ( m i Γ ) 1 / 4 , where f I = I Γ 1 / 4 2 2 .

From this formula we can derive a general semi-empirical formula for the interaction energy m_X, where the kinematic factor is dimensionless [ I Γ ] = 1 ,

m X = m i 2 2 ( I Γ m i Γ ) 1 / 4 (10)

where I Γ = I Γ 2 ( m 1 , m 2 , m 3 ) for 2-body decay,

I Γ = I Γ 3 ( m 1 , m 2 , m 3 , m 4 ) for 3-body decay (see 1.8, 1.9)

For k = 1 m i 4 m X 4 = 64   G ˜ 2 I Γ | P l m ( x ) | 2 with the phenomenological formula

Γ = G ˜ 2 m i | P l m ( x ) | 2 (10a)

For k = 5 1 m X 4 = 64   G ˜ 2 I Γ | P l m ( x ) | 2 with Γ = G ˜ 2 m i 5 ( 1 − x 2 ) 4 = G F 2 m i 5 192 π 3 ( 1 − x 2 ) 4

(10b)

For k = 3 1 m X 2 = 64   G ˜ 2 I Γ | P l m ( x ) | 2 with Γ = G ˜ 2 m i 3 ( 1 − x 2 ) 3 (10c)

Laplace operators in spherical coordinates reads [1]

Δ ψ = 1 r 2 ( ∂ ∂ r ( r 2 ∂ ψ ∂ p ) + 1 sin θ ∂ ∂ θ ( sin θ ∂ ψ ∂ θ ) + 1 sin 2 θ ∂ 2 ψ ∂ φ 2 )

quantum kinetic energy is E k i n ( ψ ) = − ℏ 2 Δ ψ 2 m

with the ansatz ψ = R ( r ) Θ ( θ , φ )

for rigid rotator r=const and we obtain the angular momentum spectrum

1 sin θ ∂ ∂ θ ( sin θ ∂ ψ ∂ θ ) + 1 sin 2 θ ∂ 2 ψ ∂ φ 2 = l ( l + 1 ) with kinetic energy E k i n = L 2 2 I = h 2 l ( l + 1 ) 8 π 2 I

where l is the angular momentum quantum number

with eigenfunctions

ψ l m = N l m P l | m | ( cos θ ) exp ( i m φ ) , | ψ l m | 2 = 1 4 π ( l − | m | ) ! ( l + | m | ) ! ( 2 l + 1 ) P l | m | ( cos θ ) 2

and decay width Γ = A | ψ l m | 2 = A 4 π ( l − | m | ) ! ( l + | m | ) ! ( 2 l + 1 ) P l | m | ( cos θ ) 2

with x = cos θ = p z | p → | and x = ∑ m f m i , m = l or m = l − 1

Legendre functions P l m ( x ) = 1 Γ ( 1 − m ) ( 1 + x 1 − x ) m / 2 F 2 1 ( − l , l + 1 ; 1 − m ; 1 − x 2 )

with hypergeometric function F 2 1

associated Legendre polynomials

P l m ( x ) = ( − 1 ) m 2 l ( 1 − x 2 ) m / 2 ∑ k = m l k ! ( k − m ) ! x k − m ( l k ) ( ( l + k − 1 ) / 2 l )

P l + 1 l ( x ) = x ( 2 l + 1 ) P l l ( x ) , P l l ( x ) = ( − 1 ) l ( 2 l − 1 ) ! ! ( 1 − x 2 ) 1 / 2 , P l + 1 l + 1 ( x ) = − ( 2 l + 1 ) 1 − x 2 P l l ( x )

P 0 0 ( x ) = 1 , P 1 0 ( x ) = x , P 1 − 1 ( x ) = 1 2 ( 1 − x 2 ) 1 / 2 , P 1 1 ( x ) = − ( 1 − x 2 ) 1 / 2 ,

P 2 0 ( x ) = 1 2 ( 3 x 2 − 1 )

and the decay width becomes Γ l , α ( x ) = C x 2 ( α − 1 ) ( 1 − x 2 ) l , α = 1 , 2

In the following, the angular momentum L is replaced by the isospin I.

The analytical formula from the Feynman diagram is

Γ ( μ → e   ν e   ν μ ) = G F 2 m μ 5 192 π 3 [6] , G_F Fermi-constant (11)

exact formula with corrections [10]

Γ ( μ → e   ν ¯ e   ν μ ) = G F 2 m μ 5 192 π 3 f ( m e 2 m μ 2 ) ( 1 + f r c ( α π ) ) ( 1 + 3 5 m μ 2 m W 2 + ⋯ ) (11a)

f r c ( x ) = x 2 ( 25 4 − π 2 ) ( 1 + x ( 2 3 log m μ m e − 3.7 ) + x 2 ( 4 9 log 2 m μ m e − 2 log m μ m e + C ) + ⋯ ) (11b)

and the phenomenological formula

Γ ( μ → e   ν e   ν μ ) = G F 2 m μ 5 192 π 3 ( 1 − m e 2 m μ 2 ) 4 = G F 2 m μ 5 192 π 3 ( 1 − x 2 ) 4 [1] , l = 4 (11c)

This is the general formula for a leptonic weak 3-body decay, setting initial mass m i = m μ .

The (charged) weak interaction in the Feynman-Gell-Mann form reads

H w e a k = G F 2 J μ J μ † [6] (9.1), where J_μ is the charged leptonic-hadronic current

J μ = L μ + H μ and L μ ( x ) = 2 e ¯ L ( x ) γ μ ν e ( x ) + ⋯ is the leptonic current,

H μ ( x ) = 2 u ¯ L ( x ) γ μ d L ( x ) + ⋯ is the analogous hadronic current.

In the standard model, the (charged) weak interaction is mediated by the massive W-boson W_μ with mass M_W for the charged current, with the Lagrangian

L w e a k = − g 2 2 ( J μ W ¯ μ + W μ J ¯ μ ) (12)

where the effective interaction constant is G = G F 2 = g 2 8 m W 2 .

We can use the total current and use an excited intermediate W-boson, which includes the hadronic part, with the total mass m_X > M_W and calculate it from

the effective measured coupling constant G = G F 2 , setting g = 1:

G 2 = 1 64 m X 4

The isospin numbers are l = Δ I = I f + I i = 3 + 1 = 4 and m = l = 4 .

The analytical formula from the Feynman diagram is

Γ ( π → μ ν μ ) = G F 2 m π 8 π f π 2 V u d 2 m μ 2 ( 1 − m μ 2 m π 2 ) 2 ≈ G F 2 m π 5 8 π m μ 2 m π 2 ( 1 − m μ 2 m π 2 ) 2 [ [11] , (13.26)] (13)

G_F Fermi-constant, and the phenomenological formula

Γ ( π → μ ν μ ) = G 2 m π 4 π m μ 2 m π 2 ( 1 − m μ 2 m π 2 ) 2 = G 2 m π 4 π x 2 ( 1 − x 2 ) 2 [1] (13a)

The (charged) weak interaction has the form

H w e a k = G F 2 J λ ( μ , ν μ ) J λ ( u , d ) with the leptonic current

J λ ( μ , ν μ ) = μ ¯ γ λ ( 1 − γ 5 ) ν and the hadronic current J λ ( u , d ) = u ¯ γ λ d for π − = u ¯ d ( [6] , (13.6)).

Using the same procedure as above with the excited intermediate W-boson, we calculate M_X from the above two formulas:

G F 2 8 π ( m π m X ) 4 = G 2 4 π and G F 2 = g F 2 8 M X 2

and setting g_F = 1 and initial mass m i = m π

we obtain ( m i m X ) 4 = 64 G 2 .

The isospin numbers are l = Δ I = I f + I i = 2 + 1 = 3 and m = l − 1 = 2 .

The kaon-pion decay is shown in Figure 9.

The generalized and isospin-adapted 3-body semi-leptonic formula (from the muon) decay is

Γ ( K ± → π ±   π +   π − ) = g 4 m i 32 × 192 π 3 ( m i 4 m X 4 ) ( 1 − m f 2 m i 2 ) 4 (14)

and the phenomenological formula

Γ ( K ± → π ±   π +   π − ) = G 2 m K + 4 π ( 1 − m π 2 m K + 2 ) 2 [1] , where G = 2 g 1 α (14a)

From these two formulas setting g = 1 we obtain for m_x:

m i 4 m X 4 = 8 × 192 π 2 G 2 = 32 × 192 π 2 g 1 2 α

The interaction is mediated by W-boson and a gluon: it is a weak-hadronic transformation.

The isospin numbers are l = Δ I = I f − I i = 3 − 1 = 2 and m = l = 2 .

The quark process of the neutron decay is shown in Figure 10.

The analytical formula from the Feynman diagram is [8]

Γ ( n → p   e   ν e ) = ( G V 2 + 3 G A 2 ) m e 5 2 π 3 f R = G F 2 V u d 2 ( 1 + 3 λ 2 ) m e 5 2 π 3 f R (15)

G V = G F V u d , G A = G F V u d λ , λ = 1.255 , V u d = 0.974 , G F = 1.166 × 10 − 5   GeV − 1

with the phase-space term [9]

f R = 1 60 ( 2 ξ 4 − 9 ξ 2 − 8 ) ξ 2 − 1 + 1 4 ξ log ( ξ + ξ 2 − 1 ) with ξ ≡ m n − m p m e = 2.53 ,

so f R = 1.6332

and the phenomenological formula for decay width is

Γ ( n → p   e   ν e ) = G 2 m i 5 ( 1 − m f 2 m i 2 ) 4 [1] , l = 4 (15a)

with initial mass m i = m n and final mass m f = m p + m e and obtain with the same ansatz as for Γ ( μ → e   ν e   ν μ ) :

G 2 = 1 192 π 3   32 M X 4

The neutron decay involves in fact only 2 quarks Γ ( n → p   e   ν e ) = Γ ( d d → u d   e   ν e )

so the isospin numbers are l = Δ I = I f + I i = 3 + 1 = 4 and m = l = 4 with I f = I ( u d ) + I ( e ) + I ( ν ) = 1 + 1 + 1 = 3 and I i = I ( d d ) = 1 .

The eta-pion decay is shown in Figure 11.

The generalized and isospin-adapted 3-body semi-leptonic formula (from the muon) decay is

Γ ( η → π 0   π +   π − ) = g 4 m i 32 × 192 π 3 ( m i 4 m X 4 ) ( 1 − m f 2 m i 2 ) 4 (16)

and the phenomenological formula [1]

Γ ( η → π 0   π +   π − ) = G 2 4 π m i ( 1 − m f 2 m i 2 ) 4 (16a)

From this setting g = 1 we obtain for m_x:

m i 4 m X 4 = 8 × 192 π 2 G 2

The decay is mainly hadronic, but the kinematics is one of a 3-body decay, so we can use the generalized 3-body semi-leptonic formula from above.

The intermediate boson here is π 0 , so m X ∝ m ( π 0 ) and the isospin numbers are l = Δ I = I f + I i = 3 + 1 = 4 and m = l = 4 .

The formula for the radiative 2-photon meson decay is:

Γ ( π 0 → γ   γ ) = α 2 m i 3 64 π 3 m X 2 (17)

where m i = m ( π 0 ) and m X = F π , F π = 2 m u Z π pseudoscalar weak decay constant [20] .

the phenomenological formula is

Γ ( π 0 → γ   γ ) = G 2 m i 3 4 π ( 1 − m f 2 m i 2 ) 3 (17a)

From this we obtain for m_x: m X = 4 π α G .

The intermediate boson here is the strongly excited π 0 , so m X ∝ 10 m ( π 0 ) and the isospin numbers are l = Δ I = I f + I i = 2 + 1 = 3 and m = l = 3 .

The photon hyperon decay is shown in Figure 12.

The interaction becomes for the transition s → d   γ

H s d γ = d ¯ σ μ ν ( a + b γ 5 ) s   q μ A ν α , where A ν , q μ are the photon and its momentum.

For the analytical formula we can use the extended isospin-adapted expression from the pion decay (here Γ ( Σ + → p   γ ) )

G F 2 m π 5 8 π m μ 2 m π 2 ( 1 − m μ 2 m π 2 ) 2

Γ ( Σ + → p   γ ) = f π G F 2 α 2 m i 5 8 π ( 1 − m f 2 m i 2 ) 2 (18)

where m i = m ( Σ + ) and m f = m ( p ) , f π is the hadronic correction factor the phenomenological formula is

Γ ( Σ + → p   γ ) = G 2 m i 5 4 π ( 1 − m f 2 m i 2 ) 2 (18a)

From this we obtain for m_x: m X 4 = α 2 64 G 2 or m X = α 8 G

and the isospin numbers are l = Δ I = I f + I i = ( 1 / 2 + 1 ) + 1 / 2 = 2 and m = l = 2 .

We have seen in 2.3 for the muon decay that the decay interaction has the Feynman-Gell-Mann form

H w e a k = G F 2 J ¯ μ J μ where G F / 2 = g 2 / ( 8 M W 2 )

or in generalized form (in natural units)

H int = g 2 8 m X 2 J ¯ 1 μ ( J 2 ) μ (19)

where g is the (dimensionless) interaction constant, m_X is the interaction energy (excitation energy of the intermediate boson), J₁ and J₂ are the currents involved, e.g. the lepton current ( J 2 ) μ = e ¯ ( x ) γ μ ( 1 − γ 5 ) ν e ( x ) .

The current has dimension length-³, so the formula in cgs units reads

H int = ( ℏ c ) 3 g 2 8 m X 2 J ¯ 1 μ ( J 2 ) μ , so H_int has dimension energy/length³, i.e. energy

density, as it should be.

The decay width (energy) becomes then

Γ = ∫ H int ( x ) d 3 x (19a)

Here we have [23]

Γ = C | ψ 1 , 1 | 2 = G 2 4 π m i − 1 ( 1 − x 2 ) , | Δ S | = 1 , l = Δ I = 1 , m = 1 , x = m f m i , k = − 1

with g h 1 = 5.81 × 10 − 7   GeV

Λ → p π − , Λ → n π 0 , G = g h 1 resp. G = g h 1 / 2

Σ + → p π 0 , Σ + → n π + , Σ − → n π − , G = g h 1

Ξ 0 → Λ π 0 , Ξ − → Λ π − , G = g h 1 resp. G = g h 1 / 2

The data for the strange hyperon decays with pions are shown in Table 6.

The decays can be roughly ordered according to the interaction energy.

Lambda into nucleon pion m X ≈ 400   GeV .

Sigma into nucleon pion m X ≈ 400   GeV .

Γ = C | ψ 3 , 2 | 2 = G 2 4 π m i x 2 ( 1 − x 2 ) , Δ S = 0 , l = Δ I = 3 , m = 2 , x = m f m i , k = 1

π ± → l ν , G = g 0

K ± → l ν , G = g 1

K S 0 → π + π − , G = 2 g 1 α   443 , K S 0 → π 0 π 0 , G = 2 g 1 α   443 / 2

K ± → π ± π 0 , G = 2 g 1 α 443

K L 0 → π + π − , G = 2 g 1 α , K L 0 → π 0 π 0 , G = 2 g 1 α / 2

The data for the non-strange two-body meson decays are shown in Table 7.

The decays can be roughly ordered according to the interaction energy

Pion-lepton m X ≈ 100   GeV ,

Kaon-lepton: m X ≈ 400   GeV ,

Kaon-pion: m X ≈ 600   GeV ,

Short-lived K_s0-pion: m X ≈ 150   GeV ,

Long-lived K_L0-pion: m X ≈ 3200   GeV .

Γ = C | ψ l , m | 2 = G 2 4 π m i ( 1 − x 2 ) l , | Δ S | = 1 , l = Δ I = 2 , m = 2 , x = m f m i , k = 1

except K ± → π ± π + π − γ , where Δ I = 1 , m = 1

K ± → π 0 l ν , G = g 1 α 2

K L 0 → π ± l ν , G = g 1 α

K ± → π ± π + π − , G = 2.53 g 1 α

K ± → π ± π 0 π 0 , G = 2.53 g 1 α 2

K L 0 → π 0 π 0 π 0 , G = 2 g 1 α

K L 0 → π + π − π 0 , G = 2 g 1 α / 3 / 2

K L 0 → π 0 π + e − ν e , G = g 1 α π

K ± → π ± π ∓ l ± ν , K ± → π 0 π 0 l ± ν , G = g 1 α π

K ± → π 0 π ± γ , G = g 1 ( α 2 )

K ± → π ± π + π − γ , G = g 1 α 2

The data for the three-four-body decays of strange mesons are shown in Table 8.

The decays can be roughly ordered according to the interaction energy

K⁺, K L 0 into pi 2lepton m X ≈ 1.8   TeV

K⁺, K L 0 into 3 pi m X ≈ 150   GeV

K⁺, K L 0 into 2 pi 2lepton m X ≈ 7...10   TeV

K⁺ into 2pi photon m X ≈ 5.7   TeV

K⁺ into 3 pion photon m X ≈ 3.5   TeV

Γ = C | ψ l , m | 2 = G 2 4 π m i 4 ( 1 − x 2 ) l , | Δ S | = 1 , l = Δ I = 2 , m = 2 , x = m f m i , k = 4

with g h 2 = 1.4 × 10 − 8   GeV − 3 / 2

Λ → p l   ν , G = g h 2 3

Σ − → n l   ν , G = g h 2   2

The data for the three-body decays of strange hyperons are shown in Table 9.

The decays can be roughly ordered according to the interaction energy

Λ into pi 2lepton m X ≈ 1.2   TeV

Σ into pi 2lepton m X ≈ 1.7   TeV

A ' → A e   ν ( Δ S = 0 )

Γ = C | ψ 4 , 4 | 2 = G 2 m i 5 ( 1 − x 2 ) 4 , l = Δ I = 4 , m = 4 , k = 5

π ± → π 0 e ± ν , G = G 0 / 192 × 50 π 3

n → p e − ν ¯ e , G = G 0 / 192 × 175 π 3

Σ ± → Λ e ± ν , G = G 0 / 192 × 65 π 3

μ − → e − ν ¯ e ν μ , τ − → e − ν ¯ e ν τ , τ − → μ − ν ¯ μ ν τ , G = G 0 / 192 π 3

The data for non-strange leptonic three-body decays are shown in Table 10.

Here pure-leptonic transitions are bi-quark transitions

n → p e ν ¯ becomes d u → u u   e ν ¯

Σ + → Λ e ¯ ν becomes u u → u d   e ¯ ν

The decays can be roughly ordered according to the interaction energy

lepton into lepton 2 neutrino m X ≈ 700   GeV

pi into pi 2 lepton m X ≈ 300   GeV

neutron decay n ->p e νe m X ≈ 100   GeV

Σ into Λ 2lepton m X ≈ 800   GeV

The decay width is [24]

Γ = C | ψ 4 , 4 | 2 = G 2 m i 4 π ( 1 − x 2 ) 4 , l = Δ I = 4 , m = 4 , k = 1

η → π 0 π 0 π 0 , η → π + π − π 0 , G = 0.0145

η → π + π − γ , G = 0.00213

The data for three-body eta-pion decays are shown in Table 11.

The decays can be roughly ordered according to the interaction energy

eta into 3 pion m X ≈ 0.3   GeV

eta into 2 pion photon m X ≈ 1   GeV

The decay width is [24] [25] [26]

Γ = C | ψ 3 , 3 | 2 = G 2 m i 3 4 π ( 1 − x 2 ) 3 , l = Δ I = 3 , m = 3 , k = 3

with α = e 2 4 π , g p h ' = 0 .138

pseudoscalar mesons P → γ γ , P = π 0 , η , η ′ theory Γ ( P → γ γ ) = e 4 g p h 2 64 π m P 3 ,

x = 0 , G = 2 π α   C 1 g p h ' , C 1 = 1   , 5 / 4   , 5 / 3

Γ = C | ψ 2 , 2 | 2 = G 2 m i 5 4 π ( 1 − x 2 ) 2 , l = Δ I = 2 , m = 2

g p h = 9.769 × 10 − 9 , G = C 1 g p h

hyperons Λ → n γ , Σ + → p γ , Ξ 0 → Λ γ , Ξ 0 → Σ 0 γ , Ξ − → Σ − γ , C 1 = 7 / 2   , 2   , 1   , 8   , 1 / 2

The data for photon-radiative decays are shown in Table 12.

The decays can be roughly ordered according to the interaction energy

pi, eta into 2 photon m X ≈ 20   GeV

Λ, Σ into nucleon photon m X ≈ 130   GeV

Xi into Λ photon m X ≈ 180   GeV

Xi into Σ photon m X ≈ 100 , ... , 200   GeV

In the following Table 13, are shown the collected decay data from chap.5.

In the above table, the decays are grouped according to type and interaction energy m_X.

Consider the general decay

P i n → P 1 + P 2 + P 3 + P 4

In the table above, the column P_in contains the structure of the original particle, the column P_i contains the structures of the outgoing particles, separated by

slash, the rows m₁, ···, m₄ and m_X contain the respective mass.

The configuration is described either by quarks (like Λ = uds) or by l (lepton) or by Z, W.

The scheme in the last column describes the QHCD/QCD model of the interaction energy with number of active hc-bosons, e.g. sd’(2h) → Z → π⁰(2h) for the decay Ξ → Λ π.

E.g. the generic decay Λ/Σ → n π has the incoming configuration P_in = uds and the outgoing generic configuration P₁₂ = (n = udd)/(π⁰ = (uu’-dd’)), with the interaction energy m_X ≈ 400 GeV, and the decay scheme sd’(2h) → Z → π⁰, where the significant incoming current is s d ¯ interacting via 2 hc-bosons, the intermediate boson is the Z-boson, and the outgoing current is π 0 = ( u u ¯ − d d ¯ ) / 2 . The number of active hc-bosons (or active gluons, in the pion-mediated decays) determines roughly the energy level.

Discussion of the results

The table reveals a simple principle for the scheme:

q 1 q ¯ 2 → b → p or q ¯ 1 q 2 → b → p , where q₁, q₂ are quarks in the incoming quark-current, b is the mediating boson b = W ,   Z ,   π 0 , p are the outgoing particles, p = π 0 ,   π   ,   W   ,   γ , where p can be represented as one or more quark-currents except for the photon γ, which is itself the electromagnetic current.

The resulting interaction energy m_X in the table above is not distributed uniformly, but accumulates around certain values, the energy classes.

E h 1 ≈ 150   GeV for 1 hc-boson

E h 2 ≈ 400   GeV for 2 hc-bosons

E h 4 ≈ 700   GeV for 4 hc-bosons

E h 6 ≈ 1500   GeV for 6 hc-bosons

E h 12 ≈ 3500   GeV for non-diagonal 12 hc-bosons outgoing W (1hcb)

E h 12 , 3 h ≈ 5700   GeV for non-diagonal 12 hc-bosons outgoing W (3hcb)

E h 15 ≈ 7500   GeV for all 15 hc-bosons outgoing W (3hcb)

E h 15 , 3 h ≈ 9000   GeV for all 15 hc-bosons outgoing W (6hcb)

E c 1 ≈ 0.3   GeV for 3 gluons (color interaction, factor 1000 weaker than hc-interaction)

E c 6 ≈ 7   GeV for 6 non-diagonal gluons; E c 8 ≈ 20   GeV for all 8 gluons

For weak decays the energy span in m_X is roughly: E h 15 E h 1 = 9000   GeV 150   GeV = 60 ,

so the energy span scales like E h 15 E h 1 = 60 ≈ ( n h ) 3 / 2

In 2.1 a general relationship between the interaction energy m_X and the decay width Γ was derived:

m X m i = f I ( m i Γ ) 1 / 4 (21)

where f I = I Γ 1 / 4 2 2

The following plot in Figure 13 depicts this relationship for all 54 decays of the quarks u, d, s and all leptons, dealt with in this chapter [2] .

The x-axis is x = ( m i Γ ) 1 / 4 , the y-axis is y = m X m i , the labels consist of the first

3 characters of the name of the corresponding decay, followed by the number in the total decay table, e.g. pi0 ->γ γ has the number 47, and the label “pi047”.

One sees immediately, that the decays separate in two large groups: those with x > 1000 are weak, i.e. hypercolor decays, those with x < 60 are strong (pure color) decays.

If there are 1 or 2 photons on the right side, then the electromagnetic Lagrangian component is used in the calculation in chap.7.

In the pure-color decays only the color SU(3)-Lagrangian is used, in the weak decays both the SU(3) and the hypercolor SU(4)-Lagrangian is used.

Every basic particle of the SM is assigned a preon and a hc-boson configuration [2] [29] [32] .

The preon configuration of a fermion (leptons and quarks) occupies two of the 4 positions in a hc-quadruplet by a Dirac-bispinor, e.g. for electron with

index pair (1,3) we have ( r L − 0 ) in position 1 and ( r R − 0 ) in position 3,

according to the hc-charge. The hc-quadruplet has the hc-charges (L−, L+, R−, R+).

There are 3 possible hc-boson configurations for an index-pair (i,j), which are consistent with the SU(4)-symmetry: 1 hc-boson Aij corresponding to first generation of flavor = 1, 4 hc-bosons A i j + A ¯ i j + A k l + A ¯ k l corresponding to flavor = 2 (the bar specifies the conjugate coupler, and (k,l) is the complementary index pair, e.g. for electron it is (2,4)), and finally all 15 hc-bosons corresponding to flavor = 3.

The fermions (leptons and quarks) have two independent preon-components u1 and u2, they form a bispinor with spin S = 1/2.

The bosons (weak boson W, Z, H) have only one independent preon-component u1, which is a linear combination of two preons, the spins add up to S = 1 for W and Z, or to S = 0 for H, e.g. for Z = Z0

u 1 = ( ( r L − ) + ( r R − ) ) / 2 and Z 0 = ( ( u 1 0 ) ,   ( 0 u 1 ) ,   ( u 1 0 ) , ( 0 u 1 ) ) / 2 . The weak

bosons W and Z0 are carrier of the residual weak interaction.

In the following, we present the basics of the SU(3)-color, the SU(4)-hypercolor, and its Yukawa weak Pauli force in three tables Tables 14(a)-(c) [29] .

Example 1: neutron decay n → p   e   ν ¯

The basic idea of the Fermi model of weak 3-body decay in the Feynman picture mediated by the weak boson W is explained at the example of the neutron decay n → p   e   ν ¯ with the decay scheme d u ¯ ( 1 h , 3 g ) → W → W ( 1 h ) .

The incoming Lagrangian is L ( d u ¯ ) = L Q H C D ( x μ , { u 1 , u 2 } , { A g 4 } ) + L Q C D ( x μ , { u 1 , u 2 , 0 } , { A c 1 , A c 2 , A c 3 } ) with the quark wavefunctions u 1 = d = r − q + , u 2 = u ¯ = r − q − in the hypercolor-SU(4)-preon model, and one hc-boson A g 4 corresponding to the SU(4) generalized Gell-Mann matrix λ 4 and the SU(4) index pair {1,3} and the interaction r L − ↔ r R − in the hc-charge-quadruple ( r L − , r L + , r R − , r R + ) [2] . Furthermore, there are 3 gluons { A c 1 , A c 2 , A c 3 } , which carry the color interaction.

We recall that both L_QHCD and L_QCD have the generic form.

Dirac part L D = u ¯ ( i   ℏ   D μ γ μ − m c ) u , covariant derivative D μ = ∂ μ − i g A a μ λ a ,

with field A a μ , field part L g f = − 1 4 F a μ ν F a μ ν , field tensor

F a μ ν = ∂ μ A a ν − ∂ ν A a μ + g f a b c A b μ A c ν , where λ a are the Gell-Mann matrices, with the structure constants f a b c of the respective Lie algebra (SU(3) or SU(4)) and λ a are the generators of the algebra,

From the preon composition of d u ¯ results the following form of the SU(4) quadruple wavefunction

u 11 = ( r L − + q L + ) / 2 , u 12 = ( r R − + q R + ) / 2 , d = ( ( u 11 0 )   ,   ( 0 0 )   ,   ( u 12 0 )   ,   ( 0 0 ) )

u 21 = ( r L − + q L − ) / 2 , u 22 = ( r R − + q R − ) / 2 , u ¯ = ( ( u 21 u 21 )   ,   ( 0 0 )   ,   ( u 21 u 21 )   ,   ( 0 0 ) ) / 2

The outgoing Lagrangian is L ( W ) = L Q H C D ( x μ , { u 3 , u 3 } , { A g ' 4 } ) with the weak boson W u 3 = r L − r R − and another hc-boson A g ' 4 .

u 31 = r L − , u 32 = r R − , W = ( ( u 31 0 )   ,   ( 0 0 )   ,   ( u 31 0 )   ,   ( 0 0 ) )

The interaction Lagrangian is the Fermi current-current interaction with the mediating exchange boson

L J J ( J ( d u ¯ ) , J ( W ) ) = ( u 1 + γ μ u 2 ) ( u 3 + γ μ u 4 ) m X 2 , with the notation Dirac-conjugate

u 1 +

The interaction energy is

m X = E o u t = E ( u 3 ) + m W + E ( A g ' 4 ) = E i n = E ( u 1 ) + E ( u 2 ) + m u + m d + E ( A g 4 ) + ∑ i = 1 3 A c i

So we have in total two particle configurations, the incoming n p ¯ and the outgoing e ν ¯ , each with an interaction Lagrangian, coupled by the Fermi current-current interaction, and mediated by the corresponding W-boson W = r L − r R − .

In the incoming system d u ¯ we have to take into account the color interaction of the quarks L C ( d u ¯ ) = L Q C D ( x μ , { u 1 , u 2 } , { A c 2 , A c 5 , A c 7 } ) in the basic gluon configuration with 3 rgb-gluons.

Feynman diagram of the decay n → p   e   ν ¯ , with the notation of the antiparticle p ¯ = p c (conjugate), is shown in Figure 14.

The quark-hc-boson-gluon decay-scheme in the SU4PM model d u ¯ ( 1 h , 3 g ) → W → W ( 1 h ) , where the mediating boson W = r L − r R − acts via the current-current-interaction L_JJ is shown in Figure 15.

In the decay-scheme the weak (SU(4)) interaction is carried on the left side by 1 h = 1 hypercolor SU(4) boson Ag₄, and the color (SU(3)) interaction by

3 g = 3 (anticoupler) gluons Ac₂ Ac₅ Ac₇.

The incoming color Lagrangian is

L Q C D ( x μ , { u 1 , u 2 , 0 } , { A c 1 , A c 2 , A c 3 } ) , where the color triple is { u 1 , u 2 , 0 } , on which act the 3x3 color Gell-Mann matrices λ i .

On the right side, the weak (SU(4)) interaction is carried by 1 h = 1 hypercolor SU(4) boson Ag₄ and there is no color interaction, as the mediating boson W has only a weak charge, no color charge.

Example 2: 4-body kaon-pion photonic decay K + → π +   π +   π −   γ

We illustrate the calculation ansatz in more detail in the more complicated and computationally much more challenging example of the 4-body kaon-pion photonic decay K + → π +   π +   π −   γ with the quark-hcboson-gluon decay scheme u s ¯ ( 12 h , 3 g ) → W → π +   π +   π − ( 6 h , 3 g , 1 γ ) .

The Feynman diagram of the process is shown in Figure 16.

The corresponding decay-scheme in the SU4PM model is shown in Figure 17.

The incoming Lagrangian is

L ( u s ¯ ) = L Q H C D ( x μ , { u 1 , u 2 } , { A g n } ) + L Q C D ( x μ , { u 1 , u 2 , 0 } , { A c 2 , A c 5 , A c 7 } ) , with K + = u s ¯ , with the quark wavefunctions u = r + q + , s ¯ = r + q − ,

u 11 = ( r L + + q L + ) / 2 , u 12 = ( r R + + q R + ) / 2 , u = ( ( 0 0 )   , ( u 11 u 11 ) ,   ( 0 0 )   ,   ( u 12 u 12 ) ) / 2

u 21 = ( r L + + q L − ) / 2 , u 22 = ( r R + + q R − ) / 2 , s ¯ = ( ( 0 0 )   , ( 0 u 21 )   ,   ( 0 0 )   ,   ( u 21 u 21 ) )

and 12 non-diagonal hc-bosons A g n corresponding to the non-diagonal SU(4) generator matrices λ i and the SU(4) indices n = { 1 , 2 , 4 , 5 , 6 , 7 , 9 , 10 , 11 , 12 , 13 , 14 } .

It contains also 3 gluons { A c 2 , A c 5 , A c 7 } which carry the color interaction.

The outgoing Lagrangian is

L ( π + π − π + ) = L Q H C D ( x μ , { u 3 , u 4 , u 5 } , { A g ' i } )   + L Q C D ( x μ , { u 3 , u 4 , u 5 } , { A c ' 2 , A c ' 5 , A c ' 7 } ) + L e ( π + π − π + ) , indices

i = { 1 , 4 , 6 , 9 , 11 , 13 } , with the pions and their corresponding wavefunctions

π + π − = u d ¯ d u ¯ = r L − + r R + + q L − + q R +

u 31 = u 41 = ( r L − + q L − ) / 2 , u 32 = u 42 = ( r R + + q R + ) / 2 ,

π + π − = ( ( u 31 − u 41 u 31 − u 41 ) , ( 0 0 )   ,   ( 0 0 )   ,   ( u 32 − u 42 u 32 − u 42 ) ) / 2

π + = u d ¯ = r L + + r R + + q L − + q R + , u 51 = ( r L + + q L − ) / 2 , u 52 = ( r R + + q R − ) / 2 ,

π + = ( ( 0 u 51 ) , ( 0 u 51 )   ,   ( 0 u 52 )   ,   ( 0 u 52 ) ) / 2

and the 6 hc-bosons A g ' i , which are the 6 couplers of SU(4).

It contains also 3 diagonal gluons { A c ' 2 , A c ' 5 , A c ' 7 } which carry the color interaction.

The interaction Lagrangian is

L J J ( J ( u s ¯ ) , J ( π + π − π + ) ) = ( u 1 + γ μ u 2 ) ( u 3 + γ μ u 3 + u 4 + γ μ u 4 + u 5 + γ μ u 5 ) m X 2 , with the

notation Dirac-conjugate u 1 + .

The interaction energy is

m X = E o u t = E ( u 3 ) + E ( u 4 ) + E ( u 5 ) + 3 m π + ∑ i E ( A g ' i ) + ∑ i E ( A c ' i ) + A e 0 = E i n = E ( u 1 ) + E ( u 2 ) + m u + m s + + ∑ n E ( A g n ) + ∑ i E ( A c i )

So we have in total two particle configurations, the incoming K + = u s ¯ and the outgoing π + π − π + , each with an interaction Lagrangian, coupled by the Fermi current-current interaction, and mediated by the corresponding W-boson W = r L − r R − ,

In the incoming system K + = u s ¯ and the outgoing π + π − π + we have to take into account the color interaction of the quarks L C ( u s ¯ ) = L Q C D ( x μ , { u 1 , u 2 , 0 } , { A c 2 , A c 5 , A c 7 } ) and L C ( π + π − π + ) = L Q C D ( x μ , { u 3 , u 4 , u 5 } , { A c ' 2 , A c ' 5 , A c ' 7 } ) in the basic gluon configuration with 3 rgb-gluons.

The outgoing photon is active in the additional third electromagnetic Lagrangian L e ( π + π − π + ) = L e ( x μ , { u 3 , u 4 , u 5 } , { A e 0 } ) .

Now we minimize the action S = ∫ L ( x μ , u i , A g i )   r 2 sin θ   d t   d r   d θ   d φ for the total Lagrangian L ( x μ , u i , A g i ) = L ( d u ¯ ) + L J J ( W ) + L ( J ( d u ¯ ) , J ( W ) ) + L C ( d u ¯ ) under the constraint of energy conservation E ( d u ¯ ) = E ( W ) , as required in the Feynman diagram of the process.

We have for the particle wavefunctions { u 1 , u 2 , u 3 } the normalization condition ∫ | u i | 2 d 3 x = 1 and for the field bosons we set up a boundary condition for r = r₀ A g i ( r 0 ) = 0 and A c i ( r 0 ) = 0 and the Lorenz-gauge-condition ∂ μ ( A g i ) μ = 0 and ∂ μ ( A c i ) μ = 0 .

The energy, length, and time are made dimensionless by using the units:

E( E 0 = ℏ c 1   am = 0.196   TeV ), r(fm), t(am/c) am = 10⁻¹⁸ m. We can assume axial

symmetry, so we can set φ = 0 and use the spherical coordinates ( t , r , θ ) .

We choose the equidistant lattice for the intervals ( t , r , θ ) ∈ [ 0 , 1 ] × [ 0 , 1 ] × [ 0 , π ] with 21 × 21 × 11 points and, for the minimization n_sub in parallel, n_sub random sublattices of length l_sub, where n_sub = 8 or 16, and l_sub = 25 or 50 or 100 according to the complexity of the corresponding Lagrangian.

l [ i x , j ] = { { ( t i 1 , r i 2 , t i 3 ) | ( i 1 , i 2 , i 3 ) = r a n d o m ( l a t t i c e ,   j = 1 , ⋯ , l s u b ) } | i x = 1 , ⋯ , n s u b } .

For the Ritz-Galerkin expansion we use the 12 functions f k ( r , θ ) = { b f u n c ( r , r 0 , d r 0 ) r k 1 , k 1 = 0 , ⋯ , n r } × { ( cos k 2 θ , cos k 2 θ sin θ ) , k 2 = 0 , ⋯ , n θ }

The action S = ∫ L ( x μ , u i , A g i )   r 2 sin θ   d t   d r   d θ   d φ becomes a mean-value on the sublattice l [ i x ]

S ˜ [ i x ] = 1 N ( l [ i x ] ) ∑ x ∈ l [ i x ] s u b L ( x ,   u i , A g i )   2 π V t r θ , where V t r θ = π the

( t , r , θ ) -volume and l s u b = N ( l [ i x ] ) is the number of points. We impose the boundary condition for A g i ( r = r 0 ) = 0 via penalty-function (imposing exact conditions is possible, but slows down the minimization process enormously).

S ˜ is minimized n_sub x in parallel with the Mathematica-minimization method “simulated annealing”.

The proper parameters of the particles u i and the hc-bosons A g i are:

p a r ( p i ) = { E u i ,   a i ,   r u i ,   θ u i ,   d r u i } , p a r ( A g i ) = { E A i ,   a A i } , p a r ( A c i ) = { E A c i ,   a A c i }

The complexities and execution times (on a 2.7 GHz Xeon E5 work-station) differ greatly for different decays.

For the neutron decay n → p   e   ν ¯ with the scheme d u ¯ ( 1 h ) → W → W ( 1 h ) (1hc-boson on both sides) and color interaction L ( d u ¯ , 3 g ) with basic 3 gluons complexity (Lagrangian) = (3.7 + 4.8) × 10⁶ terms, minimization time t (minimization) = 111 s.

The mathematical details of the calculation, and the results can be studied in depth in the corresponding Mathematica programs [34] .

Table 15 of decays with calculation results mX and experimental values mXexp according to the formula in 2.1 from the observed decay width Γobs, is as follows [34] .

Table description

The scheme (last) column describes the model of the decay, on which the calculation is based, where the notation q’ is used for the antiparticle q ¯ .

Here the calculation result (m_Xcal) and the value from decay time (m_Xexp) are given in GeV.

m_X(er) is the calculated m_X-value with uncertainty er in GeV.

E_col specifies the calculated color interaction energy in GeV and the number of active gluons on left side of the process, e.g. 250 (3 g), E_em is the electromagnetic energy of the involved photons, if any.

12 > and 12 > are the mean radius in am-units (1 am = 10 ⁻¹⁸ m) and its quantum “smear-out” in the left-side (incoming) part of the scheme.<> <>

The mean boson amplitude (hypercolor, color, electromagnetic) of the incoming and outgoing system Agi Aci Aei expressed in units am⁻¹ is given in column four.

m X = E ( B m e d ) , where for weak decays the mediating boson is B m e d = W or B m e d = Z .

Classification according to m_X: strong decays

There are here 3 strong (color) decays: pion and eta decays, with scales m X ≈ ( 0.3 ,   7.5 ,   20 ) GeV , mediated by a pion

η -> π0 π0 π0, η -> π0 π0 γ, π0/η -> γ γ

Strong decays have an assessed upper limit of interaction energy m_X for strong decays: E c , m a x = N c o m p M m a x ,where M m a x is the maximum energy-mass, for 1+2-generation M m a x = m c = 1.3   GeV for charm-quark, and maximum number of components N c o m p = 3 + 15 , where 3 stands for 3 quarks, and 15 stands for 15 hc-bosons, so E c , m a x = 18 × 1.3 = 23.4   GeV .

Classification according to m_X: weak decays

The minimum interaction energy m_X for weak decays is E w , m i n = m W = 80.4   GeV .

The weak decays considered here can be put into 3 categories.

- low interaction energy 100 - 400 GeV

n → p   e   ν ¯ , m X ~ 200   GeV

schematic photonic Λ / Σ / Ξ → γ ... , m X = 130...260   GeV

schematic pionic Λ / Σ / Ξ → π ... , m X ~ 400   GeV

schematic one-pion K → π , m X ~ 160   GeV

schematic leptonic K / π → l   ν , m X = 130...180   GeV

-middle interaction energy 700 - 1700 GeV

schematic nucleonic Λ / Σ → ( p / n ) ν , m X = 1300...1700   GeV

schematic pure leptonic μ / τ → l   2 ν , m X ~ 700   GeV

schematic leptonic Σ → Λ   l   ν , m X ~ 700   GeV

- high interaction energy kaon 3400 - 9200 GeV

schematic pionic-leptonic K → π l   ν , m X ~ 2000   GeV

schematic pionic K L 0 → π   π , m X ~ 3400   GeV

schematic pionic-leptonic K → π π l   ν , m X = 7600...9200   GeV

schematic pionic-photonic K → 3 π   γ , m X ~ 3600   GeV

schematic pionic-photonic K → 2 π   γ , m X ~ 5700   GeV

Characterization of radius

The calculated radius r of decaying particle is a parameter of the ingoing Lagrangian, and is measured in am = 10⁻¹⁸ m.

For weak decays we obtain values r = 0.25...1.3   am and quantum smear-out Δ r ~ r .

For strong decays we obtain r ~ 0.3   am .

The following plot Figure 18 presents the measured and calculated interaction energy described in the above table [2] [34] .

The decays above 80 GeV (=m_W) are weak (hypercolor) decays, those below 25 GeV (=6m_b, m_b = 4.2 GeV) are strong (color) decays, the observed m_X is dark-blue, the calculated m_X is red (with calculation error bar), the color energy for weak decays, respectively electromagnetic energy for strong decays is cyan.

Another interesting decay parameter is the mean radius 12> of the incoming system on the left side of the scheme, e.g. for the neutron decay<>

Figure 18. Measured and calculated interaction energy.

Figure 19. Mean radius in dependence of interaction energy.

Figure 20. Mean field boson amplitude in dependence of interaction energy.

n->p e νe, the incoming system is n p ¯ , in the decay scheme it is represented by d u ¯ [2] [34] . The plot of radius is shown in Figure 19.

It is interesting to see, that the mean radius separates basically into two groups: high-energy non-leptonic kaon-pion decays with 12 >>0.9 am and the remaining decays with 12 > <0.5 am, apart from the photonic Λ/Σ -> n γ.<> <>

The other important decay parameter is the mean (hypercolor, color, electromagnetic) field boson amplitude Agi for the weak decays, Aci for the color decays, of the incoming system, expressed in units am⁻¹ [2] [34] . The mean field boson amplitude is shown in Figure 20.

Again, the amplitude separates into two groups, amplitude ≥ 0.6 for the kaon-pion decays an pure leptonic decays, and the remaining with amplitude ≤ 0.3, with the outlier K+ -> π+ π0.