19 September 2019 AD; the feast of St Januarius and Comps.; Our Lady of La Salette.

 7 November 2019 AD; St Carina, St Melassipius, St Anthony (360); St Willibrord (739); St Engelbert (1225)

 

To get the results we need a couple of things:

 

0. Approximate/exact (guessed/measured) value of CP Violation Phase Angle of Boson.

(More in this article: 68. Getting approximate/exact CP Violation Phase Angle of all four elements )

 

1. Quantum Fraction giving the factors with which we can calculate the angles.

2. Spherical correction angles for each combination of the four fundamental elements - Neutrino and Boson, Quark and Graviton.

 

3. The sum of the Mixing (Oscillation) Angles of Boson, Neutrino, Quark, and Graviton from before and after the transformation into the positive part of Real Plane (if we use just the sum of complex vectors).

 

1 - So, the Quantum Fraction for a lack of better word is a fraction different for each of the four fundamental elements, but nevertheless forming two pairs.

For Boson the Quantum Fraction equals to:

 

( 21 / 5 )  / ( ( 27 / 10 ) - ( 3 / 2) ) =  7 / 2

 

Reciprocal of each part will be used for the calculation of the CP Violation Phase Angle of Boson, in the following way:

 

( 5 / 21 ) - for calculating CP Violation Phase Angle of the following combinations - Quark, Graviton, Quark + Graviton, Neutrino + Boson, Quark + Boson + Neutrino, Graviton + Boson + Neutrino, Quark+ Graviton + Boson + Neutrino.

 

( 10 / 27 ) - for calculating CP Violation Phase Angle of the following combinations - Neutrino, Quark + Neutrino, Graviton + Neutrino, Quark + Graviton + Neutrino.

 

( 2 / 3 ) - for calculating CP Violation Phase Angle of the following combinations - Boson, Quark + Boson, Graviton + Boson, Quark + Graviton + Boson.

 

 

2 - Spherical correction angles. Those are just the sums of the angles of different elements and reduced to multiplications ( 90+ deg, or if you prefer to 22.5+ deg. "+" stands for a spherical angle).

 

Quark + Graviton = 1/2 x + 1/2 y = 90. 50998 deg

Boson + Neutrino = ( x + y ) / 4 = 90.1793 deg

 

( Quark + Graviton ) + Boson = ( 1/ 2 ) x + ( 1 / 2 ) y = 90.34464 deg

( Quark + Graviton ) + Neutrino = as above 

 

Quark + ( Boson + Neutrino ) = ( 1 / 5 ) x + ( 4 / 5 ) y = 90.245436 deg

Graviton + ( Boson + Neutrino ) = as above

This correction factor is slightly different than those used in previous articles for calculationg Quark and Neutrino CP Violation Phase angles. But the differtence is miniscule.

 

(Quark + Graviton ) + ( Boson + Neutrino ) = ( 1 / 3 ) x + ( 2 / 3 ) y = 90.289527 deg

 

Quark + Boson = ( 1 / 3 ) x + ( 2 / 3 ) y = 90.289527 deg

Quark + Neutrino = as above

Graviton + Boson = as above

Graviton + Neutrino = as above

 

3 - The sums of Mixing (Oscillation) Angles for Boson, Neutrino, Quark, and Graviton from before and after the transformation:

 

θ₁₃  = -28.1209 deg, and after the transformation θ₁₃ = 60.2637 deg

θ₂₃ = 13.3623 deg, and after the transformation θ₂₃ = 1.3316 deg

θ₁₂ = 28.0002 deg, and after the transformation θ₁₂ = 28.0002 deg

 

It'll be necessary to go back to Article "55. Boson Mixing (Oscillation) Angles Final - after the transformation" - ( 55. Boson Mixing (Oscillation) Angles Final - after the transformation )

 

Angles before the transformation are calculated using the built-in Fortran function [ ATAN2 (Im, Re) ]. This function gives Polar coordinates: Length of the vector and the Angle of this vector with the X-axis. The Angle depends on the location of the initial vector ( i.e. in which quadrant the vector is located in Cartesian coordinates ).

Here is the description from Fortran Wiki: ATAN2

Important note: The angles above θ₁₃ ; θ₂₃ ; θ₁₂ ; from before the transformation and after the transformation are different than the original angles since the Quantum Numbers were applied to them, so that the Mixing (Oscillation) Angles may be calculated.

 

Here are the original angles from before the transform:

 

Θ₁₃ = -157.4770 deg. When multiplied by Quantum Number  [ ( 1 / 3.5 ) X ( 5 / 8 ) ] then equals to the θ₁₃  = -28.1209 deg 

Θ₂₃ = 163.6881 deg. When multiplied by Quantum Number [ ( ( 1/3.5 )² ) ] then equals to the θ₂₃ = 13.3623 deg 

Θ₁₂ = 44.8004 deg. When multiplied by Quantum Number [ ( 5 / 8 ) ] then equals to the θ₁₂ = 28.0002 deg 

 

The transformation:

The Transformation computes "Final Angle" - this is a set of rules to get all the angles Θ from previous calculations and transform them into a set of angles with the following values: 

0.0 deg < Θ_FINAL < 90.0 deg and

270.0 deg < Θ_FINAL < 360.0 deg

This is how it is done:

1: If the original angle Θ is located in the first quadrant: Θ_FINAL = Θ

2: If the original angle Θ is located in the second quadrant: Θ_FINAL = ( 180 deg = π ) - | Θ |; i.e. 180 deg - absolute value of theta.

3: If the original angle Θ is located in the third quadrant: Θ_FINAL = ( 180 deg = π ) + | Θ |; i.e. 180 deg + absolute value of theta.

4: If the original angle Θ is located in the fourth quadrant: Θ_FINAL = ( 360 deg = 2π ) - | Θ |; i.e. 360 deg - absolute value of theta.

 

After that transform, the Θ_FINAL angles are multiplied by the same Quantum Numbers as in the case of Θ_{BEFORE THE TRANSFORM} ; (which was computed right before), giving the angles θ_FINAL .  

 

Here are the Angles after the Transformation:

θ_{13 FINAL} = ( 337.4770 degrees = angle after the transformation ). When multiplied by Quantum Number [ ( 1 / ( 3.5 ) ) X ( 5 / 8 ) ] then equals to θ_{13 FINAL} = 60.2637 degrees

θ_{23 FINAL} = ( 16.3120 degrees = angle after the transformation ). When multiplied by Quantum Number [ ( 1 / ( 3.5 ) ² ) ] then equals to θ_{23 FINAL} = 1.3316 degrees

θ_{12 FINAL} = ( 44.8004 degrees = angle after the transformation ). When multiplied by Quantum Number [ ( 5 / 8 ) ] then equals to θ_{12 FINAL} = 28.0002 degrees

 

The sum of all these Mixing ( Oscillation) Angles from before and after the transformation is the Total Boson Mixing (Oscillation) Angle and equals to:

θ _{TOTAL BOSON MIXING (OSCILLATION) ANGLE} = 102.8371 deg

θ _{TOTAL BOSON MIXING (OSCILLATION) ANGLE} = [ ( θ₁₃ = - 28.1209 deg + 60.2637 deg ) + ( θ₂₃ = 13.3623 deg + 1.3316 deg ) + ( θ₁₂ = 28.0002 deg X 2 ) ] = 102.8371 degrees

 

In a similar manner we get mixing angles of the other elements:

θ _{TOTAL QUARK MIXING (OSCILLATION ) ANGLE} = 90.4706 deg

θ _{TOTAL QUARK MIXING (OSCILLATION ) ANGLE} = [ ( θ₁₃ = - 0.20134 deg + 59.7987 deg ) + ( θ₂₃ = 2.3781 deg X 2 ) + ( θ₁₂ = 13.0585 deg X 2 ) ] = 90.4706 degrees

 

θ _{TOTAL GRAVITON MIXING ANGLE} = 90.5494 deg

θ _{TOTAL GRAVITON MIXING ANGLE} = [ ( θ₁₃ = 6.1879 deg X 2 ) + ( θ₂₃ = 0.5154 deg X 2 ) + ( θ₁₂ = - 69.9459 deg + 147.0887 deg ) ] = 90.5494 degrees

 

θ _{TOTAL NEUTRINO MIXING (OSCILLATION) ANGLE} = 257.8371 deg

θ _{TOTAL NEUTRINO MIXING (OSCILLATION) ANGLE} = [ ( θ₁₃ = - 8.5770 deg + 145.7087 deg ) + ( θ₂₃ = - 47.2551 deg + 98.6837 deg ) + ( θ₁₂ = 34.6599 deg X 2 ) ] = 257.8801 degrees

 

 

θ _{BOSON MIXING ANGLE} = 102.8371 deg.

This angle is crucial in getting the CP Violation Phase Angle of Boson.

 

Calculating CP Violation Phase Angles for:

 

Singles:

 

Quark = ( 90.4706 + 5 x 90.51) x ( 5 / 21 ) = 129.2906 deg

Graviton = ( 90.5494 + 5 x 90.51 ) x ( 5 / 21 ) = 129.3094 deg

Boson = ( 102.8371 + 90.1793 ) x ( 2 / 3 ) = 128.6776 deg

Neutrino = ( 257.8801 + 90.1793 ) x ( 10 / 27 ) = 128.9109 deg

 

Arithmetic Mean of Singles = 516.1885 deg / 4 = 129.0471 deg

 

Pairs:

 

Quark + Neutrino = ( 90.4706 + 257.8801 ) x ( 10 / 27 ) = 129.0188 deg

Graviton + Neutrino = ( 90.5494 + 257.8801 ) x ( 10 / 27 ) = 129.0480 deg

 

Quark + Boson = ( 90.4706 + 102.8371 ) x ( 2 / 3 ) = 128.8718 deg

Graviton + Boson = ( 90.5494 + 102.8371 ) x (2 / 3 ) = 128.8924 deg

 

Quark + Graviton = (0.4706 + 90.5494 + 4 x 90.5100 ) x ( 5 / 21 ) = 129.3000 deg

 

Boson + Neutrino = (102.8371 + 257.8801 + 2 x 90.1793 ) x ( 5 / 21 ) = 128.8276 deg

 

Arithmetic Mean for Pairs = 773.9586 deg / 6 = 128.9931 deg

 

Triplets:

 

Quark + Graviton + Neutrino = 90.4706 + 90.5494 + 257.8801 - 1 x 90.3446 ) x ( 10 / 27 ) = 129.0946 deg

Quark + Graviton + Boson = ( 90.4706 + 90.5494 + 102.8371 - 1 x 90.3446 ) x (2 / 3 ) = 129.0083 deg

Quark + Neutrino + Boson = ( 90.4706 + 257.8801 + 102.8371 + 1 x 90.2454 ) x ( 5 / 21 ) = 128.9127 deg

Graviton + Neutrino + Boson = ( 90.5494 + 257.8801 + 102.8371 + 1 x 90.2454 ) x ( 5 / 21 ) = 128.9314 deg

 

Arithmetic Mean for Triplets = 515.9470 / 4 = 128.9868 deg

 

Quadruplet (Pentagon - sum of internal angles equal to 540 degrees):

 

Quark + Graviton + Boson + Neutrino = δ _{CP VIOLATION PHASE ANGLE BOSON}  = ( 90.4706 + 90.5494 + 102.8371 + 257.8801 ) x ( 5 / 21 ) = 128.985038 deg = 2.25121360 radians

 

 

Arithmetic Mean of Singles + Pairs + Triplets + Quadruplet = δ _{CP VIOLATION PHASE ANGLE BOSON}  = 1935.07920 deg / 15 = 129.0053 deg = 2.25156689 radians

 

I think the result from just the Quadruplet ( all four elements ) is better = δ _{CP VIOLATION PHASE ANGLE BOSON}  = 128.985038 deg = 2.25121360 radians

 

Next articles - calculating the CP Violation Phase Angle of Neutrino (Revisited); Quark (Revisited) and Graviton.