24 September 2019 AD; Our Lady of Ransom; St Pacific of San Severino

11 November 2019 AD; St Martin of Tours (397); St Mennas (3rd Century)

This is an updated version of how to compute the CP Violation Phase Angle of Neutrino.

To get the results we need a couple of things:

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

(more in this article: 68. Getting CP Violation Phase Angles of Neutrino, Boson, Quark, and Graviton )

**1. Quantum Fraction** from which we get factors to calculate the angles.

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

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

**0 -** We need some **approximation of a CP Violation Phase Angle of Neutrino** or a measured value (see article 68, with the link above) or an official estimate which is now about

**δ _{CP VIOLATION PHASE ANGLE NEUTRINO} = approx. 234 deg (+43 deg; -31 deg)**

Having this angle will speed up the computations.

**1 - Quantum Fraction** which is a fraction different for each of the four fundamental elements, but nevertheless forming two pairs.

For Neutrino the Quantum Fraction is:

**( 21 / 9 ) / ( ( 3 / 2 ) - ( 5 / 6 ) ) = 7 / 2**

**The reciprocal of each part will be used** for calculation of the CP Violation Phase Angle of Neutrino, in the following way:

**( 9 / 21 )** for calculation CP Violation Phase Angle of the following combinations:

Quark, Graviton, Quark + Graviton, Boson + Neutrino, Quark + Boson + Neutrino, Graviton + Boson + Neutrino, and Quark + Graviton + Boson + Neutrino.

**( 2 / 3 )** for the following combinations:

Neutrino, Quark + Neutrino, Graviton + Neutrino, Quark + Graviton + Neutrino.

**( 6 / 5 )** for 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 of 90+ deg (or if you prefer multiplications of 22.5+ deg).

Correction Angles for combinations of the elements:

**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**

**( 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**

These values are necessary to get the Arithmetic Mean of the CP Violation Angle for any element.

**3 - The sum of Mixing (Oscillation) Angles of Neutrino** from before and after the transformation:

**θ _{13} = -8.5770 deg, and after the transformation θ_{ 13 }= 145.7087 deg**

**θ _{23} = -47.2551 deg, and after the transformation θ _{23} = 98.6837 deg**

**θ _{12} = 34.6599 deg, and after the transformation θ _{12} = 34.6599 deg**

It'll be necessary to go to the article "56. Neutrino Mixing (Oscillation) Angles Final - after the Transformation": ( 56. Neutrino Mixing (Oscillation) Angles Final - after the Transformation )

Angles before the transformation are calculated using **built-in Fortran function [ ATAN2 (Im, Re) ]**. This function gives Polar coordinates: Length of the vector and the Angle of this vector with 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 θ_{13} ; θ_{23} ; θ_{12} ; 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:**

**Θ _{13} = -20.0130 deg. When multiplied by Quantum Number [ ( 1 / 3.5 ) X ( 3 / 2 ) ] then equals to the θ_{13 } = -8.5770 deg**

**Θ _{23} = -165.3929 deg. When multiplied by Quantum Number [ ( 1 / 3.5 ) ] then equals to the θ_{23} = -47.2551 deg**

**Θ _{12} = 23.1066 deg. When multiplied by Quantum Number [ ( 3 / 2 ) ] then equals to the θ_{12} = 34.6599 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 }= (339.9870 degrees = angle after the transformation). When multiplied by the Quantum Number [ ( 1 / 3.5 ) X ( 3 / 2 ) ] then equals to θ_{13 FINAL }= 145.7087 degrees**

**θ _{23 FINAL }= (345.3929 degrees = angle after the transformation). When multiplied by the Quantum Number [ ( 1 / 3.5 ) ] then equals to θ_{23 FINAL }= 98.6837 degrees**

**θ _{12 FINAL }= (23.1066 degrees = angle after the transformation). When multiplied by the Quantum Number [ ( 3 / 2 ) ] then equals to θ_{12 FINAL }= 34.6599 degrees**

**The sum of all these Mixing ( Oscillation ) Angles is the Total Neutrino Mixing (Oscillation) Angle** and equal to:

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

**θ _{TOTAL NEUTRINO}_{ MIXING (OSCILLATION) ANGLE }= [ ( θ_{13} = - 8.5770 deg + 145.7087 deg ) + ( θ_{23 }= - 47.2551 deg + 98.6837 deg ) + ( θ_{12 }= 34.6599 deg X 2 ) ] = 257.8801 degrees**

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

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

**θ _{TOTAL }_{BOSON MIXING (OSCILLATION) ANGLE }= [ ( θ_{13} = - 28.1209 deg + 60.2637 deg ) + ( θ_{23 }= 13.3623 deg + 1.3316 deg ) + ( θ_{12 }= 28.0002 deg X 2 ) ] = 102.8371 degrees**

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

**θ _{TOTAL QUARK MIXING (OSCILLATION ) ANGLE }= [ ( θ_{13} = - 0.20134 deg + 59.7987 deg ) + ( θ_{23 }= 2.3781 deg X 2 ) + ( θ_{12 }= 13.0585 deg X 2 ) ] = 90.4706 degrees**

**θ _{TOTAL }_{GRAVITON MIXING ANGLE }= 90.5494 deg**

**θ _{TOTAL }_{GRAVITON MIXING ANGLE }= [ ( θ_{13} = 6.1879 deg X 2 ) + ( θ_{23 }= 0.5154 deg X 2 ) + ( θ_{12 }= - 69.9459 deg + 147.0887 deg ) ] = 90.5494 degrees**

Now, we can proceed with getting the CP Violation Phase Angle of Neutrino.

**Singles:**

**Quark = (90.4706 deg + 5 X 90.5100 deg) X ( 9 / 21 ) = 543.0205 deg X ( 9 / 21 ) = 232.7231 deg**

**Graviton = (90.5494 deg + 5 X 90.5100 deg ) X ( 9 / 21 ) = 543.0993 deg X ( 9 / 21 ) = 232.7568 deg**

**Boson = (102.8371 deg + 1 X 90.1793 deg ) X ( 6 / 5 ) = 193.0164 deg X ( 6 / 5 ) = 231.6197 deg**

**Neutrino = (257.8801 deg + 1 X 90.1793 deg ) X ( 2 / 3 ) = 348.0594 deg X ( 2 / 3 ) = 232.0396 deg**

**Arithmetic Mean of the Singles = 929.1392 deg / 4 = 232.2848 deg**

**Pairs: **

**Quark + Neutrino = (90.4706 deg + 257.8801 deg + 0 X 90+ deg) X ( 2 / 3 ) = 348.3507 X ( 2 / 3 ) = 232.2338 deg**

**Graviton + Neutrino = (90.5494 deg + 257.8801 deg + ) X 90+ deg) X ( 2 / 3 ) = 348.4295 X ( 2 / 3 ) = 232.2863 deg**

** **

**Quark + Boson = (90.4706 deg + 102.8371 deg + 0 X 90+ deg) X ( 6 / 5 ) = 193.3077 deg X ( 6 / 5 ) = 231.9692 deg**

**Graviton + Boson = 90.5494 deg + 102.8371 deg + 0 X 90+ deg) X ( 6 / 5 ) = 193.3865 deg X ( 6 / 5 ) = 232.0638 deg**

** **

**Quark + Graviton = (90.4706 deg + 90.5494 deg + 4 X 90.5100 deg ) X ( 9 / 21/ ) = 543.0599 deg X ( 9 / 21 ) = 232.7400 deg**

**Boson + Neutrino = (102.8371 deg + 257.8801 deg + 2 X 90.90.1793 deg ) X ( 9 / 21 ) = 541.0758 deg X ( 9 / 21 ) = 231.8896 deg**

**The arithmetic mean of the Pairs = 1393.1827 deg / 6 = 232.1971 deg**

**Triplets:**

**Quark + Graviton + Boson = (90.4706 deg + 90.5494 deg + 102.8371 deg - 1 X 90.34464 deg) X ( 6 / 5 ) = 193.5124 deg X ( 6 / 5 ) = 232.2149 deg**

**Quark + Graviton + Neutrino = (90.4706 deg + 90.5494 deg + 257.8801 deg - 1 X 90.34464 deg ) X ( 2 / 3 ) = 348.5554 X ( 2 / 3 ) = 232.3703 deg**

**Quark + Boson + neutrino = (90.4706 deg + 102.8371 deg + 257.8801 deg + 1 X 90.2454 ) X ( 9 / 21 ) = 541.4332 deg X ( 9 / 21 ) = 232.0428 deg**

**Graviton + Boson + Neutrino = (90.5494 deg + 102.8371 deg + 257.8801 deg + 1 X 90.2454 deg) X ( 9 / 21 ) = 541.5120 deg X ( 9 / 21 ) = 232.0766 deg**

**The Arithmetic Mean of the Triplets = 928.7046 deg / 4 = 232.1762 deg**

**Quadruplet** (the sum of the angles = 540 degrees i.e. Pentagon shape):

**Quark + Graviton + Boson + Neutrino = δ _{ CP VIOLATION PHASE ANGLE NEUTRINO }= (90.4706 deg + 90.5494 deg +102.8371 deg + 257.8801 deg + 0 X 90+ deg) X ( 9 / 21 ) = 541.73716 deg X ( 9 / 21 ) = 232.1731 deg = 4.05218448 radians**

**The arithmetic mean of Singles, Pairs, Triplets, and Quadruplet = δ_{ CP VIOLATION PHASE ANGLE NEUTRINO }= ( 232.2848 deg X 4 + 232.1971 deg X 6 + 232.1762 deg X 4 + 232.1731 deg ) / 15 = 3483.1997 deg / 15 = 232.2133 deg = 4.05288688 radians**

Again, the result from the quadruplet seems to be better and a lot simpler to compute, than the arithmetic mean.

Next articles Revisited: Quark.

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