 1 November 2019 AD; All Saints Day.

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

To get the results we need a couple of things:

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

(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 Quark or a measured value (see article 68, with the link above) or an official estimate which is now about

δ CP VIOLATION PHASE ANGLE QUARK = approx. 68.8 degrees ( 64.2 degrees - 73.3 degress ) = 1.200 ( +/- 0.08 ) radians

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 Quark the Quantum Fraction is:

( 21 / 4 ) / ( ( 15 / 4 ) - ( 3 / 2 ) ) = 7 / 3

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

( 4 / 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.

( 4 / 15 ) for the following combinations:

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

( 2 / 3 ) 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 Arithmetic Mean of the CP Violation Angle for any element.

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

θ 13 = - 0.20134 deg, and after the transformation θ 13 = 59.7987 deg

θ 23 = 2.3781 deg, and after the transformation θ 23 = 2.3781 deg

θ 12 = 13.0585 deg, and after the transformation θ 12 = 13.0585 deg

It will be necessary to go back to the article "54.Quark mixing angles final - after the transformation": Quark mixing angles

( I did some small changes to this article, so it is easier to read. It will be useful later now, in the main article about Quark ).

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 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 = - 1.20805 deg. When multiplied by Quantum Number  [ ( 1 / 6 ) ] then equals to the θ13  = - 0.20134 deg

Θ23 = 8.3234 deg. When multiplied by Quantum Number [  ( 1 / 3.5 ) ] then equals to the θ23 = 2.3781 deg

Θ12 = 26.1171 deg. When multiplied by Quantum Number [ ( 1 / 2 ) ] then equals to the θ12 = 13.0585 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 < ΘFINAL < 90.0 deg and

270.0 deg < ΘFINAL < 360.0 deg

This is how it is done:

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

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

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

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

After that transform, the ΘFINAL angles are multiplied by the same Quantum Numbers as in the case of ΘBEFORE THE TRANSFORM ; (which was computed right before), giving the angles θFINAL .

Here are the angles after the transformation:

θ13 FINAL = ( 358.7920 degrees = angle after the transformation ). When multplied by a Quantum Number [ ( 1 / 6 ) ] then equals to θ13 FINAL = 59.7987 degrees

θ23 FINAL = ( 8.3234 degrees = angle after the transformation ). When multplied by a Quantum Number [ ( 1 / ( 3.5 ) ) ] then equals to θ23 FINAL = 2.3781 degrees

θ12 FINAL = ( 26.1171 degrees = angle after the transformation ). When multplied by a Quantum Number [ ( 1 / 2 ) ] then equals to θ12 FINAL = 13.0585 degrees

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

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

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

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 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

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

Now, we can proceed with getting the Cp Violation Phase Angle of Quark.

Singles:

Quark = (90.4706 deg + 3 X 90.4706 deg) X ( 4 / 21 ) = 361.8822 deg X ( 4 / 21 ) = 68.9300 deg

Graviton = (90.5494 deg + 3 X 90.5494 deg ) X ( 4 / 21 ) = 362.1976 deg X ( 4 / 21 ) = 68.9900 deg

Boson = (102.8371 deg + 0 X 90.1793 deg ) X ( 2 / 3 ) = 102.8371 deg X ( 2 / 3 ) = 68.5581 deg

Neutrino = (257.8801 deg + 0 X 90.1793 deg ) X ( 4 / 15 ) = 257.8801 deg X ( 4 / 15 ) = 68.7680 deg

Arithmetic Mean of the Singles = 275.2461 deg / 4 = 68.81153 deg

Pairs:

Quark + Neutrino = (90.4706 deg + 257.8801 deg - 1 X 90+ deg) X ( 4 / 15 ) = 258.0611 deg X ( 4 / 15 ) = 68.8163 deg

Graviton + Neutrino = (90.5494 deg + 257.8801 deg - 1 X 90+ deg) X ( 4 / 15 ) = 258.1400 deg X ( 4 / 15 ) = 68.8373 deg

Quark + Boson = (90.4706 deg + 102.8371 deg - 1 X 90+ deg) X ( 2 / 3 ) = 103.0181 deg X ( 2 / 3 ) = 68.6788 deg

Graviton + Boson = 90.5494 deg + 102.8371 deg - 1 X 90+ deg) X ( 2 / 3 ) = 103.3097 deg X ( 2 / 3 ) = 68.7313 deg

Quark + Graviton = (90.4706 deg + 90.5494 deg + 2 X 90.5100 deg ) X ( 4 / 21/ ) = 362.0399 deg X ( 4 / 21 ) = 68.9600 deg

Boson + Neutrino = (102.8371 deg + 257.8801 deg + 0 X 90.1793 deg ) X ( 4 / 21 ) = 360.7172 deg X ( 4 / 21 ) = 68.7080 deg

The arithmetic mean of the Pairs = 412.73173 deg / 6 = 68.78862 deg

Triplets:

Quark + Graviton + Boson = (90.4706 deg + 90.5494 deg + 102.8371 deg - 2 X 90.34464 deg) X ( 2 / 3 ) = 103.1679 deg X ( 2 / 3 ) = 68.7785 deg

Quark + Graviton + Neutrino = (90.4706 deg + 90.5494 deg + 257.8801 deg - 2 X 90.34464 deg ) X ( 4 / 15 ) = 258.2108 X ( 4 / 15 ) = 68.8562 deg

Quark + Boson + Neutrino = (90.4706 deg + 102.8371 deg + 257.8801 deg - 1 X 90.2454 ) X ( 4 / 21 ) = 360.9423 deg X ( 4 / 21 ) = 68.7509 deg

Graviton + Boson + Neutrino = (90.5494 deg + 102.8371 deg + 257.8801 deg - 1 X 90.2454 deg) X ( 4 / 21 ) = 361.0212 deg X ( 4 / 21 ) = 68.7659 deg

The Arithmetic Mean of the Triplets = 275.15159 deg / 4 = 68.7879 deg

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

Quark + Graviton + Boson + Neutrino = (90.4706 deg + 90.5494 deg +102.8371 deg + 257.8801 deg - 2 X 90.28953 deg) X 4 / 21 ) = 361.15811 deg X ( 4 / 21 ) = δCP VIOLATION PHASE ANGLE QUARK  = 68.79202 deg = 1.20064725 radians

The arithmetic mean of Singles, Pairs, Triplets, and Quadruplet = ( 68.81153 deg X 4 + 68.78862 deg X 6 + 68.7879 deg X 4 + 68.79202 deg ) / 15 = 1031.92146 deg / 15 = δQUARK CP VIOLATION PHASE ANGLE = 68.79476 deg = 18.0104271 radians / 15 = 1.20069514 radians

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

Next articles - Graviton