18-19 March 2025 AD, Tuesday, Wednesday

St. Cyril of Jerusalem (386 AD); Our Lady of Mercy; St. Joseph (1^st century AD)

 

In this article, I will show how to calculate three mixing angles of quarks using angles from main equations (‘visible’ and ‘invisible’ parts) and applying multiplication factors.

I take into consideration two sets of measurements, one from the year 2008 (Wikipedia) and the more recent one 2022 PDG (Particle Data Group).

The first set was beautifully simple, giving multiplication factors in the form of simple fractions (m/n). However, the 2022 measurements published by PDG complicated this simplicity. Transcendental fractions had to be introduced.

How to calculate the mixing angles of quarks using angles from the main equations and mixing angles?

First, we need values from the sum of the given constants. Then, multiplication factors have to be used to obtain exact value of a mixing angle of a quark, Theta₁₃, Theta₂₃ and Theta₁₂ (this is the order the angles appear in the main equation). It’s very simple.

Let’s use Wikipedia Data 2008   >>>

 

https://en.wikipedia.org/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix

 

Quark theta₁₃ = 0.201° ±0.11

Quark theta₂₃ = 2.38° ±0.06

Quark theta₁₂ = 13.04° ±0.05

Now, the printout from the main equation with the substituted values for the constant of the quark theta₁₃ and the sum for the constant CM₀, where the quark theta₁₃ is concealed.

 

ALFA C_M_0 INVISIBLE

 A + Bi

    0.31543485581699715059D+00   -0.26183391631054520055D+00

 MAGNITUDE & THETA IN RADIANS

    0.40994651845674623392D+00   -0.69281098073758540234D+00

 MAGNITUDE & THETA IN DEGREES

    0.40994651845674623392D+00   -0.39695145196583020208D+02

 MAGNITUDE & THETA FINAL IN RADIANS

    0.40994651845674623392D+00    0.55903743264420011627D+01

 MAGNITUDE & THETA FINAL IN DEGREES

    0.40994651845674623392D+00    0.32030485480341701532D+03

 

 ALFA C_M_0 VISIBLE

 C + Di

    0.86487560717547162792D+00    0.23694398624611551218D+00

 MAGNITUDE & THETA IN RADIANS

    0.89674537551377431210D+00    0.26740194047193616145D+00

 MAGNITUDE & THETA IN DEGREES

    0.89674537551377431210D+00    0.15321002622650418346D+02

 MAGNITUDE & THETA FINAL IN RADIANS

    0.89674537551377431210D+00    0.26740194047193616145D+00

 MAGNITUDE & THETA FINAL IN DEGREES

    0.89674537551377431210D+00    0.15321002622650418346D+02

 

Now, we have the sum of these two entries. We will be interested in the polar coordinates ‘magnitude and theta in degrees.’

 

SUM

  ( A + Bi ) + ( C + Di )

    0.11803104629924687785D+01   -0.24889930064429688361D-01

 MAGNITUDE & THETA IN RADIANS

    0.11805728684279122032D+01   -0.21084487866453137384D-01

 MAGNITUDE & THETA IN DEGREES

    0.11805728684279122032D+01   -0.12080521679425584303D+01

 MAGNITUDE & THETA FINAL IN RADIANS

    0.11805728684279122032D+01    0.62621008193131331154D+01

 MAGNITUDE & THETA FINAL IN DEGREES

    0.11805728684279122032D+01    0.35879194783205741714D+03

 

The multiplication factor for quark theta₁₃ is equal to () and the angle quark theta₁₃ is equal to - 1.2080521679425584303° x () = - 0.201342028° 

versus official data 0.201° with no error.

 

Now, the printout from the main equation with the substituted values for the constant of the quark theta₂₃ and the sum for the constant C₁, where the quark theta₂₃ is concealed.

 

ALFA C_1 INVISIBLE

 A + Bi

    0.11200551706560395449D+01    0.33638129160159646602D+00

 MAGNITUDE & THETA IN RADIANS

    0.11694767884198848940D+01    0.29175552460713544134D+00

 MAGNITUDE & THETA IN DEGREES

    0.11694767884198848940D+01    0.16716360209614094856D+02

 MAGNITUDE & THETA FINAL IN RADIANS

    0.11694767884198848940D+01    0.29175552460713544134D+00

 MAGNITUDE & THETA FINAL IN DEGREES

    0.11694767884198848940D+01    0.16716360209614094856D+02

 

 ALFA C_1 VISIBLE

 C + Di

    0.10656448508282325704D+01   -0.16611578721698876115D-01

 MAGNITUDE & THETA IN RADIANS

    0.10657743159995707316D+01   -0.15587025448759837462D-01

 MAGNITUDE & THETA IN DEGREES

    0.10657743159995707316D+01   -0.89307077337694673602D+00

 MAGNITUDE & THETA FINAL IN RADIANS

    0.10657743159995707316D+01    0.62675982817308266704D+01

 MAGNITUDE & THETA FINAL IN DEGREES

    0.10657743159995707316D+01    0.35910692922662309456D+03

 

Now, we have the sum of these two entries. We will be interested in the polar coordinates ‘magnitude and theta in degrees.’

 

SUM

  ( A + Bi ) + ( C + Di )

    0.21857000214842718933D+01    0.31976971287989758297D+00

 MAGNITUDE & THETA IN RADIANS

    0.22089674631355795675D+01    0.14527022249462359649D+00

 MAGNITUDE & THETA IN DEGREES

    0.22089674631355795675D+01    0.83233706378683649518D+01

 MAGNITUDE & THETA FINAL IN RADIANS

    0.22089674631355795675D+01    0.14527022249462359649D+00

 MAGNITUDE & THETA FINAL IN DEGREES

    0.22089674631355795675D+01    0.83233706378683649518D+01

 

The multiplication factor for quark theta₂₃ is equal to () and the angle quark theta₂₃ is equal to 8.3233706378683649518° x () = 2.378105897° 

versus official data 2.38° with almost no error.

 

Now, the printout from the main equation with the substituted values for the constant of the quark theta₁₂ and the sum for the constant C₂, where the quark theta₁₂ is concealed.

 

ALFA C_2 INVISIBLE

 A + Bi

    0.48510386600283728065D-01    0.16605512949939024714D+00

 MAGNITUDE & THETA IN RADIANS

    0.17299584862408765029D+00    0.12865713697983593988D+01

 MAGNITUDE & THETA IN DEGREES

    0.17299584862408765029D+00    0.73715109531811108923D+02

 MAGNITUDE & THETA FINAL IN RADIANS

    0.17299584862408765029D+00    0.12865713697983593988D+01

 MAGNITUDE & THETA FINAL IN DEGREES

    0.17299584862408765029D+00    0.73715109531811108923D+02

 

 ALFA C_2 VISIBLE

 C + Di

    0.31928985128290165996D+01    0.14230911743130176816D+01

 MAGNITUDE & THETA IN RADIANS

    0.34956815363578743749D+01    0.41927655536197777630D+00

 MAGNITUDE & THETA IN DEGREES

    0.34956815363578743749D+01    0.24022777071024531637D+02

 MAGNITUDE & THETA FINAL IN RADIANS

    0.34956815363578743749D+01    0.41927655536197777630D+00

 MAGNITUDE & THETA FINAL IN DEGREES

    0.34956815363578743749D+01    0.24022777071024531637D+02

 

Now, we have the sum of these two entries. We will be interested in the polar coordinates ‘magnitude and theta in degrees.’

 

SUM

  ( A + Bi ) + ( C + Di )

    0.32414088994293002166D+01    0.15891463038124078455D+01

 MAGNITUDE & THETA IN RADIANS

    0.36100024415809062184D+01    0.45582856957724077196D+00

 MAGNITUDE & THETA IN DEGREES

    0.36100024415809062184D+01    0.26117053218261293779D+02

 MAGNITUDE & THETA FINAL IN RADIANS

    0.36100024415809062184D+01    0.45582856957724077196D+00

 MAGNITUDE & THETA FINAL IN DEGREES

    0.36100024415809062184D+01    0.26117053218261293779D+02

 

The multiplication factor for quark theta₁₂ is equal to () and the angle quark theta₁₂ is equal to 26.117053218261293779° x () = 13.058526609° 

versus official data 13.04°, with slight error.

 

Next article will be about multiplication factors and quark angles for the new 2022 PDG Data.