17 July 2019 AD; St Alexis the Beggar; Bl. martyrs of Compiegne

To obtain the values of the Final Quark Mixing Angles I will use the transformation of a full plane into a positive part of the real plane:

The transformation is as follows:

first quadrant: theta final = theta

second quadrant: theta final = 180 degrees - absolute value |theta|

third quadrant: theta final = 180 degrees + absolute value |theta|

fourth quadrant: theta final = 360 degrees - absolute value |theta|

This is a boring part - just calculations but it has to be done so you know where the Final Angles come from.

There will be three different angles, with the same quantum numbers as before - for "regular"  quark mixing angles.

These printouts are the results of substituting for Real value X constant C_M_0, C_1 and C_2 in the main program (more about it later):

First final angle:

These printouts are the results of substituting for Real value X constant C_M_0 in the main program (more about it later):

ALFA C_M_0 INVISIBLE
A + Bi
0.31543485581699715059D+00 -0.26183391631054520055D+00
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
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

PRODUCT
( A + Bi ) * ( C + Di )
0.33485188431408513532D+00 -0.15171337520996638903D+00
0.36761764463405932624D+00 -0.42540904026564924090D+00
MAGNITUDE & THETA IN DEGREES
0.36761764463405932624D+00 -0.24374142573932598310D+02
MAGNITUDE & THETA FINAL IN RADIANS
0.36761764463405932624D+00 0.58577762669139374907D+01
MAGNITUDE & THETA FINAL IN DEGREES
0.36761764463405932624D+00 0.33562585742606745498D+03

DIVISION
( A + Bi ) / ( C + Di )
0.26210451532486189752D+00 -0.37454866607077447194D+00
0.45714929750474003756D+00 -0.96021292120952161930D+00
MAGNITUDE & THETA IN DEGREES
0.45714929750474003756D+00 -0.55016147819233438554D+02
MAGNITUDE & THETA FINAL IN RADIANS
0.45714929750474003756D+00 0.53229723859700648347D+01
MAGNITUDE & THETA FINAL IN DEGREES
0.45714929750474003756D+00 0.30498385218076657566D+03

SUM
( A + Bi ) + ( C + Di )
0.11803104629924687785D+01 -0.24889930064429688361D-01
0.11805728684279122032D+01 -0.21084487866453137384D-01
MAGNITUDE & THETA IN DEGREES
0.11805728684279122032D+01 -0.12080521679425584303D+01 This is angle θ13  from before the transformation
MAGNITUDE & THETA FINAL IN RADIANS
0.11805728684279122032D+01 0.62621008193131331154D+01
MAGNITUDE & THETA FINAL IN DEGREES
0.11805728684279122032D+01 0.35879194783205741714D+03 This is angle θ13 after the transformation

θ13 = (- 0.120805216794...D+01) * ( 1 / 6 ) = - 0.20134 degrees

θ13 Final = (0.35879194783205741714D+03) * (1/6) = 59.7987 degrees

i.e ("ordinary" angle) * (quantum number) = final angle

Second final angle is the same as for ordinary mixing angle:

These printouts are the results of substituting for Real value X constant C_1 in the main program (more about it later):

ALFA C_1 INVISIBLE
A + Bi
0.11200551706560395449D+01 0.33638129160159646602D+00
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
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

PRODUCT
( A + Bi ) * ( C + Di )
0.11991688495590924202D+01 0.33985710667019281006D+00
0.12463983242555776165D+01 0.27616849915837560214D+00
MAGNITUDE & THETA IN DEGREES
0.12463983242555776165D+01 0.15823289436237150341D+02
MAGNITUDE & THETA FINAL IN RADIANS
0.12463983242555776165D+01 0.27616849915837560214D+00
MAGNITUDE & THETA FINAL IN DEGREES
0.12463983242555776165D+01 0.15823289436237150341D+02

DIVISION
( A + Bi ) / ( C + Di )
0.10458838456949364204D+01 0.33196338645390927047D+00
0.10973024690720318830D+01 0.30734255005589528054D+00
MAGNITUDE & THETA IN DEGREES
0.10973024690720318830D+01 0.17609430982991042924D+02
MAGNITUDE & THETA FINAL IN RADIANS
0.10973024690720318830D+01 0.30734255005589528054D+00
MAGNITUDE & THETA FINAL IN DEGREES
0.10973024690720318830D+01 0.17609430982991042924D+02

SUM
( A + Bi ) + ( C + Di )
0.21857000214842718933D+01 0.31976971287989758297D+00
0.22089674631355795675D+01 0.14527022249462359649D+00
MAGNITUDE & THETA IN DEGREES
0.22089674631355795675D+01 0.83233706378683649518D+01 This is angle θ23 before the transformation
MAGNITUDE & THETA FINAL IN RADIANS
0.22089674631355795675D+01 0.14527022249462359649D+00
MAGNITUDE & THETA FINAL IN DEGREES
0.22089674631355795675D+01 0.83233706378683649518D+01 This is angle θ23 after the transformation

θ23 = ( 0.83233706378683649518D+01  ) * ( 1 / ( 3.5 )) = 2.3781 degrees

θ23 Final = (0.83233706378683649518D+01) * (1/(3.5)) = 2.3781 degrees

i.e ("ordinary" angle) * (quantum number) = final angle

Now the third angle, the same as for "ordinary" quark mixing angle:

These printouts are the results of substituting for Real value X constant C_2 in the main program (more about it later):

ALFA C_2 INVISIBLE
A + Bi
0.48510386600283728065D-01 0.16605512949939024714D+00
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
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

PRODUCT
( A + Bi ) * ( C + Di )
-0.81422848007180914642D-01 0.59923187905960917377D+00
0.60473839390178496522D+00 0.17058479251603371196D+01
MAGNITUDE & THETA IN DEGREES
0.60473839390178496522D+00 0.97737886602835629901D+02
MAGNITUDE & THETA FINAL IN RADIANS
0.60473839390178496522D+00 0.14357447284294559964D+01
MAGNITUDE & THETA FINAL IN DEGREES
0.60473839390178496522D+00 0.82262113397164370099D+02

DIVISION
( A + Bi ) / ( C + Di )
0.32013672048757012278D-01 0.37738986962879875198D-01
0.49488446480262265892D-01 0.86729481443638167804D+00
MAGNITUDE & THETA IN DEGREES
0.49488446480262265892D-01 0.49692332460786573733D+02
MAGNITUDE & THETA FINAL IN RADIANS
0.49488446480262265892D-01 0.86729481443638156701D+00
MAGNITUDE & THETA FINAL IN DEGREES
0.49488446480262265892D-01 0.49692332460786566628D+02

SUM
( A + Bi ) + ( C + Di )
0.32414088994293002166D+01 0.15891463038124078455D+01
0.36100024415809062184D+01 0.45582856957724077196D+00
MAGNITUDE & THETA IN DEGREES
0.36100024415809062184D+01 0.26117053218261293779D+02 This is angle θ12 before the transformation
MAGNITUDE & THETA FINAL IN RADIANS
0.36100024415809062184D+01 0.45582856957724077196D+00
MAGNITUDE & THETA FINAL IN DEGREES
0.36100024415809062184D+01 0.26117053218261293779D+02 This is angle θ12 after the transformation

θ12 = (0.26117053218261293779D+02) * (1/2) = 13.0585 degrees

θ12 Final = (0.26117053218261293779D+02) * (1/2) = 13.0585 degrees

i.e ("ordinary" angle) * (quantum number) = final angle

Now, the sum of all the six quark mixing angles (three "ordinary" and three after the transformation) is:

(-.0.20134 + 59.7987) + ( 2.3781 + 2.3781) + (13.0585 + 13.0585) = 90.4706 degrees

As you can see, the result is slightly bigger than 90 degrees, suggesting the spherical angle.