# Neutrino Mixing Matrix

Please look at equation 3 and 4 of this paper

http://arxiv.org/abs/0707.2481v1

I am facing problem to write the matrix Us

Can anybody help me to write the complete matrix?

Hepth
Gold Member
I think I understand how to write it, but I feel like I'm getting some different minus signs. Basically you want the SO(5) rotation group, and matrices for one direction about another (plane, or whatever its called).

If you look at http://reference.wolfram.com/mathematica/ref/RotationMatrix.html
Under Applications, they show how to generate the matrix form for a rotation in SO[N]. Then you take these and multiply them how they have it in the paper, order obviously matters.

I do:
Code:
SO[n_] := Map[RotationMatrix[\[Theta], #] &,    Subsets[Table[UnitVector[n, i], {i, n}], {2}]];
SO5MAP = Map[ MatrixForm, SO];
\[Theta][a_, b_] := Subscript[\[CapitalTheta], a, b]
R12 = SO5MAP[] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      1, 2]};
R13 = SO5MAP[] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      1, 3]};
R23 = SO5MAP[] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      2, 3]};
R14 = SO5MAP[] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      1, 4]};
R15 = SO5MAP[] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      1, 5]};
R24 = SO5MAP[] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      2, 4]};
R25 = SO5MAP[] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      2, 5]};
R34 = SO5MAP[] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      3, 4]};
R35 = SO5MAP[] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      3, 5]};
R45 = SO5MAP[] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      4, 5]};
ROT = R45.(R35.(R34.(R25.(R24.(R15.(R14.(R23.(R13.R12))))))));
ROT /. {Cos[Subscript[\[CapitalTheta], a_, b_]] -> Subscript[c, a, b],     Sin[Subscript[\[CapitalTheta], a_, b_]] -> Subscript[s, a, b]} //   Simplify // MatrixForm

The output looks like

$$\left( \begin{array}{ccccc} c_{1,2} c_{1,3} c_{1,4} c_{1,5} & -c_{1,3} c_{1,4} c_{1,5} s_{1,2} & -c_{1,4} c_{1,5} s_{1,3} & -c_{1,5} s_{1,4} & -s_{1,5} \\ c_{2,3} c_{2,4} c_{2,5} s_{1,2}-c_{1,2} \left(c_{2,4} c_{2,5} s_{1,3} s_{2,3}+c_{1,3} \left(c_{2,5} s_{1,4} s_{2,4}+c_{1,4} s_{1,5} s_{2,5}\right)\right) & c_{1,2} c_{2,3} c_{2,4} c_{2,5}+s_{1,2} \left(c_{2,4} c_{2,5} s_{1,3} s_{2,3}+c_{1,3} \left(c_{2,5} s_{1,4} s_{2,4}+c_{1,4} s_{1,5} s_{2,5}\right)\right) & -c_{1,3} c_{2,4} c_{2,5} s_{2,3}+s_{1,3} \left(c_{2,5} s_{1,4} s_{2,4}+c_{1,4} s_{1,5} s_{2,5}\right) & -c_{1,4} c_{2,5} s_{2,4}+s_{1,4} s_{1,5} s_{2,5} & -c_{1,5} s_{2,5} \\ c_{3,5} \left(s_{1,2} \left(c_{3,4} s_{2,3}-c_{2,3} s_{2,4} s_{3,4}\right)+c_{1,2} \left(c_{2,3} c_{3,4} s_{1,3}+\left(-c_{1,3} c_{2,4} s_{1,4}+s_{1,3} s_{2,3} s_{2,4}\right) s_{3,4}\right)\right)-\left(c_{1,2} c_{1,3} c_{1,4} c_{2,5} s_{1,5}+\left(c_{2,3} c_{2,4} s_{1,2}-c_{1,2} \left(c_{2,4} s_{1,3} s_{2,3}+c_{1,3} s_{1,4} s_{2,4}\right)\right) s_{2,5}\right) s_{3,5} & c_{3,5} \left(c_{1,2} c_{3,4} s_{2,3}+s_{1,2} \left(c_{1,3} c_{2,4} s_{1,4}-s_{1,3} s_{2,3} s_{2,4}\right) s_{3,4}-c_{2,3} \left(c_{3,4} s_{1,2} s_{1,3}+c_{1,2} s_{2,4} s_{3,4}\right)\right)-\left(c_{2,4} \left(c_{1,2} c_{2,3}+s_{1,2} s_{1,3} s_{2,3}\right) s_{2,5}+c_{1,3} s_{1,2} \left(-c_{1,4} c_{2,5} s_{1,5}+s_{1,4} s_{2,4} s_{2,5}\right)\right) s_{3,5} & s_{1,3} \left(c_{2,4} c_{3,5} s_{1,4} s_{3,4}+\left(c_{1,4} c_{2,5} s_{1,5}-s_{1,4} s_{2,4} s_{2,5}\right) s_{3,5}\right)+c_{1,3} \left(c_{2,3} c_{3,4} c_{3,5}+s_{2,3} \left(c_{3,5} s_{2,4} s_{3,4}+c_{2,4} s_{2,5} s_{3,5}\right)\right) & c_{2,5} s_{1,4} s_{1,5} s_{3,5}+c_{1,4} \left(-c_{2,4} c_{3,5} s_{3,4}+s_{2,4} s_{2,5} s_{3,5}\right) & -c_{1,5} c_{2,5} s_{3,5} \\ c_{4,5} \left(s_{1,2} \left(c_{2,3} c_{3,4} s_{2,4}+s_{2,3} s_{3,4}\right)+c_{1,2} \left(c_{1,3} c_{2,4} c_{3,4} s_{1,4}+s_{1,3} \left(-c_{3,4} s_{2,3} s_{2,4}+c_{2,3} s_{3,4}\right)\right)\right)-\left(s_{1,2} \left(c_{3,4} s_{2,3} s_{3,5}+c_{2,3} \left(c_{2,4} c_{3,5} s_{2,5}-s_{2,4} s_{3,4} s_{3,5}\right)\right)+c_{1,2} \left(s_{1,3} \left(-c_{2,4} c_{3,5} s_{2,3} s_{2,5}+\left(c_{2,3} c_{3,4}+s_{2,3} s_{2,4} s_{3,4}\right) s_{3,5}\right)+c_{1,3} \left(c_{1,4} c_{2,5} c_{3,5} s_{1,5}-s_{1,4} \left(c_{3,5} s_{2,4} s_{2,5}+c_{2,4} s_{3,4} s_{3,5}\right)\right)\right)\right) s_{4,5} & c_{4,5} \left(c_{3,4} \left(-c_{1,3} c_{2,4} s_{1,2} s_{1,4}+\left(c_{1,2} c_{2,3}+s_{1,2} s_{1,3} s_{2,3}\right) s_{2,4}\right)+\left(-c_{2,3} s_{1,2} s_{1,3}+c_{1,2} s_{2,3}\right) s_{3,4}\right)-\left(c_{3,5} \left(c_{2,4} \left(c_{1,2} c_{2,3}+s_{1,2} s_{1,3} s_{2,3}\right) s_{2,5}+c_{1,3} s_{1,2} \left(-c_{1,4} c_{2,5} s_{1,5}+s_{1,4} s_{2,4} s_{2,5}\right)\right)+\left(c_{1,2} c_{3,4} s_{2,3}+s_{1,2} \left(c_{1,3} c_{2,4} s_{1,4}-s_{1,3} s_{2,3} s_{2,4}\right) s_{3,4}-c_{2,3} \left(c_{3,4} s_{1,2} s_{1,3}+c_{1,2} s_{2,4} s_{3,4}\right)\right) s_{3,5}\right) s_{4,5} & c_{4,5} \left(-c_{3,4} \left(c_{2,4} s_{1,3} s_{1,4}+c_{1,3} s_{2,3} s_{2,4}\right)+c_{1,3} c_{2,3} s_{3,4}\right)-\left(c_{3,5} \left(-c_{1,4} c_{2,5} s_{1,3} s_{1,5}+\left(-c_{1,3} c_{2,4} s_{2,3}+s_{1,3} s_{1,4} s_{2,4}\right) s_{2,5}\right)+\left(c_{2,4} s_{1,3} s_{1,4} s_{3,4}+c_{1,3} \left(c_{2,3} c_{3,4}+s_{2,3} s_{2,4} s_{3,4}\right)\right) s_{3,5}\right) s_{4,5} & c_{2,5} c_{3,5} s_{1,4} s_{1,5} s_{4,5}+c_{1,4} \left(c_{3,5} s_{2,4} s_{2,5} s_{4,5}+c_{2,4} \left(c_{3,4} c_{4,5}+s_{3,4} s_{3,5} s_{4,5}\right)\right) & -c_{1,5} c_{2,5} c_{3,5} s_{4,5} \\ c_{4,5} \left(s_{1,2} \left(c_{3,4} s_{2,3} s_{3,5}+c_{2,3} \left(c_{2,4} c_{3,5} s_{2,5}-s_{2,4} s_{3,4} s_{3,5}\right)\right)+c_{1,2} \left(s_{1,3} \left(-c_{2,4} c_{3,5} s_{2,3} s_{2,5}+\left(c_{2,3} c_{3,4}+s_{2,3} s_{2,4} s_{3,4}\right) s_{3,5}\right)+c_{1,3} \left(c_{1,4} c_{2,5} c_{3,5} s_{1,5}-s_{1,4} \left(c_{3,5} s_{2,4} s_{2,5}+c_{2,4} s_{3,4} s_{3,5}\right)\right)\right)\right)+\left(s_{1,2} \left(c_{2,3} c_{3,4} s_{2,4}+s_{2,3} s_{3,4}\right)+c_{1,2} \left(c_{1,3} c_{2,4} c_{3,4} s_{1,4}+s_{1,3} \left(-c_{3,4} s_{2,3} s_{2,4}+c_{2,3} s_{3,4}\right)\right)\right) s_{4,5} & c_{4,5} \left(c_{3,5} \left(c_{2,4} \left(c_{1,2} c_{2,3}+s_{1,2} s_{1,3} s_{2,3}\right) s_{2,5}+c_{1,3} s_{1,2} \left(-c_{1,4} c_{2,5} s_{1,5}+s_{1,4} s_{2,4} s_{2,5}\right)\right)+\left(c_{1,2} c_{3,4} s_{2,3}+s_{1,2} \left(c_{1,3} c_{2,4} s_{1,4}-s_{1,3} s_{2,3} s_{2,4}\right) s_{3,4}-c_{2,3} \left(c_{3,4} s_{1,2} s_{1,3}+c_{1,2} s_{2,4} s_{3,4}\right)\right) s_{3,5}\right)+\left(c_{3,4} \left(-c_{1,3} c_{2,4} s_{1,2} s_{1,4}+\left(c_{1,2} c_{2,3}+s_{1,2} s_{1,3} s_{2,3}\right) s_{2,4}\right)+\left(-c_{2,3} s_{1,2} s_{1,3}+c_{1,2} s_{2,3}\right) s_{3,4}\right) s_{4,5} & c_{4,5} \left(c_{3,5} \left(-c_{1,4} c_{2,5} s_{1,3} s_{1,5}+\left(-c_{1,3} c_{2,4} s_{2,3}+s_{1,3} s_{1,4} s_{2,4}\right) s_{2,5}\right)+\left(c_{2,4} s_{1,3} s_{1,4} s_{3,4}+c_{1,3} \left(c_{2,3} c_{3,4}+s_{2,3} s_{2,4} s_{3,4}\right)\right) s_{3,5}\right)+\left(-c_{3,4} \left(c_{2,4} s_{1,3} s_{1,4}+c_{1,3} s_{2,3} s_{2,4}\right)+c_{1,3} c_{2,3} s_{3,4}\right) s_{4,5} & -c_{4,5} \left(c_{2,5} c_{3,5} s_{1,4} s_{1,5}+c_{1,4} \left(c_{3,5} s_{2,4} s_{2,5}+c_{2,4} s_{3,4} s_{3,5}\right)\right)+c_{1,4} c_{2,4} c_{3,4} s_{4,5} & c_{1,5} c_{2,5} c_{3,5} c_{4,5} \end{array} \right)$$

again, im not sure where there is a sign difference.

Thanks a loooooooooooot ! :)

Hepth
Gold Member
Ah actually what you need to do is take the transpose of each of those matrices, then you get whats right:

R12 = Transpose[SO5MAP[]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][1, 2]};

The transpose is what you want, and then apply those in order. The signs will be correct then.

Thanks again. I realized that too. So did not check recently.