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Neutrino Mixing Matrix

  1. Sep 23, 2012 #1
    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?
     
  2. jcsd
  3. Sep 24, 2012 #2

    Hepth

    User Avatar
    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 (Text):

    SO[n_] := Map[RotationMatrix[\[Theta], #] &,    Subsets[Table[UnitVector[n, i], {i, n}], {2}]];
    SO5MAP = Map[ MatrixForm, SO[5]];
    \[Theta][a_, b_] := Subscript[\[CapitalTheta], a, b]
    R12 = SO5MAP[[1]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      1, 2]};
    R13 = SO5MAP[[2]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      1, 3]};
    R23 = SO5MAP[[5]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      2, 3]};
    R14 = SO5MAP[[3]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      1, 4]};
    R15 = SO5MAP[[4]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      1, 5]};
    R24 = SO5MAP[[6]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      2, 4]};
    R25 = SO5MAP[[7]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      2, 5]};
    R34 = SO5MAP[[8]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      3, 4]};
    R35 = SO5MAP[[9]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      3, 5]};
    R45 = SO5MAP[[10]] /. {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

    [tex]
    \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)
    [/tex]

    again, im not sure where there is a sign difference.
     
  4. Sep 24, 2012 #3
    Thanks a loooooooooooot ! :)
     
  5. Sep 25, 2012 #4

    Hepth

    User Avatar
    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[[1]]] /. {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.
     
  6. Sep 28, 2012 #5
    Thanks again. I realized that too. So did not check recently.
     
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