Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

180deg Rotation in Isospin space

  1. Feb 15, 2009 #1

    malawi_glenn

    User Avatar
    Science Advisor
    Homework Helper

    The G-partiy of the pion is -1, which is rotation in isospin space around y-axis by angle [itex] \pi [/itex] followed by C-conjugation.

    The rotation matrix around y-axis , with angle [itex] \pi [/itex], is: (e.g. Sakurai, Halzen ..)

    [tex]\begin{pmatrix}0 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & 0 \end{pmatrix} [/tex]

    Thus on [itex] \pi^+ [/itex]: T = 1, Tz = +1:
    [tex]\begin{pmatrix}0 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0\\ 0 \\ 1 \end{pmatrix} = \pi^-[/tex]

    But this is not [itex] -\pi^- [/itex], which is required (see e.g. http://arxiv.org/PS_cache/hep-ph/pdf/9903/9903256v1.pdf page 14)

    I feel really stupid now, I don't get the required minus sign for pi- and pi+, only for pi0....

    The rotation matrix I use is:
    [tex]\begin{pmatrix} \tfrac{1}{2}(1+\cos \beta ) & - \tfrac{1}{\sqrt{2}}\sin \beta & \tfrac{1}{2}(1 - \cos \beta ) \\ -\tfrac{1}{\sqrt{2}}\sin \beta& \cos \beta & - \tfrac{1}{\sqrt{2}}\sin \beta \\ \tfrac{1}{2}(1-\cos \beta ) & \tfrac{1}{\sqrt{2}}\sin \beta & \tfrac{1}{2}(1 + \cos \beta ) \end{pmatrix}[/tex]
     
  2. jcsd
  3. Feb 19, 2009 #2

    malawi_glenn

    User Avatar
    Science Advisor
    Homework Helper

    I found that the author has made an error, one should rotate around x-axis with that convention of C-parity.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: 180deg Rotation in Isospin space
  1. Photon Isospin (Replies: 3)

  2. Isospin question (Replies: 6)

  3. Isospin question (Replies: 3)

  4. Isospin asymmetry (Replies: 1)

  5. Decomposed isospin (Replies: 9)

Loading...