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CP on pion (combinations)

  1. Jun 7, 2010 #1

    I have a question regarding the CP operator on pion systems.
    1) CP [itex] \mid \pi^0 \rangle [/itex]
    2) CP [itex] \mid \pi^+ \pi^- \rangle [/itex]
    3) CP [itex] \mid \pi^0 \pi^0 \rangle [/itex]

    I'd like to solve this in the above ket notation and apply the operators as is on the different parts of the represented wave function. My solution for 2) is:
    CP [itex] \mid \pi^+ \pi^- \rangle [/itex]
    [itex] = C \mid \pi^- \pi^+ \rangle [/itex] (switch pions physically in e.g. x-coordinate)
    [itex] = \mid \pi^+ \pi^- \rangle [/itex] (invert charges)
    Thus CP is +1 for [itex] \mid \pi^+ \pi^- \rangle [/itex]. This does not seem to work for 1). Note I have somehow lost the notion of [itex] (-)^l [/itex] that should be present somewhere :S

    Any help is appreciated.
    Last edited: Jun 7, 2010
  2. jcsd
  3. Jun 7, 2010 #2
    write the state for the pi0 in terms quarks
  4. Jun 7, 2010 #3
    OK, this is where I get:
    [tex] CP \mid \pi^0 \rangle = CP \frac{\mid u \bar{u} \rangle - \mid d \bar{d} \rangle}{\sqrt{2}}[/tex]
    [tex]= \frac{ CP \mid u \bar{u} \rangle - CP \mid d \bar{d} \rangle}{\sqrt{2}}[/tex]
    [tex]= \frac{C \mid \bar{u} u \rangle - C \mid \bar{d} d \rangle}{\sqrt{2}}[/tex]
    [tex]= \frac{ \mid u \bar{u} \rangle - \mid d \bar{d} \rangle}{\sqrt{2}} = \mid \pi^0 \rangle[/tex]
    This would mean CP for a [tex]\pi^0[/tex] is +1, while I kind of remember it being -1... What do I do wrong?
  5. Jun 7, 2010 #4
    sorry for my first message, I did not pay attention.

    Together with the flavor wavefunction [itex]\frac{1}{\sqrt{2}}\left( |u\bar{u}\rangle - |d\bar{d}\rangle \right)[/itex]
    one has to take into account spin [itex]\frac{1}{\sqrt{2}}\left(|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle \right)[/itex] as well
    \frac{1}{\sqrt{2}}\left( |u_\uparrow\bar{u}_\downarrow\rangle - |d_\uparrow\bar{d}_\downarrow\rangle \right)
    - \frac{1}{\sqrt{2}}\left( |u_\downarrow\bar{u}_\uparrow\rangle -|d_\downarrow\bar{d}_\uparrow\rangle \right) \right]
    |u_\uparrow\bar{u}_\downarrow\rangle -
    |u_\downarrow\bar{u}_\uparrow\rangle -
    |d_\uparrow\bar{d}_\downarrow\rangle +
    |d_\downarrow\bar{d}_\uparrow\rangle \right]
    Last edited: Jun 8, 2010
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