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4-momentum dot product help!

  1. Sep 18, 2014 #1
    1. The problem statement, all variables and given/known data

    So a kaon moving at some speed in the +x direction spontaneously decays into one pion and one anti-pion. The anti-pion moves away with velocity of 0.8c, and the pion moves away with velocity of 0.9c.

    Mass of kaon = 498 MeV/c^2
    Mass of pion/anti-pion = 140 MeV/c^2

    2. Relevant equations

    I understand that momentum of the kaon qualms the momentums of the two pions. p(kaon) = p(pion) + p(anti-pion). I can then square both sides and use the principle of invariance.

    3. The attempt at a solution

    What I'm having issues with is how to calculate the term of +2 p(pion)•p(anti-pion).

    I don't understand how to multiply the vector parts. p(pion) = ( E(pion)/c , vector p(pion) )
    p(anti-pion) = ( E(anti-pion)/c , vector p(anti-pion) )

    I'll get a cos(theta) term out of this dot product on the vector side but how do I use the velocities I have to get a dot product of those two momentum vectors? They're both moving so I am guessing I have to substitute in (gamma)(mass of the particle)(velocity vector) but I just don't understand how to do the math after that.

    Help please!
     
  2. jcsd
  3. Sep 19, 2014 #2

    vanhees71

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    2016 Award

    What's the question? I guess you should get the original speed of the kaon. Let's also set [itex]c=1[/itex], which makes everything a lot easier. Now we measure masses, energies, and momenta in the same unit, MeV.

    You have energy-momentum conservation
    [tex]p=p_1'+p_2',[/tex]
    where [itex]p[/itex] is the four-momentum of the kaon in the initial state, and [itex]p_1'[/itex], [itex]p_2'[/itex] are the four momenta of the pions in the final state.

    Further you have the on-shell conditions
    [tex]p^2=m_{\text{K}}^2, \quad p_1'^2=p_2'^2=m_{\pi}^2.[/tex]
    The relation between three-velocity and energy and momentum is
    [tex]\vec{v}=\frac{\vec{p}}{E},[/tex]
    for all the particles.

    Now you can calculate the momenta of the pions and then use the above properties of the four-vectors. From this you should be able to get the three-momentum of the kaon and then its speed.
     
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