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Center of mass energy problem

  1. Mar 2, 2017 #1
    1. The problem statement, all variables and given/known data
    While not explicitly a homework question, I am having some trouble with deriving expressions for the center of mass energy in a fixed target experiment versus a collider experiment. The question is effectively, "Derive an expression for the center of mass energy in a fixed target experiment and compare this to the center of mass energy in a collider experiment."

    2. Relevant equations
    The momentum 4-vector. Also, the formula for center of mass energy ##\sqrt{s}=\sqrt{(p_1+p_2)^2}##.

    3. The attempt at a solution
    For a fixed target experiment, we have the two momentum 4-vectors, ##(\frac{E_b}{c},p_b)##, and ##(m_tc,0)##, for the beam particle and target particle respectively. Then, $$s=\frac{E_b^2}{c^2}+m_t^2c^2+2E_bm_t-p_b^2.$$

    We can group the first and last term together to obtain ##m_b^2c^2+m_t^2c^2+2E_bm_t##. However, my textbook at this point claims that this is equivalent to ##2m^2c^2+2Em##. My question is then, would this not only hold true for ##m_b\approx m_t##?

    Similarly, for a collider experiment, we have, ##s=(\frac{E_A}{c}+\frac{E_B}{c})^2\rightarrow s=\frac{(E_A+E_B)^2}{c^2}##. Again, my textbook claims that this is equivalent to ##\frac{4E^2}{c^2}##, which again I think, should only hold true for ##E_A\approx E_B##.

    If I am not understanding incorrectly, why can these approximations be made?
     
  2. jcsd
  3. Mar 4, 2017 #2

    Orodruin

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    They indeed seem to be assuming the masses to be equal.
     
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