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Greiner Exercise 2.11

  1. Dec 27, 2016 #1
    1. The problem statement, all variables and given/known data
    Proof the leptonic-hadronic tensor multiplication, with ##p^\mu=(M,\textbf{0})## and ##p'^\mu=(E',\textbf{p}')## is rest target and final target momentum respectively, and ##k^\mu=(\omega,\textbf{k})##, ##k'\mu=(\omega'
    ,\textbf{k}')## is momenta of incoming and outgoing electron, hence we get

    $$L_{\mu\nu}T^{\mu\nu}=8\Big(2(k\cdot p)(k'\cdot p)+\frac{q^2}{2M^2}\Big)$$

    2. Relevant equations

    The hadronic tensor,
    $$T^{\mu\nu}=(p+p')^\mu(p+p')^\nu$$

    The leptonic tensor,
    $$L_{\mu\nu}=2\Big(k'_\mu k\nu+k'_\nu k_\mu + \frac{1}{2}q^2g_{\mu\nu}\Big)$$.

    3. The attempt at a solution

    I try something like this

    $$ L_{\mu\nu}T^{\mu\nu} = 2\Big(k'_\mu k_\nu+k'_\nu k_\mu + \frac{1}{2}q^2g_{\mu\nu}\Big) (p+p')^\mu(p+p')^\nu $$

    Expanding the second term in the rhs

    $$ L_{\mu\nu}T^{\mu\nu} = 2\Big(k'_\mu k_\nu+k'_\nu k_\mu + \frac{1}{2}q^2g_{\mu\nu}\Big) (p^\mu p^\nu + p^\mu p'^\nu + p'^\mu p^\nu + p'^\mu p'^\nu) $$

    hence,

    $$ L_{\mu\nu}T^{\mu\nu} = 2 \Big[ (k' \cdot p)(k \cdot p) + 2 (k'\cdot p)(k \cdot p') + (k' \cdot p')(k \cdot p') + (k' \cdot p)(k \cdot p) + 2 (k'\cdot p')(k \cdot p) + (k' \cdot p')(k \cdot p') + \frac{1}{2}q^2 (p^\mu p^\nu + p^\mu p'^\nu + p'^\mu p^\nu + p'^\mu p'^\nu) \Big] $$
     
  2. jcsd
  3. Dec 29, 2016 #2
    I think Greiner missprint the result. I've finally got the contraction result:

    $$ L_{\mu\nu}=8\Big(2(k\cdot p)(k'\dot p)+\frac{1}{2}q^2 M^2\Big) ,$$

    and if I use the relativistic limit, my result is confirmed.
     
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