Leptonic-Hadronic Tensor Multiplication Proof with Rest and Final Target Momenta

[Muh. Fauzi M.]

Homework Statement

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

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

Homework Equations

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

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

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,

$$ L_{\mu\nu}T^{\mu\nu} = 2 \Big[ (k' \cdot p)(k \cdot p) + 2 (k'\cdot p)(k \cdot p') + (k' \cdot p')(k \cdot p') + (k' \cdot p)(k \cdot p) + 2 (k'\cdot p')(k \cdot p) + (k' \cdot p')(k \cdot p') + \frac{1}{2}q^2 (p^\mu p^\nu + p^\mu p'^\nu + p'^\mu p^\nu + p'^\mu p'^\nu) \Big] $$

 

[Muh. Fauzi M.]

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.