Question 3.7: The Scattering Tensor for Scalar Particles Repeat the steps ......
The Scattering Tensor for Scalar Particles
Repeat the steps leading to (3.18) for scalar particles.
W_{\mu\nu}^{\mathrm{incl.}}=\left(-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{q^{2}}\right)W_{1}(\,Q^{2},\,\nu)\\ +\left(P_{\mu}-q_{\mu}\frac{P\cdot q}{q^{2}}\right)\left(P_{\nu}-q_{\nu}\frac{P\cdot q}{q^{2}}\right)\frac{W_{2}(Q^{2},\nu)}{M_{\mathrm{N}}^{2}} (3.18)
We start with (3.6). In the case of scalar particles all terms containing γ matrices vanish:
\Gamma_{\mu}=A\gamma_{\mu}+B P_{\mu}^{\prime}+C P_{\mu}+{\mathrm i}D P^{\prime\nu}\sigma_{\mu\nu}+{\mathrm i}E P^{\nu}\sigma_{\mu\nu}~, (3.6)
\Gamma_{\mu}=B P_{\mu}^{\prime}+C P_{\mu}\,\,\,. (1)
The requirement of gauge invariance, (3.8), then yields
q^{\mu}\,\bar{u}(P^{\prime})\,\Gamma_{\mu}\,u(P)=0\ . (3.8)
q^{\mu}{\Gamma}_{\mu}=(P^{\prime}{}^{\mu}-P^{\mu})\Gamma_{\mu}=B(P^{\prime}{}^{\mu}-P^{\mu})P^{\prime}{}_{\mu}+C(P^{\prime}{}^{\mu}-P^{\mu})P_{\mu}=B P^{\prime\mu}P_{\ \mu}^{\prime}-C P^{\mu}P_{\mu}=(B-C)M_{\mathrm{N}}^{2}=0\quad\Rightarrow C=B~, (2)
\Gamma_{\mu}=B(P_{\mu}^{\prime}+P_{\mu})\,\,\,. (3)
This is rewritten by replacing P_{\mu}^{\prime} by P_{\mu} and {\mathcal{q}}_{\mu}{\dot{\cdot}}
\Gamma_{\mu}=B(P_{\mu}+q_{\mu}+P_{\mu}) \\ =2B\left(P_{\mu}+\frac{1}{2}q_{\mu}\right) \\ =2B\left(P_{\mu}-q_{\mu}\frac{-q^{2}/2}{q^{2}}\right)\\ =2\left(P_{\mu}-q_{\mu}\frac{P\cdot q}{q^{2}}\right)B(Q^{2})~, (4)
where relation (3.7) has been employed in the last step. For a free scalar field one simply has
P^{\prime}\cdot q=P^{\prime2}-P^{\prime}\cdot P=\frac{1}{2}q^{2}\ . (3.7)
u^{*}(P)u(P)=\mathrm{const}\ . (5)
Therefore W_{\mu\nu} follows directly from \Gamma_{\mu} as
W_{\mu\nu}=\mathrm{const}\times \Gamma_{\mu}\Gamma_{\nu}=\mathrm{const}\times\left(P_{\mu}-q_{\mu}\frac{P\cdot q}{q^{2}}\right)\left(P_{\nu}-q_{\nu}\frac{P\cdot q}{q^{2}}\right)B^{2}(Q^{2})\;. (6)
Up to now we have considered the elastic process. To obtain the equation equivalent
to (6) for the inclusive inelastic case, we again have to replace B^{2}(\,Q^{2}\,) by B^{2}(\,O^{2},\,{\nu})=:W_{2}(\,Q^{2},\,\nu).
W_{\mu\nu}=\mathrm{const}\times\left(P_{\mu}-q_{\mu}\frac{P\cdot q}{q^{2}}\right)\left(P_{\nu}-q_{\nu}\frac{P\cdot q}{q^{2}}\right)W_{2}(Q^{2},\nu)\ . (7)
Comparing this result with (3.18) shows that there is only a \textstyle W_{2} function and no \textstyle W_{1} function for scalar particles. Consequently (3.66) then becomes
\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}x\,\mathrm{d}y}\bigg|_{\mathrm{eN}}=\frac{4\pi s\alpha^{2}}{Q^{4}}\Biggl[x y^{2}F_{1}^{\mathrm{eN}}(Q^{2},x)+\Biggl(1-y-\frac{x y M_{\mathrm{N}}^{2}}{s}\Biggr)F_{2}^{\mathrm{eN}}(Q^{2},x)\Biggl]. (3.66)
\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}x\mathrm{d}y}=\frac{4\pi s\alpha^{2}}{Q^{4}}(1-y)F_{2}(Q^{2},x)~, (8)
with a corresponding structure function F_{2}({Q^{2},x}). Here, in the last step, the term x y M_{\mathrm{N}}^{2}/s has been neglected for large s.