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## Main Question or Discussion Point

I want to derive the Callan-Gross relation from the parton model but I am having some problems obtaining the textbook result. I am following

Starting from the hard scattering coefficient obtained from the partonic scattering amplitude for [itex]\gamma^\ast q_i\rightarrow q_i[/itex] (eq. 32.32),

$$\hat{W}^{\mu\nu}(z,Q^2)=2\pi Q^2_i\delta(1-z)\left[A^{\mu\nu}+\frac{4z}{Q^2}B^{\mu\nu}\right],$$

where [itex]A^{\mu\nu}:=-g^{\mu\nu}+\frac{q^\mu q^\nu}{Q^2}[/itex], [itex]B^{\mu\nu}:=\left(p^\mu+\frac{pq}{Q^2}q^\mu\right)\left(p^\nu+\frac{pq}{Q^2}q^\nu\right)[/itex], and the convolution formula for the hardonic tensor [itex]W^{\mu\nu}(x,Q^2)[/itex] obtained from factorisation, we arrive at

\begin{align*}

W^{\mu\nu}(x,Q^2)

&=2\pi\int^1_x\frac{d\xi}{\xi}\sum_if_i(\xi)Q^2_i\delta(1-\frac{x}{\xi})\left[A^{\mu\nu}+\frac{4x}{Q^2\xi}B^{\mu\nu}\right]\\

&=2\pi\int^1_xd\xi\sum_if_i(\xi)Q^2_i\delta(\xi-x)\left[A^{\mu\nu}+\frac{4x}{Q^2\xi}B^{\mu\nu}\right]\\

&=2\pi\sum_if_i(x)Q^2_i\left[A^{\mu\nu}+\frac{4}{Q^2}B^{\mu\nu}\right],\end{align*}

such that [itex]W_1(x,Q^2)=2\pi\sum_if_i(x)Q^2_i=\frac{Q^2}{4}W_2(x,Q^2)[/itex].

Now, the textbook says that the result should be [itex]W_1(x,Q^2)=\frac{Q^2}{4x^2}W_2(x,Q^2)[/itex] (eq. 32.23, 32.24). Did I make a mistake somewhere in my calculations?

*M.D. Schwartz: Quantum Field Theory and the Standard Model*(pp.672, 675, 678).Starting from the hard scattering coefficient obtained from the partonic scattering amplitude for [itex]\gamma^\ast q_i\rightarrow q_i[/itex] (eq. 32.32),

$$\hat{W}^{\mu\nu}(z,Q^2)=2\pi Q^2_i\delta(1-z)\left[A^{\mu\nu}+\frac{4z}{Q^2}B^{\mu\nu}\right],$$

where [itex]A^{\mu\nu}:=-g^{\mu\nu}+\frac{q^\mu q^\nu}{Q^2}[/itex], [itex]B^{\mu\nu}:=\left(p^\mu+\frac{pq}{Q^2}q^\mu\right)\left(p^\nu+\frac{pq}{Q^2}q^\nu\right)[/itex], and the convolution formula for the hardonic tensor [itex]W^{\mu\nu}(x,Q^2)[/itex] obtained from factorisation, we arrive at

\begin{align*}

W^{\mu\nu}(x,Q^2)

&=2\pi\int^1_x\frac{d\xi}{\xi}\sum_if_i(\xi)Q^2_i\delta(1-\frac{x}{\xi})\left[A^{\mu\nu}+\frac{4x}{Q^2\xi}B^{\mu\nu}\right]\\

&=2\pi\int^1_xd\xi\sum_if_i(\xi)Q^2_i\delta(\xi-x)\left[A^{\mu\nu}+\frac{4x}{Q^2\xi}B^{\mu\nu}\right]\\

&=2\pi\sum_if_i(x)Q^2_i\left[A^{\mu\nu}+\frac{4}{Q^2}B^{\mu\nu}\right],\end{align*}

such that [itex]W_1(x,Q^2)=2\pi\sum_if_i(x)Q^2_i=\frac{Q^2}{4}W_2(x,Q^2)[/itex].

Now, the textbook says that the result should be [itex]W_1(x,Q^2)=\frac{Q^2}{4x^2}W_2(x,Q^2)[/itex] (eq. 32.23, 32.24). Did I make a mistake somewhere in my calculations?