Deriving the Callan-Gross Relation via the Parton Model

[weningth]

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 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 $\gamma^\ast q_i\rightarrow q_i$ (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 $A^{\mu\nu}:=-g^{\mu\nu}+\frac{q^\mu q^\nu}{Q^2}$, $B^{\mu\nu}:=\left(p^\mu+\frac{pq}{Q^2}q^\mu\right)\left(p^\nu+\frac{pq}{Q^2}q^\nu\right)$, and the convolution formula for the hardonic tensor $W^{\mu\nu}(x,Q^2)$ 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 $W_1(x,Q^2)=2\pi\sum_if_i(x)Q^2_i=\frac{Q^2}{4}W_2(x,Q^2)$.

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

 

[AndyW]

Hi there,

Thank you for sharing your work and question. It seems like you have made a mistake in your final step where you equate the integrals. The correct result should be:

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*}

However, when you substitute the expressions for A^{\mu\nu} and B^{\mu\nu}, you should get:

W^{\mu\nu}(x,Q^2)
&=2\pi\sum_if_i(x)Q^2_i\left[-g^{\mu\nu}+\frac{q^\mu q^\nu}{Q^2}+\frac{4}{Q^2}\left(p^\mu p^\nu+\frac{p^\mu q^\nu}{Q^2}+\frac{q^\mu p^\nu}{Q^2}+\frac{q^\mu q^\nu}{Q^2}\right)\right]\\
&=2\pi\sum_if_i(x)Q^2_i\left[-g^{\mu\nu}+\frac{q^\mu q^\nu}{Q^2}+\frac{4}{Q^2}\left(p^\mu p^\nu+\frac{2p^\mu q^\nu}{Q^2}+\frac{q^\mu p^\nu}{Q^2}\right)\right]\\
&=2\pi\sum_if_i(x)Q^2_i\left[-g^{\mu\nu}+\frac{q^\mu q^\nu}{Q^2}