How Schwartz switch to the new variables in integration but left some of old variables untouched?

Question

In Section 5.1.2 of Schwartz's when he changes variables from Equation 5.27 to Equation 5.29 he left a piece of old variables and treat it as constant. Shouldn't $p_f$ also be replaced with some expression $p_f(x)$?

The relevant equations are reproduced below; $E_3=\sqrt{m_3^2+p_f^2}$ and $E_4=\sqrt{m_4^2+p_f^2}$ \begin{align} \tag{5.27}\mathrm d\Pi_{LIPS}&=\frac{1}{16\pi^2}\mathrm d\Omega \int\mathrm dp_f \frac{p_f^2}{E_3E_4}\delta(E_3+E_4-E_\text{CM})\\ \text{Introducing}\qquad x(p_f)&=E_3(p_f)+E_4(p_f)-E_\text{CM}\\ \tag{5.28}\frac{\mathrm dx}{\mathrm dp_f}=\frac{\mathrm d\ \ }{\mathrm dp_f}(E_3+E_4-E_\text{CM})&=\frac{p_f}{E_3}+\frac{p_f}{E_4}=\frac{E_3+E_4}{E_3E_4}p_f\qquad\text{(Jacobian)}\\ \mathrm d\Pi_{LIPS}&=\frac{1}{16\pi^2}\mathrm d\Omega\int\mathrm dx \frac{p_f}{E_{CM}}\delta(x)\\ \tag{5.29}&=\frac1{16\pi^2}\mathrm d\Omega\frac{p_f}{E_\text{CM}}\theta(E_\text{CM}-m_3-m_4) \end{align}

Answer 1

It's elementary calculus, or, rather, notation. $p_f$ is not a real variable, as it is constraint to the constant enforced by the δ-function constraint: the integration is fake/illusory! The author is aiming to collapse the δ-function, so he switches, locally invertibly, to $x(p_f)$ from $p_f$. The expression he left, as you noticed, is actually its inverse function, $p_f(x)$, a function of x in reality, with the original name retained, but, as originally, it is fixed to a constant by the vanishing of the δ-function argument.

He can now collapse the δ-function, given the transformed lower limit of the integral, into a step function (some xs are disallowed as being out of that range!). If you wish, you may now "remember" the function $p_f(0)$ as the original constant $p_f$, as the variable x has disappeared with the integration, and you need not recall it anymore. You may now luxuriate in the simplicity of equation (5.32).