# Numerator's algebraic manipulation in Peskin & Schroeder; Vertex function correction pg 191-192

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• mdb71
mdb71
Hi all, I have a problem working out the algebra of the following expression in Peskin & Schroeder in a smart way to give the result. It is on page 191, regarding the numerator of the vertex correction function.
We want to get from the LHS to the RHS of the following expression
$$\bar{u}(p')[-\frac{1}{2}l^2\gamma^\mu + (z\not{p}-y\not{q})\gamma^{\mu}(z\not{p}+(1-y)\not{q}) + m^2\gamma^{\mu}-2m((1-2y)q^{\mu}+2zp^{\mu})]u(p)$$ $$= \bar{u}(p')[\gamma^{\mu}(-\frac{1}{2}l^2+(1-x)(1-y)q^2+(1-2z-z^2)m^2)+(p'^{\mu}+p^{\mu})mz(1-z)+q^{\mu}m(z-2)(x-y)]u(p)$$
using
$$q \equiv p'-p, \not{p}u(p) = mu(p), \bar{u}(p')\not{p'} = m\bar{u}(p'), \bar{u}(p')\not{q}u(p) = 0,\\ x+y+z = 1$$

In the LHS you have ##m^2##, while in the RHS you have ##(1-2z-z^2)m^2##. What is the relation between ##\bar{u}(p')## (LHS) and ##u(p')## (RHS)?

mdb71
In the LHS you have ##m^2##, while in the RHS you have ##(1-2z-z^2)m^2##. What is the relation between ##\bar{u}(p')## (LHS) and ##u(p')## (RHS)?
The argument of u is p, not p'. What do you mean what is their relation? They are spinors coming from the vertices which the propagators are attached. However, this is irrelevant; what is important here is the on shell identity - coming from the Dirac equation - by which we may simplify terms...

mdb71
Just solved; I'll post the scan of the working in the next few days just for reference as it is quite an annoying passage. Other than that, feel free to post your answers, it is an opportunities to see some smart tricks at play