# The use of the renormalisation constant $Z_2$ of the electron self-energy in Peskin & Schroeder

+ 5 like - 0 dislike
173 views

The use of renormalisation constants often puzzles me. A good example is the use of Z_2 in the equation (7.58) of Peskin Schroeder. Z_2 is defined in equation (7.26). as $Z_2^{-1} = 1-\frac{d\Sigma}{dp}$. Later in equation (7.31) it is said:

$Z_2-1 = d\Sigma/dp$ although this term is supposed to be infinite. But $d\Sigma/dp$ is treated of being smaller than 1. Okay, in this example the Pauli-Villars renormalisation is used where a rather high $\Lambda$ is needed to make $d\Sigma/dp$ larger than 1. But what would be if $Z_2$ were computed with dimensional regularization ?

Shouldn't be at least : $d\Sigma/dp + (d\Sigma/dp)^2 + (d\Sigma/dp)^3 + \dots$

I know that $d\Sigma/dp$ is of order $\alpha$ and there should be counter term to make the sum of both small (to order \alpha). On the other I am almost sure that when the counter terms of the next order \alpha^2 are calculated that it already forgotten that there was also a term $(d\Sigma/dp)^2$ which also need a counter term and for $\alpha^3$ order again and so on. Could somebody explain it to me ? Thank you.

This post imported from StackExchange Physics at 2014-06-10 21:34 (UCT), posted by SE-user Frederic Thomas
asked Jun 8, 2014
Hi Frederic, may I recommend you to write your equations with MathJax, see here for more info.

This post imported from StackExchange Physics at 2014-06-10 21:34 (UCT), posted by SE-user Hunter
@Hunter: He did, but he forgot the dollar signs :)

This post imported from StackExchange Physics at 2014-06-10 21:34 (UCT), posted by SE-user JamalS
@JamalS look at his previous questions, for instance here and here. The OP always seems to "forget" the dollar signs, so I thought someone should make him/here aware of this so that we don't constantly have to improve his/her posts.

This post imported from StackExchange Physics at 2014-06-10 21:34 (UCT), posted by SE-user Hunter

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsO$\varnothing$erflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.