note on renormalization

work in the $\bar{\rm{MS}}$ scheme with dimension regularization $d=4-2\epsilon$ .
First every operator with $n_i$ fields of type $i$ could be written as

${\cal O}_R = \prod_{i} Z_i^{\frac{n_i}{2}} Z^{-1}_{\cal O} \Gamma \Phi_{R,i}^{n_i}$ .

Thus, generally the correction to the operator will be of the form.

$\frac{A_1}{\bar{\epsilon}^2_{\rm UV}} + \frac{A_2}{\bar{\epsilon}_{\rm UV}} + \frac{B_1}{\bar{\epsilon}^2_{\rm IR}} + \frac{B_2}{\bar{\epsilon}_{\rm IR}} + {\rm finit \ terms} + \sum_i \frac{n_i}{2}\delta Z_i - \delta Z_{\cal O}$ .

The ultra-violet term should be canceled by the counter-term, in this way $Z_{\cal O}$ could be determined order by order as long as we know $Z_{i}$ .

Secondly, since ${\cal O}_0 = Z_{\cal O} {\cal O}_R$ and ${\cal O}_0$ is scale
independent. Thus $\mu \mathrm{d} ( Z_{\cal O} {\cal O}_R )/ \mathrm{d} \mu = 0$ .

Third, the correction to the propagator is denoted as $-i\Sigma$ , thus the full propagator is $i/({\bar p} -m -\Sigma[p]) = iR/({\bar p} - m) +{\rm analytic }$ . Once on shell, only the residual part will survive due to the LSZ reduction formula and give a contribution $\sqrt{R}$ .
The fermion self energy has the form

$\Sigma_i[p] = (A_i[p^2]+B_i[p^2]) m_i + B_i[p^2] ({\bar p} - m_i)$ .

and its counter term

$\Sigma_{ct} = -\delta Z_i({\bar p} - m_i) + \delta Z_m m_i$ .

Therefore the contribution to $R$ is given by

$\delta R_i = B_i[m_i^2] + 2m_i^2 \frac{\mathrm{d}(A_i+B_i)}{\mathrm{d}p^2}|_{p^2=m_i^2} -\delta Z_i$ .

Last, in effective theory in some cases the integrations involved will be scaleless, thus zero. Hence the operator with $n_j$ external legs of type $j$ will have corrections (all infrared divergence):

$- \sum_j \frac{n_j}{2}\delta Z_j + \sum_i \frac{n_i}{2}\delta Z_i - \delta Z_{\cal O}= - \sum_j \frac{n_j}{2}\delta Z_j - \frac{A_1}{\bar{\epsilon}^2} -\frac{A_2}{\bar{\epsilon}}$ .

This should reproduce the IR poles in the full theory.

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账本

note on renormalization

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