In Effective Field Theory video lectures found here, the professor explained power counting in effective field theories and the difficulties of power counting associated with loop diagrams. He then mentions that introducing a cutoff ($\Lambda_{UV}$) to regulate our divergences does not preserve power counting due to the new scale that we are introducing. To see this he uses four-fermi theory with the diagram,
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We do our power counting (i.e., Taylor expansions) in powers of $m^2/M^2$ and then go on to consider the mass correction through,
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Using a cutoff this gives a mass correction, \begin{align} a\frac{ m }{ M ^2 } \int _0^{\Lambda_{UV}}\frac{ \,d^4k _E }{ (2\pi)^4} \frac{1}{ k _E ^2 + m ^2 } & = a\frac{ m }{ ( 4 \pi ) ^2 } \left[ \frac{ \Lambda _{ UV } ^2 }{ M ^2 } + \frac{ m ^2 }{ M ^2 } \log \frac{ m ^2 }{ \Lambda _{ UV } ^2 } - \frac{ m ^4 }{ M ^2 \Lambda _{ UV } ^2 } + ... \right] \end{align}
If I understand correctly this breaks the power counting because even if $\Lambda_{UV} \sim M$, the first term is an order 1 correction since its not proportional to $m^2/M^2$. So far so good. However, then the professor says that you can still use power counting with a cutoff if you fix the power counting order by order and that this can be done by introducing an intermediate scale, $\Lambda$. But I don't how this fixes anything...
With an intermediate scale ($\Lambda$) we have, \begin{equation} a\frac{m}{M^2}\int _{ \Lambda } ^{ \Lambda _{ UV }} \frac{ \,d^4k _E }{ (2\pi)^4 } \frac{1}{ k _E ^2 + m ^2 } = \frac{a\,m}{ (4\pi)^2M^2 } \left\{ \left(\Lambda ^2 + m ^2 \log \frac{ m ^2 }{ \Lambda ^2 + m ^2 } + ... \right) + \left( \Lambda _{ UV } ^2 - \Lambda + m ^2 \log \frac{ \Lambda ^2 + m ^2 }{ \Lambda ^2 _{ UV }} \right) \right\} \end{equation} but how does this fix anything?
For more context see my lecture notes here under Effective Field Theory (starts around equation 4.6)