# Conversion of results between cutoff regularization and dimensional regularization

Generally it would be expected that a renormalizable/physical quantum field theory (QFT) would be regularization independent. For this I would first fix my regularization scheme and then compute stuff.

I'm interested to know whether there's a general way to convert results obtained from one regularization scheme to another, particularly those quantum corrections that affect the beta functions. More specifically, say I want to compare the coefficients of the $\ln{\Lambda}$ terms for computations done at the upper critical dimension $d = d_c$ of the QFT, with the coefficient of $\epsilon^{-1}$ terms obtained via dimensional regularization at $d = d_c - \epsilon$. In general, I would expect that the coefficients of $\ln{\Lambda}$ terms at $d = d_c$ and the $\epsilon^{-1}$ terms at $d < d_c$ would be different. But, is there a simple relationship between these numbers, such as a multiplicative factor?

• Were you able to find a nice answer to this question? – NoethersOneRing May 24 '19 at 10:18
• No, I don't know a general way to answer this. But, it seems, at one-loop order these coefficients are usually easy to relate, and most of the time they are identical with proper redefinitions. – vik Jun 2 '19 at 17:07

Eqs (3) and (4) of the arXiv paper state relations connecting the quadratic and logarithmic divergences in a 4-momentum cutoff scheme with those calculated using dimensional regularization. They arrive at this result by matching the one and two point Passarino-Veltman functions. $$4 \pi \mu^2 \left(\frac{1}{\epsilon-1}+1\right) = \Lambda^2,$$ and $$\frac{1}{\epsilon}- \gamma_{E} +\log(4 \pi \mu^2) +1 = \log \Lambda^2.$$
Here, $\mu$ is the mass-scale of dimensional regularization and $\gamma_E$ is the Euler-Macheroni constant.