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I thought the whole point was that with Pauli-Villars regularization you can perform the integral over the momentum and be left with a single parameter [itex]M[/itex]? Can anyone explain what I'm missing?

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- Thread starter copernicus1
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- #1

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I thought the whole point was that with Pauli-Villars regularization you can perform the integral over the momentum and be left with a single parameter [itex]M[/itex]? Can anyone explain what I'm missing?

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Avodyne

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$$\Pi_{\text{ren}}(M^2)=\Pi(M^2)-\Pi(\mu^2)-(M^2-\mu^2) \Pi'(\mu^2)+C_1 +C_2 M^2,$$

where the constants ##C_1## and ##C_2## are determined by the appropriate renormalization conditions. ##\mu^2## is the mass-subtraction scale. Indeed, using your schematic expression leads to

$$\Pi(M^2)-\Pi(\mu^2)-(M^2-\mu^2) \Pi'(\mu^2)=\int_0^{\infty} \mathrm{d} q \frac{q^3(M^2-\mu^2)^2}{(q^2+\mu^2)^2(q^2+M^2)},$$

which is convergent.

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http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

Pauli-Villars is one of many regularization methods, which is usually quite tedious. A much more convenient regularization is dimensional regularization.

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