A Renormalized vertex functions in terms of bare ones

A
Thread starter Siupa
Start date Sep 12, 2023
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Quantum field theory Renormalization Renormalization group

Sep 12, 2023

Siupa

Let ##\Gamma[\varphi] = \Gamma_0[\sqrt{Z}\varphi ] = \Gamma_0[\varphi_0]## be the generating functional for proper vertex functions for a massless ##\phi##-##4## theory. The ##0## subscripts refer to bare quantities, while the quantities without are renormalized. Then
$$\tilde{\Gamma}^{(n)}(p_i, \mu, \lambda) = Z^{\frac{n}{2}}\left( \tfrac{\Lambda}{\mu}, \lambda\right) \tilde{\Gamma}_0^{(n)}(p_i, \Lambda, \lambda_0)$$
Where the ##\tilde{\Gamma}^{(n)}## are the ##n##-point proper vertex functions in Fourier space (bare and renormalized), ##\Lambda## is the Pauli-Villars cutoff, ##\mu## an arbitrary scale, ##p_i## external momenta, ##\lambda## the ##\phi##-##4## couplings (bare and renormalized). How does one show this?

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