A Renormalized vertex functions in terms of bare ones

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

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The discussion centers on the relationship between renormalized and bare vertex functions in massless φ-4 theory, specifically through the generating functional for proper vertex functions. It establishes that the n-point proper vertex functions in Fourier space can be expressed in terms of bare quantities using a renormalization factor Z. The formula provided connects these functions with external momenta, the Pauli-Villars cutoff, and arbitrary scales. The main inquiry is about the methodology to demonstrate this relationship. Understanding this connection is crucial for analyzing quantum field theories and their renormalization processes.

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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