Renormalizability of the continuous chain model

For d 4 in the theorem of renormalizability we can take the naive continuous chain limit, addressed here as NCL for shortness. 

We know that the additively renormalized bare theory (the left hand side in Eq. (11.10)) exists in that limit. Also the -factors attain finite limits Z(ti), Zu(u), Zn(u). Indeed they are constructed as power series in u, the coefficients for t 0,6 0 taking the form of polynomials in / r) . Thus no problem results from setting t = 0. Since the renormalized theory is finite for d = 4, whereas the bare continuous chain model diverges for d — 4, showing poles in 5, also the Z-factors must diverge for d — 4. In the NCL we can therefore formulate the theorem of renormalizability as follows  

The renormalization factors can be chosen to absorb all the pole terms of the dimensionally regularized bare theory to yield a renormalized theory finite for d 4. 

Using a power series ansatz analogous to (11.2) with. 4  

We know that the additively renormalized bare theory (the left hand side in Eq. (11.10)) exists in that limit. Also the. Z-factors attain finite limits Z u)yZu u),Zn u). Indeed they are constructed as power series in u, the coefficients for 0,6 0 taking the form of polynomials in Thus 

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