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The assumption of a full eigensystem is usually made for convenience. But it is not always satisfied. If it is not satisfies one gets additional logarithmic contributions to the scaling laws. This is discussed, e.g., in the paper by Wegner and Riedel.

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A typical renormalization group flow can be thought of as a smooth vector field $\vec V(\mu)$ defined on parameter space. Starting with parameters $\vec\mu(\ell)$ at scale $\ell$, you obtain parameters at scale $\ell'$ by solving the differential equation $\frac{d\vec\mu}{d\ell}=\vec V(\vec\mu(\ell))$. The function $R$ referred to above can be thought of as ...

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I think Andrey Grozin's http://arxiv.org/pdf/hep-ph/0508242.pdf works quite well enough if you are looking for a general strategy to calculate the anomalous dimension of an operator. You need to somehow define $Z$, i.e. you need to develop a scheme. Now let's say you have defined your scheme or you have simply tried one of the conventional ones. The rest is ...

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There are attempts to use nonstandard analysis (e.g., Albeverio) or Colombeau algebras, but these haven't been developed very far. I haven't seen anything in terms of surreal numbers, but they may probably substitute for the infinitesimals in nonstandard analysis.

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