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In Polchinski's string theory text he asserts (volume 1, section 3.4) that the trace of the energy-momentum tensor of a classically scale -invariant theory becomes proportional in the quantum theory to the beta function of the coupling, as a general point of QFT. This makes a kind of intuitive sense, but does anyone know of a reference that demonstrates the proportionality more precisely?

Can I also ask if anyone is aware of a similar connection in general between the trace of the energy-momentum tensor and the beta function of an arbitrary (i.e. non-conformal) theory?

Thanks in advance.