Anomalous dimension ($\eta$)

An anomalous dimension is the interaction-generated difference between a full scaling dimension and its engineering dimension. For a scalar field, momentum-dependent self-energy corrections require wave-function renormalization and produce ${\Delta }_{\varphi }=(d-2+\eta )/2$.

If ${\varphi }_B=Z_{\varphi }^{1/2}{\varphi }_R$, differentiating a renormalized $n$-point function at fixed bare parameters gives $(\mu {\partial }_{\mu }+\beta {\partial }_g+n{\gamma }_{\varphi })G_{n,R}=0$. With this convention the field scaling dimension at a fixed point is its engineering dimension plus ${\gamma }_{\varphi }$. The critical exponent $\eta$ for scalar two-point decay is $2{\gamma }_{\varphi }$.

Under a length blocking factor $b$, a scalar scaling field transforms as ${\varphi }^{\prime }(x^{\prime })=b^{x_{\varphi }}{\varphi }_{<}(b{x}^{\prime })$. Its critical connected correlation function consequently decays as $r^{-2x_{\varphi }}$. Comparing with $r^{-(D-2+\eta )}$ identifies the anomalous part of the field dimension. The uniform conjugate field has scaling exponent $y_h=D-x_{\varphi }$. Wave-function factors must be defined explicitly before assigning a sign to their logarithmic derivatives.