 Field scaling and anomalous dimension

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.