Anomalous Dimensions and the Breakdown of Scale Invariance in Perturbation Theory

Canonical field theory predicts that a zero-mass scalar field theory with a $\lambda {\phi }^4$ interaction is scale invariant. It is shown here that the renormalized perturbation expansion of the $\lambda {\phi }^4$ theory is not scale invariant in order ${\lambda }^2$. Matrix elements of the divergence of the dilation current $D_{\mu }\mathrm{}\left(x\right)\mathrm{}$ are computed in order ${\lambda }^2$ using Ward identities; it is found that ${\nabla }^{\mu }{D}_{\mu }\mathrm{}\left(x\right)\mathrm{}$ is proportional to ${\lambda }^2{\phi }^4\mathrm{}\left(x\right)\mathrm{}$. It is also shown that the dimension of the field ${\phi }^4$ differs from the canonical value in order $\lambda$, and that this result leads one to expect a ${\lambda }^2{\phi }^4$ term in ${\nabla }^{\mu }{D}_{\mu }$. It is also found that matrix elements of the composite field ${\phi }^4\mathrm{}\left(x\right)\mathrm{}$ in perturbation theory have troublesome singularities at short distances which force one to give careful definitions for equal-time commutators and Fourier transforms of $T$ products in the Ward identities involving this field.