Anomalous Dimensions and the Breakdown of Scale Invariance in Perturbation Theory

Canonical field theory predicts that a zero-mass scalar field theory with a $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ interaction is scale invariant. It is shown here that the renormalized perturbation expansion of the $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ theory is not scale invariant in order ${\ensuremath{\lambda}}^{2}$. Matrix elements of the divergence of the dilation current ${D}_{\ensuremath{\mu}}(x)$ are computed in order ${\ensuremath{\lambda}}^{2}$ using Ward identities; it is found that ${\ensuremath{\nabla}}^{\ensuremath{\mu}}{D}_{\ensuremath{\mu}}(x)$ is proportional to ${\ensuremath{\lambda}}^{2}{\ensuremath{\varphi}}^{4}(x)$. It is also shown that the dimension of the field ${\ensuremath{\varphi}}^{4}$ differs from the canonical value in order $\ensuremath{\lambda}$, and that this result leads one to expect a ${\ensuremath{\lambda}}^{2}{\ensuremath{\varphi}}^{4}$ term in ${\ensuremath{\nabla}}^{\ensuremath{\mu}}{D}_{\ensuremath{\mu}}$. It is also found that matrix elements of the composite field ${\ensuremath{\varphi}}^{4}(x)$ 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.