On the Question of an Ultraviolet Zero of the Beta Function of the $\lambda ({\vec{\varphi }}^2{)}_4^2$ Theory

Abstract

We investigate the possibility of an ultraviolet (UV) zero in the $n$-loop beta function of a $\lambda ({\vec{\varphi }}^2{)}^2$ field theory with an $N$-component scalar field, $\vec{\varphi }$, in four spacetime dimensions, up to the level of $n=5$ loops. Although the two-loop beta function has a UV zero, we find that the $n$-loop beta function for $n=3,4,5$ either does not have a UV zero or does not have one at a value of $\lambda$ in approximate agreement with the two-loop calculation. Similar results are obtained after application of scheme transformations to the beta function and via calculation of Padé approximants. We thus conclude that in the range of $\lambda$ where the perturbative calculation of the $n$-loop beta function is reliable, the theory does not exhibit robust evidence of a UV zero up to the level of $n=5$ loops.

16 pages latex with four figures

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• An Integral Formalism for the Construction of Scheme Transformations in Quantum Field Theory
• Study of the Six-Loop Beta Function of the $\lambda {\varphi }_4^4$ Theory
• On the Question of a Possible Infrared Zero in the Beta Function of the Finite-$N$ Gross-Neveu Model
• Study of the Question of an Ultraviolet Zero in the Six-Loop Beta Function of the O($N$) $\lambda |\vec{\varphi }{|}^4$ Theory
• Effect of Scheme Transformations on a Beta Function with Vanishing One-Loop Term
• Study of the Ultraviolet Behavior of an O($N$) $|\vec{\varphi }{|}^6$ Theory in $d=3$ Dimensions
• Search for an Ultraviolet Zero in the Seven-Loop Beta Function of the $\lambda {\varphi }_4^4$ Theory
• The a-theorem for the four-dimensional vector model