Renormalization Scheme Dependence, RG Flow and Borel Summability in ${\varphi }^4$ Theories in $d<4$

Abstract

Renormalization group (RG) and resummation techniques have been used in $N$-component ${\varphi }^4$ theories at fixed dimensions below four to determine the presence of non-trivial IR fixed points and to compute the associated critical properties. Since the coupling constant is relevant in $d<4$ dimensions, the RG is entirely governed by renormalization scheme-dependent terms. We show that the known proofs of the Borel summability of observables depend on the renormalization scheme and apply only in "minimal" ones, equivalent in $d=2$ to an operatorial normal ordering prescription, where the $\beta$-function is trivial to all orders in perturbation theory. The presence of a non-trivial fixed point can be unambiguously established by considering a physical observable, like the mass gap, with no need of RG techniques. Focusing on the $N=1$, $d=2$ ${\varphi }^4$ theory, we define a one-parameter family of renormalization schemes where Borel summability is guaranteed and study the accuracy on the determination of the critical exponent $\nu$ as the scheme is varied. While the critical coupling shows a significant sensitivity on the scheme, the accuracy in $\nu$ is essentially constant. As by-product of our analysis, we improve the determination of $\nu$ obtained with RG methods by computing three more orders in perturbation theory.

arXiv commentv1: 24 pages, 4 figures, 2 tables; v2: Minor improvements, Eq.(2.3) corrected

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• Self-Dualities and Renormalization Dependence of the Phase Diagram in 3d $O(N)$ Vector Models
• Transitioning from equal-time to light-front quantization in ${\varphi }_2^4$ theory
• Critical behavior of the 2d scalar theory: resumming the ${\mathrm{N}}^8\mathrm{LO}$ perturbative mass gap
• The Breakdown of Resummed Perturbation Theory at High Energies
• Tadpoles and vacuum bubbles in light-front quantization
• Resurgence and $1/N$ Expansion in Integrable Field Theories
• Exact $\beta$-function of Yang-Mills theory in 2+1 dimensions
• Nonperturbative dynamics of (2+1)d ${\varphi }^4$-theory from Hamiltonian truncation