Two-loop scale-invariant scalar potential and quantum effective operators

D. M. Ghilencea; Z. Lalak; P. Olszewski

doi:10.1140/epjc/s10052-016-4475-0

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2024 Impact factor 4.8

Particles and Fields

Open Access

Eur. Phys. J. C (2016) 76: 656
https://doi.org/10.1140/epjc/s10052-016-4475-0

Regular Article - Theoretical Physics

Two-loop scale-invariant scalar potential and quantum effective operators

D. M. Ghilencea¹^,2^*, Z. Lalak³ and P. Olszewski³

¹ Theoretical Physics Department, National Institute of Physics and Nuclear Engineering (IFIN-HH), 077125, Bucharest, Romania
² Theory Division, CERN, 1211, Geneva 23, Switzerland
³ Faculty of Physics, Institute of Theoretical Physics, University of Warsaw, ul. Pasteura 5, 02-093, Warsaw, Poland

^* e-mail: dumitru.ghilencea@cern.ch

Received: 19 August 2016
Accepted: 2 November 2016
Published online: 29 November 2016

Abstract

Spontaneous breaking of quantum scale invariance may provide a solution to the hierarchy and cosmological constant problems. In a scale-invariant regularization, we compute the two-loop potential of a Higgs-like scalar $\phi$ in theories in which scale symmetry is broken only spontaneously by the dilaton ( $\sigma$ ). Its VEV $\langle \sigma \rangle$ generates the DR subtraction scale ( $\mu \sim \langle \sigma \rangle$ ), which avoids the explicit scale symmetry breaking by traditional regularizations (where $\mu$ $$=$$ fixed scale). The two-loop potential contains effective operators of non-polynomial nature as well as new corrections, beyond those obtained with explicit breaking ( $\mu$ $$=$$ fixed scale). These operators have the form $\phi ^6/\sigma ^2$ , $\phi ^8/\sigma ^4$ , etc., which generate an infinite series of higher dimensional polynomial operators upon expansion about $\langle \sigma \rangle \gg \langle \phi \rangle$ , where such hierarchy is arranged by one initial, classical tuning. These operators emerge at the quantum level from evanescent interactions ( $\propto \epsilon$ ) between $\sigma$ and $\phi$ that vanish in $$d=4$$ but are required by classical scale invariance in $d=4-2\epsilon$ . The Callan–Symanzik equation of the two-loop potential is respected and the two-loop beta functions of the couplings differ from those of the same theory regularized with $\mu =$ fixed scale. Therefore the running of the couplings enables one to distinguish between spontaneous and explicit scale symmetry breaking.