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Asymptotic Safety Guaranteed PHYSICAL REVIEW D 97, 085008 (2018) More asymptotic safety guaranteed † Andrew D. Bond* and Daniel F. Litim Department of Physics and Astronomy, U Sussex, Brighton BN1 9QH, United Kingdom (Received 2 February 2018; published 10 April 2018) We study interacting fixed points and phase diagrams of simple and semisimple quantum field theories in four dimensions involving non-Abelian gauge fields, fermions and scalars in the Veneziano limit. Particular emphasis is put on new phenomena which arise due to the semisimple nature of the theory. Using matter field multiplicities as free parameters, we find a large variety of interacting conformal fixed points with stable vacua and crossovers inbetween. Highlights include semisimple gauge theories with exact asymptotic safety, theories with one or several interacting fixed points in the IR, theories where one of the gauge sectors is both UV free and IR free, and theories with weakly interacting fixed points in the UV and the IR limits. The phase diagrams for various simple and semisimple settings are also given. Further aspects such as perturbativity beyond the Veneziano limit, conformal windows, and implications for model building are discussed. DOI: 10.1103/PhysRevD.97.085008 I. INTRODUCTION An important new development in the understanding of Asymptotic freedom is a key feature of non-Abelian asymptotic safety has been initiated in [32] where it was gauge theories [1,2]. It predicts that interactions weaken shown that certain four-dimensional quantum field theories with growing energy due to quantum effects, thereby involving SUðNÞ gluons, quarks, and scalars can develop reaching a free ultraviolet (UV) fixed point under the weakly coupled UV fixed points. Results have been renormalization group. Asymptotic safety, on the other extended beyond classically marginal interactions [33]. hand, stipulates that running couplings may very well Structural insights into the renormalization of general asymptote into an interacting UV fixed point at highest gauge theories have led to necessary and sufficient con- energies [3,4]. The most striking difference between ditions for asymptotic safety, alongside strict no go asymptotically free and asymptotically safe theories relates theorems [34,35]. Asymptotic safety invariably arises as to residual interactions in the UV. Canonical power count- a quantum critical phenomenon through cancellations at loop level for which all three types of elementary degrees ing is modified, whence establishing asymptotic safety in a — — reliable manner becomes a challenging task [5]. of freedom scalars, fermions, and gauge fields are Rigorous results for asymptotic safety at weak coupling required. Findings have also been extended to cover have been known since long for models including either supersymmetry [36] and UV conformal windows [37]. scalars, fermions, gauge fields or gravitons, and away Throughout, it is found that suitable Yukawa interactions from their respective critical dimensionality [4,6–16].In are pivotal [34,35]. In this paper, we are interested in fixed points of these toy models asymptotic safety arises through the semisimple gauge theories. Our primary motivation is cancellation of tree level and leading order quantum the semisimple nature of the standard model, and the terms. Progress has also been made to substantiate the prospect for asymptotically safe extensions thereof [38]. asymptotic safety conjecture beyond weak coupling [5]. We are particularly interested in semisimple theories where This is of particular relevance for quantum gravity interacting fixed points and asymptotic safety can be where good evidence has arisen in a variety of different established rigorously [34]. More generally, we also wish settings [17–31]. to understand how low- and high-energy fixed points are generated dynamically, what their features are, and whether *[email protected] novel phenomena arise owing to the semisimple nature of † [email protected] the underlying gauge symmetry. Understanding the stabil- ity of a Higgs-like ground state at interacting fixed points Published by the American Physical Society under the terms of is also of interest in view of the “near-criticality” of the the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to standard model vacuum [39,40]. the author(s) and the published article’s title, journal citation, We investigate these questions for quantum field theories 3 and DOI. Funded by SCOAP . with SUðNCÞ × SUðNcÞ local gauge symmetry coupled to 2470-0010=2018=97(8)=085008(35) 085008-1 Published by the American Physical Society ANDREW D. BOND and DANIEL F. LITIM PHYS. REV. D 97, 085008 (2018) massless fermionic and singlet scalar matter. Our models A. Fixed points in perturbation theory alsohaveaglobalUðNFÞL × UðNFÞR × UðNf ÞL × UðNfÞR We are interested in the renormalization of general flavor symmetry, and are characterized by up to nine gauge theories coupled to matter fields, with or without independent couplings. Matter field multiplicities serve as Yukawa couplings. The running of the gauge couplings free parameters. We obtain rigorous results from the leading α 2 4π 2 μ i ¼ gi =ð Þ with the renormalization group scale is orders in perturbation theory by adopting a Veneziano limit. determined by the beta functions of the theory. Expanding We then provide a comprehensive classification of quantum them perturbatively up to two loop we have field theories according to their UV and IR limits, their fixed points, and eigenvalue spectra. Amongst these, we find 2 4 μ∂μα ≡β ¼ α ð−B þC α −2Y4 ÞþOðα Þ; ð1Þ semisimple gauge theories with exact asymptotic safety in i i i i ij j ;i the UV. We also find a large variety of theories with where a sum over gauge group factors j is implied. The crossover- and low-energy fixed points. Further novelties one- and two-loop gauge contributions B and C and the include theories with inequivalent yet fully attractive IR i ij two-loop Yukawa contributions Y4 are known for general conformal fixed points, theories with weakly interacting ;i gauge theories, see [34,41–44] for explicit expressions. fixed points in both the UVand the IR, and massless theories While B and C may take either sign, depending on the with a nontrivial gauge sector which is UV free and IR free. i ii matter content, the Yukawa contribution Y4;i and the off- We illustrate our results by providing general phase diagrams ≠ for simple and semisimple gauge theories with and without diagonal gauge contributions Cij (i j) are strictly positive Yukawa interactions. in any quantum field theory. Scalar couplings do not play The paper is organized as follows. General aspects of any role at this order in perturbation theory. The effect of weakly interacting fixed points in 4d gauge theories are Yukawa couplings is incorporated by projecting the gauge β 0 laid out in Sec. II, together with first results and expressions beta functions (1) onto the Yukawa nullclines ð Y ¼ Þ, for universal exponents. In Sec. III we introduce concrete leading to explicit expressions for Y4;i in terms of the families of semisimple gauge theories coupled to elemen- gauge couplings gj. Moreover, for many theories the tary singlet “mesons” and suitably charged massless Yukawa contribution along nullclines can be written as α ≥ 0 fermions. Perturbative RG equations for all gauge, Y4;i ¼ Dij j with Dij [34]. We can then go one Yukawa and scalar couplings and masses in a Veneziano step further and express the net effect of Yukawa couplings → 0 limit are provided to the leading non-trivial orders in as a shift of the two loop gauge contribution, Cij Cij ¼ perturbation theory. Section IV presents our results for Cij − 2Dij ≤ Cij. Notice that the shift will always be by all interacting perturbative fixed points and their universal some negative amount provided at least one of the Yukawa scaling exponents. Particular attention is paid to new effects couplings is nonvanishing. It leads to the reduced gauge which arise due to the semisimple nature of the models. beta functions Section V provides the corresponding fixed points in the scalar sector. It also establishes stability of the quantum β α2 − 0 α α4 i ¼ i ð Bi þ Cij jÞþOð Þ: ð2Þ vacuum whenever a physical fixed point arises in the gauge sector. Using field multiplicities as free parameters, Sec. VI Fixed points solutions of (2) are either free or interacting provides a complete classification of distinct models with and αà ¼ 0 for some or all gauge factors is always a self- asymptotic freedom or asymptotic safety in the UV, or consistent solution. Consequently, interacting fixed points without UV completions, together with their scaling in the are solutions to deep IR. In Sec. VII, the generic phase diagrams for simple and semisimple gauge theories with and without Yukawas 0 αà αà 0 Bi ¼ Cij j ; subject to i > ; ð3Þ are discussed. The phase diagrams, UV—IR transitions, and aspects of IR conformality are analysed in more where only those rows and columns are retained where depth for sample theories with asymptotic freedom and gauge couplings are interacting. asymptotic safety. Further reaching topics such as exact Next we discuss the role of Yukawa couplings for the perturbativity, extensions beyond the Veneziano limit, and fixed point structure. In the absence of Yukawa couplings, conformal windows are discussed in Sec. VIII. Section IX 0 the two-loop coefficients remain unshifted Cij ¼ Cij.An closes with a brief summary. immediate consequence of this is that any interacting fixed point must necessarily be IR. The reason is as follows: for II. FIXED POINTS OF GAUGE THEORIES an interacting fixed point to be UV, asymptotic freedom In this section, we discuss general aspects of interacting cannot be maintained for all gauge factors, meaning that fixed points in semisimple gauge theories which are weakly some Bi < 0. However, as has been established in [34], coupled to matter, with or without Yukawa interactions, Bi ≤ 0 necessarily entails Cij ≥ 0 in any 4d quantum gauge following [34,35].
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