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Safety Versus Triviality on the Lattice University of Southern Denmark Safety versus triviality on the lattice Leino, Viljami; Rindlisbacher, Tobias; Rummukainen, Kari; Tuominen, Kimmo; Sannino, Francesco Published in: Physical Review D DOI: 10.1103/PhysRevD.101.074508 Publication date: 2020 Document version: Final published version Document license: CC BY Citation for pulished version (APA): Leino, V., Rindlisbacher, T., Rummukainen, K., Tuominen, K., & Sannino, F. (2020). Safety versus triviality on the lattice. Physical Review D, 101(7), [074508]. https://doi.org/10.1103/PhysRevD.101.074508 Go to publication entry in University of Southern Denmark's Research Portal Terms of use This work is brought to you by the University of Southern Denmark. Unless otherwise specified it has been shared according to the terms for self-archiving. If no other license is stated, these terms apply: • You may download this work for personal use only. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying this open access version If you believe that this document breaches copyright please contact us providing details and we will investigate your claim. Please direct all enquiries to [email protected] Download date: 04. Oct. 2021 PHYSICAL REVIEW D 101, 074508 (2020) Safety versus triviality on the lattice Viljami Leino* Physik Department, Technische Universität München, 85748 Garching, Germany † ‡ Tobias Rindlisbacher , Kari Rummukainen , and Kimmo Tuominen§ Department of Physics & Helsinki Institute of Physics, University of Helsinki, P.O. Box 64, FI-00014 Helsinki, Finland ∥ Francesco Sannino CP3-Origins & Danish IAS, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark (Received 19 September 2019; accepted 20 March 2020; published 10 April 2020) We present the first numerical study of the ultraviolet dynamics of nonasymptotically free gauge-fermion theories at large number of matter fields. As test bed theories, we consider non-Abelian SU(2) gauge theories with 24 and 48 Dirac fermions on the lattice. For these numbers of flavors, asymptotic freedom is lost, and the theories are governed by a Gaussian fixed point at low energies. In the ultraviolet, they can develop a physical cutoff and therefore be trivial, or achieve an interacting safe fixed point and therefore be fundamental at all energy scales. We demonstrate that the gradient flow method can be successfully implemented and applied to determine the renormalized running coupling when asymptotic freedom is lost. Additionally, we prove that our analysis is connected to the Gaussian fixed point as our results nicely match with the perturbative beta function. Intriguingly, we observe that it is hard to achieve large values of the renormalized coupling on the lattice. This might be an early sign of the existence of a physical cutoff and imply that a larger number of flavors is needed to achieve the safe fixed point. A more conservative interpretation of the results is that the current lattice action is unable to explore the deep ultraviolet region where safety might emerge. Our work constitutes an essential step toward determining the ultraviolet fate of nonasymptotically free gauge theories. DOI: 10.1103/PhysRevD.101.074508 I. INTRODUCTION low number of matter fields are time-honored examples of asymptotically free gauge theories. These theories Asymptotically free theories [1,2] are fundamental feature a single four-dimensional marginal coupling according to Wilson [3,4] since they are well defined from induced by the gauge dynamics, and no further interactions low to arbitrary high energies. This remarkable property are needed to render the theories asymptotically free. stems simply from the fact that asymptotically free theories On the other hand, purely scalar and scalar-fermion are obviously conformal (and therefore scale invariant) at theories are not asymptotically free. Adding elementary short distances, given that all interaction strengths vanish in scalars, and upgrading gauge-fermion theories to gauge- that limit. This is one of the reasons why asymptotic Yukawa systems, one discovers that scalars render the freedom has played such an important role when building existence of asymptotic freedom less guaranteed. In par- extensions of the Standard Model (SM). Non-Abelian ticular, complete asymptotic freedom in all marginal cou- gauge-fermion theories, like QCD, with a sufficiently plings is no longer automatically ensured by requiring a sufficiently low number of scalar and fermion matter fields. To determine the asymptotically free conditions on the *[email protected] † low energy values of the accidental couplings that may lead [email protected] ‡ [email protected] to complete asymptotic freedom, a one-loop analysis in all §[email protected] marginal couplings is sufficient. One discovers that the ∥ [email protected] gauge interactions are essential to tame the unruly behavior of the accidental couplings provided the latter start running Published by the American Physical Society under the terms of within a specific region in coupling space at low energies. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to Nonasymptotically free theories can belong either to the the author(s) and the published article’s title, journal citation, trivial or the safe category. Triviality occurs when the and DOI. Funded by SCOAP3. theories develop a physical cutoff and can therefore be 2470-0010=2020=101(7)=074508(15) 074508-1 Published by the American Physical Society VILJAMI LEINO et al. PHYS. REV. D 101, 074508 (2020) viewed as low energy effective descriptions of a more Interesting hints come from the knowledge of the beta fundamental but typically unknown quantum field theory. function for Abelian and non-Abelian gauge-fermion Triviality literally means that the only way to make sense of theories at leading order in 1=Nf [27–30]. This result these theories as fundamental theories (when trying to suggests, as we shall review below, the potential existence remove the cutoff) is by turning off the interactions. In of short distance conformality [30,31]. However, due to the truth, it is rather difficult to demonstrate that a theory is fact that the UV zero in the beta function stems from a trivial beyond perturbation theory since the couplings logarithmic singularity and more generally that the beta become large in the UV and therefore one can imagine a function is not a physical quantity, careful consistency nonperturbative UV fixed point to emerge leading to checks of these results are crucial [32]. nonperturbative asymptotic safety. Nevertheless, we can For these reasons, and because it is important to uncover be confident that certain theories are indeed trivial. Perhaps the phase diagram of four-dimensional quantum field 4 the best known example is the four-dimensional λϕ theory, theories, we initiate here a consistent lattice investigation which was studied on the lattice in a series of papers by of the ultraviolet fate of non-Abelian gauge-fermion theories Lüscher and Weisz [5–7]. The triviality here was estab- at a small number of colors but at a large number of flavors lished using large order hopping parameter expansion with where asymptotic freedom is lost. These parameters are perturbative renormalization group evolution, and corrobo- currently inaccessible with other methods. Specifically, we rated with several lattice Monte Carlo simulations (see, will investigate the SU(2) gauge theory with 24 and 48 for example, Refs. [8,9]). Related to our study here, the massless Dirac flavors. These two numerical values are triviality versus conformality has also been investigated in substantially larger than the value where asymptotic free- large-Nf QCD with staggered fermions [10]. dom is lost, which is 11, and they are also very roughly Analytically, using a-maximization and violation of the estimated to be close to the region where the 1=Nf squared a-theorem, one can also demonstrate that certain non- corrections become relevant [29,31]. This was taken to be a asymptotically free supersymmetric gauge-Yukawa theo- sign of where one would expect the UV fixed point to ries such as super QCD with(out) a meson and their disappear [31] and the theory develop a cutoff. generalizations are trivial once asymptotic freedom is lost To numerically investigate the UV properties of these by adding enough super matter fields [11]. two theories, in our lattice analysis, we focus on the We move now to safe theories. These achieve UV determination of the renormalized running gauge coupling. conformality while remaining interacting, meaning that This is achieved by implementing the Yang-Mills gradient the interaction strengths freeze in the UV without vanish- flow method at finite volume and with Dirichlet boundary ing. The first four-dimensional safe field theories were conditions [33], enabling us to perform lattice simulations discovered in Ref. [12] within a perturbative study of at vanishing fermion mass. We are able to demonstrate that gauge-Yukawa theories in the Veneziano limit of a large our simulations are performed in the physical region number of flavors and colors. This discovery opened the connected to the Gaussian fixed point. This is corroborated door to new ways to generalize the Standard Model as by the fact that our results match onto the scheme- envisioned first in Ref. [13] and then investigated in independent two-loop perturbative beta function in the Refs. [14–20] with impact in dark matter physics [21,22] infrared. We also discover that it is difficult to achieve large and cosmology [18]. Additionally, it allowed us to use the values of the renormalized coupling on the lattice. This large charge [23,24] method to unveil new controllable might be interpreted as an early sign of the existence of a conformal field theory (CFT) properties for four-dimen- physical cutoff both for 24 and 48 flavors and that a larger sional nonsupersymmetric quantum field theories [25].
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