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
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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]. number of flavors would be needed to achieve the safe fixed Although both safe and free theories share the common point. However, a closer look at the large-Nf predicted UV feature of having no cutoff, the respective mechanisms and behavior of these theories, if applicable, would suggest a dynamics for becoming fundamental field theories are safe fixed point to occur at a much stronger value of the dramatically different [12]. For example, within perturbation gauge coupling1 than achievable with the current simula- theory, it is impossible to achieve safety with gauge-fermion tions. This suggests a more conservative interpretation, that theories [26]. Yukawa interactions and the consequent need we are not yet able to explore the deep ultraviolet region for elementary scalars is an essential ingredient to tame the where safety might emerge. Nevertheless, we feel that our UV behavior of these theories. Of course, this is a welcome work constitutes a necessary stepping stone toward unveil- discovery given that it provides a pleasing theoretical ing the ultraviolet fate of nonasymptotically free gauge justification for the existence of elementary scalars, such theories. as the Higgs, and Yukawa interactions without the need to The paper is organized as follows. In Sec. II, we briefly introduce baroque symmetries such as supersymmetry. review the analytical results for (non-)Abelian gauge Beyond perturbation theory, however, little is known, and it is therefore worth asking whether scalars are needed to 1 We observe that the leading Nf beta function is scheme achieve asymptotic safety. independent.
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2 theories at leading order in 1=N . We compare the large-N ∂ ln α α g f f βðαÞ ≡ ¼ −b1 þ ; α ¼ ; ð1Þ beta function with two-loop perturbation theory and discuss ∂ ln μ π 4π the conformal window 2.0 [31]. It is straightforward to g N specialize the results of this section to the case of SU(2) with the gauge coupling. At large f, it is convenient to A ≡ N T α=π gauge theory investigated on the lattice. Section III is work in terms of the normalized coupling f R . 1 constituted by several subsections with the goal to make it Expanding in =Nf, we can write easier for the reader to focus on the relevant aspects of the X∞ 3 βðAÞ H ðAÞ lattice setup and results. We begin with an introduction to ¼ 1 þ i ; ð2Þ 2 A Ni the lattice action its features. We then discuss how the i¼1 f gauge coupling and its running is defined through the where the identity term corresponds to the one-loop result Yang-Mills gradient flow method. Then, we summarize and constitutes the zeroth order term in the 1=N expansion. the lattice results for N ¼ 24 and N ¼ 48. In Sec. IV,we f f f If the functions jH ðAÞj were finite, then in the large-N offer our conclusions and directions for further studies. i f limit, the zeroth order term would prevail, and the Landau pole would be inevitable. This, however, is not the case due II. CONFORMAL WINDOW 2.0: REVIEW to the occurrence of a divergences in the H ðAÞ functions. OF THE ANALYTIC RESULTS i According to the large-Nf limit, each function HiðAÞ Consider an SUðNcÞ gauge theory with Nf fermions resums an infinite set of Feynman diagrams at the same transforming according to a given representation of the gauge order in Nf with A kept fixed. To make this point explicit, group. Assume that asymptotic freedom is lost, meaning that consider the leading H1ðAÞ term. The nth-loop beta AF ≥ 2 the number of flavors is larger than N ¼ 11C =ð4T Þ, function coefficients bn for n are polynomials of order f G R − 1 where the first coefficient of the beta function changes sign. n in TRNf: In the fundamental representation, the relevant group Xn−1 2 − 1 2 k theory coefficients are CG ¼ Nc, CR ¼ðNc Þ= Nc and bn ¼ bn;kðTRNfÞ : ð3Þ TR ¼ 1=2. At the one-loop order, the theory is free in the k¼0 infrared, i.e., non-Abelian QED, and simultaneously trivial. The coefficient with the highest power of T N will be As discussed in the Introduction, this means that the theory R f has a sensible continuum limit, by sending the Landau pole bn;n−1, and this is the coefficient contributing to H1ðAÞ at induced cutoff to infinity, only if the theory becomes non- the nth-loop order. Moreover, it was shown in Ref. [34] interacting at all energy scales. that the bn;n−1 terms are invariant under the scheme At two loops, in a pioneering work, Caswell [26] transformations that are independent of Nf (as appropriate demonstrated that the sign of the second coefficient of for the large-Nf limit). the gauge beta function is such that an UV interacting fixed Now, the nth-loop beta function will have an interacting point, which would imply asymptotic safety, cannot arise UV fixed point (UVFP) when the following equation has a when the number of flavors is just above the value for physical zero [30]: which asymptotic freedom is lost. This observation Xn β0 11C 2T N immediately implies that for gauge-fermion theories trivi- αk−1 0 G − R f b1 þ bk ¼ where b1 ¼ 2 ¼ 6 3 : ð4Þ ality can be replaced by safety only above a new critical k¼2 number of flavors. In order to investigate this possibility, This expression simplifies at large N . Truncating at a consider the large-Nf limit at fixed number of colors. The f leading order large-Nf beta functions for QED and non- given perturbative order nmax, one finds that the highest-
Abelian gauge theories were constructed in Refs. [27,28], loop beta function coefficient bnmax contains just the n −1 while a summary of the results and possible investigation highest power of ðTRNfÞ max multiplied by the coef- for the existence of UV fixed points appeared first in ficient bnmax;nmax−1, as can be seen from Eq. (3). Since this n −1 Refs. [29,30] with the scaling exponents computed first in highest power of ðTRNfÞ max dominates in the Nf → ∞ Ref. [12]. Although in this work we will concentrate on limit and since in this limit b1 < 0, the criterion for the non-Abelian gauge theories, we now briefly comment on existence of an UV zero in the nmax-loop beta function the status of QED. Even though the large-Nf beta function becomes [30] develops a nontrivial zero, it was demonstrated in Ref. [31] that at the alleged UV fixed point the fermion mass → ∞ β α 0 for Nf ; ð Þ has an UVFP only if bnmax;nmax−1 > : anomalous dimension violates the unitarity bound and hence the UV fixed point is unphysical. At this order in In perturbation theory, only the first few coefficients 1 =Nf, we conclude that QED is trivial. bn;n−1 are known, but, remarkably, it is possible to resum For the non-Abelian case, using the conventions of the perturbative infinite sum to obtain H1ðAÞ. From the Refs. [27,29], the standard beta function reads results in Refs. [27,28],
074508-3 VILJAMI LEINO et al. PHYS. REV. D 101, 074508 (2020) Z 11 A=3 − CG achieves the perturbative Banks-Zaks infrared fixed point H1ðAÞ¼ þ I1ðxÞI2ðxÞdx; α − 4 TR 0 (IRFP), that at two loops yields ¼ b1=b2. This analysis 2 2 3 2 has been extended to the maximum known order in ð1þxÞð2x−1Þ ð2x−3Þ sinðπxÞ Γðx−1Þ Γð−2xÞ I1 x ð Þ¼ −2 π3 perturbation theory in Refs. [30,42,43]. ðx Þ As the number of flavors decreases, the IRFP becomes 20−43 32 2 −14 3 4 4 IRFP CR ð xþ x x þ x ÞCG strongly coupled and at some critical Nf is lost. The I2ðxÞ¼ þ 2 : ð5Þ TR 4ð2x−1Þð2x−3Þð1−x Þ TR lower boundary of the conformal window has been esti- mated analytically in different ways [44–47] and summa- By inspecting I1ðxÞ and I2ðxÞ, one notices that the CG term rized in Ref. [48]. Combining these analytic results in I2 has a pole in the integrand at x ¼ 1 (A ¼ 3). This with nonperturbative lattice studies [49–60] defines the corresponds to a logarithmic singularity in H1ðAÞ, and will current state of the art. See also the overviews [61,62] and cause the beta function to have an UV zero already at references therein. this order in the 1=Nf expansion, and by solving Just above the loss of asymptotic freedom, as already 1 þ H1ðAÞ=Nf ¼ 0, this nontrivial UV fixed point occurs mentioned, Caswell [26] demonstrated that no perturbative at [12] UVFP can emerge. By continuity, there should be a region in color-flavor space where the resulting theory is non- N Abelian QED with an unavoidable Landau pole. This is the 3 − − f A ¼ exp k þ l ; ð6Þ unsafe QCD region. The theories in this region are low Nc energy effective field theories featuring a trivial IRFP. This where k ¼ 16TR and l ¼ 18.49 − 5.26CR=CG. means that one can expect the existence of a critical value Safe Performing a Taylor expansion of the integrand in Eq. (5) of number of flavors Nf above which safety emerges. and integrating term by term, we can obtain the nth-loop This region extends to infinite values of Nf, i.e., the safe coefficients bn;n−1 and check our criteria above for the QCD region [31]. existence of the safe fixed point [34,35]. A preliminary For the fundamental representation, the leading 1=Nf investigation was performed in Ref. [34] up to 18th-loop expansion is applicable only for Nc ≲ Nf=10, while for order, where it was also checked that the first four loops the adjoint representation, we find Nf ≳ 7 for any Nc. agree with the known perturbative results. It was found that, Following Ref. [31], it is sensible to use these values as a even though up to the 12th-loop order the resulting first rough estimate of the lower boundary of the safe QCD coefficients are scattered between the positive and negative region. Altogether, these constraints allowed the authors to values, starting from the 13th-loop order all bn;n−1 are draw the corresponding phase diagrams in Ref. [31]. For the positive for the fundamental representation, two-index reader’s convenience, we draw in Fig. 1 again the phase representations, and symmetric or antisymmetric rank-3 diagrams presented first in Ref. [31] both for the funda- tensors. Unfortunately, the positivity of the coefficients is mental (panel a) and adjoint representations (panel b). insufficient to prove the stability of the series and determine Before specializing to the theories that we will inves- its radius of convergence. The first complete study of the tigate on the lattice, let us comment also on the safe status analytic properties of the leading nontrivial large-Nf of supersymmetric gauge theories. An UV safe fixed point expansion appeared recently in Ref. [35]. Here, it was can, in principle, flow to either a Gaussian IR fixed point demonstrated that an analysis of the expansion coefficients (noninteracting) or to an interacting IR fixed point. So far, to roughly 30 orders is required to establish the radius for the nonsupersymmetric case, we discussed the first class of convergence accurately, and to characterize the (loga- of theories because it is theoretically and phenomenologi- rithmic) nature of the first beta function singularity. cally important to assess whether nonasymptotically free These studies agree with the existence of a singular theories can be UV complete, up to gravity. We provided an structure at leading order in 1=Nf leading to a zero in the affirmative answer for gauge-Yukawa theories that are beta function. Although not a proof, see e.g., Ref. [36],it remarkably similar in structure to the Standard Model in can be viewed as lending support for the possible existence Ref. [12]. The situation for gauge fermions is more of an UV fixed point in these theories. These results have involved, and this is the reason we further investigate it been confirmed when extended to theories with Yukawa here via first-principle lattice simulations. Given the above, interactions [37–39] and employed to build realistic the general conditions that must be (nonperturbatively) asymptotically safe extensions of the SM [15,17,18,40,41]. abided by nonasymptotically free supersymmetric theories Using the results above, we can sketch a complete phase to achieve safety were put forward in Ref. [11] generalizing diagram, as a function of Nc and Nf, for an SUðNcÞ gauge and correcting previous results of Ref. [63]. To make the theory with fermionic matter in a given representation. story short, at least one chiral superfield must achieve a A robust feature of this phase diagram is the line where large-R charge at the safe fixed point to ensure that the AF 11 4 asymptotic freedom is lost, i.e., Nf ¼ CG=ð TRÞ.Asis variation of the a-function between the safe and Gaussian well known, decreasing Nf slightly below this value, one fixed points is positive as better elucidated in Ref. [64].
074508-4 SAFETY VERSUS TRIVIALITY ON THE LATTICE PHYS. REV. D 101, 074508 (2020)
70 10 60 Safe QCD Safe Adjoint QCD 50 8
40 f
f 6 N 30 N 4 20 IR conformal 10 2 IR conformal
0 0 2 3 4 5 6 7 2 3 4 5 6 7
Nc Nc
FIG. 1. For ease for the reader, we summarize here the phase diagram of SUðNcÞ gauge theories with fermionic matter in the fundamental (left panel) and adjoint (right panel) representation put forward in Refs. [30,31]. The shaded areas depict the corresponding conformal windows where the theories develop an IRFP (light red region) or an UVFP (light blue region). The estimate of the lower boundary of the IRFP conformal window is taken from Ref. [30].
Models of this type were shown to exist in Refs. [63–65]. theories we choose to investigate on the lattice, i.e., an Another possible way to elude the constraints [11,64] is to SU(2) gauge theory with either 24 or 48 flavors trans- consider UV fixed points flowing to IR interacting fixed forming according to the fundamental representation of the points. Within perturbation theory, nonsupersymmetric underlying gauge group. theories of this type were discovered in Ref. [66], and supersymmetric theories were discovered in Refs. [65,67]. III. SAFETY ON THE LATTICE Summarizing the status for supersymmetric safety, we can say that theories abiding the constraints of Ref. [11] A. Lattice formulation exist. Nevertheless, specializing Ref. [11] to the case in In this section, we will define the model we consider and ≥ 3 which asymptotic freedom is lost (Nf Nc) for super discuss the methods we use. Our treatment of the general QCD with(out) a meson [11,63], one can show that these features will be brief here as more detailed description can theories are unsafe for any number of matter fields. This is be found in Refs. [49,69]. The model is defined by the 1 in agreement with the =Nf studies of the supersymmetric lattice action beta functions [63,68] as explained in Ref. [35]. The unsafety of super QCD with(out) a meson should be contrasted with QCD at large number of flavors for which, S ¼ SGðUÞþSFðVÞþcSWSSWðVÞ; ð7Þ as we argued above, safety is possible [31] and QCD with a meson for which safety is a fact [12]. where U is the standard SU(2) gauge link matrix, V is After the supersymmetric parentheses, it is time to smeared gauge link defined by hypercubic truncated stout resume the investigation of the nonsupersymmetric con- smearing (HEX smearing) [70], SG is the single plaquette formal window 2.0. What has been reviewed so far clearly gauge action, and SF is the Wilson fermion action with the motivates a nonperturbative study of the large number of clover term SSW. We set the Sheikholeslami-Wohlert flavor dynamics via lattice simulations. Recalling that the 1 coefficient cSW ¼ , which is a good approximation with rough estimate for the lower boundary of the safe side of HEX smeared fermions [69]. 10 the conformal window 2.0 requires Nf > Nc, it is clear The coupling constant is measured using the gradient that to minimize the number of flavors we should use flow method with Dirichlet boundary conditions [33]. This 2 20 Nc ¼ , with Nf > . Therefore, in Fig. 2, we present the method has been used successfully to measure the evolu- leading Nf beta function for an SU(2) gauge theory with tion of the coupling constant in SU(2) gauge theory with either 24 or 48 flavors. The large-Nf beta function is shown Nf < 10, motivating its use also here [49,69]. On a lattice by solid curves, while the dotted curves show the five-loop of size L4, we use periodic boundary conditions at the perturbative beta function. The upper panel is for the Nf ¼ spatial boundaries. At the temporal boundaries x0 ¼ 0, L, 24 48 case, and the lower for the Nf ¼ one. At leading we use Dirichlet boundary conditions by setting the gauge order in 1=Nf, the beta functions support the presence of an link matrices to U ¼ V ¼ 1 and the fermion fields to zero. UVFP for both flavors. It is instructive to show also the These boundary conditions enable simulations at vanishing four- and five-loop perturbative results. These are the two fermion mass.
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FIG. 2. Comparison of the leading order large-Nf beta function (solid curve) and the perturbative beta function at two-loop (dash- dotted curve), four-loop (dashed curve), and five-loop (dotted curve) orders. The upper panels are for Nf ¼ 24, and the lower ones are 2 2 for Nf ¼ 48. The left-hand panels show the beta function as defined in (1) as function of A ¼ NfTRg =ð4π Þ, while the right-hand panels show the beta function as function of g2 and rescaled with an additional factor of g2.
We run the simulations using the hybrid Monte Carlo observed with staggered fermions [10]. We note that SU(2) algorithm with the fifth order Omelyan integrator [71,72] pure gauge lattice theory has a smooth bulk crossover at 2 and chronological initial values for the fermion matrix g0 ∼ 1.7–1.8, which we do not observe in the large-Nf case inversions [73]. The step length is tuned to have an within the range of couplings used. acceptance rate of the order of 80% or higher. We run To define the running coupling, we apply the Yang-Mills the simulations with bare couplings varying within the gradient flow method [33,77,78]. This method defines a β ≡ 4 2 ∈ −1 6 β range L =g0 ½ ; , and for each value of L,we flow that smooths the gauge fields, removes UV divergen- κ κ β tune the hopping parameter to its critical value cð LÞ, for ces, and automatically renormalizes gauge invariant objects which the absolute value of the PCAC fermion mass [74] is [79]. The method is set up by introducing a fictitious flow of the order of 10−4 on a lattice of size 244. The obtained time t and studying the evolution of the flow gauge field critical hopping parameter values, κcðβLÞ, are then used for Bμðx; tÞ according to the flow equation all used lattice sizes L=a ¼ 12, 18, 24, and 30. A summary of the simulation parameters and corresponding PCAC ∂tBμ ¼ DνGνμ; ð8Þ quark masses, as well as the acceptance rates and accu- mulated statistics for each simulation, is given in the where Gμνðx; tÞ is the field strength of the flow field Bμ and Tables I and II. Dμ ¼ ∂μ þ½Bμ; · . The initial condition is Bμðx; t ¼ 0Þ¼ Because we need to use strong coupling in the UV, we AμðxÞ, where Aμ is the original continuum gauge field. In are forced to use small values for β . We include even L the lattice formulation, the lattice link variable U replaces negative values for β in our study. This is to compensate L the continuum flow field, which is then evolved using the for the effective positive β shift induced by the Wilson L tree-level improved Lüscher-Weisz pure gauge action [80]. fermions [75,76]. To leading order, this shift is proportional to the number of flavors and can therefore be substantial at B. What to expect large Nf. As a consequence, even at the smallest βL ¼ −1 at Nf ¼ 48, the lattice gauge field observables (for exam- The gradient flow equation has the form of a heat ple, the plaquette) behave as if the effective gauge coupling equation, which, in the case of a free gluon field in the remains positive. A qualitatively similar effect has been Landau gauge, is for example solved by the heat kernel [81]
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TABLE I. The table shows for Nf ¼ 24 for each of our simulation points the value of βL and κ, as well as the resulting PCAC quark −4 mass mqa (in units of 10 ), determined on the L ¼ 24a lattice. Furthermore, the acceptance rates (acc.) of the hybrid Monte Carlo (HMC) trajectories and the accumulated statistics (stat.), i.e., number of sampled configurations, are shown for each system size.
mqaL¼ 12aL¼ 18aL¼ 24aL¼ 30a −4 βL κ ð10 Þ acc. stat. acc. stat. acc. stat. acc. stat. −0.3 0.131578 8.3(4) 0.88 5k 0.78 5k 0.83 4k 0.86 2.6k −0.25 0.131354 9.5(5) 0.89 5k 0.78 5k 0.84 4k 0.88 2.6k −0.2 0.131139 9.1(5) 0.89 5k 0.8 5k 0.84 4k 0.87 2.6k −0.15 0.13093 8.1(5) 0.9 5k 0.8 5k 0.84 4k 0.88 2.6k −0.1 0.130725 7.0(4) 0.9 5k 0.82 5k 0.86 4k 0.88 2.7k −0.05 0.130519 5.3(6) 0.91 5k 0.82 5k 0.85 4k 0.9 2.6k 0.001 0.130322 0.1(4) 0.93 10k 0.84 4.9k 0.73 4k 0.89 2.7k 0.05 0.130129 1.2(4) 0.94 10k 0.84 4.8k 0.73 4.3k 0.9 2.3k 0.1 0.129944 −0.7ð4Þ 0.94 10k 0.86 4.6k 0.74 4.2k 0.92 2.4k 0.15 0.129758 −0.3ð4Þ 0.94 10k 0.86 4.5k 0.74 4.1k 0.91 2.7k 0.2 0.129579 −0.7ð4Þ 0.94 10k 0.86 4.5k 0.77 3.8k 0.92 2.5k 0.25 0.129403 0.3(3) 0.94 10k 0.86 4.4k 0.78 4.1k 0.92 2.4k 0.3 0.129232 0.7(3) 0.95 10k 0.87 4.4k 0.78 4k 0.92 2.4k 1. 0.127336 −0.1ð3Þ 0.97 10k 0.94 4.4k 0.85 4k 0.96 2.4k 3. 0.125525 −0.1ð1Þ 0.99 10k 0.98 5k 0.98 4k 0.98 2k 4. 0.125355 1.0(1) 0.99 10k 0.98 5k 0.96 4k 0.99 2.7k 6. 0.125224 −0.2ð1Þ 0.99 10k 0.98 5k 0.95 4k 0.99 3k
TABLE II. The table shows for Nf ¼ 48 for each simulation point the value of βL and κ, as well as the resulting PCAC quark mass mq (in units of 10−4), determined on the L ¼ 24a lattice. Furthermore, the acceptance rates (acc.) of the hybrid Monte Carlo (HMC) trajectories and the accumulated statistics (stat.), i.e., number of sampled configurations, are shown for each system size.
mqaL¼ 12aL¼ 18aL¼ 24aL¼ 30a −4 βL κ ð10 Þ acc. stat. acc. stat. acc. stat. acc. stat. −1. 0.129534 2.2(6) 0.85 5k 0.82 5k 0.83 2.1k 0.78 1.5k −0.5 0.128463 0.1(3) 0.88 5k 0.84 4.4k 0.85 1.9k 0.82 1.1k −0.4 0.128269 0.5(3) 0.88 5k 0.86 4.9k 0.86 1.9k 0.83 1.1k −0.3 0.128084 0.4(4) 0.9 4.6k 0.86 4.7k 0.86 1.8k 0.83 1.1k −0.25 0.127996 −0.6ð3Þ 0.9 4.1k 0.87 4.4k 0.86 1.8k 0.84 1.1k −0.2 0.127905 0.1(3) 0.9 4.5k 0.87 4.1k 0.88 1.6k 0.87 1.3k −0.15 0.12782 −0.1ð3Þ 0.9 4.1k 0.88 4.4k 0.88 1.5k 0.86 1.2k −0.1 0.127737 −0.4ð3Þ 0.91 4.5k 0.89 4.8k 0.9 1.6k 0.84 1.2k −0.05 0.127651 0.1(3) 0.91 4k 0.88 4.5k 0.89 1.6k 0.87 1.2k 0.001 0.12757 −0.6ð2Þ 0.92 4k 0.88 4.8k 0.83 2.4k 0.88 1k 0.05 0.127487 1.8(3) 0.92 4k 0.89 5.1k 0.82 2.2k 0.9 1k 0.1 0.127413 0.4(2) 0.92 4k 0.89 4.6k 0.83 2.6k 0.87 1k 0.15 0.127344 −1.7ð2Þ 0.93 4k 0.89 5.4k 0.82 2.3k 0.89 1k 0.2 0.127265 0.5(3) 0.94 4k 0.9 5k 0.83 2.1k 0.88 1k 0.25 0.127191 1.5(3) 0.94 4k 0.91 4.7k 0.85 2.4k 0.9 1k 0.3 0.127125 −0.1ð2Þ 0.94 4k 0.91 4.4k 0.86 2.3k 0.9 1k 1. 0.126332 0.5(2) 0.97 4k 0.95 5k 0.92 2k 0.95 1.1k 3. 0.12544 −0.1ð1Þ 0.98 2k 0.99 2.4k 0.97 2k 0.97 1.3k 6. 0.125209 −0.3ð1Þ 0.98 2k 0.98 2.3k 0.98 1.9k 0.98 1.3k
ffiffiffiffi 2 p 1 ðy − xÞ with smearing radius σ ¼ 2t, which means that physical Hðy − x; tÞ¼ exp − : ð9Þ pffiffiffiffi ð4πtÞd=2 4t processes at length scales below 2σ ¼ 8t ≕ λðtÞ get effectively integrated out. The scale μðtÞ¼1=λðtÞ can Hence, by evolving a gauge configuration with the gradient therefore be interpreted as renormalization scale of gauge flow for some time t, one performs a Gaussian smearing observables measured at flow time t.
074508-7 VILJAMI LEINO et al. PHYS. REV. D 101, 074508 (2020)
μ The nonperturbative renormalized coupling at scale with Landau pole [81] is then defined via energy measurement as 2 2 10 g0 >> gcr
) 2 2 2 2 0 2 −1 2 2 g >g μ N 2 0 cr g0 =gcr g ð Þ¼ t hEðtÞijt 1= 8μ ; ð10Þ GF ¼ ð Þ ,g 0
/ 5
N ( where is a normalization factor that has been calculated 2
g 2 2 in Ref. [82] to match the MS coupling at tree level and the g0 074508-8 SAFETY VERSUS TRIVIALITY ON THE LATTICE PHYS. REV. D 101, 074508 (2020) λ 2 2 8 scale ðtÞ, whereas for g0 2 2 0 system when g0 crosses g . For a theory which is IR— 4 cr / 2 2 –7 — ( g0 =gcr –10 instead of UV free, the dependency of the lattice spacing 2 g 2 a on the bare lattice parameter βL is unfortunately not necessarily such that a can become arbitrarily small, and it λ 0 2 2 might therefore not be possible for a ¼ 0 to reach values g0 = 2.5 g0 =1. g2 ≥ g2 for which 0 cr. 0 1 2 3 4 5 6 7 8 The discovery of an interacting UV fixed point on the / lattice in this way is possible only if the true, nonperturbative 0 2 2 beta function is such that the discussion in connection with FIG. 4. The running coupling g ðλ=λ0;g0Þ for the leading order Fig. 3 applies. As discussed above, the four-loop beta large-Nf beta function from Eq. (2) with Nf ¼ 48 (cf. lower function from Fig. 2 was used to model the UV fixed point panels of Fig. 2) as a function of increasing (relative) renorma- behavior in Fig. 3. This four-loop beta function is smooth lizaion length scale λ=λ0 for different choices of initial coupling 2 2 2 everywhere, and its slope in the neighborhood of the UV g0 ¼ g ð1;g0Þ and corresponding reference scale λ0. The red, fixed point is moderate. However, if the nonperturbative beta horizontal dotted line indicates the location of the UV fixed point, 2 4 9348022… 48 function behaves more like the leading order large-Nf beta gcr ¼ . . In contrast to the four-loop Nf ¼ beta function (red curve in lower panels of Fig. 2), the situation is function used for the discussion in Fig. 3, which had a moderate significantly more complicated, as the UV fixed point arises slope at the fixed point, the slope of the leading order large-Nf 2 due to a singularity as β → −∞. This abrupt change makes beta function at the fixed point gcr is very steep. This makes it 2 2 almost impossible to draw the line of constant running coupling, the running coupling for g0 074508-9 VILJAMI LEINO et al. PHYS. REV. D 101, 074508 (2020) L = - 0.3 L = 0.001 τ 8 L=12 8 L=12 method used in Refs. [49,83] to optimize .However,asit ) L=18 ) L=18 turns out that the gradient flow coupling obtained by our L 6 L 6 , L=24 , L=24 simulations are quite small, we can expect the scheme ( 4 L=30 ( 4 L=30 GF GF 2 2 independent perturbative two-loop beta function to be a g g 2 2 very accurate approximation to the true beta function at β 2.5 5 7.5 10 12.5 15 2.5 5 7.5 10 12.5 15 large enough flow time (cf. Fig. 2). For each L, we tune τ =c L =c L by matching the largest volume L ¼ 30a gradient flow L =3 L =6 coupling to the two-loop perturbative coupling over the 1.2 L=12 L=24 L=12 L=24 interval λ ∈ ½3a; 5a . We then use this value for τ for 0.6 ) L=18 L=30 ) L=18 L=30 L 1.1 L smaller volumes L=a ¼ 12, 18, 24. Examples for (11) at , 1 , 0.5 τ β 24 ( ( optimal ð LÞ are shown in Fig. 7 (Nf ¼ ) and Fig. 8 0.9 2 GF 2 GF g g 48 β 0.8 0.4 (Nf ¼ ) for two different values of L, together with the 0.7 corresponding two-loop running coupling. 2.5 5 7.5 10 12.5 15 2.5 5 7.5 10 12.5 15 In Fig. 9, we show the step scaling function =c L =c L 2 24 Σðu; s; L; cÞ¼g ðcsL; βLÞjβ ∶ 2 β ; ð12Þ FIG. 5. The gradient flow couplings for Nf ¼ , measured at GF;sL L gGF;LðcL; LÞ¼u different values of the bare inverse lattice coupling β ,as pffiffiffiffi L functions of the flow scale λ ¼ 8t (where t is the flow time which tells us how the coupling constant evolves when the in lattice units). length scale where it is evaluated changes from cL to scL. In the following, we use c 0.22; i.e., the gradient flow pffiffiffiffi ¼ 2 λ β time is fixed so that 8t ¼ 0.22L. We are forced to use a coupling gGFð ; LÞ can be expected to behave like the 2 2 relatively small value for c (in comparison with conven- running coupling g ðλ=λ0;g Þ of the continuum theory, is 0 0 4…0 5 approximately given by λ ∈ ½3a; 0.25L , where the lower tional c ¼ . . , used in Ref. [49], for example) in bound of 3a is somewhat optimistic. This renders the order to avoid excessive finite volume effects. The step smallest lattices L ≤ 18 of very limited use. We can optimize the gradient flow coupling by adding a shift τ to the flow time [83]: 2 −1 2 λ β N τ 2 gGFð ; LÞ¼ t hEðt þ Þijt¼λ =8: ð11Þ The effect of τ vanishes in the continuum limit (if it exists), and it can be tuned to get rid of most Oða2Þ lattice artifacts. Due to the very large finite size artifacts, we cannot use the L =-1 L = 0.001 6 L=12 2.5 L=12 L=24 ) L=18 ) L=18 L=30 L L 2 4 , L=24 , 1.5 ( L=30 ( 2 2 GF 2 GF g g 1 0 0.5 0 2.5 5 7.5 10 12.5 15 0 2.5 5 7.5 10 12.5 15 =c L =c L L =3 L =6 1 L=12 L=24 0.5 L=12 L=24 ) L=18 L=30 ) L=18 L=30 L L 0.45 0.8 , , 0.4 ( ( 0.6 2 GF 2 GF g g 0.35 0.4 0.3 0 2.5 5 7.5 10 12.5 15 0 2.5 5 7.5 10 12.5 15 =c L =c L FIG. 7. The gradient flow coupling for Nf ¼ 24 and bare inverse lattice coupling βL ¼ −0.3 (upper panel) and βL ¼ 6 FIG. 6. The gradient flow couplings for Nf ¼ 48, measured at (lower panel) as functions of the flow scale λ after optimization of different values of the bare inverse lattice coupling β ,as τ β 2 λ λ 1 2 pffiffiffiffi L the parameter ð LÞ. The value of g0 ¼ gð = 0 ¼ ;g0Þ, stated in functions of the flow scale λ ¼ 8t (where t is the flow time the panel, can be used as a definition of an effective lattice-scale in lattice units). coupling for the lattice theory. 074508-10 SAFETY VERSUS TRIVIALITY ON THE LATTICE PHYS. REV. D 101, 074508 (2020) L=24, c=0.22 L=24, c=0.22 1.15 s=1/2 1.25 s=1/2 s=3/4 s=3/4 1.2 1.1 s=5/4 s=5/4 1.15 1.05 1.1 (u,s,L,c)/u (u,s,L,c)/u 1.05 1 1 0.95 0.95 0.5 0.6 0.7 0.8 0.9 1 0.35 0.4 0.45 0.5 0.55 u u FIG. 9. Nf ¼ 24 (left) and Nf ¼ 48 (right) step scaling Σðu; s; L; cÞ=u as functions of u, with Σðu; s; L; cÞ¼ 2 g ðscL; βLÞjβ ∶ 2 β defined in terms of the gradient flow GF L gGFðcL; LÞ¼u 2 β coupling gGFðcL; LÞ. The latter can be interpreted as a non- perturbative definition of the renormalized coupling at scale λ ¼ cLa. For fixed c and βL, the ratio of the gradient flow couplings obtained with system sizes L and L0 ¼ sL, i.e., 2 0 β 2 β gGFðcL ; LÞ=gGFðcL; LÞ, can then be identified with the ratio of renormalized couplings g2ðλ0Þ=g2ðλÞ at scales λ0 and λ, where λ0=λ ¼ s. The dashed lines show the corresponding perturbative two-loop results. FIG. 8. As in Fig. 7 but for Nf ¼ 48. well. As we can observe, the lattice result approaches the scaling is particularly well suited for doing a continuum two-loop result as the volume increases. extrapolation (if it exists) of the running coupling. For a From the results above, we see that the measured conventional extrapolation, one would, however, need a set gradient flow couplings are very small in the region where we can trust the measurements, at most g2 < 1.75. The of lattice sizes A ¼fL1; …;L g, from which one can form GF n λ several pairs ðL0;LÞ that correspond to the same ratio couplings grow as the distance is reduced ( decreased), λ ≲ 3 L0=L ¼ s. The lattice volumes available to us do not allow until lattice artifacts take over at a. In order to reach this. Furthermore, very large lattice artifacts at smaller strong couplings, we use small values for the inverse bare β volumes would make the extrapolation unreliable. coupling L (even negative, as discussed in Sec. III A). β What we can do instead is compare the step scaling However, it turns out that if L is small enough, the gradient function for different s directly with the corresponding two- flow coupling becomes almost independent of its value. loop results in Fig. 9. The agreement is reasonable, except This behavior is compatible with the Landau-pole-like for s ¼ 1=2, which involves the smallest system size behavior, as we will discuss in Sec. III C 1 below. L ¼ 12 for which the finite size and finite volume effects are particularly strong. A more transparent comparison of data from pairs of c=0.22 c=0.22 0 0 system size ðL ;LÞ with different s ¼ L =L can be obtained 0.2 s=3/2 0.1 s=3/2 with the discrete beta function s=2 s=2 0.08 0.15 s=5/3 s=5/3 2 0.06 g ðscL;β Þ−u (s) L 0.1 (s) L βðsÞ − GF;sL L L ðu;L;cÞ¼ 2 ; ð13Þ 0.04 2 logðs Þ β ∶g ðcL;β Þ¼u 0.05 L GF;L L 0.02 0 0 which approaches the conventional beta function in the → 0 2 → 0 2 0.5 0.75 1 1.25 1.5 0.3 0.4 0.5 0.6 0.7 limits s or g . Because gGF is small, this is a good u u approximation of the true beta function. The results are 24 48 shown in Fig. 10 for c ¼ 0.22 and s ¼ 3=2 ¼ 18=12, FIG. 10. Nf ¼ (left) and Nf ¼ (right) discrete beta 2 24 12 30 18 5 3 βðsÞ − 2 β − 2 β 2 s ¼ ¼ = , and s ¼ = ¼ = . For comparison, function L ðu;L;cÞ¼ ðgGF;sLðscL; LÞ gGF;LðcL; LÞÞ=logðs Þ 2 β the corresponding results for the discrete beta function, as functions of u ¼ gGFðcL; LÞ. The dashed lines show the evaluated with the two-loop running coupling, are shown as corresponding perturbative two-loop results. 074508-11 VILJAMI LEINO et al. PHYS. REV. D 101, 074508 (2020) 2 1. Effective g0 I As mentioned earlier, the pure gauge system undergoes As we saw above, the gradient flow coupling makes sense a smooth crossover to a lattice bulk phase around 2 ≈ 1 7–1 8 only when the flow scale is large enough, i.e. larger than about g0;eff . . . This artifact, not present in our simulations 2 2 3 with fermions, does not invalidate the use of g0 as proxy a. Nevertheless, it is of interest to relate gGF to some lattice ;eff β 4=g2 for the coupling at the lattice cutoff scale, but the functional ultraviolet scale effective coupling. The inverse of L ¼ 0 2 does not make much sense, because it can be negative. dependency of the gradient flow running coupling on g0;eff 2 Here, we use the value of the plaquette (trace of the 1 × 1 can start to deviate from the expected behavior when g0;eff 2 1 7–1 8 Wilson loop) as a proxy; it is an UV quantity on the lattice, enters the bulk region, i.e., when g0;eff > . . . This is and its expectation value is readily measurable. To convert shown by vertical lines in Fig. 11. Nevertheless, the the plaquette values to effective couplings, we use an excellent match of the lattice measurements with the “inverse Monte Carlo” procedure: we perform new simu- perturbative curve indicates that the effect is small. 2 → 0 lations using a pure gauge SU(2) lattice theory with Wilson We observe the following behavior: as g0;eff , the plaquette action [i.e., using only S ðUÞ in Eq. (7)] and tune 2 2 → 1 G ratio gGF=g0;eff , as dictated by universality at weak the pure gauge inverse coupling βPG so that the plaquette 2 2 L coupling. On the other hand, as g0;eff grows, gGF seems to expectation value matches that of the original theory. The approach a constant. This is characteristic behavior for an effective bare coupling is obtained by inversion UV Landau pole; as the UV scale approaches the Landau pole, the UV scale coupling diverges. Because g2 is g2 4=βPG: GF 0;eff ¼ L ð14Þ evaluated at length scales which are by a constant factor larger than the UV length scale, the value of g2 will This quantity is positive for all of our lattices. Because GF instead approach a constant value. the plaquette expectation value is, in practice, independent 2 2 The couplings g and g have been obtained using g2 β N 0;eff GF of the volume, 0;eff is only a function of L and f.It different schemas, and thus their values cannot be com- β increases monotonically as L is decreased. At our smallest pared without matching. Nevertheless, we obtain a reason- β −0 3 24 −1 0 48 L values, . (Nf ¼ ) and . (Nf ¼ ), it reaches able fit of the data in Fig. 11 with the two-loop perturbative 2 2 2 2 maximum values of about 6 and 5.3, respectively. running coupling g λ=λ0;g by identifying g ≡ g and 2 ð 0Þ 0 0;eff 2 2 2 2 In Fig. 11, we plot the gradient flow couplings gGF, λ λ ≡ λ 2 g ð = 0;g0Þ gGFð =a; g0;effÞ. A good qualitative agree- measured at fixed flow scale λ ¼ 4a, against g0 . The ;eff ment is obtained with the matching coefficients λ0 ¼ a=9 gradient flow coupling is almost independent of the for Nf ¼ 24 and λ0 ¼ a=18 for Nf ¼ 48. This implies that volume, excluding L=a ¼ 12, where flow scale 4a is large 2 the effective lattice coupling g0 corresponds to a refer- enough so that finite volume effects become important. We ;eff 12 ence scale, which is about 9, respectively 18, times smaller shall ignore L=a ¼ in the following. than the lattice scale a. The fit is compatible with the existence of the Landau pole, because the two-loop beta function also features one. However, we naturally cannot exclude the existence of an UVFP at stronger UV coupling than reached here. The possible existence of the Landau pole gives a natural explanation for the small value of the gradient flow coupling, the lattice UV length scale is approximately a, 2 whereas the gradient flow coupling gGF is evaluated at 3 2 length scale a or larger. Thus, in terms of energy, gGF is μ μ 3 μ evaluated at scale GF < LP= , where LP is the scale where Landau pole appears. This gives ample room for the coupling to decrease. The actual value of the coupling depends on the details of the scheme. FIG. 11. The relation between the gradient flow coupling 2. Effective g2 II 2 β λ 4 0 gGFðL; L;cÞ at flow scale ¼ cL ¼ a and the effective bare 2 β 24 48 Another possibility to define an effective coupling for coupling g0;eff ð LÞ at Nf ¼ (left) and Nf ¼ (right). The dashed line is the perturbative (pert.) two-loop result, where the the lattice theory comes as a side product of the method we 2 2 2 used to determine the optimal value of the τ parameter in the continuum couplings g0 and g ðλ=λ0;g0Þ are, respectively, 2 2 λ 2 definition (11) of the improved gradient flow coupling, i.e., identified with the g0;eff and gGFð =a; g0;eff Þ from the lattice. g2 g2 The 0;eff ¼ GF line is shown with dots, and the vertical dashed 2 τ 2 2 t hEðt þ Þi line indicates the value of g0 above which the pure gauge g λ; β ; τ : ;eff GFð L Þ¼ N τ ð15Þ system is in the lattice bulk phase. ðt þ Þ t¼λ2=8 074508-12 SAFETY VERSUS TRIVIALITY ON THE LATTICE PHYS. REV. D 101, 074508 (2020) X χ2 τ λ τ − λ 2 τ 2 τ λ τ 2 ð Þ¼ ð ið Þ ptðgGF;ið Þ;g0ð Þ; 0ð ÞÞÞ ð19Þ i τ λ τ 2 τ with respect to , where ð 0ð Þ;gGF;0ð ÞÞ is given by the pair in (17) for which the corresponding original λi, i ¼ 1; 2; …, is closest to the middle of the fitting interval λ ∈ ½3; 5 . After having determined in this way the optimal τ, one λ 2 λ τ 2 τ also has fixed the values ð 0;g0Þ¼ð 0ð Þ;gGF;0ð ÞÞ, for which (18) has within the given fit interval the best overlap with the τ-shifted lattice data. Solving now λ 2 2 λ 1 ptðg ;g0; 0Þ¼ ð20Þ for g2, one obtains an effective coupling for the lattice theory 2 β at the cutoff scale a, which we call g0;eff2ð LÞ. In Figs. 7 and 2 β 8, the value of g0;eff2ð LÞ can be read off from the value of the perturbative two-loop curve (dashed line) at λ ¼ 1. 2 β From the derivation of the effective coupling g0;eff2ð LÞ, it is not a big surprise that one finds in Fig. 12 excellent agreement between lattice data and two-loop result, when plotting the gradient flow coupling at fixed flow scale λ 2 β (in lattice units) as function of g0;eff2ð LÞ, identifying the 2 λ λ 2 2 λ β λ ≡ perturbative g ð = 0;g0Þ and gGFð ; LÞ, with 0 a and 2 g2 ≡ g2 β . FIG. 12. The relation between gGF and effective lattice coupling 0 0;eff2ð LÞ 2 β 24 48 g0;eff2ð LÞ at Nf ¼ (left) and Nf ¼ (right), evaluated at flow scale λ ¼ cL ¼ 4a (upper panels) and 6a (lower panels). IV. CONCLUSIONS The dashed line is the perturbative (pert.) two-loop result, where 2 2 2 the continuum couplings g0 and g ðλ=λ0;g0Þ are, respectively, The UV behavior of gauge-fermion theories as a function 2 2 λ 2 of the number of matter fields poses a fundamental problem identified with the g0;eff and gGFð =a; g0;eff2Þ from the lattice. The 2 2 on our understanding of quantum field theory, both g0;eff2 ¼ gGF line is shown with dots. perturbatively and nonperturbatively. Recent theoretical The procedure is as follows. From the lattice data for developments suggest that at inifinite number of flavors 2 λ β 2 λ β ;0 an interacting UV fixed point could exist making these the gradient flow coupling gGFð ; LÞ¼gGFð ; L Þ as function of flow scale λ, we produce data pairs ðλ ;g2 Þ, theories asymptotically safe. i GF;i In this paper, we have discussed in detail the categoriza- i ¼ 1; 2; …, with g2 ¼ g2 ðλ ; β Þ and λ ∈ ½3; 5 . Then, GF;i GF i L i tion of gauge-fermion theories based on their possible UV using that behaviors. Then, we described our computational setting to λ4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi investigate the renormalization group evolution of SU(2) 2 λ β ; τ 2 λ2 8τ β gauge theory with 24 or 48 Dirac fermions. Finally, we gGFð ; L Þ¼ 2 2 gGF þ ; L ; ð16Þ ðλ þ 8τÞ presented the results of our first pioneering analysis. We found that with the methodologies developed in our τ we form the corresponding pairs for the -shifted gradient past work, the nonperturbative computations can be suc- flow coupling (15): cessfully carried out. We have demonstrated that our results qffiffiffiffiffiffiffiffiffiffiffiffiffiffi match well with perturbation theory. On the other hand, we ðλ2 − 8τÞ2 λ τ ;g2 τ λ2 − 8τ; i g2 : were unable to reach lattices where strong renormalized ð ið Þ GF;ið ÞÞ ¼ i λ4 GF;i ð17Þ i couplings could be controllably obtained. We used the gradient flow procedure to determine the renormalized We then use the solution to the differential equation (1) in the coupling on the lattice. By construction, gradient flow form coupling can be evaluated at length scales which are at Z least a few lattice spacings. This leaves room for a Landau g2 λ 2 2 λ λ − du pole or otherwise large couplings to exist at shorter distances ptðg ;g0; 0Þ¼ 0 exp ; ð18Þ 2 β g0 u ðuÞ (in the lattice ultraviolet limit). We observed indications of this behavior by defining an effective lattice ultraviolet with g2βðg2Þ being the perturbative two-loop beta function, coupling. The observed behavior is compatible with the shown in Fig. 2, and minimize existence of a Landau pole, but we cannot exclude an 074508-13 VILJAMI LEINO et al. PHYS. REV. D 101, 074508 (2020) ultraviolet fixed point at even stronger couplings ACKNOWLEDGMENTS than reached here. Consequently, the true continuum The support of the Academy of Finland Grants behavior of these theories in the deep ultraviolet remain No. 308791, No. 310130, and No. 320123 is acknowl- undetermined. edged. The authors wish to acknowledge CSC - IT Center Nevertheless, our study provides an important milestone for Science, Finland, for computational resources. V. L. is toward establishing the ultaviolet fate of gauge-fermion supported by the DFG cluster of excellence Origins. The theories when asymptotic freedom is lost. The results work of F. 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