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REVIEW 3 major objections 5 minor 98 references

Safety versus triviality on the lattice

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper demonstrates that the Yang–Mills gradient flow extracts the renormalized running coupling of SU(2) with 24 or 48 Dirac flavors and that the coupling stays small at accessible scales—a hint of triviality, not certainty.

desk verdict First lattice look at SU(2) with 24/48 flavors: the saturation signal is real, but the claimed 'proof' of perturbative matching is partly built into the fitting, so treat it as an exploratory step, not a settled result. read the letter →

arxiv 1908.04605 v2 pith:DZSUHUSC submitted 2019-08-13 hep-lat hep-phhep-th

Viljami Leino , Tobias Rindlisbacher , Kari Rummukainen , Francesco Sannino , Kimmo Tuominen This is my paper · ORCID
classification hep-lathep-phhep-th
keywords asymptoticsafetytrivialitygradientflowlargenumberofflavorslatticegaugetheoryrenormalizedrunningcouplingconformalwindow2.0SU(2)
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper reports the first lattice study of the ultraviolet behaviour of a non-abelian gauge theory with a very large number of fermion flavours, where asymptotic freedom is lost. Using the Yang–Mills gradient flow on SU(2) gauge theory with 24 and 48 massless Dirac fermions, it extracts a renormalized running coupling and shows that the result agrees with the two-loop perturbative beta function, tying the lattice data to the Gaussian infrared fixed point. The paper's key observation is that the renormalized coupling refuses to grow large on the lattice even as the bare coupling is pushed toward strong values; this is compatible with the theory developing a physical cutoff (triviality), while leaving open the alternative that the lattice action simply cannot reach the deep ultraviolet region where an interacting safe fixed point might live. A sympathetic reader takes this as a proof of principle that the ultraviolet fate of such theories is now numerically accessible, not as a settlement of the triviality-versus-safety question.

What carries the argument

The central object is the Yang–Mills gradient flow: a fictitious flow time $t$ smooths the gauge field by the heat equation, so that observables measured at flow time $t$ probe the theory at renormalization scale $\mu = 1/\sqrt{8t}$. On the lattice, with Dirichlet temporal boundary conditions and the clover-improved Wilson action, the paper defines the renormalized coupling $g_{\mathrm{GF}}^2 = \mathcal{N}^{-1}\, t^2 \langle E(t)\rangle$ at the central time slice, with $\mathcal{N}$ fixed to match the $\overline{\mathrm{MS}}$ scheme. Two devices carry the argument: a flow-time shift $\tau$ that removes most $\mathcal{O}(a^2)$ lattice artefacts by matching the $L=30a$ data to the two-loop perturbative running, and two alternative definitions of an effective bare coupling (one via plaquette matching to pure gauge theory, one from inverting the perturbative running at the lattice cutoff), which together allow the authors to compare lattice data against perturbative evolution and to display saturation of the renormalized coupling as the bare coupling grows.

What would settle it

Measure the discrete $\beta$ function at $N_f = 48$ on larger volumes ($L/a = 36, 42$) and at stronger bare couplings ($\beta_L < -1$): if the gradient-flow coupling at fixed $\lambda/a$ continues to saturate at $g_{\mathrm{GF}}^2 \sim 1.75$ while the effective bare coupling grows past 6, the physical-cutoff (triviality) interpretation survives; if the coupling instead bends upward toward the large-$N_f$ fixed-point value $g^2_{\mathrm{cr}} \sim 4.93$, the data would point to an ultraviolet fixed point that the current action cannot reach. A sharper test is to run the same analysis at $N_f = 96$: a rising plateau value with $N_f$ would indicate the safe region is being approached, while a $N_f$-independent plateau would support triviality.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the gradient flow method—originally developed for asymptotically free theories—remains a valid tool when asymptotic freedom is lost. For SU(2) with 24 and 48 fundamental Dirac flavours, the lattice gradient-flow coupling $g_{\mathrm{GF}}^2$, measured at flow scales between about three lattice spacings and a quarter of the lattice size, matches the scheme-independent two-loop perturbative running coupling after a small flow-time shift $\tau$ is tuned on the largest volume. This match anchors the simulations to the Gaussian fixed point in the infrared. The authors then find that even at the strongest accessible bare couplings (including negative $\beta_L$ to compensate the large positive shift induced by many Wilson fermions), the renormalized coupling measured at flow scales $\lambda \in [3a,\, 0.25L]$ stays below $g_{\mathrm{GF}}^2 \lesssim 1.75$, and two complementary effective bare couplings (from plaquette matching and from the $\tau$-fit) indicate that the coupling would have to rise sharply at scales below the lattice spacing if it is to reach an interacting fixed point. They read this as compatible with a Landau pole—an early sign of a physical cutoff—while explicitly not ruling out an ultraviolet fixed point at stronger coupling than the simulations reach.

Load-bearing premise

The analysis assumes that the two-loop perturbative beta function is an accurate description of the nonperturbative running at the flow scales and couplings probed, so that tuning the flow-time shift to match it is a legitimate improvement rather than imposing the answer.

Editorial extensions

If this is right

  • The gradient flow method is now available as a probe of non-perturbative running in theories that are infrared-free, opening the same analysis to other large-flavour gauge theories.
  • Matching to the two-loop perturbative beta function confirms that both $N_f=24$ and $N_f=48$ SU(2) theories lie in the basin of the Gaussian fixed point at the scales reached.
  • The saturation of the measured coupling as the bare coupling grows is compatible with a Landau pole below the lattice cutoff, i.e. with a physical cutoff and triviality in the ultraviolet.
  • If the saturation is physical, reaching an interacting ultraviolet fixed point would require even larger numbers of flavours than 48, or a lattice action able to probe deeper into the ultraviolet.
  • The step-scaling data for $s>1/2$ approach the two-loop discrete beta function as the volume increases, providing a quantitative starting point for future continuum extrapolations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension is to simulate $N_f = 64$ or $96$: if the renormalized coupling at fixed $\lambda/a$ still saturates at the same plateau value, the Landau-pole/triviality interpretation is strengthened; if it begins to rise, that points to a safe fixed point moving into reach.
  • Because the saturation could depend on the lattice action (Wilson clover with HEX smearing), repeating the measurement with staggered fermions or different smearing schemes would separate a physical cutoff from an action artefact.
  • If these theories are indeed trivial, they cannot serve as ultraviolet completions on their own; asymptotically safe extensions of the Standard Model built on large-flavour gauge sectors would then need elementary scalars and Yukawa couplings, as in the original gauge-Yukawa constructions.
  • The successful two-loop matching also suggests the flow-time shift can be repurposed as a scale-setting tool for infrared-free theories, where the usual chiral-scale-setting observables are absent.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper presents a lattice study of SU(2) gauge theory with 24 and 48 massless Dirac flavors, for which asymptotic freedom is lost. The authors use the Yang-Mills gradient flow with Dirichlet boundary conditions to define a renormalized running coupling, at bare couplings beta_L in [-1,6] and volumes L/a = 12, 18, 24, 30. They introduce a flow-time shift tau(beta_L) that is tuned so that the L=30a data match the two-loop perturbative running on the interval lambda in [3a,5a], and then compare step-scaling functions and discrete beta functions with the same two-loop curves. Two effective UV couplings are also constructed, one from a plaquette-based inverse Monte Carlo procedure and one from the tau-fitting procedure itself. The authors observe that the gradient-flow coupling saturates as the effective bare coupling grows, which they interpret as compatible with a Landau pole or physical cutoff, while explicitly stating that an ultraviolet fixed point at stronger coupling cannot be excluded.

Significance. If correct, this would be the first lattice evidence about the ultraviolet dynamics of large-N_f gauge-fermion theories, a regime not accessible by other methods. The paper is transparent about many limitations: it states that no continuum extrapolation is performed, that L=12 and L=18 suffer from strong finite-size effects, and that the results do not settle the safety-versus-triviality question. It also provides detailed simulation parameters and statistics. The most robust new observation, the saturation of g_GF as a function of the plaquette-based effective coupling in Fig. 11, is cautiously worded. However, the central claim of agreement with the two-loop perturbative beta function is weakened by the fact that two of the comparison schemes are defined by fitting to that same curve; the word 'prove' in the abstract is therefore not supported by the evidence presented. The paper is a useful exploratory first step, but the central claim needs reframing and additional cross-checks.

major comments (3)
  1. [III.C, Eq. (11), Figs. 7-10] The central claim that the lattice gradient-flow coupling matches the two-loop perturbative beta function is partly generated by the fitting procedure. In Sec. III.C the authors state that for each beta_L they 'tune tau by matching the largest volume L=30a gradient flow coupling to the 2-loop perturbative coupling over the interval lambda in [3a,5a]', and the same two-loop curve is then displayed alongside the tau-shifted data in Figs. 7 and 8 and used as the reference in the step-scaling and discrete-beta-function comparisons in Figs. 9 and 10. Agreement on [3a,5a] is therefore a consistency condition rather than an independent test. The independent content is limited to whether L=18 and L=24 with the same tau continue to match, and this is weakened by the absence of error bars and by the paper's own statement that the smaller volumes are of limited use. I request an independent cross-check, for example a tau fixed by a criterion that does not reference the perturbative curve, or a quantitative comparison of tuned and untuned running, before the abstract's 'prove' language can be retained.
  2. [III.C.2, Eqs. (15)-(19), Fig. 12] The second effective-coupling construction is explicitly a fit to the same two-loop running over the same lambda in [3a,5a] interval: chi^2(tau) in Eq. (19) minimizes the deviation of the tau-shifted data from lambda_pt(g^2,g0^2,lambda0) obtained from the two-loop beta function. Consequently the 'excellent agreement' displayed in Fig. 12 is by construction and cannot corroborate the claim that the lattice running is perturbative. This section should be rewritten to state that the construction defines g_{0,eff2} rather than tests the beta function, and the interpretation of Fig. 12 should be adjusted accordingly.
  3. [III.C, Figs. 9-11] The quantitative basis for the claimed agreement is thin: no continuum extrapolation is performed, the key figures show no statistical error bars, and the paper itself acknowledges that L=12 and L=18 are strongly affected by finite-size effects. Since the bulk of the evidence rests on L=24 and L=30, with c=0.22 chosen to avoid finite-volume effects, the manuscript should either provide an estimate of the residual finite-volume and systematic uncertainty or temper the claims of agreement accordingly. As it stands, Figs. 9 and 10 do not allow the reader to assess whether the deviations, in particular the s=1/2 points, are statistically significant.
minor comments (5)
  1. [Abstract and Conclusions] The word 'prove' is used for the matching with perturbation theory; given the fitting-based definitions of tau and g_{0,eff2}, 'provide evidence' or 'is consistent with' is more appropriate.
  2. [Figs. 5-10 and Fig. 12] Several figures do not include statistical error bars; either error bars should be added or a statement explaining that they are smaller than the symbol size.
  3. [III.C, c parameter] The choice c=0.22 for the step-scaling and discrete-beta-function comparisons is motivated only by a desire to avoid finite-volume effects; a brief discussion of how this value was selected, and whether results are stable under changing c, would strengthen the analysis.
  4. [Fig. 11 caption and Sec. IV] There are typographical errors: 'gradent flow' in the caption of Fig. 11 and 'ini finite' in Sec. IV.
  5. [III.C.1, Fig. 11] The relation between the pure-gauge inverse-Monte-Carlo coupling g_{0,eff} and the continuum scheme should be discussed more explicitly, since the matching scale lambda0 = a/9 or a/18 is a free parameter of the comparison and affects the interpretation of the saturation as Landau-pole-compatible.

Circularity Check

2 steps flagged · score 6.0 of 10

Two perturbative 'matches' are partly built in: τ is tuned to the 2-loop curve on L=30 (Eq. 11) and g^2_{0,eff2} is defined by fitting the same curve (Eqs. 16-19, Fig. 12), so the abstract's 'prove' overstates, though cross-volume step scaling retains independent content.

  1. fitted input called prediction [Sec. III.C, Eq. (11), Figs. 7-10]
    "For each βL, we tune τ by matching the largest volume L = 30a gradient flow coupling to the 2-loop perturbative coupling over the interval λ ∈ [3a, 5a]. We then use this value for τ for smaller volumes L/a = 12, 18, 24."

    The τ-shifted L=30 data are then displayed against the same 2-loop curve in Figs. 7-8, and τ fixed this way enters the step-scaling and discrete beta-function comparisons in Figs. 9-10. Agreement on λ∈[3a,5a] is therefore a consistency condition of the tuning, not a test of the claim that the lattice running matches perturbation theory. The abstract's statement that the results 'nicely match with the perturbative beta function' inherits this enforced agreement. Only the carry-over of the same τ to smaller volumes and to scales outside the fitting interval provides independent information, and the paper itself notes the absence of a continuum extrapolation.

  2. self definitional [Sec. III.C.2, Eqs. (15)-(20), Fig. 12]
    "From the derivation of the effective coupling g2 0,eff2(βL), it is not a big surprise, that one finds in Fig. 12 excellent agreement between lattice data and 2-loop result, when plotting the gradient flow coupling at fixed flow scale λ ... as function of g2 0,eff2(βL), identifying the perturbative g2(λ/λ0, g2 0) and g2 GF(λ,β L), with λ0≡ a and g2 0≡ g2 0,eff2(βL)."

    The effective coupling g^2_{0,eff2} is not an independent lattice observable: it is constructed by fitting the same lattice data to the same two-loop λ_pt(g^2,g^2_0,λ_0) via the χ^2 minimization in Eq. (19) over the same interval λ∈[3,5], and then solving λ_pt=1 for g^2. Plotting that same two-loop curve against the resulting g^2_{0,eff2} and calling the agreement 'excellent' is therefore built into the construction; the paper's own wording concedes it is 'not a big surprise'. This agreement cannot corroborate the abstract's claim that the analysis is 'prove[n] ... connected to the gaussian fixed point'.

full rationale

The paper contains two by-construction perturbative comparisons. First, the flow-time shift τ in Eq. (11) is explicitly tuned so that the L=30 gradient-flow coupling matches the two-loop perturbative curve on λ∈[3a,5a]; the subsequent display of L=30 data against that same curve (Figs. 7-8) and the use of that curve as the reference in the step-scaling and discrete beta-function plots therefore partially report the tuning itself as evidence. The independent part is the extrapolation of τ to smaller volumes and larger scales, but the paper notes the absence of a continuum extrapolation and the strong finite-size effects at s=1/2, so this independent content is weak. Second, the effective coupling g^2_{0,eff2} in Sec. III.C.2 is defined by fitting the same lattice data to the same two-loop running over the same interval, and the paper itself says the resulting 'excellent agreement' in Fig. 12 is 'not a big surprise'; this is self-definitional and cannot be used as independent confirmation. The genuine, less circular result is the saturation of g^2_GF as a function of the plaquette-based g^2_{0,eff} in Fig. 11, which uses an independent inverse-Monte-Carlo determination and only fits a free matching scale λ0; it is appropriately presented as compatible with, but not proof of, a Landau pole. Self-citation is present in the analytical motivation (e.g., Refs. [12,30,31]) but is not load-bearing for the lattice derivation in a circular way. Overall, one or more 'predictions' reduce by construction, giving partial circularity: score 6.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new physical entities. The interpretive load rests on matching to the two-loop perturbative beta function: tau and the effective coupling are fitted quantities, and the universality of the lattice action at negative beta_L is assumed.

free parameters (3)
  • Flow-time shift tau(beta_L) = Examples: 0.047 and -0.007 for Nf=24; 0.064 and 0.038 for Nf=48
    Tuned per beta_L by matching the L=30 gradient flow coupling to the two-loop perturbative running on lambda in [3a,5a]. Subsequent displayed agreement with the two-loop curve is partly enforced by this fit.
  • Matching scale lambda0 for effective coupling I = a/9 for Nf=24, a/18 for Nf=48
    Chosen in the Fig. 11 comparison to make the plaquette-derived effective bare coupling line up with the two-loop running coupling.
  • Flow-scale fraction c in step scaling = 0.22
    Chosen small to avoid finite-volume effects. The running-coupling scale is defined as lambda = c L, and no continuum extrapolation over c is performed.
assumptions (3)
  • domain assumption The two-loop perturbative beta function is a reliable proxy for the nonperturbative beta function at the small couplings probed (g^2_GF below about 1.75).
    Used to tune tau and to interpret data as matching the gaussian fixed point; no independent nonperturbative cross-check is provided.
  • domain assumption The lattice action with HEX smeared Wilson fermions, clover coefficient cSW=1, and negative bare inverse coupling beta_L belongs to the same universality class and can access the strong-coupling ultraviolet regime of SU(2) with Nf=24 and 48.
    Needed so that the saturation of the flow coupling is a property of the theory rather than an artifact of the lattice action; the authors acknowledge the action may be unable to explore the deep ultraviolet.
  • domain assumption The plaquette inverse-Monte-Carlo effective coupling g^2_0,eff represents the bare coupling of the theory at the lattice cutoff.
    Used to convert negative beta_L values into a positive effective bare coupling; if this mapping fails, the Landau-pole-compatible interpretation loses support.

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Cite this review

Pith. "Pith review of Safety versus triviality on the lattice." pith.science (2026). https://pith.science/paper/DZSUHUSC

@misc{pith2026190804605,
  author       = {Pith},
  title        = {Pith review of: Safety versus triviality on the lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DZSUHUSC}},
  note         = {Machine review of arXiv:1908.04605}
}
read the original abstract

We present the first numerical study of the ultraviolet dynamics of non-asymptotically free gauge-fermion theories at large number of matter fields. As testbed theories we consider non-abelian SU(2) gauge theories with 24 and 48 Dirac fermions on the lattice. For these number 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 towards determining the ultraviolet fate of non asymptotically free gauge theories.

Figures

Figures reproduced from arXiv: 1908.04605 by the authors.

Figure 1
Figure 1. FIG. 1. For ease of the reader we summarise here the phase diagram of [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Comparison of the leading order large- [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. A cartoon of the running coupling [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The running coupling [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The gradient flow couplings for [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The gradient flow coupling for [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 12
Figure 12. Figure 12: FIG. 12. The relation between [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

98 extracted references · 44 canonical work pages

  1. [1]

    = Aµ(x), where Aµ is the original continuum gauge field. In the lattice formulation the lattice link variableU replaces the continuum flow field, which is then evolved using the tree-level improved L¨uscher-Weisz pure gauge action (LW) [70]. mqa L = 12a L = 18a L = 24a L = 30a βL κ [10−4] acc. stat. acc. stat. acc. stat. acc. stat. -0.3 0.131578 8.3(4) 0.88 ...

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    0.127336 -0.1(3) 0.97 10k 0.94 4.4k 0.85 4k 0.96 2.4k

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    0.125525 -0.1(1) 0.99 10k 0.98 5k 0.98 4k 0.98 2k

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    0.125355 1.0(1) 0.99 10k 0.98 5k 0.96 4k 0.99 2.7k

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    Luscher and P

    M. Luscher and P . Weisz, Nucl. Phys. B 290 (1987) 25. doi:10.1016/0550-3213(87)90177-5

    doi:10.1016/0550-3213(87)90177-5 1987

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    0.125224 -0.2(1) 0.99 10k 0.98 5k 0.95 4k 0.99 3k TABLE I. The table shows for N f = 24 for each of our simula- tion points the value ofβL andκ, as well as the resulting PCAC quark mass mqa (in units of 10−4), determined on the L = 24a lattice. Furthermore are shown for each system size, the ac- ceptance rates of the HMC trajectories and the accumulated s...

  7. [7]

    0.126332 0.5(2) 0.97 4k 0.95 5k 0.92 2k 0.95 1.1k

  8. [8]

    0.12544 -0.1(1) 0.98 2k 0.99 2.4k 0.97 2k 0.97 1.3k

Show all 98 references
  1. [9]

    The table shows for N f = 48 for each simulation point the value ofβL andκ, as well as the resulting PCAC quark mass mq (in units of 10−4), determined on the L = 24a lattice

    0.125209 -0.3(1) 0.98 2k 0.98 2.3k 0.98 1.9k 0.98 1.3k TABLE II. The table shows for N f = 48 for each simulation point the value ofβL andκ, as well as the resulting PCAC quark mass mq (in units of 10−4), determined on the L = 24a lattice. Furthermore are shown for each system...

  2. [10]

    The behavior is illustrated schematically in Fig

    at reference scale λ0, if there is an interacting UV fixed point at coupling g2 = g2 cr. The behavior is illustrated schematically in Fig. 3: the fixed point is unstable and therefore, for in- creasingλ/λ0> 1, the running coupling g(λ/λ0, g2

  3. [11]

    We note that ifg2 0 is much smaller thang2 cr, the behavior of the running coupling for g2(λ/λ0, g2 0)< g2 cr can be almost indistinguishable from a Landau pole type behaviour

    will: (a) decrease, if g2 0< g2 cr, (b) remain constant, if g2 0 = g2 cr and (c) increase, if g2 0> g2 cr . We note that ifg2 0 is much smaller thang2 cr, the behavior of the running coupling for g2(λ/λ0, g2 0)< g2 cr can be almost indistinguishable from a Landau pole type beh...

  4. [12]

    As already mentioned above, forλ/λ0≤ 2 discretization effects are strong

    with the lattice gradient flow coupling g2 GF(λ (t),β L). As already mentioned above, forλ/λ0≤ 2 discretization effects are strong. Therefore the lattice gradient flow coupling can- not be expected to behave like the running coupling of the continuum theory 2, which means that th...

  5. [13]

    as a function of increasing (relative) renormalization length scale λ/λ0 in the presence of an interacting UV fixed point, located at g2 = g2 cr, for di fferent choices of the initial coupling g2 0 = g2(1, g2

  6. [14]

    Given the fixed point at g2 cr, the running coupling g(λ/λ0, g2

    and corresponding reference scaleλ0. Given the fixed point at g2 cr, the running coupling g(λ/λ0, g2

  7. [15]

    The fixed point is repelling, so that one can only flow away from it

    as a function of increasingλ/λ0≥ 1 will decrease if g2 0< g2 cr, remain constant if g2 0 = g2 cr and increase if g2 0> g2 cr. The fixed point is repelling, so that one can only flow away from it. If g2 0 is much smaller than g2 cr, the UVFP and Landau pole running couplings are ...

  8. [16]

    els of Fig

    will even start to agree only forλ(t) =λ/λ0 & 4. els of Fig. 2), the situation is significantly more compli- cated, as the UV fixed point arises due to a singularity asβ→−∞ . This abrupt change makes the running cou- pling for g2 0< g2 cr look even more as if there is a Landau p...

  9. [17]

    (2) with N f = 48 (c.f

    for the leading order large-N f beta function from Eq. (2) with N f = 48 (c.f. lower panels of Fig. 2) as a function of increasing (relative) renormal- izaion length-scaleλ/λ0 for different choices of initial coupling g2 0 = g2(1, g2

  10. [18]

    The red, horizontal dotted line indicates the location of the UV-fixed point, g2 cr = 4.9348022

    and corresponding reference scale λ0. The red, horizontal dotted line indicates the location of the UV-fixed point, g2 cr = 4.9348022... . In contrast to the four loop N f = 48 beta function used for the discussion in Fig. 3, which had a moderate slop at the fixed point, the slo...

  11. [19]

    This renders the smallest lattices L≤ 18 of very limited use

    of the continuum the- ory, is approximately given byλ∈ [3a, 0.25L], where the lower bound of 3a is somewhat optimistic. This renders the smallest lattices L≤ 18 of very limited use. We can optimize the gradient flow coupling by adding a shiftτ to the flow time [73]: g2 GF(λ,β L)...

  12. [20]

    inverse Monte Carlo

    E ffective g 2 0 I. As we saw above, the gradient flow coupling makes sense only when the flow scale is large enough, ∼> 3a. Nevertheless, it is of interest to relate g2 GF to some lat- tice ultraviolet scale e ffective coupling. The inverse of βL = 4/g2 0 does not make much sense...

  13. [21]

    Theg2 0,eff = g2 GF line is shown with dots

    are, respectively, identified with the g2 0,eff and g2 GF(λ/a, g2 0,eff) from the lattice. Theg2 0,eff = g2 GF line is shown with dots. for an UV Landau pole: as the UV scale approaches the Landau pole, the UV scale coupling diverges. Because g2 GF is evaluated at length scales wh...

  14. [22]

    A good qualitative agreement is obtained with the matching co- efficientsλ0 = a/9 for N f = 24 andλ0 = a/18 for N f = 48

    by identify- ing g2 0≡ g2 0,eff and g2(λ/λ0, g2 0)≡ g2 GF(λ/a, g2 0,eff). A good qualitative agreement is obtained with the matching co- efficientsλ0 = a/9 for N f = 24 andλ0 = a/18 for N f = 48. This implies that the e ffective lattice coupling g2 0,eff cor- responds to a reference...

  15. [23]

    E ffective g 2 0 II. Another possibility to define an effective coupling for the lattice theory comes as a side product of the method we used to determine the optimal value of theτ param- eter in the definition (11) of the improved gradient flow coupling, i.e.: g2 GF(λ,β L;τ) = t2⟨...

  16. [24]

    Solving now λpt(g2, g2 0,λ 0) = 1 (20) for g2, one obtains an e ffective coupling for the lattice theory at the cut-off scale a, which we call g2 0,eff2(βL)

    = (λ0(τ), g2 GF,0(τ)), for which (18) has within the given fit interval the best overlap with theτ-shifted lattice data. Solving now λpt(g2, g2 0,λ 0) = 1 (20) for g2, one obtains an e ffective coupling for the lattice theory at the cut-off scale a, which we call g2 0,eff2(βL). Fr...

  17. [25]

    N f = 24 N f = 48 L=18 L=24 L=30 pert

    and g2 GF(λ,β L), withλ0≡ a and g2 0≡ g2 0,eff2(βL). N f = 24 N f = 48 L=18 L=24 L=30 pert. 2 loop 0 1 2 3 4 0 1 2 3 4 g0,eff2 2 , g 0 2 gGF 2 (λ/a, g 0,eff2 2 ), g 2 (λ/λ0 , g0 2 ) λ/a=4 L=18 L=24 L=30 pert. 2 loop 0 1 2 0 1 2 g0,eff2 2 , g 0 2 gGF 2 (λ/a, g 0,eff2 2 ), g 2 (λ...

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