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Asymptotic gauge symmetry and UV extension of the nonperturbative coupling in holographic QCD

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Pith's one-line read This paper imposes asymptotic-freedom boundary conditions on the analytic holographic QCD coupling and shows that for SU(3) color they force a UV scaling exponent $r = 1/2$, six quark flavors, an infrared fixed point…

desk verdict A useful UV extension of the holographic coupling, but the flavor/exponent link is a consistency check that leans hard on the inherited maximal-analyticity input. read the letter →

arxiv 2505.19545 v1 pith:B5HW2Z7D submitted 2025-05-26 hep-ph hep-exhep-th

Guy F. de Teramond , Arpon Paul , Hans Gunter Dosch , Stanley J. Brodsky , Alexandre Deur , Tianbo Liu , Raza Sabbir Sufian This is my paper · ORCID
classification hep-phhep-exhep-th PACS 12.38.-t11.25.Tq11.15.Tk
keywords holographicQCDrunningcouplingeffectivechargeBjorkensumruleasymptoticfreedominfraredfixedpointheavy-quarkthresholdsconfinementscale
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 claims that the deep-ultraviolet behavior of the nonperturbative QCD coupling is forced by a single boundary condition: the effective coupling must merge smoothly into the perturbative, asymptotically free QCD coupling as the momentum transfer goes to infinity. To implement this, the authors promote the holographic confinement scale to a running quantity, $\kappa^2(Q^2) = \kappa_0^2 f(Q^2)$, and show that the scaling function $f(Q^2) \sim (\sigma Q^2)^r$ is fixed by analyticity together with the one-loop $\beta$-function coefficient. For the effective charge defined by the Bjorken spin sum rule, with $\alpha_{\mathrm{eff}}(0) = \pi$ and the maximal analyticity condition $\rho = \pi/2$, the boundary at infinity is reached only with six quark flavors and the scaling exponent $r = 1/2$. Including the charm, bottom, and top thresholds through two sum rules, the model describes the measured strong coupling from the GeV scale to multi-TeV scales and predicts $\alpha_{\overline{\mathrm{MS}}}(M_Z) = 0.1161 \pm 0.0017$. This matters because the whole description rests on one dimensionful input, the confinement scale $\kappa_0$ fixed by hadron spectroscopy; the flavor content and the Z-pole coupling emerge from analyticity and the boundary condition rather than being put in by hand.

What carries the argument

The load-bearing object is the rescaled effective coupling, $\alpha_{\mathrm{eff}}(Q^2,\kappa(Q^2)) = \alpha_{\mathrm{eff}}(0)\exp\left[-\int_0^{Q^2/4\kappa^2(Q^2)} \frac{dv}{1 + v\ln(\rho v)}\right]$, obtained by replacing $u$ with $u/f(u)$ in the predecessor model's integral, which is equivalent to letting $\kappa_0^2 \to \kappa_0^2 f(Q^2)$. This rescaling preserves the singularity flow of the original model and, by construction, the maximal analyticity condition $\kappa_0^2 = (\pi/8)\Lambda^2$, namely $\rho = \pi/2$. The relation that carries the argument is the asymptotic boundary identity $\alpha_{\mathrm{eff}}(0)\,\beta_0\,K_{w\to\infty}(\rho) = 1 - r$, which converts the one-loop $\beta$-function coefficient into a constraint linking the ultraviolet scaling exponent $r$, the number of flavors $n_f$, and the infrared fixed point; for $N_c = 3$, $\rho = \pi/2$, and $K = 2/7$ it becomes the Diophantine relation $(11N_c - 2n_f)/42 = 1 - r$. Flavor dynamics enter through the threshold function $f(Q^2) = \sum_f C_f\left(1 + Q^2/m_f^2\right)^r$ with the sum rules $\sum_f C_f = 1$ and $\sum_f C_f \sigma_f^r = \sigma^r$; imposing equal bottom and top weights at zero charm weight fixes $\sigma = 1/(4\mu^2)$ with $1/\mu = 1/m_b + 1/m_t$, determining the ultraviolet scale from the two heavy-quark masses alone.

What would settle it

A decisive test is a measurement of $\alpha_s$ at the Z pole with total uncertainty below about $5 \times 10^{-4}$: the paper predicts $0.1161 \pm 0.0017$, so a settled value at or above the current world average ($0.1180 \pm 0.0009$) would rule the prediction out. A second check targets the parameter-free scale $\sigma$: an independent fit of the multi-hundred-GeV to TeV world data to the threshold function (50) must return $\sigma \approx 0.015$ GeV$^{-2}$ within the locus band of Fig. 3, and a best-fit outside that band would falsify the threshold sum rules (51). A third probe is a lattice computation of the effective charge's analytic continuation: if the first singularity of $\alpha_{\mathrm{eff}}$ were located at a value of $\rho$ different from $\pi/2$, the chain from maximal analyticity to $r = 1/2$ would break at its root.

Watch

Extended reading notes

Core claim

The central claim is that gauge symmetry at the asymptotic boundary fixes the ultraviolet running of the holographic confinement strength, and through it the flavor content of QCD. Imposing $\lim_{Q^2\to\infty}\alpha_{\mathrm{eff}}(Q^2,\kappa(Q^2))/\alpha_s(Q^2) = 1$ on the analytically continued effective coupling yields the identity $\alpha_{\mathrm{eff}}(0)\,\beta_0\,K_{w\to\infty}(\rho) = 1 - r$, where $r$ is the power of $Q^2$ in the scaling function and $K$ is a constant computed from the singularity structure of the model. For color SU(3), the Bjorken-scheme fixed point $\alpha_{\mathrm{eff}}(0) = \pi$, and the maximal analyticity value $\rho = \pi/2$ at which $K$ converges to $2/7$, the identity reduces to $(11N_c - 2n_f)/42 = 1 - r$: six open flavors select the exponent $r = 1/2$ observed in heavy-quark holographic spectroscopy, and conversely $r = 1/2$ requires $n_f = 6$. The paper states the outcome directly: the deep-UV boundary conditions require six flavors and an infrared fixed point $\alpha_{\mathrm{eff}}(0) = \pi$. With threshold weights satisfying $\sum_f C_f = 1$ and $\sum_f C_f \sigma_f^r = \sigma^r$, and setting the bottom and top weights equal with zero charm weight, the ultraviolet scale is fixed to $\sigma = 0.01501\ \mathrm{GeV}^{-2}$ from the two quark masses alone, giving $\alpha_{\overline{\mathrm{MS}}}(M_Z) = 0.1161 \pm 0.0017$ and an accurate description of the measured coupling at all scales.

Load-bearing premise

The argument's load-bearing premise is that the maximal analyticity relation $\kappa_0^2 = (\pi/8)\Lambda^2$, equivalently $\rho = \pi/2$, holds exactly, because it sets the constant $K = 2/7$ that converts the one-loop $\beta$-function coefficient into the scaling exponent $r = 1/2$; if $\rho$ deviates from $\pi/2$ even slightly, the six-flavor requirement and the predicted $\alpha_{\overline{\mathrm{MS}}}(M_Z) = 0.1161$ both shift.

Editorial extensions

If this is right

  • A single dimensionful scale, the confinement scale $\kappa_0 \simeq 0.534$ GeV from hadron spectroscopy, now drives the strong coupling from its infrared fixed point $\alpha_{\mathrm{eff}}(0) = \pi$ through the transition region to the highest measured virtualities.
  • The deep-UV boundary condition fixes the flavor content: six open flavors are required for $N_c = 3$ with $r = 1/2$, matching the three-generation structure seen in the $Z$ width, CKM unitarity, and cosmological bounds on the effective number of neutrino species.
  • The ultraviolet scaling exponent of the confinement strength is pinned to $r = 1/2$, the same power law previously found in holographic heavy-quark spectroscopy, so two independent observables point to the same ultraviolet behavior.
  • Equal bottom and top threshold weights with zero charm weight determine the ultraviolet scale $\sigma = 0.01501$ GeV$^{-2}$ from the quark masses alone, and the model then predicts $\alpha_{\overline{\mathrm{MS}}}(M_Z) = 0.1161 \pm 0.0017$, compatible with the current world average.
  • For other gauge groups the same relation $(11N_c - 2n_f)/42 = 1 - r$ admits exactly 42 discrete scaling exponents, so the framework specifies how the ultraviolet scaling of the coupling must depend on the color group and flavor content.

Reading between the lines

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

  • The derivation reverses the usual role of flavor thresholds: rather than inputting the measured flavor content and evolving the coupling through thresholds, the model derives $n_f = 6$ at infinity from an infrared datum ($\alpha_{\mathrm{eff}}(0) = \pi$) plus maximal analyticity. If this is right, a QCD-like theory with a different infrared fixed point would be forced to a different flavor count,
  • The threshold construction leaves the charm weight $C_c$ free in the interval $[0, 0.15]$, which translates into a band of possible couplings at intermediate $Q^2$ of order 10-100 GeV$^2$. A high-precision measurement of the Bjorken-sum effective charge in that window, from future polarized deep-inelastic data, would effectively measure $C_c$, the one parameter the model does not fix.
  • Because the identity $(11N_c - 2n_f)/42 = 1 - r$ ties the ultraviolet scaling to the particle content, any extension of the strong gauge group that changes the effective flavor count at high energies would have to announce itself as a change in the measured scaling of $\kappa(Q^2)$; confronting this relation with high-energy jet data is a quantitative way to look for beyond-Standard-Model color dy
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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. This manuscript extends the holographic light-front QCD effective coupling of Ref. [1] into the deep UV by imposing asymptotic-freedom boundary conditions. The authors introduce a scale-dependent confinement strength kappa^2(Q^2) = kappa_0^2 f(Q^2) with f(Q^2) -> 1 in the IR and f(Q^2) ~ (sigma Q^2)^r in the UV (Eqs. 30 and 38), and show that the integral form of the coupling is preserved (Eq. 33). Imposing lim_{Q^2->infty} alpha_eff/alpha_s = 1 (Eq. 39) yields the central relation alpha_eff(0) beta_0 K_{w->infty}(rho) = 1 - r (Eq. 40). With alpha_eff(0) = pi (Bjorken-g1 scheme), maximal analyticity rho = pi/2 inherited from Ref. [1] (Eqs. 19 and 23), and (N_c, n_f) = (3, 6), this gives r = 1/2; conversely, assuming r = 1/2 reproduces n_f = 6 (Eqs. 41-43 and Fig. 1). Heavy-quark thresholds are encoded in f(Q^2) = sum_f C_f (1 + Q^2/m_f^2)^r (Eq. 50), with two sum rules (Eq. 51) leaving a one-parameter locus of solutions; fixing the locus by C_b = C_t, C_c = 0 determines sigma = 0.01501 GeV^-2 (Eqs. 53-54) without a UV fit. The model is then compared with world data at all scales (Figs. 4-5) and yields alpha_MS(MZ) = 0.1161 +/- 0.0017 (Eq. 55).

Significance. If the robustness issues below are addressed, this is a useful contribution: it is an explicit construction in which one holographic scale kappa_0 interpolates between the IR fixed point and the perturbative regime, with an all-scale description of the strong coupling data. The analytic chain from Eq. (29) to Eq. (40) is internally consistent; I have re-derived Eq. (33), verified that Eq. (40) follows from the large-Q^2 forms, and checked the algebraic content of Eqs. (53)-(54) and the n_f = 6 crossing in Fig. 1. The near-equality of the data-independent value sigma = 0.01501 GeV^-2 with the fitted value 0.015 GeV^-2 is a genuine internal consistency result, and Eq. (55) is a concrete, falsifiable target. However, the headline claim, advertised in the abstract and in Sec. IX as 'the deep UV boundary conditions require six flavors', overstates what the model actually delivers: the connection among r, alpha_eff(0), and n_f is a consistency relation contingent on the exact inherited input rho = pi/2, and the quoted alpha_s(MZ) lacks a complete error budget. The paper is best framed as a successful consistency check together with an all-scale phenomenological description.

major comments (3)
  1. [Sec. VI A; Eqs. (19), (21), (40); Table I] Eq. (40) is the pivot of the central claim, and it inherits the exact input rho = pi/2 from Eq. (19), which is taken from Ref. [1] and not re-derived here. Table I shows that K_w(rho) is extremely sensitive to rho: K(pi/2) = 0.2858 versus K(pi/4) = 0.0571, a factor of five over the allowed range stated in Eq. (21). Inserting rho = pi/4 into Eq. (40) with alpha_eff(0) = pi, N_c = 3, n_f = 6 gives 1 - r ~ 0.100 (r ~ 0.90) instead of 1/2; conversely, keeping r = 1/2 at rho = pi/4 would require 11N_c - 2n_f ~ 105, i.e., a negative n_f. Since Eq. (21) permits any rho in (1/e, pi/2] and the asymptotic boundary condition (39) does not by itself select rho, the advertised r = 1/2 and n_f = 6 results are only as secure as the unquantified maximal-analyticity input. The manuscript should either quantify the robustness (for example, the shift in alpha_s(MZ) as rho moves within its allowed interval, propagated through Eqs. (40)-(43) and (48)-(55)) or explicitly delimit the claim to the exact rho = pi/2 case.
  2. [Sec. VI A (Eqs. 40-43, Fig. 1) and Sec. IX] Eq. (40) is used in both directions: with n_f = 6 as input to derive r = 1/2, and then with r = 1/2 as input to derive n_f = 6; the Conclusions then state that 'the deep UV boundary conditions require six flavors.' This is a consistency loop rather than an independent derivation. The boundary condition (39) determines only the product combination in Eq. (40); with alpha_eff(0) = pi, N_c = 3, and n_f = 6 all being established external inputs, the genuinely model-derived content is the near-equality K_{w->infty}(pi/2) ~ 2/7 (Eq. 42), which makes those inputs mutually consistent. Because n_f = 6 is already known from direct observation, the 'requires six flavors' wording should be replaced by an explicit statement of mutual consistency of r = 1/2, alpha_eff(0) = pi, and n_f = 6 within the model. The comparison with Neff from BBN/CMB in Sec. IX concerns the number of light neutrino generations and is not an independent constraint on the six active quark flavors.
  3. [Sec. VII B (Eqs. 53-55) and Sec. VIII (Fig. 5)] The quoted prediction alpha_MS(MZ) = 0.1161 +/- 0.0017 has no stated error budget. Figure 5 shows two distinct bands, the threshold-locus spread (C_c in [0, 0.15], narrow gray band) and the kappa_0 uncertainty (wide band), but the text does not state which uncertainties are propagated into Eq. (55). Moreover, the central value is computed for one particular point of the locus, C_b = C_t, C_c = 0, while the 'red curve' compared with data in Figs. 4-5 corresponds to a different point (C_c = 0.075, C_b = 0.250, C_t = 0.675); the manuscript should state which trajectory underlies Eq. (55) and report the spread of alpha_s(MZ) across the locus. The condition C_b = C_t is an unmotivated choice that converts a one-parameter family of fits into a 'data-independent' determination (Sec. VII B); the claim in Sec. IX that the model describes the data 'without any additional free parameter' is therefore overstated, since sigma is either fitted to UV data (Fig. 2) or fixed by an ad hoc equality of weights.
minor comments (5)
  1. [Sec. VII and Fig. 2 caption] The value Lambda_MS^{n_f=6} = 87 GeV appears inconsistent with the standard n_f = 6 QCD scale (Lambda_MS ~ 90 MeV) and with the behavior implied by the NLO formula (27); please verify whether 0.087 GeV (87 MeV) is intended and correct all occurrences.
  2. [Sec. VII A and Fig. 3 discussion] The inequality '0 >= Cc >= 0.15' is written backwards in two places and should read 0 <= Cc <= 0.15; the symbols 'Cs = 0' and 'Cs = 0.15' should read 'Cc = 0' and 'Cc = 0.15'.
  3. [Fig. 4 caption] The symbols 'Cq' and 'sigma_q' are introduced without definition; they should be replaced by the corresponding coefficients (C_c or the heavy-quark weights) and sigma (or sigma_f).
  4. [Sec. II C] There is a duplicated word in the sentence 'its usefulness has been limited in practice limited to a few predictions'; one 'limited' should be removed.
  5. [Sec. V A] The assertion that the rescaling (28) preserves the singularity-flow analysis of Ref. [1] and hence the maximal-analyticity condition (19) is stated rather than demonstrated; since Eq. (19) is load-bearing for the central result, a short argument or an explicit reference showing that the branch cuts of f(Q^2) do not overlap the singularity flow would be valuable.

Circularity Check

1 steps flagged · score 4.0 of 10

One internal consistency loop in the K=2/7 rationalization of r=1/2; the central nf=6 and alpha_s(MZ) results are otherwise conditional on external inputs, not circular.

  1. self definitional [Section VI.B, Eqs. (41)-(47); Eq. (42)]
    "From (41) Kw→∞(π/2) = 2/7 = 0.285714···, (42)... Substituting the power solution (45) in (37) we obtain βeff_0 = (1−r)/(αeff(0)Kw→∞(ρ)) , (46) which we compare with the leading coefficient of the QCD β-function β0 (26) with six open flavors β0 = 7/4π . (47) Thus, we recover the solution r = 1/2 obtained above for maximal analyticity, ρ=π/2, K(π/2)=2/7 from (42), and αeff(0)=π."

    Equation (41) is Eq. (40) with r=1/2 already inserted ('start with the exponent r=1/2'). Inserting the chosen nf=6 and αeff(0)=π into (41) fixes K=2/7; this is not an independent evaluation of the integral (22). This K is then fed into (46) and compared with β0=7/(4π) for nf=6, algebraically returning 1−r=1/2. Thus the 'recovery' of r=1/2 is an identity: r=1/2 entered through K and was pulled out again. The exact rational relation (43) and the Diophantine table in Appendix B consequently rest on this circularly fixed K, while the numerical integral gives K(π/2)=0.285785 and 1−r≈0.500124, not exactly 1/2.

full rationale

The main derivation is not centrally circular. Equation (40) follows algebraically from the asymptotic boundary condition (39) plus the power-law ansatz (38), and the numerical branch using K(π/2)=0.285785 independently gives r≈1/2 for nf=6, αeff(0)=π and ρ=π/2. The second branch that yields nf=6 uses r=1/2 from external heavy-quark spectroscopy and αeff(0)=π from the Bjorken sum rule, so the advertised nf=6 result is a genuine conditional inference rather than a restatement of its own conclusion. The alpha_s(MZ) prediction is also a legitimate external benchmark: σ can be obtained without fitting via Eq. (54), and the comparison with the PDG value is data outside the model. The circularity that does exist is localized: the rational value K=2/7 in Eq. (42) is derived from Eq. (41) after r=1/2 has already been assumed, and is then used in Eqs. (46)-(47) to 'recover' r=1/2. This is an algebraic identity, not a new result, and it feeds the exact relation (43) and the Diophantine uniqueness claim in Appendix B. The conclusion in Section IX that the deep UV boundary conditions 'require six flavors' also omits the external r=1/2 premise, overstating what the boundary conditions alone imply. Because the approximate numerical route independently fixes r≈1/2 and the nf=6 claim has independent support, the paper is only partially circular, not globally reducible to its inputs.

Assumptions & free parameters 5 free parameters · 5 assumptions · 1 invented entities

The model relies on several inputs from outside the paper (kappa_0, alpha_g1(0) = pi, nf = 6, rho = pi/2) plus one new ansatz for f(Q^2). The numerical predictions are largely consistency conditions among these inputs; no machine-checked proof or independent computational artifact is supplied.

free parameters (5)
  • kappa_0 = 0.534 GeV (0.5345 in places), from hadron spectroscopy [84]
    Confinement scale fixed by light hadron spectra in prior work; all Q^2 dependences scale through it and its uncertainty drives the wider band in Fig. 5.
  • sigma = 0.015 GeV^-2 (0.01501 via Eq. 54)
    Main parameter of the scaling function f(Q^2) = (1 + sigma Q^2)^r. Initially fitted to the UV data around 1000 GeV (Sec. VII A); later re-derived from quark masses only under the choice Cb = Ct, Cc = 0.
  • r = 1/2
    Scaling exponent of f. Determined by the combination Nc = 3, nf = 6, alpha_eff(0) = pi, rho = pi/2, not by a direct fit; it also matches an exponent used in earlier holographic heavy-quark spectroscopy.
  • Cc = 0.075 for the central curve; 0 in the independent-sigma route
    Charm threshold weight. The sum rules leave a one-parameter family of solutions; a specific member is chosen to draw the red curves in Figs. 4 and 5.
  • Lambda_MS (nf = 6) = 87 GeV
    Used in the NLO perturbative curve and the comparison with data; the text states the UV data around 1000 GeV are used to find its value together with sigma.
assumptions (5)
  • standard math Asymptotic freedom and the one- and two-loop RG coefficients beta_0, beta_1 for an SU(Nc) gauge theory (Eqs. 24 to 27)
    The asymptotic boundary condition (Eq. 39) is imposed against this standard RG structure.
  • domain assumption Maximal analyticity relation kappa_0^2 = (pi/8) Lambda^2 and rho = pi/2 (Eq. 19) from the prior paper [1]
    All numerical values K(pi/2) = 2/7 and the exponent/flavor relation depend on this exact value; it is inherited, not re-derived here.
  • domain assumption The g1 effective charge is a physical observable with IR fixed point alpha_g1(0) = pi (Eq. 15)
    This external input anchors the value of alpha_eff(0) used in the asymptotic condition and the nf/r connection.
  • domain assumption Six active quark flavors in the deep UV (nf = 6)
    Imported from Standard Model observations; it is used to derive r = 1/2 and is later presented as a predicted consequence, which is the main circular step.
  • ad hoc to paper The scaling function f(Q^2) = (1 + sigma Q^2)^r and its threshold decomposition (Eqs. 48 and 50) preserve the holomorphy and maximal analyticity of alpha_eff
    This ansatz is introduced ad hoc and the no-overlap of threshold cuts with the singularity flow is asserted, not proven.
invented entities (1)
  • Scale-dependent confinement strength kappa(Q^2) = kappa_0 sqrt(f(Q^2))
    purpose: Interpolates between the fixed IR confinement scale and the UV power-law scaling needed to satisfy the asymptotic boundary condition.
    No independent measurement fixes the functional form or its trajectory in the complex plane; it is a model function tuned to alpha_s data and partly to quark masses.

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Cite this review

Pith. "Pith review of Asymptotic gauge symmetry and UV extension of the nonperturbative coupling in holographic QCD." pith.science (2026). https://pith.science/paper/B5HW2Z7D

@misc{pith2026250519545,
  author       = {Pith},
  title        = {Pith review of: Asymptotic gauge symmetry and UV extension of the nonperturbative coupling in holographic QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B5HW2Z7D}},
  note         = {Machine review of arXiv:2505.19545}
}
abstract

We extend our recent analytic study of the strong coupling $\alpha_{\rm eff}$ in the nonperturbative and near-perturbative regimes~\cite{deTeramond:2024ikl} by imposing rigorous renormalization-group results from asymptotically free gauge theories at $Q^2 \to \infty$. The asymptotic boundary conditions modify the scaling properties of $\alpha_{\rm eff}$ at large values of the momentum transfer $Q^2$, and lead to a scale-dependent confinement strength $\kappa(Q^2)$. This requires that both $\kappa(Q^2)$ and $\alpha_{\rm eff}\left(Q^2, \kappa(Q^2)\right)$ remain holomorphic in the complex $Q^2$ plane, except at the physical cuts associated with the heavy-quark thresholds and the singularity flow trajectory studied in~\cite{deTeramond:2024ikl}. For color $SU(3)$, a precise connection is found between the scaling exponent of $\kappa(Q^2)$ in the ultraviolet, the value of the infrared fixed point of the strong coupling, and the number of flavors in agreement with observations. The nonperturbative analytic model gives an accurate description of the strong coupling at all scales, up to the highest available data.

Figures

Figures reproduced from arXiv: 2505.19545 by the authors.

Figure 1
Figure 1. FIG. 1. The light blue curve represents the value of the infrared fixed point value [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. World data for [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The locus of possible values of the charm, bottom and top weight coefficients, [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The present model prediction from ( [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The model predictions from ( [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The behavior of the [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

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  1. Insight on confinement from the QCD effective charge

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