REVIEW
Analytic Model for the QCD Running Coupling with Universal $\bar{\alpha}_s(0)$ Value
High Energy Physics - Phenomenology · Submitted 16 Apr 1997 · 14 references
Pith's verdict: The one-loop analytic coupling is exact and parameter-free, but the title's 'universal' is a property of their specific unsubtracted-spectral prescription, not of QCD.
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Referee report
The paper
The paper claims
Imposing analyticity on the QCD running coupling replaces its ghost pole with a fixed infrared value of about 1.4, set by colour-group factors alone.
The reading
What carries the argument
The load-bearing device is the 'analytization procedure': (I) take the standard renormalization-group-improved coupling in the Euclidean region, (II) continue it to $4$ and define the spectral density as the imaginary part, (III) reconstruct the coupling from the Källén–Lehmann representation $4$. The one-loop spectral density $4$ is explicit, and the resulting integral gives the closed form $4$. The second term is what does the work: it is a nonperturbative contribution of order $4$ that cancels the ghost pole and supplies the finite infrared limit. A second piece of machinery carries the higher-loop argument: the difference between the one- and two-loop values at $4$ is written as a contour integral $4$ whose integrand is analytic in the half-plane below the contour, hence zero. The authors also offer practitioners an approximate two-loop formula, Eq. (12), with a single fitted-looking constant $4$ that is in fact computed, accurate to better than 0.5 percent in the intermediate region.
Extended reading notes
Core claim
The central claim is that the ghost pole of the perturbative QCD running coupling can be removed not by hand but by a principled procedure, and that the resulting object has an infrared limit fixed by symmetry. Starting from the one-loop leading-log expression and its spectral density, the authors apply the three-step 'analytization' recipe borrowed from 1950s QED work: write the Euclidean running coupling, continue it to the Minkowskian region to read off a spectral density, then rebuild the coupling from a spectral representation. The rebuilt coupling, Eq. (5), has no pole at $4$; instead the pole is cancelled by the term $4$, and at $4$ the coupling tends to $4$ for three flavours. Because the construction is scale-invariant in the appropriate sense, this limit does not depend on $4$. For the two-loop case no closed form is available, but the authors prove with a contour integral, $4$, that the ghost-removing and cut-removing compensation terms contribute exactly $4$ each, so the same universal value emerges. Numerically they report that the two-loop analytic coupling differs from the one-loop one by under 10 percent and the three-loop from the two-loop by under 1 percent across the whole Euclidean region, and that substituting the analytic coupling into $4$ gives values in the range 0.48 to 0.55 at $4$, to be compared with the jet-physics estimate $4$.
Load-bearing premise
The whole construction rests on the strong running coupling actually obeying a Källén–Lehmann spectral representation with a positive spectral density in the working scheme; the paper raises this itself, noting 'there is a question about validity of the spectral representation', settles it by citing prior work rather than deriving it here, and additionally assumes the MS scheme (or MOM in transverse gauge) so that gauge running does not contaminate the coupling.
Editorial extensions
If this is right
- The infrared behaviour of the strong coupling becomes a prediction rather than a fit: $4$ depends only on $4$, so measurements of $4$ or of the scale $4$ cannot move it.
- The long-standing practice of 'freezing' the coupling by hand is replaced by a derivation, and the frozen value is pinned at roughly 1.4 for three active flavours.
- Because the two- and three-loop analytic couplings stay within 10 percent and 1 percent of the one-loop one, low-energy QCD phenomenology becomes insensitive to the truncation order of the perturbative series.
- The nonperturbative correction term slows the evolution of the coupling even at moderate momenta, contributing about 4 percent near 3 GeV, which the authors suggest could matter for reconciling low-energy and $4$ determinations.
- Dispersive treatments of the $4$ hadronic width, which rely on analyticity in the complex $4$ plane, become internally consistent rather than requiring a patch.
Reading between the lines
- If the universal value holds, it gives a parameter-free target for lattice and Dyson–Schwinger determinations of the coupling in the sub-GeV region, turning a long-standing scheme-comparison muddle into a sharp numerical test.
- The construction suggests a general diagnostic: any renormalization-group resummation whose spectral density is not positive-definite signals an unphysical singularity that analyticity would remove, which could be used to screen truncations in other theories with infrared freedom.
- The paper's own remark that the procedure is not unique, combined with the observation that threshold matching breaks analyticity, hints that the 'universal' value may be universal only within the class of analytizations that preserve the one-loop spectral weight; a smooth mass-dependent matching scheme would be the natural place to see whether the number shifts.
- The mechanism is close in spirit to spectral reconstructions used elsewhere in physics where a perturbative expansion is continued from a cut plane, so the same ghost-cancellation trick may transfer to other asymptotically free couplings with a similar analytic structure.
Original abstract
We discuss the new model expression $\bar{\alpha}_{an}(Q^2)$ recently obtained for the QCD running coupling with a regular ghost-free behavior in the "low $Q^2$" region. Being deduced from the standard "asymptotic-freedom" expression by imposing the $Q^2$-analyticity -- without any adjustable parameters -- it obeys nice features: (i) The universal limiting value $\bar{\alpha}_{an}(0)=4\pi /\beta_0 \simeq 1.4$ expressed only via group symmetry factors and independent of experimental estimates on the running coupling $\bar{\alpha}_s(Q^2)$ (of QCD scale parameter $\Lambda$). This value turns out to be stable with respect to higher order corrections; (ii) Stability of IR behavior with respect to higher-loop effects; (iii) Coherence between the experimental $\bar{\alpha}_{an}(M_{\tau}^2)$ value and integral information on IR $\bar{\alpha}_s(Q^2)$ behavior as extracted from jet physics data.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We discuss the new model expression $\bar{\alpha}_{an}(Q^2)$ recently obtained for the QCD running coupling with a regular ghost-free behavior in the "low $Q^2$" region. Being deduced from the standard "asymptotic-freedom" expression by imposing the $Q^2$-analyticity -- without any adjustable parameters -- it obeys nice features: (i) The universal limiting value $\bar{\alpha}_{an}(0)=4\pi /\beta_0 \simeq 1.4$ expressed only via group symmetry factors and independent of experimental estimates on the running coupling $\bar{\alpha}_s(Q^2)$ (of QCD scale parameter $\Lambda$). This value turns out to be stable with respect to higher order corrections; (ii) Stability of IR behavior with respect to higher-loop effects; (iii) Coherence between the experimental $\bar{\alpha}_{an}(M_{\tau}^2)$ value and integral information on IR $\bar{\alpha}_s(Q^2)$ behavior as extracted from jet physics data.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We discuss the new model expression $\bar{\alpha}_{an}(Q^2)$ recently obtained for the QCD running coupling with a regular ghost-free behavior in the "low $Q^2$" region. | contribution 0.97 | 0.39 | 0.12 | asserted 0.92 |
| s1 | Being deduced from the standard "asymptotic-freedom" expression by imposing the $Q^2$-analyticity -- without any adjustable parameters -- it obeys nice features: (i) The universal limiting value $\bar{\alpha}_{an}(0)=4\pi /\beta_0 \simeq 1.4$ expressed only via group symmetry factors and independent of experimental estimates on the running coupling $\bar{\alpha}_s(Q^2)$ (of QCD scale parameter $\Lambda$). | result 0.67 | 0.75 | 0.58 | asserted 1.00 |
| s2 | This value turns out to be stable with respect to higher order corrections; (ii) Stability of IR behavior with respect to higher-loop effects; (iii) Coherence between the experimental $\bar{\alpha}_{an}(M_{\tau}^2)$ value and integral information on IR $\bar{\alpha}_s(Q^2)$ behavior as extracted from jet physics data. | result 0.95 | 0.73 | 0.22 | asserted 0.99 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:hep-ph/9704333 by the authors.
References
How the literature uses it
35 arXiv papers cite this in 81 sentences
Papers whose main result relates to this one's
States the same result 3 papers
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It does not contain any adjustable parameter and has a finite IR limit $\bar{\alpha}_s(0)$ which depends only on group symmetry factors.
Analytic QCD running coupling with finite IR behaviour and universal ${\bar \alpha}_s(0)$ value · 1.00
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This value as well as the $\bar{\alpha}_{an}$ behavior in the whole IR region $Q^2 \leq \Lambda^2$ turns out to be stable with respect to higher loop corrections; %\item~ (iii) Coherence between observed $\bar{\alpha}_s(M_{\tau}^2)$ value and integral information on the IR $\bar{\alpha}_s(Q^2)$ behavior extracted from jet physics.
On the Analytic "Causal" Model for the QCD Running Coupling · 0.91
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This integral form manifestly accounts for the universal value of the infrared limit $\bar\alpha_t(Q^2=0)= 4 \pi /\beta_0$.
Papers citing it · 44 sentences without a confident label
Contests it 1 paper
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The minimal analytic (MA) model [ref] , with the value of $Λ$ such that high-energy QCD observables are reproduced, turns out to give for this quantity too low values $r_τ ≈0.14$ [ref] unless the (current) masses of the light quarks are taken to be unrealistically large ($m_q ≈0.25$-$0.45$ GeV) or strong threshold effects are introduced [ref] .
Reconciling the analytic QCD with the ITEP operator product expansion philosophy · Evaluation of low-energy observables · contest 0.71
Uses its method or tool 12 papers
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In this work, we apply two anQCD models of $(a_pt)_an. ≡_1$, namely APT of Refs. [ref] , and the 2$δ$anQCD model of Refs. [ref] , to evaluations of the perturbation series of the binding energy $E_q q̅$ of heavy quarkonia ($Υ(1S)$ and $J/ψ(1S)$) and of the quark pole mass $m_q$.
Calculation of binding energies and masses of quarkonia in analytic QCD models · Introduction · 0.87
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The model involves the insertion, inside standard threshold resummation formulae, of an effective QCD coupling $α̃_S(k^2)$, based on an analyticity requirement and resumming absorptive effects in gluon cascades [ref] .
Vub extraction using the Analytic Coupling model · Introduction · 0.84
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The model basically involves the insertion, inside standard threshold resummation formulae, of an effective QCD coupling $α̃_S(k^2)$; such coupling is based on an analyticity requirement and includes resumming absorptive effects in gluon cascades [ref] .
Inclusive Measure of |V_ub| with the Analytic Coupling Model · Introduction · 0.85
Uses its data or values 1 paper
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On the other hand, in the MA model [ref] , the value $Λ_MA = 0.260$ GeV (at $n_f=5$) is the one that reproduces the high energy QCD phenomenology (see also [ref] ).
Analogs of noninteger powers in general analytic QCD · Numerical calculations · 0.94
Compares with it 4 papers
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Furthermore, we will now prove that the obtained result ([ref]) (equivalent to equations ([ref]) and ([ref])) reproduces, in the specific case of the Minimal Analytic model (MA) of [ref] at one-loop level, the explicit result obtained in [ref] [ Note that $a$ in [ref] corresponds to our $β_0 a$ (with our $β_0 = (1/4)(11 - 2 n_f/3)$); their $_ν+1$ corresponds to our $β_0^ν+1 _ν+ 1$; they use the transcendental Lerch function notation $z Φ(z,ν^',1) ≡F(z, ν^')$ for the polylogarithm function $Li_ν^'(z)$.
Analogs of noninteger powers in general analytic QCD · Logarithmic noninteger derivatives of Euclidean coupling in any analytic QCD model · 0.99
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It turns out that in the (fractional) APT model of Refs. [ref] , the (fractional) power analogs $_ν^(APT)$, Eq. ([ref]), constructed entirely from the discontinuities of the pQCD coupling $a^ν$, coincide with the result of the approach described here, for the corresponding special (APT) case: $ρ(σ) = ρ^(pt)(σ)$, i.e., when $Im (-σ- i ϵ) = Im a(-σ- i ϵ)$.
Techniques of evaluation of QCD low-energy physical quantities with running coupling with infrared fixed point · Analogs of $a^n$ and $a_n$ for noninteger $n=ν$ in IR fixed point scenarios · 0.97
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The attractiveness of the obtained holomorphic (analytic) QCD models is that they are perturbative, i.e., beta function $β(a)$ is fully described by the Taylor series in powers of $a$, it has no nonperturbative contributions such as $exp(-K/a(Q^2)) ∼1/(Q^2)^M$, in contrast to the presently known analytic QCD models Refs. [ref] .
Perturbative QCD in acceptable schemes with holomorphic coupling · Conclusions · 0.84
Cites it as background 13 papers
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The nonanalyticity of $a(Q^2)$ in low-momentum regimes in the usual pQCD schemes was addressed in the seminal works of Shirkov, Solovtsov et al. [ref] , where a holomorphic version $^(APT)(Q^2)$ of the pQCD coupling $a(Q^2)$ (in any mass independent scheme) was constructed, via a use of the Cauchy theorem and dispersion integral in which the offending (Landau) cut of $a(Q^2)$ was eliminated and the cut of $a(Q^2)$ for $Q^2 < 0$ was left unchanged; in a sense, this is a “minimal” analytization approach, widely referred in the literature as Analytic Perturbation Theory (APT).
Perturbative QCD in acceptable schemes with holomorphic coupling · Introduction · 0.83
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The problem of Landau poles in the QCD coupling was first addressed in a systematic manner about 15 years ago by the authors of [ref] , who constructed and used an analytic QCD coupling parameter $_1^(MA)(Q^2)$ closely based on the perturbative coupling parameter $a_pt(Q^2)$: in the dispersive integral expression for $a_pt(Q^2)$, they removed the integration over the offending spacelike discontinuity cut (i.e., at $-Q^2 = σ< 0$), while keeping the discontinuity (spectral) function $ρ_1^(pt)(σ) ≡Im a_pt(Q^2 = -σ- i ϵ)$ unchanged on the timelike discontinuity cut, i.e., at $σ≥0$.
Extended analytic QCD model with perturbative QCD behavior at high momenta · Introduction · 0.88
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Several attempts at achieving such corrections have been recorded during the last 14 years starting from (what we call) the minimal analytic (MA) QCD of Shirkov and Solovtsov [ref] .
Reconciling the analytic QCD with the ITEP operator product expansion philosophy · Introduction · 0.95
Discussion
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