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Techniques of evaluation of QCD low-energy physical quantities with running coupling with infrared fixed point
High Energy Physics - Phenomenology · Submitted 6 Sep 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We argue that an alternative series in terms of logarithmic derivatives of A(Q^2) should be used.
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Perturbative QCD (pQCD) running coupling a(Q^2) (=alpha_s(Q^2)/pi) is expected to get modified at low spacelike momenta 0 < Q^2 < 1 GeV^2 so that, instead of having unphysical (Landau) singularities it remains smooth and finite there, due to infared (IR) fixed point. This behavior is suggested by: Gribov-Zwanziger approach, Dyson-Schwinger equations (DSE) and other functional methods, lattice calculations, light-front holographic mapping AdS/CFT modified by a dilaton background, and by most of the analytic (holomorphic) QCD models. All such couplings, A(Q^2), differ from the pQCD couplings a(Q^2) at |Q| > 1 GeV by nonperturbative (NP) terms, typically by some power-suppressed terms ~1/(Q^2)^N. Evaluations of low-energy physical QCD quantities in terms of such A(Q^2) couplings (with IR fixed point) at a level beyond one-loop are usually performed with (truncated) power series in A(Q^2). We argue that such an evaluation is not correct, because the NP terms in general get out of control as the number of terms in the power series increases. The series consequently become increasingly unstable under the variation of the renormalization scale, and have a fast asymptotic divergent behavior compounded by the renormalon problem. We argue that an alternative series in terms of logarithmic derivatives of A(Q^2) should be used. Further, a Pad\'e-related resummation based on this series gives results which are renormalization scale independent and show very good convergence. Timelike low-energy observables can be evaluated analogously, using the integral transformation which relates the timelike observable with the corresponding spacelike observable.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Perturbative QCD (pQCD) running coupling a(Q^2) (=alpha_s(Q^2)/pi) is expected to get modified at low spacelike momenta 0 < Q^2 < 1 GeV^2 so that, instead of having unphysical (Landau) singularities it remains smooth and finite there, due to infared (IR) fixed point. | background 1.00 | 0.05 | 0.04 | hedged possibility 0.88 |
| s1 | This behavior is suggested by: Gribov-Zwanziger approach, Dyson-Schwinger equations (DSE) and other functional methods, lattice calculations, light-front holographic mapping AdS/CFT modified by a dilaton background, and by most of the analytic (holomorphic) QCD models. | background 1.00 | 0.04 | 0.02 | hedged possibility 0.97 |
| s2 | All such couplings, A(Q^2), differ from the pQCD couplings a(Q^2) at |Q| > 1 GeV by nonperturbative (NP) terms, typically by some power-suppressed terms ~1/(Q^2)^N. | background 1.00 | 0.07 | 0.03 | asserted 0.85 |
| s3 | Evaluations of low-energy physical QCD quantities in terms of such A(Q^2) couplings (with IR fixed point) at a level beyond one-loop are usually performed with (truncated) power series in A(Q^2). | background 0.97 | 0.07 | 0.04 | asserted 0.93 |
| s4 | We argue that such an evaluation is not correct, because the NP terms in general get out of control as the number of terms in the power series increases. | contribution 0.49 | 0.38 | 0.46 | asserted 0.96 |
| s5 | The series consequently become increasingly unstable under the variation of the renormalization scale, and have a fast asymptotic divergent behavior compounded by the renormalon problem. | interpretation 0.56 | 0.32 | 0.10 | asserted 1.00 |
| s6 | We argue that an alternative series in terms of logarithmic derivatives of A(Q^2) should be used. | method 0.61 | 0.61 | 0.46 | asserted 0.78 |
| s7 | Further, a Pad\'e-related resummation based on this series gives results which are renormalization scale independent and show very good convergence. | result 0.75 | 0.84 | 0.30 | asserted 1.00 |
| s8 | Timelike low-energy observables can be evaluated analogously, using the integral transformation which relates the timelike observable with the corresponding spacelike observable. | method 0.78 | 0.41 | 0.05 | asserted 0.61 |
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Original abstract
Perturbative QCD (pQCD) running coupling a(Q^2) (=alpha_s(Q^2)/pi) is expected to get modified at low spacelike momenta 0 < Q^2 < 1 GeV^2 so that, instead of having unphysical (Landau) singularities it remains smooth and finite there, due to infared (IR) fixed point. This behavior is suggested by: Gribov-Zwanziger approach, Dyson-Schwinger equations (DSE) and other functional methods, lattice calculations, light-front holographic mapping AdS/CFT modified by a dilaton background, and by most of the analytic (holomorphic) QCD models. All such couplings, A(Q^2), differ from the pQCD couplings a(Q^2) at |Q| > 1 GeV by nonperturbative (NP) terms, typically by some power-suppressed terms ~1/(Q^2)^N. Evaluations of low-energy physical QCD quantities in terms of such A(Q^2) couplings (with IR fixed point) at a level beyond one-loop are usually performed with (truncated) power series in A(Q^2). We argue that such an evaluation is not correct, because the NP terms in general get out of control as the number of terms in the power series increases. The series consequently become increasingly unstable under the variation of the renormalization scale, and have a fast asymptotic divergent behavior compounded by the renormalon problem. We argue that an alternative series in terms of logarithmic derivatives of A(Q^2) should be used. Further, a Pad\'e-related resummation based on this series gives results which are renormalization scale independent and show very good convergence. Timelike low-energy observables can be evaluated analogously, using the integral transformation which relates the timelike observable with the corresponding spacelike observable.
Human review
How the literature uses it
13 arXiv papers cite this in 29 sentences
Papers whose main result relates to this one's
States the same result 1 paper
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I argue that in such QCD frameworks the spacelike quantities should not be evaluated as a power series, but rather as a series in derivatives of the coupling with respect to the logarithm of the squared momenta.
Evaluations of low-energy physical quantities in QCD with IR freezing of the coupling · 0.99
Papers citing it · 25 sentences without a confident label
Cites it as support 4 papers
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If the truncated series are resummed by the mentioned approach $d_res^[2 M](Q^2)$, with any holomorphic QCD coupling, the resulting sequence converges [ref] as the number of terms increases, it converges for any spacelike $Q^2$, and it converges to the known large-$β_0$ result [ref] .
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This problem does not appear in QCD with holomorphic couplings (AQCD), as noted in [ref] , because $(κ_j Q^2)$ coupling has no Landau singularities.
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In such cases, even if the truncation index $N$ in the relations ([ref]) is very high, we do not have $_n(Q^2) ≈(Q^2)^n$ at low values $|Q^2| ≲1 GeV^2$.[ At high $|Q^2| > 1 GeV^2$ we have in general $_n(Q^2) ≈_n(Q^2) ≈(Q^2)^n ≈a(Q^2)^n$, due to the relation ([ref]), i.e., ($2δ$ and $3δ$) $$QCD in the high-momentum regime is practically indistinguishable from the underlying pQCD.] In [ref] it was argued that if in Eq. ([ref]) the naive powers $(Q^2)^n$ were used instead of $_n(Q^2)$, this would bring into the series spurious uncontrollable nonperturbative contributions at $|Q^2| ≲1 GeV^2$.
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What it cites · 86 sentences without a confident label
Uses the method or tool of 4 papers
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We vary the renormalization scheme $μ^2 = κQ^2$, and perform evaluations in various IR fixed point frameworks, with $N_f=3$: * A representative case of freezing – the case with constant effective gluon mass $m$, Refs. [ref] , Eq. ([ref]), applied to the coupling ([ref]) in the spirit of Ref. [ref] ^(m)(μ^2) = a(μ^2+m^2) ,
Asymptotic freedom and IR freezing in QCD: the role of gluon paramagnetism · Renormalization scale dependence of truncated series · 0.83
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We vary the renormalization scheme $μ^2 = κQ^2$, and perform evaluations in various IR fixed point frameworks, with $N_f=3$: * A representative case of freezing – the case with constant effective gluon mass $m$, Refs. [ref] , Eq. ([ref]), applied to the coupling ([ref]) in the spirit of Ref. [ref] ^(m)(μ^2) = a(μ^2+m^2) ,
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We vary the renormalization scheme $μ^2 = κQ^2$, and perform evaluations in various IR fixed point frameworks, with $N_f=3$: * A representative case of freezing – the case with constant effective gluon mass $m$, Refs. [ref] , Eq. ([ref]), applied to the coupling ([ref]) in the spirit of Ref. [ref] ^(m)(μ^2) = a(μ^2+m^2) ,
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Uses data or values from 2 papers
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The input values of the model are the central ones used in Ref. [ref] (among them: $c_2=-4.76$, $Λ_L. = 0.260$ GeV) and give the value $_1^(2 δ)(m_τ^2)=0.291/π$.
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It was obtained in Ref. [ref] as a consequence of the use of nonperturbative QCD background, and is $m ∼1$ GeV.
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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 $Im (-σ- i ϵ) = Im a(-σ- i ϵ)$.
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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 $Im (-σ- i ϵ) = Im a(-σ- i ϵ)$.
Higher-order QCD perturbation theory in different schemes: From FOPT to CIPT to FAPT · Analogs of $a^n$ and $a_n$ for noninteger $n=ν$ in IR fixed point scenarios · 0.97
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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 $Im (-σ- i ϵ) = Im a(-σ- i ϵ)$.
QCD Analytic Perturbation Theory. From integer powers to any power of the running coupling · Analogs of $a^n$ and $a_n$ for noninteger $n=ν$ in IR fixed point scenarios · 0.97
Cites as support 2 papers
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As shown in Refs. [ref] , these approximants work very well in practice in the analytic QCD frameworks.
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Cites as background 8 papers
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Another analytic model, based on the minimal analytization of the function $d lna(Q^2)/d lnQ^2$, Refs. [ref] , leads to a coupling with no freezing in the IR.
Quark-antiquark potential in the analytic approach to QCD · IR fixed point scenarios · 0.81
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Another case is the Analytic Perturbation Theory (APT) coupling [ref] , which is obtained by “minimally” analytizing the perturbative ($n$-loop) coupling $a(Q^2)$.
Analytic Model for the QCD Running Coupling with Universal $\bar{\alpha}_s(0)$ Value · IR fixed point scenarios · 0.95
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Another case is the Analytic Perturbation Theory (APT) coupling [ref] , which is obtained by “minimally” analytizing the perturbative ($n$-loop) coupling $a(Q^2)$.
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