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Determination of perturbative QCD coupling from ALEPH $\tau$ decay data using pinched Borel-Laplace and Finite Energy Sum Rules
High Energy Physics - Phenomenology · Submitted 1 May 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The extracted value of the ${\overline {\rm MS}}$ coupling is $\alpha_s(m_{\tau}^2) = 0.3116 \pm 0.0073$ [$\alpha_s(M_Z^2)=0.1176 \pm 0.0010$] for the average of the FOPT and PV methods, which we regard as our main result.
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We present a determination of the perturbative QCD (pQCD) coupling using the V+A channel ALEPH $\tau$-decay data. The determination involves the double-pinched Borel-Laplace Sum Rules and Finite Energy Sum Rules. The theoretical basis is the Operator Product Expansion (OPE) of the V+A channel Adler function in which the higher order terms of the leading-twist part originate from a model based on the known structure of the leading renormalons of this quantity. The applied evaluation methods are contour-improved perturbation theory (CIPT), fixed-order perturbation theory (FOPT), and Principal Value of the Borel resummation (PV). All the methods involve truncations in the order of the coupling. In contrast to the truncated CIPT method, the truncated FOPT and PV methods account correctly for the suppression of various renormalon contributions of the Adler function in the mentioned sum rules. The extracted value of the ${\overline {\rm MS}}$ coupling is $\alpha_s(m_{\tau}^2) = 0.3116 \pm 0.0073$ [$\alpha_s(M_Z^2)=0.1176 \pm 0.0010$] for the average of the FOPT and PV methods, which we regard as our main result. On the other hand, if we include in the average also the CIPT method, the resulting values are significantly higher, $\alpha_s(m_{\tau}^2) = 0.3194 \pm 0.0167$ [$\alpha_s(M_Z^2)=0.1186 \pm 0.0021$].
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We present a determination of the perturbative QCD (pQCD) coupling using the V+A channel ALEPH $\tau$-decay data. | contribution 1.00 | 0.95 | 0.04 | asserted 1.00 |
| s1 | The determination involves the double-pinched Borel-Laplace Sum Rules and Finite Energy Sum Rules. | method 0.95 | 0.83 | 0.03 | asserted 1.00 |
| s2 | The theoretical basis is the Operator Product Expansion (OPE) of the V+A channel Adler function in which the higher order terms of the leading-twist part originate from a model based on the known structure of the leading renormalons of this quantity. | method 0.95 | 0.37 | 0.02 | asserted 1.00 |
| s3 | The applied evaluation methods are contour-improved perturbation theory (CIPT), fixed-order perturbation theory (FOPT), and Principal Value of the Borel resummation (PV). | method 1.00 | 0.77 | 0.02 | asserted 1.00 |
| s4 | All the methods involve truncations in the order of the coupling. | method 0.60 | 0.58 | 0.02 | asserted 1.00 |
| s5 | In contrast to the truncated CIPT method, the truncated FOPT and PV methods account correctly for the suppression of various renormalon contributions of the Adler function in the mentioned sum rules. | interpretation 0.65 | 0.74 | 0.09 | asserted 1.00 |
| s6 | The extracted value of the ${\overline {\rm MS}}$ coupling is $\alpha_s(m_{\tau}^2) = 0.3116 \pm 0.0073$ [$\alpha_s(M_Z^2)=0.1176 \pm 0.0010$] for the average of the FOPT and PV methods, which we regard as our main result. | result 1.00 | 0.97 | 0.94 | asserted 1.00 |
| s7 | On the other hand, if we include in the average also the CIPT method, the resulting values are significantly higher, $\alpha_s(m_{\tau}^2) = 0.3194 \pm 0.0167$ [$\alpha_s(M_Z^2)=0.1186 \pm 0.0021$]. | result 0.99 | 0.94 | 0.04 | asserted 0.99 |
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The paper
Original abstract
We present a determination of the perturbative QCD (pQCD) coupling using the V+A channel ALEPH $\tau$-decay data. The determination involves the double-pinched Borel-Laplace Sum Rules and Finite Energy Sum Rules. The theoretical basis is the Operator Product Expansion (OPE) of the V+A channel Adler function in which the higher order terms of the leading-twist part originate from a model based on the known structure of the leading renormalons of this quantity. The applied evaluation methods are contour-improved perturbation theory (CIPT), fixed-order perturbation theory (FOPT), and Principal Value of the Borel resummation (PV). All the methods involve truncations in the order of the coupling. In contrast to the truncated CIPT method, the truncated FOPT and PV methods account correctly for the suppression of various renormalon contributions of the Adler function in the mentioned sum rules. The extracted value of the ${\overline {\rm MS}}$ coupling is $\alpha_s(m_{\tau}^2) = 0.3116 \pm 0.0073$ [$\alpha_s(M_Z^2)=0.1176 \pm 0.0010$] for the average of the FOPT and PV methods, which we regard as our main result. On the other hand, if we include in the average also the CIPT method, the resulting values are significantly higher, $\alpha_s(m_{\tau}^2) = 0.3194 \pm 0.0167$ [$\alpha_s(M_Z^2)=0.1186 \pm 0.0021$].
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How the literature uses it
11 arXiv papers cite this in 55 sentences
Papers citing it · 33 sentences without a confident label
Contests it 1 paper
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The UV SL case ($-p=-1; j=2$ SL) was already given in Table II of [ref] , and the IR SL ($p=2, j=0$ SL) in Table 1 of [ref] .[There is a minor typo there, the ratio $d^IR_2,-1/^IR_2,-1$ there is given as $(+2.00 ±0.002)$, it should be $(+2.00 ±0.02)$.] We recall that the IR case $(p=2, j=0)$ is given in Table [ref] and corresponds to $B[](u) = π^IR_2,0 (-1) ln(1 - u/2)$; while the SL case of this, given here, corresponds to $B[](u) = π^IR_2,-1 (2-u) ln(1 - u/2)$.
Borel-Laplace Sum Rules with $\tau$ decay data, using OPE with improved anomalous dimensions · Results for the subleading cases of renormalon contributions · contest 0.80
Builds on it 2 papers
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This work can be regarded as a continuation and improvement of our previous analysis [ref] , where now the improved form ([ref]) of the OPE terms is used; further, the variation of the higher order perturbation coefficients of the renormalon-motivated extension of the Adler function is performed now in a more realistic way.
Using improved Operator Product Expansion in Borel-Laplace Sum Rules with ALEPH $\tau$ decay data, and determination of pQCD coupling · Introduction · 0.99
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In this work we improved on our previous works [ref] of application of the double-pinched Borel-Laplace sum rules to the (V+A)-channel semihadronic strangeless $τ$ decay data of ALEPH.
Borel-Laplace Sum Rules with $\tau$ decay data, using OPE with improved anomalous dimensions · Final results and conclusions · 0.94
Uses its method or tool 1 paper
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In the sum rules ([ref])-([ref]), we will consider the weight functions $g(Q^2)$ of the double-pinched Borel-Laplace transforms $B(M^2)$ [ref] , which are functions of a complex squared energy parameter $M^2$
Borel-Laplace Sum Rules with $\tau$ decay data, using OPE with improved anomalous dimensions · Sum rules and Adler function · 0.89
Uses its data or values 2 papers
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We employ the known QCD corrections up to $𝒪(^4)$ and use an estimate for the 6-loop coefficient $c̅_5,1$, consistent with the values used in the recent analyses [ref] .
The strong coupling constant: State of the art and the decade ahead · Reconciling the fixed order and contour improved perturbative series in hadronic tau decays. [Authors: M. A. Benitez-Rat · 0.97
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(We remind the reader that the ALEPH and OPAL analyses had to rely on Monte Carlo simulated data for these subleading channels.) Realistic phenomenological analyses are either based on the 2013 update of the ALEPH spectral functions [ref] , which allow for the analysis of the “more inclusive” $V+A$ channel, or on the new purely data based $V$ spectral function, which has significantly smaller uncertainties near the end-point of the spectrum [ref] .
Reconciling the Contour-Improved and Fixed-Order Approaches for $\tau$ Hadronic Spectral Moments II: Renormalon Norm and Application in $\alpha_s$ Determinations · Impact on Strong Coupling Determinations · 0.89
Compares with it 4 papers
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In addition, in Sec. [ref] we compare the obtained results with our previous results [ref] and the results obtained by other groups from the semihadronic $τ$-decays.
Borel-Laplace Sum Rules with $\tau$ decay data, using OPE with improved anomalous dimensions · Introduction · 0.99
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We refer to [ref] for discussion and comparison of various of the approaches and results presented in the Table.
Using improved Operator Product Expansion in Borel-Laplace Sum Rules with ALEPH $\tau$ decay data, and determination of pQCD coupling · Final results and conclusions · 0.88
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In our previous work [ref] we used the OPE with simple $D=6,8$ terms, and took for $d_4$ higher values $d_4=338 ±63$ than here Eq. ([ref]).
The strong coupling constant: State of the art and the decade ahead · Extraction of alphas using Borel–Laplace sum rules for tau decay data [Authors: C. Ayala (U. Tarapacá), G. Cvetič (UTFSM · 0.82
Cites it as support 2 papers
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As argued in detail in our previous work, the FO, $FO$ and PV methods of evaluation of the sum rules lead to truncated series[In the PV case, the truncated series refers to the series $δd(e^i ϕ)^[N_t]_D=0$ in Eq. ([ref]) which is free from the leading renormalon singularities and gives the corresponding contour integrals (sum rules) also free from the leading renormalon singularities.] that have the leading renormalon contribution of the Adler function (double UV renormalon pole at $u=+1$) suppressed in them; and that the CI method in the sum rules does not have this property.[These arguments, presented in a very general way in [ref] (especially Appendix A there), are also compatible with somewhat related arguments presented in [ref] .
Using improved Operator Product Expansion in Borel-Laplace Sum Rules with ALEPH $\tau$ decay data, and determination of pQCD coupling · Final results and conclusions · 0.98
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Furthermore, the CI approach does not respect the renormalon structure of the considered sum rules when truncated [ref] .
Borel-Laplace Sum Rules with $\tau$ decay data, using OPE with improved anomalous dimensions · Introduction · 0.98
Cites it as background 5 papers
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Such OPE form is usually used in numerical analyses, cf. [ref] .
Using improved Operator Product Expansion in Borel-Laplace Sum Rules with ALEPH $\tau$ decay data, and determination of pQCD coupling · Sum rules and Adler function · 0.80
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The non-strange $V+A$ inclusive channel has been thoroughly studied in the literature as a QCD laboratory [ref] .
Semileptonic tau decays beyond the Standard Model · V+A · 1.00
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Strong efforts are currently being made aimed to improve our understanding of the perturbative series, e.g. see [ref] .
Violations of Quark-Hadron Duality in Low-Energy Determinations of $\alpha_s$ · Theoretical formalism · 0.92
What it cites · 78 sentences without a confident label
Builds on 1 paper
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The above arguments also show that the sum rules (which are timelike quantities) have FOPT perturbation expansions ([ref]) for which the same renormalon-related arguments [ref] can be applied as for the perturbation expansions of spacelike quantities such as the Adler function Eq. ([ref]), except that now, instead of in general complex (and nonnegative) $Q^2$, we have $Q^2 = >0$.[In Ref. [ref] these arguments were presented for spacelike quantities, but are valid also for (FOPT expansion) of timelike quantities, as shown here.] Specifically, if $()$ has a double-pole (DP) or a single-pole (SP) $u=-1$ UV renormalon, the 'lpt'-expansion coefficients for large $n$ behave as: $_n ∼(n+1)!
Renormalon-motivated evaluation of QCD observables · Relation with the full expansion coefficients · 0.85
Uses the method or tool of 2 papers
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The result Eq. ([ref]), at the canonical scale $Q^2=M_Z^2$ (and $N_f=5$) was then obtained by RGE-evolution using the five-loop $$ $β$-function [ref] and the corresponding four-loop quark threshold matching [ref] .[The threshold matching was performed at the scales $Q^2_thr = κm̅_q^2$ with $κ=2$, and $m̅_q ≡m̅_q( m̅_q^2)$ equal to $4.2$ GeV ($q=b$) and $1.27$ GeV ($q=c$).]
Five-Loop Running of the QCD coupling constant · Summary of the results and comparison with literature · 0.97
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In practice, we chose 11 equidistant points along each of the three rays, i.e., the sum ([ref]) contains 33 terms.[The fit results are practically unchanged if the number of points $M^2_α$ is increased beyond 33.] Further, $δ_B(M^2_α)$ is the experimental standard deviation of $B_exp(M^2_α;)$.[The construction of $δ_B(M^2_α)$ involves the covariance matrix of the ALEPH data and the weight function $f(σ_j; M^2)$ (where $σ=σ_j$ is in the $j$'th bin of ALEPH data) corresponding to the Borel-Laplace sum rule, as explained in App. C of Ref. [ref] .
Nearly perturbative lattice-motivated QCD coupling with zero IR limit · Double-pinched Borel-Laplace sum rules · 0.80
Uses data or values from 6 papers
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The latter are principally from ALEPH Collaboration [ref] , where the spectral function $ω(σ)$ was measured with high precision.[$ω(σ) ∝Im Π(-σ- i ϵ)$, where $Π(Q^2)$ is the polarisation function of the quark current correlator.
The Determination of alpha_s from Tau Decays Revisited · Introduction · 0.99
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The latter are principally from ALEPH Collaboration [ref] , where the spectral function $ω(σ)$ was measured with high precision.[$ω(σ) ∝Im Π(-σ- i ϵ)$, where $Π(Q^2)$ is the polarisation function of the quark current correlator.
Update of the ALEPH non-strange spectral functions from hadronic $\tau$ decays · Introduction · 0.99
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which was measured by OPAL [ref] and ALEPH Collaborations [ref] in strangeless semihadronic $τ$ decays.
An updated determination of \alpha_s from tau decays · Sum rules and Adler function · 0.83
Compares with 10 papers
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The works of [ref] (cf. also [ref] where $R_ee(s)$ is used), on the other hand, give for (sufficiently pinched) FESRs the solutions of OPE with considerably larger absolute values of the condensates $⟨O_D ⟩_V+A$, for many terms (up to $D=16$).[ The works [ref] and [ref] use for the $D=0$ Adler function the series truncated at $∼a^5$ (i.e., $N_t=5$), with $d_4 = 283 ±283$ [ref] and $d_4=275 ±400$ [ref] .] This shows that there are at least two very different sets of OPE solutions to the $τ$-decay data, which correspond to the same or approximately same spectral function $ω(σ)_V+A$ (ALEPH) for (sufficiently pinched) FESRs.
Determination of the QCD Coupling from ALEPH $\tau$ Decay Data · Double-pinched Borel-Laplace sum rules · 0.95
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The works of [ref] (cf. also [ref] where $R_ee(s)$ is used), on the other hand, give for (sufficiently pinched) FESRs the solutions of OPE with considerably larger absolute values of the condensates $⟨O_D ⟩_V+A$, for many terms (up to $D=16$).[ The works [ref] and [ref] use for the $D=0$ Adler function the series truncated at $∼a^5$ (i.e., $N_t=5$), with $d_4 = 283 ±283$ [ref] and $d_4=275 ±400$ [ref] .] This shows that there are at least two very different sets of OPE solutions to the $τ$-decay data, which correspond to the same or approximately same spectral function $ω(σ)_V+A$ (ALEPH) for (sufficiently pinched) FESRs.
The strong coupling from the revised ALEPH data for hadronic $\tau$ decays · Double-pinched Borel-Laplace sum rules · 0.95
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The works of [ref] (cf. also [ref] where $R_ee(s)$ is used), on the other hand, give for (sufficiently pinched) FESRs the solutions of OPE with considerably larger absolute values of the condensates $⟨O_D ⟩_V+A$, for many terms (up to $D=16$).[ The works [ref] and [ref] use for the $D=0$ Adler function the series truncated at $∼a^5$ (i.e., $N_t=5$), with $d_4 = 283 ±283$ [ref] and $d_4=275 ±400$ [ref] .] This shows that there are at least two very different sets of OPE solutions to the $τ$-decay data, which correspond to the same or approximately same spectral function $ω(σ)_V+A$ (ALEPH) for (sufficiently pinched) FESRs.
The strong coupling from hadronic $\tau$ decays: a critical appraisal · Double-pinched Borel-Laplace sum rules · 0.95
Cites as support 7 papers
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The results of the works [ref] and [ref] suggest that the duality violations are well suppressed in the $V+A$ channel for FESRs which are at least doubly-pinched.[The $n$-pinched FESR weight functions $g(Q^2)$ behave in the timelike limit, i.e., when $Q^2 →-$, as: $g(Q^2) ∼(1 + Q^2/)^n$.] In our global fit analysis, we used doubly-pinched Borel-Laplace sum rules, whose weight functions $g_M^2(Q^2)$, Eq. ([ref]), are additionally suppressed in the timelike limit by the exponential factor $exp(cos(Ψ) Q^2/|M^2|) →exp(- cos(Ψ) /|M^2|)$ (where we took: $Ψ≡arg(M^2) = 0, π/6, π/4$).
The strong coupling from the revised ALEPH data for hadronic $\tau$ decays · Double-pinched Borel-Laplace sum rules · 0.97
-
The results of the works [ref] and [ref] suggest that the duality violations are well suppressed in the $V+A$ channel for FESRs which are at least doubly-pinched.[The $n$-pinched FESR weight functions $g(Q^2)$ behave in the timelike limit, i.e., when $Q^2 →-$, as: $g(Q^2) ∼(1 + Q^2/)^n$.] In our global fit analysis, we used doubly-pinched Borel-Laplace sum rules, whose weight functions $g_M^2(Q^2)$, Eq. ([ref]), are additionally suppressed in the timelike limit by the exponential factor $exp(cos(Ψ) Q^2/|M^2|) →exp(- cos(Ψ) /|M^2|)$ (where we took: $Ψ≡arg(M^2) = 0, π/6, π/4$).
Determination of the QCD Coupling from ALEPH $\tau$ Decay Data · Double-pinched Borel-Laplace sum rules · 0.97
-
The results of the works [ref] and [ref] suggest that the duality violations are well suppressed in the $V+A$ channel for FESRs which are at least doubly-pinched.[The $n$-pinched FESR weight functions $g(Q^2)$ behave in the timelike limit, i.e., when $Q^2 →-$, as: $g(Q^2) ∼(1 + Q^2/)^n$.] In our global fit analysis, we used doubly-pinched Borel-Laplace sum rules, whose weight functions $g_M^2(Q^2)$, Eq. ([ref]), are additionally suppressed in the timelike limit by the exponential factor $exp(cos(Ψ) Q^2/|M^2|) →exp(- cos(Ψ) /|M^2|)$ (where we took: $Ψ≡arg(M^2) = 0, π/6, π/4$).
The strong coupling from hadronic $\tau$ decays: a critical appraisal · Double-pinched Borel-Laplace sum rules · 0.97
Cites as background 17 papers
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The integration is usually performed by Taylor-expanding the integrand around $Q^2=σ_max >0$ (fixed order [FI]) or RGE-evolving the integrand along the countour $Q^2=σ_max e^i ϕ$ (contour improved [CIPT]) [ref] .
Renormalization group analysis of the tau-lepton decay within QCD · Introduction · 0.85
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The integration is usually performed by Taylor-expanding the integrand around $Q^2=σ_max >0$ (fixed order [FI]) or RGE-evolving the integrand along the countour $Q^2=σ_max e^i ϕ$ (contour improved [CIPT]) [ref] .
Higher-order QCD perturbation theory in different schemes: From FOPT to CIPT to FAPT · Introduction · 0.85
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We will use the notation $Q^2 ≡- q^2 = - (q^0)^2 + q⃗^2$.] The extraction of the value of the running QCD coupling $α_s(Q^2)$ at such low momenta $Q^2 ≈m^2_τ$ represents a test of QCD, especially when comparing, via renormalisation group equation (RGE) evolution, with the extraction of the running coupling from experiments at higher energies $Q^2 ≫m^2_τ$ where the perturbative methods of evaluation work very well [ref] .
The QCD Running Coupling · Introduction · 0.82
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