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Small-momentum expansion of heavy-quark correlators in the large-$\beta_0$ limit and $\alpha_s$ extractions
High Energy Physics - Phenomenology · Submitted 10 Jun 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that these ratios benefit from a partial cancellation of the leading renormalon singularities.
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The abstract, sentence by sentence
We calculate the small-momentum expansion of vector, axial-vector, scalar, and pseudo-scalar heavy-quark current correlators in the large-$\beta_0$ limit of QCD, extending the analysis of Grozin and Sturm beyond the vector current. Our results are used to study the higher-order behaviour of dimensionless ratios of vector and pseudo-scalar moments used for the precise extraction of the strong coupling, $\alpha_s$, from relativistic quarkonium sum rules and lattice data, respectively. We show that these ratios benefit from a partial cancellation of the leading renormalon singularities. Our results can guide the design of combinations of moments with improved perturbative behaviour.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We calculate the small-momentum expansion of vector, axial-vector, scalar, and pseudo-scalar heavy-quark current correlators in the large-$\beta_0$ limit of QCD, extending the analysis of Grozin and Sturm beyond the vector current. | contribution 1.00 | 0.96 | 0.10 | asserted 1.00 |
| s1 | Our results are used to study the higher-order behaviour of dimensionless ratios of vector and pseudo-scalar moments used for the precise extraction of the strong coupling, $\alpha_s$, from relativistic quarkonium sum rules and lattice data, respectively. | contribution 0.88 | 0.83 | 0.15 | asserted 1.00 |
| s2 | We show that these ratios benefit from a partial cancellation of the leading renormalon singularities. | result 0.93 | 0.92 | 0.43 | asserted 1.00 |
| s3 | Our results can guide the design of combinations of moments with improved perturbative behaviour. | interpretation 0.66 | 0.44 | 0.20 | hedged possibility 0.94 |
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The paper
Original abstract
We calculate the small-momentum expansion of vector, axial-vector, scalar, and pseudo-scalar heavy-quark current correlators in the large-$\beta_0$ limit of QCD, extending the analysis of Grozin and Sturm beyond the vector current. Our results are used to study the higher-order behaviour of dimensionless ratios of vector and pseudo-scalar moments used for the precise extraction of the strong coupling, $\alpha_s$, from relativistic quarkonium sum rules and lattice data, respectively. We show that these ratios benefit from a partial cancellation of the leading renormalon singularities. Our results can guide the design of combinations of moments with improved perturbative behaviour.
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How the literature uses it
4 arXiv papers cite this in 7 sentences
Papers citing it · 6 sentences without a confident label
Cites it as background 1 paper
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A large-$β_0$ renormalon-based analysis for the low energy moments of the current correlators can be found in Ref. [ref] .
Renormalization group improved determination of $\alpha_s$, $m_c$, and $m_b$ from the low energy moments of heavy quark current correlators · Theoretical inputs · 0.93
What it cites · 90 sentences without a confident label
Uses the method or tool of 7 papers
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From the definition of the moments $M_q,n^δ$ given in Eq. [ref], one obtains [equation] For the calculation of $C_n^δ(μ)$ in the large-$β_0$ limit, given that renormalisation is required, we rely on the formalism described in detail in Ref. [ref] , which was employed in the original calculation of the small-momentum expansion of the vector correlator in this limit [ref] .
Correlator of heavy-quark currents at small q^2 in the large-beta_0 limit · The moments Mndelta in the large-beta0 limit · 0.99
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From the definition of the moments $M_q,n^δ$ given in Eq. [ref], one obtains [equation] For the calculation of $C_n^δ(μ)$ in the large-$β_0$ limit, given that renormalisation is required, we rely on the formalism described in detail in Ref. [ref] , which was employed in the original calculation of the small-momentum expansion of the vector correlator in this limit [ref] .
Renormalons: technical introduction · The moments Mndelta in the large-beta0 limit · 0.99
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In the case of the axial-vector current, $Π_L^A$ can be obtained by applying the projector $q^μq^ν$ or using the axial Ward identity, which relates this contribution to the pseudo- scalar correlator [ref] .[The axial-vector moments are defined with respect to the small momentum expansion of the transverse contribution.] When using dimensional regularisation for loop computations, the currents that contain $γ_5$ must be carefully extended to $d$ dimensions; we employ the prescription described in Ref. [ref] .[No finite renormalisation of the axial and pseudo-scalar currents is required in our case.]
Moments of Heavy Quark Current Correlators at Four-Loop Order in Perturbative QCD · Theory overview · 0.82
Uses data or values from 6 papers
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For the strong coupling we use the reference value $α_s^(n_f=5)(m_Z) = 0.1179$, with $m_Z=91.1876$GeV [ref] , which yields $α_s^(n_f=4)(_b)=0.2245$ and $α_s^(n_f=3)(_c)=0.3865$ using the five-loop running coupling [ref] and four-loop matching [ref] at the thresholds, both in full QCD.
Five-loop renormalisation of QCD in covariant gauges · Higher order behavior of the perturbative series · 0.83
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For the strong coupling we use the reference value $α_s^(n_f=5)(m_Z) = 0.1179$, with $m_Z=91.1876$GeV [ref] , which yields $α_s^(n_f=4)(_b)=0.2245$ and $α_s^(n_f=3)(_c)=0.3865$ using the five-loop running coupling [ref] and four-loop matching [ref] at the thresholds, both in full QCD.
Five-Loop Running of the QCD coupling constant · Higher order behavior of the perturbative series · 0.83
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For the strong coupling we use the reference value $α_s^(n_f=5)(m_Z) = 0.1179$, with $m_Z=91.1876$GeV [ref] , which yields $α_s^(n_f=4)(_b)=0.2245$ and $α_s^(n_f=3)(_c)=0.3865$ using the five-loop running coupling [ref] and four-loop matching [ref] at the thresholds, both in full QCD.
The five-loop beta function of Yang-Mills theory with fermions · Higher order behavior of the perturbative series · 0.83
Compares with 8 papers
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When compared to $R_c,2^V$ shown in Fig. [ref], from which the main results of Refs. [ref] are based, we see that the perturbative series of $R_c^V(-1/3, 1, -1/3)$ approaches faster the true value given by the Borel sum, has a weaker dependence on the renormalisation scale, and does not present a run-away behaviour typically seen in series dominated by IR renormalons.
Precise $\alpha_s$ determination from charmonium sum rules · Combined ratios of moments · 1.00
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When compared to $R_c,2^V$ shown in Fig. [ref], from which the main results of Refs. [ref] are based, we see that the perturbative series of $R_c^V(-1/3, 1, -1/3)$ approaches faster the true value given by the Borel sum, has a weaker dependence on the renormalisation scale, and does not present a run-away behaviour typically seen in series dominated by IR renormalons.
Precise determination of $\alpha_s$ from relativistic quarkonium sum rules · Combined ratios of moments · 1.00
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For the vector correlator, we computed the functions $S_n^V(u)$ up to $n=12$, finding agreement with the results presented in Ref. [ref] , which were given up to $n=2$.
Correlator of heavy-quark currents at small q^2 in the large-beta_0 limit · Results · 0.99
Cites as support 3 papers
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The knowledge of the gluon-condensate contribution shows that this is an excellent approximation for the bottom and a very good one for the charm [ref]
Precise determination of $\alpha_s$ from relativistic quarkonium sum rules · The ratios Rndelta in the large-beta0 limit · 0.96
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The knowledge of the gluon-condensate contribution shows that this is an excellent approximation for the bottom and a very good one for the charm [ref]
Precise $\alpha_s$ determination from charmonium sum rules · The ratios Rndelta in the large-beta0 limit · 0.96
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This is particularly true for bottom quark moments, where the non-perturbative contributions can be neglected for all practical purposes [ref] .
Bottom and Charm Mass Determinations with a Convergence Test · Results · 0.99
Cites as background 1 paper
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The result for the small-momentum expansion of the vector correlator in the large-$β_0$ limit is available since the work of Grozin and Sturm [ref] .
Correlator of heavy-quark currents at small q^2 in the large-beta_0 limit · Introduction · 0.81
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