The strong coupling constant: state of the art and the decade ahead

2.1.1. Overview of the current situation

FLAG stands for ‘Flavour Lattice Averaging Group’ and constitutes an effort by the lattice QCD community to supply qualified information on lattice results for selected physical quantities to the wider particle physics community. These include the QCD parameters, i.e. α_S and the quark masses. An extensive report aimed at a general particle physics audience is published every 2–3 years [11, 12] and in the meantime occasional updates are made to the online version maintained at the University of Bern ⁴⁸ . We strongly encourage readers to explore this report, in particular, the α_S chapter. For a complementary pedagogical introduction see the review [13].

In the FLAG α_S working group we have recently provided the updated lattice QCD average

$$\begin{eqnarray}&&{\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1184(8),\end{eqnarray} \tag{ 2.1 }$$

based on results published ⁴⁹ in [14–21]. This represents a minimal change from FLAG 2019 [12], and no reduction in the error. The FLAG criteria for α_S have remained unchanged since FLAG 2019 and there are now some indications that the criteria may need to be revised in the future [11].

Lattice determinations of α_S use up, down, and strange quarks (and sometimes the charm quark) in the sea, and perturbatively evolve across the charm and bottom thresholds to obtain α_S in the 5-flavour theory. The perturbative matching across quark thresholds has been put to a nonperturbative test in [22] which demonstrates that the perturbative description of decoupling (known to 4-loop order [23–28]) provides an excellent quantitative description even for the charm quark. Hence, one may avoid the potentially large cutoff effects associated with the charm quark mass, which is usually not so small compared to the lattice cutoff scale 1/a (a denotes the lattice spacing).

A lattice determination of α_S starts with the choice of an observable O(μ) depending on a single scale μ with a perturbative expansion of the form

$$\begin{eqnarray}&&O(\mu )={c}_{0}+{c}_{1}{\alpha }_{\overline{\mathrm{MS}}}(\mu )+{c}_{2}{\alpha }_{\overline{\mathrm{MS}}}^{2}(\mu )+\ldots \end{eqnarray} \tag{ 2.2 }$$

It is convenient to normalize the observable as an effective coupling

$$\begin{eqnarray}&&{\alpha }_{\mathrm{eff}}=(O(\mu )-{c}_{0})/{c}_{1}={\alpha }_{\overline{\mathrm{MS}}}+{d}_{1}{\alpha }_{\overline{\mathrm{MS}}}^{2}\,+\ldots \end{eqnarray} \tag{ 2.3 }$$

We then refer to d₁ = c₂/c₁ as the 1-loop matching coefficient, even though some choices of O(μ) may require a 1-loop-diagram to obtain c₀ and thus 2 more loops to obtain d₁. It should be clear that each choice of O(μ) leads to a different lattice determination of α_S , in close analogy to phenomenological determinations. The main difference therefore is whether the original data are produced by a lattice simulation or taken from experiment. A bonus of the lattice setup is the access to observables which are not experimentally measurable, for instance, one may study QCD in a finite Euclidean space-time volume L⁴ and define the observable O through a finite volume effect [29]. However, many renormalization schemes for the coupling assume large volume, i.e. in practice one needs to show that the necessarily finite L is causing negligible effects on the chosen observable. Note also that quark masses can be varied, and one may naturally define mass-independent couplings (such as the $\overline{\mathrm{MS}}$ coupling) by imposing renormalization conditions in the chiral limit [30].

A common problem of most lattice determinations of α_S has been dubbed the ‘window problem’: in order to match to hadronic physics, the spatial volume L³ must be large enough to avoid significant finite volume effects due to pion polarization ‘around the world’. On the other hand, the matching to the coupling requires perturbative expansions to be reliable, so one needs to reach as high a scale μ as possible, but still significantly below the cutoff scale 1/a such as to avoid large lattice distortions. If taken together this means

$$\begin{eqnarray}&&\displaystyle \frac{1}{L}\ll {m}_{\pi }\ll \mu \ll \displaystyle \frac{1}{a}\quad \Rightarrow \quad \displaystyle \frac{L}{a}={ \mathcal O }({10}^{3}),\end{eqnarray} \tag{ 2.4 }$$

(where the hadronic scale m_π is the pion mass) which is just a reflection of the fact that very different energy scales cannot be resolved simultaneously on a single lattice of reasonable size ⁵⁰ . In addition, the continuum limit a → 0 requires a range of lattice sizes satisfying the above constraint, and one would like to have a range of scales μ such as to verify that the perturbative regime has been reached. This window problem enforces various compromises; in most cases the energy scales reached for perturbative matching to the $\overline{\mathrm{MS}}$-coupling is therefore rather low. As a consequence, even in the best cases (with 3-loop matching to the $\overline{\mathrm{MS}}$-coupling) systematic errors due to truncation of the perturbative series and/or contributions from nonperturbative effects are dominant [11, 13].

A solution to the window problem is however known since the 1990s, in the form of the step-scaling method [31]. The method is based on a finite volume renormalization scheme with μ = 1/L. It is then possible to recursively step up the energy scale by a fixed scale factor s = 2, and a scale difference of ${ \mathcal O }({10}^{2})$ is thus covered in 5 to 6 steps. The window problem is by-passed, as the approach uses multiple (pairs of) lattices with size L/a and 2L/a, covering a wide range of physical scales μ = 1/L without the need to represent simultaneously any hadronic scale, except at the lowest scale reached, ${\mu }_{\mathrm{had}}=1/{L}_{\max }$. Once the scale μ_had is matched to a hadronic quantity such as the proton mass, all the higher scales are known too, as the scale ratios are powers of two. At the high energy end, now orders of magnitude above the hadronic scales, perturbation theory can be safely applied to match to the $\overline{\mathrm{MS}}$-coupling, or, equivalently to extract the 3-flavour Λ-parameter. The method has been applied in [18], with the result ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(3)}=341(12)$ MeV, which translates to ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1185(8)$. It is important to realize that this is the only lattice determination of α_S where the error is still statistics dominated. For this reason, we quote this error for the FLAG average (2.1) as a conservative estimate of the uncertainty, instead of combining the (mostly systematic) errors in quadrature.

2.1.2. Future prospects and conclusions

The determination of the strong coupling using lattice QCD uses a nonperturbatively defined quantity O(q) that depends on a single short distance scale 1/q. Using the perturbative expression for this quantity ⁵²

$$\begin{eqnarray}&&O(q)\mathop{\sim }\limits^{q\to \infty }{\alpha }_{S}(q)+\displaystyle \sum _{n=2}^{N}{c}_{n}{\alpha }_{S}^{n}(q)+{ \mathcal O }\left({\alpha }_{S}^{N+1}(q)\right)+{ \mathcal O }{\left(\displaystyle \frac{{\rm{\Lambda }}}{q}\right)}^{p}+...,\end{eqnarray} \tag{ 2.5 }$$

where N is the number of known coefficients of the perturbative series, one can estimate the value of α_S (q). On the lattice, apart from the value of O(q), one needs to determine the value of the scale q in units of some well-measured hadronic quantity (e.g. the ratio q/m_p with m_p being the proton mass).

There are two types of corrections in equation (2.5). First, we have the nonperturbative (‘power’) corrections. They are of the form (Λ/q)^p . Second, we have the perturbative corrections. Their origin is the truncation of the perturbative series to a finite order N, and parametrically these corrections are of the form ${\alpha }_{S}^{N+1}(q)$.

In principle, both kind of uncertainties decrease by taking q → ∞ . Unfortunately, though, a single lattice simulation can only cover a limited range of energy scales (figure 2). Thus, if one insists on determining the hadronic scale (e.g. m_p ) and the value of the observable O(q) using the same lattice simulation, the volume has to be large, L ≳ 1/m_π , and therefore the energy scales q that can be reached are at most a few GeV. Power corrections decrease quickly with the energy scale q, but due to the logarithmic dependence of the strong coupling with the energy scale,

$$\begin{eqnarray}&&{\alpha }_{S}(q)\mathop{\sim }\limits^{q\to \infty }\displaystyle \frac{1}{\mathrm{log}(q/{{\rm{\Lambda }}}_{\mathrm{QCD}})},\end{eqnarray} \tag{ 2.6 }$$

perturbative uncertainties decrease very slowly. In fact, most lattice QCD extractions of the strong coupling are dominated by the truncation uncertainties of the perturbative series.

Zoom In Zoom Out Reset image size

What can be said about such perturbative uncertainties? First, it has to be noted that due to the asymptotic nature of perturbative expansions, it is in general very difficult to estimate the difference between the truncated series for O(q) and its nonperturbative value (see the original works [38, 39] and the review [13]). Second, an idea of the size of such uncertainties can be obtained by using the scale-variation method. If one assumes that at scales q power corrections are negligible, one can use an arbitrary renormalization scale in the perturbative expansion equation (2.5),

$$\begin{eqnarray}&&O(q)\mathop{\sim }\limits^{q\to \infty }{\alpha }_{S}(\mu )+\displaystyle \sum _{n=2}^{N}{c}_{n}(\mu /q){\alpha }_{S}^{n}(\mu )+{ \mathcal O }\left({\alpha }_{S}^{N+1}(\mu )\right)+....\end{eqnarray} \tag{ 2.7 }$$

The dependence on μ on the r.h.s. is spurious, and due to the truncation of the perturbative series. This dependence can be exploited to estimate the truncation effects: the value of α_S can be extracted for different choices of μ, and the differences among these extractions will give us an estimate of the effects due to missing higher-order pQCD corrections. Moreover, following [13], the estimate of the truncation uncertainties can be obtained from the known perturbative coefficients alone. No nonperturbative data for O(q) is needed to estimate these uncertainties.

In [13] a detailed analysis of several lattice methods to extract the strong coupling is performed along these lines. What are their conclusions? Most ‘large volume’ approaches (those that aim at computing the scale q in physical units and the observable O(q) using the same lattices) have perturbative uncertainties between about 1% and 3% in ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$.

It is important to emphasize that this generic approach cannot say what the errors of a specific determination are. However, given the fact that scale uncertainties can underestimate the true truncation errors (see [38] for a concrete example), this exercise draws a clear picture: a substantial reduction in the uncertainty of the strong coupling will only come from dedicated approaches, where the multiscale problem discussed above is solved. In this contribution, we will summarize the efforts of the ALPHA collaboration in solving this challenging problem (see [34, 40] for a review).

2.2.1. Finite-volume schemes

2.2.2. Step-scaling strategy

2.2.3.  α_S from a nonperturbative determination of ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}}=3)}$

2.2.4. How accurate is N_f = 3 QCD?

2.2.5. Effective theory of decoupling and perturbative matching

For the ease of presentation, we define our fundamental theory as N_f-flavour QCD (${\mathrm{QCD}}_{{N}_{{\rm{f}}}}$) with N_ℓ massless quarks, and N_h = N_f − N_ℓ mass-degenerate heavy quarks of mass M. In the limit where M is larger than any other scale in the problem, this theory can be approximated by an effective theory (EFT) with Lagrangian [43]

$$\begin{eqnarray}&&{{ \mathcal L }}_{\mathrm{dec}}={{ \mathcal L }}_{{\mathrm{QCD}}_{{N}_{{\ell }}}}+\displaystyle \frac{1}{{M}^{2}}\displaystyle \sum _{i}{\omega }_{i}{{\rm{\Phi }}}_{i}+{ \mathcal O }({M}^{-4}).\end{eqnarray} \tag{ 2.8 }$$

The leading order term in the 1/M expansion is the Lagrangian ${{ \mathcal L }}_{{\mathrm{QCD}}_{{N}_{{\ell }}}}$ of massless ${\mathrm{QCD}}_{{N}_{{\ell }}}$, while the effect of the heavy quarks is represented by nonrenormalizable interactions Φ_i suppressed by higher powers of 1/M. Massless ${\mathrm{QCD}}_{{N}_{{\ell }}}$ has a single parameter, the gauge coupling ${\bar{g}}^{({N}_{{\ell }})}$. The EFT is hence predictive once its coupling is given as a proper function of the coupling of the fundamental theory, ${\bar{g}}^{({N}_{{\rm{f}}})}$, and the quark masses M. In this situation, we say that the couplings are matched,

$$\begin{eqnarray}&&{\bar{g}}^{({N}_{{\ell }})}(\mu )={F}_{O}({\bar{g}}^{({N}_{{\rm{f}}})}(\mu ),M/\mu ).\end{eqnarray} \tag{ 2.9 }$$

The function F_O depends in principle on the observable O that is used to establish the matching. At leading order in the EFT, however, it is consistent to drop any correction of ${ \mathcal O }({M}^{-2})$ in the relation (2.9), which thus becomes universal, i.e. it only depends on the renormalization scheme chosen for the couplings. In the $\overline{\mathrm{MS}}$-scheme, the so-called decoupling relation (2.9) is known at 4-loop order [23–28], and it is usually evaluated at the scale μ = m_⋆, where ${m}_{\star }={\overline{m}}_{\overline{\mathrm{MS}}}({m}_{\star })$ with ${\overline{m}}_{\overline{\mathrm{MS}}}(\mu )$ the mass of the heavy quarks in the $\overline{\mathrm{MS}}$-scheme. In formulas,

$$\begin{eqnarray}&&{\bar{g}}_{\overline{\mathrm{MS}}}^{({N}_{{\ell }})}({m}_{\star })={g}_{\star }\,\xi ({g}_{\star }),\ \ {g}_{\star }\equiv {\bar{g}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}({m}_{\star }),\ \ \xi (g)=1+{c}_{2}{g}^{4}+{c}_{3}{g}^{6}+{c}_{4}{g}^{8}+{ \mathcal O }({g}^{10}).\end{eqnarray} \tag{ 2.10 }$$

The relation between the couplings can be reexpressed in terms of the corresponding Λ-parameters of the two theories,

$$\begin{eqnarray}&&{P}_{{\ell },{\rm{f}}}\left(M/{{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}\right)={{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\ell }})}/{{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}={\varphi }_{\overline{\mathrm{MS}}}^{({N}_{{\ell }})}({g}_{\star }\,\xi ({g}_{\star }))/{\varphi }_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}({g}_{\star }),\end{eqnarray} \tag{ 2.11 }$$

where the RGI-parameters are given by ${{\rm{\Lambda }}}_{{\rm{X}}}^{({N}_{{\rm{f}}})}=\mu \,{\varphi }_{{\rm{X}}}^{({N}_{{\rm{f}}})}({\bar{g}}_{{\rm{X}}}^{({N}_{{\rm{f}}})}(\mu ))$ and $M={\overline{m}}_{{\rm{X}}}(\mu ){\varepsilon }_{{\rm{X}}}^{({N}_{{\rm{f}}})}({\bar{g}}_{{\rm{X}}}^{({N}_{{\rm{f}}})}(\mu ))$. Explicit expressions for the functions ${\varphi }_{{\rm{X}}}^{({N}_{{\rm{f}}})}$ and ${\varepsilon }_{{\rm{X}}}^{({N}_{{\rm{f}}})}$ in terms of the β- and τ-functions in a generic scheme X can be found in e.g. [40].

In figure 3 we present the perturbative results from [22] for ${P}_{{\ell },{\rm{f}}}\left({M/{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}\right)$, for the phenomenological relevant cases of N_ℓ = 3, N_f = 4 and N_ℓ = 4, N_f = 5. More precisely, we show the relative deviation with respect to the 1-loop approximation, ${P}_{{\ell },{\rm{f}}}^{(1)}$, for different orders of the perturbative expansion of P_ℓ,f. Focusing on the case of P_3,4, we see how the perturbative corrections at 4- and 5-loop order are very small already for values of M comparable to that of the charm. Judging from the perturbative behavior alone, the series thus appears to be well within its regime of applicability. As a result, any estimate for the perturbative truncation errors on P_ℓ,f based on the last-known contributions to the series leads to very small uncertainties. When translated to the coupling we find, for instance, ${\alpha }_{\overline{\mathrm{MS}}}^{(5)}({m}_{{\rm{Z}}})=0.1185{(8)(3)}_{\mathrm{PT}\mathrm{dec}}$, where the second error is estimated as the sum of the full 4- and 5-loop contributions due to the decoupling of both charm and bottom quarks [18]. As anticipated, the perturbative error estimate is well-below the uncertainties from other sources.

Zoom In Zoom Out Reset image size

2.2.6. How perturbative are heavy quarks?

Above we have shown how, within perturbation theory, the perturbative decoupling of the charm appears to be very accurate. A natural question to ask at this point is how reliable this picture is at the nonperturbative level. In other words, can we quantify the accuracy of perturbative decoupling for the charm by comparing it directly to nonperturbative results, thus getting an estimate for the size of nonperturbative corrections? In order to answer this question, we start by formulating the matching of Λ-parameters (2.11) in nonperturbative terms [22, 44],

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\ell }})}}{{m}_{\mathrm{had},1}^{({N}_{{\ell }})}}={P}_{{\ell },{\rm{f}}}^{\mathrm{had},1}\left(M/{{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}\right)\displaystyle \frac{{{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}}{{m}_{\mathrm{had},1}^{({N}_{{\rm{f}}})}(M)}.\end{eqnarray} \tag{ 2.12 }$$

We thus say that the EFT is matched to the fundamental theory if its Λ-parameter ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\ell }})}\equiv {{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\ell }})}\left(M,{{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}\right)$ in units of a hadronic quantity ${m}_{\mathrm{had},1}^{({N}_{{\ell }})}$, is a proper function of the heavy quark masses M, and the Λ-parameter of the fundamental theory ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}$ in units of the same hadronic quantity ${m}_{\mathrm{had},1}^{({N}_{{\rm{f}}})}(M)$, computed in ${\mathrm{QCD}}_{{N}_{{\rm{f}}}}$ where N_h quarks are heavy with mass M ≫ Λ_QCD. ⁵⁸ Once the theories are matched, decoupling predicts that for any other hadronic quantity ${m}_{\mathrm{had},2}^{({N}_{{\ell }})}$ computed in the EFT, we should expect: ${m}_{\mathrm{had},2}^{({N}_{{\ell }})}={m}_{\mathrm{had},2}^{({N}_{{\rm{f}}})}(M)+{ \mathcal O }({M}^{-2})$ (see section 2.2.5). As anticipated by our notation, the function ${P}_{{\ell },{\rm{f}}}^{\mathrm{had}}$ depends on the hadronic quantity considered for the matching. If we were to choose a different quantity, we expect: ${P}_{{\ell },{\rm{f}}}^{\mathrm{had},1}\left({M/{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}\right)\sim {P}_{{\ell },{\rm{f}}}^{\mathrm{had},2}\left({M/{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}\right)+{ \mathcal O }({M}^{-2})$.

The relation (2.12) leads to the interesting factorization formula [22, 44],

$$\begin{eqnarray}&&\displaystyle \frac{{m}_{\mathrm{had}}^{({N}_{{\rm{f}}})}(M)}{{m}_{\mathrm{had}}^{({N}_{{\rm{f}}})}(0)}={Q}_{{\ell },{\rm{f}}}^{\mathrm{had}}\times {P}_{{\ell },{\rm{f}}}^{\mathrm{had}}\left(M/{{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}\right),\qquad {Q}_{{\ell },{\rm{f}}}^{\mathrm{had}}=\displaystyle \frac{{m}_{\mathrm{had}}^{({N}_{{\ell }})}/{{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\ell }})}}{{m}_{\mathrm{had}}^{({N}_{{\rm{f}}})}(0)/{{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}}.\end{eqnarray} \tag{ 2.13 }$$

On the l.h.s. of this equation we have the hadronic quantity ${m}_{\mathrm{had}}^{({N}_{{\rm{f}}})}(M)$ computed in ${\mathrm{QCD}}_{{N}_{{\rm{f}}}}$ where N_h quarks have mass M, over the same hadronic quantity ${m}_{\mathrm{had}}^{({N}_{{\rm{f}}})}(0)$ computed in the chiral limit, i.e. where all N_f quarks are massless. This ratio can be expressed as the product of a nonperturbative, M-independent factor ${Q}_{{\ell },{\rm{f}}}^{\mathrm{had}}$, and the function ${P}_{{\ell },{\rm{f}}}^{\mathrm{had}}\left({M/{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}\right)$, which encodes all the M-dependence. As we recalled in section 2.2.5, for M → ∞ the function ${P}_{{\ell },{\rm{f}}}^{\mathrm{had}}$ admits an asymptotic perturbative expansion. Hence, by measuring nonperturbatively on the lattice the l.h.s. of equation (2.13), we can compare the M-dependence of several such ratios with what is predicted by a perturbative approximation for ${P}_{{\ell },{\rm{f}}}^{\mathrm{had}}$. In this way, we can assess the typical size of nonperturbative effects in ${P}_{{\ell },{\rm{f}}}^{\mathrm{had}}$ as a function of M. In [22] a careful study was carried out considering the case of the decoupling of two heavy quarks with mass M ∼ M_c , for the case of N_ℓ = 0, N_f = 2. ⁵⁹ Under very reasonable assumptions, it is possible to extract from these results quantitative information on the nonperturbative corrections in the phenomenological interesting case of ${P}_{\mathrm{3,4}}\left({{M}_{c}/{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(4)}\right)$. The conclusions of [22] are that these effects are very much likely below the per-cent level. This means that it is safe to use a perturbative estimate for ${P}_{\mathrm{3,4}}\left({{M}_{c}/{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(4)}\right)$ in transitioning from ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(3)}\to {{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(4)}$, as long as, say, $\delta {{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(3)}\gtrsim 1.5 \%$ or so.

If we now consider ratios of hadronic quantities, where the dependence on the Λ-parameters drops out, we immediately conclude that

$$\begin{eqnarray}&&{m}_{\mathrm{had},1}^{({N}_{{\ell }})}/{m}_{\mathrm{had},2}^{({N}_{{\ell }})}={m}_{\mathrm{had},1}^{({N}_{{\rm{f}}})}(M)/{m}_{\mathrm{had},2}^{({N}_{{\rm{f}}})}(M)+{ \mathcal O }({M}^{-2}).\end{eqnarray} \tag{ 2.14 }$$

In this case, one can obtain estimates for the typically size of ${ \mathcal O }({M}^{-2})$ effects in these ratios by comparing both sides of the above equation computed through lattice simulations. In [45, 46] careful studies have been conducted considering both the case of the decoupling of two heavy quarks of mass M ∼ M_c with N_ℓ = 0, N_f = 2, and more recently for the decoupling of a single charm quark with N_ℓ = 3, N_f = 3 + 1, i.e. in the presence of three mass-degenerate lighter quarks. From these calculations, the authors conclude that the typical effects due to the charm on dimensionless ratios of low-energy quantities are well-below the per-cent level. This means that these effects are not relevant for a per-cent precision determinations of ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(3)}$.

In summary, thanks to recent dedicated studies, we are able to conclude that it is safe to rely on perturbative decoupling for the charm quark in connecting ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(3)}$ and ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(4)}$, as long as $\delta {{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(3)}\gtrsim 1.5 \%$. The 0.7% precision extraction of α_S from [18] is based on a determination of ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(3)}$ with an uncertainty of $\delta {{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(3)}\approx 3.5 \%$ (see section 2.2.3). Hence, there is still about a factor of two of possible improvement within the N_f = 3 flavour theory.

2.2.7. The strong coupling from the decoupling of heavy quarks

The previous section suggests a method to relate the Λ-parameters in theories with a different number of quark flavours (see equation (2.12)). Taking this relation to the extreme, one should be able to determine ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}$ from the pure-gauge one, ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(0)}$. Of course this requires to decouple N_f heavy quarks with $M\gg {{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}$. The physical world is very different from this limit, but lattice QCD can simulate such unphysical situation.

A possible strategy for the determination of the strong coupling based on this idea is the following:

• 1.  
  Starting from a scale μ_dec in ${\mathrm{QCD}}_{{N}_{{\rm{f}}}}$, one determines the value of a nonperturbatively defined coupling at such scale in a massless renormalization scheme, ${\bar{g}}_{O}^{({N}_{{\rm{f}}})}({\mu }_{\mathrm{dec}})$.
• 2.  
  By performing lattice simulations at increasing values of the quark masses, one is able to determine the value of the coupling in a massive scheme, ${\bar{g}}_{O}^{({N}_{{\rm{f}}})}({\mu }_{\mathrm{dec}},M)$.
• 3.  
  For M larger than any other scale in the problem (i.e. M ≫ Λ_QCD, μ_dec), the massive coupling is, up to heavy-mass corrections, the same as the corresponding coupling in the pure-gauge theory, i.e.

  $$\begin{eqnarray}&&{\bar{g}}_{O}^{({N}_{{\rm{f}}})}({\mu }_{\mathrm{dec}},M)={\bar{g}}_{O}^{(0)}({\mu }_{\mathrm{dec}}),\end{eqnarray} \tag{ 2.15 }$$

  where corrections of ${ \mathcal O }(1/{M}^{2})$ have been dropped.
• 4.  
  From the running of the coupling ${\bar{g}}_{O}^{(0)}({\mu }_{\mathrm{dec}})$ in the pure-gauge theory we can determine the pure-gauge Λ-parameter in units of μ_dec,

  $$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(0)}}{{\mu }_{\mathrm{dec}}}=\displaystyle \frac{{{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(0)}}{{{\rm{\Lambda }}}_{O}^{(0)}}\times {\varphi }_{O}^{(0)}({\bar{g}}_{O}^{(0)}({\mu }_{\mathrm{dec}})).\end{eqnarray} \tag{ 2.16 }$$

  (Note that the ratio of Λ-parameters in different schemes can be exactly determined via a perturbative, 1-loop computation (see e.g. [40]).)
• 5.  
  Since all N_f quarks are heavy, one can employ the decoupling relations to estimate the N_f ≥ 3 flavour Λ-parameter as (see equation (2.12)),

  $$\begin{eqnarray}&&\,{{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}={\mu }_{\mathrm{dec}}\times \displaystyle \frac{{{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(0)}}{{{\rm{\Lambda }}}_{O}^{(0)}}\times {\varphi }_{O}^{(0)}({\bar{g}}_{O}^{(0)}({\mu }_{\mathrm{dec}}))\times \displaystyle \frac{1}{{P}_{0,{N}_{{\rm{f}}}}^{(5-\mathrm{loop})}\left(M/{{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}\right)}+{ \mathcal O }({\alpha }_{S}^{4}({m}_{\star }))+{ \mathcal O }({M}^{-2}).\end{eqnarray} \tag{ 2.17 }$$

Equation (2.17) is our master formula to determine ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}}=3)}$ from precise results for the nonperturbative running of the gauge coupling in the pure-gauge theory (i.e. for the function ${\varphi }_{O}^{(0)}$). Several comments are in order at this point:

• There are two types of corrections in equation (2.17). First, ‘perturbative’ ones of ${ \mathcal O }({\alpha }_{S}^{4}({m}_{\star }))$. They come from using perturbation theory for the function ${P}_{0,{N}_{{\rm{f}}}}$. Second, we have nonperturbative ‘power’ corrections. Their origin is the decoupling condition, equation (2.15), as well as the use of a perturbative approximation for the function ${P}_{0,{N}_{{\rm{f}}}}$.
• Both perturbative and power corrections vanish for M → ∞. In fact, the following relation is exact
  $$\begin{eqnarray}&&{{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}=\mathop{\mathrm{lim}}\limits_{M\to \infty }{\mu }_{\mathrm{dec}}\times \displaystyle \frac{{{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(0)}}{{{\rm{\Lambda }}}_{O}^{(0)}}\times {\varphi }_{O}^{(0)}({\bar{g}}_{O}^{(0)}({\mu }_{\mathrm{dec}}))\times \displaystyle \frac{1}{{P}_{0,{N}_{{\rm{f}}}}^{(n-\mathrm{loop})}\left(M/{{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}\right)},\,\end{eqnarray} \tag{ 2.18 }$$
  with ${\bar{g}}_{O}^{(0)}({\mu }_{\mathrm{dec}})={\bar{g}}_{O}^{({N}_{{\rm{f}}})}({\mu }_{\mathrm{dec}},M)$.
• The situation and challenges in this approach might look similar to those present in more ‘conventional’ extractions of the strong coupling (see equation (2.5)). The subtle difference however is that, in the present case, perturbative corrections are very small even for scales ∼M_c (see section 2.2.5). In particular, if one is considering quarks with masses of a few GeV, they are completely negligible in practice, and one has only to deal with the power corrections, which decrease much faster with the relevant energy scale.

This method to extract the strong coupling was proposed in [33] (for a recent review see [34]). Here we present first results using this strategy [37]. ⁶⁰ We follow closely the strategy described above, skipping the technical details. The reader interested in the details is encouraged to look at the original references [33, 34, 37].

• 1.  
  A scale μ_dec = 789(15) MeV is determined in a finite-volume renormalization scheme in three-flavour QCD using results from [42]. This corresponds to a value of the renormalized nonperturbative coupling ${\bar{g}}^{(3)}({\mu }_{\mathrm{dec}})\approx 1.9872$.
• 2.  
  For technical reasons the massive coupling is determined in a slightly different renormalization scheme than the massless one. The value of the massive coupling is then determined by keeping the value of the bare coupling (and therefore the lattice spacing) fixed and increasing the value of the quark masses. This determination is performed for several values of the quark masses, z = M/μ_dec = 1.972, 4, 6, 8, 10, 12, and several values of the lattice spacing with 1/(a μ_dec) = 12, …, 48. The results are extrapolated to the continuum limit for each value of the quark masses labeled by z (see figure 4). We refer the reader to [37] for a detailed discussion.
• 3.  
  The values of the massive coupling are used as matching condition between ${\mathrm{QCD}}_{{N}_{{\rm{f}}}}$ and the pure-gauge theory (see equation (2.15)). The running in the pure-gauge theory is known very precisely from the literature [35]. The ratio ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(0)}/{\mu }_{\mathrm{dec}}$ can thus be accurately determined.
• 4.  
  Given this result, by applying the master formula equation (2.17), one obtains the estimates, ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}},\mathrm{eff}}^{(3)}$, for ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(3)}$ given in table 1. These estimates should approach the correct three-flavour Λ-parameter in the limit M → ∞. Figure 5 shows that this is indeed the case. Our data are consistent with a linear extrapolation in ${\mu }_{\mathrm{dec}}^{2}/{M}^{2}$, which results in

  $$\begin{eqnarray}&&{{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{(3)}=336(10)(6)(3)\mathrm{MeV},\end{eqnarray} \tag{ 2.19 }$$

  where the first uncertainty is statistical, the second is due to an estimate of the ${ \mathcal O }(a)$ effects present in our setup, and the last is due to different parameterizations of the M → ∞ limit.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Table 1. Values of the massive three-flavour coupling ${\bar{g}}^{(3)}({\mu }_{\mathrm{dec}},M)$ and the corresponding values of ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}},\mathrm{eff}}^{(3)}$ obtained by applying our master formula equation (2.17) and ignoring perturbative and power corrections.

z	${[{\bar{g}}^{(3)}({\mu }_{\mathrm{dec}},M)]}^{2}$	${{\rm{\Lambda }}}_{\overline{\mathrm{MS}},\mathrm{eff}}^{(3)}$ [MeV]
1.972	5.076(56)	426(14)
4	5.316(70)	388(13)
6	5.408(69)	363(12)
8	5.530(76)	351(12)
10	5.713(90)	349(12)
12	5.80(10)	343(12)

This result for the three-flavour Λ-parameter translates, after crossing the charm and bottom thresholds, into

$$\begin{eqnarray}&&{\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.11823(84).\end{eqnarray} \tag{ 2.20 }$$

The result has a 0.7% uncertainty, which is in fact dominated by the statistical uncertainties in the scale μ_dec = 789(15) MeV and in the running of the pure-gauge theory. These statistical uncertainties can be reduced with a modest computational effort. The effects of the truncation of the perturbative series are subdominant in our approach. A nonperturbative determination of the relevant improvement parameter would eliminate entirely the systematics related to ${ \mathcal O }(a)$ effects, and make possible a further reduction of the uncertainty by a factor of two with this approach (see [37] for a full discussion). Going beyond this precision would require including charm effects nonperturbatively and also some serious thinking on how to include electromagnetic effects in the determination of both the scale and the running.

A lattice method conceptually similar to the determination of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ or heavy-quark masses from the R-ratio via quarkonium sum rules [48, 49] uses heavy-quark two-point correlators; for a recent review see [50]. Renormalization group invariant pseudoscalar correlators G(t) and their nth time moments G_n are defined as

$$\begin{eqnarray}&&\,\begin{array}{l}{G}_{n}=\displaystyle \sum _{t}{t}^{n}(G(t)+G({N}_{t}-t)),\quad G(t)={a}^{6}\displaystyle \sum _{{\boldsymbol{x}}}{{{am}}_{h0}}^{2}\langle {j}_{5}({\boldsymbol{x}},t){j}_{5}(0,0)\rangle ,\\ \quad {j}_{5}({\boldsymbol{x}},t)=\bar{\psi }({\boldsymbol{x}},t){\gamma }_{5}\psi ({\boldsymbol{x}},t).\end{array}\end{eqnarray} \tag{ 2.21 }$$

The symmetrization accounts for (anti)periodic boundary conditions in time. Here, am_h0 is the corresponding bare heavy-quark mass in the lattice scheme; am_h0, resp. m_h , can be varied quite liberally for valence quarks in the charm- and bottom-quark regions and in between. Since sea quark mass variation is expensive, most results have partially quenched heavy quarks, i.e. heavy-quark masses in sea and valence sectors can be different in (2+1+1)-flavour QCD, or heavy quarks are left out from the sea in (2+1)-flavour QCD.

The weak-coupling series of G_n , which are finite for n ≥ 4, is known up to N³LO, resp. ${ \mathcal O }({\alpha }_{S}^{3}({\nu }_{h}))$, for N_f massless and one massive quark flavour [51–53],

$$\begin{eqnarray}&&\begin{array}{l}{G}_{n}=\tfrac{{g}_{n}({\alpha }_{S}(\nu ),\nu /{m}_{h})}{{{am}}_{h}^{n-4}({\nu }_{m})},\end{array}\end{eqnarray} \tag{ 2.22 }$$

where ν = xm_h , proportional to the heavy-quark mass m_h , is the $\overline{\mathrm{MS}}$ renormalization scale of the coupling; and ν_m = x_m m_h , the scale at which the $\overline{\mathrm{MS}}$ heavy-quark mass m_h (ν_m ) is defined, could be different from ν [48]. In published weak-coupling results, heavy-quark masses on internal (sea) and external (valence) quark lines must agree [51–53]. This restriction has profound implications for α_S extractions in (2+1+1)-flavour QCD.

As the bare heavy-quark mass in lattice units, am_h0, is a large parameter, improved quark actions are needed; so far most calculations have employed the highly improved staggered quarks (HISQ) [16, 17, 54–57], while domain-wall fermions have been used as well [58]. Some data sets involved values of am_h0 corresponding to different heavy-quark masses m_h [16, 17, 56, 57]; for this reason even (2+1+1)-flavour QCD lattice results still involve partially quenched heavy quarks [17]. Enforcing an upper bound am_h0 ≲ 1 to limit the size of lattice artifacts implies that fewer independent ensembles can constrain the data at larger m_h and entail larger errors of the respective continuum limit.

The so-called (tree-level) reduced moments, known perturbatively at ${ \mathcal O }({\alpha }_{S}^{3}({\nu }_{h}))$,

$$\begin{eqnarray}&&{R}_{4}=\left(\tfrac{{G}_{4}^{\mathrm{QCD}}}{{G}_{4}^{0}}\right),\qquad \,{R}_{4}\left({\alpha }_{S}(\nu ),\tfrac{\nu }{{m}_{h}}\right)=1+\displaystyle \sum _{j=1}^{3}{r}_{{nj}}\left(\tfrac{\nu }{{m}_{h}}\right){\left(\tfrac{{\alpha }_{S}(\nu )}{\pi }\right)}^{j}\,\end{eqnarray} \tag{ 2.23 }$$

$$\begin{eqnarray}&&{R}_{n}={\left(\tfrac{{G}_{n}^{\mathrm{QCD}}}{{G}_{n}^{0}}\right)}^{\tfrac{1}{(n-4)}},\qquad {R}_{n}\left({\alpha }_{S}(\nu ),\tfrac{\nu }{{m}_{h}}\right)=\left(\tfrac{{m}_{h0}}{{m}_{h}({\nu }_{m})}\right)\left(1+\displaystyle \sum _{j=1}^{3}{r}_{{nj}}\left(\tfrac{\nu }{{m}_{h}}\right){\left(\tfrac{{\alpha }_{S}(\nu )}{\pi }\right)}^{j}\right),\,n\gt 4,\end{eqnarray} \tag{ 2.24 }$$

eliminate the tree-level contribution, thus cancelling the leading lattice artifacts [54]. The coefficients r_nj are numbers of order 1 without any obvious pattern, see, e.g. table 1 of [50]. On the one hand, the lowest reduced moment R₄ and ratios of higher reduced moments, i.e. R₆/R₈ or R₈/R₁₀, are dimensionless; their respective continuum extrapolations turned out to be challenging, in particular due to $\mathrm{log}(a{{\rm{\Lambda }}}_{\mathrm{QCD}})$ and $\mathrm{log}({{am}}_{h0})$ dependence [57]. Such $\mathrm{log}(a)$ dependence is manifest in slopes that decrease for larger am_h0. On the other hand, higher moments R_n /m_h0, n ≥ 6, are dimensionful and scale with 1/m_h0 = a/(am_h0); thus, because the scale uncertainty and the uncertainty of the tuned bare heavy-quark mass (am_h0) strongly impact the results, these are insensitive to any $\mathrm{log}(a)$ effects, and continuum extrapolations are straightforward. For the improved actions used in published results, lattice spacing dependence could be parameterized for functions of the reduced moments, i.e. R(m_h ) = {R₄(m_h ), R_n (m_h )/m_h0, R_n (m_h )/R_n+2(m_h ), }, n ≥ 6, as

$$\begin{eqnarray}&&\begin{array}{l}R({m}_{h})={R}^{\mathrm{cont}}({m}_{h})+\displaystyle \sum _{i=1}^{N}\displaystyle \sum _{j=1}^{{M}_{i}}{b}_{{ij}}^{(R)}{\left({\alpha }_{S}^{b}\right)}^{i}\left[1+\displaystyle \sum _{k=1}^{i}{d}_{{ijk}}^{(R)}{\mathrm{ln}}^{k}({{am}}_{h0})\right]{({{am}}_{h0})}^{2j},\end{array}\end{eqnarray} \tag{ 2.25 }$$

where ${\alpha }_{S}^{b}={g}_{0}^{2}/(4\pi {u}_{0}^{4})$ is the boosted lattice coupling [59]; the tadpole factor u₀ is defined in terms of the plaquette, ${u}_{0}^{4}=\langle \mathrm{Tr}\,{U}_{\mu \nu }{N}_{c}\rangle$, and partially accounts for the expected $\mathrm{log}(a{{\rm{\Lambda }}}_{\mathrm{QCD}})$ dependence.

For R₄ and R₆/R₈, separate continuum extrapolations for each heavy-quark mass proved feasible only for m_h ≤ 1.5m_c [56]. Continuum extrapolation for m_h ≥ 2m_c required joint fits including m_h < 2m_c [57]; similar joint fits with Bayesian priors were used in [16, 17]. The published continuum results for R₄ and R₆/R₈ at m_h = m_c or 1.5m_c are consistent among each other [50, 57]; any significant deviations can be traced back to known deficiencies in the respective analyses [55, 58], see e.g. table 65 of [11]. For R_n /m_h0, n ≥ 6 fits with N = 1, M₁ = 2 are sufficient for any m_h , and published results for R_n (m_c )/m_c0, n ≥ 6 are consistent. Severe finite volume effects affect R₈/R₁₀; R₈/R₁₀ at m_c is systematically low (and inconsistent with R₄), while continuum extrapolation with joint fits proved reliable for m_h ≥ 1.5m_c [57]. With the aforementioned exceptions the continuum results in (2+1)-flavour QCD span the region m_h = m_c , …, 4m_c for valence quarks, see tables 1 and 4 of [57]; with increasing heavy-quark mass m_h , R₄ and the ratios decrease towards 1, while the errors of the lattice calculations increase. Continuum results in (2+1+1)-flavour QCD have not been published.

Comparing R₄(m_h ) to R₄(α_S (ν), ν/m_h ) permits extraction of α_S (ν); truncation errors, estimated via inclusion of terms $\pm 5{r}_{n3}\,{\alpha }_{S}^{4}(\nu )$, turn out to be too large for the ratios to provide more than a cross-check [57]. Once α_S (ν) is given, R_n /m_h0 permits obtaining the $\overline{\mathrm{MS}}$ heavy-quark mass m_h (ν_m ); combining both yields ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}$. Whether or not the two scales ν_h = xm_h and ν_m = x_m m_h , should be varied jointly (x = x_m ) or independently (x ≠ x_m ) is under investigation; the latter has yielded in a reanalysis of published lattice results at m_h = m_c about 50% larger estimates of truncation errors [49]. α_S (ν) or m_h (ν_m ) for different am_h0 = x_lat am_c0 are consistent with perturbative running [57]. Nonperturbative contributions—parametrized in terms of QCD condensates—are negligible for m_h ≥ 1.5m_c due to suppression by powers of at least $1/{m}_{h}^{4};$ similarly, truncation errors diminish dramatically for m_h ≥ 1.5m_c [57].

A recent analysis has exposed that the choice of the lattice scale ratio, i.e. x_lat = m_h /m_c = (am_h0)/(am_c0), and the perturbative scale ratio, x_pert = ν/m_h , both have a significant and systematic impact on the extracted ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}^{({N}_{{\rm{f}}})}$, and consequently on ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ [57], while the composition of the error budget is very different (table 2). Neglecting the spread due to varying either of these two scale ratios led in most past determinations to significantly underestimated errors [16, 17, 54–56]. Nonetheless, the central value of [16] is in good agreement with the corresponding entry of table 2.

The current FLAG sub-average—taking the latest results [57] partially into account, and using error estimates from independent scale variation [49]—reports

$$\begin{eqnarray}&&\begin{array}{l}\,{\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.11826(200)\,\ {\rm{(FLAG\; sub-average\; for\; heavy-quark\; two-point\; correlators)}}\,[11].\end{array}\end{eqnarray} \tag{ 2.26 }$$

Table 2 suggests that a viable approach on the lattice side to reducing the errors in the next decade may be by performing more accurate lattice calculations using masses m_h ≥ 2m_c , where the truncation errors are subleading in current results. The corresponding continuum extrapolations could be made more robust in two ways. First, by including more intermediate heavy-quark mass values (e.g. m_h /m_c = 1.75, 2.25, 2.5, 2.75, 3.25, etc) in the joint fits, equation (2.25), one may hope to significantly reduce the lattice errors of the continuum results for m_h ≥ 2m_c . Second, by relying on one-loop instead of tree-level reduced moments at finite lattice spacing as suggested in [60], one may simplify the approach to the continuum limit; while cumbersome, these calculations in lattice perturbation theory are in principle straightforward and are expected to eliminate all i = 1 terms from the series corresponding to equation (2.25). The availability of lattice results in (2+1+1)-flavour QCD with partially quenched heavy quarks suggests that a viable approach on the perturbative side to reducing the errors may be to permit partially quenched heavy quarks, i.e. different heavy-quark masses on internal (sea) and external (valence) lines. In particular, application of this method in (2+1+1)-flavour QCD permits no α_S extraction for valence quarks at m_h > m_c , the only readily available strategy to alleviate the truncation error, on the basis of the currently available weak-coupling calculations that require m_h,sea = m_h,val. If this deficiency were remedied, the constraining power could be improved in a joint analysis of partially quenched lattice calculations with different m_h in (2+1+1)-flavour QCD similar to the recent analysis in (2+1)-flavour QCD [57], or by even combining (2+1)- and (2+1+1)-flavour QCD continuum results in a joint analysis that assumes perturbative decoupling of the heavy quark. Last but not least, the expected accuracy would obviously benefit from N⁴LO, resp. ${ \mathcal O }({\alpha }_{S}^{4}({\nu }_{h}))$, calculations.

Acknowledgments—PP was supported by US Department of Energy under Contract No. DE-SC0012704. JHW’s research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)–Projektnummer 417 533 893/GRK2575 ‘Rethinking Quantum Field Theory’.

QCD observables that have been computed with high precision in perturbative- and lattice-QCD with 2 + 1 or 2 + 1 + 1 flavours are well suited to provide determinations of α_S in the kinematic regions where pQCD applies. The advantage of looking at observables is that continuum analytical and lattice results may be compared without having to deal with renormalization issues and change of scheme. Moreover, if in the kinematic regions where pQCD is used the perturbative series converges well, and nonperturbative corrections turn out to be small, and if lattice data can be extrapolated to continuum, then a very precise extraction of α_S is possible even by comparing few lattice data with the perturbative expression. The comparison provides ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}$ times the lattice scale. By supplying a precise determination of the lattice scale, one obtains ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}$. Finally, ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}$ may be traded with α_S conventionally expressed at the mass of the Z, ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$.

2.4.1. The QCD static energy

The QCD static energy E₀(r), i.e. the energy between a static quark and a static antiquark separated by a distance r, is one of these golden observables for the extraction of α_S and it is also a basic object to study the strong interactions [61]. The QCD static energy, E₀(r), is defined (in Minkowski spacetime) as

$$\begin{eqnarray}&&{E}_{0}(r)=\mathop{\mathrm{lim}}\limits_{T\to \infty }\displaystyle \frac{i}{T}\mathrm{ln}\left\langle \mathrm{Tr}\,{\rm{P}}\,\exp \left\{{ig}{\oint }_{r\times T}{{dz}}^{\mu }\,{A}_{\mu }\right\}\right\rangle \equiv \mathop{\mathrm{lim}}\limits_{T\to \infty }\displaystyle \frac{i}{T}\mathrm{ln}\left\langle {W}_{r\times T}\right\rangle ,\end{eqnarray} \tag{ 2.27 }$$

where the integral is over a rectangle of spatial length r, the distance between the static quark and the static antiquark, and time length T; 〈...〉 stands for the path integral over the gauge fields A_μ and the light quark fields, P is the path-ordering operator of the colour matrices and g is the SU(3) gauge coupling (α_S = g²/(4π)); W_r×T is the static Wilson loop. The above definition of E₀(r) is valid at any distance r.

On the lattice side, the Wilson loop is one of the most accurately known quantities that has been computed since the establishment of lattice QCD. In the short distance range, rΛ_QCD ≪ 1 for which α_S (1/r) ≪ 1, E₀(r) may be computed in pQCD and expressed as a series in α_S (computed at a typical scale of order 1/r):

$$\begin{eqnarray}&&{E}_{0}(r)={{\rm{\Lambda }}}_{s}-\displaystyle \frac{4{\alpha }_{S}}{3r}(1+...),\end{eqnarray} \tag{ 2.28 }$$

where Λ_s is a constant that accounts for the normalization of the static energy and the dots stand for higher-order terms. The expansion of E₀(r) in powers of α_S has been computed, in the continuum in the $\overline{\mathrm{MS}}$ scheme, using perturbative and effective field theory techniques, in particular potential Nonrelativistic QCD [62]. It is currently known at next-to-next-to-next-to-leading-logarithmic (N³LL) accuracy, i.e. including terms up to order ${\alpha }_{S}^{4+n}{\mathrm{ln}}^{n}{\alpha }_{S}$ with n ≥ 0. At three loops, a contribution proportional to $\mathrm{ln}{\alpha }_{S}$ appears for the first time. This three-loop logarithm has been computed in [63]. The complete three-loop contribution has been computed in [64, 65]. The leading logarithms have been resummed to all orders in [66], providing, among others, also the four-loop contribution proportional to ${\alpha }_{S}^{5}{\mathrm{ln}}^{2}{\alpha }_{S}$. The four-loop contribution proportional to ${\alpha }_{S}^{5}\mathrm{ln}{\alpha }_{S}$ has been computed in [67]. Next-to-leading logarithms have been resummed to all orders in [68]. However the constant coefficient of the ${\alpha }_{S}^{5}$ term is not yet known. E₀(r) is, therefore, one of the best known quantities in pQCD lending to a perfect playground for the extraction of α_S .

The $\mathrm{ln}{\alpha }_{S}$ terms in E₀(r) signal the cancellation of contributions coming from the soft energy scale 1/r and the ultrasoft (US) energy scale α_S /r. The ultrasoft scale is generated in loop diagrams by the emission of virtual ultrasoft gluons changing the colour state of the quark-antiquark pair [63, 69]. The resummation of these logarithms accounts for the running from the soft to the ultrasoft scale.

The soft and ultrasoft energy scales may be effectively factorized in potential Nonrelativistic QCD [62, 70]. The factorization of scales leads to the formula [62, 63]:

$$\begin{eqnarray}&&\begin{array}{l}{E}_{0}(r)={{\rm{\Lambda }}}_{s}+{V}_{s}(r,\mu )-i\displaystyle \frac{{V}_{A}^{2}}{3}{\displaystyle \int }_{0}^{\infty }{dt}\,{e}^{-{it}({V}_{o}-{V}_{s})}\langle \mathrm{Tr}\,\{{\bf{r}}\cdot g{\bf{E}}(t,0)\,{\bf{r}}\cdot g{\bf{E}}(0,0)\}\rangle (\mu )+...,\end{array}\end{eqnarray} \tag{ 2.29 }$$

where μ is the US renormalization scale, V_s (r, μ) = −4α_S /(3r) + ... is the colour-singlet static potential, V_o (r, μ) = α_S /(6r) + ... is the colour-octet static potential, ${V}_{A}=1+{ \mathcal O }({\alpha }_{S}^{2})$ is a Wilson coefficient giving the chromoelectric dipole coupling, and E is the chromoelectric field. The colour-singlet and colour-octet static potentials encode the contributions from the soft scale 1/r, whereas the low-energy contributions are in the term proportional to the two chromoelectric dipoles. While at short distances, rΛ_QCD ≪ 1, the potentials V_s and V_o may be computed in perturbation theory, low-energy contributions include nonperturbative contributions.

The perturbative expansion of V_s is affected by a renormalon ambiguity of order Λ_QCD. The ambiguity reflects in the poor convergence of the perturbative series. A first method to cure the poor convergence of the perturbative series of V_s consists in subtracting a (constant) series in α_S from V_s and reabsorb it into a redefinition of the normalization constant Λ_s . This is the strategy we followed, for instance, in [71]. A second possibility consists in considering the force

$$\begin{eqnarray}&&F(r,\nu )=\displaystyle \frac{d}{{dr}}{E}_{0}(r).\end{eqnarray} \tag{ 2.30 }$$

It does not depend on Λ_s and is free from the renormalon of order Λ_QCD [72, 73]. Once integrated upon the distance, the force gives back the static energy

$$\begin{eqnarray}&&{E}_{0}(r)={\int }_{{r}_{* }}^{r}{dr}^{\prime} \,F(r^{\prime} ,1/r^{\prime} ),\end{eqnarray} \tag{ 2.31 }$$

up to an irrelevant constant determined by the arbitrary distance r*, which can be reabsorbed in the overall normalization when comparing with lattice data. This is the strategy followed, for instance, in [19]. We note that equation (2.29) provides also the explicit form of the nonperturbative contributions encoded in the chromoelectric correlator. They are proportional to r³ at very short distances and to r² at somewhat larger distances.

In summary, E₀(r) is one of the best known quantities in pQCD lending an ideal observable for the extraction of α_S by comparing lattice data and perturbative calculation in the appropriate short distance window. This way of extracting of α_S has been developed in a series of papers [71, 74, 75]. Here we report about our best determination from [19]. The method provides one of the most precise low energy determinations of α_S . The strong coupling constant extracted in this way relies typically on low energy data because the lattice cannot explore too small distances. It therefore provides a precise check of the running of the coupling constant and a determination of it that is complementary to high-energy determinations.

Concerning the power counting of the perturbative series a remark is in order. Upon inspection of the numerical size of the contributions coming from the soft and the US scale at each order, in the analysis of [19] it was decided to count the leading US resummed terms along with the three loop terms, since the ${\alpha }_{S}^{4}\,\mathrm{log}\,{\alpha }_{S}$ terms appear to be of the same size as the ${\alpha }_{S}^{4}$ terms, and, moreover, to partially cancel each other. It was also decided not to include subleading US logarithms in the analysis, as the finite four loop contribution is unknown and a cancellation similar to the one happening at three loops may also happen at four loops. Nevertheless, it may be also legitimate to count leading US resummed terms as if they were parametrically of order ${\alpha }_{S}^{3}$ and count subleading US logarithms of order ${\alpha }_{S}^{5}\,\mathrm{log}\,{\alpha }_{S}$ as if they were parametrically of order ${\alpha }_{S}^{4}$, including them in the analysis. This is the procedure adopted in [20]. Most of the difference between the central value of α_S obtained in the analysis of [19] and in the one of [20] is due to this different counting of the perturbative series. The two analyses are consistent once errors, in particular those due to the truncation of the perturbative series, are accounted for.

The static energy can be computed on the lattice as the ground state of Wilson loops or temporal Wilson line correlators in a suitable gauge, typically in Coulomb gauge. Polyakov loops at sufficiently low temperatures could be employed as well. All energy levels from any of these correlators are affected by a constant, lattice spacing dependent self-energy contribution that diverges in the continuum limit. It can be removed by matching the static energy at each finite lattice spacing to a finite value at some distance. In calculations with an improved action, all these correlators, which are obtained from spatially extended operators, are affected by nonpositive contributions at very small distance and time, which cannot be resolved on coarse lattices or with insufficient suppression of the lowest excited states. Although Wilson line correlators retain an advantage in terms of the excited state suppression, the relative disadvantage of Wilson loops could be alleviated to some extent with smeared spatial links.

The ground state energy E₀( r , a) can be extracted from such correlators, e.g. via multi-exponential fits, in the large Euclidean time region for each lattice three-vector r /a = (n₁, n₂, n₃), where 0 ≤ n_i ≤ L_i /2a. Obviously, small lattice spacing a is indispensable in order to access small distances ∣ r ∣ ≲ 0.15 fm [75]. Yet simulations with periodic boundary conditions fail to sample the different topological sectors of the QCD vacuum properly at small a. This topological freezing, which is known to be a quantitatively small but significant problem in low-energy hadron physics, does not lead in high-energy quantities (such as E₀( r , a) at small distances) to statistically significant effects due to small changes of the topological charge [76]. Although significant effects in E₀( r , a) due to large changes of the topological charge cannot be completely ruled out, they seem very unlikely, given that the topology does not contribute at all in the weak-coupling calculations used in the comparison. Furthermore, sea quark mass effects due to light or strange quarks do not play a role in this range [76, 77], while dynamical charm effects are significant [78, 79]. At distance larger than ∣ r ∣ ≳ 0.15 fm light sea quark mass effects become nonnegligible [76, 77].

Continuum extrapolation is only possible for somewhat larger distances probed by multiple lattice spacings, or if the functional form of E₀( r , a) were known to sufficient accuracy to predict the shape at small r /a on the fine lattices. Due to the breaking of the continuous O(3)-symmetry group to the discrete W₃-symmetry group the lattice gluon propagator, and hence E₀( r , a), is a non-smooth function of ∣ r ∣/a; geometrically inequivalent combinations of the n_i , i.e. belonging to different representations of W₃ but corresponding to the same geometric distance $| {\boldsymbol{r}}| /a=\sqrt{{n}_{1}^{2}+{n}_{2}^{2}+{n}_{3}^{2}}$, e.g. (3, 0, 0) or (2, 2, 1), yield inconsistent E₀(∣ r ∣, a) due to lattice artifacts. Moreover, the same ∣ r ∣ accessed through different lattice spacings a is generally affected by different types of non-smooth lattice artifacts corresponding to the different underlying r /a. For this reason, a continuum extrapolation of E₀(∣ r ∣, a) at fixed ∣ r ∣ utilizing a parametrization of lattice artifacts in terms of a smooth function in ∣ r ∣ is incapable of describing the small ∣ r ∣/a region, see e.g. [50]. These inconsistencies are much larger than the statistical errors at small ∣ r ∣/a, but covered by the statistical errors at large ∣ r ∣/a. A tree-level correction (TLC) procedure defines the (tree-level) improved distance r_I /a = f(n₁, n₂, n₃) and alleviates these inconsistencies somewhat: at ∣ r ∣/a ≳ 3 this is sufficient, while further effort is needed at smaller ∣ r ∣/a. A nonperturbative correction (NPC) procedure heuristically estimates the lattice artifacts remaining in E₀(r_I , a) by comparing to a suitable smooth function, either obtained at a finer lattice spacing, or in a continuum calculation, which, however, potentially introduces systematic errors. Both approaches have been used yielding consistent results; for details see [19].

For Wilson line correlators, all combinations of 0 ≤ n₁, n₂, n₃ ≤ L/2a are accessible, thus permitting access to noninteger distances in units of the lattice spacing; this entails all of the aforementioned complications. After the nonperturbative correction, ${E}_{0}^{\mathrm{NPC}}({r}_{I},a)$ from Wilson line correlators computed in (2+1)-flavour QCD on lattices with highly improved staggered quarks (HISQ) was shown to exhibit no statistically significant lattice artifacts anymore [19] (figure 6). With this data set the range of interest can be probed using a rather large number of r_I /a values and many lattice spacings (up to six spacings in the range a ∈ (0.024, …, 0.06)fm), where the underlying high statistics ensembles had been generated for studies of the QCD equation of state [77, 82]. The uncertainty due to the lattice scale, lattice spacing dependence, estimates of the uncertainty due to treating residual lattice artifacts with the nonperturbative correction, or due to changes of the fit range are within the the statistical error and subleading in the error budget. Instead it was found that estimates of the continuum perturbative truncation error dominate the error budget. In [19] these have been estimated by a scale variation between 2/r and 1/2r, inclusion of a parametric estimate of a higher order term $\pm 4/3{\alpha }_{S}^{5}/r$, and variation between resummation or no resummation of the leading ultrasoft logarithms ${\alpha }_{S}^{3+n}{\mathrm{ln}}^{n}{\alpha }_{S}/r$. The scale dependence becomes nonmonotonic below $1/\sqrt{2}r$ at large r, which makes robust error estimates challenging unless the range is restricted to $\max (| {\boldsymbol{r}}| )\leqslant 0.1\,\mathrm{fm}$. As lattice data at larger ∣ r ∣ are discarded, the statistical error increases while the truncation error decreases. Eventually, for $\max (| {\boldsymbol{r}}| )\ll 0.1\,\mathrm{fm}$, the nonperturbative correction to the lattice data becomes essential to having enough data, while the central value hardly changes. For our joint fit using nonperturbatively corrected HISQ data at five lattice spacings we report the best compromise between the different contributions to the error budget as

$$\begin{eqnarray}&&{\alpha }_{S}({m}_{{\rm{Z}}}^{2})={0.11660}_{-0.00056}^{+0.00110},\end{eqnarray} \tag{ 2.32 }$$

where the total, symmetric lattice error amounts to 0.00047 for ∣ r ∣ ≤ 0.073 fm.

Zoom In Zoom Out Reset image size

In [20], a reanalysis of a subset of these data (a = 0.024 fm, $\sqrt{8}\leqslant | {\boldsymbol{r}}| /a\leqslant 4$) was carried out that included resummation of next-to-leading ultrasoft logarithms ${\alpha }_{S}^{4+n}{\mathrm{ln}}^{n}{\alpha }_{S}/r$, i.e. the full N³LL accuracy, and hyperasymptotic expansion, resulting in ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1181(9)$. Concerning the central value, we have already commented that it differs from (2.32) mostly because of the inclusion of the subleading ultrasoft logarithms. Concerning the error budget, it may possibly increase by including an estimate of the lattice spacing dependence and a variation of $\min (| {\boldsymbol{r}}| )$ or $\max (| {\boldsymbol{r}}| )$ to smaller values. Nevertheless, even inside the quoted errors the result is consistent with (2.32).

For Wilson loops, spatial Wilson lines connecting the temporal ones entail additional, r /a-dependent self-energy divergences. The static energy from Wilson loops is usually computed only for few specific geometries, and spatial link smearing is applied to suppress these divergences. The static energy has been obtained from Wilson loops for two different geometries r /a ∝ (1, 0, 0) or (1, 1, 0) [80, 81] in (2+1)-flavour QCD on lattices with Möbius domain-wall fermions with a pion mass of m_π ≈ 300 MeV and three lattice spacings a ∈ (0.04, …, 0.08)fm that had been generated by the JLQCD collaboration [83]. Two separate analyses were performed: a two-step analysis (I) with a continuum extrapolation at large distances sequentially followed by the α_S extraction, and another one-step analysis (II) using a single joint fit to achieve both at once. Both analyses relied on a particular form of operator product expansion, and used data in the range 0.24 fm ≤ ∣ r ∣ ≤ 0.6 fm or 0.044 fm ≤ ∣ r ∣ ≤ 0.36 fm, respectively. The reported values from the two analyses are ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})={0.1166}_{-0.0020}^{+0.0021}$ and ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})={0.1179}_{-0.0014}^{+0.0015}$. They are both dominated by the estimate of the truncation errors. As both analyses extend far into ranges where E₀(∣ r ∣, a) is known to be sensitive to the pion mass [76], the authors had to include condensate terms, while reporting no significant mass dependence when assuming only a perturbative contribution from massive light quarks. The first analysis has no data for ∣ r ∣ ≲ 0.10 fm. The second analysis does have data in that region, but has to fit ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}$ simultaneously with the lattice artifacts at small r /a. As a result the continuum extrapolated static energy from this analysis has large errors for ∣ r ∣ ≲ 0.10 fm and lies systematically below the HISQ result from [19] (figure 6).

2.4.2. Static force

Another possibility consists of computing the force directly from the lattice, i.e. not as the slope of the static energy. The force, F, between a static quark located in r and a static antiquark located in 0 can be defined as [84]

$$\begin{eqnarray}&&\begin{array}{l}F(r)={\partial }_{r}{E}_{0}(r)=\mathop{\mathrm{lim}}\limits_{T\to \infty }-i\displaystyle \frac{\left\langle \mathrm{Tr}\left\{{W}_{r\times T}\,\hat{{\bf{r}}}\cdot g{\bf{E}}({\bf{r}},t* )\right\}\right\rangle }{\langle \mathrm{Tr}\{{W}_{r\times T}\}\rangle }.\end{array}\end{eqnarray} \tag{ 2.33 }$$

The chromoelectric field E(r, t*) is located at the quark line of the Wilson loop.

This definition permits to obtain the force directly from the lattice instead of reconstructing it, after interpolation, from the lattice data of the static energy. The perturbative calculation of the force in continuum QCD is free of the leading renormalon and it is known at N³LL. Therefore the force provides a clean way to extract α_S . In [85], we have used the multilevel algorithm to perform a preliminary quenched lattice study of the chromoelectric insertion in a static Wilson loop given by equation (2.33) both with smeared Wilson loops and with Polyakov loops. The result is consistent with the force obtained via derivative of the static energy upon multiplication with a constant renormalization factor Z_E that encodes the very slow convergence of lattice operators containing gluonic operators. Recently in [86] we have performed the same calculation with gradient flow, which eliminates the necessity of Z_E and makes the lattice calculation more efficient. We plan to go to very fine lattice spacings and perform an extraction of ${{\rm{\Lambda }}}_{\overline{\mathrm{MS}}}{r}_{0}$.

2.4.3. Static singlet free energy

One reason for which it is challenging to reach the fine lattice spacings needed for the best extraction of α_S is that one has to simultaneously maintain the control over finite volume effects from the propagation of the lightest hadronic modes, namely, the pions. A lattice simulation at high enough temperature avoids this infrared problem, and thus enables reaching much finer lattice spacings using smaller volumes. In [19], we considered the extraction of the strong coupling from the singlet free energy at nonzero temperature, as it is expected that at small distances medium effects are small. The singlet free energy in terms of the correlation function of two thermal Wilson lines in Coulomb gauge is given by (T is now the temperature)

$$\begin{eqnarray}&&{F}_{S}(r,T)=-T\mathrm{ln}\left(\displaystyle \frac{1}{{N}_{c}}\langle \mathrm{Tr}[W(r){W}^{\dagger }(0)]\rangle \right).\end{eqnarray} \tag{ 2.34 }$$

At distances much smaller than the inverse temperature, rT ≪ 1, we can write using potential nonrelativistic QCD [87]

$$\begin{eqnarray}&&{F}_{S}(r,T)={V}_{s}(r,\mu )+\delta {F}_{S}(r,T,\mu ).\end{eqnarray} \tag{ 2.35 }$$

The form of the US correction depends on the scale hierarchy that is now featuring also the temperature and the Debye mass m_D ∼ gT: if 1/r ≫ α_S /r ≫ T ≫ m_D then μ ∼ α_S /r and δ F_S (r, T, μ) = δ E_US (μ) + ΔF_S (r, T), with δ E_US (μ) being the US contribution to E₀ in vacuum that we already discussed. The singlet free energy has been studied on the lattice in [88] using a wide temperature range and several lattice spacings, i.e. several temporal extents N_τ . The shortest distance that we can access, due to a single lattice spacing on our finest lattice at T > 0, is 0.00814 fm. For our analysis the relevant data correspond to N_τ = 10, 12 and 16, since rT has to be small. From the analysis of Ref. [88], we see that thermal effects are small for rT ≲ 0.3. In particular we see that for $| {\boldsymbol{r}}| /a\leqslant \sqrt{6}$ the difference between the singlet free energy at (T > 0) and the static energy at (T = 0) approaches a constant proportional to the temperature. No temperature effects beyond a constant can be seen in this range within the errors of the lattice results. Therefore we treat the finite temperature data in this range as if they were at zero temperature and fit them with the three-loop plus leading ultrasoft resummed result of the static energy at T = 0.

A drawback of this analysis is the restriction to very small ∣ r ∣/a, which implies that the nonperturbative correction (NPC) is indispensable. Yet the main advantage of the analysis is that at very small distances, $\max (| {\boldsymbol{r}}| )\leqslant 0.03\,\mathrm{fm}$, the truncation error becomes very small. Beyond what has been necessary at zero temperature, one must verify that T > 0 effects do not become significant when including larger ∣ r ∣/a or when varying the temperature T = 1/aN_τ at fixed lattice spacing. We confirmed the statistical irrelevance in these two measures of T > 0 effects by comparing ∣ r ∣/a ≤ 2 with ∣ r ∣/a ≤ 3 and N_τ = 12, 16, and, where possible, 64, and included these error estimates with the other lattice errors in the error budget. The key ingredient to making this extraction work with substantial constraining power is that a wide range in ∣ r ∣ can be covered using only narrow windows in ∣ r ∣/a by performing a joint analysis using multiple lattice spacings a that cover ∣ r ∣ in multiple, overlapping segments. For our joint fit using nonperturbatively corrected T > 0 HISQ data at eight lattice spacings and an ∣ r ∣ region and set of lattice spacings nonoverlapping with the zero temperature analysis we report

$$\begin{eqnarray}&&{\alpha }_{S}({m}_{{\rm{Z}}}^{2})={0.11638}_{-0.00087}^{+0.00095}\end{eqnarray} \tag{ 2.36 }$$

(the total, symmetric lattice error amounts to 0.000 85) for ∣ r ∣ ≤ 0.03 fm [19]. This result is marginally lower than the result from our zero temperature analysis. Remarkably, however, the zero temperature result is still compatible with the T > 0 data over a much wider range in ∣ r ∣/a, a factor of sixteen in ∣ r ∣, and over almost a factor of ten in lattice spacings (figure 7).

Zoom In Zoom Out Reset image size

2.4.4. Outlook

Every method to determine the strong coupling α_S starts with an observable that depends on a short distance, 1/Q (or high energy Q). The notion of ‘short’ or ‘high’ relates to other scales in the problem. The observable can be multiplied by the appropriate power of Q to obtain a dimensionless quantity, ${ \mathcal R }(Q)$, which can be written

$$\begin{eqnarray}&&{ \mathcal R }(Q)=R(Q)+{Q}^{-p}S(M,Q),\end{eqnarray} \tag{ 2.37 }$$

where M is another (energy) scale, M ≪ Q, the first term R does not depend on M, and the power p > 0 is something like 2, 4, or 1. The separation can be justified with tools such as the operator product expansion (OPE) [89] or an effective field theory (EFT) [90, 91]. (Sometimes equation (2.37) is posited on the basis of arguments or assumptions.) With the OPE or an EFT, the power p corresponds to the dimension of an operator. For example, R might be the short-distance (or ‘Wilson’) coefficient of the unit operator, and S the product of another coefficient and the matrix element of a dimension-4 operator, in which case p = 4.

In an asymptotically free quantum field theory, such as QCD, the short-distance part can be well approximated with perturbation theory:

$$\begin{eqnarray}&&R(Q)=\displaystyle \sum _{l=0}^{\infty }{R}_{l}(Q/\mu ){\alpha }_{S}^{l}(\mu )=\displaystyle \sum _{l=0}^{\infty }{R}_{l}{\alpha }_{S}^{l}(Q),\end{eqnarray} \tag{ 2.38 }$$

where μ is a separation scale introduced by the OPE, EFT, or other consideration; in the last expression, R_l = R_l (1). Because μ is unphysical, it can be chosen, at least after thorough analysis of the scale-separation details, equal or proportional to Q. At this stage, it is useful to introduce (with μ = Q)

$$\begin{eqnarray}&&{\alpha }_{{ \mathcal R }}(Q)=\displaystyle \frac{{ \mathcal R }(Q)-{R}_{0}}{{R}_{1}},\end{eqnarray} \tag{ 2.39 }$$

which is known as the effective charge (or effective coupling) for the observable ${ \mathcal R }$. It may be more apt to note that ${\alpha }_{{ \mathcal R }}$ is simply a physical, regularization-independent choice of renormalization scheme.

It is tempting to identify ‘short distance’ with ‘perturbative’ and ‘long distance’ with ‘nonperturbative’. In the OPE or an EFT, however, the home for small instantons (for example) of diameter 1/Q is in short-distance coefficients. Fortunately, small-instanton contributions scale with a large power, 11 − 2N_f/3 ≥ 7 (where N_f is the number of active quark flavours), so they are small enough that neglecting them is safer than other compromises that must be made. Further complications in separating scales arise, such as renormalons. In the end, the point of equation (2.37) is that short-distance quantities depend on Q logarithmically with power-law corrections.

A practical version, then, of equation (2.37) is

$$\begin{eqnarray}&&{\alpha }_{{ \mathcal R }}(Q)=\displaystyle \sum _{l=1}^{{n}_{\mathrm{loop}}}{r}_{l}{\alpha }_{S}^{l}(Q)+\displaystyle \sum _{l=1+{n}_{\mathrm{loop}}}{r}_{l}{\alpha }_{S}^{l}(Q)+\displaystyle \sum _{p}{d}_{p}{(M/Q)}^{p},\quad {r}_{1}=1,\end{eqnarray} \tag{ 2.40 }$$

presupposing a preferred renormalization scheme (such as $\overline{\mathrm{MS}}$), truncating the perturbative series at n_loop terms (those available from explicit perturbative calculation), acknowledging that the remaining terms do not vanish, and allowing for several power corrections. The renormalization group provides more information, in particular showing how to relate ${\alpha }_{{ \mathcal R }}(Q)$ at one scale to ${\alpha }_{{ \mathcal R }}({Q}_{0})$ at a fiducial scale Q₀. Instead of using the r_l to relate ${\alpha }_{{ \mathcal R }}$ to ${\alpha }_{\overline{\mathrm{MS}}}$, they can be used to convert the coefficients of the β function in the $\overline{\mathrm{MS}}$ scheme to the ${ \mathcal R }$ scheme. Thus, perturbation theory predicts the logarithmic Q dependence of ${\alpha }_{{ \mathcal R }}(Q)$ without explicit reference to the ultraviolet regularization.

Equation (2.40) provides a guide to controlling a determination of α_S . The quantity ${ \mathcal R }$ should be something that can be measured (in a laboratory experiment) or computed (with numerical lattice QCD) precisely. The higher the order in α_S , n_loop, the better. One wants Q to be as large as possible, both to reduce the power corrections and to reduce the truncation error. Even better is a wide range of Q, both to verify the running of ${\alpha }_{{ \mathcal R }}$ and to separate the logarithmic dependence on Q from the power-law behavior of the remainder. An observable is better suited when p can be proven, argued, or demonstrated to be large. Another desirable feature is to have several similar quantities, especially if the power corrections are related.

The quantity ${ \mathcal R }$ can be computed in lattice QCD or measured in high-energy scattering or heavy-particle decays. Many reviews separate the two as if they are completely different objects. Table 3 compares the ingredients in the two approaches.

Table 3. Ingredients of α_S determinations, with nonperturbative ‘measurements’ directly from lattice gauge theory (LGT), from the continuum limit of lattice QCD, or (to choose one example) scaling violations of the moments of structure functions in deep inelastic scattering (DIS). Effects beyond the Standard Model (BSM) may be present in experimental data (no one knows) and are omitted (as a rule) from the theory.

Ingredient	Small Wilson loops	LGT with a → 0	DIS scaling violation
Obtain ${ \mathcal R }(Q)$	Compute from QCD Lagrangian		Measure e−p scattering
Large energy scale	a−1	L−1, 2mQ , ...	Momentum transfer Q
Scale separation	OPE	Various	OPE
Perturbation theory	Lattice (NLO, maybe NNLO)	Dimensional regularization (NNLO, N3LO)
Number of quantities	Several	one or few	Several
Electroweak	Omitted by construction		Included in data and theory
BSM	Omitted by construction		Unknown/omitted
Units in GeV	Hadronic quantity, viz., Q = MPDG[(Qa)/(Ma)]lat		Detector calibration

The scaling violations of moments of deep-inelastic structure functions are taken as a textbook example of high-energy scattering (further columns could be added without much work). Table 3 shows two classes of methods based on lattice gauge theory: small Wilson loops and every observable for which the continuum limit is taken (including everything else discussed in the subsections below). Even if one starts with a spacetime lattice as an ultraviolet regularization, the continuum limit is the same QCD as probed by high-energy experiments. In particular, the same methods for perturbation theory—based on dimensional regularization—apply and, thus, the issues related to truncating the series, the size of and range in Q, etc, are the same. Indeed, moments of quarkonium correlation functions can be calculated with lattice QCD or measured in e⁺e⁻ → hadrons: the perturbative series are exactly the same. We have here an example of a lattice-QCD method that has more in common with a high-energy-scattering method than it has with other lattice-QCD methods (see sections 2.3 and 9.1 for details).

Because of the similarities, criteria for assessing issues such as truncation of perturbation theory and the range in Q should be the same for both. Ideally, the ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ averaging approaches used by the PDG from the world data [92] and by the FLAG collaboration from lattice results [12] should be more closely aligned. Some remarks on the criteria are provided below.

In table 3, small Wilson loops are listed separately because they are defined at the scale of the lattice spacing, i.e. at the ultraviolet cutoff. There are further such quantities, including the bare coupling. They are a different object because the lattice—including details of the chosen lattice action—is present. The lattice is like ‘new physics’ at the highest energies, except that the action of the new physics is exactly known. Determinations of α_S from small Wilson loops warrant discussion here in order to clarify discussions in [11, 13, 50].

In continuum language, a Wilson loop is a path-ordered exponential integrating dz · A around a closed loop [61]:

$$\begin{eqnarray}&&{W}_{P}={\mathbb{P}}\,\exp \,{ig}{\oint }_{P}{dz}\cdot A(z),\end{eqnarray} \tag{ 2.41 }$$

where A = t^a A^a is the gauge potential, and ${\mathbb{P}}$ denotes path ordering. For small loops of linear size a, this operator admits an OPE:

$$\begin{eqnarray}&&{W}_{P}\doteq {Z}_{P}\left[{\mathbb{1}}+{C}_{P}^{({FF})}{a}^{4}{\alpha }_{S}{F}^{2}+{C}_{P}^{(\bar{q}q)}{a}^{4}m\bar{q}q+{ \mathcal O }({a}^{6})\right],\end{eqnarray} \tag{ 2.42 }$$

where the equivalence ≐ is in the weak sense of matrix elements between low-energy states. Here, Z_P , ${Z}_{P}{C}_{P}^{({FF})}$, and ${Z}_{P}{C}_{P}^{(\bar{q}q)}$, are short distance coefficients, so they can be calculated in perturbation theory. Note that the operators that appear do not depend on the path: when taking matrix elements, the same quantities enter over and over again. Note also the high power 4 in the power corrections.

Equation (2.42) applies equally well in a lattice gauge theory. Indeed, in pure gauge theory, the OPE is on a very solid footing [93, 94]. The short-distance coefficients must be calculated in lattice perturbation theory, which is less developed than perturbation theory with dimensional regularization. It is instructive to show the tree-level expression for the coefficient ${Z}_{P}{C}_{P}^{({FF})}$ of a planar Wilson loop of size ma × na,

$$\begin{eqnarray}&&{Z}_{m\times n}{C}_{m\times n}^{({FF})}=-\displaystyle \frac{\pi {({mn})}^{2}}{36},\end{eqnarray} \tag{ 2.43 }$$

where a is now the lattice spacing. The condensate contribution to the 1 × 1 loop is 16 times smaller than that of the 2 × 2 loop.

Vacuum expectation values ${{ \mathcal R }}_{P}\equiv \tfrac{1}{3}\langle \mathrm{Re}\,\mathrm{tr}{W}_{P}\rangle$ or ${ \mathcal R }{{\prime} }_{P}\equiv -\mathrm{ln}\tfrac{1}{3}\langle \mathrm{Re}\,\mathrm{tr}{W}_{P}\rangle$, and combinations thereof, satisfy equation (2.40). They can be computed very precisely. (The bare coupling, mentioned above, is known exactly.) Fits of the precise data can include several orders beyond the ${n}_{\mathrm{loop}}^{\mathrm{th}}$ term [14, 95, 96], which is not a shortcoming but a strength of the method. Such fitting could be applied to any quantity ${ \mathcal R }$ with per mil uncertainties, because assuming that the higher-order terms vanish is obviously wrong. In practice, the fits include terms that cannot be determined by the data, but the correct interpretation of these parameters is a marginalization over terms whose Q dependence is known, even if their strength is not. For determining α_S , one is not interested in the values (and errors) of higher-order or power-law coefficients. One is interested is how imperfect knowledge of these terms propagates to uncertainty in ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$.

Incorporating the next few terms via fit parameters, with suitable priors, can be seen as more conservative and more robust than the popular method of varying the scale by a factor of two up and down. The popular method sets μ = sQ and moves $\mathrm{ln}s\in [-\mathrm{ln}2,+\mathrm{ln}2]$. It picks out a one-dimensional curve in a multidimensional space, rather than allowing a data-guided exploration of the space. Reference [13] uses the scale variation method to estimate the truncation uncertainty in analyses such as those in [14, 95, 96], not discussing that the additional fitted terms could absorb such a variation.

The FLAG collaboration [11] is considering making its quality criteria stricter. It is worth scrutinizing the criteria and asking whether they are the most apt. One of the criteria requires ${\alpha }_{{ \mathcal R }}$ to be sufficiently small. The bare coupling of lattice QCD satisfies the 2021 criterion and probably any future one, but it has been deprecated (for well-known reasons) as a route to α_S . Another criterion demands that the truncation error, ${\alpha }_{{ \mathcal R }}^{{m}_{\mathrm{loop}}+1}$, be smaller than the statistical (and systematic) error. This criterion, as it stands, can exclude quantities that are precise enough to verify higher-order perturbative behavior by fitting.

Because small Wilson loops are defined at the scale of the lattice spacing, effects of QED and strong isospin breaking (m_u ≠ m_d ) are often very small. The leading effect arises not in the effective α_S but in the conversion from lattice units to GeV. The ensembles of lattice gauge fields best suited to a future study of small Wilson loops are those being used for the hadronic-vacuum-polarization contribution to the muon’s anomalous magnetic moment, because they have the widest range in a, highest statistics, and include QED and strong isospin contributions in the determination of the lattice spacing. A typical target is δ a/a ≲ 0.5%, leading to a δ α_S /α_S that is β₀ α_S times smaller, or ≲0.1%.

The inclusive distribution of the final hadrons in τ decay provides the needed information to perform a clean low-energy determination of the strong coupling [97]. The relevant dynamical quantities are the two-point correlation functions for the vector ${V}^{\mu }=\overline{u}{\gamma }^{\mu }d$ and axial-vector ${A}^{\mu }=\overline{u}{\gamma }^{\mu }{\gamma }_{5}d$ colour-singlet charged currents:

$$\begin{eqnarray}&&i\int {d}^{4}x\ {e}^{{iqx}}\ \langle 0| T[{{ \mathcal J }}^{\mu }(x){{ \mathcal J }}^{\nu \dagger }(0)]| 0\rangle =(-{g}^{\mu \nu }{q}^{2}+{q}^{\mu }{q}^{\nu })\ {{\rm{\Pi }}}_{{ \mathcal J }}({q}^{2})+{g}^{\mu \nu }{q}^{2}\ {{\rm{\Pi }}}_{{ \mathcal J }}^{(0)}({q}^{2}),\end{eqnarray} \tag{ 3.1 }$$

with ${ \mathcal J }=V,A$. If the tiny up and down quark masses are neglected, ${q}^{2}\ {{\rm{\Pi }}}_{{ \mathcal J }}^{(0)}({q}^{2})=0$ and only the first term needs to be considered. The correlators ${{\rm{\Pi }}}_{{ \mathcal J }}(s)$ are analytic functions in the entire complex s plane, except for a cut along the positive real axis where their imaginary parts (spectral functions) have discontinuities. This implies the exact mathematical identity [98]

$$\begin{eqnarray}&&{A}_{{ \mathcal J }}^{\omega }({s}_{0})\ \equiv \ {\int }_{{s}_{\mathrm{th}}}^{{s}_{0}}\displaystyle \frac{{ds}}{{s}_{0}}\ \omega (s)\mathrm{Im}{{\rm{\Pi }}}_{{ \mathcal J }}(s)\ =\ \displaystyle \frac{i}{2}\ {\oint }_{| s| ={s}_{0}}\displaystyle \frac{{ds}}{{s}_{0}}\ \omega (s){{\rm{\Pi }}}_{{ \mathcal J }}(s),\end{eqnarray} \tag{ 3.2 }$$

where ω(s) is any weight function analytic in ∣s∣ ≤ s₀, s_th is the hadronic mass-squared threshold, and the complex integral in the right-hand side runs counter-clockwise around the circle ∣s∣ = s₀. The inclusive Cabibbo-allowed hadronic decay width of the τ just corresponds to the weight ${\omega }_{\tau }(x)={(1-x)}^{2}(1+2x)$ with x = s/s₀ and ${s}_{0}={m}_{\tau }^{2}$. The measured invariant-mass distribution determines then the left-hand-side integral for ${s}_{0}\leqslant {m}_{\tau }^{2}$.

For large-enough values of s₀, the operator product expansion (OPE) [99],

$$\begin{eqnarray}&&{{\rm{\Pi }}}_{{ \mathcal J }}^{\mathrm{OPE}}(s)\ =\ \displaystyle \sum _{D}\displaystyle \frac{1}{{(-s)}^{D/2}}\displaystyle \sum _{\dim \,{ \mathcal O }=D}{C}_{D,{ \mathcal J }}(-s,\mu )\ \langle 0| { \mathcal O }(\mu )| 0\rangle \ \equiv \ \displaystyle \sum _{D}\ \displaystyle \frac{{{ \mathcal O }}_{D,{ \mathcal J }}}{{(-s)}^{D/2}},\end{eqnarray} \tag{ 3.3 }$$

can be used to expand the contour integral in inverse powers of s₀. The first term (D = 0) contains the perturbative QCD contribution, which is known to ${ \mathcal O }({\alpha }_{S}^{4})$ [100], while nonperturbative power corrections have D ≥ 4. The small differences between the physical values of the integrated moments ${A}_{{ \mathcal J }}^{\omega }({s}_{0})$ and their OPE approximations are known as quark-hadron duality violations. They are very efficiently minimized by taking ‘pinched’ weight functions which vanish at s = s₀, suppressing in this way the contributions from the region near the real axis where the OPE is not valid [98, 101, 102].

The high sensitivity of the τ hadronic width to the strong coupling follows from four important facts:

• 1.  
  The perturbative contribution amounts to a sizeable 20% effect because ${\alpha }_{S}({m}_{\tau }^{2})\sim 0.3$ is large.
• 2.  
  The OPE can be safely used at ${s}_{0}\sim {m}_{\tau }^{2}$. The weight ω_τ (x) contains a double zero at s = s₀, heavily suppressing the numerical impact of duality violations.
• 3.  
  Since ω_τ (x) = 1 − 3x² + 2x³, the contour integral is only sensitive to OPE corrections with D = 6 and 8, which are strongly suppressed by the corresponding powers of m_τ . Moreover, these power corrections appear with opposite signs in the vector and axial-vector correlators [98, 103, 104], which implies an additional numerical cancellation in the total vector + axial (V + A) decay width.
• 4.  
  The opening of high-multiplicity hadronic thresholds dilutes very soon the prominent ρ(2π) and a₁(3π) resonances, leading to a quite flat V + A spectral distribution that approaches very fast the perturbative QCD predictions.

The small nonperturbative contributions can be extracted from data, analysing moments more sensitive to power corrections [101]. The detailed analyses performed by ALEPH [105], CLEO [106] and OPAL [107] confirmed a long time ago that these corrections are smaller than the perturbative uncertainties. The latest and more precise experimental determination of the strong coupling obtains ${\alpha }_{S}^{({N}_{{\rm{f}}}=3)}({m}_{\tau }^{2})=0.332\pm {0.005}_{\exp }\pm {0.011}_{\mathrm{th}}$ [103]. Taking as input their measured nonperturbative correction, the strong coupling can be directly extracted from the total τ hadronic width (and/or lifetime), which results in ${\alpha }_{S}^{({N}_{{\rm{f}}}=3)}({m}_{\tau }^{2})\,=0.331\pm 0.013$ [108].

An exhaustive reanalysis with the updated ALEPH τ data [103] has been performed in [104], in order to carefully assess any possible source of systematic uncertainties. A large variety of methodologies, including all previously considered strategies, have been explored, looking for potential hidden weaknesses and testing the stability of the fitted results under slight variations of the assumed inputs. The most reliable determinations of ${\alpha }_{S}({m}_{\tau }^{2})$, obtained with the total V + A distribution, are summarized in table 4. The dominant uncertainties are the perturbative errors associated with the unknown higher-order corrections. This is clearly illustrated in the table, which shows the results obtained under two different procedures, either performing the contour integrals with a running α_S (−s), by solving numerically the five-loop β-function equation (contour-improved perturbation theory, CIPT) [109, 110], or naively expanding them in powers of ${\alpha }_{S}({m}_{\tau }^{2})$ (fixed-order perturbation theory, FOPT). FOPT generates a somewhat larger perturbative contribution and, therefore, leads to a slightly smaller fitted value of α_S . Within each procedure, the perturbative error has been estimated taking a very conservative range for the fifth-order coefficient of the Adler series, K₅ = 275 ± 400, and varying the renormalization scale in the interval μ²/(−s) ∈ (0.5, 2). The values obtained with the two procedures are finally combined, adding quadratically half their difference as an additional systematic error.

Table 4. Determinations of ${\alpha }_{S}^{({N}_{{\rm{f}}}=3)}({m}_{\tau }^{2})$ from τ decay data, in the V + A channel [104].

Method	${\alpha }_{S}^{({N}_{{\rm{f}}}=3)}({m}_{\tau }^{2})$
	CIPT	FOPT	Average
ωkl (x) weights	$0.339{}_{-\,0.017}^{+\,0.019}$	$0.319{}_{-\,0.015}^{+\,0.017}$	$0.329\,{}_{-\,0.018}^{+\,0.020}$
${\hat{\omega }}_{{kl}}(x)$ weights	$0.338{}_{-\,0.012}^{+\,0.014}$	$0.319{}_{-\,0.010}^{+\,0.013}$	$0.329{}_{-\,0.014}^{+\,0.016}$
ω(2,m)(x) weights	$0.336{}_{-\,0.016}^{+\,0.018}$	$0.317{}_{-\,0.013}^{+\,0.015}$	$0.326{}_{-\,0.016}^{+\,0.018}$
s0 dependence	0.335 ± 0.014	0.323 ± 0.012	0.329 ± 0.013
${\omega }_{a}^{(1,m)}(x)$ weights	$0.328{}_{-\,0.013}^{+\,0.014}$	$0.318{}_{-\,0.012}^{+\,0.015}$	$0.323{}_{-\,0.013}^{+\,0.015}$
Average	0.335 ± 0.013	0.320 ± 0.012	0.328 ± 0.013

The different rows in the table correspond to different choices of pinched weights:

$$\begin{eqnarray}\begin{array}{ll}{\omega }_{{kl}}(x)={(1-x)}^{(2+k)}\,{x}^{l}\,(1+2x), & \,(k,l)=\{(0,0),(1,0),(1,1),(1,2),(1,3)\},\\ {\hat{\omega }}_{{kl}}(x)={(1-x)}^{(2+k)}\,{x}^{l}, & \,(k,l)=\{(0,0),(1,0),(1,1),(1,2),(1,3)\},\\ {\omega }^{(2,m)}(x)=1-(m+2){x}^{m+1}+(m+1){x}^{m+2}, & \,1\leqslant m\leqslant 5,\\ {\omega }_{a}^{(1,m)}(x)=(1-{x}^{m+1}){{\rm{e}}}^{-{ax}}, & \,0\leqslant m\leqslant 6.\end{array}\end{eqnarray} \tag{ 3.4 }$$

In all cases, we find a high sensitivity to the strong coupling and a very small sensitivity to power corrections, which gets reflected in large uncertainties for the fitted condensates. The first set of weights was adopted by ALEPH and allows for a direct use of the measured distribution. With the five indicated moments, we have fitted ${\alpha }_{S}({m}_{\tau }^{2})$ and the nonperturbative corrections ${{ \mathcal O }}_{\mathrm{4,6,8}}$. The impact on α_S from neglected condensates of higher dimensions (the highest moment involves corrections with D ≤ 16) has been estimated including ${{ \mathcal O }}_{10}$ in the fit and taking the difference as an additional uncertainty. Additional power corrections with D > 10 would certainly modify the poorly determined values of the fitted condensates, specially those with higher dimensions, but they have a negligible impact on α_S , compared with the errors already included. Our results (first line in the table) are in good agreement with [103]. Very similar values are obtained with the modified weights ${\hat{\omega }}_{{kl}}(x)$ (second line), which eliminate the highest-D power correction from every moment, showing that these contributions do indeed play a minor numerical role.

Moments constructed with the optimized (double-pinched) weights ω^(2,m)(x) only receive power corrections with D = 2(m + 2) and 2(m + 3). The third line in the table shows that they give values of ${\alpha }_{S}({m}_{\tau }^{2})$ in very good agreement with those in the first two lines. Similar results (not shown in the table) are obtained from a fit to four moments, based on the weights ${\omega }^{(n,0)}(x)={(1-x)}^{n}$, with 0 ≤ n ≤ 3, which have a different sensitivity to power corrections and (for n=0,1) are less protected against duality violations and experimental uncertainties. The values of ${\alpha }_{S}({m}_{\tau }^{2})$, ${{ \mathcal O }}_{2(m+2)}$ and ${{ \mathcal O }}_{2(m+3)}$ can also be extracted from a fit to the s₀ dependence of a single A^(2,m)(s₀) moment, above some ${\hat{s}}_{0}\geqslant 2.0\ {\mathrm{GeV}}^{2}$. One finds a quite poor sensitivity to power corrections, as expected, but a surprising stability in the extracted values of ${\alpha }_{S}({m}_{\tau }^{2})$ at different ${\hat{s}}_{0}$. Combining the information from three different moments (m = 0, 1, 2), and adding as an additional theoretical error the fluctuations with the number of fitted bins, one gets the ${\alpha }_{S}({m}_{\tau }^{2})$ values given in the fourth line of table 4. This determination is much more sensitive to violations of quark-hadron duality because the s₀ dependence of consecutive bins feels the local structure of the spectral function. ⁶⁵ The agreement with the determinations in the other lines of the table confirms the small size of duality violations in the V + A distribution above ${\hat{s}}_{0}$.

A completely different sensitivity to nonperturbative corrections is achieved with the weights ${\omega }_{a}^{(1,m)}$. Their exponential suppression nullifies the high-s region, strongly reducing the violations of quark-hadron duality, at the price of enhancing the exposition to power corrections of any dimensionality. In a pure perturbative analysis the neglected power corrections should manifest as large instabilities of α_S under variations of s₀ and a ≠ 0; however, stable results are found for a broad range of both s₀ and a, which indicates that power corrections are small. The last line in the table has been obtained combining the information from seven different moments with ${\omega }_{a}^{(1,m)}$ (0 ≤ m ≤ 6) weights.

Fully compatible results with slightly larger uncertainties are also obtained from the separate V and A distributions [104]. The excellent overall agreement among determinations obtained with a broad variety of numerical approaches that have very different sensitivities to nonperturbative corrections, and the many complementary tests successfully performed in [104], demonstrate the robustness and reliability of the results shown in table 4. The final average value

$$\begin{eqnarray}&&{\alpha }_{S}^{({N}_{{\rm{f}}}=3)}({m}_{\tau }^{2})=0.328\pm 0.013\end{eqnarray} \tag{ 3.5 }$$

is in good agreement with the results of [103] and with a recent determination (see section 3.3) based on Borel–Laplace sum rules and a renormalon-motivated model [114]. After evolution up to the scale m_Z , the strong coupling decreases to

$$\begin{eqnarray}&&{\alpha }_{S}^{({N}_{{\rm{f}}}=5)}({m}_{{\rm{Z}}}^{2})=0.1197\pm 0.0015,\end{eqnarray} \tag{ 3.6 }$$

in excellent agreement with the direct N³LO determination at the Z peak.

A much better control of the small nonperturbative contributions could be achieved with more precise data, specially at the highest energy bins. The high-statistics expected from Belle-II or a future TeraZ facility should make that possible. A reduction of the dominant perturbative error requires a better theoretical understanding of higher-order corrections (CIPT versus FOPT, renormalons, etc). In the long term, an explicit calculation of the K₅ term in the Adler series would have a major impact.

Acknowledgments—This work has been supported by MCIN/AEI/10.13039/501100011033, Grant No. PID2020-114473GB-I00, by the Generalitat Valenciana, Grant No. Prometeo/2021/071, and by the Agence Nationale de la Recherche (ANR), Grant ANR-19-CE31-0012 (project MORA).

We review the sum-rule framework for determining the strong coupling, α_S , from inclusive spectral functions measured in hadronic τ decays. We then discuss a new inclusive vector isovector spectral function, obtained from combining the dominant exclusive-mode spectral-function data from ALEPH and OPAL with BaBar data for τ → K⁻ K⁰ ν_τ and several CVC-converted R-ratio data sets for small exclusive-mode contributions that had previously been estimated using Monte Carlo methods. We summarize our most recent results for α_S , and discuss prospects for future improvements.

3.2.1. Review

Finite-energy sum rules (FESRs) allow for the extraction of the strong coupling at the τ mass, ${\alpha }_{S}({m}_{\tau }^{2})$, from inclusive vector (V) and/or axial (A) non-strange spectral functions, which can and have been measured through hadronic τ decays [105, 107, 115]. Such sum rules have the form

$$\begin{eqnarray}&&{I}^{(w)}({s}_{0})\equiv \displaystyle \frac{1}{{s}_{0}}{\int }_{0}^{{s}_{0}}{ds}\,w(s)\rho (s)=-\displaystyle \frac{1}{2\pi {{is}}_{0}}{\oint }_{| z| ={s}_{0}}{dz}\,w(z){\rm{\Pi }}(z),\end{eqnarray} \tag{ 3.7 }$$

where Π(z) is a vacuum polarization, $\rho (s)=\tfrac{1}{\pi }{\rm{Im}}\,{\rm{\Pi }}(s)$, with s = q², is the corresponding spectral function, s₀ > 0, and w(s) is typically a polynomial in s/s₀. The left-hand side of equation (3.7) can be determined from data for ${s}_{0}\leqslant {m}_{\tau }^{2}$, while, for s₀ large enough, the right-hand side can be represented in QCD perturbation theory, with nonperturbative (NP) corrections. The latter are needed because of the relatively low value of m_τ .

The perturbative expansion for Π(z) in powers of α_S (μ²) and $\mathrm{log}(-z/{\mu }^{2})$ is known to order ${\alpha }_{S}^{4}$ [100], where μ is the renormalization-group scale. In most of the literature, two scale-choice prescriptions have been considered, either FOPT, in which μ² = s₀, or CIPT [109, 110], in which μ² is set equal to z when evaluating the perturbative contribution to the contour integral on the right-hand side of equation (3.7).

NP corrections (away from the positive real axis, see below) are incorporated through the operator product expansion (OPE), which schematically takes the form

$$\begin{eqnarray}&&{{\rm{\Pi }}}_{\mathrm{OPE}}(z)=\displaystyle \sum _{k=0}^{\infty }{(-1)}^{k}\displaystyle \frac{{C}_{2k}(z)}{{z}^{k}}.\end{eqnarray} \tag{ 3.8 }$$

The k = 0 term represents purely perturbative contributions, the k = 1 term perturbative mass corrections (in the non-strange channels, C₂ can be set to zero because of the smallness of the up and down quark masses); NP condensate contributions start at k = 2. The presence of a cut in the complex z = q² plane extending to infinity along the positive real axis, means the OPE is not convergent; it is, at best, an asymptotic series. It is also generally believed that the D = 2k terms (for k > 1) are related to renormalon ambiguities in the perturbative expansion [116–118]. While the higher-dimension coefficients C_2k in principle depend logarithmically on z, they are generally taken as constants, as their z dependence is suppressed by two powers of α_S , in addition to the 1/z^k suppression.

Recently, it was shown that the Borel sums of the FOPT and CIPT versions of the perturbative expansion of the right-hand side of equation (3.7) are not the same and, related, that the OPE (3.8) does not reflect the renormalon ambiguities for CIPT [119, 120], creating a mismatch between the use of CIPT and the OPE in the form (3.8). Because of this, we will always employ FOPT [121]. The recent work of [122] (see section 3.4) paves the way for a reconciliation between the FOPT and CIPT series but, until this is realized in practice, averaging the results obtained using FOPT and CIPT should be avoided.

The nonconvergent, at-best-asymptotic, nature of the OPE noted above has two implications. First, it is advisable to restrict the use of the OPE to low orders in the expansion (3.8), as little is known about where it starts to diverge (at the energies relevant in τ decays). Second, one expects NP corrections beyond the OPE, just as the OPE represents NP corrections beyond perturbation theory. This phenomenon, known as the violation of quark-hadron duality [123–125], manifests itself physically in the presence of overlapping resonance contributions in ρ(s). These duality violations (DVs) need to be modeled (as long as a NP solution to QCD is not known). We have developed a theoretical framework, based on generally accepted assumptions about QCD in the framework of hyperasymptotics [126], which leads to the following large-s asympotic form for the duality-violating part of the spectral function:

$$\begin{eqnarray}&&{\rho }_{\mathrm{DV}}(s)={e}^{-\delta -\gamma s}\sin (\alpha +\beta s)\left[1+{ \mathcal O }\left(\displaystyle \frac{1}{\mathrm{log}s};\displaystyle \frac{1}{s};\displaystyle \frac{1}{{N}_{c}}\right)\right].\end{eqnarray} \tag{ 3.9 }$$

With this ansatz, the sum rule (3.7) can be reformulated as [127]

$$\begin{eqnarray}&&\displaystyle \frac{1}{{s}_{0}}{\int }_{0}^{{s}_{0}}{ds}\,w(s)\rho (s)=-\displaystyle \frac{1}{2\pi {{is}}_{0}}{\oint }_{| z| ={s}_{0}}{dz}\,w(z){{\rm{\Pi }}}_{\mathrm{OPE}}(z)-\displaystyle \frac{1}{{s}_{0}}{\int }_{{s}_{0}}^{\infty }{ds}\,w(s){\rho }_{\mathrm{DV}}(s).\end{eqnarray} \tag{ 3.10 }$$

This introduces a new set of parameters into the theory representation of the right-hand side of equation (3.7). A separate set of DV parameters is needed for each of the V and A channels. The main assumption is that the form (3.9) can be used at values of s below ${m}_{\tau }^{2}$—whether this is the case can be tested by a variety of fits to data. For a more detailed version of this very brief review, and many more references, we refer to [121].

At this point, a quick look at data is instructive. Figure 8 shows a rescaled version of the sum of the non-strange V and A spectral functions from [103], with the α_S -independent parton-model contribution subtracted. Resonance oscillations are immediately evident, with an amplitude comparable in size to the α_S -dependent pQCD contribution, represented by the dashed curve. The figure makes clear that DVs are important, though, because of the large errors, it is difficult to tell whether oscillations above $s={m}_{\tau }^{2}$ (relevant to determining DV contributions to the sum rules (3.10) with ${s}_{0}={m}_{\tau }^{2}$) will be numerically relevant or not. Clearly, DVs need to be accounted for, and their contributions quantitatively assessed, even in the V + A channel.

Two different strategies, which we refer to as the ‘truncated OPE’ (tOPE) [101] and the ‘DV-model’ strategies [128, 129], have been developed to deal with the NP contributions discussed above.

In the tOPE approach, s₀ is taken equal to ${m}_{\tau }^{2}$, and, to suppress contributions to the contour integral from the region near the positive real axis, and thus from DVs, weights with a double or triple zero at $s={s}_{0}={m}_{\tau }^{2}$ are employed. The set of such multiple-zero (‘pinched’) polynomial weights typically extends up to degree k = 7, and thus probes the OPE up to dimension D = 2k + 2 = 16. The number of OPE parameters to be fitted (α_S and the condensates C_D , D = 4, 6, ⋯,16) necessarily exceeds the number of such independent weights, and hence the number of independent ${s}_{0}={m}_{\tau }^{2}$ spectral integrals available to fix these parameters. Some subset of the C_D in principle present must thus be set to zero by hand to leave a fit with fewer parameters than data points. DVs are also neglected (in terms of equation (3.9), δ is set to ∞) under the assumption that the use of pinched moments sufficiently suppressess these contributions.

The main problem with this strategy, employed most recently in [103, 104], is the neglect of these higher-D OPE contributions. Several tests of the tOPE strategy were carried out in [130, 131], exposing clear inconsistencies. Such inconsistencies can arise because integrated dimension-D OPE contributions scale as $1/{s}_{0}^{D/2}$. If contributions from omitted higher-D condensates are, in fact, not numerically negligible, then when these are absorbed into the values of the lower-D OPE parameters obtained in a fixed-s₀ tOPE fit, the resulting theory representation will, in general, have an s₀ dependence which is incorrect. The tOPE truncation can thus be tested for selfconsistency by comparing spectral-integral predictions obtained using the theory parameters obtained from the single-s₀ tOPE fit to the experimental values of these same spectral integrals over a range of s₀ values. Such tests, designed to take into account the impact of the very strong correlations between (i) spectral integrals at different s₀, (ii) theory integrals at different s₀, and (iii) fitted theory parameters and spectral integrals, were carried out in [131], with related tests also carried out in [130]. The s₀ dependence of the theory predictions was found to be in clear disagreement with that of the corresponding experimental spectral integral combinations. ⁶⁷

Zoom In Zoom Out Reset image size

In a recent publication [113], Pich and Rodriguez-Sanchez have addressed the relative merits of the tOPE and DV-model based approaches, commenting on [130, 131]. In this brief overview, there is not enough space to discuss, in sufficient detail, a number of the ongoing misconceptions and/or mis-statements concerning the tOPE and DV model approaches contained in both this new work and in section 3.1 of this review. Neither, in fact, addresses the points raised in [131] beyond a few vague comments, while the discussion of [130] in [113] is based on a number of assumptions/prejudices about the OPE and misinterpretations of the results presented in [130]. Some further discussion of these shortcomings can be found in the Mattermost forum provided for the recent ECT* Trento α_S (2022) Workshop on Precision Measurements of the QCD Coupling Constant. ⁶⁸ We leave further discussion, including of [113], to a future publication.

In the alternate DV-model strategy, recent implementations employ weights sensitive to terms in the OPE only up to D = 8, avoiding potential problems with higher-D OPE contributions. ⁶⁹ DVs cannot, in general, be ignored, and are instead modeled by equation (3.9), with fits performed to spectral integrals for a range of s₀ between a minimum value s₀ ^min and ${m}_{\tau }^{2}$, where s₀ ^min is determined by the quality of the fits and turns out to be ∼1.5 GeV². Results are found to be very stable against variations of s₀ ^min.

To determine both DV parameters and α_S , the analysis should include at least one weight with good sensitivity to DVs. The choice w = 1, which produces no DV suppression, is ideal for this purpose. Stability of the DV-model approach has been tested using various combinations of weight functions in analyses of purely V-channel τ data, both V- and A-channel τ data [132], and the KNT [133, 134] compilation of R-ratio (inclusive e⁺e⁻ → hadrons) data [135]. [121] contains our most recent application of this strategy. Earlier applications, laying out the framework in more detail, can be found in [128]. The DV-model strategy uses a range of s₀ as it is important to check the s₀ dependence of the match between experiment and theory (left-hand and right-hand sides of equation (3.10), respectively), to test the validity of the theory representation employed for Π(z). One also needs to check that oscillations in the V and A (hence also V + A) spectral functions are well represented by the ansatz (3.9), for at least some range of s below ${m}_{\tau }^{2}$.

3.2.2. Data

Traditionally, in most papers since the appearance of the perturbative result to order ${\alpha }_{S}^{4}$ in [100], the τ-based α_S has been obtained from the ALEPH data [103, 104, 114, 132, 136]. However, the value obtained from OPAL data [128, 129] is consistent with the ALEPH-based result [132], and it thus makes sense to combine the ALEPH and OPAL data sets prior to integration, following the approach now routinely employed for electroproduction data when obtaining dispersive estimates for the leading hadronic contribution to the muon anomalous magnetic moment. Moreover, while the bulk of the inclusive spectral functions comes from the 2-pion and the 4-pion modes (in the V channel), and the 3-pion and 5-pion modes (in the A channel), most of the residual exclusive channels (including some of the 5-pion channels) were obtained from Monte Carlo simulations rather than from experimental data in [105, 107, 115]. In the V channel, this can now be remedied by using, in addition to BaBar results for τ → K⁻ K⁰ ν_τ [137], recent high-precision results for higher-multiplicity, G-parity positive (hence I = 1) exclusive-mode electroproduction cross sections. CVC allows these to be converted to the equivalent higher-multiplicity contributions to the I = 1, V τ-based spectral function. ⁷⁰ The detailed construction of the combined inclusive I = 1, V spectral function along these lines is described in [121]. With 98% of the inclusive total by branching fraction (BF) coming from the combined ALEPH and OPAL 2-pion and 4-pion modes, and the support for the residual modes lying well above the ρ/ω meson interference region, isospin-breaking corrections, which should in principle be applied to the CVC relations, will be safely negligible.

The new V spectral function is shown in figure 9. The increased precision in the large-s region, coming from the replacement of Monte Carlo data with electroproduction data (which are not kinematically limited by the τ mass) is immediately evident. A more quantitative measure of the improved precision is provided in table 5, which shows a dramatically reduced statistical error on the w = 1 spectral moment I^(w=1)(s₀) resulting from the improved precision of the combined spectral function, compared to the ALEPH or OPAL spectral functions, in the larger s region.

Zoom In Zoom Out Reset image size

Table 5. Comparison of the left-hand side of equation (3.7) for w = 1, at two values of s₀, for the combined, ALEPH, and OPAL versions of the V spectral function. s₀* is a value close to 1.5 GeV² for each case, and s₀** is close to 2.9 GeV² for each case. Note that, because the values of s₀* and s₀** are slightly different for the three cases, the central values cannot be directly compared. The errors can, however, be compared, as they vary slowly with s₀. For details, see [121].

	Combined	ALEPH	OPAL
s0* ≈ 1.5 GeV2	0.03137(14)	0.03145(17)	0.03140(46)
s0** ≈ 2.9 GeV2	0.02952(29)	0.03133(65)	0.03030(170)

3.2.3. Results

The increased precision of the new non-strange I = 1 V spectral function constructed in [121] produces an improved determination of ${\alpha }_{S}({m}_{\tau }^{2})$, with the CVC-based improvement of residual-mode contributions playing a particularly important role. Since electroproduction data are purely vector, a similar improvement is not possible for the axial channel. Higher precision is thus now obtainable from V-channel than from V + A-channel analyses. ⁷¹

Our result for α_S at the τ-mass is [121]

$$\begin{eqnarray}&&\,{\alpha }_{S}({m}_{\tau }^{2})=0.3077\pm {0.0065}_{\exp }\pm {0.0038}_{\mathrm{theory}}=0.3077\pm 0.0075\qquad ({N}_{{\rm{f}}}=3,{\rm{FOPT}}),\end{eqnarray} \tag{ 3.11 }$$

where the first error is the fit error, and the second an estimate of the systematic uncertainty associated with the truncation of perturbation theory and the use of equation (3.9) to model DVs. For comparison, analyses of ALEPH or OPAL data alone produce combined errors of± 0.010 and ± 0.018, respectively.

Converting equation (3.11) to the N_f = 5, Z-mass scale using five-loop running and four-loop matching at the charm and bottom thresholds, one finds

$$\begin{eqnarray}&&{\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1171\pm 0.0010\qquad (\tau ,{N}_{{\rm{f}}}=5,{\rm{FOPT}}).\end{eqnarray} \tag{ 3.12 }$$

This is in good agreement with an independent determination from electroproduction data using the same theoretical framework [135]:

$$\begin{eqnarray}&&{\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1158\pm 0.0022\qquad ({\rm{EM}},{N}_{{\rm{f}}}=5,{\rm{FOPT}}).\end{eqnarray} \tag{ 3.13 }$$

The latter determination was obtained from FESRs with 3.25 GeV${}^{2}\leqslant {s}_{0}^{{\min }}\leqslant 4$ GeV², significantly higher than the interval ∼1.5 GeV${}^{2}\leqslant {s}_{0}^{{\min }}\leqslant {m}_{\tau }^{2}$ available from τ decays. The wider range of accessible s₀ allowed tests of our model for DVs, equation (3.9), to be performed by comparing values of ${\alpha }_{S}({m}_{\tau }^{2})$ obtained from the w = 1 FESR (which is maximally sensitive to DV effects, and hence provides a good probe of their effect) with and without DV contributions included. The effect found was small (of order 0.005) ⁷² for 3.25 GeV${}^{2}\leqslant {s}_{0}^{{\min }}\leqslant 4$ GeV² but increased rapidly as s₀ ^min was lowered below ${m}_{\tau }^{2}$. ⁷³

3.2.4. Future improvements

The application of double-pinched Borel–Laplace sum rules to ALEPH τ-decay data is discussed. For the leading-twist (D = 0) Adler function a renormalon-motivated extension is used, and the 5-loop coefficient is taken to be d₄ = 275 ± 63. Two D = 6 terms appear in the truncated OPE (D ≤ 6) to enable cancellation of the corresponding renormalon ambiguities. Two variants of the fixed order perturbation theory, and the inverse Borel transform, are applied to the evaluation of the D = 0 contribution. The truncation index N_t is fixed by the requirement of local insensitivity of the momenta a^(2,0) and a^(2,1) under variation of N_t . The averaged value of the coupling obtained is ${\alpha }_{S}({m}_{\tau }^{2})={0.3235}_{-0.0126}^{+0.0138}$ (corresponding to ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1191\pm 0.0016$). The theoretical uncertainties are significantly larger than the experimental ones.

The sum rule corresponding to the application of the Cauchy theorem to a contour integral containing the (u-d) quark V + A correlator Π(Q²) (Q² ≡ − q²) and a weight function g(Q²), ${\oint }_{{C}_{1}+{C}_{2}}{{dQ}}^{2}g({Q}^{2}){\rm{\Pi }}({Q}^{2})=0$, gives the sum rule

$$\begin{eqnarray}&&{\int }_{0}^{{\sigma }_{{\rm{m}}}}d\sigma g(-\sigma ){\omega }_{\exp }(\sigma )=\displaystyle \frac{1}{2\pi }{\int }_{-\pi }^{\pi }d\phi \ {{ \mathcal D }}_{\mathrm{th}}({\sigma }_{{\rm{m}}}{e}^{i\phi })G({\sigma }_{{\rm{m}}}{e}^{i\phi }),\end{eqnarray} \tag{ 3.14 }$$

where ${\sigma }_{{\rm{m}}}\equiv {\sigma }_{\max }$ is the maximal used energy in the data, and $\omega {(\sigma )}_{\exp }$ is the ALEPH-measured discontinuity (spectral) function of the (V + A)-channel polarization function

$$\begin{eqnarray}&&\omega (\sigma )\equiv 2\pi \ \mathrm{Im}\ {\rm{\Pi }}({Q}^{2}=-\sigma -i\epsilon ).\end{eqnarray} \tag{ 3.15 }$$

The function g(Q²) is the double-pinched Borel–Laplace weight function

$$\begin{eqnarray}&&{g}_{{M}^{2}}({Q}^{2})={\left(1+\displaystyle \frac{{Q}^{2}}{{\sigma }_{{\rm{m}}}}\right)}^{2}\displaystyle \frac{1}{{M}^{2}}\exp \left(\displaystyle \frac{{Q}^{2}}{{M}^{2}}\right),\end{eqnarray} \tag{ 3.16 }$$

G(Q²) is the integral of g

$$\begin{eqnarray}&&G({Q}^{2})={\int }_{-{\sigma }_{{\rm{m}}}}^{{Q}^{2}}{{dQ}}^{{\prime} 2}g({Q}^{{\prime} 2}),\end{eqnarray} \tag{ 3.17 }$$

and ${{ \mathcal D }}_{\mathrm{th}}({Q}^{2})$ is the full Adler function ${ \mathcal D }({Q}^{2})\equiv -2{\pi }^{2}d{\rm{\Pi }}({Q}^{2})/d\mathrm{ln}{Q}^{2}$, whose OPE truncated at dimension D = 6 terms has the form

$$\begin{eqnarray}&&{{ \mathcal D }}_{\mathrm{th}}({Q}^{2})=d({{Q}^{2})}_{D=0}+1+4{\pi }^{2}\displaystyle \frac{\langle {O}_{4}\rangle }{{({Q}^{2})}^{2}}+\displaystyle \frac{6{\pi }^{2}}{{({Q}^{2})}^{3}}\left[\displaystyle \frac{\langle {O}_{6}^{(2)}\rangle }{a({Q}^{2})}+\langle {O}_{6}^{(1)}\rangle \right].\end{eqnarray} \tag{ 3.18 }$$

Here, a(Q²) ≡ α_S (Q²)/π. The two terms of D = 6 in the above OPE are needed to enable the cancellation of the corresponding u = 3 IR renormalon ambiguities originating from the D = 0 contribution $d{({Q}^{2})}_{D\,=\,0}$. The latter contribution has the perturbation expansion

$$\begin{eqnarray}&&d({{Q}^{2})}_{D=0,\mathrm{pt}}={d}_{0}a(\kappa {Q}^{2})+{d}_{1}(\kappa )\ a{(\kappa {Q}^{2})}^{2}+\ldots +{d}_{n}(\kappa )\ a{(\kappa {Q}^{2})}^{n+1}+\ldots ,\end{eqnarray} \tag{ 3.19 }$$

where κ ≡ μ²/Q² is the renormalization scale parameter (0 < κ ≲ 1; usually κ = 1), the first four terms (d₀ = 1; d₁, d₂, d₃) are exactly known [100], and for the coefficient d₄ [≡d₄(κ) with κ = 1 and N_f = 3] we take the following values based on various specific estimates in the literature [100, 117, 143–145]:

$$\begin{eqnarray}&&{d}_{4}=275\pm 63.\end{eqnarray} \tag{ 3.20 }$$

The expansion of the Borel transform of $d{({Q}^{2})}_{D\,=\,0}$ is ${ \mathcal B }[d]{(u;\kappa )}_{\mathrm{ser}.}\,={\sum }_{n\geqslant 0}{d}_{n}(\kappa ){u}^{n}/n!/{\beta }_{0}^{n}$. The extension of $d{({Q}^{2})}_{D\,=\,0}$ beyond ∼a⁵ is performed with a renormalon-motivated model [145] in which the Borel transform is constructed first for an auxiliary quantity $\widetilde{d}({Q}^{2})$ of the Adler function [145], resulting in the Borel transform ${ \mathcal B }[d](u)$ having terms $\sim 1/{(2-u)}^{{\widetilde{\gamma }}_{2}}$, $1/{(3-u)}^{{\widetilde{\gamma }}_{3}\,+\,1}$, $1/{(3-u)}^{{\widetilde{\gamma }}_{3}}$ and $1/{(1+u)}^{1+{\overline{\gamma }}_{1}}$, and similar terms with lesser powers, where ${\widetilde{\gamma }}_{p}=1+p{\beta }_{1}/{\beta }_{0}^{2}$ (${\overline{\gamma }}_{p}=1-p{\beta }_{1}/{\beta }_{0}^{2})$, and β₀ = (11 − 2N_f /3)/4 and β₁ = (1/16)(102 − 38N_f /3) are the first two β-function coefficients (N_f = 3). ⁷⁵ This extension gives, for the choice d₄ = 275. , the coefficients of the expansion (3.19): d₅ = 3159.5; d₆ = 16136.; d₇ = 3.4079 × 10⁵; d₈ = 3.7816 × 10⁵; d₉ = 6.9944 × 10⁷; d₁₀ = − 5.8309 × 10⁸; etc.

The cancellation of the IR renormalon ambiguity requires: (i) for u = 2 IR renormalon term ${ \mathcal B }[d](u)\sim 1/{(2-u)}^{{\widetilde{\gamma }}_{2}}$, the D = 4 OPE term of the Adler function to be of the form $1/{({Q}^{2})}^{2};$ (ii) for the u = 3 IR renormalon term ${ \mathcal B }[d]{(u)\sim 1/(3-u)}^{{\widetilde{\gamma }}_{3}}$ to be of the form $1/{({Q}^{2})}^{3};$ (iii) and for the u = 3 IR renormalon term ${ \mathcal B }[d]{(u)\sim 1/(3-u)}^{{\widetilde{\gamma }}_{3}\,+\,1}$ to be of the form $1/{({Q}^{2})}^{3}/a({Q}^{2})$. These three terms (D = 4, 6) are taken into account in the OPE (3.18).

The D = 0 contribution $d{({Q}^{2})}_{D\,=\,0}$ to the Adler function in the sum rule contour integral (3.14) is evaluated in three different ways. We apply two variants of fixed order perturbation theory (FOPT). In the first variant, the powers of $a{(\kappa {\sigma }_{{\rm{m}}}{e}^{i\phi })}^{n}$ are expressed as truncated Taylor series in powers of a(κ σ_m) (FO). In the second variant, $d{({Q}^{2})}_{D\,=\,0}$ is expressed as the sum of the logarithmic derivatives ${\widetilde{a}}_{n}({Q}^{2})$ $[\propto {(d/d\mathrm{ln}{Q}^{2})}^{n-1}a({Q}^{2})$], and then ${\widetilde{a}}_{n}(\kappa {\sigma }_{{\rm{m}}}{e}^{i\phi })$ are expressed as truncated Taylor series of ${\widetilde{a}}_{k}(\kappa {\sigma }_{{\rm{m}}})$ ($\widetilde{\mathrm{FO}}$). The third way of evaluation is the use of the inverse Borel transformation of $d{({Q}^{2})}_{D\,=\,0}$, where the Borel integral is evaluated with the Principal Value (PV) prescription; in the integrand, the Borel transform ${ \mathcal B }[d](u)$ is taken as as a series consisting of the mentioned (renormalon-related) inverse powers ∼(p − u)^k /(p − u)^γ (k = 0, 1, …), where the series is truncated; this truncation requires for $d{({\sigma }_{{\rm{m}}}{e}^{i\phi })}_{D\,=\,0}$ introduction of an additional correction polynomial $\delta d({\sigma }_{{\rm{m}}}{e}^{i\phi }{)}_{D\,=\,0}^{[{N}_{t}]}$ in powers of a(Q²). In all the three methods, a truncation index N_t is involved, i.e. only the terms up to the power a^N _t (or ${\widetilde{a}}_{{N}_{t}}$ in $\widetilde{\mathrm{FO}}$) are taken into account.

We apply the Laplace−Borel sum rules, with the weight function (3.16), to the ALEPH V + A data with ${\sigma }_{\max }(\equiv {\sigma }_{{\rm{m}}})$ =2.8 GeV² (i.e. the last two bins are excluded due to large uncertainties). In the sum rule (3.14), this gives on both sides the Borel–Laplace sum rule quantity B(M²; σ_m). In practice, the rule is applied to the real parts only, $\mathrm{Re}{B}_{\exp }({M}^{2};{\sigma }_{{\rm{m}}})=\mathrm{Re}{B}_{\mathrm{th}}({M}^{2};{\sigma }_{{\rm{m}}})$, and for the scale parameters M² along rays in the first quadrant: ${M}^{2}=| {M}^{2}| \exp (i{\rm{\Psi }})$ with 0 ≤ Ψ < π/2. We minimise (with respect to α_S , 〈O₄〉, $\langle {O}_{6}^{(1)}\rangle$ and $\langle {O}_{6}^{(2)}\rangle$) the following sum of squares:

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \sum _{\alpha =0}^{n}{\left(\displaystyle \frac{\mathrm{Re}{B}_{\mathrm{th}}({M}_{\alpha }^{2};{\sigma }_{{\rm{m}}})-\mathrm{Re}{B}_{\exp }({M}_{\alpha }^{2};{\sigma }_{{\rm{m}}})}{{\delta }_{B}({M}_{\alpha }^{2})}\right)}^{2},\end{eqnarray} \tag{ 3.21 }$$

where $\{{M}_{\alpha }^{2}\}$ was taken as a dense set of points along the chosen rays with Ψ = 0, π/6, π/4 and 0.9 GeV² ≤ ∣M_α ∣² ≤ 1.5 GeV². We chose 11 equidistant points along each of the three rays, and the series (3.21) thus contains 33 terms (the fit results remain practically unchanged when the number of points is increased). In the sum (3.21), the quantities ${\delta }_{B}({M}_{\alpha }^{2})$ are the experimental standard deviations of $\mathrm{Re}{B}_{\exp }({M}_{\alpha }^{2};{\sigma }_{{\rm{m}}})$, with the ALEPH covariance matrix for the (V + A)-channel taken into account (see appendix C of [147] for more explanation). For each evaluation method (FO, $\widetilde{\mathrm{FO}}$, PV) and for each chosen truncation index N_t , the fit procedure gives us results, and the fit is usually of good quality, χ² ≲ 10⁻³ (figure 10).

Zoom In Zoom Out Reset image size

The truncation index N_t is then fixed by considering the first two double-pinched momenta a^(2,0)(σ_m) and a^(2.1)(σ_m) ⁷⁶ and requiring local stability of their values under the variation of N_t . The resulting extracted values of the coupling are

$$\begin{eqnarray}\begin{array}{rcl}{\alpha }_{S}{\left({m}_{\tau }^{2}\right)}^{(\mathrm{FO})} & = & 0.3228\pm 0.0003{(\exp )}_{+0.0070}^{-0.0026}{(\kappa )}_{+0.0079}^{-0.0103}{({d}_{4})}_{-0.0057}^{+0.0081}({N}_{t})\\ & = & {0.3228}_{-0.0121}^{+0.0134}\approx {0.323}_{-0.012}^{+0.013}\qquad ({N}_{t}={8}_{-3}^{+2}).\end{array}\end{eqnarray} \tag{ 3.22 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\alpha }_{S}{\left({m}_{\tau }^{2}\right)}^{(\widetilde{\mathrm{FO}})} & = & 0.3209\pm 0.0003{(\exp )}_{+0.0201}^{-0.0038}{(\kappa )}_{+0.0047}^{-0.0039}{({d}_{4})}_{-0.0084}^{+0.0293}({N}_{t})\\ & = & {0.3209}_{-0.0100}^{+0.0359}\approx {0.321}_{-0.010}^{+0.036}\qquad ({N}_{t}=5\pm 2).\end{array}\end{eqnarray} \tag{ 3.23 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\alpha }_{S}{\left({m}_{\tau }^{2}\right)}^{(\mathrm{PV})} & = & 0.3269\pm 0.0003{(\exp )}_{+0.0102}^{+0.0007}(\kappa {)}_{+0.0155}^{-0.0064}{({d}_{4})}_{-0.0006}^{+0.0092}({N}_{t}{)}_{-0.0067}^{+0.0167}(\mathrm{amb})\\ & = & {0.3269}_{-0.0093}^{+0.0266}\approx {0.327}_{-0.009}^{+0.027}\qquad ({N}_{t}={8}_{-3}^{+2}).\end{array}\end{eqnarray} \tag{ 3.24 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\alpha }_{S}{\left({m}_{\tau }^{2}\right)}^{(\mathrm{CI})} & = & 0.3488\pm 0.0005{(\exp )}_{+0.0004}^{+0.0078}(\kappa )\pm 0.0000{({d}_{4})}_{+0.0119}^{-0.0027}({N}_{t})\\ & = & {0.3488}_{-0.0028}^{+0.0142}\approx {0.349}_{-0.003}^{+0.014}\qquad ({N}_{t}={4}_{-1}^{+2}).\end{array}\end{eqnarray} \tag{ 3.25 }$$

The uncertainties are presented as separate terms. The variation of the renormalization scale parameter κ ≡ μ²/Q² was taken in the range 2/3 ≤ κ ≤ 2 (κ = 1 for the central values). The truncation index is N_t = 8, 5, 8 for the central cases of FO, $\widetilde{\mathrm{FO}}$ and PV.

The (truncated) Contour Improved perturbation theory (CIPT) results are also included in the above results, for comparison. However, the truncated CIPT approach for $d{({Q}^{2})}_{D\,=\,0}$ evaluation appears to require a different type of OPE in the D > 0 part of the contributions, because the renormalon structure and the related renormalon ambiguities are not reflected in the truncated CIPT series [120]. We thus include only FO, $\widetilde{\mathrm{FO}}$ and PV results in the average

$$\begin{eqnarray}&&\begin{array}{l}\,{\alpha }_{S}({m}_{\tau }^{2})={0.3235}_{-0.0126}^{+0.0138}\qquad (\mathrm{FO}+\widetilde{\mathrm{FO}}+\mathrm{PV})\\ \Rightarrow \ {\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1191\pm 0.0016.\end{array}\end{eqnarray} \tag{ 3.26 }$$

In table 6 we compare these results with some other in the literature.

Table 6.  ${\alpha }_{S}({m}_{\tau }^{2})$ values extracted by various groups from ALEPH τ-decay data applying sum rules and other methods.

Group	Sum rule	FO	CI	PV	Average
Baikov et al. 2008 [100]	a(2,1) = rτ	0.322 ± 0.020	0.342 ± 0.011	—	0.332 ± 0.016
Beneke and Jamin, 2008 [117]	a(2,1) = rτ	${0.320}_{-0.007}^{+0.012}$	—	0.316 ± 0.006	0.318 ± 0.006
Caprini, 2020 [148]	a(2,1) = rτ	—	—	0.314 ± 0.006	0.314 ± 0.006
Davier et al. 2013 [103]	a(i,j)	0.324	0.341 ± 0.008	—	0.332 ± 0.012
Pich and Sánchez, 2016 [104]	a(i,j)	0.320 ± 0.012	0.335 ± 0.013	—	0.328 ± 0.013
Boito et al. 2014 [132]	DV in a(i,j)	0.296 ± 0.010	0.310 ± 0.014	—	0.303 ± 0.012
Our prev. work, 2021 [136]	BL (O6, O8)	0.308 ± 0.007	—	${0.316}_{-0.006}^{+0.008}$	0.312 ± 0.007
This work, 2022 (also [114])	BL (${O}_{6}^{(1)},{O}_{6}^{(2)}$)	${0.323}_{-0.012}^{+0.013}$(FO)	—	${0.327}_{-0.009}^{+0.027}$	0.324 ± 0.013
		${0.321}_{-0.030}^{+0.021}$($\widetilde{\mathrm{FO}}$)

The results (3.26) can be significantly affected when the assumptions or methods are changed. For example, if we chose, instead of the central value d₄ = 275. , the upper upper bound d₄ = 338. of equation (3.20) as the central value, the results would decrease somewhat, to ${\alpha }_{S}({m}_{\tau }^{2})\approx 0.320\pm 0.015$ (corresponding to ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})\approx {0.1187}_{-0.0019}^{+0.0016}$), i.e. $\delta {\alpha }_{S}({m}_{\tau }^{2})\approx -0.0004$. If we took, instead of the two mentioned D = 6 terms in the OPE, the simple D = 6 and D = 8 OPE terms $[\sim 1/{({Q}^{2})}^{3}$ and $\sim 1/{({Q}^{2})}^{4}$], the central value would decrease by about $\delta {\alpha }_{S}({m}_{\tau }^{2})\approx -0.008$. In our previous work [136] we used the OPE with simple D = 6,8 terms, and took for d₄ higher values d₄ = 338 ± 63 than here equation (3.20). If we took N_t = 5 in all three methods (i.e. no extension of Adler function beyond d₄ a⁵), then the central value of ${\alpha }_{S}({m}_{\tau }^{2})$ in FO changes from 0.3288 to 0.3171, and in PV from 0.3269 to 0.3277 ⇒ for the average of the three methods the central value changes from 0.3235 to 0.3219 (correspondingly, ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ goes from 0.1191 to 0.1189), i.e. $\delta {\alpha }_{S}({m}_{\tau }^{2})=-0.0016\approx -0.002$, smaller.

According to the results (3.22)–(3.24), Borel–Laplace sum rules indicate that the theoretical uncertainties dominate over the experimental ones. Even if maximally strong correlations were assumed among the experimental Borel–Laplace sum rule values at different ${M}_{\alpha }^{2}$, the presented experimental uncertainty of α_S would increase by a factor of less than five. Part of these theoretical uncertainties would be reduced by: (1) the calculation of the five-loop Adler function coefficient d₄; (2) the use of the more complicated structure of the D = 6 OPE terms [146] and the corresponding terms in the D = 0 u = 3 IR renormalon structure; (3) the use of a variant of the QCD coupling a(Q²) without the Landau singularities in the D = 0 contribution, because this would allow for the resummation to all orders (no truncation) of the renormalon-motivated contribution D = 0 and would eliminate the renormalization scale ambiguity (κ). The high precision ALEPH determination of the τ spectral function represents an important source of data for better understanding the behaviour of QCD at the limit between the perturbative and nonperturbative regimes.

In [122, 149] an approach was proposed to reconcile the long-standing discrepancy between the fixed order (FOPT) and contour improved (CIPT) pQCD expansions of hadronic spectral function moments relevant for the precise determination of α_S from τ decays. This is achieved by a simple change of scheme of the gluon condensate matrix element so that it becomes renormalon-free. The technical aspects of the scheme change are similar to the well-known implementation of short-distance mass schemes for massive-quark-sensitive observables. The scheme relies on external knowledge about the gluon condensate renormalon norm and is capable of resolving the long-standing discrepancy between FOPT and CIPT predictions, under the assumption that the gluon condensate renormalon gives a sizeable contribution to QCD perturbative coefficients at the accessible intermediate orders. The approach is briefly outlined in the following. For details we refer to [122, 149].

As outlined in section 3.2, strong coupling determinations from hadronic τ decay data are based on weighted integrals of the experimental spectral functions with an upper bound ${s}_{0}\leqslant {m}_{\tau }^{2}$ and to integrals of the QCD Adler function over a closed contour in the complex momentum plane. For the latter one must set the renormalization scale in α_S when performing the contour integrals. The two widely used prescriptions are based on FOPT and CIPT. In FOPT, the renormalization scale is fixed at μ² = s₀, which leads to a power series in α_S (s₀). In CIPT, the scale is tied to the contour's momentum variable such that logarithms related to the QCD β-function are summed, so that the resulting series is not any more a power series in α_S . These two expansion methods exhibit a sizeable discrepancy at orders ${ \mathcal O }({\alpha }_{S}^{4})$ and ${ \mathcal O }({\alpha }_{S}^{5})$ that is larger than the individual renormalization scale variations. As a consequence, α_S determinations based on CIPT tend to yield higher values than those based on FOPT. This discrepancy has been one of the main sources of theory uncertainty in α_S from τ decays.

In [119, 120], Hoang and Regner suggested that the discrepancy is of infrared (IR) origin and largely dominated by the leading IR renormalon of the massless-quark QCD Adler function, which is associated with the gluon condensate (GC). This entails that, once this renormalon is subtracted from the Adler function's perturbation series, the truncated series of the two expansion methods should yield more consistent values already at intermediate orders, reducing the theoretical uncertainty in the extractions of α_S . In [122, 149] a concrete subtraction method of the gluon condensate renormalon was devised and its practical value was demonstrated.

The subtraction method is based on two premises. First, it is assumed that the GC renormalon gives a sizeable contribution to the Adler function's perturbative coefficients at intermediate orders (${ \mathcal O }({\alpha }_{S}^{3})$ to ${ \mathcal O }({\alpha }_{S}^{5})$). This assumption can be considered as natural, since the GC represents the leading IR sensitivity. It is also supported by all-order results in the large-β₀ limit and by renormalon models [117, 118] built for the Adler function in QCD. Second, our framework relies on the fact that the OPE condensate corrections not only provide nonperturbative corrections but also compensate for the associated factorial renormalon growth of the perturbative series coefficients, thus leading to an unambiguous theoretical description. The first premise is important. If it were not true, the CIPT-FOPT discrepancy problem would be unrelated to IR renormalons and no information on the norm of the GC renormalon could be gained from the known QCD corrections for the Adler function. The second premise is an established characteristics of multiloop QCD calculations in the limit of vanishing IR cutoff, where sensitivities to IR momenta are a cause of QCD perturbation series to be asymptotic. Since this is the common approach for the loop corrections for inclusive quantities based on dimensional regularization and the $\overline{\mathrm{MS}}$ scheme for the strong coupling, we call this approach to regularize IR momenta ‘$\overline{\mathrm{MS}}$ scheme’ as well. This should not be confused with the $\overline{\mathrm{MS}}$ scheme for α_S , which refers to the regularization of UV momenta. The concrete analytic expressions shown in this subsection below actually use the strong coupling in the C scheme (for the concrete value C = 0) [150], referred to as $\bar{{\alpha }_{S}}$ below. (See the appendix of [122] for concrete formulae to obtain the results for the strong coupling in the $\overline{\mathrm{MS}}$ scheme.)

In heavy-quark physics, it is well established that the leading (linear) IR sensitivity that arises in the pole mass renormalization scheme can be eliminated by switching to short-distance mass schemes [8, 151]. In [122, 149] a similar approach is proposed for the GC matrix element. One starts from the operator product expansion (OPE) for the massless quark Adler function. Using the $\overline{\mathrm{MS}}$ scheme (to regularize IR momenta), it can be cast in the form (using μ² = − s as renormalization scale for the strong coupling)

$$\begin{eqnarray}&&D(s)=\displaystyle \sum _{n=1}^{\infty }\,{\bar{c}}_{n,1}{\left(\displaystyle \frac{\bar{{\alpha }_{S}}(-s)}{\pi }\right)}^{n}+\displaystyle \frac{{C}_{\mathrm{4,0}}(\bar{{\alpha }_{S}}(-s))}{{s}^{2}}\langle {\bar{{ \mathcal O }}}_{\mathrm{4,0}}\rangle +\displaystyle \sum _{d=6}^{\infty }\displaystyle \frac{1}{{(-s)}^{d/2}}\displaystyle \sum _{i}{C}_{d,i}(\bar{{\alpha }_{S}}(-s))\langle {\bar{{ \mathcal O }}}_{d,{\gamma }_{i}}\rangle ,\end{eqnarray} \tag{ 3.27 }$$

where the first term is the perturbative contribution, and the 1/s OPE power corrections, starting from the term with d = 4, are the nonperturbative corrections. The perturbative coefficients ${\bar{c}}_{n,1}$ are exactly known up ${ \mathcal O }({\alpha }_{S}^{4})$ [100, 152]. It is customary to also include an estimate for the ${ \mathcal O }({\alpha }_{S}^{5})$ coefficient ${\bar{c}}_{\mathrm{5,1}}$ in phenomenological analyses [104, 121, 132, 136]. The terms $\langle {\bar{{ \mathcal O }}}_{d,{\gamma }_{i}}\rangle$ are nonperturbative vacuum matrix elements (condensates) and the C_d,i are the respective Wilson coefficients which are a power series in α_S (−s). The bar over the operator indicates that the condensates are defined in the $\overline{\mathrm{MS}}$ scheme (to regularize IR momenta).

The leading d = 4 OPE power correction arises from the GC and is central to this work. It can be cast in terms of the renormalization scale invariant GC matrix element $\langle {\bar{G}}^{2}\rangle$ as ($\bar{a}(-s)\equiv {\beta }_{0}\,\bar{{\alpha }_{S}}(-s)/(4\pi )$, β₀ = 11 − 2N_f/3)

$$\begin{eqnarray}&&\begin{array}{l}\delta {D}_{4,0}^{\mathrm{OPE}}(s)=\tfrac{1}{{s}^{2}}\displaystyle \frac{2{\pi }^{2}}{3}\,\left[1-\displaystyle \frac{22}{81}\,\bar{a}(-s)\right]\,\langle {\bar{G}}^{2}\rangle .\end{array}\end{eqnarray} \tag{ 3.28 }$$

Associated with the GC OPE correction is a renormalon singularity in the Borel function of the Adler function that has the form

$$\begin{eqnarray}&&\begin{array}{l}{B}_{\mathrm{4,0}}(u)=\left[1-\displaystyle \frac{22}{81}\,\bar{a}(-s)\right]\,\displaystyle \frac{{N}_{\mathrm{4,0}}}{{(2-u)}^{1+4{\hat{b}}_{1}}},\end{array}\end{eqnarray} \tag{ 3.29 }$$

where N_4,0 is the Adler function's GC renormalon norm and ${\hat{b}}_{1}={\beta }_{1}/2{\beta }_{0}^{2}$, with β₁ being the two-loop QCD β-function coefficient. This GC renormalon singularity contributes to the Adler function's perturbative series coefficients ${\bar{c}}_{n,1}$ in the form

$$\begin{eqnarray}&&\delta {\hat{D}}_{\mathrm{4,0}}(s)={N}_{\mathrm{4,0}}\,\left[1-\displaystyle \frac{22}{81}\,\bar{a}(-s)+\ldots \right]\,\displaystyle \sum _{n=1}^{\infty }\,{r}_{n}^{(4,0)}\,\bar{a}{(-s)}^{n},\end{eqnarray} \tag{ 3.30 }$$

where the terms ${r}_{n}^{(4,0)}={2}^{-(n+4{\hat{b}}_{1})}{\rm{\Gamma }}(n+4{\hat{b}}_{1})/{\rm{\Gamma }}(1+4{\hat{b}}_{1})$ diverge factorially. The use of the C-scheme [150] is convenient because the term $1/{(2-u)}^{1\,+\,4{\hat{b}}_{1}}$ in B_4,0(u) and the coefficients ${r}_{n}^{(4,0)}$ in $\delta {\hat{D}}_{\mathrm{4,0}}(s)$ are exact, i.e. they do not receive any further higher-order corrections. The first premise entails that N_4,0 is sufficiently sizeable, such that the series in $\delta {\hat{D}}_{\mathrm{4,0}}(s)$ makes up for a sizeable contribution in the coefficients ${\bar{c}}_{n,1}$ at the intermediate orders relevant for phenomenological analyses. In the context of this assumption it has already been shown in [119, 120], that the FOPT-CIPT discrepancy can be caused by the diverging Adler function series contributions given in $\delta {\hat{D}}_{\mathrm{4,0}}(s)$.

Let us now turn to the construction of the renormalon-free (RF) gluon condensate scheme. The RF scheme only deals with the GC renormalon; all other renormalons are strictly unaltered. The starting point is by imposing that the order-dependence of $\langle {\bar{G}}^{2}\rangle$ that compensates the factorial growth of the coefficients in equation (3.30) is made explicit, while maintaining the generic form of the GC OPE correction in equation (3.28). The relation between the original, order dependent, $\overline{\mathrm{MS}}$ GC matrix element and a new renormalon-free GC matrix element, 〈G²〉(R²), is then

$$\begin{eqnarray}&&\langle {\bar{G}}^{2}{\rangle }^{(n)}\equiv \,\langle {G}^{2}\rangle ({R}^{2})-\,{R}^{4}\,\displaystyle \sum _{{\ell }=1}^{n}\,{N}_{g}\,{r}_{{\ell }}^{(4,0)}\,{\bar{a}}_{R}^{{\ell }},\end{eqnarray} \tag{ 3.31 }$$

where ${\bar{a}}_{R}\equiv {\beta }_{0}\,\bar{{\alpha }_{S}}({R}^{2})/(4\pi )$ and the order dependence on the left-hand side has been made explicit. The term N_g is the universal GC renormalon norm, related to N_4,0 by N_g = 3N_4,0/(2π²). The condensate 〈G²〉(R²) depends explicitly on the scale R, which sets its parametric size to be ${ \mathcal O }({R}^{4})$ (instead of ${ \mathcal O }({{\rm{\Lambda }}}_{\mathrm{QCD}}^{4})$ for the $\overline{\mathrm{MS}}$ GC $\langle {\bar{G}}^{2}{\rangle }^{(n)}$). The scale R plays the role of an IR factorization scale which can in principle be chosen arbitrarily. In practice, its value should be chosen smaller, but still of the order of, the typical dynamical scale of the observable of interest to avoid the appearance of (or, equivalently, to sum potentially) large logarithms. The dependence of 〈G²〉(R²) on R is controlled by an R-evolution equation [153] (see [122] for its precise form).

From a practical perspective, the R dependence of 〈G²〉(R²) is somewhat inconvenient. Therefore, in a second constructive step, a scale-invariant renormalon-free GC matrix element is defined. This is achieved by the addition of a function which obeys the same R-evolution equation as 〈G²〉(R²). This is possible because the R-derivative of the subtraction series on the r.h.s. of equation (3.31) is convergent for n → ∞ . Such a function is obtained from the Borel sum of the subtraction series, defined through the principal value (PV) prescription:

$$\begin{eqnarray}&&\begin{array}{l}{\bar{c}}_{0}({R}^{2})\equiv {R}^{4}\,\mathrm{PV}\,{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{{\rm{d}}u}{{(2-u)}^{1+4{\hat{b}}_{1}}}\,{e}^{-\tfrac{u}{{\bar{a}}_{R}}}.\end{array}\end{eqnarray} \tag{ 3.32 }$$

The result for ${\bar{c}}_{0}({R}^{2})$ can be given in closed analytic form [122]. Thus one can define a scale invariant renormalon-free GC matrix element, denoted as 〈G²〉^RF, by

$$\begin{eqnarray}&&\begin{array}{l}\langle {G}^{2}\rangle ({R}^{2})\equiv \langle {G}^{2}{\rangle }^{\mathrm{RF}}+{N}_{g}\,{\bar{c}}_{0}({R}^{2}).\end{array}\end{eqnarray} \tag{ 3.33 }$$

We denote as 〈G²〉^RF the GC in the ‘RF scheme’. The particular forms of the subtraction series in equation (3.31) and the function c₀(R²) are particular choices to define the RF scheme, but may in principle be chosen differently. In general, the subtraction must have the correct large order behavior to eliminate the factorial growth of the coefficients ${r}_{n}^{(4,0)}$, but could have additional convergent contributions. This would then imply a different form for ${\bar{c}}_{0}({R}^{2})$. Moreover, one could also add an additional constant to ${\bar{c}}_{0}({R}^{2})$. The RF scheme entails that the difference between the original $\overline{\mathrm{MS}}$ GC and the new scale-independent RF GC, 〈G²〉^RF, is formally ${ \mathcal O }({\bar{a}}_{R}^{n+1})$. In this sense, the modifications to the GC in our new RF scheme are minimal.

Using equations (3.33) and (3.28), it is straightforward to write down the resulting perturbation series for the Adler function in the RF GC scheme. Treating the term proportional to ${\bar{c}}_{0}({R}^{2})$, which is part of the definition of 〈G²〉^RF, like a tree-level contribution, the perturbation series has the form: (${\bar{c}}_{n}\equiv 4{\bar{c}}_{n,1}/{\beta }_{0}$ for n = 1, 2, …)

$$\begin{eqnarray}\begin{array}{rcl}{\hat{D}}^{\mathrm{RF}}(s,{R}^{2}) & = & \tfrac{1}{{s}^{2}}\,\left[1-\displaystyle \frac{22}{81}\bar{a}(-s)\right]\,{N}_{\mathrm{4,0}}\,{\bar{c}}_{0}({R}^{2})+\,\displaystyle \sum _{n=1}^{\infty }\,{\bar{c}}_{n}\,{\bar{a}}^{n}(-s)\\ & & -\left[1-\displaystyle \frac{22}{81}\,\bar{a}(-s)\right]\,{N}_{\mathrm{4,0}}\,\displaystyle \frac{{R}^{4}}{{s}^{2}}\displaystyle \sum _{n=1}^{\infty }{r}_{n}^{(4,0)}{\bar{a}}_{R}^{{\ell }}.\end{array}\end{eqnarray} \tag{ 3.34 }$$

It is essential to reexpand and truncate the perturbative series terms (excluding the ${\bar{c}}_{0}$ term) coherently, using the strong coupling at a common renormalization scale. Only then the cancellation of the GC renormalon is realized is a consistent way. The treatment of the ‘tree-level’ term proportional to ${\bar{c}}_{0}({R}^{2})$ entails that the GC OPE correction retains its form of equation (3.28) with $\langle {\bar{G}}^{2}\rangle$ replaced by 〈G²〉^RF. Furthermore, it is possible to show that the Borel sum of the perturbation series for the RF scheme Adler function ${\hat{D}}^{\mathrm{RF}}(s,{R}^{2})$ based on the PV prescription (as shown in equation (3.32)) remains the same as that of the original $\overline{\mathrm{MS}}$ scheme Adler function $\hat{D}(s)={\sum }_{n\,=\,1}^{\infty }{\bar{c}}_{n}{\bar{a}}^{n}(-s)$ independently of the value of N_g (or N_4,0) and the choice for R. Variations of R therefore vanish in the limit of larger-order truncations. The value of norm N_g is an input to the RF scheme that must be supplemented independently.

In [122], the effectiveness of the RF scheme for the perturbation series of the hadronic τ decay width has been demonstrated. In the following we quote some results given in [122], where the known QCD corrections up to ${ \mathcal O }({\alpha }_{S}^{4})$ have been included and an estimate for the 6-loop coefficient ${\bar{c}}_{\mathrm{5,1}}$, consistent with the values used in the recent analyses [104, 121, 132, 136] has been employed. In addition, corrections from beyond ${ \mathcal O }({\alpha }_{S}^{5})$ were considered from a renormalon Borel model for the Adler function, based on the works of [117, 118], which has been shown there to be realistic within the first premise mentioned above. The model provides the concrete value N_g = 0.64, so that the decay width series in the RF scheme can be studied in a concrete way. In the left panel of figure 11, the results for FOPT (red) and CIPT (blue) are shown in the $\overline{\mathrm{MS}}$ scheme for the GC. The vertical dashed line indicates the order up to which current state-of-the phenomenological analyses are carried out (including the concrete perturbative coefficients up to 6-loop), and for orders beyond the series relies on the renormalon model. The FOPT and CIPT series exhibit the well-known discrepancy at intermediate orders, which does not diminish with the successive inclusion of higher-order terms predicted by the model. In the context of the first premise, the results show that the discrepancy is systematic and not related to missing higher orders. The two series eventually run into the divergent behavior expected for asymptotic series. The right panel displays the results in the RF GC scheme for R = 0.8 m_τ . The FOPT series in the RF scheme is almost unaltered. This can be understood, since the decay width receives only a tiny contribution from the GC through the s-dependence of the Wilson coefficient's 1-loop correction and because the dominant part of the subtraction series in FOPT remains real-valued and factors out of the contour integral. The CIPT series, however, changes dramatically. This modification is by far larger than the tiny size of the GC OPE corrections itself. This is because the subtraction series needs to be expanded according to the CIPT prescription such that the contour integration modifies the contributions from the subtraction series in a nontrivial way. The fact that the resulting numerical effect is so large, corroborates the finding of [119, 120] that the CIPT series (in the $\overline{\mathrm{MS}}$ GC scheme) is not compatible with the standard analytic form of OPE corrections shown in equation (3.27). Remarkably, in the RF GC scheme the discrepancy between the two series diminishes order by order and at ${ \mathcal O }({\alpha }_{S}^{5})$ it is already significantly reduced. For even higher orders (in the context of the Borel model), the two series become fully compatible and approach essentially the same value. The same observations are made for any other spectral function moment with a suppressed GC. At the same time, the convergence of the perturbation series for moments with an enhanced GC is substantially improved.

Zoom In Zoom Out Reset image size

In [149] the practical value of the RF GC scheme in phenomenological analyses using the perturbative coefficients up to ${\bar{c}}_{\mathrm{5,1}}$ was analyzed in detail. Is was shown that N_g can be determined with a relative uncertainty of 40%. Following the two recent state-of-the-art strong coupling determination analyses at ${ \mathcal O }({\alpha }_{S}^{5})$ of [104] (in the tOPE approach) and [121] (in the DV-model approach) it was demonstrated that the RF GC scheme can successfully reconcile the extractions of ${\alpha }_{S}({m}_{\tau }^{2})$ based on CIPT and FOPT for both approaches to treat the nonperturbative corrections. The additional uncertainties that arise in the RF GC scheme due to variations of R and the uncertainty of N_g only lead to a small or moderate increase of the final uncertainty of ${\alpha }_{S}({m}_{\tau }^{2})$, and affect mainly the CIPT expansion results. The RF GC scheme thus constitutes a powerful new ingredient for future analyses of τ hadronic spectral function moments. Delicate issues such as the adequate treatment of nonperturbative corrections can now be studied without having to also deal with the CIPT-FOPT discrepancy problem.

Acknowledgments—DB and MJ would like to thank the Particle Physics Group of the University of Vienna for hospitality. We acknowledge partial support by the FWF Austrian Science Fund under the Doctoral Program ‘Particles and Interactions’ No. W1252-N27 and under the Project No. P32383-N27. DB's work was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001 and by the São Paulo Research Foundation (FAPESP) Grant No. 2021/06756-6.

We describe the measurement of the strong coupling constant ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ based on the scheme-invariant evolution of unpolarized and polarized flavour non-singlet structure functions at N³LO accuracy with a precision at the subpercent level. Measurements of this kind can be performed at future facilities such as the EIC and LHeC, provided both proton and deuteron targets are used. The theory framework for this is already available. The measurement requires excellent control of the experimental systematics.

Currently, the values of the strong coupling constant ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ as determined in different classes of measurement, and even inside these classes, is obtained at different values differing by several standard deviations [154–156], reaching an individual accuracy of ∼1%. One reason for this lies in the experimental systematics but also in different theoretical approaches being applied. We will consider the measurement of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ using deep-inelastic scattering (DIS) data. To minimize theoretical uncertainties, the measurement method has to be as simple as possible and widely free of effects that are difficult to control or are even widely unknown. Usually one performs mixed twist-2 flavour non-singlet/singlet analyses on a wide host of DIS and other hard scattering data, which are required because, besides the flavour non-singlet parton distribution functions, those of the different sea quark and gluon distributions are needed as well. Furthermore, the range of Q² usually includes also the region in which higher-twist contributions have to be fitted in addition. Also, the treatment of heavy-quark effects varies in the different approaches, although there is no such freedom in general. Some of the cross sections used in the fits may not have the same higher-order correction as the massless DIS cross sections. The presence of all these different distribution functions will cost a large part of the statistics power to be determined and various parameter correlations, and potentially introduces theoretical biases [157, 158]. All this complicates the measurement of the strong coupling constant in using approaches of this kind.

The situation is completely different in the case of scheme-invariant flavour non-singlet evolution of the structure functions ${F}_{2}^{\mathrm{NS}}(x,{Q}^{2})$ or ${g}_{1}^{\mathrm{NS}}(x,{Q}^{2})$. The latter structure function usually needs a much higher luminosity, since in addition the longitudinal polarization difference has to be carried out. To form the corresponding data sets, DIS off proton and deuteron targets has to be measured and a reliable description of the deuteron wave function effects is needed. In both cases the input distribution is a measured structure function: ${F}_{2}^{\mathrm{NS}}(x,{Q}_{0}^{2})$ or ${g}_{1}^{\mathrm{NS}}(x,{Q}_{0}^{2})$, with experimental error bands. Both the shape of these quantities and their experimental uncertainties can be parameterized at sufficient precision, forming the input for the one-dimensional evolution equation [159]. What remains to be determined in the fit of the data for ${Q}^{2}\gt {Q}_{0}^{2}$ is the strong coupling constant ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$, the precision of which receives also contributions from the experimental uncertainty of the measured input distribution.

In the following we describe the theoretical basis of the scheme-invariant measurement of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ and illustrate a few relevant aspects numerically, following [160], which may be performed in future experiments at EIC [161] or the LHeC [162]. We consider the combination of structure functions

$$\begin{eqnarray}&&{F}_{2}^{\mathrm{NS}}(x,{Q}^{2})={F}_{2}^{p}-\displaystyle \frac{1}{2}{F}_{2}^{d}=\displaystyle \frac{1}{6}{{xC}}_{2}^{\mathrm{NS}}(x,{Q}^{2})\otimes {v}_{3}(x,{Q}^{2}),\end{eqnarray} \tag{ 4.1 }$$

where ⨂ denotes the Mellin convolution, ${C}_{2}^{\mathrm{NS}}(x,{Q}^{2})$ is the corresponding unpolarized flavour non-singlet Wilson coefficient and

$$\begin{eqnarray}&&{v}_{3}(x,{Q}^{2})={u}_{v}(x,{Q}^{2})-{d}_{v}(x,{Q}^{2}),\end{eqnarray} \tag{ 4.2 }$$

the difference of valence u- minus d-quark distributions. Nor the flavour singlet, gluon, or special sea-quark distributions enter the partonic input distribution. However, the different flavours contribute through virtual QCD corrections, including charm and bottom quarks.

The scale evolution of ${F}_{2}^{\mathrm{NS}}(x,{Q}^{2})$ can be described by an evolution operator E_NS(x, Q²)

$$\begin{eqnarray}&&{F}_{2}^{\mathrm{NS}}(x,{Q}^{2})={E}_{\mathrm{NS}}(x,{Q}^{2},{Q}_{0}^{2};{m}_{c},{m}_{b})\otimes {F}_{2}^{\mathrm{NS}}(x,{Q}_{0}^{2}),\end{eqnarray} \tag{ 4.3 }$$

which reads in Mellin space

$$\begin{eqnarray}&&\begin{array}{l}{E}_{\mathrm{NS}}({Q}^{2},{Q}_{0}^{2})={\left(\displaystyle \frac{a}{{a}_{0}}\right)}^{-\tfrac{{P}_{0}}{2{\beta }_{0}}}\left\{1+\displaystyle \frac{a-{a}_{0}}{2{\beta }_{0}^{2}}\right.\\ \,\times \ \left\{\left[1+{a}^{2}{C}_{2}^{{Q}^{2}}-{a}_{0}^{2}{C}_{2}^{{Q}_{0}^{2}}\right]\left(2{\beta }_{0}^{2}{C}_{1}-{\beta }_{0}{P}_{1}+{\beta }_{1}{P}_{0}\right)-\displaystyle \frac{\left({a}^{2}-{a}_{0}^{2}\right)}{4{\beta }_{0}^{3}}\right.\\ \,\times \,\left(2{\beta }_{0}^{2}{C}_{1}-{\beta }_{0}{P}_{1}+{\beta }_{1}{P}_{0}\right)\left[2{\beta }_{0}^{3}{C}_{1}^{2}+{\beta }_{0}^{2}{P}_{2}-{\beta }_{0}{\beta }_{1}{P}_{1}+\left({\beta }_{1}^{2}-{\beta }_{0}{\beta }_{2}\right){P}_{0}\right]+\displaystyle \frac{\left({a}^{2}+{{aa}}_{0}+{a}_{0}^{2}\right)}{3{\beta }_{0}^{2}}\\ \,\times \,\left[2{\beta }_{0}^{4}{C}_{1}^{3}-{\beta }_{0}^{3}{P}_{3}+{\beta }_{0}^{2}{\beta }_{1}{P}_{2}+\left({\beta }_{0}^{2}{\beta }_{2}-{\beta }_{0}{\beta }_{1}^{2}\right){P}_{1}+\left({\beta }_{0}^{2}{\beta }_{3}-2{\beta }_{0}{\beta }_{1}{\beta }_{2}+{\beta }_{1}^{3}\right){P}_{0}\right]+\displaystyle \frac{a-{a}_{0}}{4{\beta }_{0}^{2}}\left(2{\beta }_{0}^{2}{C}_{1}\right.\\ {\,-\,{\beta }_{0}{P}_{1}+{\beta }_{1}{P}_{0})}^{2}+\displaystyle \frac{({a-{a}_{0})}^{2}}{24{\beta }_{0}^{4}}{\left(2{\beta }_{0}^{2}{C}_{1}-{\beta }_{0}{P}_{1}+{\beta }_{1}{P}_{0}\right)}^{3}-\displaystyle \frac{a+{a}_{0}}{2{\beta }_{0}}\left[2{\beta }_{0}^{3}{C}_{1}^{2}+{\beta }_{0}^{2}{P}_{2}-{\beta }_{0}{\beta }_{1}{P}_{1}\right.\\ \left.\left.\left.\,+\,{P}_{0}\left({\beta }_{1}^{2}-{\beta }_{0}{\beta }_{2}\right)\right]\Space{0ex}{3.2ex}{0ex}\right\}+{a}^{2}{C}_{2}^{{Q}^{2}}-{a}_{0}^{2}{C}_{2}^{{Q}_{0}^{2}}-{C}_{1}\left[{a}^{3}{C}_{2}^{{Q}^{2}}-{a}_{0}^{3}{C}_{2}^{{Q}_{0}^{2}}\right]+{a}^{3}{C}_{3}^{{Q}^{2}}-{a}_{0}^{3}{C}_{3}^{{Q}_{0}^{2}}\Space{0ex}{3.6ex}{0ex}\right\}.\end{array}\end{eqnarray} \tag{ 4.4 }$$

Here a = a(Q²) denotes the strong coupling constant and ${a}_{0}=a({Q}_{0}^{2})$, P_i are the non-singlet splitting functions and C_i = c_i + h_i the expansion coefficients of the Wilson coefficient, where c_i is the massless contribution and h_i the massive contribution, where ${h}_{1}=0,{h}_{2}={h}_{2}^{c}+{h}_{2}^{b}$ and ${h}_{3}={h}_{3}^{c}+{h}_{3}^{b}+{h}_{3}^{{cb}}$, i.e. there are mixed charm and bottom contributions from three-loops onward [160] for details. The dependence on the heavy-quark mass is logarithmic, with highest logarithmic powers ${\mathrm{ln}}^{k}({Q}^{2}/{m}_{h}^{2})$ for h_k . While the non-singlet splitting functions to three-loop order are known [163, 164], the four-loop non-singlet splitting function P₃ is not yet known, but it can be very well described by a Padé approximation. Eight Mellin moments are known, with the earliest calculation [165], and the presently available set [166]. The massless three-loop Wilson coefficients were calculated in [167] and the single and double mass three-loop heavy flavour corrections in [168]. At two-loop order the corrections are even available for the whole kinematic region [169, 170]. Different proposals to perform scheme-invariant evolution both in the non-singlet and singlet case have been made since 1979, see [92–100] of [160].

To illustrate the potential of the ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ measurement using the present method, we show in figure 12 (left) the ratio of ${F}_{2}^{\mathrm{NS}}(x,{Q}^{2})$ to ${F}_{2}^{\mathrm{NS}}(x,{Q}_{0}^{2})$ for ${Q}_{0}^{2}=10$ GeV² up to scales Q² = 10⁴ GeV². In the lower x region positive corrections up to a factor of 1.5 are reached, while at large x the corrections are negative. In figure 12 (right) we illustrate the impact of the heavy flavour corrections due to charm and bottom quark effects. In the region below x = 0.6 they are bound to ±1.5% and grow for larger values of x. This shows their importance, because the future measurements will be performed at subpercent experimental accuracy. In [160] we also performed related studies for the case of the polarized structure function g₁(x, Q²).

Zoom In Zoom Out Reset image size

A precision determination of the strong coupling constant ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ requires a high luminosity measurement of a sufficiently simple inclusive observable. The measurement must be carried out under stringent systematic control. Such a measurement would have been possible in the past, if the proposal [171] would have been carried out. It has not been possible at HERA, since deuterons have not been probed [172] and the reconstruction of a non-singlet structure function from charged current data has not been precise enough. Given a sufficient preparation, the measurement can be carried out at the EIC using proton and deuteron targets. LHeC may also perform such a measurement, provided also deuteron data will be available and the statistics for the non-singlet measurement is high enough. The theoretical analysis method is then scheme-invariant evolution in the flavour non-singlet case for the structure function ${F}_{2}^{\mathrm{NS}}(x,{Q}^{2})$. Both the light and heavy flavour corrections for this quantity are known at the level of the twist-2 approximation for Q² ≥ 10 GeV², W² ≥ 15 GeV² [173], to measure ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ at an accuracy well below the 1% level. This is one way to decide what is the correct value of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$.

To summarize, a high luminosity measurement of F₂ ^p and F₂ ^d at the EIC would allow to perform the N³LO measurement of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ using the scheme-invariant method discussed here.

Acknowledgments—Support from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 764 850, SAGEX, is gratefully acknowledged.

We scrutinize the DIS F₂(x, Q²) structure functions (SFs) measured by the SLAC, NMC, and BCDMS experiments [174–179] at NNLO accuracy in massless perturbative QCD in order to extract ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$. The so-called deep-inelastic scattering (DIS) scheme of the SFs [180] is considered, which leads to effective resummation of large-x logarithms into the Wilson coefficient function. The study presented here is a continuation of investigations carried out in closely related papers [181–185].

The function F₂(x, Q²) is represented as a sum of the leading twist (LT) ${F}_{2}^{\mathrm{pQCD}}(x,{Q}^{2})$ and the twist-four terms (hereinafter, the superscripts pQCD and LT denote the twist-two approximation with and without target-mass corrections):

$$\begin{eqnarray}&&{F}_{2}(x,{Q}^{2})={F}_{2}^{\mathrm{pQCD}}(x,{Q}^{2})\left(1+\displaystyle \frac{{\tilde{h}}_{4}(x)}{{Q}^{2}}\right).\end{eqnarray} \tag{ 4.5 }$$

For large x values, gluons do not nearly contribute and the Q² evolution of the twist-two DIS F₂(x, Q²) SF is well determined by the so-called nonsinglet (NS) part. In this approximation, there is a direct relation between the moments of the DIS F₂(x, Q²) SF and the moments of the NS parton distribution function f(x, Q²)

$$\begin{eqnarray}&&{M}_{n}({Q}^{2})={\int }_{0}^{1}{{dxx}}^{n-2}{F}_{2}^{\mathrm{LT}}(x,{Q}^{2}),\,\,{\bf{f}}(n,{Q}^{2})\,=\,{\int }_{0}^{1}{{dxx}}^{n-1}\,{\bf{f}}(x,{Q}^{2})\end{eqnarray} \tag{ 4.6 }$$

which can be expressed as follows

$$\begin{eqnarray}&&{M}_{n}({Q}^{2})=R(f)\times C(n,{a}_{s}({Q}^{2}))\times {\bf{f}}(n,{Q}^{2}),\end{eqnarray} \tag{ 4.7 }$$

where the strong coupling constant

$$\begin{eqnarray}&&{a}_{s}({Q}^{2})=\displaystyle \frac{{\alpha }_{S}({Q}^{2})}{4\pi }\end{eqnarray} \tag{ 4.8 }$$

and the Wilson coefficient function is denoted as C(n, a_s (Q²)). The constant R(f) = 1/6 for f = 4 [186].

4.2.1. Strong coupling constant derivation

The strong coupling constant is found from the corresponding renormalization group equation. At the NLO level, ${a}_{s}^{\mathrm{NLO}}({Q}^{2})\equiv {a}_{1}({Q}^{2})$, the latter looks like

$$\begin{eqnarray}&&\displaystyle \frac{1}{{a}_{1}({Q}^{2})}-\displaystyle \frac{1}{{a}_{1}({m}_{{\rm{Z}}}^{2})}+{b}_{1}\mathrm{ln}\left[\displaystyle \frac{{a}_{1}({Q}^{2})}{{a}_{1}({m}_{{\rm{Z}}}^{2})}\displaystyle \frac{(1+{b}_{1}{a}_{1}({m}_{{\rm{Z}}}^{2}))}{(1+{b}_{1}{a}_{1}({Q}^{2}))}\right]={\beta }_{0}\mathrm{ln}\left(\displaystyle \frac{{Q}^{2}}{{m}_{{\rm{Z}}}^{2}}\right).\end{eqnarray} \tag{ 4.9 }$$

At NNLO level, ${a}_{s}^{\mathrm{NNLO}}({Q}^{2})\equiv {a}_{2}({Q}^{2})$, the strong coupling constant is derived from the following equation:

$$\begin{eqnarray}\begin{array}{rcl} & & \displaystyle \frac{1}{{a}_{2}({Q}^{2})}-\displaystyle \frac{1}{{a}_{2}({m}_{{\rm{Z}}}^{2})}+{b}_{1}\mathrm{ln}\left[\displaystyle \frac{{a}_{2}({Q}^{2})}{{a}_{2}({m}_{{\rm{Z}}}^{2})}\sqrt{\displaystyle \frac{1+{b}_{1}{a}_{2}({m}_{{\rm{Z}}}^{2})+{b}_{2}{a}_{2}^{2}({m}_{{\rm{Z}}}^{2})}{1+{b}_{1}{a}_{2}({Q}^{2})+{b}_{2}{a}_{2}({Q}^{2})}}\right]\\ & & \,+\left({b}_{2}-\displaystyle \frac{{b}_{1}^{2}}{2}\right)\times \left(I({a}_{s}({Q}^{2}))-I({a}_{s}({m}_{{\rm{Z}}}^{2}))\right)={\beta }_{0}\mathrm{ln}\left(\displaystyle \frac{{Q}^{2}}{{m}_{{\rm{Z}}}^{2}}\right).\end{array}\end{eqnarray} \tag{ 4.10 }$$

The expression for I is

$$\begin{eqnarray*}&&I({a}_{s}({Q}^{2}))=\left\{\begin{array}{c}\displaystyle \frac{2}{\sqrt{{\rm{\Delta }}}}\arctan \displaystyle \frac{{b}_{1}+2{b}_{2}{a}_{2}({Q}^{2})}{\sqrt{{\rm{\Delta }}}}\hspace{1.9cm}{\rm{for}}\,\,f=3,4,5;\,\,\,{\rm{\Delta }}\gt 0,\\ \displaystyle \frac{1}{\sqrt{-{\rm{\Delta }}}}\mathrm{ln}\left[\displaystyle \frac{{b}_{1}+2{b}_{2}{a}_{2}({Q}^{2})-\sqrt{-{\rm{\Delta }}}}{{b}_{1}+2{b}_{2}{a}_{2}({Q}^{2})+\sqrt{-{\rm{\Delta }}}}\right]\,\,{\rm{for}}\,f=6;\hspace{1.08cm}{\rm{\Delta }}\lt 0,\end{array}\right.\end{eqnarray*}$$

where ${\rm{\Delta }}=4{b}_{2}-{b}_{1}^{2}$ and b_i = β_i /β₀ are read off from the QCD β-function:

$$\begin{eqnarray*}&&\beta ({a}_{s})\,=\,-{\beta }_{0}{a}_{s}^{2}-{\beta }_{1}{a}_{s}^{3}-{\beta }_{2}{a}_{s}^{4}+\ldots .\end{eqnarray*}$$

The coefficient function C(n, a_s (Q²)) is then expressed in terms of the coefficients B_j (n), which are exactly known (for the odd n values, B_j (n) and Z_j (n) can be obtained by using the analytic continuation [187–189])

$$\begin{eqnarray}&&C(n,{a}_{s}({Q}^{2}))=1+{a}_{s}({Q}^{2}){B}_{1}(n)+{a}_{s}^{2}({Q}^{2}){B}_{2}(n)+{ \mathcal O }({a}_{s}^{3}).\end{eqnarray} \tag{ 4.11 }$$

The Q²-evolution of the PDF moments can be calculated within the framework of perturbative QCD:

$$\begin{eqnarray}&&\displaystyle \frac{{\bf{f}}(n,{Q}^{2})}{{\bf{f}}(n,{Q}_{0}^{2})}={\left[\displaystyle \frac{{a}_{s}({Q}^{2})}{{a}_{s}({Q}_{0}^{2})}\right]}^{\tfrac{{\gamma }_{0}(n)}{2{\beta }_{0}}}\times \displaystyle \frac{h(n,{Q}^{2})}{h(n,{Q}_{0}^{2})},\end{eqnarray} \tag{ 4.12 }$$

where

$$\begin{eqnarray}&&h(n,{Q}^{2})=1+{a}_{s}({Q}^{2}){Z}_{1}(n)+{a}_{s}^{2}({Q}^{2}){Z}_{2}(n)+{ \mathcal O }\left({a}_{s}^{3}\right),\end{eqnarray} \tag{ 4.13 }$$

and

$$\begin{eqnarray}&&{Z}_{1}(n)=\displaystyle \frac{1}{2{\beta }_{0}}[{\gamma }_{1}(n)-{\gamma }_{0}(n){b}_{1}],\ {Z}_{2}(n)=\displaystyle \frac{1}{4{\beta }_{0}}\left[{\gamma }_{2}(n)-{\gamma }_{1}(n){b}_{1}+{\gamma }_{0}(n)({b}_{1}^{2}-{b}_{2})\right]+\displaystyle \frac{1}{2}{Z}_{1}^{2}(n)\end{eqnarray} \tag{ 4.14 }$$

are combinations of the NLO and NNLO anomalous dimensions γ₁(n) and γ₂(n).

For large n (this corresponds to large x values), the coefficients ${Z}_{j}(n)\sim \mathrm{ln}n$ and ${B}_{j}(n)\sim {\mathrm{ln}}^{2j}n$. So, the terms ∼B_j (n) can lead to potentially large contributions and, therefore, should be resummed.

4.2.2. Scale dependence

We are going to consider the dependence of the results on the factorization μ_F scale caused by the truncation of a perturbative series [181]. This way, equation (4.7) takes the form:

$$\begin{eqnarray*}&&{M}_{n}({Q}^{2})=R(f)\times \hat{C}(n,{a}_{s}({k}_{F}{Q}^{2}))\times {\bf{f}}(n,{k}_{F}{Q}^{2}).\end{eqnarray*}$$

The function $\hat{C}$ is to be obtained from C by modifying the r.h.s. of equation (4.11) as follows:

$$\begin{eqnarray}&&\begin{array}{l}{a}_{s}({Q}^{2})\to {a}_{s}({k}_{F}{Q}^{2}),\,\,{B}_{1}(n)\to {B}_{1}(n)+\displaystyle \frac{1}{2}{\gamma }_{0}(n)\mathrm{ln}{k}_{F},\\ {B}_{2}(n)\to {B}_{2}(n)+\displaystyle \frac{1}{2}{\gamma }_{1}(n)\mathrm{ln}{k}_{F}+\left(\displaystyle \frac{1}{2}{\gamma }_{0}+{\beta }_{0}\right){B}_{1}\mathrm{ln}{k}_{F}+\displaystyle \frac{1}{8}{\gamma }_{0}({\gamma }_{0}+2{\beta }_{0}){\mathrm{ln}}^{2}{k}_{F}.\end{array}\end{eqnarray} \tag{ 4.15 }$$

Taking a special form for the coefficient k_F , we can decrease contributions coming from the terms ∼B_j (n). To accomplish this task, we consider the DIS-scheme [180], where the NLO corrections to the Wilson coefficients are completely cancelled by changes in the factorization scale.

In the NLO case

$$\begin{eqnarray}&&{a}_{s}({Q}^{2})\to {a}_{s}({k}_{\mathrm{DIS}}(n){Q}^{2})\equiv {a}_{n}^{\mathrm{DIS}}({Q}^{2}),\,\,{k}_{\mathrm{DIS}\ }(n)=\exp \left(\displaystyle \frac{-2{B}_{1}(n)}{{\gamma }_{0}(n)}\right)=\exp \left(\displaystyle \frac{-{r}_{1}^{\mathrm{DIS}\ }(n)}{{\beta }_{0}}\right),\end{eqnarray} \tag{ 4.16 }$$

where

$$\begin{eqnarray}&&{r}_{1}^{\mathrm{DIS}\ }(n)=\displaystyle \frac{2{B}_{1}(n){\beta }_{0}}{{\gamma }_{0}}\,\,\,{\rm{and}}\,\,\,{B}_{1}(n)\to {B}_{1}^{\mathrm{DIS}}=0.\end{eqnarray} \tag{ 4.17 }$$

The NLO coupling ${a}_{n}^{\mathrm{DIS}}({Q}^{2})$ obeys the following equation

$$\begin{eqnarray}&&\displaystyle \frac{1}{{a}_{n}^{\mathrm{DIS}}({Q}^{2})}-\displaystyle \frac{1}{{a}_{1}({m}_{{\rm{Z}}}^{2})}+{b}_{1}\mathrm{ln}\left[\displaystyle \frac{{a}_{n}^{\mathrm{DIS}}({Q}^{2})}{{a}_{1}({m}_{{\rm{Z}}}^{2})}\displaystyle \frac{(1+{b}_{1}{a}_{1}({m}_{{\rm{Z}}}^{2}))}{(1+{b}_{1}{a}_{n}^{\mathrm{DIS}}({Q}^{2}))}\right]={\beta }_{0}\mathrm{ln}\left(\displaystyle \frac{{Q}^{2}}{{m}_{{\rm{Z}}}^{2}}\right)-{r}_{1}^{\mathrm{DIS}}(n).\end{eqnarray} \tag{ 4.18 }$$

In the NNLO case, in addition to equations (4.16) and (4.17), there is also the following modification

$$\begin{eqnarray}&&{B}_{2}(n)\to {B}_{2}^{\mathrm{DIS}}(n)={B}_{2}(n)-\left(\displaystyle \frac{1}{2}+\displaystyle \frac{{\beta }_{0}}{{\gamma }_{0}(n)}\right){B}_{1}^{2}(n)-\displaystyle \frac{{\gamma }_{1}(n)}{{\gamma }_{0}(n)}\,{B}_{1}(n),\end{eqnarray} \tag{ 4.19 }$$

that leads to the cancellation of the large terms $\sim {\mathrm{ln}}^{4}(n)$ in ${B}_{1}^{\mathrm{DIS}}(n)$.

The NNLO coupling ${a}_{n}^{\mathrm{DIS}}({Q}^{2})$ obeys the equation

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{1}{{a}_{n}({Q}^{2})}-\displaystyle \frac{1}{{a}_{2}({m}_{{\rm{Z}}}^{2})}+{b}_{1}\mathrm{ln}\left[\displaystyle \frac{{a}_{n}({Q}^{2})}{{a}_{2}({m}_{{\rm{Z}}}^{2})}\sqrt{\displaystyle \frac{1+{b}_{1}{a}_{2}({m}_{{\rm{Z}}}^{2})+{b}_{2}{a}_{2}^{2}({m}_{{\rm{Z}}}^{2})}{1+{b}_{1}{a}_{n}({Q}^{2})+{b}_{2}{a}_{n}({Q}^{2})}}\right]\\ \,+\,\left({b}_{2}-\displaystyle \frac{{b}_{1}^{2}}{2}\right)\times \left(I({a}_{n}({Q}^{2}))-I({a}_{s}({m}_{{\rm{Z}}}^{2}))\right)={\beta }_{0}\mathrm{ln}\left(\displaystyle \frac{{Q}^{2}}{{m}_{{\rm{Z}}}^{2}}\right)-{r}_{1}^{\mathrm{DIS}}(n).\end{array}\end{eqnarray} \tag{ 4.20 }$$

4.2.3. Fit results

Our analysis is carried out for the moments of the F₂(x, Q²) SF defined in equation (4.6). Then, for each Q², the F₂(x, Q²) SF is recovered using the Jacobi polynomial decomposition method [190–192]:

$$\begin{eqnarray}&&{F}_{2}(x,{Q}^{2})={x}^{a}{(1-x)}^{b}\displaystyle \sum _{n=0}^{{N}_{\max }}{{\rm{\Theta }}}_{n}^{a,b}(x)\displaystyle \sum _{j=0}^{n}{c}_{j}^{(n)}(\alpha ,\beta ){M}_{j+2}({Q}^{2}),\end{eqnarray} \tag{ 4.21 }$$

where ${{\rm{\Theta }}}_{n}^{a,b}$ are the Jacobi polynomials, a, b are the parameters to be fit. The program MINUIT [193] is used to minimize the difference between experimental data and theoretical predictions for the F₂(x, Q²) SF.

We use free data normalizations for various experiments. As a reference set, the most stable data of hydrogen BCDMS are used at the value of the initial beam energy E₀ = 200 GeV. Contrary to previous analyses [181, 182], the cut Q² ≥ 2 GeV² is used throughout, since for smaller Q² values equations (4.18) and (4.20) have no real solutions.

The starting point of Q²-evolution is taken at ${Q}_{0}^{2}$ = 90 GeV². This value of ${Q}_{0}^{2}$ is close to the average values of Q² covering the corresponding data. Based on studies done in [184, 185], it is sufficient to take the maximum number of moments ${N}_{\max }=8$, and the cut 0.25 ≤ x ≤ 0.8 is applied on the data.

We work within the variable-flavour-number scheme (VFNS) [181]. To strengthen the effect of changing the sign of twist-four corrections, we also present results obtained in the fixed-flavour-number scheme with N_f = 4.

As one can see from table 7, the central values of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ are mostly close to each other upon taking into account total experimental and theoretical errors [181, 182]:

$$\begin{eqnarray}&&\pm 0.0022\,\,\,{\rm{(total}}\,{\rm{exp.}}\,{\rm{error)}},\,\,\,\left\{\begin{array}{c}+0.0028\\ -0.0016\end{array}\right.\,\,\,{\rm{(theor.}}\,{\rm{error)}}.\end{eqnarray} \tag{ 4.22 }$$

We plan to study these errors in more detail and present them in an upcoming publication.

Table 7. Parameter values of the twist-four term in different cases obtained in the analysis of data (314 points: Q² ≥ 2 GeV²) carried out within VFNS (FFNS).

	NLO	NLO	NNLO	NNLO
x	$\overline{\mathrm{MS}}$ scheme	DIS scheme	$\overline{\mathrm{MS}}$ scheme	DIS scheme
	χ2 = 246 (259)	χ2 = 238 (251)	χ2 = 241 (254)	χ2 = 242(249)
	${\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1195$	${\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1177$	${\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1177$	${\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1178$
	(0.1192)	(0.1179)	(0.1170)	(0.1171)
0.275	−0.245 ( − 0.264)	−0.187 ( − 0.174)	−0.188 ( − 0.204)	−0.141 ( − 0.170)
0.35	−0.243 ( − 0.252)	−0.111 ( − 0.134)	−0.188 ( − 0.193)	−0.133 ( − 0.149)
0.45	−0.191 ( − 0.187)	−0.040 ( − 0.094)	−0.172 ( − 0.158)	−0.110 ( − 0.104)
0.55	−0.116 (0.096)	−0.106 ( − 0.088)	−0.174 ( − 0.137)	−0.121 ( − 0.084)
0.65	0.054 (0.118)	−0.167 ( − 0.094)	−0.145 ( − 0.051)	−0.223 ( − 0.100)
0.75	0.337 (0.477)	−0.568 ( − 0.442)	0.115 (0.648)	−0.587 ( − 0.314)

From the table, it can also be seen that after resumming at large x values (i.e. in the DIS scheme), the twist-four corrections change sign at large values of x. Thus, unlike the standard analyses performed in [194], in this case, when twist-four corrections change the sign, (part of) the powerlike terms can be said to be swallowed up into a QCD analytic constant [195] just much like as it was observed at low-x values [196] in the framework of the so-called double asymptotic scaling approach [197, 198].

In previous papers [199, 200], where resummation at large values of x was performed within the framework of the Grunberg approach [201, 202], we saw only a decrease in the twist-four contribution, since the terms were not studied in detail. That is why we plan to include Grunberg’s approach in an analysis similar to the present one and study in some detail the values of the twist-four corrections.

Data from deep-inelastic scattering (DIS) experiments collected during the last few decades contributes significantly to the knowledge about the structure of the proton and the extraction of parton distributions (PDFs). The theory description builds on QCD factorization, which allows to express the structure functions in the charged-lepton- (or neutrino-) proton hard scattering with large momentum transfer Q schematically as

$$\begin{eqnarray}&&{F}_{a}={f}_{i}^{}\,\otimes {c}_{a,i},\qquad a=2,3,L,\end{eqnarray} \tag{ 4.23 }$$

where the process-dependent coefficient functions are denoted by c_a,i and the PDFs for a given fraction of the proton momentum x by f_i (x, μ²). Their dependence on the (renormalization and factorization) scale μ is governed by the well-known evolution equations

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial \mathrm{ln}{\mu }^{2}}\,{f}_{i}^{}(x,{\mu }^{2})=\left[{P}_{{ik}}^{}({\alpha }_{S}({\mu }^{2}))\otimes {f}_{k}^{}({\mu }^{2})\right](x),\end{eqnarray} \tag{ 4.24 }$$

where the standard convolution is abbreviated by ⨂. Both, the splitting functions ${P}_{{ik}}^{}$ and the coefficient functions by c_a,i are calculable in QCD perturbation theory in an expansion in powers of the strong coupling constant a_s ≡ α_S (μ²)/(4π),

$$\begin{eqnarray}&&P\,={a}_{s}\,{P}^{(0)}+\,{a}_{s}^{2}\,{P}^{(1)}+\,{a}_{s}^{3}\,{P}^{(2)}+\,{a}_{s}^{4}\,{P}^{(3)}+\,\ldots ,\end{eqnarray} \tag{ 4.25 }$$

$$\begin{eqnarray}&&{c}_{a,i}={c}_{a,i}^{(0)}+\,{a}_{s}\,{c}_{a,i}^{(1)}+\,{a}_{s}^{2}\,{c}_{a,i}^{(2)}\,+\,{a}_{s}^{3}\,{c}_{a,i}^{(3)}+\,\ldots .\end{eqnarray} \tag{ 4.26 }$$

Here, the first three terms define the NNLO predictions for DIS, which is currently the standard approximation [163, 203] for the splitting functions and [204–207] for the DIS coefficient functions at this accuracy. Form the perturbative expansion in equations (4.25) and (4.26) as well as from the PDF evolution in equation (4.24) it is obvious that there is, in general, significant correlation between the PDFs and the value of strong coupling constant ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ [158]. In the ABM PDF fits, this information can be obtained from the (positive-definite) covariance matrix [208].

Equation (4.23) holds up to power corrections suppressed by 1/Q², and the accessible range of kinematics in the momentum transfer Q and Bjorken x requires careful considerations of the respective kinematic regions. Typical cuts in the application to DIS data restrict the invariant mass of the hadronic system ${W}^{2}={m}_{p}^{2}+{Q}^{2}(1-x)/x$ (with the proton mass m_p ) to W² ≥ 12.5 GeV² and Q² ≥ 2.5–10 GeV² [208]. Depending on these cuts, an improved theoretical description of ${F}_{a}x,{Q}^{2}$ to account for higher-twist and target-mass corrections becomes necessary. Higher-twist corrections arise from the infinite tower of power corrections ${(1/{Q}^{2})}^{n}$, in the operator product expansion (with n = 1, 2, 3, ...). Higher-twist terms have a physical interpretation as multiparton correlations and modify the structure functions in equation (4.23) as

$$\begin{eqnarray}&&{F}_{a}^{\mathrm{ht}}(x,{Q}^{2})={F}_{a}^{\mathrm{TMC}}(x,{Q}^{2})+\displaystyle \frac{{H}_{a}^{\tau =4}(x)}{{Q}^{2}},\qquad \qquad a=2,T,\end{eqnarray} \tag{ 4.27 }$$

where the additive terms ${H}_{a}^{\tau =4}(x)$ models the twist-four contribution and ${F}_{a}^{\mathrm{TMC}}$ denotes the leading-twist structure function with target mass corrections, which account for the finite nucleon mass [208].

Another important aspect in the theoretical description of the leading-twist structure functions from equation (4.23) entering in equation (4.27) concerns the active number of flavours and the treatment of DIS heavy quark production [209] A fixed-flavour number scheme and the use the standard decoupling relations for heavy quarks in QCD in the transition from N_f = 3 to N_f = 5 is justified given the currently available kinematics in x and Q for DIS charm quark data [158]. Variable-flavour-number schemes attempting the resummation of large logarithms in the ratio ${Q}^{2}/{m}_{h}^{2}$ with the heavy-quark mass squared m² _h introduce additional theoretical uncertainties from the matching, which manifest themselves predominantly at small Q² [210].

The impact of higher-twist terms and target-mass corrections in equation (4.27) on the extracted value of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ to NNLO accuracy from world DIS data has been illustrated in figure 13, where it is shown that variants of the fits with no higher twist lead to systematically larger values of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$. The combined fit of PDFs and ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ to the world data shown in figure 13 delivers in the ABMP16 analysis at NNLO in QCD the value of [208]

$$\begin{eqnarray}&&{\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1145\pm 0.0009\,\qquad \mathrm{for}\,{N}_{{\rm{f}}}=5,\end{eqnarray} \tag{ 4.28 }$$

while the full ABMP16 analysis (also including LHC data on top-quark hadroproduction) finds the value

$$\begin{eqnarray}&&{\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1147\pm 0.0008\,\qquad \mathrm{for}\,{N}_{{\rm{f}}}=5.\end{eqnarray} \tag{ 4.29 }$$

Zoom In Zoom Out Reset image size

Higher-order QCD corrections beyond NNLO become important whenever high precision is needed for benchmark processes at the LHC and for the novel accurate DIS measurements expected at the future Electron Ion Collider (EIC). Then, determinations of PDFs and α_S at N³LO accuracy requires the calculation of the N³LO corrections in equations (4.25) and (4.26). Those for the DIS coefficient functions are already available [167, 218–220], including effects of massive quarks [168]. Work on the four-loop splitting functions in equation (4.25) to ensure QCD evolution equations at N³LO accuracy is ongoing [166, 217, 221] and the gain in the theoretical accuracy in the solution of equation (4.24) for the evolution of the PDFs is illustrated in figure 14 for a number of Mellin moments (N = 2 to N = 6) available from [217]. With the full N³LO contributions to DIS the residual theoretical uncertainty due the scale variation and the truncation of the perturbative series will be limited to 1% in the range of parton kinematics relevant for the current world DIS data and the EIC. With respect to the theory error, inclusive DIS data offer the most precise method to measure ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$.

Zoom In Zoom Out Reset image size

The CT18 global QCD analysis [222] updates the previous CTEQ-TEA PDFs, CT14 [223] and CT14HERA2 [224], adding a variety of high-precision data from the Large Hadron Collider (LHC). New measurements of single-inclusive jet production, Drell–Yan lepton pairs, top-quark pairs, as well as high transverse momentum (p_T) Z bosons from ATLAS, CMS and LHCb, at the center-of-mass energies of 7 and 8 TeV, provide significant additional sensitivity to the PDF determination. A new family of PDF sets, CT18, CT18A, CT18X, and CT18Z, are released at both NLO and NNLO. The nominal PDF set, CT18, is recommended for general collider phenomenology studies, while CT18A includes an additional ATLAS 7 TeV W/Z precision data set [225], which is found to be in tension with other data sets in the global fit. An alternative PDF set, CT18X, has adopted an x-dependent DIS scale, which captures the small-x behavior at low Q² and improves the QCD description of HERA DIS data. The CT18Z PDF set includes the features of CT18A and CT18X, in addition to having a slightly larger charm pole mass, m_c = 1.4 GeV versus m_c = 1.3 GeV. CT18Z maximizes the difference from CT18 PDFs, but preserves a similar goodness-of-fit. More details can be found in [222].

The final product—the published error PDF sets such as CT18—takes the strong coupling at a scale m_Z as the world average value ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.118$ [226]. Alternative error PDF sets are produced with a series of fixed ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ values for the use in the estimation of combined PDF+α_S uncertainties. As shown in CT10 [227], a change in α_S is partially compensated by changes in the PDF parameters. An α_S uncertainty can be defined, which quantifies the allowed variation of α_S when allowed to be freely varied in the PDF fit. In general, the ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ sensitivity to a specific data set is introduced either through radiative corrections, such as in the Drell–Yan pair production, or through scaling violations, such as in DIS.

Simultaneous fits of the α_S and PDFs were also performed during the CT18 study. An optimal way to explore experimental constraints on ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ is to examine the corresponding χ² variations with the Lagrangian multiplier (LM) technique [228]. In figures 15, 16 and 17, we adapt this technique to plot a a series of curves for Δχ² = χ²(α_S ) − χ²(α_S,0) for CT18 NNLO, CT18NLO, and CT18Z NNLO, respectively, where α_S,0 corresponds to the global χ² minimum in the PDF+α_S fit. The black solid curves are for the total Δχ², and the other curves are for Δχ² from the individual experiments with the highest sensitivities to α_S . From the figures, it is clear that the data sets and even groups of data sets have different preferences to ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ in terms of both the central values and uncertainties. The spread of the pulls on α_S by the data sets is broader than it would be normally expected from random fluctuations of data around theory, suggesting that a more conservative prescription for the estimation of the α_S uncertainty is necessary than the straightforward averaging over the data sets [229].

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

From the figures, we see that the HERA I and II combined DIS data [215] provide the strongest constraints on ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ in CT18 at both NLO and NNLO. The BCDMS proton data [177] play a slightly stronger role in the CT18Z PDF fit. These two DIS data sets prefer a lower value of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ than the world average, with a value about 0.114−0.116, but with larger uncertainties. On the other hand, we see that the hadron collider data, such as inclusive jet, top-quark pairs, and Drell–Yan production, pull the ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ value higher, with a preferred value of 0.117−0.119 for the CT18 NNLO case. The strongest pull comes ATLAS 8 TeV Z p_T data [230], followed by the LHCb 8 TeV W/Z measurement [231]. We also see significant pulls from inclusive jet production at ATLAS 7 TeV [232] and D0 Run II [233], as well as the measurements of the ${p}_{{\rm{T}}}^{t}$ and ${m}_{t\bar{t}}$ distributions in top-quark pair production at ATLAS 8 TeV [234]. In contrast to the ATLAS case, the CMS measurements of inclusive jet production at both 7 TeV [235] and 8 TeV [236] prefer a smaller ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$, reflecting a tension between these measurements from the two experiments.

Within the 68% probability level, the global average values are

$$\begin{eqnarray}&&{\alpha }_{S}{({m}_{{\rm{Z}}}^{2})}_{\mathrm{NNLO}}=0.1164\pm 0.0026,\quad {\alpha }_{S}{({m}_{{\rm{Z}}}^{2})}_{\mathrm{NLO}}=0.1187\pm 0.0026,\end{eqnarray} \tag{ 4.30 }$$

while CT18Z NNLO gives 0.1169 ± 0.0027, all consistent with the world average [226]. In comparison with the CT14 global fit [223], the CT18 data sets prefer a larger central ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ with a marginally smaller uncertainty. Note that the uncertainties quoted here are larger than those in the contemporary analyses done by other two groups, MSHT and NNPDF. This is primarily a result of a more conservative prescription for the uncertainty adopted in CT18, despite similar constraining power of the experiments included by the three groups. The CT18 prescription reflects in part some inconsistency between the fitted data sets, as discussed above. In the presence of such inconsistencies due to unidentified systematic effects, which would lead to enlarged variations in α_S when the selection of experimental data sets is varied, the final uncertainty should be generally increased [237]. In addition, the CT18 uncertainty incorporates outcomes from the fits with comparable χ² that are obtained with more than 250 alternative functional forms of the PDFs and with alternative QCD scale choices in inclusive jet and high-p_T production. To achieve this, the α_S and PDF uncertainties are estimated using the tolerance chosen so as to cover a large number of candidate fits that are made with such alternative choices and pass the goodness-of-fit criteria accepted for the final fit [229, 238].

Acknowledgments—The work of KX is supported by the US Department of Energy, under Grant No. DE-SC0007914, U.S. National Science Foundation under Grant No. PHY-2112829, and in part by the PITT PACC. The work of PN is supported by U.S. Department of Energy under Grant No. DE-SC0010129.

Deep inelastic scattering (DIS) of electrons on protons, e–p, at centre-of-mass energies of up to $\sqrt{s}\approx 320\,$ GeV at HERA has been central to the exploration of proton structure and quark–gluon dynamics as described by perturbative Quantum Chromodynamics (pQCD). The combination of H1 and ZEUS data on inclusive e–p scattering and the subsequent pQCD analysis, introducing the family of parton density functions (PDFs) known as HERAPDF2.0 [215], was a milestone for the exploitation of the HERA data. The work presented here represents a completion of the HERAPDF2.0 family with a fit at NNLO to HERA combined inclusive data and jet production data published separately by the ZEUS and H1 collaborations. This was not possible at the time of the original introduction of HERAPDF2.0 because a treatment at NNLO of jet production in e−p scattering was not available then.

The name HERAPDF stands for a pQCD analysis within the DGLAP formalism [239–241], where predictions from pQCD are fitted to data. These predictions are obtained by solving the DGLAP evolution equations at NNLO in the $\overline{\mathrm{MS}}$ scheme. The inclusive and dijet production data which were already used for HERAPDF2.0Jets NLO were again used for the analysis presented here. A new data set [242] published by the H1 collaboration on jet production in low Q² events, where Q² is the four-momentum-transfer squared, was added as input to the fits.

The fits presented here were done in the same way as for all other members of the HERAPDF2.0 family, for full details see [243] and references therein. All parameter settings were the same as for the HERAPDF2.0Jets NLO fit, with the exception of the heavy quark masses m_c and m_b and their uncertainty ranges, which were reevaluated using the recently published HERA combined charm and beauty data [244]. The default minimum Q² of data entering the fit is Q² = 3.5 GeV² and the starting scale for DGLAP evolution is ${Q}_{0}^{2}\,=1.9$ GeV². Note that all HERA data are at very large W so that higher twist effects at small Q² and large x are not relevant.

As for previous HERAPDF analyses, model and parametrization uncertainties were evaluated. For the present analysis the uncertainties on the hadronization corrections for the jet data were included as systematic uncertainties, 50% correlated and 50% uncorrelated, together with the experimental systematic uncertainties.

The jet data were included in the fits at NNLO by calculating predictions for the jet cross sections using the NNLOJet [245] extension of NLOjet++ interfaced to the Applfast framework in order to achieve the speed necessary for iterative PDF fits. The predictions were supplied with uncertainties which were also input to the fit as 50% correlated and 50% uncorrelated systematic uncertainties.

The NNLO analysis of the jet data was applied to a slightly reduced phase space compared to HERAPDF2.0NLOJets. All data points with $\sqrt{\langle {p}_{{\rm{T}}}^{2}\rangle +{Q}^{2}}\leqslant 10\,\mathrm{GeV}$ were excluded to keep the level of scale uncertainties in the predictions for the data points to ≲10%. Six data points, the lowest 〈p_T〉 bin for each Q² region, were excluded from the ZEUS dijet data set because predictions for these points were not fully NNLO. The trijet data which were used as input to HERAPDF2.0NLOJets had to be excluded as their treatment at NNLO is not available. In addition six extra data points for H1 inclusive jet data at high Q² but low p_T, which were published more recently [242], were included.

The choice of scales was also adjusted for the NNLO analysis. At NLO, the factorization scale was chosen as ${\mu }_{{\rm{f}}}^{2}={Q}^{2}$, while the renormalization scale was linked to the transverse momenta, p_T, of the jets by ${\mu }_{{\rm{r}}}^{2}=({Q}^{2}+{p}_{{\rm{T}}}^{2})/2$. For the NNLO analysis, ${\mu }_{{\rm{f}}}^{2}\,={\mu }_{{\rm{r}}}^{2}={Q}^{2}+{p}_{{\rm{T}}}^{2}$, was chosen for both factorization and renormalization scales.

Jet production data are essential for the determination of the strong coupling constant, ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$. In pQCD fits to inclusive DIS data alone the value of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ is strongly correlated to the shape of the gluon PDF. Data on jet production cross sections provide an independent constraint on the gluon distribution since inclusive jet and dijet production are directly sensitive to ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$. Thus such data allow for an accurate simultaneous determination of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ and the PDFs.

The HERAPDF2.0Jets NNLO fit with free ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ and ${Q}_{\min }^{2}=3.5$ GeV² gives a value of

$$\begin{eqnarray*}&&\,{\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1156\pm 0.0011\,(\exp {)}_{-0.0002}^{+0.0001}\,(\mathrm{model}/\mathrm{parameterization})\pm 0.0029\,(\mathrm{scale}).\end{eqnarray*}$$

Note that the the label experimental uncertainty covers the full experimental uncertainties of the simultaneous PDF and free ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ fit, which thus includes contributions from the PDF parametrization, the uncertainty on the hadronization corrections and the small uncertainties on the theoretical predictions. Additional model uncertainties come from variation of the input assumptions such as the the value of ${Q}_{0}^{2}$, the starting scale for QCD evolution, the value of the minimum Q² of data entering the fit and the values of heavy quark masses. Additional parametrization uncertainties come from the consideration of fits which have additional parameters that do not change the χ² of the fit significantly, but which can sometimes alter PDF shape. A further cross-check on the parametrization comes from modifying the gluon parametrization such that it must remain positive definite even at very low-x, x < 10⁻⁴, and Q² below 2 GeV². The largest uncertainty comes from the scale uncertainty, as discussed later.

The HERAPDF2.0Jets NNLO fit with free ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ uses 1363 data points and has a goodness-of-fit per degree of freedom of χ²/n_dof = 1614/1348 = 1.197. This can be compared to the χ²/n_dof = 1363/1131 = 1.205 for HERAPDF2.0 NNLO based on inclusive data only. The similarity of the χ²/n_dof values indicates that the data on jet production do not introduce any tension. Data/fit comparisons are shown in [243].

A scan of the fit χ² for fits with varying ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ shown in figure 18 and confirms the value of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ and its experimental uncertainty found in the simultaneous ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ and PDF fit. The model, parameterization and scale uncertainties are also shown in the figure. The PDFs associated with this NNLO analysis are described in [243], the input of the jet data reduces the gluon PDF uncertainties.

Zoom In Zoom Out Reset image size

The question whether data with relatively low Q² bias the determination of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ arises in the context of the HERA data analysis for which low Q² is also low x. A treatment beyond DGLAP may be necessary because of low-x higher-twist terms, $\mathrm{ln}(1/x)$ terms or even parton saturation. To check for such bias the minimum Q² entering the fit was varied to ${Q}_{\min }^{2}=10$ and 20 GeV². Thus the HERAPDF analysis considers the possible impact of nonperturbative and beyond DGLAP effects on the result for ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$. These variations do not alter the value of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ obtained significantly, the value of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ obtained for ${Q}_{\min }^{2}=10$ GeV² is

$$\begin{eqnarray*}&&{\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1156\pm 0.0011\,(\exp )\pm 0.0002\,(\mathrm{model}/\mathrm{parameterization})\pm 0.0021\,(\mathrm{scale}).\end{eqnarray*}$$

The scale uncertainty is reduced with this harder cut on Q² but the largest uncertainty still comes from the scale uncertainty.

The scale uncertainty was evaluated by varying the renormalization and factorization scales by a factor of two, both separately and simultaneously (7-point variation), and taking the maximal positive and negative deviations. Scale uncertainties were assumed to be 100% correlated between bins and data sets.

A strong motivation to determine ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ at NNLO was the hope to substantially reduce scale uncertainties from those determined at NLO. However, for our previous NLO result we had treated scale uncertainties as 50 % correlated and 50 % uncorrelated. If we repeat this treatment then the scale uncertainty for the present NNLO analysis (with ${Q}_{\min }^{2}=3.5$ GeV², comparable to the NLO analysis) would be ±0.0022, very significantly lower than the +0.0037, − 0.0030 previously observed for the HERAPDF2.0NLOJets analysis. The absolute value of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ at NNLO ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1156$ is also lower than that observed at NLO ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1183$. However, the analyses were done at different scales and with somewhat different data sets. In fact, if these choices were harmonized, the difference in the values of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ would be even greater: at NLO ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1186\pm 0.0014$ (exp) and at NNLO ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1144\pm 0.0013$ (exp), where the scale choices are both ${Q}^{2}+{p}_{{\rm{T}}}^{2}$, the cuts on the data sets for both orders are the harder cuts introduced for the present analysis and the H1 low Q² jet data set [242] is excluded (this choice is made because these data cannot be well fitted at NNLO). The main reason for the decrease in the central value at NNLO is exclusion of these low-Q² H1 inclusive and dijet data.

It is clear that the main limiting factor today is the theoretical uncertainty, which we estimate from the scale uncertainty. The decrease in scale uncertainty between NLO and NNLO gives some hope for an increase in precision at N³LO. Once theoretical uncertainties are reduced, the experimental uncertainty will need to be reduced. One may hope for progress from analyses at future e−p colliders such as the EIC/LHeC/FCC-eh. Experimental accuracy on ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})\sim 0.0001$ is projected at the LHeC [246].

Acknowledgments—AMC-S would like to thank the Leverhulme Trust.

MSHT20 [247] represented a significant step forward in the global determination of PDFs within the MSHT collaboration (previously MRST/MSTW/MMHT), with more data of greater precision across more channels and more differential in nature now incorporated alongside the inclusion of full NNLO QCD theoretical predictions (and NLO EW corrections where relevant [248]). This has been further complemented by progress on the methodological side, including in particular the extension of the PDF parameterization. The overall result was a significant improvement in our knowledge of the PDFs, including the central values and a general reduction of the PDF uncertainties. Within this context we undertook a follow-up study [249] examining the effects of varying the strong coupling and heavy quark masses within the global fit. This updated previous work with MMHT14 [250]. As a result of this new analysis we determined the preferred value of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ and its associated uncertainty and we report on this in this brief review.

The default PDFs within MSHT are provided at ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.118$ at NNLO in order to provide a common value between different PDF fitting groups which is also consistent with the world average value of ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})=0.1179\pm 0.0009$ [251]. However, ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ can also be left as a free parameter and fit alongside the PDFs, we are then able to utilize the global nature of the PDF fits to extract ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$. Performing the fit with ${\alpha }_{S}({m}_{{\rm{Z}}}^{2})$ free within MSHT20 at NLO and NNLO obtains the best fit values of 0.1203 and 0.1174 respectively, with the excellent quadratic χ² profiles shown in figure 19, demonstrating the clear sensitivity at the global fit level.