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Bjorken polarized sum rule and infrared-safe QCD couplings

  • César Ayala
  • Gorazd CvetičEmail author
  • Anatoly V. Kotikov
  • Binur G. Shaikhatdenov
Open Access
Regular Article - Theoretical Physics

Abstract

Experimental data obtained for the polarized Bjorken sum rule (BSR) \(\Gamma _1^{p-n}(Q^2)\) are fitted by using predictions derived within a truncated operator product expansion (OPE) approach to QCD. Four QCD versions are considered: perturbative QCD (pQCD) in the \({\overline{\mathrm{MS}}}\) scheme, Analytic Perturbation Theory (APT), and 2\(\delta \) and 3\(\delta \) analytic QCD versions. In contrast to pQCD, these QCD variants do not have Landau singularities at low positive \(Q^2\), which facilitates the fitting procedure significantly. The fitting procedure is applied first to the experimental data of the inelastic part of BSR, and the known elastic contributions are added after the fitting. In general, when 2\(\delta \) and 3\(\delta \) QCD coupling is used the fitted curves give the best results, within the \(Q^2\)-range of the fit as well as in extended \(Q^2\)-intervals. When the fitting procedure is applied to the total BSR, i.e., to the sum of the experimental data and the elastic contribution, the quality of the results deteriorates significantly.

1 Introduction

The polarized Bjorken sum rule (BSR) \(\Gamma _1^{p-n}(Q^2)\) [1, 2] is an important spacelike QCD observable for various reasons. It is a difference of the first moment of the spin-dependent structure functions of proton and neutron, therefore its isovector nature makes it easier to describe it theoretically, in pQCD in terms of OPE, than the separate integrals of the two nucleons. Further, high quality experimental results for this quantity, obtained in polarized deep inelastic scattering (DIS), are now available in a large range of spacelike squared momenta \(-q^2 \equiv Q^2\): \(0.054 \ \mathrm{GeV}^2 \le Q^2 < 5 \ \mathrm{GeV}^2\) [3, 4, 5, 6, 7, 8, 9, 10, 11]. In particular, the newer experimental results [5] with highly reduced statistical uncertainties, extracted mainly from the Jefferson Lab CLAS EG1-DVCS experiment [12] taken on polarized targets made of protons and deuterons, render BSR an attractive quantity to test on it various extensions of pQCD to low \(Q^2 \lesssim 1 \ \mathrm{GeV}^2\).

Theoretically, pQCD (with OPE), in \(\overline{\mathrm{MS}}\) scheme, has been the usual approach to describe such quantities, cf. [3, 4, 5]. This approach, however, has the theoretical disadvantage that the running coupling \(a(Q^2)\) [\(\equiv \alpha _s(Q^2)/\pi \)] possesses Landau singularities at low positive \(Q^2 \lesssim 0.1 \ \mathrm{GeV}^2\), and this makes it inconvenient for evaluation of spacelike observables \(\mathcal{D}(Q^2)\) at low \(Q^2\), such as BSR. In recent years, an extension of pQCD couplings to low \(Q^2\), without Landau singularities, called (Fractional) Analytic Perturbation Theory [(F)APT)] [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] has been applied in the fitting of the theoretical OPE expression to the experimental inelastic contributions to BSR [28, 29, 30, 31], with good results.

In this work we fit the theoretical OPE expressions to the experimental BSR results in pQCD, in (F)APT, and two additional extensions of QCD to low \(Q^2\), namely the 2\(\delta \) [32, 33] and 3\(\delta \) [34, 35] \({\mathcal {A}}\)QCD. The latter two extensions have the coupling \({\mathcal {A}}(Q^2)\) [the analog of the pQCD coupling \(a(Q^2)\)] which is free of Landau singularities and physically motivated in the entire relevant regime of \(Q^2\) in the complex plane, \(Q^2 \in \mathbb {C} \backslash (-\infty , -M_\mathrm{thr}^2]\), where \(M_\mathrm{thr}^2 \lesssim 1 \ \mathrm{GeV}^2\) is a positive threshold scale. The present work is an extension of our previous work on BSR [36, 37]; in comparison with the latter work, we now vary the \(Q^2\)-range of the fit, include the consideration of the elastic contribution, and estimate the uncertainties of the values of the extracted fit parameters due to systematic (in addition to statistical) uncertainties of the experimental BSR data.

The mentioned QCD variants (F)APT, 2\(\delta \) and 3\(\delta \) \({\mathcal {A}}\)QCD were constructed directly by imposing certain physically motivated conditions on the QCD coupling. In this context, we point out that there exist several other approaches to the construction of the QCD coupling, among them those using Dyson–Schwinger equations which involve various versions of the dynamically generated gluon mass (the mentioned 2\(\delta \) \({\mathcal {A}}\)QCD gives a coupling with similar properties). For a recent review of various approaches, we refer to [38].

In Sect. 2 we present the theoretical leading-twist and higher-twist OPE contributions for the considered quantity BSR, as well as the parametrization of the elastic contribution to BSR. In Sect. 3 we present the results of various fits of theoretical expressions to the experimental results. Finally, in Sect. 4 we summarize our conclusions.

Most of the formal aspects of the calculations are relegated to Appendices: in Appendix A we present the form of the leading-twist perturbation coefficients for a general renormalization scale and scheme; in Appendix B we present construction of \({\mathcal {A}}_n\), the analogs of pQCD powers \(a^n\), in extensions of pQCD without Landau singularities; in Appendix C we summarize such extensions [(F)APT, 2\(\delta \) and 3\(\delta \)]; in Appendix D we explain how the statistical and systematic uncertainties of the experimental data are reflected in the corresponding uncertainties of the parameters extracted in the fits; and in Appendix E we estimate the effects of the finiteness of the charm quark mass in our evaluations.

2 Bjorken sum rule: theoretical expressions

The polarized Bjorken sum rule (BSR), \(\Gamma _1^{p-n}\), is defined as the difference between proton and neutron polarized structure functions \(g_1\) integrated over the whole x-Bjorken interval
$$\begin{aligned} \Gamma _1^{p-n}(Q^2)=\int _0^1 dx \left[ g_1^p(x,Q^2)-g_1^n(x,Q^2) \right] . \end{aligned}$$
(1)
Based on the various measurements of these and the related structure functions, the inelastic part of the above quantity, \(\Gamma _1^{p-n}(Q^2)_\mathrm{inel.}\), has been extracted at various values of squared momenta \(Q_j^2\) (\(0.054 \ \mathrm{GeV}^2 \le Q_j^2 < 5 \ \mathrm{GeV}^2\)): [3, 4, 5] (Jefferson Lab), [8, 9] (SLAC), [6, 7] (COMPASS at CERN).1
Theoretically, this quantity can be written in the Operator Product Expansion (OPE) form [1, 2]
$$\begin{aligned} \Gamma _1^{p-n, \mathrm {OPE}}(Q^2)= {\Big |}\frac{g_A}{g_V} {\Big |} \frac{1}{6}(1 - \mathcal{D}_\mathrm{BS}(Q^2))+\sum _{i=2}^\infty \frac{\mu _{2i}(Q^2)}{Q^{2i-2}}.\nonumber \\ \end{aligned}$$
(2)
Here, \(|g_A/g_V|\) is the ratio of the nucleon axial charge, \((1-\mathcal{D}_\mathrm{BS})\) is the perturbation expansion for the leading-twist contribution, and \(\mu _{2i}/Q^{2i-2}\) are the higher-twist contributions. The value obtained from neutron \(\beta \) decay is known to a high accuracy, \(|g_A/g_V|=1.2723 \pm 0.0023\) [39], and we will use the central value. In the higher-twist terms, we will take only the terms \(\sim 1/Q^2\) and \(1/(Q^2)^2\).

2.1 Perturbation expansion of the leading-twist

The leading-twist term has the canonical part \(\mathcal{D}_\mathrm{BS}(Q^2)\) whose perturbation expansion in \(a \equiv \alpha _s/\pi \) is known up to \(\hbox {N}^3 \hbox {LO}\) (\(\sim a^4\))
$$\begin{aligned} \mathcal{D}_\mathrm{BS}(Q^2)_\mathrm{pt}= & {} {\bar{a}} + {\bar{d}}_1 {\bar{a}}^2 + {\bar{d}}_2 {\bar{a}}^3 + {\bar{d}}_3 {\bar{a}}^4 + \mathcal{O}({\bar{a}}^5), \end{aligned}$$
(3)
where the bar indicates that the expansion is in the \(\overline{\mathrm{MS}}\) scheme, and the renormalization scale \(\mu ^2\) is implicitly understood to be equal to the physical scale \(Q^2\). The NLO, N\(^2\)LO and N\(^3\)LO coefficients \({\bar{d}}_j\) (\(j=1,2,3\)) were obtained in [40, 41, 42], respectively. In the considered range of momentum transfer \(0< Q^2 < 5 \ \mathrm{GeV}^2\), we will assume that the effective number of active quark flavors is \(N_f=3\), and therefore only the nonsinglet (NS) contributions appear.2
When the renormalization scale is changed from \(\mu ^2=Q^2\) to a general value \(\mu ^2 = k Q^2\) (\(0 < k \sim 1\)), and when the renormalization scheme parameters are changed from the \(\overline{\mathrm{MS}}\) values \({\bar{c}}_j \equiv {\bar{\beta }}_j/\beta _0\) to general scheme parameter values \(c_j\) (\(j \ge 2\)), the perturbation expansion changes accordingly
$$\begin{aligned} \mathcal{D}_\mathrm{BS}(Q^2)_\mathrm{pt}= & {} a(k Q^2) + d_1(k) a(k Q^2)^2 + d_2(k;c_2) a(k Q^2)^3 \nonumber \\&+\, d_3(k; c_2, c_3) a(k Q^2)^4 + \mathcal{O}(a^5). \end{aligned}$$
(4)
The expressions for the new coefficients \(d_1(k)\), \(d_2(k; c_2)\) and \(d_3(k; c_2, c_3)\) are obtained on the basis of independence of the observable \(\mathcal{D}_\mathrm{BS}(Q^2)_\mathrm{pt}\) from k and \(c_j\) (\(j \ge 2\)), and are given in Appendix A.
In those versions of \({\mathcal {A}}\)QCD where the coupling is a holomorphic function \({\mathcal {A}}(Q^2)\) [instead of the nonholomorphic \(a(Q^2)\)] with nonperturbative contributions, the power expansion (4) becomes a nonpower expansion where \(a^n\) get replaced by \({\mathcal {A}}_n\) (\(\ne {\mathcal {A}}^n\))
$$\begin{aligned} \mathcal{D}_\mathrm{BS}(Q^2)_\mathrm{{{\mathcal {A}}}QCD}= & {} {\mathcal {A}}(k Q^2) + d_1(k) {\mathcal {A}}_2(k Q^2) \nonumber \\&+\, d_2(k;c_2) {\mathcal {A}}_3(k Q^2) \nonumber \\&+\, d_3(k; c_2, c_3) {\mathcal {A}}_4(k Q^2) + \mathcal{O}({\mathcal {A}}_5). \end{aligned}$$
(5)
The construction of the power analogs \({\mathcal {A}}_n\) of \(a^n\) were obtained in Ref. [45, 46] for integer n and in Ref. [47] for general real \(n > -1\). A brief description for the construction of \({\mathcal {A}}_n\) is given in Appendix B. These expressions are based on the renormalization group equation (RGE), in close analogy with the RGE in the perturbation theory. The couplings \({\mathcal {A}}_n(Q^2)\) can be obtained once the coupling \({\mathcal {A}}(Q^2)\) is known. The construction of \({\mathcal {A}}(Q^2)\) coupling is summarized in Appendix C for various variants of QCD with holomorphic coupling: (F)APT, 2\(\delta \) and 3\(\delta \) \({\mathcal {A}}\)QCD, and we refer to that Appendix for more details.

2.2 Higher-twist

In the theoretical OPE expression (2), the term with the dimension \(D=2\) (i.e., \(\propto 1/Q^2\)) has the coefficient
$$\begin{aligned} \mu _4 = \frac{M_N^2}{9} (a_2^{p-n} + 4 d_2^{p-n} + 4 f_2^{p-n}(Q^2)), \end{aligned}$$
(6)
where \(M_N \approx 0.94\) GeV is the nucleon mass, \(a_2^{p-n}\) is the (twist-2) target mass correction, and \(d_2^{p-n}\) is a twist-3 matrix element
$$\begin{aligned} d_2^{p-n} = \int _0^1 dx x^2 (2 g_1^{p-n} + 3 g_2^{p-n}). \end{aligned}$$
(7)
At \(Q^2=1 \ \mathrm{GeV}^2\), we have \(a_2^{p-n}=0.031 \pm 0.010\) and \(d_2^{p-n}=0.008 \pm 0.0036\) [5]. We will neglect \(Q^2\)-dependence of these two quantities [as was done also in Ref. [5]], and will take the central values, i.e., \(a_2^{p-n} + 4 d_2^{p-n}=0.063\). On the other hand, the coefficient \(f_2^{p-n}(Q^2)\) will be a parameter of the fit, and its \(Q^2\)-dependence will not be neglected, its evolution is known [48, 49] in pQCD
$$\begin{aligned} f_2^{p-n}(Q^2)=f_2^{p-n}(Q_\mathrm{in}^2) \left( \frac{a(Q^2)}{a(Q_\mathrm{in}^2)}\right) ^{\gamma _0/8\beta _0}, \end{aligned}$$
(8)
where \(\nu \equiv \gamma _0/(8\beta _0) =32/81\) when \(N_f=3\), and the reference scale will be taken \(Q_\mathrm{in}^2 = 1 \ \mathrm{GeV}^2\). In QCD with holomorphic coupling \({\mathcal {A}}(Q^2)\), the power \(a^{\gamma _0/8\beta _0}\) gets replaced by \({\mathcal {A}}_{\gamma _0/8\beta _0}\) (which is in general not equal to the simple power \({\mathcal {A}}^{\gamma _0/8\beta _0}\), as mentioned above, cf. also Appendix B,
$$\begin{aligned} f_2^{p-n}(Q^2)=f_2^{p-n}(Q_\mathrm{in}^2) \left( \frac{{\mathcal {A}}_{\gamma _0/8\beta _0}(Q^2)}{{\mathcal {A}}_{\gamma _0/8\beta _0}(Q_\mathrm{in}^2)}\right) , \end{aligned}$$
(9)
In addition, in some of the fits we will also include the \(D=4\) term in the theoretical OPE expression (2) \(\mu _6/(Q^2)^2\), where we will consider the coefficient \(\mu _6\) as \(Q^2\)-independent. Thus, our theoretical expression for BSR will be the truncated (at \(D=4\), or \(D=2\)) OPE expression
$$\begin{aligned}&\Gamma _1^{p-n,\mathrm{OPE[4]}}(Q^2; k, f_2^{p-n}(1); \mu _6) = {\Big |} \frac{g_A}{g_V} {\Big |} \frac{1}{6}(1 - \mathcal{D}_\mathrm{BS}(Q^2))\nonumber \\&\quad +\, \frac{M_N^2}{Q^2} \frac{1}{9} \left( a_2^{p-n} +4 d_2^{p-n} + 4 f_2^{p-n}(Q^2) \right) +\frac{\mu _6}{(Q^2)^2}, \end{aligned}$$
(10)
where the leading-twist truncated expressions are given in Eqs. (4) and (5), and the \(Q^2\)-dependent part of the \(D=2\) term (twist-4) in Eqs. (8) and (9), for the pQCD and \({\mathcal {A}}\)QCD version of QCD, respectively. We will regard the renormalization scale parameter \(k \equiv \mu ^2/Q^2\) and the higher-twist coefficients \(f_2^{p-n}(Q^2_\mathrm{in})\) (with \(Q^2_\mathrm{in}=1 \ \mathrm{GeV}^2\)) and \(\mu _6\) as the free parameters to be determined in the fitting procedure.

In our approach, we will include in the leading-twist part of the OPE all the known terms (i.e., up to order \(\sim a^4 \sim {\mathcal {A}}_4\)). The order of the leading-twist terms in general affects the fitted higher-twist contributions (cf. [50, 51, 52, 53, 54, 55, 56, 57] for the fit of truncated OPE to structure functions).

2.3 Elastic contribution

The OPE evaluation is, in principle, for inclusive observables; in the case of BSR, this means that the OPE fit should be applied to the experimental values of the sum of the inelastic and elastic contribution.

The elastic contribution to BSR can be expressed [58, 59] in terms of the proton and neutron electromagnetic form factors \(F_1\) and \(F_2\)
$$\begin{aligned} \Gamma _1^{p-n}(Q^2)_\mathrm{el.}= & {} \frac{1}{2} F_1^{p}(Q^2) \left( F_1^{p}(Q^2) + F_2^{p}(Q^2) \right) \nonumber \\&-\, \frac{1}{2} F_1^{n}(Q^2) \left( F_1^{n}(Q^2) + F_2^{n}(Q^2) \right) . \end{aligned}$$
(11)
The most recent parametrization of these form factors was performed in [60] from light-front holographic QCD (LFH QCD). Namely, these form factors can be expressed in terms of the inverse power expressions
$$\begin{aligned} \mathcal{F}_{\tau }\equiv & {} \left( 1 + \frac{Q^2}{M_{\rho }^2} \right) ^{-1} \times \cdots \times \left( 1 + \frac{Q^2}{(2 \tau -3 ) M_{\rho }^2} \right) ^{-1}, \nonumber \\&\quad (\tau =2,3,\ldots ), \end{aligned}$$
(12)
which are products of \(\tau -1\) poles along the vector meson Regge radial trajectory in terms of the \(\rho \)-vector mass \(M_{\rho }=0.755\) GeV and its radial excitations. We have
$$\begin{aligned} F_1^p(Q^2)= & {} \mathcal{F}_{\tau =3}(Q^2), \nonumber \\ F_2^p(Q^2)= & {} \chi _p \left[ (1 - \gamma _p) \mathcal{F}_{\tau =4}(Q^2) + \gamma _p\mathcal{F}_{\tau =6}(Q^2) \right] , \nonumber \\\end{aligned}$$
(13a)
$$\begin{aligned} F_1^n(Q^2)= & {} -\frac{1}{3} r \left[ \mathcal{F}_{\tau =3}(Q^2) - \mathcal{F}_{\tau =4}(Q^2) \right] , \nonumber \\ F_2^n(Q^2)= & {} \chi _n \left[ (1 - \gamma _n) \mathcal{F}_{\tau =4}(Q^2) + \gamma _n\mathcal{F}_{\tau =6}(Q^2) \right] . \nonumber \\ \end{aligned}$$
(13b)
Here: \(\chi _p = \mu _p-1 = 1.793\) and \(\chi _n = \mu _n=-1.913\) are the anomalous moments of p and n, respectively; \(\gamma _{n,p}\) are the higher Fock probabilities for the \(L=0 \rightarrow L=1\) spin-flip electromagnetic form factors, and r is a phenomenological factor. The values of these three parameters are obtained by fitting to polarization data for the form factors [60]: \(\gamma _p=0.27\); \(\gamma _n=0.38\); \(r=2.08\).
As a consequence, the elastic contribution (11) can be represented as a combination of inverse powers \(( 1 + Q^2/M_{\rho }^2/(2 m -1) )\), and for large \(Q^2\) this can be expanded in inverse powers of \(Q^2\)
$$\begin{aligned} \Gamma _1^{p-n}(Q^2)_\mathrm{el.}= & {} 2.3368 \left( \frac{M_{\rho }^2}{Q^2}\right) ^4 \nonumber \\&+\, 13.8851 \left( \frac{M_{\rho }^2}{Q^2}\right) ^5 + \mathcal{O}(1/Q^{12}). \end{aligned}$$
(14)
This means that at high \(Q^2\) the elastic contribution starts with the dimension \(D=8\) term \(\sim (1/Q^2)^4\). Theoretically, it is not included in the truncated OPE expression (10) in pQCD.3\(^{,}\)4 Further, in 3\(\delta \) and 2\(\delta \) \({\mathcal {A}}\)QCD, the higher dimensional terms up to (and including) the \(D=8\) term are not included in the leading-twist contribution (5) because in these approaches \({\mathcal {A}}(Q^2) - a(Q^2) \sim (\Lambda ^2/Q^2)^5\) as explained in Appendix C.2 Eq. (C7). Therefore, the truncated OPE expression (10) does not contain the dimension \(D=8\) term \(\sim (1/Q^2)^4\) also in 3\(\delta \) and 2\(\delta \) \({\mathcal {A}}\)QCD approaches. The only exception is the (F)APT where \({\mathcal {A}}(Q^2) - a(Q^2) \sim (\Lambda ^2/Q^2)^1\), cf. Appendix C.1, and where the truncated OPE (10) could contain, in principle, all the elastic contributions, including the \(\sim (1/Q^2)^4\) term.

Therefore, it looks more natural to first fit the truncated OPE expression (10), with pQCD and (3\(\delta \) and 2\(\delta \)) \({\mathcal {A}}\)QCD, to the BSR experimental results [3, 4, 5, 7, 8] which are obtained for the inelastic contribution; and after such a fit, add the elastic contribution (11) [with the parametrization (12)–(13b)]. We note that the theoretical expression obtained for the total BSR in this way is the sum of the expressions (10) and (11), which is again an OPE expression as it should be for such an inclusive spacelike observable as the total BSR.

The other approach would be to fit the truncated OPE expression (10) with the experimental points for the total BSR, i.e., fit with the experimental points for inelastic BSR with the elastic contribution added to them. Such an approach seems less natural, at least in pQCD and 3\(\delta \) and 2\(\delta \) \({\mathcal {A}}\)QCD, because the truncated expression (10) in these cases in principle does not contain the leading elastic contribution \(\sim 1/(Q^2)^4\), cf. Eq. (C7) in Appendix C.2.

Nonetheless, below we will apply, for completeness, both approaches in our numerical fitting procedures.

3 Numerical fits

In the numerical fits, we will consider the following parameters to be fitted in the truncated OPE expression (10): (i) the renormalization scale parameter \(k \equiv \mu ^2/Q^2\) of the (truncated) leading-twist contribution (4) [for pQCD] or (5) [for \({\mathcal {A}}\)QCD]; (ii) the parameter \(f_2^{p-n}(1 \ \mathrm{GeV}^2)\) appearing in the \(D=2\) term of Eq. (10) [cf. Eqs. (8) and (9)]; (iii) and in some fits the parameter \(\mu _6\) in the \(D=4\) term of Eq. (10) will be included. We will take for the experimental data points for the inelastic contribution to BSR the data from [3, 4, 5, 7, 8];5 they are in the momentum interval \(0.054 \ \mathrm{GeV}^2 \le Q_j^2 < 5 \ \mathrm{GeV}^2\). Among these data points, we exclude from the fit the four points with \(Q_j^2 \ge 3 \ \mathrm{GeV}^2\) (three from [5], at \(Q_j^2=3.223\), 3.871 and \(4.739 \ \mathrm{GeV}^2\); and one from [7] at \(Q^2=3 \ \mathrm{GeV}^2\)) because they tend to decrease the quality parameter \(\chi ^2/\mathrm{(d.o.f.)}\) of the fit significantly. Nonetheless, we will show both quality parameters \(\chi ^2/\mathrm{(d.o.f.)}\) obtained from the mentioned fits, i.e., the one with the four points excluded, and included.6 The number of active flavors will be kept all the time at \(N_f=3\). The fits will be performed by the least squares method, taking into account the statistical uncertainties \(\sigma _{j,\mathrm{stat}}\) in the points \(Q_j^2\), we will consider them independent of each other. These uncertainties \(\sigma _{j,\mathrm{stat}}\) are in general significantly smaller than the systematic uncertainties \(\sigma _{j,\mathrm{sys}}\). The latter are strongly correlated, and we will consider them as completely correlated. The uncertainties in the values of the extracted parameters \(k \equiv \mu ^2/Q^2\), \(f_2^{p-n}(Q^2_\mathrm{in})\) and \(\mu _6\) are then due to statistical (small) and systematic (larger) uncertainties of the data. We refer to Appendix D on how we obtained the uncertainties of the extracted values of the fit parameters.
Fig. 1

Fits of the OPE expression (10) truncated at \(D=2\) term, i.e., \(\mu _6=0\), to the experimental data for \(\Gamma _1^{p-n}(Q^2)_\mathrm{inel.}\), in four different QCD variants, where in the fits: a \(Q^2 \ge 0.66~\mathrm{GeV}^2\); b \(Q^2 \ge 0.268~\mathrm{GeV}^2\). See the text for details. The respective lower bound of the fitting interval, \(Q^2_\mathrm{min}=0.66\) or \(0.268~\mathrm{GeV}^2\), is included as the thin dotted vertical line

Using specific values of the renormalization scale parameter \(k \equiv \mu ^2/Q^2\) may allow us to incorporate in our evaluations (5) at least a part of the contributions of higher orders of the series. It is expected that \(k \sim 1\), and usually it is taken in the literature in the range \(1/2< k < 2\), sometimes \(1/4< k < 4\). In the considered work, after replacing the powers of the pQCD coupling by their analytic counterparts, cf. Eqs. (4)–(5), a spacelike observable depends usually weakly on the contributions of higher orders. Still, in order not to miss the possibly relevant influence of higher orders, we decided to increase the range of possible k values to: \(1/16< k < 16\). This choice still avoids very large or very small renormalization scales where the corresponding coefficients \(d_n(k)\) at the powers \(a(k Q^2)^{n+1}\), or at their analytic counterparts \({\mathcal {A}}_{n+1}(k Q^2)\), contain powers of the large terms \(\sim \ln k\) [cf. Eq. (A3)] which may destroy the convergence of the series already at low n. We will see that the results of the fitting will rarely give the extreme values \(k=16\) or \(k=1/16 \approx 0.063\), mostly in the cases of (\(\overline{\mathrm{MS}}\)) pQCD and (F)APT.

We present below the results of the fits of the inelastic contributions to the truncated OPE expression (10), first in Sect. 3.1 with \(\mu _6=0\), and then in Sect. 3.2 for \(\mu _6 \ne 0\) included as a fit parameter; the elastic contribution is added afterwards. In Sect. 3.3 we present the corresponding fits for the case when the higher-twist term in the OPE has a mass parameter. In Sect. 3.4 we include the fits for two different ansätze for BSR at very low \(Q^2\). Finally, in Sect. 3.5 we present the fits of the truncated OPE expression (10), with \(\mu _6 \ne 0\) included as a fit parameter, applied to the total BSR, i.e., to the sum of the inelastic data points (at \(Q_j^2\)) and the corresponding elastic contribution \(\Gamma _1^{p-n}(Q_j^2)_\mathrm{el.}\).

In each case, the fits are performed for four variants of QCD: in pQCD (in \(\overline{\mathrm{MS}}\)), in (F)APT (in \(\overline{\mathrm{MS}}\)), and in 2\(\delta \) [33] and 3\(\delta \) \({\mathcal {A}}\)QCD [35]. Each of these fits is performed by excluding among the data points those with \(Q^2 < Q^2_\mathrm{min}\), where \(Q^2_\mathrm{min}=0.268\) or \(0.66 \ \mathrm{GeV}^2\) (and sometimes also: \(Q^2_\mathrm{min}=0.47 \ \mathrm{GeV}^2\)). As mentioned earlier, the four points at high \(Q^2 \ge 3 \ \mathrm{GeV}^2\) are excluded from the fit as well.

It is reasonable to assume that the number of active quark flavors in the considered interval \(0.054 \ \mathrm{GeV}^2< Q^2 < 5 \ \mathrm{GeV}^2\) is \(N_f=3\), cf. Appendix E. All the fits will be performed in \(\overline{\mathrm{MS}}\) pQCD approach, in the Fractional Analytic Perturbation Theory [(F)APT], and in 2\(\delta \) and 3\(\delta \) \({\mathcal {A}}\)QCD approach, cf. Appendix C where also the values of the parameters of these three QCD variants are presented. In \(\overline{\mathrm{MS}}\) pQCD and in 2\(\delta \) and 3\(\delta \) \({\mathcal {A}}\)QCD the (underlying) pQCD running coupling \(a(Q^2)\) is determined by the requirement \(\pi a(M_Z^2;\overline{\mathrm{MS}}) = 0.1185\) [39, 61], and in (F)APT we use \({\overline{\Lambda }}_3=0.45\) GeV (at the end of Sect. 3.2 we also comment on the case \({\overline{\Lambda }}_3=0.40\) GeV); we refer to Appendix C for more details.
Table 1

The values of the extracted fit parameters \(k = \mu ^2/Q^2\) and \(f_2^{p-n}(Q^2_\mathrm{in})\) (\(Q^2_\mathrm{in} = 1 \ \mathrm{GeV}^2\)), obtained by fitting the OPE expression (10), truncated at \(D=2\) (\(\mu _6=0\)), for the considered four QCD variants, and for the minimal \(Q^2\) values of the fit being 0.66, 0.47 and \(0.268~ \mathrm{GeV}^2\). The uncertainties of the extracted \(f_2^{p-n}\) values are separated to those coming from the statistical (small) and systematic (large) uncertainties of the experimental data (cf. Appendix D). The resulting fit quality parameters \(\chi ^2/\mathrm{d.o.f.}\) are given as well (see the text for explanation)

QCD variant

\(Q^2_\mathrm{min}(\mathrm{fit})\)

k

\(f_2^{p-n}(1.)\)

\(\chi ^2/\mathrm{d.o.f}\)

\(\chi ^2_\mathrm{ext}/\mathrm{d.o.f}\)

\(\chi ^2_{0.268}/\mathrm{d.o.f.}\)

\(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\)

\(\overline{\mathrm{MS}}\) pQCD

0.66

6.04

\(-\,0.103 \pm 0.001 \pm 0.011\)

44.5

161

\(6.81 \times 10^4\)

\(\infty \)

(F)APT

0.66

2.74

\(-\,0.143 \pm 0.001 \pm 0.021\)

23.3

85.6

63.0

970

2\(\delta \)

0.66

0.397

\(-\,0.077 \pm 0.001 \pm 0.024\)

8.34

52.2

11.1

406

3\(\delta \)

0.66

2.41

\(-\,0.064 \pm 0.001 \pm 0.023\)

6.85

56.1

6.17

130

\(\overline{\mathrm{MS}}\) pQCD

0.47

6.35

\(-\,0.094 \pm 0.000 \pm 0.009\)

66.1

177

\(5.71 \times 10^4\)

\(\infty \)

(F)APT

0.47

3.03

\(-\,0.137 \pm 0.000 \pm 0.015\)

30.5

88.4

51.2

873

2\(\delta \)

0.47

0.395

\(-\,0.077 \pm 0.000 \pm 0.023\)

7.91

45.2

10.9

402

3\(\delta \)

0.47

2.58

\(-\,0.064 \pm 0.000 \pm 0.023\)

6.49

50.2

5.92

129

\(\overline{\mathrm{MS}}\) pQCD

0.268

16

\(-\,0.160 \pm 0.001 \pm 0.045\)

58.7

145

58.7

\(1.20 \times 10^{3}\)

(F)APT

0.268

3.71

\(-\,0.134 \pm 0.000 \pm 0.102\)

49.6

103

49.6

827

2\(\delta \)

0.268

0.338

\(-\,0.074 \pm 0.000 \pm 0.022\)

10.6

46.6

10.6

372

3\(\delta \)

0.268

2.55

\(-\,0.064 \pm 0.000 \pm 0.018\)

5.91

44.5

5.91

129

3.1 Fits with \(\mu _6=0\)

In Fig. 1a, b, we present the curves in the mentioned four QCD variants, for \(Q^2_\mathrm{min}=0.66\) and \(0.268 \ \mathrm{GeV}^2\), respectively. The corresponding results and four fit quality parameters \(\chi ^2/\mathrm{d.o.f}\) are given in Table 1, for \(Q^2_\mathrm{min} = 0.66\), 0.47 and \(0.268 \ \mathrm{GeV}^2\). As mentioned earlier, the fit is performed for the data in the momentum interval \(Q^2_\mathrm{min} \le Q_j^2 \le 3 \ \mathrm{GeV}^2\) (with the COMPASS data point [7] excluded), and the corresponding quality parameter for that interval is denoted as \(\chi ^2/\mathrm{d.o.f}\). In addition, \(\chi ^2_\mathrm{ext}/\mathrm{d.o.f.}\) is the quality parameter for the wider interval \(Q^2_\mathrm{min} \le Q_j^2 < 5 \ \mathrm{GeV}^2\); \(\chi ^2_{0.268}/\mathrm{d.o.f.}\) is the parameter when \(0.268 \ \mathrm{GeV}^2 \le Q_j^2 \le 3 \ \mathrm{GeV}^2\); and \(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\) is the parameter when all the experimental points [3, 4, 5, 7, 8] are included, \(0.054 \ \mathrm{GeV}^2 \le Q_j^2 < 5 \ \mathrm{GeV}^2\). In these quantities \(\chi ^2\), the central values of k and \(f_2^{p-n}(Q^2_\mathrm{in})\) (\(Q^2_\mathrm{in} = 1 \ \mathrm{GeV}^2\)) parameters obtained from the mentioned fit interval \(Q^2_\mathrm{min} \le Q_j^2 \le 3 \ \mathrm{GeV}^2\) are used; we take here “d.o.f” as \((N-p)\) where N is the number of fitted data points entering the considered \(\chi ^2\) and p is the number of parameters of the fit (\(p=2\) here)
$$\begin{aligned}&\chi ^2(Q_{K+1}^2 \le Q^2 \le Q_{K+N}^2)/\mathrm{d.o.f.} \nonumber \\&\quad =\frac{1}{(N\!-\!p)}\!\sum _{j=K+1}^{K+N} \frac{1}{\sigma ^2_{j,\mathrm{stat}}} \left( \Gamma _1^{p-n,OPE[4]}(Q_j^2; k, f_2^{p-n}(1); \mu _6=0)\right. \nonumber \\&\qquad \left. - \Gamma _1^{p-n}(Q_j^2)_\mathrm{exp} \right) ^2. \end{aligned}$$
(15)
The values of these quantities are always large, in the best cases between 1 and 10. This is so because the statistical uncertainties of the newer JLAB data [5] are very small, \(\sigma _{j,\mathrm{stat}} \lesssim 10^{-3}\), and simultaneously, our fit function (truncated OPE) is different from the ideal function which we do not know. Nonetheless, they decrease when analytic QCD variants are employed, especially 2\(\delta \) and 3\(\delta \) \({\mathcal {A}}\)QCD.

We wish to point out that the approach of (\(\overline{\mathrm{MS}}\)) QCD in the case of \(Q_\mathrm{min}^2=0.268 \ \mathrm{GeV}^2\) is, in principle, not applicable. This is so because the corresponding coupling \(a(Q^2)\) has a Landau branching point at \(Q^2_\mathrm{branch}=0.371 \ \mathrm{GeV}^2\), which makes the \(D=2\) running coefficient \(f_2^{p-n}(Q^2)\), Eq. (8), undefined at \(Q^2 \le 0.371 \ \mathrm{GeV}^2\). Nonetheless, in order to be able to present a curve, we applied in the fitting case \(Q_\mathrm{min}^2=0.268 \ \mathrm{GeV}^2\) in the \(\overline{\mathrm{MS}}\) pQCD approach a restriction on the leading-twist renormalization scale \(\mu ^2 = k Q^2\), namely \(k > 1.383\); and this not just in the leading-twist contribution, but we also made an ad hoc replacement in the \(D=2\) running coefficient \(f_2^{p-n}(Q^2)\), Eq. (8): \(f_2^{p-n}(Q^2) \mapsto f_2^{p-n}(k Q^2)\). We applied this also in the case when \(\mu _6 \ne 0\) (Sect. 3.2, for \(\overline{\mathrm{MS}}\) pQCD approach with \(Q_\mathrm{min}^2=0.268 \ \mathrm{GeV}^2\)). In other approaches (APT and \({\mathcal {A}}\)QCD’s) this is not necessary, as there are no Landau singularities. Further, we note that in the last column in Table 1 we have \(\chi ^2/\mathrm{d.o.f.} = \infty \) in the case of \(\overline{\mathrm{MS}}\) pQCD for \(Q^2_\mathrm{min}=0.66\) and \(0.47 \ \mathrm{GeV}^2\). This is so because at the lowest available experimental point, \(Q_{j=1}^2=0.054 \ \mathrm{GeV}^2\), the coupling is \(a(k Q_1^2; \overline{\mathrm{MS}}) = \infty \) because \(k Q_1^2 < Q^2_\mathrm{branch}=0.371 \ \mathrm{GeV}^2\) and thus we hit Landau singularities there.

We can deduce from Fig. 1 and Table 1: (a) the best results in the considered approach (\(\mu _6=0\)) are obtained in 3\(\delta \) \({\mathcal {A}}\)QCD; (b) the quality of extrapolation of the obtained fitted curves from the fitting interval, \(Q^2_\mathrm{min} \le Q^2 \le 3 \ \mathrm{GeV}^2\), to the entire interval, \(0.054 \ \mathrm{GeV}^2 \le Q^2 < 5 \ \mathrm{GeV}^2\), does not improve significantly when the fitted interval is extended (i.e., when \(Q^2_\mathrm{min}\) is lowered), cf. also the last column in Table 1. This indicates that the behavior of the curves in the extrapolated regions, \(0.054 \ \mathrm{GeV}^2< Q^2 < 0.268 \ \mathrm{GeV}^2\) and \(3 \ \mathrm{GeV}^2 < Q^2 \le 5 \ \mathrm{GeV}^2\), is of similar quality in the cases of different values of \(Q^2_\mathrm{min}(\mathrm{fit})\), and gives the dominant part of \(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\). The same can be observed for the values of \(f_2^{p-n}(1)\): they do not depend much on the value of \(Q^2_\mathrm{min}(\mathrm{fit})\), but only on the QCD variant used in the fit. This appears to be related with the fact that only one parameter beyond the leading-twist contribution is used here (\(\mu _4 \leftrightarrow f_2^{p-n}\)), representing a truncated OPE ansatz with a significantly restricted freedom.

3.2 Fits with \(\mu _6 \ne 0\)

When we include \(\mu _6\) in the fit as the third parameter, the resulting curves are presented in Fig. 2a, b, for \(Q^2_\mathrm{min}=0.66\) and \(0.47 \ \mathrm{GeV}^2\), respectively. Further, when \(Q^2_\mathrm{min}= 0.268 \ \mathrm{GeV}^2\), the results are shown in Fig. 3a, b, at the higher \(Q^2\) and the lower \(Q^2 < 1 \ \mathrm{GeV}^2\) momenta, respectively. The corresponding results are given in Table 2, for \(Q^2_\mathrm{min} = 0.66\), 0.47 and \(0.268 \ \mathrm{GeV}^2\), with the same notations as in Table 1. The various versions of \(\chi ^2/\mathrm{d.o.f.}\) are those as explained in the previous Sect. 3.1, except that now in the relation (15) the factor in front of the sum is \(1/(N-3)\) (\(p=3\), d.o.f. is \(N-3\)).
Fig. 2

Fits of the OPE expression (10) truncated at \(D=4\) term (i.e., \(\mu _4 \ne 0\)), to the experimental data for \(\Gamma _1^{p-n}(Q^2)_\mathrm{inel.}\), done in four different QCD variants, where in the fits: a left plot, \(Q^2 \ge 0.66~\mathrm{GeV}^2\); b right plot, \(Q^2 \ge 0.47~\mathrm{GeV}^2\). See the text for details. The respective lower bound of the fitting interval, \(Q^2_\mathrm{min}=0.66\) and \(0.47~\mathrm{GeV}^2\), is included as the thin dotted vertical line

Fig. 3

As Fig. 2, but for \(Q^2_\mathrm{min} = 0.268~\mathrm{GeV}^2\): a for larger \(Q^2\); b for \(Q^2 < 1~\mathrm{GeV}^2\). The lower bound of the fitting interval, \(Q^2_\mathrm{min}=0.268~\mathrm{GeV}^2\), is included as the thin dotted vertical line

Table 2

The values of the extracted fit parameters \(k = \mu ^2/Q^2\), \(f_2^{p-n}(1 \ \mathrm{GeV}^2)\) and \(\mu _6\) (in \(\mathrm{GeV}^4\)), obtained by fitting the OPE expression (10), truncated at \(D=4\) (\(\mu _6 \ne 0\)). The notations are as in Table 1. See the text in Sect. 3.1 for explanation of the various \(\chi ^2/\mathrm{d.o.f.}\)’s

QCD variant

\(Q^2_\mathrm{min}(\mathrm{fit})\)

k

\(f_2^{p-n}(1.)\)

\(\mu _6\)

\(\chi ^2/\mathrm{d.o.f}\)

\(\chi ^2_\mathrm{ext}/\mathrm{d.o.f}\)

\(\chi ^2_{0.268}/\mathrm{d.o.f.}\)

\(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\)

\(\overline{\mathrm{MS}}\) pQCD

0.66

16

\(-\,0.219 \pm 0.002 \pm 0.111\)

\(0.032 \pm 0.001 \pm 0.055\)

7.31

53.4

\(2.92 \times 10^5\)

\(1.68 \times 10^{6}\)

(F)APT

0.66

0.063

\(-\,0.198 \pm 0.002 \pm 0.106\)

\(0.018\pm 0.001 \pm 0.028\)

7.85

44.2

17.7

\(1.04 \times 10^{4}\)

2\(\delta \)

0.66

0.999

\(-\,0.116 \pm 0.002 \pm 0.094\)

\(0.014\pm 0.001 \pm 0.023\)

6.99

51.8

19.5

\(5.92 \times 10^{3}\)

3\(\delta \)

0.66

4.46

\(-\,0.101 \pm 0.002 \pm 0.114\)

\(0.013 \pm 0.001 \pm 0.025\)

6.45

57.5

28.0

\(7.27 \times 10^{3}\)

\(\overline{\mathrm{MS}}\) pQCD

0.47

16

\(-\,0.216 \pm 0.002 \pm 0.120\)

\(0.031 \pm 0.001 \pm 0.045\)

7.34

47.3

\(2.85 \times 10^5\)

\(1.66 \times 10^{6}\)

(F)APT

0.47

0.063

\(-\,0.194 \pm 0.002 \pm 0.098\)

\(0.016\pm 0.001 \pm 0.021\)

7.65

35.6

10.0

\(6.95 \times 10^{3}\)

2\(\delta \)

0.47

0.772

\(-\,0.098 \pm 0.002 \pm 0.078\)

\(0.008\pm 0.001 \pm 0.017\)

6.88

48.2

7.42

\(1.40 \times 10^{3}\)

3\(\delta \)

0.47

3.10

\(-\,0.072 \pm 0.002 \pm 0.138\)

\(0.003 \pm 0.001 \pm 0.026\)

6.64

53.5

6.56

218

\(\overline{\mathrm{MS}}\) pQCD

0.268

3.00

\(-\,0.180 \pm 0.001 \pm 0.237\)

\(0.023 \pm 0.000 \pm 0.013\)

6.52

38.7

6.52

\(\infty \)

(F)APT

0.268

5.27

\(-\,0.181 \pm 0.001 \pm 0.194\)

\(0.013\pm 0.000 \pm 0.011\)

7.77

37.0

7.77

\(3.91 \times 10^{3}\)

2\(\delta \)

0.268

0.149

\(-\,0.089 \pm 0.001 \pm 0.166\)

\(0.005\pm 0.000 \pm 0.010\)

6.27

41.9

6.27

293

3\(\delta \)

0.268

2.72

\(-\,0.007 \pm 0.001 \pm 0.262\)

\(0.001\pm 0.000 \pm 0.015\)

5.99

46.5

5.99

47.3

Comparing Table 2 and Figs. 2 and 3 with Table 1 and the Fig. 1 of the previous Sect. 3.1 where \(\mu _6=0\) was kept, we can see that in the cases of \(Q^2(\mathrm{fit}) \ge 0.66 \ \mathrm{GeV}^2\) and \(Q^2(\mathrm{fit}) \ge 0.47 \ \mathrm{GeV}^2\) the \(\mu _6=0\) fits give in general better \(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\) (the last columns of Tables 1 and 2). This means that the extrapolation down to the lowest experimental point \(Q^2=Q_{j=1}^2=0.054 \ \mathrm{GeV^2}\) is better when \(\mu _6=0\) (with the exception of \(\overline{\mathrm{MS}}\) pQCD case where problems with Landau singularities appear). This occurs because the inclusion of the \(\mu _6/(Q^2)^2\) term in the truncated OPE makes this expression less stable at very low \(Q^2\).

On the other hand, when \(Q^2(\mathrm{fit}) \ge 0.268 \ \mathrm{GeV}^2\), some of the fits (2\(\delta \) and 3\(\delta \) \({\mathcal {A}}\)QCD) give better extrapolation when \(\mu _6 \ne 0\). Table 2 and Fig. 3 also show that when \(\mu _6 \ne 0\) and \(Q^2(\mathrm{fit}) \ge 0.268 \ \mathrm{GeV}^2\), the best extrapolation to low \(Q^2\) is obtained in 3\(\delta \) \({\mathcal {A}}\)QCD, followed by 2\(\delta \) \({\mathcal {A}}\)QCD. As Fig. 3b suggests, the fitted curve in pQCD \(\overline{\mathrm{MS}}\) extrapolated to low \(Q^2\) appears to be almost as good; in this case, however, we should keep in mind that the renormalization scale is \(\mu ^2 = k Q^2\) (\(k =3.00\)) and that this scale was used also in \(f^{p-n}_2(Q^2)\), i.e., the ad hoc replacement \(f^{p-n}_2(Q^2) \mapsto f^{p-n}_2(k Q^2)\) was performed in order to avoid the Landau singularities in the \(D=2\) term at \(Q^2 \ge 0.268 \ \mathrm{GeV}^2\) (cf. also the discussion about that point in Sect. 3.1). Despite this replacement, in \(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\) Landau singularities are hit, because the three lowest experimental points give \(k Q_{j}^2 < Q^2_\mathrm{branch}=0.371 \ \mathrm{GeV}^2\) (\(j=1,2,3\)) and are thus in the Landau singularity region.7

In the case of (F)APT, in contrast to 2\(\delta \) and 3\(\delta \) \({\mathcal {A}}\)QCD, the coupling \({\mathcal {A}}(Q^2)\) differs from the underlying pQCD coupling \(a(Q^2)\) nonnegligibly at high \(|Q^2| > 1 \ \mathrm{GeV}^2\) [the index in Eq. (B1) is \(N=1\) in (F)APT; \(N=5\) in 2\(\delta \) and 3\(\delta \) \({\mathcal {A}}\)QCD]. This implies that (F)APT has a certain ambiguity when “normalizing” the strength of \({\mathcal {A}}(Q^2)\). As mentioned, we fixed the strength of the coupling \({\mathcal {A}}(Q^2)\) in (F)APT to the value \({\overline{\Lambda }}_{N_f=3}=0.45\) GeV, since in such a case (F)APT reproduces approximately the QCD phenomenology at high energies (cf. also [17, 18]). A question appears whether the results of our fits in (F)APT depend significantly on this value. We repeated the analysis in (F)APT with the value \({\overline{\Lambda }}_{N_f=3}=0.40\) GeV, and it turned out that the results of the fits did not change significantly. For example, when \(Q^2_\mathrm{min}=0.268 \ \mathrm{GeV}^2\) and \(\mu _6=0\), we obtained \(k=2.86\) and the central value \(f_2^{p-n}(1.)= -0.137\), and for the three quality parameters \(\chi ^2_\mathrm{ext}/\mathrm{d.o.f.}\), \(\chi ^2_{0.268}/\mathrm{d.o.f.}\) and \(\chi ^2_\mathrm{all}/\mathrm{d.o.f}\) the values 120., 54.7 and 878., respectively (to be compared with the corresponding values in Table 1, the third line from below). When \(\mu _6 \ne 0\) in the fit, we obtained \(k=2.62\), \(f_2^{p-n}(1.)=-0.187\), \(\mu _6=0.013 \ \mathrm{GeV}^4\) and for the mentioned three quality parameters the values 42.3, 7.94 and \(4.43 \times 10^3\), respectively (to be compared with the corresponding values in Table 2, the third line from below).

When we add the parametrized elastic contribution of BSR, Eqs. (11)–(13), to the experimental points and to the theoretical curves of Fig. 3, we obtain the results presented in Fig. 4a, b. In comparison with Fig. 3, the values of BSR are shifted to significantly higher values at low \(Q^2\). With this approach, the quality of fits (\(\chi ^2/\mathrm{d.o.f}\)) does not change, as we consider the elastic contribution as known (and parametrized) and added here simultaneously to the (inelastic) theoretical fitting curves and to the data points. This means that the results of Table 2 remain unchanged under this subsequent addition of \(\Gamma _1^{p-n}(Q^2)_\mathrm{el.}\).
Fig. 4

As Fig. 3, but shifted upwards by the parametrized elastic contribution (11) [cf. also Eqs. (12)–(13)]

We can observe in the results of Tables 1 and 2 that the values of the higher-twist parameters, \(f_2^{p-n}(1)\) and \(\mu _6\), are in the analytic variants of QCD smaller than in pQCD, this reduction being especially strong in the 3\(\delta \) QCD variant. It has been noted in the literature that in pQCD OPE there is a duality between the order of truncation of the leading-twist series and the higher-twist contribution [28, 29, 30, 31, 50, 51, 52, 53, 54, 55, 56, 57]: higher-twist contribution often significantly decreases with the inclusion of higher orders in the leading-twist part. This effect and ambiguity become stronger in the ranges where the perturbation theory becomes questionable (for example, at the large and low values of the Bjorken variable x, as it was shown in Refs. [62, 63], respectively). It has been observed that the higher-twist contribution is smaller, but also more stable (under the inclusion of more terms in the leading-twist), in QCD variants with infrared modifications of the coupling (various modifications lead to quite similar results [64, 65]). The latter probably incorporate a part of the higher-twist contributions (which are rather cumbersome [66]) into (formally) the leading-twist contribution for small x range at moderately small \(Q^2\) values (\(\lesssim 1 \ \mathrm{GeV}^2\)) (see Ref. [67] and more recent studies [68, 69, 70] of the precise combined H1 and ZEUS data [71] for the DIS structure function \(F_2\)).

Following the above observations in Tables 1 and 2, we can conclude that the applications of the 2\(\delta \) and especially 3\(\delta \) \({\mathcal {A}}\)QCD are very appropriate frameworks for the BSR studies because they appear to resum effectively a large part of the perturbative contribution into the leading-twist part (5). In this context, we wish to recall that 3\(\delta \) \({\mathcal {A}}\)QCD is significantly different from the other two \({\mathcal {A}}\)QCD variants [(F)APT and 2\(\delta \) \({\mathcal {A}}\)QCD] in the infrared region, because its coupling is not just finite there but goes to zero, \({\mathcal {A}}(Q^2) \sim Q^2 \rightarrow 0\), as motivated by large-volume lattice calculations, cf. [72, 73, 74, 75, 76] and Appendix C.2. Further, we wish to point out that the 2\(\delta \) and 3\(\delta \) \({\mathcal {A}}\)QCD couplings \({\mathcal {A}}\) (and thus \({\widetilde{\mathcal {A}}}_n\) and \({\mathcal {A}}_n\)) are at large \(Q^2\) indistinguishable from their underlying pQCD couplings a (\({\widetilde{a}}_n\), \(a^n\)), cf. Eq. (C7), in contrast to (F)APT which satisfies the relation Eq. (B1) with \(N=1\). Therefore, theoretically, neither the higher-twist contribution of order \(1/(Q^2)^N\) with \(N \le 4\) (\(D \le 8\)), nor a part of it, is incorporated in the leading-twist contribution (5) in the 2\(\delta \) and 3\(\delta \) \({\mathcal {A}}\)QCD, in contrast to (F)APT. This indicates that the higher-twist terms extracted here with 2\(\delta \) and 3\(\delta \) \({\mathcal {A}}\)QCD (in truncated OPE) represent an effective form for the true higher-twist contribution with dimension \(D \le 8\) [and a part of the other (\(D \ge 10\)) presumably small contribution]. In pQCD this is definitely not so, because of the mentioned duality there between the order of truncation of the leading-twist series and the extracted higher-twist contribution. As a consequence, the extracted effective higher-twist contribution in pQCD represents a sum of the true higher-twist contribution and a significant part of the perturbative (leading-twist) contribution; this effective higher-twist contribution appears to be in general larger than the true higher-twist contribution.
Fig. 5

As Fig. 3, but for the fit the OPE expression (16) with “massive” higher-twist term was used

Table 3

As in Table 2, but now the OPE form is “massive”, Eq. (16). The fits were made only for \(Q^2_\mathrm{min}(\mathrm{fit})=0.268 \ \mathrm{GeV}^2\). The extracted values of the squared mass \(M^2\) are in \(\mathrm{GeV}^2\)

QCD variant

\(Q^2_\mathrm{min}(\mathrm{fit})\)

k

\(f_2^{p-n}(1.)\)

\(M^2\)

\(\chi ^2/\mathrm{d.o.f}\)

\(\chi ^2_\mathrm{ext}/\mathrm{d.o.f}\)

\(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\)

\(\overline{\mathrm{MS}}\) pQCD

0.268

16.0

\(-\,0.286 \pm 0.003 \pm 1.035\)

\(0.623 \pm 0.017 \pm 0.471\)

7.94

35.4

130

(F)APT

0.268

16.0

\(-\,0.209 \pm 0.002 \pm 0.744\)

\(0.439\pm 0.012 \pm 0.463\)

7.48

31.5

31.2

2\(\delta \)

0.268

0.689

\(-\,0.097 \pm 0.002 \pm 0.155\)

\(0.336\pm 0.023 \pm 0.371\)

6.25

41.1

36.9

3\(\delta \)

0.268

2.71

\(-\,0.065 \pm 0.001 \pm 0.462\)

\(0.036\pm 0.017 \pm 0.377\)

5.99

46.6

55.0

3.3 Fits with “massive” OPE

For comparison, we performed a similar fit, but now with a “massive” higher-twist term instead of the truncated OPE expression (10)
$$\begin{aligned}&\Gamma _1^{p-n,\mathrm{mOPE[4]}}(Q^2; k, f_2^{p-n}(1); M^2) \nonumber \\&\quad = {\Big |}\frac{g_A}{g_V} {\Big |} \frac{1}{6}(1 - \mathcal{D}_\mathrm{BS}(Q^2)) \nonumber \\&\qquad +\, \frac{M_N^2}{(Q^2 + M^2)} \frac{1}{9} \left( a_2^{p-n} + 4 d_2^{p-n} + 4 f_2^{p-n}(Q^2) \right) ,\nonumber \\ \end{aligned}$$
(16)
where the squared mass \(M^2\) in the denominator of the higher-twist part8 is taken to be constant (not running), and is expected to be \(0 < M^2 \lesssim 1 \ \mathrm{GeV}^2\). Now, instead of \(f^{p-n}_2(1)\) and \(\mu _6\), the fit parameters are \(f_2^{p-n}(1)\) and \(M^2\). The resulting curves, for \(Q^2_\mathrm{min}=0.268 \ \mathrm{GeV}^2\), are given in Fig. 5a, b, at the higher \(Q^2\) and the lower \(Q^2 < 1 \ \mathrm{GeV}^2\) momenta, respectively. The corresponding results are given in Table 3. These curves are analogous to those in the previous Fig. 3a, b in which the truncated OPE (10) was used. Numerically, the behavior at low \(Q^2\) in the “massive” case is significantly influenced by the \(Q^2\)-dependence of \(f_2^{p-n}(Q^2)\). In the \(\overline{\mathrm{MS}}\) pQCD case, as in the \(\overline{\mathrm{MS}}\) pQCD cases of the analyses in all the Sections, we replaced in the higher-twist running parameter \(f_2^{p-n}(Q^2)\) [cf. Eq. (8)] the scale \(Q^2\) in an ad hoc way by the renormalization scale \(k Q^2\) used in the leading-twist part (\(k=16\) resulted here), in order to artificially avoid the problem of Landau singularities in the pQCD coupling \(a(Q^2)\). Comparing Fig. 5 and Table 3 with the corresponding “nonmassive” case Fig. 3 and Table 2, we see that the results and extrapolations in the case of 3\(\delta \) \({\mathcal {A}}\)QCD are now comparably good in the “massive” and the \( \mu _4 \& \mu _6\) approaches. Stated differently, the corresponding \(\chi ^2_{ \{... \}}/\mathrm{d.o.f.}\) values in Tables 2 and 3 (with \(Q^2_\mathrm{fit}=0.268 \ \mathrm{GeV}^2\)) are very similar. In the case of 2\(\delta \) \({\mathcal {A}}\)QCD and (F)APT, the extrapolations are better in the “massive” than in the \( \mu _4 \& \mu _6\) approach, i.e., \(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\) is significantly reduced in the “massive” case. One reason for this lies perhaps in the fact that the massive higher-twist term is under control at very low \(Q^2\), unlike the separate \(\mu _4(Q^2)/Q^2\) and \(\mu _6/(Q^2)^2\) terms. Further, the extracted values of \(f_2^{p-n}(1)\) are in general similar in the \(\mu _6=0\), \( \mu _4 \& \mu _6\) and the “massive” approaches, although the uncertainties of the extracted parameters are quite high in the “massive” approach.9
The results of Table 3 show that the QCD variants with infrared-finite analytic coupling, and especially 3\(\delta \) \({\mathcal {A}}\)QCD, give smaller values of higher-twist parameters \(f_2^{p-n}(1)\) and \(M^2\) than pQCD. On the one hand, at low \(Q^2\), the smaller values of \(M^2\) compensate partially the decreased value of \(f_2^{p-n}\) in the higher-twist contribution. On the other hand, smaller values of \(M^2\) and \(f_2^{p-n}(1)\) mean that at higher values of \(Q^2\) the higher-twist contribution is significantly reduced; this can be seen also by expanding the massive higher-twist term of Eq. (16) in powers of \(M^2/Q^2\). Such effect is in full agreement with one observed in Refs. [28, 29, 30, 31]; the effect can be considered as a stabilization of the higher-twist contribution. We also recall that 3\(\delta \) \({\mathcal {A}}\)QCD is significantly different from (F)APT and 2\(\delta \) \({\mathcal {A}}\)QCD in the infrared region, since its coupling goes to zero there, \({\mathcal {A}}(Q^2) \sim Q^2 \rightarrow 0\).
Table 4

As in Table 3, but now the squared mass in the higher-twist term in Eq. (16) is running according to Eq. (17)

QCD variant

\(Q^2_\mathrm{min}(\mathrm{fit})\)

k

\(f_2^{p-n}(1.)\)

\(M^2(1 \ \mathrm{GeV}^2)\)

\(\chi ^2/\mathrm{d.o.f}\)

\(\chi ^2_\mathrm{ext}/\mathrm{d.o.f}\)

\(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\)

\(\overline{\mathrm{MS}}\) pQCD

0.268

11.9

\(-\,0.214 \pm 0.002 \pm 0.651\)

\(0.240 \pm 0.010 \pm 0.272\)

5.91

48.1

252

(F)APT

0.268

16.0

\(-\,0.171 \pm 0.002 \pm 0.186\)

\(0.152\pm 0.007 \pm 0.100\)

6.65

43.2

38.8

2\(\delta \)

0.268

9.26

\(-\,0.160 \pm 0.002 \pm 0.496\)

\(0.277 \pm 0.011 \pm 0.318\)

5.88

49.0

48.0

3\(\delta \)

0.268

2.65

\(-\,0.064 \pm 0.001 \pm 0.097\)

\(0.010 \pm 0.009 \pm 0.162\)

5.99

46.5

48.3

Fig. 6

As Fig. 3, but now with the low-\(Q^2\) expressions (18) and (19) included. b is a zoomed-in version of a, for \(Q^2 < 1 \ \mathrm{GeV}^2\). The lower bound of the fitting interval, \(Q^2_\mathrm{min}=0.268~\mathrm{GeV}^2\), is included as the thin dotted vertical line

It is possible to choose for the higher-twist term a massive form with a running mass, in the spirit of a dynamical effective gluon mass of the gluon propagator at low \(Q^2\) [82]. Such masses appear in the literature often in definitions of QCD couplings at low \(Q^2\), and are responsible for the freezing (finiteness) of the coupling at \(Q^2 \rightarrow 0\). The couplings in (F)APT and 2\(\delta \) \({\mathcal {A}}\)QCD have a freezing which could be described also via a running effective gluon mass. Our view is that the OPE higher-twist terms represent a new contribution not contained in the QCD coupling itself. The squared mass \(M^2\) in such terms, Eq. (16), is considered constant, in the spirit of the approach of Ref. [77] (cf. also [78, 79]), where the basic component (delta function) in the spectral function of the higher-twist contribution gives such a mass term. Nonetheless, we repeated the aforementioned analysis for the case of a running squared mass \(M^2(Q^2)\) representative of a dynamical effective gluon mass, chosen with a simple parametrization of Ref. [83]
$$\begin{aligned} M^2(Q^2) = \frac{m_0^2}{1 + (Q^2/\mathcal{M})^{1+p}} \quad (\mathcal{M}=0.5 \ \mathrm{GeV}, p=0.1), \nonumber \\ \end{aligned}$$
(17)
where we chose for the parameters \(\mathcal{M}\) and p values within the expected regions [83]. The adjustable squared mass scale was taken (instead of \(m_0^2\)) to be \(M^2(1 \ \mathrm{GeV}^2)\). The same analysis then gave the results presented in Table 4. We can see that the results are qualitatively similar to those with the constant squared mass \(M^2\), Table 3, except that the values of \(M^2(1 \ \mathrm{GeV}^2)\) are now lower. Nonetheless, a reasonable definition of the average squared mass value in the present analysis, for the considered fit interval \(0.268 \ \mathrm{GeV}^2< Q^2 < 3 \ \mathrm{GeV}^2\) may be \(\langle M^2(Q^2) \rangle = (1/2) \times ( M^2(0.268) + M^2(3) ) \approx 1.52 \times M^2(1 \ \mathrm{GeV}^2)\). Or, if we regard that the squared mass is most relevant only in the low-\(Q^2\) part \(0.268 \ \mathrm{GeV}^2< Q^2 < 1 \ \mathrm{GeV}^2\) of the fit interval, a reasonable definition of the average squared mass would be \(\langle M^2(Q^2) \rangle = (1/2) \times ( M^2(0.268) + M^2(1) ) \approx 1.85 \times M^2(1 \ \mathrm{GeV}^2)\).

3.4 Testing low-\(Q^2\) regime ansätze

At low \(Q^2\), the inelastic contribution to BSR behaves as \(\sim Q^2\), according to Gerasimov–Drell–Hearn sum rule [84, 85] as pointed out and used in [28, 29, 30, 31, 86, 87, 88, 89]. Based on this, an expansion [4] motivated by chiral perturbation theory (\(\chi \)PT) can be constructed
$$\begin{aligned} \Gamma _1^{p-n}(Q^2)_\mathrm{inel.}= & {} \frac{\chi _n^2 - \chi _p^2}{8 M_N^2} Q^2 + A (Q^2)^2+ B (Q^2)^3 \nonumber \\&\quad (Q^2 \lesssim 0.5 \ \mathrm{GeV}^2), \end{aligned}$$
(18)
where, according to Gerasimov–Drell–Hearn sum rule [84, 85], \(\chi _n\) and \(\chi _p\) are anomalous magnetic moments of nucleons [which appear also in the elastic BSR contributions, cf. Eq. (13)]; the parameters A and B are determined in the fit. When we fit with this expression the inelastic BSR data [3, 4, 5] for \(Q^2 \le Q^2_\mathrm{max}=0.2 \ \mathrm{GeV}^2\), we obtain \(A=0.765 \ \mathrm{GeV}^{-2}\) and \(B= 0.678 \ \mathrm{GeV}^{-4}\), with \(\chi ^2(Q^2 \le Q^2_\mathrm{max})/\mathrm{d.o.f.} = 0.720\). On the other hand, if we take \(Q^2_\mathrm{max}=0.5 \ \mathrm{GeV}^2\) in the fitting, we obtain \(A=0.744 \ \mathrm{GeV}^{-2}\) and \(B= -1.033 \ \mathrm{GeV}^{-4}\), \(\chi ^2(Q^2 \le Q^2_\mathrm{max})/\mathrm{d.o.f.} = 1.313\).
Another possible ansatz for the inelastic contribution to BSR at low \(Q^2\) is the form of the light-front holographic (LFH) effective charge \({\mathcal {A}}^\mathrm{(LFH)}\) in the BSR (\(g_1\)) scheme [90, 91] [\({\mathcal {A}}(0)_{g_1}=1\)]
$$\begin{aligned}&\Gamma _1^{p-n}(Q^2)_\mathrm{inel.}\nonumber \\&\quad = \left| \frac{g_A}{g_V} \right| \frac{1}{6} \left[ 1 - {\mathcal {A}}^\mathrm{(LFH)}(Q^2) \right] \nonumber \\&\quad = \left| \frac{g_A}{g_V} \right| \frac{1}{6} \left[ 1 - \exp \left( - \frac{Q^2}{2 \kappa ^2} \right) \right] ,\quad (Q^2 \lesssim 1 \mathrm{GeV}^2). \end{aligned}$$
(19)
When we fit with this expression the inelastic BSR data [3, 4, 5] for \(Q^2 < Q^2_\mathrm{max}=0.5 \ \mathrm{GeV}^2\), we obtain \(\kappa =0.479\) GeV.10
Fig. 7

As Fig. 3, but now the truncated (at \(D=4\)) OPE expression (10) is fitted to the data for the total contribution \(\Gamma _1^{p-n}(Q_j^2)_\mathrm{inel.+el.}\). The lower bound of the fitting interval, \(Q^2_\mathrm{min}=0.268~\mathrm{GeV}^2\), is included as the thin dotted vertical line

In Fig. 6a, b, we present these low-\(Q^2\) expressions. The \(\overline{\mathrm{MS}}\) pQCD and 3\(\delta \) \({\mathcal {A}}\)QCD curves (obtained from fit with \(Q^2 \ge Q^2_\mathrm{min} = 0.268 \ \mathrm{GeV}^2\)) are also included for comparison. In Fig. 6b we can see that the theoretical fitted curves of the presented QCD variants connect smoothly with the nonperturbative low-\(Q^2\) curves (\(\chi \)PT and LFH, both fitted to BSR up to \(Q^2=0.5 \ \mathrm{GeV}^2\)) at the values of \(Q^2\) around \(0.3 \ \mathrm{GeV}^2\). Nonetheless, we recall that the apparent success of the (\(\overline{\mathrm{MS}}\)) pQCD curve, down to about \(0.3 \ \mathrm{GeV}^2\), was achieved due to the ad hoc change of scale in \(f^{p-n}_2(Q^2)\) to \(k Q^2\) (with \(k=3\)), to avoid the Landau singularities in the \(D=2\) term at \(Q^2 \ge 0.268 \ \mathrm{GeV}^2\) (cf. also the discussion about that point in Sect. 3.2). We further notice that the 3\(\delta \) QCD curve agrees well with the mentioned \(\chi \)PT curve in a broader interval, \(0.17 \ \mathrm{GeV}^2< Q^2 < 0.3 \ \mathrm{GeV}^2\). Similar analyses with the goal of connecting the curves of pQCD (or of specific QCD variants) with nonperturbative curves at low \(Q^2\) were performed in some of the references [28, 29, 30, 31], in [92], and was discussed also in [93].

In Refs. [94, 95], BSR at low \(Q^2 < 0.3 \ \mathrm{GeV}^2\) was calculated with baryon chiral perturbation theory (B\(\chi \)PT) up to NLO. The authors of [94, 95] did not give the values of their parameters for BSR. However, careful visual comparison of their obtained \(Q^2\)-dependence [cf. curves and bands in their Fig. 6(b) in Ref. [95]] with our \(\chi \)PT curve (18) with \(Q^2_\mathrm{fit} \le 0.5 \ \mathrm{GeV}^2\) [the long-dashed curve in our Fig. 6b] shows very good agreement between them.
Table 5

As in Table 2, but now the fit is performed on the sum which includes the elastic contribution, i.e., on \(\Gamma _1^{p-n}(\mathrm{inel.+el.})\)

QCD variant

\(Q^2_\mathrm{min}(\mathrm{fit})\)

k

\(f_2^{p-n}(1.)\)

\(\mu _6\)

\(\chi ^2/\mathrm{d.o.f}\)

\(\chi ^2_\mathrm{ext}/\mathrm{d.o.f}\)

\(\chi ^2_{0.268}/\mathrm{d.o.f.}\)

\(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\)

\(\overline{\mathrm{MS}}\) pQCD

0.66

16

\(-\,0.253 \pm 0.002 \pm 0.125\)

\(0.101 \pm 0.001 \pm 0.064\)

7.11

55.6

\(3.65 \times 10^5\)

\(1.26 \times 10^{6}\)

(F)APT

0.66

16

\(-\,0.227 \pm 0.002 \pm 0.105\)

\(0.086 \pm 0.001 \pm 0.028\)

7.53

46.3

697

\(3.54 \times 10^{5}\)

2\(\delta \)

0.66

0.487

\(-\,0.121 \pm 0.002 \pm 0.057\)

\(0.073 \pm 0.001 \pm 0.011\)

7.18

52.0

555

\(2.60 \times 10^{5}\)

3\(\delta \)

0.66

2.01

\(-\,0.095 \pm 0.002 \pm 0.145\)

\(0.067 \pm 0.001 \pm 0.033\)

7.04

55.8

509

\(2.27 \times 10^{5}\)

\(\overline{\mathrm{MS}}\) pQCD

0.47

1.37

\(-\,0.194 \pm 0.002 \pm 0.111\)

\(0.114 \pm 0.001 \pm 0.056\)

8.82

39.0

\(\infty \)

\(\infty \)

(F)APT

0.47

1.67

\(-\,0.191 \pm 0.002 \pm 0.086\)

\(0.072 \pm 0.001 \pm 0.021\)

19.4

63.4

364

\(2.46 \times 10^{5}\)

2\(\delta \)

0.47

0.194

\(-\,0.089 \pm 0.002 \pm 0.052\)

\(0.059 \pm 0.001 \pm 0.093\)

14.8

57.8

250

\(1.67 \times 10^{5}\)

3\(\delta \)

0.47

0.775

\(-\,0.076 \pm 0.002 \pm 0.135\)

\(0.053 \pm 0.001 \pm 0.029\)

12.6

44.9

193

\(1.38 \times 10^{5}\)

\(\overline{\mathrm{MS}}\) pQCD

0.268

1.82

\(-\,0.153 \pm 0.001 \pm 0.081\)

\(0.086 \pm 0.000 \pm 0.010\)

28.0

82.3

28.0

\(\infty \)

(F)APT

0.268

1.98

\(-\,0.096 \pm 0.001 \pm 0.054\)

\(0.044\pm 0.000 \pm 0.011\)

131

254

131

\(8.84 \times 10^{4}\)

2\(\delta \)

0.268

0.063

\(0.018 \pm 0.001 \pm 0.042\)

\(0.028\pm 0.000 \pm 0.006\)

37.9

113

37.9

\(3.91 \times 10^{4}\)

3\(\delta \)

0.268

0.846

\(0.000 \pm 0.001 \pm 0.091\)

\(0.030\pm 0.000 \pm 0.016\)

68.7

145

68.7

\(4.61 \times 10^{4}\)

3.5 Fitting to total (inelastic \(+\) elastic) BSR data

We perform also the fitting of the truncated (at \(D=4\)) OPE expression (10) to the data for the total BSR \(\Gamma _1^{p-n}(Q^2)_\mathrm{inel.+el.}\). As argued at the end of Sect. 2.3 [after Eq. (14)], such a fit has problematic aspects. These data are obtained by adding to the experimental data \(\Gamma _1^{p-n}(Q_j^2)_\mathrm{inel.}\) of Refs. [3, 4, 5] the parametrized elastic contribution Eqs. (11)–(13) of Ref. [60]. The uncertainties of the parametrized elastic form factors are considered to be less than \(10 \%\) [60]. Since the elastic contribution (11) is quadratic in the form factors (and is numerically dominated by the proton contribution), the relative uncertainties of the elastic contribution are less than \(5 \%\). Since we do not have more information about these uncertainties, we will neglect them in this analysis. This means that in the total BSR, we will consider that the statistical and the systematic uncertainties \(\sigma _{j, \mathrm{stat}}\) and \(\sigma _{j, \mathrm{sys}}\) are those of the inelastic contribution. Otherwise, the fitting is performed as in the previous Sect. 3.2.

The resulting curves are presented in Fig. 7a, b, for \(Q^2 \ge Q^2_\mathrm{min}\) with \(Q^2_\mathrm{min} = 0.268 \ \mathrm{GeV}^2\). The obtained results are given in Table 5, for fits with various \(Q^2_\mathrm{min} = 0.66\), 047 and \(0.268 \ \mathrm{GeV}^2\). Comparing the obtained results in Fig. 7 and Table 5 with the corresponding results in Fig. 4 and Table 2 (where the elastic part was not included in the fit procedure), we see that the inclusion of the elastic contribution in the fit procedure significantly deteriorates (increases) the values of the various fit quality parameters \(\chi ^2\), especially when the fit is performed in the larger interval \((Q^2_\mathrm{min}=) 0.268 \ \mathrm{GeV}^2 \le Q^2 < 3 \mathrm{GeV}^2\). In particular, the differences in quality are clearly visible when comparing Fig. 7b with Fig. 4b at low \(Q^2\). Only when \(Q^2_\mathrm{min}\) is relatively high, \(Q^2_\mathrm{min}=0.66 \ \mathrm{GeV}^2\), are some of the \(\chi ^2\) parameters comparable in the two cases (but not the extrapolation quality parameters \(\chi ^2_{0.268}\) and \(\chi ^2_\mathrm{all}\)). Further, the inclusion of the elastic contribution in the fit in general does not change significantly the extracted (mostly negative) values of \(f_2^{p-n}(1 \mathrm{GeV}^2)\), but increases significantly the (positive) values of \(\mu _6\).

For the fit analysis with the “massive” truncated OPE, Eq. (16), the results are presented in Table 6 for \(Q^2_\mathrm{min}=0.268 \ \mathrm{GeV}^2\). In general, the results are worse than in the corresponding “massive” OPE case fitted to the inelastic contribution (cf. Table 3 in Sect. 3.2). Namely, the extracted values of the parameter \(M^2\) become negative (\(M^2 \sim -0.1 \ \mathrm{GeV}^2\)) and the fit quality values \(\chi ^2/\mathrm{d.o.f.}\) are in general significantly higher than those in Table 3. The uncertainties of the extracted values of \(f_2^{p-n}(1)\) and \(M^2\) become reduced (in comparison to the case when the elastic part is not included in the fit); this is so because the elastic parts increase significantly BSR \(\Gamma _1^{p-n}(Q^2)\) at low \(Q^2\) where they are represented mostly by the “massive” higher-twist term, and hence the relative uncertainties of BSR at low \(Q^2\) become smaller.

In addition to the massive case with constant squared mass \(M^2\), we include also the case of \(Q^2\)-dependent squared mass Eq. (17) in the analysis with the elastic contribution included in the fit. The results for this case are presented in Table 7. These results are qualitatively similar to those with constant squared mass, Table 6; and the comparison of Table 7 with its counterpart Table 4 without the elastic contribution is similar to the above comparison when the squared mass is constant. We recall that an average mass in the \(Q^2\)-independent case can be regarded to be \(\langle M^2(Q^2) \rangle = (1/2) \times ( M^2(0.268) + M^2(1) ) \approx 1.85 \times M^2(1 \ \mathrm{GeV}^2)\).
Table 6

As in Table 3, i.e., the OPE form is “massive”, Eq. (16), but now the elastic part of BSR is included in the fit

QCD variant

\(Q^2_\mathrm{min}(\mathrm{fit})\)

k

\(f_2^{p-n}(1.)\)

\(M^2\)

\(\chi ^2/\mathrm{d.o.f}\)

\(\chi ^2_\mathrm{ext}/\mathrm{d.o.f}\)

\(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\)

\(\overline{\mathrm{MS}}\) pQCD

0.268

1.47

\(0.117 \pm 0.000 \pm 0.042\)

\(-\,0.247 \pm 0.000 \pm 0.017\)

384

795

\(\infty \)

(F)APT

0.268

3.39

\(0.041 \pm 0.001 \pm 0.032\)

\(-\,0.207\pm 0.002 \pm 0.063\)

395

760

\(4.63 \times 10^{3}\)

2\(\delta \)

0.268

0.063

\(0.086 \pm 0.001 \pm 0.044\)

\(-\,0.156\pm 0.002 \pm 0.032\)

96.6

245

\(1.39 \times 10^{4}\)

3\(\delta \)

0.268

1.13

\(0.083 \pm 0.001 \pm 0.082\)

\(-\,0.151\pm 0.003 \pm 0.071\)

143

320

\(1.05 \times 10^{4}\)

Table 7

As in Table 6, but with \(Q^2\)-dependent mass Eq. (17) in the higher-twist part of Eq. (16)

QCD variant

\(Q^2_\mathrm{min}(\mathrm{fit})\)

k

\(f_2^{p-n}(1.)\)

\(M^2(1 \ \mathrm{GeV}^2)\)

\(\chi ^2/\mathrm{d.o.f}\)

\(\chi ^2_\mathrm{ext}/\mathrm{d.o.f}\)

\(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\)

\(\overline{\mathrm{MS}}\) pQCD

0.268

4.88

\(0.113 \pm 0.001 \pm 0.053\)

\(-\,0.065 \pm 0.001 \pm 0.070\)

461

932

\(\infty \)

(F)APT

0.268

3.22

\(0.052 \pm 0.001 \pm 0.030\)

\(-\,0.072 \pm 0.001 \pm 0.034\)

436

825

\(3.25 \times 10^{3}\)

2\(\delta \)

0.268

0.063

\(0.102 \pm 0.001 \pm 0.034\)

\(-\,0.051 \pm 0.001 \pm 0.020\)

119

288

\(6.58 \times 10^{5}\)

3\(\delta \)

0.268

1.14

\(0.098 \pm 0.001 \pm 0.044\)

\(-\,0.048 \pm 0.001 \pm 0.023\)

167

366

\(5.49 \times 10^{4}\)

4 Summary

Experimental results for the polarized Bjorken sum rule (BSR) \(\Gamma _1^{p-n}(Q^2)\) were fitted, for various ranges of \(Q^2\), with OPE theoretical expressions using QCD couplings obtained in four different approaches: perturbative QCD (pQCD) in \(\overline{\mathrm{MS}}\) scheme; (Fractional) Analytic Perturbation Theory [(F)APT]; Two-delta \({\mathcal {A}}\)QCD (2\(\delta \)); and Three-delta lattice-motivated \({\mathcal {A}}\)QCD (3\(\delta \)). The QCD running coupling \({\mathcal {A}}(Q^2)\) in the latter three QCD variants does not have Landau singularities, in contrast to the pQCD coupling \(a(Q^2)\) [\(\equiv \alpha _s(Q^2)/\pi \)].

In the fit of the inelastic experimental BSR results, up to two higher-twist terms [\(\sim 1/Q^2, 1/(Q^2)^2\)] were added to the theoretical leading-twist contribution. The elastic contributions, which are \(\sim 1/(Q^2)^n\) with typically \(n \ge 4\) at \(Q^2 > 1 \ \mathrm{GeV}^2\), were then added by using the parametrization obtained from the literature. The fits were performed for the ranges \(Q^2_\mathrm{min} \le Q^2 \le 3 \ \mathrm{GeV}^2\), where \(Q^2_\mathrm{min} = 0.66\), 0.47 and \(0.268 \ \mathrm{GeV}^2\). In general, the best curves were obtained when 2\(\delta \) or 3\(\delta \)-couplings were used. When only \(D=2\) (\(\sim 1/Q^2\)) higher-twist term was included in the fit, the quality of the fitted curves, in the range of the fit and in the extrapolated ranges of \(Q^2\), in general did not depend significantly on \(Q^2_\mathrm{min}\) of the fit. On the other hand, when both \(D=2\) and \(D=4\) terms were included, the quality in the extrapolated ranges of \(Q^2\) was in general better for the lowest \(Q^2_\mathrm{min}\) value (\(0.268 \ \mathrm{GeV}^2\)), i.e., when the \(Q^2\)-range of the fit was the largest. Comparably good results were obtained when “massive” higher-twist term was used in the OPE and the QCD coupling was either from (F)APT or 2\(\delta \) or 3\(\delta \) \({\mathcal {A}}\)QCD.

When the range of fit had \(Q^2_\mathrm{min}=0.268 \ \mathrm{GeV}^2\), the pQCD \(\overline{\mathrm{MS}}\) coupling approach worked and gave acceptable results only if the renormalization scale of the coupling was maintained everywhere at sufficiently high values, and the coefficient \(f_2^{p-n}(Q^2)\) [\(\sim a(Q^2)^{\gamma _0/8 \beta _0}\)] at the \(D=2\) term had an (ad hoc) increased scale \(Q^2 \mapsto k Q^2\), in order to avoid the problem of the Landau singularities.

When the fit procedure was performed by fitting the theoretical OPE, truncated at \(D=4\) (\(\sim 1/(Q^2)^2\)) terms, to the sum of (experimental) inelastic and (parametrized) elastic BSR, the quality of the results turned out to be significantly worse in all the cases of the theoretical curves, something expected by the arguments presented at the end of Sect. 2.3. Namely, the elastic contribution is dominated by terms which behave at high values of \(Q^2\) as \(\sim 1/(Q^2)^{(D/2)}\) where usually \(D \ge 8\), and these terms are not contained in the theoretical expressions for BSR which are usually OPE series truncated at \(1/(Q^2)^2\).

The results of this work can be interpreted as an additional indication of the following important property: the evaluation of the (truncated) leading-twist contribution of spacelike low-\(Q^2\) QCD observables such as inelastic BSR, in QCD variants 2\(\delta \) and in particular 3\(\delta \) \({\mathcal {A}}\)QCD [both have infrared finite and holomorphic coupling \({\mathcal {A}}(Q^2)\)], appear to resum effectively a large part of the perturbative contribution of the observables, and leads to reduced extracted values of the higher-twist terms (\(D=2, 4\)) in the truncated OPE. This property was noted earlier, for different observables, in Refs. [34, 35, 96]. In this context, it appears to be important that in 2\(\delta \) and 3\(\delta \) \({\mathcal {A}}\)QCD the coupling practically merges with the underlying pQCD coupling \(a(Q^2)\) at higher values of \(Q^2 \gg \Lambda ^2_\mathrm{QCD}\). This property is not shared by the (F)APT holomorphic coupling where the leading-twist series contains parts of the higher-twist contribution of as low dimensionality as \(D=2\). The extracted parameters in the higher-twist contribution, including those in the “massive” OPE, are especially reduced in 3\(\delta \) \({\mathcal {A}}\)QCD. This suggests the possibility that the true higher-twist contribution is small, including the (sum of) terms of high dimension; and that the (truncated) OPE with 3\(\delta \) \({\mathcal {A}}\)QCD leading-twist gives, through fitting, an extracted value which is a good approximation to this true value of the higher-twist contribution. Numerically, the significantly reduced extracted value in (truncated) OPE with 3\(\delta \) \({\mathcal {A}}\)QCD is probably partly related with the fact that 3\(\delta \) \({\mathcal {A}}\)QCD differs from both 2\(\delta \) and (F)APT \({\mathcal {A}}\)QCD variants in that its coupling goes to zero in the deep infrared regime, \({\mathcal {A}}^{(3 \delta )}(Q^2) \sim Q^2 \rightarrow 0\). The latter property, we recall, is suggested by the large-volume lattice calculations of the dressing functions of the Landau-gauge gluon and ghost propagators at low \(Q^2\) values.

Footnotes

  1. 1.

    The index j in \(Q_j^2\) indicates from here on that these are the scales at which the experimental values are given.

  2. 2.

    The singlet contribution appears for the first time at \(\sim a^4\), and only if \(N_f \ne 3\) [43, 44].

  3. 3.

    However, the last coefficient in the truncated OPE, e.g. \(\mu _6\), is sometimes in the literature regarded to include, in a certain effective way, the contributions from the higher dimensional terms \(D \ge 6\), cf. [5] for a discussion of these aspects.

  4. 4.

    The elastic contributions parametrized in a way different from that of Ref. [60] could in general contain terms of dimension \(D<8\). Nonetheless, the fitted expressions of Ref. [59] give for the elastic contributions an expansion similar to Eq. (14), where the first nonzero term is a very small \(D=6\) contribution which is negligible in comparison with \(D=8\) term (the coefficient at \(D=6\) term is by about a factor of \(10^{-4}\) smaller than at \(D=8\) term). All these terms there are divided by a large weakly \(Q^2\)-dependent factor \([\ln (27.81 + 1.72 \; Q^2/M_{\rho }^2)]^{4.296}\).

  5. 5.

    We will not take into account the SLAC E155, SMC and HERMES points [9, 10, 11], at \(Q^2=5 \ \mathrm{GeV}^2\), \(10 \ \mathrm{GeV}^2\), and 2.5 (and 5) \(\mathrm{GeV}^2\), respectively, as they were obtained by NLO pQCD \(Q^2\)-evolution from data distributed across a wide range of \(Q^2\). Among the points of SLAC E143 we exclude the point at \(Q^2=5 \ \mathrm{GeV}^2\) for the same reason.

  6. 6.

    If we perform the fit with the four points included, the results differ somewhat, but not significantly.

  7. 7.

    We have \(Q_1^2=0.054 \ \mathrm{GeV}^2\), \(Q_2^2=0.078 \ \mathrm{GeV}^2\), and \(Q_3^2=0.101 \ \mathrm{GeV}^2\).

  8. 8.

    Similar higher-twist expressions were used in the analyses of BSR in [77, 78, 79] where the leading-twist contribution was evaluated with the “Massive” Perturbation Theory (MPT) [80, 81]. MPT is an extension of APT [13, 14, 15, 16, 17, 18] where, in contrast to APT, the coupling \({\mathcal {A}}(Q^2)\) is analytic in the point \(Q^2=0\); nonetheless, the index N of Eq. (B1) (Appendix B) remains in MPT at the minimal value as in APT, i.e., \(N=1\).

  9. 9.

    The systematic uncertainties of the extracted parameters in Table 3 are large and should therefore be regarded as crude estimates only, cf. comments in Appendix D.2.

  10. 10.

    This is not far from the universal nonperturbative scale \(\kappa = M_{\rho }/\sqrt{2} = 0.548\) GeV, cf. [91].

  11. 11.

    There is another scheme parameter, the scale \(\Lambda ^2\), such that \(a(\mu ^2) = f(\mu ^2/\Lambda ^2)\). However, the change of \(\Lambda ^2\) can be regarded as the change in the definition of the renormalization scale \(\mu ^2\).

  12. 12.

    This is in principle not necessary; if, for example, a physical quantity \(\mathcal{D}(Q^2)\) is calculated up to \(\mathcal{O}(a^4) \mapsto \mathcal{O}({\mathcal {A}}_4)\), then only terms up to \({\widetilde{\mathcal {A}}}_4\) are in principle needed on the right-hand sides of Eq. (B7).

  13. 13.

    This means that in Eq. (B1) the index N is large; as a consequence, the well demonstrated success of pQCD at high momenta is reproduced in \({\mathcal {A}}\)QCD.

  14. 14.

    If \(\rho _{{\mathcal {A}}}(\sigma ) \ge 0\) for \(\sigma > 0\), then \({\mathcal {A}}(Q^2)\) and \(\Delta {\mathcal {A}}(Q^2)\) are Stieltjes functions.

  15. 15.

    Higher-twist contributions in \(r_{\tau }\) decay ratio are known to be strongly suppressed.

  16. 16.

    This is the canonical Adler function with \(N_f=3\), i.e., it is normalized in such a way that \(d(Q^2)_\mathrm{pt} = a(Q^2) + \mathcal{O}(a^2)\).

  17. 17.

    The calculation in [72, 73] were performed for \(N_f=0\) case. They are similar to the results when \(N_f=2, 4\) [74, 75, 76] although these results are not so precise. If the running is defined via the mentioned dressing functions, the mentioned conditions, with \({\mathcal {A}}(0)=0\), follow. Different definitions involving, in addition, a dynamical gluon mass [107, 108, 109], would imply \({\mathcal {A}}(0) >0\), which holds also in 2\(\delta \) \({\mathcal {A}}\)QCD and in (F)APT, with the values \({\mathcal {A}}(0) = 2.0713, 1.3963\), respectively (for \(N_f=3\)). For a discussion on these issues, we refer to [35].

  18. 18.

    This difference, of a few percent (\(\le 5\%\)), is presumably principally due to the effect of the running of \(\mu _4(Q^2)\) with \(Q^2\), the effect not accounted for in the formula (D6).

  19. 19.

    We denote, from here on, by \({\hat{f}}_2\) the value of \(f_2^{p-n}(1\mathrm{GeV}^2)\) extracted by the fit procedure.

  20. 20.

    Equations (D19b) and (D18) give the same result for \(\sigma ({\hat{\mu }}_6)_\mathrm{stat}\). For \(\sigma ({\hat{\mu }}_4)_\mathrm{stat}\) [and \(\sigma ({\hat{f}}_2)_\mathrm{stat}\), cf. Eq. (D8)], the result of Eq. (D19a) differs from that of Eq. (D18) by a few percent (\(\le 5 \%\)), principally due the effect of the running of \(f_2^{p-n}(Q^2)\) with \(Q^2\).

  21. 21.

    This is somewhat different from the unnormalized averages \(\overline{A}\) appearing in Eqs. (D16)–(D18) which are over all the experimental points of the fit, i.e., over more than two experiments.

Notes

Acknowledgements

This work was supported by FONDECYT Postdoctoral Grant no. 3170116 (C.A.), by FONDECYT Regular Grant no. 1180344 (G.C. and C.A.), and by the RFBR Foundation through Grant no. 16-02-00790-a (A.V.K. and B.G.S.). We thank A. Deur, J. Blümlein and A. L. Kataev for helpful suggestions.

References

  1. 1.
    J.D. Bjorken, Applications of the chiral \(U(6) \times (6)\) algebra of current densities. Phys. Rev. 148, 1467 (1966)ADSGoogle Scholar
  2. 2.
    J.D. Bjorken, Inelastic scattering of polarized leptons from polarized nucleons. Phys. Rev. D 1, 1376 (1970)ADSGoogle Scholar
  3. 3.
    A. Deur et al., Experimental determination of the evolution of the Bjorken integral at low \(Q^2\). Phys. Rev. Lett. 93, 212001 (2004). arXiv:hep-ex/0407007 ADSGoogle Scholar
  4. 4.
    A. Deur, Experimental study of isovector spin sum rules. Phys. Rev. D 78, 032001 (2008). arXiv:0802.3198 [nucl-ex]ADSGoogle Scholar
  5. 5.
    A. Deur et al., High precision determination of the \(Q^2\) evolution of the Bjorken Sum. Phys. Rev. D 90, 012009 (2014). arXiv:1405.7854 [nucl-ex]ADSGoogle Scholar
  6. 6.
    C. Adolph et al. (COMPASS Collaboration), The spin structure function \(g_1^{\rm p}\) of the proton and a test of the Bjorken sum rule. Phys. Lett. B 753, 18 (2016). arXiv:1503.08935 [hep-ex]
  7. 7.
    C. Adolph et al. (COMPASS Collaboration), Final COMPASS results on the deuteron spin-dependent structure function \(g_1^{\rm d}\) and the Bjorken sum rule. Phys. Lett. B 769, 34 (2017). arXiv:1612.00620 [hep-ex]
  8. 8.
    K. Abe et al. (E143 Collaboration), Measurements of the proton and deuteron spin structure functions \(g_1\) and \(g_2\). Phys. Rev. D 58, 112003 (1998). arXiv:hep-ph/9802357
  9. 9.
    P.L. Anthony et al. (E155 Collaboration), Measurements of the \(Q^2\) dependence of the proton and neutron spin structure functions \(g_1^p\) and \(g_1^n\). Phys. Lett. B 493, 19 (2000). arXiv:hep-ph/0007248
  10. 10.
    D. Adams et al. (Spin Muon (SMC) Collaboration), Spin structure of the proton from polarized inclusive deep inelastic muon-proton scattering. Phys. Rev. D 56, 5330 (1997). arXiv:hep-ex/9702005
  11. 11.
    A. Airapetian et al. (HERMES Collaboration), Precise determination of the spin structure function \(g_{1}\) of the proton, deuteron and neutron. Phys. Rev. D 75, 012007 (2007). arXiv:hep-ex/0609039
  12. 12.
    Y. Prok et al. (CLAS Collaboration), Precision measurements of \(g_1\) of the proton and the deuteron with 6 GeV electrons. Phys. Rev. C 90(2), 025212 (2014). arXiv:1404.6231 [nucl-ex]
  13. 13.
    D.V. Shirkov, I.L. Solovtsov, Analytic QCD running coupling with finite IR behaviour and universal \(\bar{\alpha }_s(0)\) value. JINR Rapid Commun. 2(76), 5–10 (1996). arXiv:hep-ph/9604363 Google Scholar
  14. 14.
    D.V. Shirkov, I.L. Solovtsov, Analytic model for the QCD running coupling with universal \(\alpha _s(0)\) value. Phys. Rev. Lett. 79, 1209 (1997). arXiv:hep-ph/9704333 ADSGoogle Scholar
  15. 15.
    K.A. Milton, I.L. Solovtsov, Analytic perturbation theory in QCD and Schwinger’s connection between the beta function and the spectral density. Phys. Rev. D 55, 5295 (1997). arXiv:hep-ph/9611438 ADSGoogle Scholar
  16. 16.
    I.L. Solovtsov, D.V. Shirkov, Analytic approach to perturbative QCD and renormalization scheme dependence. Phys. Lett. B 442, 344 (1998). arXiv:hep-ph/9711251 ADSGoogle Scholar
  17. 17.
    D.V. Shirkov, Analytic perturbation theory for QCD observables. Theor. Math. Phys. 127, 409 (2001). arXiv:hep-ph/0012283 MathSciNetzbMATHGoogle Scholar
  18. 18.
    D.V. Shirkov, Analytic perturbation theory in analyzing some QCD observables. Eur. Phys. J. C 22, 331 (2001). arXiv:hep-ph/0107282 ADSzbMATHGoogle Scholar
  19. 19.
    A.P. Bakulev, S.V. Mikhailov, N.G. Stefanis, QCD analytic perturbation theory: from integer powers to any power of the running coupling. Phys. Rev. D 72, 074014 (2005). arXiv:hep-ph/0506311 [Phys. Rev. D 72, 119908(E) (2005)]
  20. 20.
    A.P. Bakulev, A.I. Karanikas, N.G. Stefanis, Analyticity properties of three-point functions in QCD beyond leading order. Phys. Rev. D 72, 074015 (2005). arXiv:hep-ph/0504275 ADSGoogle Scholar
  21. 21.
    A.P. Bakulev, S.V. Mikhailov, N.G. Stefanis, Fractional Analytic Perturbation Theory in Minkowski space and application to Higgs boson decay into a \(b {\bar{b}}\) pair. Phys. Rev. D 75, 056005 (2007).  https://doi.org/10.1103/PhysRevD.75.056005. arXiv:hep-ph/0607040 [Erratum: Phys. Rev. D 77, 079901 (2008)]
  22. 22.
    A.P. Bakulev, S.V. Mikhailov, N.G. Stefanis, Phys. Rev. D 77, 079901(E) (2008). arXiv:hep-ph/0607040
  23. 23.
    A.P. Bakulev, S.V. Mikhailov, N.G. Stefanis, Higher-order QCD perturbation theory in different schemes: from FOPT to CIPT to FAPT. JHEP 1006, 085 (2010). arXiv:1004.4125 [hep-ph]ADSzbMATHGoogle Scholar
  24. 24.
    G.M. Prosperi, M. Raciti, C. Simolo, On the running coupling constant in QCD. Prog. Part. Nucl. Phys. 58, 387 (2007). arXiv:hep-ph/0607209 ADSGoogle Scholar
  25. 25.
    D.V. Shirkov, I.L. Solovtsov, Ten years of the analytic perturbation theory in QCD. Theor. Math. Phys. 150, 132 (2007). arXiv:hep-ph/0611229 zbMATHGoogle Scholar
  26. 26.
    A.P. Bakulev, Global fractional analytic perturbation theory in QCD with selected applications. Phys. Part. Nucl. 40, 715 (2009). arXiv:0805.0829 [hep-ph] (arXiv preprint in Russian)
  27. 27.
    N.G. Stefanis, Taming Landau singularities in QCD perturbation theory: the analytic approach. Phys. Part. Nucl. 44, 494 (2013). arXiv:0902.4805 [hep-ph]Google Scholar
  28. 28.
    R.S. Pasechnik, D.V. Shirkov, O.V. Teryaev, Bjorken sum rule and pQCD frontier on the move. Phys. Rev. D 78, 071902 (2008). arXiv:0808.0066 [hep-ph]ADSGoogle Scholar
  29. 29.
    R.S. Pasechnik, D.V. Shirkov, O.V. Teryaev, O.P. Solovtsova, V.L. Khandramai, Nucleon spin structure and pQCD frontier on the move. Phys. Rev. D 81, 016010 (2010). arXiv:0911.3297 [hep-ph]ADSGoogle Scholar
  30. 30.
    V.L. Khandramai, R.S. Pasechnik, D.V. Shirkov, O.P. Solovtsova, O.V. Teryaev, Four-loop QCD analysis of the Bjorken sum rule vs data. Phys. Lett. B 706, 340 (2012). arXiv:1106.6352 [hep-ph]ADSGoogle Scholar
  31. 31.
    V.L. Khandramai, O.P. Solovtsova, O.V. Teryaev, Polarized Bjorken sum rule analysis: revised. Nonlinear Phenom. Complex Syst. 16, 93 (2013). arXiv:1302.3952 [hep-ph]Google Scholar
  32. 32.
    C. Ayala, C. Contreras, G. Cvetič, Extended analytic QCD model with perturbative QCD behavior at high momenta. Phys. Rev. D 85, 114043 (2012). arXiv:1203.6897 [hep-ph]; in Eqs. (21) and (22) of this reference there is a typo: the lower limit of integration is written as \(s_{\rm L} - \eta \); it is in fact \(-s_{\rm L} - \eta \)
  33. 33.
    C. Ayala, G. Cvetič, anQCD: a Mathematica package for calculations in general analytic QCD models. Comput. Phys. Commun. 190, 182 (2015). arXiv:1408.6868 [hep-ph]ADSGoogle Scholar
  34. 34.
    C. Ayala, G. Cvetič, R. Kögerler, Lattice-motivated holomorphic nearly perturbative QCD. J. Phys. G 44(7), 075001 (2017). arXiv:1608.08240 [hep-ph]ADSGoogle Scholar
  35. 35.
    C. Ayala, G. Cvetič, R. Kögerler, I. Kondrashuk, Nearly perturbative lattice-motivated QCD coupling with zero IR limit. J. Phys. G 45(3), 035001 (2018). arXiv:1703.01321 [hep-ph]ADSGoogle Scholar
  36. 36.
    C. Ayala, G. Cvetič, A.V. Kotikov, B.G. Shaikhatdenov, Bjorken sum rule in QCD frameworks with analytic (holomorphic) coupling. Int. J. Mod. Phys. A 33(18n19), 1850112 (2018). arXiv:1708.06284 [hep-ph]
  37. 37.
    C. Ayala, G. Cvetič, A.V. Kotikov, B.G. Shaikhatdenov, Bjorken sum rule in QCD with analytic coupling. J. Phys. Conf. Ser. 938(1), 012055 (2017). arXiv:1712.06048 [hep-ph]Google Scholar
  38. 38.
    A. Deur, S.J. Brodsky, G.F. de Teramond, The QCD running coupling. Prog. Part. Nucl. Phys. 90, 1 (2016). arXiv:1604.08082 [hep-ph]ADSGoogle Scholar
  39. 39.
    C. Patrignani et al. (Particle Data Group Collaboration), Review of particle physics. Chin. Phys. C 40(10), 100001 (2016)Google Scholar
  40. 40.
    S.G. Gorishnii, S.A. Larin, QCD corrections to the parton model rules for structure functions of deep inelastic scattering. Phys. Lett. B 172, 109 (1986)ADSGoogle Scholar
  41. 41.
    S.A. Larin, J.A.M. Vermaseren, The \(\alpha _s^3\) corrections to the Bjorken sum rule for polarized electroproduction and to the Gross–Llewellyn Smith sum rule. Phys. Lett. B 259, 345 (1991)ADSGoogle Scholar
  42. 42.
    P.A. Baikov, K.G. Chetyrkin, J.H. Kühn, Adler function, Bjorken sum rule, and the Crewther relation to order \(\alpha _s^4\) in a general gauge theory. Phys. Rev. Lett. 104, 132004 (2010). arXiv:1001.3606 [hep-ph]ADSGoogle Scholar
  43. 43.
    P.A. Baikov, K.G. Chetyrkin, J.H. Kühn, Massless propagators, \(R(s)\) and multiloop QCD. Nucl. Part. Phys. Proc. 261–262, 3 (2015).  https://doi.org/10.1016/j.nuclphysbps.2015.03.002. arXiv:1501.06739 [hep-ph]CrossRefGoogle Scholar
  44. 44.
    S.A. Larin, The singlet contribution to the Bjorken sum rule for polarized deep inelastic scattering. Phys. Lett. B 723, 348 (2013). arXiv:1303.4021 [hep-ph]ADSzbMATHGoogle Scholar
  45. 45.
    G. Cvetič, C. Valenzuela, An approach for evaluation of observables in analytic versions of QCD. J. Phys. G 32, L27 (2006). arXiv:hep-ph/0601050 ADSGoogle Scholar
  46. 46.
    G. Cvetič, C. Valenzuela, Various versions of analytic QCD and skeleton-motivated evaluation of observables. Phys. Rev. D 74, 114030 (2006). arXiv:hep-ph/0608256 [Erratum-ibid. D 84, 019902 (2011)]
  47. 47.
    G. Cvetič, A.V. Kotikov, Analogs of noninteger powers in general analytic QCD. J. Phys. G 39, 065005 (2012). arXiv:1106.4275 [hep-ph]ADSGoogle Scholar
  48. 48.
    E.V. Shuryak, A.I. Vainshtein, Theory of power corrections to deep inelastic scattering in quantum chromodynamics. 2. \(Q^4\) effects: polarized target. Nucl. Phys. B 201, 141 (1982)Google Scholar
  49. 49.
    H. Kawamura, T. Uematsu, J. Kodaira, Y. Yasui, Renormalization of twist four operators in QCD Bjorken and Ellis–Jaffe sum rules. Mod. Phys. Lett. A 12, 135 (1997). arXiv:hep-ph/9603338 ADSGoogle Scholar
  50. 50.
    A.L. Kataev, A.V. Kotikov, G. Parente, A.V. Sidorov, Next to next-to-leading order QCD analysis of the revised CCFR data for \( x F_3\) structure function and the higher twist contributions. Phys. Lett. B 417, 374 (1998). arXiv:hep-ph/9706534 ADSGoogle Scholar
  51. 51.
    A.L. Kataev, G. Parente, A.V. Sidorov, Higher twists and \(\alpha _s(M_Z)\) extractions from the NNLO QCD analysis of the CCFR data for the \( x F_3\) structure function. Nucl. Phys. B 573, 405 (2000). arXiv:hep-ph/9905310 ADSGoogle Scholar
  52. 52.
    A.L. Kataev, G. Parente, A.V. Sidorov, Improved fits to the \(x F_3\) CCFR data at the next-to-next-to-leading order and beyond. Phys. Part. Nucl. 34, 20 (2003). arXiv:hep-ph/0106221 [Fiz. Elem. Chast. Atom. Yadra 34, 43 (2003)] Erratum: [Phys. Part. Nucl. 38(6), 827 (2007)]
  53. 53.
    S. Narison, V.I. Zakharov, Duality between QCD perturbative series and power corrections. Phys. Lett. B 679, 355 (2009). arXiv:0906.4312 [hep-ph]ADSGoogle Scholar
  54. 54.
    G. Parente, A.V. Kotikov, V.G. Krivokhizhin, Next to next-to-leading order QCD analysis of DIS structure functions. Phys. Lett. B 333, 190 (1994). arXiv:hep-ph/9405290 ADSGoogle Scholar
  55. 55.
    B.G. Shaikhatdenov, A.V. Kotikov, V.G. Krivokhizhin, G. Parente, QCD coupling constant at NNLO from DIS data. Phys. Rev. D 81, 034008 (2010). arXiv:0912.4672 [hep-ph] [Erratum: Phys. Rev. D 81, 079904 (2010)]
  56. 56.
    A.V. Kotikov, V.G. Krivokhizhin, B.G. Shaikhatdenov, Analytic and ‘frozen’ QCD coupling constants up to NNLO from DIS data. Phys. Atom. Nucl. 75, 507 (2012). arXiv:1008.0545 [hep-ph]ADSGoogle Scholar
  57. 57.
    A.V. Kotikov, V.G. Krivokhizhin, B.G. Shaikhatdenov, Improved nonsinglet QCD analysis of fixed-target DIS data. J. Phys. G 42(9), 095004 (2015). arXiv:1411.1236 [hep-ph]ADSGoogle Scholar
  58. 58.
    S. Simula, M. Osipenko, G. Ricco, M. Taiuti, Leading and higher twists in the proton polarized structure function \(g^p_1\) at large Bjorken \(x\). Phys. Rev. D 65, 034017 (2002). arXiv:hep-ph/0107036 ADSGoogle Scholar
  59. 59.
    P. Mergell, U.G. Meissner, D. Drechsel, Dispersion theoretical analysis of the nucleon electromagnetic form-factors. Nucl. Phys. A 596, 367 (1996). arXiv:hep-ph/9506375 ADSGoogle Scholar
  60. 60.
    R.S. Sufian, G.F. de Téramond, S.J. Brodsky, A. Deur, H.G. Dosch, Analysis of nucleon electromagnetic form factors from light-front holographic QCD: the spacelike region. Phys. Rev. D 95(1), 014011 (2017). arXiv:1609.06688 [hep-ph]ADSGoogle Scholar
  61. 61.
    K.A. Olive et al. (Particle Data Group Collaboration), Review of particle physics. Chin. Phys. C 38, 090001 (2014)Google Scholar
  62. 62.
    A.V. Kotikov, G. Parente, J. Sánchez Guillén, Renormalization scheme invariant analysis of the DIS structure functions \(F_2\) and \(F_L\). Z. Phys. C 58, 465 (1993)Google Scholar
  63. 63.
    V.G. Krivokhizhin, A.V. Kotikov, A systematic study of QCD coupling constant from deep-inelastic measurements. Phys. Atom. Nucl. 68, 1873 (2005) [Yad. Fiz. 68, 1935 (2005)]Google Scholar
  64. 64.
    A.V. Kotikov, A.V. Lipatov, N.P. Zotov, The longitudinal structure function \(F_L\): perturbative QCD and \(k_T\) factorization versus experimental data at fixed \(W\). J. Exp. Theor. Phys. 101, 811 (2005)ADSGoogle Scholar
  65. 65.
    A.V. Kotikov, A.V. Lipatov, N.P. Zotov, The longitudinal structure function \(F_L\): perturbative QCD and \(k_T\) factorization versus experimental data at fixed \(W\). Zh. Eksp. Teor. Fiz. 128, 938 (2005). arXiv:hep-ph/0403135 Google Scholar
  66. 66.
    A.Y. Illarionov, A.V. Kotikov, G. Parente Bermudez, Small \(x\) behavior of parton distributions. A Study of higher twist effects. Phys. Part. Nucl. 39, 307 (2008). arXiv:hep-ph/0402173
  67. 67.
    G. Cvetič, A.Y. Illarionov, B.A. Kniehl, A.V. Kotikov, Small-\(x\) behavior of the structure function \(F_2\) and its slope \(\partial \ln F_2 / \partial \ln (1/x)\) for ‘frozen’ and analytic strong-coupling constants. Phys. Lett. B 679, 350 (2009). arXiv:0906.1925 [hep-ph]ADSGoogle Scholar
  68. 68.
    A.V. Kotikov, B.G. Shaikhatdenov, \(Q^2\)-evolution of parton densities at small \(x\) values. Combined H1 and ZEUS \(F_2\) data. Phys. Part. Nucl. 44, 543 (2013). arXiv:1212.4582 [hep-ph]
  69. 69.
    A.V. Kotikov, B.G. Shaikhatdenov, \(Q^2\) evolution of parton distributions at small values of \(x\): effective scale for combined H1 and ZEUS data on the structure function \(F_2\). Phys. Atom. Nucl. 78(4), 525 (2015). arXiv:1402.4349 [hep-ph]ADSGoogle Scholar
  70. 70.
    A.V. Kotikov, B.G. Shaikhatdenov, \(Q^2\)-evolution of parton densities at small \(x\) values. Charm contribution in the combined H1 and ZEUS \(F_2\) data. Phys. Part. Nucl. 48(5), 829 (2017). arXiv:1606.07888 [hep-ph]
  71. 71.
    F.D. Aaron et al. (H1 and ZEUS Collaborations), Combined measurement and QCD analysis of the inclusive \(e^{\pm }p\) scattering cross sections at HERA. JHEP 1001, 109 (2010). arXiv:0911.0884 [hep-ex]
  72. 72.
    I.L. Bogolubsky, E.-M. Ilgenfritz, M. Müller-Preussker, A. Sternbeck, Lattice gluodynamics computation of Landau gauge Green’s functions in the deep infrared. Phys. Lett. B 676, 69 (2009). arXiv:0901.0736 [hep-lat]ADSGoogle Scholar
  73. 73.
    A.G. Duarte, O. Oliveira, P.J. Silva, Lattice gluon and ghost propagators, and the strong coupling in pure SU(3) Yang–Mills theory: finite lattice spacing and volume effects. Phys. Rev. D 94(1), 014502 (2016). arXiv:1605.00594 [hep-lat]ADSGoogle Scholar
  74. 74.
    E.-M. Ilgenfritz, M. Müller-Preussker, A. Sternbeck, A. Schiller, Gauge-variant propagators and the running coupling from lattice QCD. arXiv:hep-lat/0601027
  75. 75.
    B. Blossier et al., The strong running coupling at \(\tau \) and \(Z_0\) mass scales from lattice QCD. Phys. Rev. Lett. 108, 262002 (2012). arXiv:1201.5770 [hep-ph]ADSGoogle Scholar
  76. 76.
    B. Blossier et al., Ghost-gluon coupling, power corrections and \(\Lambda _{\bar{\rm MS}}\) from lattice QCD with a dynamical charm. Phys. Rev. D 85, 034503 (2012). arXiv:1110.5829 [hep-lat]ADSGoogle Scholar
  77. 77.
    O. Teryaev, Analyticity and higher twists. Nucl. Phys. Proc. Suppl. 245, 195 (2013). arXiv:1309.1985 [hep-ph]ADSGoogle Scholar
  78. 78.
    V.L. Khandramai, O.V. Teryaev, I.R. Gabdrakhmanov, Infrared modified QCD couplings and Bjorken sum rule. J. Phys. Conf. Ser. 678(1), 012018 (2016)Google Scholar
  79. 79.
    I.R. Gabdrakhmanov, O.V. Teryaev, V.L. Khandramai, Infrared models for the Bjorken sum rule in the APT approach. J. Phys. Conf. Ser. 938(1), 012046 (2017)Google Scholar
  80. 80.
    D.V. Shirkov, ‘Massive’ perturbative QCD, regular in the IR limit. Phys. Part. Nucl. Lett. 10, 186 (2013). arXiv:1208.2103 [hep-th]Google Scholar
  81. 81.
    D.V. Shirkov, Remarks on simple modified perturbation theory. Phys. Part. Nucl. Lett. 12(2), 205 (2015)Google Scholar
  82. 82.
    J.M. Cornwall, Dynamical mass generation in continuum QCD. Phys. Rev. D 26, 1453 (1982)ADSGoogle Scholar
  83. 83.
    A.C. Aguilar, D. Binosi, J. Papavassiliou, Renormalization group analysis of the gluon mass equation. Phys. Rev. D 89(8), 085032 (2014). arXiv:1401.3631 [hep-ph]ADSGoogle Scholar
  84. 84.
    S.B. Gerasimov, A sum rule for magnetic moments and the damping of the nucleon magnetic moment in nuclei. Sov. J. Nucl. Phys. 2, 430 (1966) [Yad. Fiz. 2, 598 (1965)]Google Scholar
  85. 85.
    M. Anselmino, B.L. Ioffe, E.Leader, On possible resolutions of the spin crisis in the parton model. Sov. J. Nucl. Phys. 49, 136 (1989); [Yad. Fiz. 49, 214 (1989)]Google Scholar
  86. 86.
    J. Soffer, O. Teryaev, The role of \(g\)-\(2\) in relating the Schwinger and Gerasimov-Drell-Hearn sum rules. Phys. Rev. Lett. 70, 3373 (1993).  https://doi.org/10.1103/PhysRevLett.70.3373
  87. 87.
    J. Soffer, O.V. Teryaev, On the \(G_2\) manifestation for longitudinally polarized particles. Phys. Rev. D 51, 25 (1995).  https://doi.org/10.1103/PhysRevD.51.25. arXiv:hep-ph/9405228
  88. 88.
    J. Soffer, O. Teryaev, QCD radiative and power corrections and generalized GDH sum rules. Phys. Rev. D 70, 116004 (2004).  https://doi.org/10.1103/PhysRevD.70.116004. arXiv:hep-ph/0410228
  89. 89.
    J. Soffer, O. Teryaev, Phys. Rev. D 70, 116004 (2004). arXiv:hep-ph/0410228 ADSGoogle Scholar
  90. 90.
    S.J. Brodsky, G.F. de Téramond, A. Deur, Nonperturbative QCD coupling and its \(\beta \)-function from light-front holography. Phys. Rev. D 81, 096010 (2010). arXiv:1002.3948 [hep-ph]ADSGoogle Scholar
  91. 91.
    A. Deur, J.M. Shen, X.G. Wu, S.J. Brodsky, G.F. de Teramond, Implications of the principle of maximum conformality for the QCD strong coupling. Phys. Lett. B 773, 98 (2017). arXiv:1705.02384 [hep-ph]ADSGoogle Scholar
  92. 92.
    A. Deur, S.J. Brodsky, G.F. de Teramond, Determination of \(\Lambda _{\overline{\rm MS}}\) at five loops from holographic QCD. J. Phys. G 44(10), 105005 (2017). arXiv:1608.04933 [hep-ph]ADSGoogle Scholar
  93. 93.
    V.D. Burkert, Comment on the generalized Gerasimov–Drell–Hearn sum rule in chiral perturbation theory. Phys. Rev. D 63, 097904 (2001). arXiv:nucl-th/0004001
  94. 94.
    V. Bernard, E. Epelbaum, H. Krebs, U.G. Meissner, New insights into the spin structure of the nucleon. Phys. Rev. D 87(5), 054032 (2013). arXiv:1209.2523 [hep-ph]ADSGoogle Scholar
  95. 95.
    V. Lensky, J.M. Alarcn, V. Pascalutsa, Moments of nucleon structure functions at next-to-leading order in baryon chiral perturbation theory. Phys. Rev. C 90(5), 055202 (2014). arXiv:1407.2574 [hep-ph]ADSGoogle Scholar
  96. 96.
    G. Cvetič, C. Villavicencio, Operator product expansion with analytic QCD in tau decay physics. Phys. Rev. D 86, 116001 (2012). arXiv:1209.2953 [hep-ph]ADSGoogle Scholar
  97. 97.
    P.M. Stevenson, Optimized perturbation theory. Phys. Rev. D 23, 2916 (1981)ADSGoogle Scholar
  98. 98.
    G. Cvetič, R. Kögerler, Scale and scheme independent extension of Pade approximants: Bjorken polarized sum rule as an example. Phys. Rev. D 63, 056013 (2001). arXiv:hep-ph/0006098 ADSGoogle Scholar
  99. 99.
    G. Cvetič, Techniques of evaluation of QCD low-energy physical quantities with running coupling with infrared fixed point. Phys. Rev. D 89(3), 036003 (2014). arXiv:1309.1696 [hep-ph]ADSGoogle Scholar
  100. 100.
    A.V. Nesterenko, C. Simolo, QCDMAPT: program package for analytic approach to QCD. Comput. Phys. Commun. 181, 1769 (2010). arXiv:1001.0901 [hep-ph]ADSMathSciNetzbMATHGoogle Scholar
  101. 101.
    A.V. Nesterenko, C. Simolo, \({\rm QCDMAPT}_{\rm F}\): Fortran version of QCDMAPT package. Comput. Phys. Commun. 182, 2303 (2011). arXiv:1107.1045 [hep-ph]ADSGoogle Scholar
  102. 102.
    C. Ayala, G. Cvetič, anQCD: Fortran programs for couplings at complex momenta in various analytic QCD models. Comput. Phys. Commun. 199, 114 (2016). arXiv:1506.07201 [hep-ph]ADSzbMATHGoogle Scholar
  103. 103.
    A.P. Bakulev, V.L. Khandramai, FAPT: a mathematica package for calculations in QCD fractional analytic perturbation theory. Comput. Phys. Commun. 184(1), 183 (2013). arXiv:1204.2679 [hep-ph]ADSMathSciNetzbMATHGoogle Scholar
  104. 104.
    G.A. Baker, P. Graves-Morris, Padé Approximants. Encyclopedia of Mathematics and Its Applications (Cambridge University Press, Cambridge 1996) (Section 5.4, Theorem 5.4.2)Google Scholar
  105. 105.
    S. Peris, Large-\(N_c\) QCD and Padé approximant theory. Phys. Rev. D 74, 054013 (2006). arXiv:hep-ph/0603190 ADSGoogle Scholar
  106. 106.
    K.G. Chetyrkin, B.A. Kniehl, M. Steinhauser, Strong coupling constant with flavour thresholds at four loops in the MSbar scheme. Phys. Rev. Lett. 79, 2184 (1997). arXiv:hep-ph/9706430 ADSGoogle Scholar
  107. 107.
    A.C. Aguilar, J. Papavassiliou, Power-law running of the effective gluon mass. Eur. Phys. J. A 35, 189 (2008). arXiv:0708.4320 [hep-ph]ADSGoogle Scholar
  108. 108.
    A.C. Aguilar, D. Binosi, J. Papavassiliou, J. Rodríguez-Quintero, Non-perturbative comparison of QCD effective charges. Phys. Rev. D 80, 085018 (2009). arXiv:0906.2633 [hep-ph]ADSGoogle Scholar
  109. 109.
    D. Binosi, C. Mezrag, J. Papavassiliou, C.D. Roberts, J. Rodríguez-Quintero, Process-independent strong running coupling. Phys. Rev. D 96(5), 054026 (2017). arXiv:1612.04835 [nucl-th]
  110. 110.
    L. von Smekal, K. Maltman, A. Sternbeck, The strong coupling and its running to four loops in a minimal MOM scheme. Phys. Lett. B 681, 336 (2009). arXiv:0903.1696 [hep-ph]ADSGoogle Scholar
  111. 111.
    P. Boucaud, F. De Soto, J.P. Leroy, A. Le Yaouanc, J. Micheli, O. Pene, J. Rodríguez-Quintero, Ghost-gluon running coupling, power corrections and the determination of Lambda(MS-bar). Phys. Rev. D 79, 014508 (2009). arXiv:0811.2059 [hep-ph]ADSGoogle Scholar
  112. 112.
    K.G. Chetyrkin, A. Retey, Three loop three linear vertices and four loop similar to MOM beta functions in massless QCD. arXiv:hep-ph/0007088
  113. 113.
    A.L. Kataev, V.S. Molokoedov, Fourth-order QCD renormalization group quantities in the V scheme and the relation of the \(\beta \) function to the Gell-Mann–Low function in QED. Phys. Rev. D 92, 054008 (2015). arXiv:1507.03547 [hep-ph]ADSGoogle Scholar
  114. 114.
    A.V. Garkusha, A.L. Kataev, V.S. Molokoedov, Renormalization scheme and gauge (in)dependence of the generalized Crewther relation: what are the real grounds of the \(\beta \)-factorization property? JHEP 1802, 161 (2018). arXiv:1801.06231 [hep-ph]ADSMathSciNetzbMATHGoogle Scholar
  115. 115.
    B.A. Arbuzov, I.V. Zaitsev, Elimination of the Landau pole in QCD with the spontaneously generated anomalous three-gluon interaction. arXiv:1303.0622 [hep-th]
  116. 116.
    P. Boucaud, F. De Soto, A. Le Yaouanc, J.P. Leroy, J. Micheli, H. Moutarde, O. Pene, J. Rodríguez-Quintero, The strong coupling constant at small momentum as an instanton detector. JHEP 0304, 005 (2003). arXiv:hep-ph/0212192 ADSGoogle Scholar
  117. 117.
    P. Boucaud, F. De Soto, A. Le Yaouanc, J.P. Leroy, J. Micheli, O. Pene, J. Rodríguez-Quintero, Modified instanton profile effects from lattice Green functions. Phys. Rev. D 70, 114503 (2004). arXiv:hep-ph/0312332 ADSGoogle Scholar
  118. 118.
    M. Baldicchi, A.V. Nesterenko, G.M. Prosperi, D.V. Shirkov, C. Simolo, Bound state approach to the QCD coupling at low energy scales. Phys. Rev. Lett. 99, 242001 (2007). arXiv:0705.0329 [hep-ph]ADSGoogle Scholar
  119. 119.
    M. Baldicchi, A.V. Nesterenko, G.M. Prosperi, C. Simolo, QCD coupling below 1 GeV from quarkonium spectrum. Phys. Rev. D 77, 034013 (2008). arXiv:0705.1695 [hep-ph]ADSGoogle Scholar
  120. 120.
    G. Cvetič, C. Valenzuela, Analytic QCD: a short review. Braz. J. Phys. 38, 371 (2008). arXiv:0804.0872 [hep-ph]ADSGoogle Scholar
  121. 121.
    A.V. Nesterenko, Quark antiquark potential in the analytic approach to QCD. Phys. Rev. D 62, 094028 (2000). arXiv:hep-ph/9912351 ADSGoogle Scholar
  122. 122.
    A.V. Nesterenko, New analytic running coupling in spacelike and timelike regions. Phys. Rev. D 64, 116009 (2001). arXiv:hep-ph/0102124 ADSGoogle Scholar
  123. 123.
    A.V. Nesterenko, Analytic invariant charge in QCD. Int. J. Mod. Phys. A 18, 5475 (2003). arXiv:hep-ph/0308288 ADSzbMATHGoogle Scholar
  124. 124.
    A.C. Aguilar, A.V. Nesterenko, J. Papavassiliou, Infrared enhanced analytic coupling and chiral symmetry breaking in QCD. J. Phys. G 31, 997 (2005). arXiv:hep-ph/0504195 ADSGoogle Scholar
  125. 125.
    K.A. Milton, I.L. Solovtsov, O.P. Solovtsova, Analytic perturbation theory and inclusive tau decay. Phys. Lett. B 415, 104 (1997). arXiv:hep-ph/9706409 ADSGoogle Scholar
  126. 126.
    K.A. Milton, I.L. Solovtsov, O.P. Solovtsova, The Adler function for light quarks in analytic perturbation theory. Phys. Rev. D 64, 016005 (2001). arXiv:hep-ph/0102254 ADSGoogle Scholar
  127. 127.
    B.A. Magradze, The gluon propagator in analytic perturbation theory. Conf. Proc. C 980518, 158 (1999). arXiv:hep-ph/9808247 Google Scholar
  128. 128.
    S. Peris, M. Perrottet, E. de Rafael, Matching long and short distances in large-\(N_c\) QCD. JHEP 9805, 011 (1998). arXiv:hep-ph/9805442 ADSGoogle Scholar
  129. 129.
    B.A. Magradze, Testing the concept of quark-hadron duality with the ALEPH \(\tau \) decay data. Few Body Syst. 48, 143 (2010). arXiv:1005.2674 [hep-ph] [Erratum: Few Body Syst. 53, 365 (2012)]
  130. 130.
    B.A. Magradze, Strong coupling constant from \(\tau \) decay within a dispersive approach to perturbative QCD. Proc. A. Razmadze Math. Inst. 160, 91–111 (2012). arXiv:1112.5958 [hep-ph]MathSciNetzbMATHGoogle Scholar
  131. 131.
    A.V. Nesterenko, J. Papavassiliou, A novel integral representation for the Adler function. J. Phys. G 32, 1025 (2006). arXiv:hep-ph/0511215 ADSGoogle Scholar
  132. 132.
    A.V. Nesterenko, Dispersive approach to QCD and inclusive tau lepton hadronic decay. Phys. Rev. D 88, 056009 (2013). arXiv:1306.4970 [hep-ph]ADSGoogle Scholar
  133. 133.
    A.V. Nesterenko, Hadronic vacuum polarization function within dispersive approach to QCD. J. Phys. G 42, 085004 (2015). arXiv:1411.2554 [hep-ph]ADSGoogle Scholar
  134. 134.
    A.V. Nesterenko, Electron-positron annihilation into hadrons at the higher-loop levels. Eur. Phys. J. C 77, 844 (2017). arXiv:1707.00668 [hep-ph]ADSGoogle Scholar
  135. 135.
    A.V. Nesterenko, Strong Interactions in Spacelike and Timelike Domains: Dispersive Approach (Elsevier, Amsterdam, 2016). eBook ISBN:9780128034484Google Scholar
  136. 136.
    J. Blümlein, G. Falcioni, A. De Freitas, The complete \(O(\alpha _s^2)\) non-singlet heavy flavor corrections to the structure functions \(g_{1,2}^{e p}(x, Q^2)\), \(F_{1,2, L}^{e p}(x, Q^2)\), \(F_{1,2,3}^{\nu (\bar{\nu })}(x, Q^2)\) and the associated sum rules. Nucl. Phys. B 910, 568 (2016). arXiv:1605.05541 [hep-ph]ADSzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of PhysicsUniversidad Técnica Federico Santa María (UTFSM)ValparaísoChile
  2. 2.Joint Institute for Nuclear ResearchDubnaRussia

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