# Bjorken polarized sum rule and infrared-safe QCD couplings

## 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

*x*-Bjorken interval

^{1}

### 2.1 Perturbation expansion of the leading-twist

^{2}

*k*and \(c_j\) (\(j \ge 2\)), and are given in Appendix A.

*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 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.

*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]: \(\gamma _p=0.27\); \(\gamma _n=0.38\); \(r=2.08\).

^{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

^{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.

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.

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 | \(Q^2_\mathrm{min}(\mathrm{fit})\) | | \(f_2^{p-n}(1.)\) | \(\chi ^2/\mathrm{d.o.f}\) | \(\chi ^2_\mathrm{ext}/\mathrm{d.o.f}\) | \(\chi ^2_{0.268}/\mathrm{d.o.f.}\) | \(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\) |
---|---|---|---|---|---|---|---|

\(\overline{\mathrm{MS}}\) 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 \(\mu _6=0\)

*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)

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 \(\mu _6 \ne 0\)

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 | \(Q^2_\mathrm{min}(\mathrm{fit})\) | | \(f_2^{p-n}(1.)\) | \(\mu _6\) | \(\chi ^2/\mathrm{d.o.f}\) | \(\chi ^2_\mathrm{ext}/\mathrm{d.o.f}\) | \(\chi ^2_{0.268}/\mathrm{d.o.f.}\) | \(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\) |
---|---|---|---|---|---|---|---|---|

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

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 \(F_2\)).

*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.

QCD variant | \(Q^2_\mathrm{min}(\mathrm{fit})\) | | \(f_2^{p-n}(1.)\) | \(M^2\) | \(\chi ^2/\mathrm{d.o.f}\) | \(\chi ^2_\mathrm{ext}/\mathrm{d.o.f}\) | \(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\) |
---|---|---|---|---|---|---|---|

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

^{8}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}

QCD variant | \(Q^2_\mathrm{min}(\mathrm{fit})\) | | \(f_2^{p-n}(1.)\) | \(M^2(1 \ \mathrm{GeV}^2)\) | \(\chi ^2/\mathrm{d.o.f}\) | \(\chi ^2_\mathrm{ext}/\mathrm{d.o.f}\) | \(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\) |
---|---|---|---|---|---|---|---|

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

*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

*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 \(B= -1.033 \ \mathrm{GeV}^{-4}\), \(\chi ^2(Q^2 \le Q^2_\mathrm{max})/\mathrm{d.o.f.} = 1.313\).

^{10}

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].

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 | \(Q^2_\mathrm{min}(\mathrm{fit})\) | | \(f_2^{p-n}(1.)\) | \(\mu _6\) | \(\chi ^2/\mathrm{d.o.f}\) | \(\chi ^2_\mathrm{ext}/\mathrm{d.o.f}\) | \(\chi ^2_{0.268}/\mathrm{d.o.f.}\) | \(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\) |
---|---|---|---|---|---|---|---|---|

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

QCD variant | \(Q^2_\mathrm{min}(\mathrm{fit})\) | | \(f_2^{p-n}(1.)\) | \(M^2\) | \(\chi ^2/\mathrm{d.o.f}\) | \(\chi ^2_\mathrm{ext}/\mathrm{d.o.f}\) | \(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\) |
---|---|---|---|---|---|---|---|

\(\overline{\mathrm{MS}}\) 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 | \(1.05 \times 10^{4}\) |

QCD variant | \(Q^2_\mathrm{min}(\mathrm{fit})\) | | \(f_2^{p-n}(1.)\) | \(M^2(1 \ \mathrm{GeV}^2)\) | \(\chi ^2/\mathrm{d.o.f}\) | \(\chi ^2_\mathrm{ext}/\mathrm{d.o.f}\) | \(\chi ^2_\mathrm{all}/\mathrm{d.o.f.}\) |
---|---|---|---|---|---|---|---|

\(\overline{\mathrm{MS}}\) 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.
The index

*j*in \(Q_j^2\) indicates from here on that these are the scales at which the experimental values are given. - 2.
- 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.
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.
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.
If we perform the fit with the four points included, the results differ somewhat, but not significantly.

- 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.
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.
- 10.
This is not far from the universal nonperturbative scale \(\kappa = M_{\rho }/\sqrt{2} = 0.548\) GeV, cf. [91].

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

- 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.
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.
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.
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.
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.

## 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.

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