Fractional Analytic QCD: The Recent Results

Abstract

In this work, we present an overview of the recent results, obtained in the framework of the fractional analytic QCD in the space-like (Euclidean) and time-like regions. The Higgs boson decays into a bottom–antibottom pair, and the polarized Bjorken sum rule is considered as an application of the obtained results.

1. Introduction

According to the general principles of (local) quantum field theory (QFT) [1], observables in a space-like region (i.e., in Euclidean space) can only have singularities for negative values of their argument, $Q^2$. However, for large $Q^2$ values, these observables are usually represented as power expansions in the running coupling ${\alpha }_s\left(Q^2\right)$, which has a ghostly singularity, the so-called Landau pole, at $Q^2={\Lambda }^2$. Therefore, to restore the analyticity of the considered expansions, this pole in the strong coupling should be removed.

The strong coupling, ${\alpha }_s\left(Q^2\right)$, obeys the renormalization group equation

$$L\equiv ln{\displaystyle \frac{Q^2}{{\Lambda }^2}}={\int }^{{\overline{a}}_s\left(Q^2\right)}{\displaystyle \frac{da}{\beta \left(a\right)}},{\overline{a}}_s\left(Q^2\right)={\displaystyle \frac{{\alpha }_s\left(Q^2\right)}{4\pi }}$$

(1)

with some boundary condition and the QCD $\beta$-function, as follows:

$$\beta \left(a_s\right)=-\sum _{i=0}{\beta }_i{\overline{a}}_s^{i+2}=-{\beta }_0{\overline{a}}_s^2\left(1+\sum _{i=1}{b}_i{a}_s^i\right),b_i={\displaystyle \frac{{\beta }_i}{{\beta }_0^{i+1}}},a_s\left(Q^2\right)={\beta }_0{\overline{a}}_s\left(Q^2\right),$$

(2)

where

$${\beta }_0=11-{\displaystyle \frac{2f}{3}},{\beta }_1=102-{\displaystyle \frac{38f}{3}},{\beta }_2={\displaystyle \frac{2857}{2}}-{\displaystyle \frac{5033f}{18}}+{\displaystyle \frac{325f^2}{54}},$$

(3)

for f active quark flavors. The first five coefficients, i.e., ${\beta }_i$ with $i\le 4$, are exactly known [2,3,4,5]. In our present consideration, we will only need $0\le i\le 2$.

Note that in Equation (2), we add the first coefficient of the QCD $\beta$-function to the $a_s$ definition, as is usually done in the analytic version of QCD (see, e.g., Refs. [6,7,8,9,10,11,12]).

So, at the leading order (LO), the next-to-leading order (NLO), and the next-to-next-to-leading order (NNLO), where $a_s\left(Q^2\right)\equiv a_s^{\left(1\right)}\left(Q^2\right)$, $a_s\left(Q^2\right)\equiv a_s^{\left(2\right)}\left(Q^2\right)=a_s^{\left(1\right)}\left(Q^2\right)+{\delta }_s^{\left(2\right)}\left(Q^2\right)$ and $a_s\left(Q^2\right)\equiv a_s^{\left(3\right)}\left(Q^2\right)=a_s^{\left(1\right)}\left(Q^2\right)+{\delta }_s^{\left(2\right)}\left(Q^2\right)+{\delta }_s^{\left(3\right)}\left(Q^2\right)$, respectively, we have the following from Equation (1):

$$a_s^{\left(1\right)}\left(Q^2\right)={\displaystyle \frac{1}{L}},{\delta }_s^{\left(2\right)}\left(Q^2\right)=-{\displaystyle \frac{b_{1}lnL}{L^2}},{\delta }_s^{\left(3\right)}\left(Q^2\right)={\displaystyle \frac{1}{L^3}}\left[b_1^2({ln}^{2}L-lnL-1)+b_2\right],$$

(4)

i.e., $a_s^{\left(i\right)}\left(Q^2\right)(i=1,2,3)$ contain poles and other singularities at $Q^2={\Lambda }^2$.

In a time-like region ($q^2>0$) (i.e., in Minkowski space), the definition of a running coupling is quite difficult. The reason for this problem is that—strictly speaking—the expansion of the perturbation theory (PT) in QCD cannot be defined directly in this region. Since the early days of QCD, much effort has been made to determine the appropriate Minkowski coupling parameter needed to describe important time-like processes, such as $e^{+}{e}^{-}$-annihilation into hadrons, the formation of quarkonia, and $\tau$-lepton decays into hadrons. Most attempts (see, for example, [13,14,15]) have relied on the analytical continuation of the strong coupling from the deep Euclidean region—where perturbative QCD calculations can be performed—to Minkowski space, where physical measurements are made. In other developments, analytical expressions for the LO coupling were obtained [16,17] directly in Minkowski space, using an integral transformation from the space-like to time-like modes from the Adler D-function.

Note that, at times, the effective argument of strong coupling reaches a region where perturbation theory becomes of little use. To extend the applicability of perturbation theory, some infrared modifications of the strong coupling are usually used. The most popular modifications are the “freezing” procedure (see, for example, Ref. [18]) and the Shirkov–Solovtsov approach [6,7,8].

The “freezing” of the strong coupling can be done in a hard or a soft way. In the hard case (see [19,20], for example), the strong coupling itself is modified: it is taken to be constant for all values of $Q^2$ below a certain threshold, $Q_0^2$, i.e., ${\alpha }_s\left(Q^2\right)={\alpha }_s\left(Q_0^2\right)$ if $Q^2\le Q_0^2$.

In the soft case [18], the argument of the strong coupling is modified. It contains a shift $Q^2\to Q^2+{\overline{M}}^2$, where $\overline{M}$ is an additional scale (the gluon effective mass) that strongly changes the infrared properties of ${\alpha }_s\left(Q^2\right)$. For massless-produced quarks, the value of M is usually taken to be the mass of the $\rho$ meson $m_{\rho }$, i.e., $\overline{M}=m_{\rho }$. In the case of massive quarks with mass $m_i$, the value ${\overline{M}}_i=2m_i$ is usually used. In some complicated cases, effective masses with more complicated shapes are used (see, for example, Ref. [21], with examples of masses obtained by solving the Schwinger–Dyson equation, and Refs. [22,23,24,25], where the coupling argument depends on the process under consideration). Moreover, at times, the elimination of the Landau pole leads to additional power-law corrections (see [26,27]).

Hereafter, we will study the analytic coupling. In Refs. [6,7,8,9], an efficient approach was developed to eliminate the Landau singularity without introducing extraneous infrared controllers, such as the gluon effective mass (see, e.g., [18,28,29,30]). (Numerically, couplings with effective mass are very close to the analytic ones (see [31])). This method is based on a dispersion relation that relates the new analytic coupling, $A_{\mathrm{MA}}\left(Q^2\right)$, to the spectral function, $r_{\mathrm{pt}}\left(s\right)$, obtained in the PT framework. In LO, this gives the following:

$$A_{\mathrm{MA}}^{\left(1\right)}\left(Q^2\right)={\displaystyle \frac{1}{\pi }}{\int }_0^{+\infty }{\displaystyle \frac{ds}{(s+t)}}{r}_{\mathrm{pt}}^{\left(1\right)}\left(s\right),r_{\mathrm{pt}}^{\left(1\right)}\left(s\right)=\mathrm{Im}{a}_s^{\left(1\right)}(-s-iϵ).$$

(5)

The [6,7,8,9] approach follows the corresponding results [32,33] obtained in the framework of quantum electrodynamics. Similarly, the analytical images of a running coupling in Minkowski space are defined using another linear operation:

$$U_{\mathrm{MA}}^{\left(1\right)}\left(s\right)={\displaystyle \frac{1}{\pi }}{\int }_s^{+\infty }{\displaystyle \frac{d\sigma }{\sigma }}{r}_{\mathrm{pt}}^{\left(1\right)}\left(\sigma \right),$$

(6)

So, we repeat the following once again: the spectral function in the dispersion relations (5) and (6) is taken directly from PT, and the analytical couplings $A_{\mathrm{MA}}\left(Q^2\right)$ and $U_{\mathrm{MA}}\left(Q^2\right)$ are restored using the corresponding dispersion relations. This approach is usually called the minimal approach (MA) (see, e.g., [34]) or the analytical perturbation theory (APT) [6,7,8,9]. (An overview of other similar approaches can be found in [35,36], including approaches that are close to APT [37,38]).

Thus, MA QCD is a very convenient approach that combines the analytical properties of QFT quantities and the results obtained in the framework of perturbative QCD, leading to the appearance of the MA couplings, $A_{\mathrm{MA}}\left(Q^2\right)$ and $U_{\mathrm{MA}}\left(s\right)$, which are close to the usual strong coupling, $a_s\left(Q^2\right)$, in the limit of large $Q^2$ values, and completely different from $a_s\left(Q^2\right)$ for small $Q^2$ values, i.e., for $Q^2\sim {\Lambda }^2$.

A further APT development is the so-called fractional APT (FAPT) [10,11,12], which extends the construction principles described above to the PT series, starting from non-integer powers of the coupling. In the framework of QFT, such series arise for quantities that have nonzero anomalous dimensions. Compact expressions for quantities within the FAPT framework were obtained mainly in LO, but this approach was also used in higher orders, mainly by re-expanding the corresponding couplings in powers of the LO coupling, as well as using some approximations.

In this review, we show the main properties of MA couplings in the FAPT framework, as obtained in Refs. [39,40] using the so-called $1/L$-expansion. Note that for an ordinary coupling, this expansion is applicable only to large $Q^2$ values, i.e., for $Q^2>>{\Lambda }^2$. However, as shown in [39,40], the situation is quite different with analytic couplings, and this $1/L$-expansion is applicable to all values of the argument. This is due to the fact that the non-leading expansion corrections vanish not only at $Q^2\to \infty$, but also at $Q^2\to 0$, (The absence of high-order corrections for $Q^2\to 0$ was also discussed in Refs. [6,7,8,9]). which leads only to nonzero (small) corrections in the region $Q^2\sim {\Lambda }^2$.

Below, we consider the representations for the MA couplings and their (fractional) derivatives obtained in [39,40] (see also [41,42,43]), which are, in principle, valid at any PT order. However, in order to avoid cumbersome formulas while still showing the main features of the approach obtained in [39,40], we restrict our consideration to only the first three PT orders.

Moreover, in this review, we present FAPT applications to the Higgs boson decay into a bottom–antibottom pair and the description of the polarized Bjorken sum rule (BSR). The results shown here were recently obtained in Ref. [40] and Refs. [44,45,46], respectively. In contrast to the formulas, the results for the Higgs boson decay and the polarized BSR will be shown in the first five PT orders, as obtained in [40,44,45,46].

This paper is organized as follows. In Section 2, we first review the basic properties of the usual strong coupling and its $1/L$-expansion. Section 3 contains fractional derivatives (i.e., $\nu$-derivatives) of the usual strong coupling, in which $1/L$-expansions can be represented as some operators acting on the $\nu$-derivatives of the LO strong coupling. In Section 4 and Section 5, we present the results for the MA couplings. Section 6 contains formulas that are convenient for $Q^2\sim {\Lambda }^2$. In Section 7 and Section 8, we present the integral representations for the MA couplings. Section 9 and Section 10 present applications of this approach to the Higgs boson decay into a bottom–antibottom pair and the Bjorken sum rule, respectively. The conclusion provides final discussions. In addition, we include several appendices, which contain the most complicated expressions.

2. Strong Coupling

As shown in the introduction, the strong coupling $a_s\left(Q^2\right)$ obeys the renormalized group Equation (1). When $Q^2>>{\Lambda }^2$, Equation (1) can be solved iteratively in the form of a $1/L$-expansion. (The $1/L$-expansion provides a good approximation for the solution of Equation (2) at $Q^2\ge$ $10 {\mathrm{GeV}}^2$ (see, for example, [47,48])). In accordance with the reasoning in the introduction, we present the first three terms of the expansion, which can be expressed in the following compact form:

$$a_{s,0}^{\left(1\right)}\left(Q^2\right)={\displaystyle \frac{1}{L_0}},a_{s,i}^{(i+1)}\left(Q^2\right)=a_{s,i}^{\left(1\right)}\left(Q^2\right)+\sum _{m=2}^i{\delta }_{s,i}^{\left(m\right)}\left(Q^2\right),(i=0,1,2,\dots ),$$

(7)

where

$$L_k=ln{t}_k,t_k={\displaystyle \frac{1}{z_k}}={\displaystyle \frac{Q^2}{{\Lambda }_k^2}}.$$

(8)

The corrections ${\delta }_{s,k}^{\left(m\right)}\left(Q^2\right)$ are represented as follows:

$${\delta }_{s,k}^{\left(2\right)}\left(Q^2\right)=-{\displaystyle \frac{b_{1}ln{L}_k}{L_k^2}},{\delta }_{s,k}^{\left(3\right)}\left(Q^2\right)={\displaystyle \frac{1}{L_k^3}}\left[b_1^2({ln}^2{L}_k-ln{L}_k-1)+b_2\right].$$

(9)

As shown in Equations (7) and (9), in any PT order, the coupling $a_s\left(Q^2\right)$ contains its dimensional transmutation parameter $\Lambda$, which is related to the normalization of ${\alpha }_s\left(M_Z^2\right)$ as follows:

$${\Lambda }_i=M_{Z}exp\left\{-{\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{1}{a_s\left(M_Z^2\right)}}+b_{1}ln{a}_s\left(M_Z^2\right)+{\int }_0^{{\overline{a}}_s\left(M_Z^2\right)}da\left({\displaystyle \frac{1}{\beta \left(a\right)}}+{\displaystyle \frac{1}{a^2({\beta }_0+{\beta }_{1}a)}}\right)\right]\right\},$$

(10)

where ${\alpha }_s\left(M_Z\right)=0.1176\pm 0.0010$ in PDG20 [49]. (Notice that the results from PDG20 [49] were used in the considered papers. Now, the new PDG24 [50] presents world average values for the strong coupling: ${\alpha }_s\left(M_Z\right)=0.1178\pm 0.0010$ and $m_c=m_c(Q^2=m_c^2)=1.230\pm 0.004$ GeV and $m_b=m_b(Q^2=m_b^2)=4.18+0.04-0.03$ GeV).

f-Dependence of the Coupling $a_s\left(Q^2\right)$

The coefficients ${\beta }_i$ (3) depend on the number f of active quarks that modify the coupling $a_s\left(Q^2\right)$ at thresholds $Q_f^2\sim m_f^2$, where an additional quark enters the game, $Q^2>Q_f^2$. Here, $m_f$ denotes the $\overline{MS}$ mass of the f quark, e.g., $m_b=4.18+0.003-0.002$ GeV and $m_c=1.27\pm 0.02$ GeV from PDG20 [49]. (Strictly speaking, the quark masses in the $\overline{MS}$ scheme depend on $Q^2$ and $m_f=m_f(Q^2=m_f^2)$. The $Q^2$-dependence is rather slow and will only be discussed for the decay $H\to b\overline{b}$ in Section 9). Thus, the coupling $a_s$ depends on f, and this f-dependence can be taken into account in $\Lambda$, i.e., it is ${\Lambda }^f$ that contributes to Equations (1) and (7).

The relationships between ${\Lambda }_i^f$ and ${\Lambda }_i^{f-1}$, i.e., the so-called matching conditions between $a_s(f,Q_f^2)$ and $a_s(f-1,Q_f^2)$, are known up to the four-loop order [51,52,53] in the $\overline{MS}$ scheme and are usually used for $Q_f^2=m_f^2$, where these relations have the simplest forms (see, e.g., [54] for a recent review).

Here, we will not consider the f-dependence of ${\Lambda }_i^f$ and $a_s(f,M_Z^2)$, since we mainly consider the range of small $Q^2$ values and, therefore, use ${\Lambda }_i^{f=3}$$(i=0,1,2,3)$ from Ref. [55]. Furthermore, since we consider the $H\to b\overline{b}$ decay as an application, we will use also the results for ${\Lambda }_i^{f=5}$, which are also taken from [55]. (The authors of [55] used the PDG20 result, ${\alpha }_s\left(M_Z\right)=0.1179\left(10\right)$. Now, there is also the PDG21 result, ${\alpha }_s\left(M_Z\right)=0.1179\left(9\right)$, which contains the same center value. Note that very close numerical relationships between ${\Lambda }_i$ were also obtained by [56] for ${\alpha }_s\left(M_Z\right)=0.1168\left(19\right)$, extracted by the ZEUS collaboration (see [57])):

$$\begin{array}{ccc}& & {\Lambda }_0^{f=3}=142\mathrm{MeV},{\Lambda }_1^{f=3}=367\mathrm{MeV},{\Lambda }_2^{f=3}=324\mathrm{MeV},{\Lambda }_3^{f=3}=328\mathrm{MeV},\hfill \\ & & {\Lambda }_0^{f=5}=87\mathrm{MeV},{\Lambda }_1^{f=5}=224\mathrm{MeV},{\Lambda }_2^{f=5}=207\mathrm{MeV},{\Lambda }_3^{f=5}=207\mathrm{MeV}.\hfill \end{array}$$

(11)

We also use ${\Lambda }_4={\Lambda }_3$, since in the highest orders, ${\Lambda }_i$ values are very similar.

In Figure 1, one can see that the strong couplings $a_{s,i}^{(i+1)}\left(Q^2\right)$ become singular at $Q^2={\Lambda }_i^2$. The values of ${\Lambda }_0$ and ${\Lambda }_j$$(j\ge 1)$ are very different (see Equation (11) below): The values of ${\left({\Lambda }_i^{f=3}\right)}^2$$(i=0,2,4)$ are also shown in Figure 1’s vertical lines.

Figure 1. The results for $a_{s,i}^{(i+1)}\left(Q^2\right)$ and ${\left({\Lambda }_i^{f=3}\right)}^2$ (vertical lines) with $i=0,2,4$. Here, and in the following figures, the ${\Lambda }_i^{f=3}$ values shown in (11) are used.

3. Fractional Derivatives

Following [58,59], we introduce the derivatives (in the $\left(i\right)$-order of PT):

$${\tilde{a}}_{n+1}^{\left(i\right)}\left(Q^2\right)={\displaystyle \frac{{(-1)}^n}{n!}}{\displaystyle \frac{d^n{a}_s^{\left(i\right)}\left(Q^2\right)}{{\left(dL\right)}^n}},$$

(12)

which are very convenient in the case of the analytical QCD (see, e.g., [60]).

The series of derivatives ${\tilde{a}}_n\left(Q^2\right)$ can successfully replace the corresponding series of $a_s$-degrees. Indeed, each derivative reduces the $a_s$ degree but is accompanied by an additional $\beta$-function $\sim a_s^2$. Thus, each application of a derivative yields an additional $a_s$, and, thus, it is indeed possible to use a series of derivatives instead of a series of $a_s$-powers.

In LO, the series of derivatives ${\tilde{a}}_n\left(Q^2\right)$ are exactly the same as $a_s^n$. Beyond LO, the relationship between ${\tilde{a}}_n\left(Q^2\right)$ and $a_s^n$ was established in [59,61], and extended to fractional cases, where $n\to$ is a non-integer $\nu$, in Ref. [62].

Now, consider the $1/L$-expansion of ${\tilde{a}}_{\nu }^{\left(k\right)}\left(Q^2\right)$. We can raise the $\nu$-power of the results from (7) and (9) and then restore ${\tilde{a}}_{\nu }^{\left(k\right)}\left(Q^2\right)$ using the relations between ${\tilde{a}}_{\nu }$ and $a_s^{\nu }$ obtained in [62] (see also Appendix A). This operation is carried out in more detail in Appendix B to [39]. Here, we present only the final results, which have the following form (The expansion (13) is similar to expansions used in Refs. [10,11] for the expansion of ${\left(a_{s,i}^{(i+1)}\left(Q^2\right)\right)}^{\nu }$ in terms of the powers of $a_{s,i}^{\left(1\right)}\left(Q^2\right)$).:

$$\begin{array}{ccc}& & {\tilde{a}}_{\nu ,0}^{\left(1\right)}\left(Q^2\right)={\left(a_{s,0}^{\left(1\right)}\left(Q^2\right)\right)}^{\nu }={\displaystyle \frac{1}{L_0^{\nu }}},{\tilde{a}}_{\nu ,i}^{(i+1)}\left(Q^2\right)={\tilde{a}}_{\nu ,i}^{\left(1\right)}\left(Q^2\right)+{\displaystyle \sum _{m=1}^i}{C}_m^{\nu +m}{\tilde{\delta }}_{\nu ,i}^{(m+1)}\left(Q^2\right),\hfill \\ & & {\tilde{\delta }}_{\nu ,i}^{(m+1)}\left(Q^2\right)={\widehat{R}}_m{\displaystyle \frac{1}{L_i^{\nu +m}}},C_m^{\nu +m}={\displaystyle \frac{\Gamma (\nu +m)}{m!\Gamma \left(\nu \right)}},\hfill \end{array}$$

(13)

where

$${\widehat{R}}_1=b_1\left[{\widehat{Z}}_1\left(\nu \right)+{\displaystyle \frac{d}{d\nu }}\right],{\widehat{R}}_2=b_2+b_1^2\left[{\displaystyle \frac{d^2}{{\left(d\nu \right)}^2}}+2{\widehat{Z}}_1(\nu +1){\displaystyle \frac{d}{d\nu }}+{\widehat{Z}}_2(\nu +1)\right]$$

(14)

and ${\widehat{Z}}_j\left(\nu \right)(j=1,2)$ are the combinations of the Euler $\Psi$-functions and their derivatives.

The representation (13) of the ${\tilde{\delta }}_{\nu ,i}^{(m+1)}\left(Q^2\right)$ corrections in terms of ${\widehat{R}}_m$-operators is very important. The results for ${\widehat{R}}_m$-operators contain the transcendental principle [63,64,65,66]: the corresponding functions ${\widehat{Z}}_k\left(\nu \right)$ ($k\le m$) involve Polygamma functions ${\Psi }_k\left(\nu \right)$ and their products, such as ${\Psi }_{k-l}\left(\nu \right){\Psi }_l\left(\nu \right)$, as well as a larger number of factors, with the same total index k. However, the importance of this property is not yet clear. This allows us to similarly present high-order results for the ($1/L$-expansion) of analytic couplings.

4. MA Couplings

We first show the LO results, and then go beyond LO, following our results (13) for the ordinary strong coupling obtained in the previous section.

4.1. LO

The LO MA coupling $A_{\mathrm{MA},\nu ,0}^{\left(1\right)}$ has the following form [10]:

$$A_{\mathrm{MA},\nu ,0}^{\left(1\right)}\left(Q^2\right)={\left(a_{\nu ,0}^{\left(1\right)}\left(Q^2\right)\right)}^{\nu }-{\displaystyle \frac{{\mathrm{Li}}_{1-\nu }\left(z_0\right)}{\Gamma \left(\nu \right)}}={\displaystyle \frac{1}{L_0^{\nu }}}-{\displaystyle \frac{{\mathrm{Li}}_{1-\nu }\left(z_0\right)}{\Gamma \left(\nu \right)}}\equiv {\displaystyle \frac{1}{L_0^{\nu }}}-{\Delta }_{\nu ,0}^{\left(1\right)},$$

(15)

where

$${\mathrm{Li}}_{\nu }\left(z\right)=\sum _{m=1}^{\infty }{\displaystyle \frac{z^m}{m^{\nu }}}={\displaystyle \frac{z}{\Gamma \left(\nu \right)}}{\int }_0^{\infty }{\displaystyle \frac{dt{t}^{\nu -1}}{(e^t-z)}}$$

(16)

is the polylogarithm.

The LO MA coupling $U_{\mathrm{MA},\nu ,0}^{\left(1\right)}$ in Minkowski space has the following form [11]:

$$U_{\mathrm{MA},\nu ,0}^{\left(1\right)}\left(\mathrm{s}\right)={\displaystyle \frac{sin\left[(\nu -1)g_0\left(s\right)\right]}{\pi (\nu -1){({\pi }^2+L_{s,0}^2)}^{(\nu -1)/2}}},(\nu >0),$$

(17)

where

$$L_{s,i}=ln{\displaystyle \frac{s}{{\Lambda }_i^2}},g_i\left(s\right)=arccos\left({\displaystyle \frac{L_{s,i}}{\sqrt{{\pi }^2+L_{s,i}^2}}}\right).$$

(18)

For $\nu =1$, we recover the famous Shirkov–Solovtsov results [6,7,8]:

$$A_{\mathrm{MA},0}^{\left(1\right)}\left(Q^2\right)\equiv A_{\mathrm{MA},\nu =1,0}^{\left(1\right)}\left(Q^2\right)={\displaystyle \frac{1}{L_0}}-{\displaystyle \frac{z_0}{1-z_0}},U_{\mathrm{MA},0}^{\left(1\right)}\left(Q^2\right)\equiv U_{\mathrm{MA},\nu =1,0}^{\left(1\right)}\left(\mathrm{s}\right)={\displaystyle \frac{g_0\left(s\right)}{\pi }}.$$

(19)

Note that the result from (19) can be taken directly for the integral forms (5) and (6), as in Refs. [6,7,8].

4.2. Beyond LO

Following Equations (15) and (17) for the LO analytic couplings, we consider the derivatives of the MA couplings as follows:

$${\tilde{A}}_{\mathrm{MA},n+1}\left(Q^2\right)={\displaystyle \frac{{(-1)}^n}{n!}}{\displaystyle \frac{d^n{A}_{\mathrm{MA}}\left(Q^2\right)}{{\left(dL\right)}^n}},{\tilde{U}}_{\mathrm{MA},n+1}\left(Q^2\right)={\displaystyle \frac{{(-1)}^n}{n!}}{\displaystyle \frac{d^n{U}_{\mathrm{MA}}\left(s\right)}{{\left(d{L}_s\right)}^n}}.$$

(20)

By analogy with the ordinary coupling, and using the results from (13) we have the following expressions for the MA couplings ${\tilde{A}}_{\mathrm{MA},\nu ,i}^{(i+1)}$ and ${\tilde{U}}_{\mathrm{MA},\nu ,i}^{(i+1)}$:

$$\begin{array}{ccc}& & {\tilde{A}}_{\mathrm{MA},\nu ,i}^{(i+1)}\left(Q^2\right)={\tilde{A}}_{\mathrm{MA},\nu ,i}^{\left(1\right)}\left(Q^2\right)+{\displaystyle \sum _{m=1}^i}{C}_m^{\nu +m}{\tilde{\delta }}_{\mathrm{A},\nu ,i}^{(m+1)}\left(Q^2\right),\hfill \\ & & {\tilde{U}}_{\mathrm{MA},\nu ,i}^{(i+1)}\left(s\right)={\tilde{U}}_{\mathrm{MA},\nu ,i}^{\left(1\right)}\left(s\right)+{\displaystyle \sum _{m=1}^i}{C}_m^{\nu +m}{\tilde{\delta }}_{\mathrm{U},\nu ,i}^{(m+1)}\left(s\right),\hfill \end{array}$$

(21)

where ${\tilde{A}}_{\mathrm{MA},\nu ,i}^{\left(1\right)}$ and ${\tilde{U}}_{\mathrm{MA},\nu ,i}^{\left(1\right)}$ are given in Equations (15) and (17), respectively, and

$${\tilde{\delta }}_{\mathrm{A},\nu ,i}^{(m+1)}\left(Q^2\right)={\tilde{\delta }}_{\nu ,i}^{(m+1)}\left(Q^2\right)-{\widehat{R}}_m\left({\displaystyle \frac{{\mathrm{Li}}_{-\nu -m+1}\left(z_i\right)}{\Gamma (\nu +m)}}\right),{\tilde{\delta }}_{\mathrm{U},\nu ,i}^{(m+1)}\left(s\right)={\widehat{R}}_m\left({\tilde{U}}_{\mathrm{MA},\nu +m,i}^{\left(1\right)}\left(s\right)\right).$$

(22)

and ${\tilde{\delta }}_{\nu ,i}^{(m+1)}\left(Q^2\right)$ and ${\widehat{R}}_m$ are given in Equations (13) and (14), respectively.

The relations (15) reflect the fact that the MA procedure (15) and the operation $d/\left(d\nu \right)$ commute. Thus, to obtain (15), we propose that the form (13), used for the usual coupling $a_s$ at high orders, is applicable (exactly in the same way) to the case of the MA coupling.

Space-like case. After some evaluations, we obtained the following expressions without operators:

$${\tilde{\Delta }}_{\nu ,i}^{(i+1)}={\tilde{\Delta }}_{\nu ,i}^{\left(1\right)}+\sum _{m=1}^i{C}_m^{\nu +m}{\overline{R}}_m\left(z_i\right)\left({\displaystyle \frac{{\mathrm{Li}}_{-\nu -m+1}\left(z_i\right)}{\Gamma (\nu +m)}}\right),$$

(23)

where

$$\begin{array}{ccc}& & {\overline{R}}_1\left(z\right)=b_1\left[{\overline{\gamma }}_{\mathrm{E}}+{\mathrm{M}}_{-\nu ,1}\left(z\right)\right],\hfill \\ & & {\overline{R}}_2\left(z\right)=b_2+b_1^2\left[{\mathrm{M}}_{-\nu -1,2}\left(z\right)+2{\overline{\gamma }}_{\mathrm{E}}{\mathrm{M}}_{-\nu -1,1}\left(z\right)+{\overline{\gamma }}_{\mathrm{E}}^2-{\zeta }_2\right]\hfill \end{array}$$

(24)

and

$${\overline{\gamma }}_{\mathrm{E}}={\gamma }_{\mathrm{E}}-1,{\mathrm{Li}}_{\nu ,k}\left(z\right)={(-1)}^k{\displaystyle \frac{d^k}{{\left(d\nu \right)}^k}}{\mathrm{Li}}_{\nu }\left(z\right)=\sum _{m=1}^{\infty }{\displaystyle \frac{z^m{ln}^{k}m}{m^{\nu }}},{\mathrm{M}}_{\nu ,k}\left(z\right)={\displaystyle \frac{{\mathrm{Li}}_{\nu ,k}\left(z\right)}{{\mathrm{Li}}_{\nu }\left(z\right)}}.$$

(25)

We see that the $\Psi \left(\nu \right)$-function and its derivatives have completely canceled out. Note that another form for ${\tilde{\Delta }}_{\nu ,i}^{(m+1)}\left(Q^2\right)$ is given in Appendix B.

So, for the MA analytic couplings, ${\tilde{A}}_{\mathrm{MA},\nu }^{(i+1)}$, we have the following expressions:

$${\tilde{A}}_{\mathrm{MA},\nu ,i}^{(i+1)}\left(Q^2\right)={\tilde{A}}_{\mathrm{MA},\nu ,i}^{\left(1\right)}\left(Q^2\right)+\sum _{m=1}^i{C}_m^{\nu +m}{\tilde{\delta }}_{\mathrm{A},\nu ,i}^{(m+1)}\left(Q^2\right)$$

(26)

where

$$\begin{array}{ccc}& & {\tilde{A}}_{\mathrm{A},\nu ,i}^{\left(1\right)}\left(Q^2\right)={\tilde{a}}_{\nu ,i}^{\left(1\right)}\left(Q^2\right)-{\displaystyle \frac{{\mathrm{Li}}_{1-\nu }\left(z_i\right)}{\Gamma \left(\nu \right)}},\hfill \\ & & {\tilde{\delta }}_{\mathrm{A},\nu ,i}^{(m+1)}\left(Q^2\right)={\tilde{\delta }}_{\nu ,i}^{(m+1)}\left(Q^2\right)-{\overline{R}}_m\left(z_i\right){\displaystyle \frac{{\mathrm{Li}}_{-\nu +1-m}\left(z_i\right)}{\Gamma (\nu +m)}}\hfill \end{array}$$

(27)

and ${\tilde{\delta }}_{\nu ,m}^{(k+1)}\left(Q^2\right)$ are given in Equation (13).

Time-like case. Using the results from (13) for the usual coupling, we have the following:

$$\begin{array}{ccc}& & {\tilde{U}}_{\nu }^{\mathrm{MA},(i+1)}\left(\mathrm{s}\right)={\tilde{U}}_{\mathrm{MA},\nu }^{\left(1\right)}\left(\mathrm{s}\right)+{\displaystyle \sum _{m=1}^i}{C}_m^{\nu +m}{\tilde{\delta }}_{\mathrm{U},\nu }^{(m+1)}\left(\mathrm{s}\right),\hfill \\ & & {\tilde{\delta }}_{\mathrm{U},\nu }^{(m+1)}\left(\mathrm{s}\right)={\widehat{R}}_m{\tilde{U}}_{\mathrm{MA},\nu +m}^{\left(1\right)}\left(\mathrm{s}\right),\hfill \end{array}$$

(28)

where ${\tilde{U}}_{\mathrm{MA},\nu }^{\left(1\right)}\left(\mathrm{s}\right)$ is given in Equation (17).

This approach allows us to express the high-order corrections in explicit form:

$${\tilde{\delta }}_{\mathrm{U},\nu }^{(m+2)}\left(\mathrm{s}\right)={\displaystyle \frac{1}{(\nu +m)\pi {({\pi }^2+L_s^2)}^{(\nu +m)/2}}}\left\{{\overline{\delta }}_{\nu +m-1}^{(m+2)}\left(\mathrm{s}\right)sin\left((\nu +m)g\right)+{\widehat{\delta }}_{\nu +m-1}^{(m+2)}\left(\mathrm{s}\right)gcos\left((\nu +m)g\right)\right\},$$

(29)

where ${\overline{\delta }}_{\nu }^{(m+2)}\left(\mathrm{s}\right)$ and ${\widehat{\delta }}_{\nu }^{(m+2)}\left(\mathrm{s}\right)$ are as follows:

$$\begin{array}{ccc}& & {\overline{\delta }}_{\nu }^{\left(2\right)}\left(\mathrm{s}\right)=b_1\left[{\widehat{Z}}_1\left(\nu \right)-G\right],{\widehat{\delta }}_{\nu }^{\left(2\right)}\left(\mathrm{s}\right)=b_1,\hfill \\ & & {\overline{\delta }}_{\nu }^{\left(3\right)}\left(\mathrm{s}\right)=b_2+b_1^2\left[{\widehat{Z}}_2\left(\nu \right)-2G{\widehat{Z}}_1\left(\nu \right)+G^2-g^2\right],{\widehat{\delta }}_{\nu }^{\left(3\right)}\left(\mathrm{s}\right)=2b_1^2\left[{\widehat{Z}}_1\left(\nu \right)-G\right]\hfill \end{array}$$

(30)

and

$$G\left(\mathrm{s}\right)={\displaystyle \frac{1}{2}}ln\left({\pi }^2+L_s^2\right).$$

(31)

4.3. The Case $\nu =1$

Here, we present only the results for the case $\nu =1$:

$$\begin{array}{ccc}& & A_{\mathrm{MA},i}^{(i+1)}\left(Q^2\right)\equiv {\tilde{A}}_{\mathrm{MA},\nu =1,i}^{(i+1)}\left(Q^2\right)=A_{\mathrm{MA},i}^{\left(1\right)}\left(Q^2\right)+{\displaystyle \sum _{m=1}^i}{\tilde{\delta }}_{\mathrm{A},\nu =1,i}^{(m+1)}\left(Q^2\right),\hfill \\ & & U_{\mathrm{MA},i}^{(i+1)}\left(s\right)\equiv {\tilde{U}}_{\mathrm{MA},\nu =1,i}^{(i+1)}\left(s\right)=U_{\mathrm{MA},i}^{\left(1\right)}\left(s\right)+{\displaystyle \sum _{m=1}^i}{\tilde{\delta }}_{\mathrm{U},\nu =1,i}^{(m+1)}\left(s\right)\hfill \end{array}$$

(32)

where $A_{\mathrm{MA},i}^{\left(1\right)}\left(Q^2\right)$ and $U_{\mathrm{MA},i}^{\left(1\right)}\left(s\right)$ are shown in Equation (19) and

$$\begin{array}{ccc}& & {\tilde{\delta }}_{\mathrm{A},\nu =1,i}^{(m+1)}\left(Q^2\right)={\tilde{\delta }}_{\nu =1,i}^{(m+1)}\left(Q^2\right)-{\displaystyle \frac{P_{m,1}\left(z_i\right)}{m!}},\hfill \\ & & {\tilde{\delta }}_{\mathrm{U},\nu =1,i}^{\left(2\right)}\left(\mathrm{s}\right)={\displaystyle \frac{b_1}{\pi {({\pi }^2+L_{s,i}^2)}^{1/2}}}\left\{g_{i}cos\left(g_i\right)-\left[1+G_i\right]sin\left(g_i\right)\right\},\hfill \\ & & {\tilde{\delta }}_{\mathrm{U},\nu =1,i}^{\left(3\right)}\left(\mathrm{s}\right)={\displaystyle \frac{1}{2\pi ({\pi }^2+L_s^2)}}\left(b_{2}sin\left(2g_i\right)+b_1^2\left[G_i^2-g_i^2-1\right]sin\left(2g_i\right)\right)\hfill \end{array}$$

(33)

with

$$\begin{array}{ccc}& & G_i\left(\mathrm{s}\right)={\displaystyle \frac{1}{2}}ln\left({\pi }^2+L_{s,i}^2\right),P_{1,\nu }\left(z\right)=b_1\left[{\overline{\gamma }}_{\mathrm{E}}{\mathrm{Li}}_{-\nu }\left(z\right)+{\mathrm{Li}}_{-\nu ,1}\left(z\right)\right],{\overline{\gamma }}_{\mathrm{E}}={\gamma }_{\mathrm{E}}-1,\hfill \\ & & P_{2,\nu }\left(z\right)=b_2{\mathrm{Li}}_{-\nu -1}\left(z\right)+b_1^2\left[{\mathrm{Li}}_{-\nu -1,2}\left(z\right)+2{\overline{\gamma }}_{\mathrm{E}}{\mathrm{Li}}_{-\nu -1,1}\left(z\right)+\left({\overline{\gamma }}_{\mathrm{E}}^2-{\zeta }_2\right){\mathrm{Li}}_{-\nu -1}\left(z\right)\right],\hfill \end{array}$$

(34)

Euler constant ${\gamma }_{\mathrm{E}}$ and

$${\mathrm{Li}}_{n,m}\left(z\right)=\sum _{m=1}{\displaystyle \frac{{ln}^{k}m}{m^n}},{\mathrm{Li}}_{-1}\left(z\right)={\displaystyle \frac{z}{{(1-z)}^2}},{\mathrm{Li}}_{-2}\left(z\right)={\displaystyle \frac{z(1+z)}{{(1-z)}^3}}.$$

(35)

Using the results in Equation (18) and transformation rules for $sin\left(ng\right)$ and $cos\left(ng\right)$, we have the following:

$$sin\left(ng\right)={\displaystyle \frac{S\left(ng\right)}{{({\pi }^2+L_s^2)}^{n/2}}},cos\left(ng\right)={\displaystyle \frac{C\left(ng\right)}{{({\pi }^2+L_s^2)}^{n/2}}},$$

(36)

where

$$\begin{array}{ccc}& & S\left(g\right)=\pi ,C\left(g\right)=L_s,S\left(2g\right)=2\pi L_s,C\left(2g\right)=L_s^2-{\pi }^2,S\left(3g\right)=\pi (3L_s^2-{\pi }^2),\hfill \\ & & C\left(3g\right)=L_s(L_s^2-3{\pi }^2).\hfill \end{array}$$

(37)

Using Equations (36) and (37), the results for ${\tilde{\delta }}_{\mathrm{U},\nu =1}^{(m+1)}\left(\mathrm{s}\right)$ in (29) can be rewritten in the following form:

$$\begin{array}{ccc}& & {\tilde{\delta }}_{\mathrm{U},\nu =1}^{\left(2\right)}\left(\mathrm{s}\right)={\displaystyle \frac{b_1}{({\pi }^2+L_s^2)}}\left({\displaystyle \frac{g}{\pi }}{L}_s-\left(1+G\right)\right),\hfill \\ & & {\tilde{\delta }}_{\mathrm{U},\nu =1}^{\left(3\right)}\left(\mathrm{s}\right)={\displaystyle \frac{1}{{({\pi }^2+L_s^2)}^2}}\left[b_2{L}_s-b_1^2\left({\displaystyle \frac{gG}{\pi }}(L_s^2-{\pi }^2)+L_s\left(1+g^2-G^2\right)\right)\right],\hfill \end{array}$$

(38)

which is similar to the results for the spectral function $r_1^{\left(i\right)}\left(\mathrm{s}\right)$ in Refs. [67,68] (see Section 6 in [39]).

5. The Behaviors of MA Couplings

Here, we show the behaviors of the MA couplings $A_{\mathrm{MA},k}^{(k+1)}\left(Q^2\right)={\tilde{A}}_{\mathrm{MA},\nu =1,k}^{(k+1)}\left(Q^2\right)$ and $U_{\mathrm{MA},k}^{(k+1)}\left(s\right)={\tilde{U}}_{\mathrm{MA},\nu =1,k}^{(k+1)}\left(s\right)$ and compare them.

5.1. Coupling ${\tilde{A}}_{\mathrm{MA},\nu ,k}^{(k+1)}\left(Q^2\right)$

From Figure 2 and Figure 3, we can see differences between $A_{\mathrm{MA},\nu =1,i}^{(i+1)}\left(Q^2\right)$ with $i=0,2,4$, which are rather small and have nonzero values around the position $Q^2={\Lambda }_i^2$. In Figure 2, the values of ${\left({\Lambda }_i^{f=3}\right)}^2$$(i=0,2,4)$ are shown by vertical lines (as seen in Figure 1).

Figure 2. The results for $A_{\mathrm{MA},\nu =1,i}^{(i+1)}\left(Q^2\right)$ and ${\left({\Lambda }_i^{f=3}\right)}^2$ (vertical lines) with $i=0,2,4$.

Figure 3. The results for $A_{\mathrm{MA},\nu =1,i}^{(i+1)}\left(Q^2\right)$ ($i=0,1,2$) but with the logarithmic scale.

Figure 4 shows the results for $A_{\mathrm{MA},\nu ,0}^{\left(1\right)}\left(Q^2\right)$ and $A_{\mathrm{MA},\nu ,1}^{\left(2\right)}\left(Q^2\right)$ and their differences ${\delta }_{\mathrm{A},\nu ,1}^{\left(2\right)}\left(Q^2\right)$, which are essentially less than the couplings themselves. From Figure 4, it is clear that for $Q^2\to 0$, the asymptotic behaviors of $A_{\mathrm{MA},\nu ,0}^{\left(1\right)}\left(Q^2\right)$, $A_{\mathrm{MA},\nu ,1}^{\left(2\right)}\left(Q^2\right)$, and $A_{\mathrm{MA},\nu ,2}^{\left(3\right)}\left(Q^2\right)$ coincide, i.e., the differences ${\delta }_{\mathrm{A},\nu =1,1}^{\left(2\right)}(Q^2\to 0)$ and ${\delta }_{\mathrm{A},\nu =1,2}^{\left(3\right)}(Q^2\to 0)$ are negligible. Also, Figure 5 shows the differences ${\delta }_{\mathrm{A},\nu =1,i}^{(i+1)}\left(Q^2\right)$ $(i\ge 2)$ are essentially less than ${\delta }_{\mathrm{A},\nu =1,1}^{\left(2\right)}\left(Q^2\right)$.

Figure 4. The results for $A_{\mathrm{MA},\nu =0.5,0}^{\left(1\right)}\left(Q^2\right)$, $A_{\mathrm{MA},\nu =0.5,1}^{\left(2\right)}\left(Q^2\right)$ and ${\delta }_{\mathrm{A},\nu =0.5,1}^{\left(2\right)}\left(Q^2\right)$.

Figure 5. The results for ${\delta }_{\mathrm{A},\nu =1,i}^{(i+1)}\left(Q^2\right)$ with $i=1,2,3,4$.

Thus, we can conclude that contrary to the case of the usual coupling, considered in Figure 1, the $1/L$-expansion of the MA coupling is a very good approximation at any $Q^2$ value. Moreover, the differences between $A_{\mathrm{MA},\nu =1,i}^{(i+1)}\left(Q^2\right)$ and $A_{\mathrm{MA},\nu =1,0}^{\left(1\right)}\left(Q^2\right)$ are small. So, the expansions of $A_{\mathrm{MA},\nu =1,i}^{(i+1)}\left(Q^2\right)i\ge 1$ through the one $A_{\mathrm{MA},\nu =1,0}^{\left(1\right)}\left(Q^2\right)$ conducted in Refs. [10,11,12] are very good approximations. Also, the approximation

$$A_{\mathrm{MA},\nu =1,i}^{(i+1)}\left(Q^2\right)=A_{\mathrm{MA},\nu =1,0}^{\left(1\right)}\left(k_i{Q}^2\right),(i=1,2),$$

(39)

introduced in [69,70,71,72] and used in [73,74] is very convenient. Indeed, since the corrections ${\delta }_{\mathrm{A},\nu =1,i}^{(i+1)}\left(Q^2\right)$ are very small, then from Equation (33), one can see that the MA couplings $A_{\mathrm{MA},\nu =1,i}^{(i+1)}\left(Q^2\right)$ are very similar to the LO ones taken with the corresponding ${\Lambda }_i$.

5.2. Coupling ${\tilde{U}}_{\mathrm{MA},\nu ,k}^{(k+1)}\left(Q^2\right)$

This subsection provides graphical results of coupling construction. Figure 6 and Figure 7 show the results for $U_{\mathrm{MA},\nu =1}^{\left(i\right)}\left(s\right)$ with $i=1,3,5$ in usual and logarithmic scales (the last one was chosen to stress the limit $U_{\mathrm{MA},\nu =1}^{\left(i\right)}(s\to 0)\to 1$). From Figure 8 and Figure 9, we can see the differences between $U_{\mathrm{MA},\nu =1}^{\left(i\right)}\left(Q^2\right)$ with $i=1,\dots ,5$, which are rather small and have nonzero values around the position $Q^2={\Lambda }_i^2$.

Figure 6. 1, 3, and 5 orders of $U_{\mathrm{MA},\nu =1}^{\left(i\right)}$. The vertical lines indicate ${\left({\Lambda }_{i-1}^{f=3}\right)}^2$.

Figure 7. 1, 3, and 5 orders of $U_{\mathrm{MA},\nu =1}^{\left(i\right)}$ with a logarithmic scale of s.

Figure 8. ${\delta }_{\mathrm{MA},\nu =1}^{\left(i\right)}$ with $i=2,3$. The vertical lines indicate ${\left({\Lambda }_{i-1}^{f=3}\right)}^2$.

Figure 9. ${\delta }_{\mathrm{U},\nu =1}^{\left(i\right)}$ with $i=4,5$. The vertical line indicates ${\left({\Lambda }_3^{f=3}\right)}^2={\left({\Lambda }_4^{f=3}\right)}^2$.

So, Figure 6, Figure 7, Figure 8 and Figure 9 show that the difference between $U_{\mathrm{MA},\nu =1}^{(i+1)}\left(s\right)$ and $U_{\mathrm{MA},\nu =1}^{\left(i\right)}\left(s\right)$ is essentially less than the couplings themselves. From Figure 7, Figure 8 and Figure 9, it is clear that for $s\to 0$, the asymptotic behaviors of $U_{\mathrm{MA},\nu =1}^{\left(1\right)}\left(s\right)$, $U_{\mathrm{MA},\nu =1}^{\left(3\right)}\left(s\right)$, and $U_{\mathrm{MA},\nu =1}^{\left(5\right)}\left(s\right)$ coincide, i.e., the differences ${\delta }_{\mathrm{U},\nu =1}^{\left(i\right)}(s\to 0)$ are negligible. Also, Figure 8 and Figure 9 show the differences ${\delta }_{\mathrm{U},\nu =1}^{(i+1)}\left(s\right)$ $(i\ge 2)$ are essentially less than ${\delta }_{\mathrm{U},\nu =1}^{\left(2\right)}\left(s\right)$. We note that the general form of these results coincides with that of the MA couplings $A_{\mathrm{MA},\nu ,i}^{(i+1)}\left(Q^2\right)$, studied in the previous subsection.

5.3. Couplings ${\tilde{A}}_{\mathrm{MA},\nu ,k}^{(k+1)}\left(Q^2\right)$ and ${\tilde{U}}_{\mathrm{MA},\nu ,k}^{(k+1)}\left(Q^2\right)$

On Figure 10, we see that $A_{\mathrm{MA},\mathrm{i}}^{(i+1)}\left(Q^2\right)$ and $U_{\mathrm{MA},\mathrm{i}}^{(i+1)}\left(Q^2\right)$ are very close to each other for $i=0$ and $i=2$. The differences between the L0 and NNLO results are nonzero only for $Q^2\sim {\Lambda }^2$.

Figure 10. The results for $A_{\mathrm{MA},\mathrm{i}}^{(i+1)}\left(Q^2\right)$ and $U_{\mathrm{MA},\mathrm{i}}^{(i+1)}\left(Q^2\right)$ with $i=0,2$.

Indeed, the similarity is shown in Figure 11 and Figure 12. In Figure 11, the results for $U_{\mathrm{MA},\nu =1}^{\left(i\right)}\left(s\right)$ and $A_{\mathrm{MA},\nu =1}^{\left(i\right)}\left(Q^2\right)$ ($i=1,3,5$) are shown in a so-called mirror form, similar to the representation previously introduced in [11]. Figure 12 contains $U_{\mathrm{MA},\nu =1}^{\left(1\right)}\left(s\right)$, $A_{\mathrm{MA},\nu =1}^{\left(1\right)}\left(Q^2\right)$, $U_{\mathrm{MA},\nu =1}^{\left(2\right)}\left(s\right)$ and $A_{\mathrm{MA},\nu =1}^{\left(2\right)}\left(Q^2\right)$, which are very close to each other but have different limit values when $Q^2\to 0$. Moreover, the differences ${\delta }_{\mathrm{A},\nu =1}^{\left(2\right)}\left(Q^2\right)$ and ${\delta }_{\mathrm{U},\nu =1}^{\left(2\right)}\left(Q^2\right)$ are almost the same, although the correction to the space-like coupling decreases more rapidly. The direct relation between $A_{\mathrm{MA},\nu =1}^{\left(i\right)}\left(Q^2\right)$ and $U_{\mathrm{MA},\nu =1}^{\left(i\right)}\left(Q^2\right)$ gives an interesting picture (see Figure 13). Obviously, we have ${\displaystyle \frac{A_{\mathrm{MA},\nu =1}^{\left(i\right)}(Q^2=0)}{U_{\mathrm{MA},\nu =1}^{\left(i\right)}(Q^2=0)}}=1$ for any order and the second similar point

$${\displaystyle \frac{A_{\mathrm{MA},\nu =1}^{\left(i\right)}(Q^2={\left({\Lambda }_{i-1}^{f=3}\right)}^2)}{U_{\mathrm{MA},\nu =1}^{\left(i\right)}(Q^2={\left({\Lambda }_{i-1}^{f=3}\right)}^2)}}=1$$

(40)

for $i=1$. Higher-order corrections break the identity (40), shifting the second point from ${\left({\Lambda }_i^{f=3}\right)}^2$. As can be seen in Figure 13, the shift is quite small. As can be seen in Figure 13, the ratio (40) asymptotically approaches 1 when ${\mathrm{Q}}^2\to \infty$.

Figure 11. One, three, and five orders of $U_{\mathrm{MA},\nu =1}^{\left(i\right)}$ and $A_{\mathrm{MA},\nu =1}^{\left(i\right)}$.

Figure 12. One and two orders of $U_{\mathrm{MA},\nu =1}^{\left(i\right)}$, $A_{\mathrm{MA},\nu =1}^{\left(i\right)}$, and ${\delta }_{\mathrm{MA},\nu =1}^{\left(2\right)}$ in Euclidean and Minkowski spaces.

Figure 13. The relation $A_{\mathrm{MA},\nu =1}^{\left(i\right)}/U_{\mathrm{MA},\nu =1}^{\left(i\right)}$ for $i=1,5$. The vertical lines indicate ${\left({\Lambda }_{i-1}^{f=3}\right)}^2$.

In Figure 6, Figure 8, Figure 9 and Figure 13 the values of ${\left({\Lambda }_i^{f=3}\right)}^2$$(i=0,2,4)$ are shown by vertical lines with color matching in each order. Note that Figure 9 contains only one vertical line since ${\left({\Lambda }_4^{f=3}\right)}^2={\left({\Lambda }_5^{f=3}\right)}^2$.

Thus, we can conclude that contrary to the case of the usual coupling, the $1/L$-expansion of the MA coupling is a very good approximation at any $Q^2\left(s\right)$ value. Moreover, the differences between $U_{\mathrm{MA},\nu =1}^{(i+1)}\left(s\right)$ and $U_{\mathrm{MA},\nu =1}^{\left(i\right)}\left(s\right)$ are smaller with the increase in order. So, the expansions of $U_{\mathrm{MA},\nu =1}^{(i+1)}\left(s\right)i\ge 1$ through the $U_{\mathrm{MA},\nu =1}^{\left(1\right)}\left(s\right)$ in Refs. [10,11,12] are very good approximations.

6. MA Coupling ${\tilde{\mathit{A}}}_{\mathbf{MA},\mathit{\nu },\mathit{k}}^{\mathbf{(}\mathit{k}\mathbf{+}\mathbf{1}\mathbf{)}}\mathbf{\left(}{\mathit{Q}}^{\mathbf{2}}\mathbf{\right)}$: The form Is Convenient for ${\mathit{Q}}^{\mathbf{2}}\mathbf{\sim }{\mathbf{\Lambda }}_{\mathit{k}}^{\mathbf{2}}$

The results from (26) for analytic coupling ${\tilde{A}}_{\mathrm{MA},\nu ,k}^{(k+1)}\left(Q^2\right)$ are very convenient both at large and small values of $Q^2$ values. For $Q^2\sim {\Lambda }_i^2$, each part—the standard strong coupling and the additional term—has singularities, which are canceled in their sum. So, some numerical applications of the results (26) can be complicated. So, here we present another form, which is very useful at $Q^2\sim {\Lambda }_i^2$ and can be used for any $Q^2$ value, except the ranges of very large and very small $Q^2$ values. As in the previous section, we will first present the LO results taken from [10] and later extend them beyond LO.

6.1. LO

The LO minimal analytic coupling $A_{\mathrm{MA},\nu }^{\left(1\right)}\left(Q^2\right)$ [6,7,8,9] also has another form [10]:

$$A_{\mathrm{MA},\nu }^{\left(1\right)}\left(Q^2\right)={\displaystyle \frac{(-1)}{\Gamma \left(\nu \right)}}\sum _{r=0}^{\infty }\zeta (1-\nu -r){\displaystyle \frac{{(-L)}^r}{r!}}(L<2\pi ),$$

(41)

where Euler functions $\zeta \left(\nu \right)$ are

$$\zeta \left(\nu \right)=\sum _{m=1}^{\infty }{\displaystyle \frac{1}{m^{\nu }}}={\mathrm{Li}}_{\nu }(z=1)$$

(42)

The result from (41) was obtained in Ref. [10], considering the property of the Lerch function, which can be considered a generalization of the polylogarithm (16). The form (41) is very convenient at low L values, i.e., at $Q^2\sim {\Lambda }^2$. Moreover, we can use the relation between $\zeta (1-\nu -r)$ and $\zeta (\nu +r)$ functions:

$$\zeta (1-\nu -r)={\displaystyle \frac{2\Gamma (\nu +r)}{{\left(2\pi \right)}^{\nu +r}}}Sin\left[{\displaystyle \frac{\pi }{2}}(1-\nu -r)\right]\zeta (\nu +r)$$

(43)

For $\nu =1$, we have the following:

$$A_{\mathrm{MA}}^{\left(1\right)}\left(L\right)=-\sum _{r=0}^{\infty }\zeta (-r){\displaystyle \frac{{(-L)}^r}{r!}}$$

(44)

with

$$\zeta (-r)={(-1)}^r{\displaystyle \frac{B_{r+1}}{r+1}}$$

(45)

and $B_{r+1}$ are Bernoulli numbers.

Using the properties of Bernoulli numbers (${\delta }_m^0$ is the Kronecker symbol), we have the following for even $r=2m$ and for odd $r=1+2l$ values:

$$\zeta (-2m)=-{\displaystyle \frac{{\delta }_m^0}{2}},\zeta (-(1+2l))=-{\displaystyle \frac{B_{2(l+1)}}{2(l+1)}}.$$

(46)

Thus, for $A_{\mathrm{MA}}^{\left(1\right)}\left(Q^2\right)$, we have the following results

$$A_{\mathrm{MA}}^{\left(1\right)}\left(Q^2\right)={\displaystyle \frac{1}{2}}\left(1+\sum _{l=0}^{\infty }{\displaystyle \frac{B_{2(l+1)}}{l+1}}{\displaystyle \frac{{(-L)}^{2l+1}}{(2l+1)!}}\right)={\displaystyle \frac{1}{2}}\left(1+\sum _{s=1}^{\infty }{\displaystyle \frac{B_{2s}}{s}}{\displaystyle \frac{{(-L)}^{2s-1}}{(2s-1)!}}\right),$$

(47)

with $s=l+1$.

6.2. Beyond LO

Now, we consider the derivatives of (minimal) analytic couplings ${\tilde{A}}_{\mathrm{MA},\nu }^{\left(1\right)}$, shown in Equation (20), as in Equation (26), i.e.,

$${\tilde{A}}_{\mathrm{MA},\nu ,i}^{(i+1)}\left(Q^2\right)={\tilde{A}}_{\mathrm{MA},\nu ,i}^{\left(1\right)}\left(Q^2\right)+\sum _{m=1}^i{C}_m^{\nu +m}{\tilde{\delta }}_{\mathrm{A},\nu ,i}^{(m+1)}\left(Q^2\right)$$

(48)

where ${\tilde{A}}_{\mathrm{MA},\nu ,i}^{\left(1\right)}=A_{\mathrm{MA},\nu }^{\left(1\right)}$ is given above in (41) with $L\to L_i$ and

$${\tilde{\delta }}_{\mathrm{A},\nu ,i}^{(m+1)}\left(Q^2\right)={\widehat{R}}_m{A}_{\mathrm{MA},\nu +m,i}^{\left(1\right)},$$

(49)

where operators ${\widehat{R}}_m$ are given above in (14).

After some calculations, we have the following:

$${\tilde{\delta }}_{\mathrm{A},\nu ,k}^{(m+1)}\left(Q^2\right)={\displaystyle \frac{(-1)}{\Gamma (\nu +m)}}\sum _{r=0}^{\infty }{\tilde{R}}_m(\nu +r){\displaystyle \frac{{(-L_k)}^r}{r!}}$$

(50)

where, in agreement with (34), we present the following:

$$\begin{array}{ccc}& & {\tilde{R}}_1(\nu +r)=b_1\left[{\overline{\gamma }}_{\mathrm{E}}\zeta (-\nu -r)+{\zeta }_1(-\nu -r)\right],\hfill \\ & & {\tilde{R}}_2(\nu +r)=b_2\zeta (-\nu -r-1)+b_1^2[{\zeta }_2(-\nu -r-1)+2{\overline{\gamma }}_{\mathrm{E}}{\zeta }_1(-\nu -r-1)\hfill \\ & & +\left[{\overline{\gamma }}_{\mathrm{E}}^2-{\zeta }_2\right]\zeta (-\nu -r-1)],\hfill \end{array}$$

(51)

with ${\overline{\gamma }}_{\mathrm{E}}={\gamma }_{\mathrm{E}}-1$ (see Equation (25)) and

$${\zeta }_n\left(\nu \right)={\mathrm{Li}}_{\nu ,n}(z=1)=\sum _{m=1}^{\infty }{\displaystyle \frac{{ln}^{n}m}{m^{\nu }}}.$$

(52)

Strictly speaking, the series representation (52) for the functions ${\zeta }_n(-m-\nu -r-k)$ is not a good definition for large r values, and we can replace them with ${\zeta }_n(m+\nu +r+k)$, using the result from (43). However, the results are long and presented in Appendix B.

6.3. The Case $\nu =1$

For the case $\nu =1$, we immediately have the following:

$$\begin{array}{ccc}& & A_{\mathrm{MA},\mathrm{i}}^{(i+1)}\left(Q^2\right)=A_{\mathrm{MA},\mathrm{i}}^{\left(1\right)}\left(Q^2\right)+{\displaystyle \sum _{m=1}^i}{\tilde{\delta }}_{\mathrm{A},\nu =1,\mathrm{i}}^{(m+1)}\left(Q^2\right),\hfill \end{array}$$

(53)

$$\begin{array}{ccc}& & {\delta }_{\mathrm{A},\mathrm{i}}^{(m+1)}\left(L\right)\equiv {\tilde{\delta }}_{\mathrm{A},\nu =1,i}^{(m+1)}\left(Q^2\right)={\displaystyle \frac{(-1)}{m!}}{\displaystyle \sum _{r=0}^{\infty }}{\tilde{R}}_m(1+r){\displaystyle \frac{{(-L_i)}^r}{r!}},\hfill \end{array}$$

(54)

where $A_{\mathrm{MA},\mathrm{i}}^{\left(1\right)}\left(Q^2\right)$ is given above in (44) (with the replacement ($L\to L_i$), and the coefficients ${\tilde{R}}_m(1+r)$ can be found in (51) when $\nu =1$.

The results from (54) can be expressed in terms of the functions ${\zeta }_n(m+\nu +r+k)$. Using the results in Appendix B and taking the even part ($r=2m$) and the odd part ($r=2s-1$) (see Equation (A17)), we have the following:

$$\begin{array}{ccc}& & {\delta }_{\mathrm{A},k}^{\left(2\right)}\left(Q^2\right)={\displaystyle \frac{2}{{\left(2\pi \right)}^2}}\left[\sum _{m=0}^{\infty }(2m+1){(-1)}^m{Q}_{1a}(2m+2){\widehat{L}}_k^{2m}-\pi {\displaystyle \sum _{s=1}^{\infty }}s{(-1)}^s{Q}_{1b}(2s+1){\widehat{L}}_k^{2s-1}\right],\hfill \\ & & {\delta }_{\mathrm{A},k}^{\left(3\right)}\left(Q^2\right)=-{\displaystyle \frac{1}{{\left(2\pi \right)}^3}}[\pi {\displaystyle \sum _{m=0}^{\infty }}(2m+1)(m+1){(-1)}^m{Q}_{2b}(2m+3){\widehat{L}}_k^{2m}\hfill \\ & & +2\sum _{s=1}^{\infty }s(2s+1){(-1)}^s{Q}_{2a}(2s+2){\widehat{L}}_k^{2s-1}],\hfill \end{array}$$

(55)

where

$${\widehat{L}}_k={\displaystyle \frac{L_k}{2\pi }}$$

(56)

and the functions $Q_{ma}$ and $Q_{mb}$ are given in Appendix B.

At the point $L_k=0$, i.e., $Q^2={\Lambda }_k^2$, we have the following:

$$\begin{array}{ccc}& & A_{\mathrm{MA}}^{\left(1\right)}={\displaystyle \frac{1}{2}},{\delta }_s^{\left(2\right)}={\displaystyle \frac{2}{{\left(2\pi \right)}^2}}{Q}_{1a}\left(2\right)=-{\displaystyle \frac{b_1}{2{\pi }^2}}\left({\zeta }_1\left(2\right)+l\zeta \left(2\right)\right),\hfill \\ & & {\delta }_s^{\left(3\right)}=-{\displaystyle \frac{\pi }{{\left(2\pi \right)}^3}}{Q}_{2b}\left(3\right)={\displaystyle \frac{b_1^2}{4{\pi }^2}}\left({\zeta }_1\left(3\right)+(2l-1)\zeta \left(3\right)\right),\hfill \end{array}$$

(57)

where ${\zeta }_k\left(\nu \right)$ are given in Equation (52) and

$$l=ln\left(2\pi \right).$$

(58)

7. Integral Representations for ${\tilde{\mathit{A}}}_{\mathrm{MA},\mathit{\nu }}^{\left(\mathit{i}\right)}\left({\mathit{Q}}^{\mathbf{2}}\right)$

As already discussed in the introduction, the MA coupling $A_{\mathrm{MA}}^{\left(1\right)}\left(Q^2\right)$ is constructed as follows: The LO spectral function is taken directly from the perturbation theory but the MA coupling $A_{\mathrm{MA}}^{\left(1\right)}\left(Q^2\right)$ itself is built using the correct integration counter. Thus, at LO, the MA coupling $A_{\mathrm{MA}}^{\left(1\right)}\left(Q^2\right)$ obeys Equation (5), as shown in the introduction.

For the $\nu$-derivative of $A_{\mathrm{MA}}^{\left(1\right)}\left(Q^2\right)$, i.e., ${\tilde{A}}_{\mathrm{MA},\nu }^{\left(1\right)}\left(Q^2\right)$, we have the following equation [62]:

$${\tilde{A}}_{\mathrm{MA},\nu }^{\left(1\right)}\left(Q^2\right)={\displaystyle \frac{(-1)}{\Gamma \left(\nu \right)}}{\int }_0^{\infty }{\displaystyle \frac{ds}{s}}{r}_{\mathrm{pt}}^{\left(1\right)}\left(s\right){\mathrm{Li}}_{1-\nu }(-sz),$$

(59)

where $r_{\mathrm{pt}}^{\left(1\right)}\left(s\right)$ is the LO spectral function defined in Equation (5) and ${\mathrm{Li}}_{1-\nu }(-sz)$ is the polylogarithmic function presented in (16).

Beyond LO, Equation (59) can be extended in two ways, which will be shown in the following subsections.

7.1. Modification of Spectral Functions

The first possibility of extending the result from (59) beyond LO is related to the modification of the spectral function. The extension is simple and the final result looks as follows:

$${\tilde{A}}_{\mathrm{MA},\nu ,k}^{(i+1)}\left(Q^2\right)={\displaystyle \frac{(-1)}{\Gamma \left(\nu \right)}}{\int }_0^{\infty }{\displaystyle \frac{ds}{s}}{r}_{\mathrm{pt}}^{(i+1)}\left(s\right){\mathrm{Li}}_{1-\nu }(-s{z}_k),$$

(60)

i.e., it is similar to (59), with the LO spectral function $r_{\mathrm{pt}}^{\left(1\right)}\left(s\right)$ replaced by the $i+1$-order one $r_{\mathrm{pt}}^{(i+1)}\left(s\right)$:

$$r_{\mathrm{pt}}^{(i+1)}\left(s\right)=r_{\mathrm{pt}}^{\left(1\right)}\left(s\right)+\sum _{m=1}^i{\delta }_{\mathrm{r}}^{(m+1)}\left(s\right)$$

(61)

and (see [37])

$$\begin{array}{c}\hfill y=lns,r_{\mathrm{pt}}^{\left(1\right)}\left(y\right)={\displaystyle \frac{1}{y^2+{\pi }^2}},{\delta }_{\mathrm{r}}^{\left(2\right)}\left(y\right)=-{\displaystyle \frac{b_1}{{(y^2+{\pi }^2)}^2}}\left[2y{f}_1\left(y\right)+({\pi }^2-y^2)f_2\left(y\right)\right],\end{array}$$

(62)

with

$$f_1\left(y\right)={\displaystyle \frac{1}{2}}ln\left(y^2+{\pi }^2\right),f_2\left(y\right)={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{\pi }}arctan\left({\displaystyle \frac{y}{\pi }}\right).$$

(63)

For the coupling itself, we have the following:

$$A_{\mathrm{MA},\mathrm{k}}^{(i+1)}\left(Q^2\right)\equiv {\tilde{A}}_{\mathrm{MA},\nu =1,k}^{(i+1)}\left(Q^2\right)={\int }_0^{+\infty }{\displaystyle \frac{ds{r}_{\mathrm{pt}}^{(i+1)}\left(s\right)}{(s+t_k)}}.$$

(64)

Numerical evaluations of the integrals in (64) can be done following the discussions in Section 4 in Ref. [67].

7.2. Modification of Polylogarithms

Beyond LO, the results from (59) can be extended by using the ${\widehat{R}}_m$ operators shown in (14). This is the path used in Section 4 and Section 5 to obtain other ${\tilde{A}}_{\mathrm{MA},\nu ,i}^{(i+1)}\left(Q^2\right)$ results.

Here, the application of the operators ${\widehat{R}}_m$ for Equation (59) leads to the following result:

$${\tilde{A}}_{\mathrm{MA},\nu ,i}^{(i+1)}\left(Q^2\right)={\int }_0^{\infty }{\displaystyle \frac{ds}{s}}{r}_{\mathrm{pt}}^{\left(1\right)}\left(s\right){\tilde{\Delta }}_{\nu ,i}^{(i+1)},$$

(65)

where the results for ${\tilde{\Delta }}_{\nu ,i}^{(i+1)}$ can be found in Equations (23) and (24) and also in Equation (34).

For MA coupling itself, we have the following beyond LO:

$$A_{\mathrm{ma},\mathrm{i}}^{(i+1)}\left(Q^2\right)\equiv {\tilde{A}}_{\mathrm{ma},\nu =1,i}^{(i+1)}\left(Q^2\right)={\int }_0^{+\infty }{\displaystyle \frac{ds}{s}}{r}_{\mathrm{pt}}^{\left(1\right)}\left(s\right){\tilde{\Delta }}_{\nu =1,i}^{(i+1)},$$

(66)

where the results for ${\tilde{\Delta }}_{\nu =1,i}^{(i+1)}$ are given in Equations (23) and (24) with $\nu =1$, i.e.,

$${\tilde{\Delta }}_{\nu =1,i}^{(i+1)}={\tilde{\Delta }}_{1,i}^{\left(1\right)}+\sum _{m=1}^i{\displaystyle \frac{P_{m,1}\left(z_i\right)}{m!}}={\Delta }_{1,i}^{\left(1\right)}+\sum _{m=1}^i{\displaystyle \frac{P_{m,1}\left(z_i\right)}{m!}},$$

(67)

where ${\Delta }_{1,i}^{\left(1\right)}={\mathrm{Li}}_0\left(z_i\right)$ and $P_{m,1}\left(z_i\right)$ are given in Equation (34).

7.3. Discussions

We considered $1/L$-expansions of $\nu$-derivatives of the strong coupling $a_s$ expressed as combinations of operators ${\widehat{R}}_m$ (14) applied to the LO coupling $a_s^{\left(1\right)}$. Applying the same operators to the $\nu$-derivatives of the LO MA coupling $A_{\mathrm{MA}}^{\left(1\right)}$, we obtained four different representations for the $\nu$-derivatives of the MA couplings, i.e., ${\tilde{A}}_{\mathrm{MA},\nu }^{\left(i\right)}$, in each i-order of perturbation theory. One form contains a combination of polylogarithms; another contains an expansion of the generalized Euler $\zeta$-function; the third is based on dispersion integrals containing the LO spectral function; the fourth representation is based on the dispersion integral containing the i-order spectral function. All results are presented up to the fifth order of perturbation theory, where the corresponding coefficients of the QCD $\beta$-function are well-known (see [2,3]).

The high-order corrections are negligible in the $Q^2\to 0$ and $Q^2\to \infty$ asymptotics and are nonzero in the vicinity of $Q^2={\Lambda }^2$. Thus, they represent only small corrections to the LO MA coupling $A_{\mathrm{MA},\nu }^{\left(1\right)}\left(Q^2\right)$. This demonstrates the possibility of expanding the high-order couplings $A_{\mathrm{MA},\nu }^{\left(i\right)}\left(Q^2\right)$ via the LO couplings $A_{\mathrm{MA},\nu }^{\left(1\right)}\left(Q^2\right)$, as was done in Ref. [12], as well as the possibility of various approximations used in [56,69,70,71,72,73,74].

8. Integral Representations for ${\tilde{\mathit{U}}}_{\mathrm{MA},\mathit{\nu }}^{\left(\mathit{i}\right)}\left(\mathit{s}\right)$

As mentioned in the introduction, the MA couplings $A_{\mathrm{MA},\nu }^{\left(1\right)}\left(Q^2\right)$ and $U_{\mathrm{MA},\nu }^{\left(1\right)}\left(s\right)$ are constructed as follows: the LO spectral function is taken directly from the perturbation theory but the MA couplings $A_{\mathrm{MA},\nu }^{\left(1\right)}\left(Q^2\right)$ and $U_{\mathrm{MA},\nu }^{\left(1\right)}\left(s\right)$ are obtained using the correct integration contours. Thus, at LO, the MA couplings $A_{\mathrm{MA},\nu }^{\left(1\right)}\left(Q^2\right)$ and $U_{\mathrm{MA},\nu }^{\left(1\right)}\left(s\right)$ obey Equations (5) and (6) presented in the introduction.

To check Equations (29) and (30), we compare them with an integral form, as follows:

$$U_1^{\left(i\right)}\left(\mathrm{s}\right)={\displaystyle \frac{1}{\pi }}\underset{s}{\overset{\infty }{\int }}{\displaystyle \frac{d\sigma }{\sigma }}{r}_{\mathrm{pt}}^{\left(i\right)}\left(\sigma \right).$$

(68)

For LO, we can take the integral form from [11]

$$U_{\nu }^{\left(1\right)}\left(\mathrm{s}\right)={\displaystyle \frac{1}{\pi }}\underset{s}{\overset{\infty }{\int }}{\displaystyle \frac{d\sigma }{\sigma }}{r}_{\nu }^{\left(1\right)}\left(\sigma \right),$$

(69)

where

$$r_{\nu }^{\left(1\right)}\left(\mathrm{s}\right)={\displaystyle \frac{sin\left[\nu g\right(s\left)\right]}{\pi {({\pi }^2+L_s^2)}^{(\nu -1)/2}}}=\nu U_{\nu +1}^{\left(1\right)}\left(\mathrm{s}\right),$$

(70)

Using our approach to obtain high-order terms from LO (69), we can extend the LO integral (69) to the following:

$${\tilde{U}}_{\nu }^{\left(i\right)}\left(\mathrm{s}\right)={\displaystyle \frac{1}{\pi }}\underset{s}{\overset{\infty }{\int }}{\displaystyle \frac{d\sigma }{\sigma }}{r}_{\nu }^{\left(i\right)}\left(\sigma \right),$$

(71)

where, obviously, we have the following:

$$r_{\nu }^{\left(i\right)}\left(\mathrm{s}\right)=\nu {\tilde{U}}_{\nu +1}^{\left(i\right)}\left(\mathrm{s}\right).$$

(72)

The spectral function $r_1^{\left(i\right)}\left(\mathrm{s}\right)$ has the following form:

$$r_1^{\left(i\right)}\left(\mathrm{s}\right)=r_1^{\left(1\right)}\left(\mathrm{s}\right)+\sum _{m=1}^i{\delta }_1^{(m+1)}\left(\mathrm{s}\right)$$

(73)

where

$$r_1^{\left(1\right)}\left(\mathrm{s}\right)=U_2^{\left(1\right)}\left(\mathrm{s}\right),{\delta }_1^{(m+1)}\left(\mathrm{s}\right)=(m+1){\tilde{\delta }}_{\nu =2}^{(m+1)}\left(\mathrm{s}\right).$$

(74)

Explicitly, we have the following:

$$\begin{array}{ccc}& & r_1^{\left(i\right)}\left(\mathrm{s}\right)={\displaystyle \frac{sin\left(g\right)}{\pi {({\pi }^2+L_s^2)}^{1/2}}}={\displaystyle \frac{1}{{\pi }^2+L_s^2}},\hfill \\ & & {\tilde{\delta }}_1^{(m+1)}\left(\mathrm{s}\right)={\displaystyle \frac{1}{\pi {({\pi }^2+L_s^2)}^{(m+1)/2}}}\left\{{\overline{\delta }}_m^{(m+1)}\left(\mathrm{s}\right)sin\left((m+1)g\right)+{\widehat{\delta }}_m^{(m+1)}\left(\mathrm{s}\right)gcos\left((m+1)g\right)\right\},\hfill \end{array}$$

(75)

where ${\overline{\delta }}_m^{(m+1)}\left(\mathrm{s}\right)$ and ${\widehat{\delta }}_m^{(m+1)}\left(\mathrm{s}\right)$ can be obtained from the results in (30) with $\nu =2$. They are as follows:

$$\begin{array}{ccc}& & {\overline{\delta }}_1^{\left(2\right)}\left(\mathrm{s}\right)=-G{b}_1,{\widehat{\delta }}_1^{\left(2\right)}\left(\mathrm{s}\right)=b_1,\hfill \\ & & {\overline{\delta }}_2^{\left(3\right)}\left(\mathrm{s}\right)=b_2+b_1^2\left[G^2-g^2-G-1\right],{\widehat{\delta }}_2^{\left(3\right)}\left(\mathrm{s}\right)=b_1^2\left[1-2G\right].\hfill \end{array}$$

(76)

Using the results from (36) and (37) for $cos\left(ng\right)$ and $sin\left(ng\right)(n\le 4)$, we see that [67,68] (see also Section 6 in [39]) give more compact results for $r_1^{\left(i\right)}\left(\mathrm{s}\right)$. We believe that Equations (75) and (76) give very compact results for $r_1^{\left(i\right)}\left(\mathrm{s}\right)$.

Note that the results from (71) for ${\tilde{U}}_{\nu }^{\left(i\right)}\left(\mathrm{s}\right)$ are exactly the same as the results in Equation (28) in the form of trigonometric factions. However, the results from (71) should be very handy in the case of non-minimal versions of analytic couplings (see Refs. [58,59,61]).

9. $\mathit{H}\to \mathit{b}\overline{\mathit{b}}$ Decay

In Ref. [39], we use the polarized Bjorken sum rule [75,76] as an example of the application of the MA coupling $A_{\mathrm{MA}}\left(Q^2\right)$, which is a popular object of study in the framework of analytic QCD (see [69,70,73,74,77,78,79,80]). Here, we consider the decay of the Higgs boson into a bottom–antibottom pair, which is also a popular application of the MA coupling $U_{\mathrm{MA}}\left(Q^2\right)$ (see, e.g., [11] and reviews in Ref. [35]).

The Higgs boson decay into a bottom–antibottom pair can be expressed in QCD by means of the correlator, i.e.,

$$\Pi \left(Q^2\right)={\left(4\pi \right)}^{2}i\int dx{e}^{iqx}<0|T\left[J_b^S\left(x\right)J_b^S\left(0\right)\right]0>$$

(77)

of two quark scalar (S) currents in terms of the discontinuity of its imaginary part, i.e., $R_S\left(s\right)=Im\Pi (-s-i{ϵ}^{)}/\left(2\pi s\right)$, so that the width reads

$$\Gamma (H\to b\overline{b})={\displaystyle \frac{G_F}{4\sqrt{2}\pi }}{M}_H{m}_b^2\left(M_H^2\right)R_s(s=M_H^2).$$

(78)

Direct multi-loop calculations were performed in the Euclidean (space-like) domain for the corresponding Adler function $D_S$ (see Refs. [81,82,83,84,85]). Hence, we write ($D_s\to {\tilde{D}}_s$ and $R_s\to {\tilde{R}}_s$ because the additional factor $m_b^2$)

$$\tilde{D}\left(Q^2\right)=3m_b^2\left(Q^2\right)\left[1+\sum _{n\ge 1}{d}_n{a}_s^n\left(Q^2\right)\right],$$

(79)

where for $f=5$, the coefficients $d_n$ are

$$d_1=2.96,d_2=11.44,d_2=50.17,d_4=260.24,$$

(80)

Taking the imagined part, one has the following:

$${\tilde{R}}_s\left(s\right)=3m_b^2\left(s\right)\left[1+\sum _{n\ge 1}{r}_n{a}^n\left(s\right)\right],$$

(81)

and for $f=5$ [83,84,85,86] (The resummation of the $r_n$ parts $\sim {\beta }_0^n$ can be found in [81] (see also [86]). The results are quite similar to those obtained in the APT case (see [11,35,87]). Such a resummation was not considered in [40] and is, thus, beyond the scope of the present study.)

$$r_1=2.96,r_2=7.93,r_3=5.93,r_4=-61.84,$$

(82)

Here, ${\overline{m}}_b^2\left(Q^2\right)$ has the following form (see Appendix C):

$${\overline{m}}_b^2\left(Q^2\right)={\widehat{m}}_b^2{a}_s^d\left(Q^2\right)\left[1+\sum _{k=1}^{k=4}{\overline{e}}_k{a}_s^k\left(Q^2\right)\right],$$

(83)

where

$${\overline{e}}_k={\displaystyle \frac{{\tilde{e}}_k}{k{\left({\beta }_0\right)}^k}}$$

(84)

and ${\tilde{e}}_k$ are conducted in Equation (A32). For $f=5$, we have the following:

$${\overline{e}}_1=1.23,{\overline{e}}_2=1.20,{\overline{e}}_3=0.55,{\overline{e}}_4=0.54.$$

(85)

The normalization constant ${\widehat{m}}_b$ can be obtained as follows (see, e.g., [35]):

$${\widehat{m}}_b={\overline{m}}_b(Q^2=m_b^2)a_s^{-d/2}\left(m_b^2\right){\left[1+\sum _{k=1}^{k=4}{\overline{e}}_k{a}_s^k\left(Q^2\right)\right]}^{-1/2}=10.814{\mathrm{GeV}}^2,$$

(86)

since ${\overline{m}}_b(Q^2=m_b^2)=m_b=4.18$ GeV.

So, we have the following:

$${\tilde{R}}_s\left(s\right)={\tilde{R}}_s^{(m=5)}\left(s\right),{\tilde{R}}_s^{(m+1)}\left(s\right)=3{\widehat{m}}_b^2{a}_s^d\left(s\right)\left[1+\sum _{k=0}^m{\overline{r}}_k{a}_s^k\left(s\right))\right],$$

(87)

where

$${\overline{r}}_k=r_k+{\overline{e}}_k+\sum _{l=1}^{k-1}{r}_l{\overline{e}}_{k-l}.$$

(88)

For $f=5$, we have the following:

$${\overline{r}}_1=4.18,{\overline{r}}_2=12.76,{\overline{r}}_3=19.76,{\overline{r}}_4=-42.25.$$

(89)

We can express all results through derivatives ${\tilde{a}}_{d+k}$ (see Appendix A):

$${\tilde{R}}_s\left(s\right)={\tilde{R}}_s^{(m=5)}\left(s\right),{\tilde{R}}_s^{(m+1)}\left(s\right)=3{\widehat{m}}_b^2\left[{\tilde{a}}_d+\sum _{k=0}^m{\tilde{r}}_k{\tilde{a}}_{d+k}\right],$$

(90)

where

$${\tilde{r}}_k={\overline{r}}_k+{\tilde{k}}_k\left(d\right)+\sum _{l=1}^{k-1}{\overline{r}}_l{\tilde{k}}_{k-l}(d+l),$$

(91)

where ${\tilde{k}}_i\left(\nu \right)$ are given in Appendix A.

For $d=24/23$ and $f=5$, we have the following:

$${\tilde{r}}_1=4.17,{\tilde{r}}_2=9.86,{\tilde{r}}_3=1.29,{\tilde{r}}_4=-71.21.$$

(92)

Performing the same analysis for the Adler function, we have the following:

$${\tilde{D}}_s={\tilde{D}}_s^{(m=5)},{\tilde{D}}_s^{(m+1)}=3{\widehat{m}}_b^2{a}_s^d\left(Q^2\right)\left[1+\sum _{k=0}^m{\overline{d}}_k{a}_s^k\left(Q^2\right)\right],$$

(93)

where

$${\overline{d}}_k=r_k+{\overline{e}}_k+\sum _{l=1}^{k-1}{d}_l{\overline{e}}_{k-l}.$$

(94)

For $f=5$, we have the following:

$${\overline{d}}_1=4.18,{\overline{d}}_2=16.27,{\overline{d}}_3=68.30,{\overline{r}}_4=337.66.$$

(95)

We express all results through derivatives ${\tilde{a}}_{d+k}$:

$${\tilde{D}}_s={\tilde{D}}_s^{(m=5)},{\tilde{D}}_s^{(m+1)}=3{\widehat{m}}_b^2\left[{\tilde{a}}_d\left(Q^2\right)+\sum _{k=0}^m{\tilde{d}}_k{\tilde{a}}_{d+k}\left(Q^2\right)\right],$$

(96)

where

$${\tilde{d}}_k={\overline{d}}_k+{\tilde{k}}_k\left(d\right)+\sum _{l=1}^{k-1}{\overline{d}}_l{\tilde{k}}_{k-l}(d+l).$$

(97)

For $f=5$ and $d=24/23$, we have the following:

$${\tilde{d}}_1=4.17,{\tilde{d}}_2=13.37,{\tilde{d}}_3=43.90,{\tilde{d}}_4=178.18.$$

(98)

As discussed earlier in [11], in FAPT, we have the following representation for ${\tilde{R}}_s$:

$${\tilde{R}}_s\left(s\right)={\tilde{R}}_s^{(m=5)}\left(s\right),{\tilde{R}}_s^{(m+1)}\left(s\right)=3{\widehat{m}}_b^2\left[{\tilde{U}}_d^{(m+1)}\left(\mathrm{s}\right)+\sum _{k=0}^m{\tilde{d}}_k{\tilde{U}}_{d+k}^{(m+1)}\left(s\right)\right],$$

(99)

The results for ${\tilde{R}}_s^{(m+1)}\left(s\right)$ are shown in Figure 14. We see that the FAPT results (99) are lower than those (90) based on conventional PT. This is in full agreement with the arguments given in [35]. But the difference becomes less notable as the PT order increases. Indeed, for ${\mathrm{N}}^3$LO, the difference is very small, which proves the assumption about the possibility of using the ${\tilde{R}}_s^{(m+1)}\left(s\right)$ expression for ${\tilde{D}}_s^{(m+1)}\left(Q^2\right)$, with $A_{\mathrm{MA}}^{\left(i\right)}\left(Q^2\right)\to U_{\mathrm{MA}}^{\left(i\right)}\left(s\right)$, as done in Ref. [11].

Figure 14. The results for ${\tilde{R}}_s^{(m+1)}\left(s\right)$ with $m=2$ and 4 in the framework of the usual PT and FAPT.

The results for ${\Gamma }^{\left(m\right)}(H\to b\overline{b})$ in the ${\mathrm{N}}^m$LO approximation using ${\tilde{R}}_s^{(m+1)}\left(s\right)$ from Equations (87) and (90) are exactly the same and have the following form:

$$\begin{array}{ccc}& & {\Gamma }^{\left(0\right)}=1.76\mathrm{MeV},{\Gamma }^{\left(1\right)}=2.27\mathrm{MeV},{\Gamma }^{\left(2\right)}=2.37\mathrm{MeV},\hfill \\ & & {\Gamma }^{\left(3\right)}=2.38\mathrm{MeV},{\Gamma }^{\left(4\right)}=2.38\mathrm{MeV}.\hfill \end{array}$$

(100)

The corresponding results for ${\Gamma }^{\left(m\right)}(H\to b\overline{b})$ with ${\tilde{R}}_s^{(m+1)}\left(s\right)$ from Equation (99) are very similar to the ones in (100). They are as follows:

$$\begin{array}{ccc}& & {\Gamma }^{\left(0\right)}=1.74\mathrm{MeV},{\Gamma }^{\left(1\right)}=2.23\mathrm{MeV},{\Gamma }^{\left(2\right)}=2.34\mathrm{MeV},\hfill \\ & & {\Gamma }^{\left(3\right)}=2.37\mathrm{MeV},{\Gamma }^{\left(4\right)}=2.38\mathrm{MeV}.\hfill \end{array}$$

(101)

So, we see good agreement between the results obtained in FAPT and in the framework of the usual PT.

It can be clearly seen that the results of FAPT are also very close to the results [88] obtained in the framework of the now very popular principle of maximum conformality [89,90,91,92,93] (for a recent review, see [94,95]). Indeed, our results are within the band obtained by varying the renormalization scale.

The standard model expectation is [96]

$${\Gamma }_{H\to b\overline{b}}^{SM}(M_H=125.1\mathrm{GeV})=2.38\mathrm{MeV}.$$

(102)

The ratios of the measured events that yield to the standard model expectations are $1.01\pm 0.12(\mathrm{stat}.)+0.16-0.15(\mathrm{syst}.)$ [97] in the ATLAS collaboration and $1.04\pm 0.14(\mathrm{stat}.)\pm 0.14(\mathrm{syst}.)$ [98] in the SMC collaboration (see also [99]).

Thus, our results—obtained in both approaches, in the standard perturbation theory, and in analytical QCD–are in good agreement with the standard model expectations [96] and the experimental data [97,98].

10. Bjorken Sum Rule

The polarized BSR [100,101] (see also [102,103]) is defined as the difference between the proton and neutron-polarized SFs, integrated over the entire interval x. (The integrals ${\Gamma }_1^p\left(Q^2\right)$ and ${\Gamma }_1^n\left(Q^2\right)$ themselves were studied in [104] but such a study is beyond the scope of this review).

$${\Gamma }_1^{p-n}\left(Q^2\right)={\int }_0^{1}dx\left[g_1^p(x,Q^2)-g_1^n(x,Q^2)\right].$$

(103)

Theoretically, the quantity can be written in the OPE form (see Refs. [105,106]):

$${\Gamma }_1^{p-n}\left(Q^2\right)={\displaystyle \frac{g_A}{6}}\left(1-D_{\mathrm{BS}}\left(Q^2\right)\right)+\sum _{i=2}^{\infty }{\displaystyle \frac{{\mu }_{2i}\left(Q^2\right)}{Q^{2i-2}}},$$

(104)

where $g_A$ = 1.2762 ± 0.0005 [49] is the nucleon axial charge, $(1-D_{BS}\left(Q^2\right))$ is the leading-twist (or twist-two) contribution, and ${\mu }_{2i}/Q^{2i-2}(i\ge 1)$ are the higher-twist (HT) contributions. (Below, in our analysis; the so-called elastic contribution will always be excluded.)

Since we plan to consider very small $Q^2$ values here, the representation (104) of the HT involves an infinite number of terms. To avoid this, it is preferable to use the so-called “massive” twist-four representation, which includes part of the HT contributions given in (104) (see Refs. [107,108,109]). (Note that Ref. [109] also contains a more complicated form of the “massive” twist-four part. It was included in our previous analysis in [44], but will not be considered here.)

$${\Gamma }_1^{p-n}\left(Q^2\right)={\displaystyle \frac{g_A}{6}}\left(1-D_{\mathrm{BS}}\left(Q^2\right)\right)+{\displaystyle \frac{{\widehat{\mu }}_4{M}^2}{Q^2+M^2}},$$

(105)

where the values of ${\widehat{\mu }}_4$ and $M^2$ were fitted in Refs. [77,79] in the different analytic QCD models. For $Q^2>>M^2$, the “massive” twist-four representation can be expanded in powers of $M^2/Q^2$, and the obtained results will have the form shown on the right-hand side of (104).

In the case of MA QCD, from [79], one can see that in (105), we have the following:

$$M^2=0.439\pm 0.012\pm 0.463,{\widehat{\mu }}_{\mathrm{MA},4}=-0.173\pm 0.002\pm 0.666,$$

(106)

where the statistical (small) and systematic (large) uncertainties are presented.

Up to the k-th order in PT, the twist-two part can be expressed as follows:

$$D_{\mathrm{BS}}^{\left(1\right)}\left(Q^2\right)={\displaystyle \frac{4}{{\beta }_0}}{a}_s^{\left(1\right)},D_{\mathrm{BS}}^{(k\ge 2)}\left(Q^2\right)={\displaystyle \frac{4}{{\beta }_0}}{a}_s^{\left(k\right)}\left(1+\sum _{m=1}^{k-1}{d}_m{\left(a_s^{\left(k\right)}\right)}^m\right),$$

(107)

where $d_1$, $d_2$, and $d_3$ are known from the exact calculations (see, [110] and references therein). The exact $d_4$ value is not known, but it was estimated in Ref. [111].

Converting the coupling powers into its derivatives, we have the following:

$$D_{\mathrm{BS}}^{\left(1\right)}\left(Q^2\right)={\displaystyle \frac{4}{{\beta }_0}}{\tilde{a}}_1^{\left(1\right)},D_{\mathrm{BS}}^{(k\ge 2)}\left(Q^2\right)={\displaystyle \frac{4}{{\beta }_0}}\left({\tilde{a}}_1^{\left(k\right)}+\sum _{m=2}^k{\tilde{d}}_{m-1}{\tilde{a}}_m^{\left(k\right)}\right),$$

(108)

where (The resummation of the $d_n$ and ${\tilde{d}}_n$ parts with renormalon singularities can be found in [112,113,114]. The results based on conventional QCD and APT are quite similar to those obtained in [44,45]. However, such a resummation was not considered in [44,45] and is, thus, beyond the scope of the present study).

$$\begin{array}{ccc}& & {\tilde{d}}_1=d_1,{\tilde{d}}_2=d_2-b_1{d}_1,{\tilde{d}}_3=d_3-{\displaystyle \frac{5}{2}}{b}_1{d}_2-\left(b_2-{\displaystyle \frac{5}{2}}{b}_1^2\right)d_1,\hfill \\ & & {\tilde{d}}_4=d_4-{\displaystyle \frac{13}{3}}{b}_1{d}_3-\left(3b_2-{\displaystyle \frac{28}{3}}{b}_1^2\right)d_2-\left(b_3-{\displaystyle \frac{22}{3}}{b}_1{b}_2+{\displaystyle \frac{28}{3}}{b}_1^3\right)d_1\hfill \end{array}$$

(109)

and $b_i={\beta }_i/{\beta }_0^{i+1}$.

In MA QCD, the results from (105) are as follows (some analyses based on other approaches can be found in [112,113,114,115,116,117]):

$${\Gamma }_{\mathrm{MA},1}^{p-n}\left(Q^2\right)={\displaystyle \frac{g_A}{6}}\left(1-D_{\mathrm{MA},\mathrm{BS}}\left(Q^2\right)\right)+{\displaystyle \frac{{\widehat{\mu }}_{\mathrm{MA},4}{M}^2}{Q^2+M^2}},,$$

(110)

where the perturbative part $D_{\mathrm{BS},\mathrm{MA}}\left(Q^2\right)$ takes the same form, however, with analytic coupling ${\tilde{A}}_{\mathrm{MA},\nu }^{\left(k\right)}$ (the corresponding expressions are taken from [39])

$$D_{\mathrm{MA},\mathrm{BS}}^{\left(1\right)}\left(Q^2\right)={\displaystyle \frac{4}{{\beta }_0}}{A}_{\mathrm{MA}}^{\left(1\right)},D_{\mathrm{MA},\mathrm{BS}}^{k\ge 2}\left(Q^2\right)={\displaystyle \frac{4}{{\beta }_0}}\left(A_{\mathrm{MA}}^{\left(1\right)}+\sum _{m=2}^k{\tilde{d}}_{m-1}{\tilde{A}}_{\mathrm{MA},\nu =m}^{\left(k\right)}\right).$$

(111)

As already discussed in Section 2, the coefficients ${\beta }_i$ depend on the number f of active quarks. So, the coupling $a_s$ is f-dependent and the f-dependence can be taken into the corresponding QCD parameter $\Lambda$ (see Equation (11)). Since we will mainly consider the region of low $Q^2$, we will use the results for ${\Lambda }_i^{f=3}$, which we need to construct the analytic coupling for small $Q^2$ values.

For the k-th order of PT, we use the results from (11) for ${\Lambda }_{k-1}^{f=3}$$(k=1,2,3,4)$ taken from the recent Ref. [55], which corresponds to the middle value of the world average ${\alpha }_s\left(M_z^2\right)=0.1179\pm 0.0009$ [49]. We also use ${\Lambda }_4={\Lambda }_3$, since in the highest orders, ${\Lambda }_i$ values become very similar. Moreover, since the results for ${\alpha }_s\left(M_z^2\right)$ and ${\Lambda }_{k-1}^{f=3}$ are taken from the range of $Q^2$ values, where the difference between the analytic and usual couplings is small, we will use these values (11) in the analytic QCD case.

For the case involving three active quark flavors ($f=3$), which is accepted in this paper, we have the following:

$$\begin{array}{ccc}& & d_1=1.59,d_2=3.99,d_3=15.42d_4=63.76,\hfill \\ & & {\tilde{d}}_1=1.59,{\tilde{d}}_2=2.73,{\tilde{d}}_3=8.61,{\tilde{d}}_4=21.52,\hfill \end{array}$$

(112)

i.e., the coefficients in the series of derivatives are slightly smaller.

10.1. Results

The fitting results of experimental data (see [118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142]) obtained only with statistical uncertainties are presented in Table 1 and shown in Figure 15 and Figure 16. For the fits, we use $Q^2$-independent $M^2$ and ${\widehat{\mu }}_4$ and the two-twist part shown in Equations (108) and (111) for regular PT and APT, respectively.

Table 1. The values of the fit parameters in (110).

	${\mathit{M}}^2$ [${\mathrm{GeV}}^2$] for ${\mathit{Q}}^2\le 5$ ${\mathrm{GeV}}^2$ (for ${\mathit{Q}}^{\mathbf{2}}\le \mathbf{0}.\mathbf{6}$ GeV2)	${\widehat{\mathit{\mu }}}_{\mathbf{MA},4}$ for ${\mathit{Q}}^2\le 5$ ${\mathrm{GeV}}^2$ (for ${\mathit{Q}}^{\mathbf{2}}\le \mathbf{0.6}$ GeV2)	${\mathit{\chi }}^2/\left(\mathbf{d}.\mathbf{o}.\mathbf{f}.\right)$ for ${\mathit{Q}}^2\le 5$ ${\mathrm{GeV}}^2$ (for ${\mathit{Q}}^{\mathbf{2}}\le \mathbf{0.6}{\mathbf{GeV}}^{\mathbf{2}}$)
LO	0.472 ± 0.035	−0.212 ± 0.006	0.667
	(1.631 ± 0.301)	(−0.166 ± 0.001)	(0.789)
NLO	0.414 ± 0.035	−0.206 ± 0.008	0.728
	(1.545 ± 0.287)	(−0.155 ± 0.001)	(0.757)
N2LO	0.397 ± 0.034	−0.208± 0.008	0.746
	(1.417 ± 0.241)	(−0.156 ± 0.002)	(0.728)
N3LO	0.394 ± 0.034	−0.209 ± 0.008	0.754
	(1.429 ± 0.248)	(−0.157 ± 0.002)	(0.747)
N4LO	0.397 ± 0.035	−0.208 ± 0.007	0.753
	(1.462 ± 0.259)	(−0.157 ± 0.001)	(0.754)

Figure 15. The results for ${\Gamma }_1^{p-n}\left(Q^2\right)$ in the first four orders of PT.

Figure 16. The results for ${\Gamma }_1^{p-n}\left(Q^2\right)$ in the first four orders of APT.

As can be seen in Figure 15, with the exception of LO, the results obtained using conventional coupling are very poor. Moreover, the discrepancy in this case increases with the order of PT (see also [69,70,71,72,77,78,79,80] for similar analyses). The LO results describe experimental points relatively well since the value of ${\Lambda }_{\mathrm{LO}}$ is quite small compared to other ${\Lambda }_i$, and disagreement with the data begins at lower values of $Q^2$ (see Figure 4 below). Thus, using the “massive” twist-four form (105) does not improve these results, since with $Q^2\to {\Lambda }_i^2$, conventional couplings become singular, which leads to large and negative results for the twist-two part (107). So, as the PT order increases, ordinary couplings become singular for ever larger $Q^2$ values, while the BSR tends to negative values at increasingly larger $Q^2$ values.

In contrast, our results obtained for different APT orders are practically equivalent. The corresponding curves become indistinguishable when $Q^2$ approaches 0 and is slightly different everywhere else. As can be seen in Figure 16, the fit quality is pretty high, which is demonstrated by the values of the corresponding ${\chi }^2/\left(\mathrm{d}.\mathrm{o}.\mathrm{f}.\right)$ (see Table 1).

10.2. Low $Q^2$ values

The full picture, however, is more complex than shown in Figure 16. The APT fitting curves become negative (see Figure 17) when we move to very low values of $Q^2$: $Q^2<0.1 {\mathrm{GeV}}^2$. So, the high quality of the fits shown in Table 1 results from their good agreement with experimental data at $Q^2>0.2 {\mathrm{GeV}}^2$. The picture improves significantly when we compare our results with experimental data for $Q^2<0.6 {\mathrm{GeV}}^2$ (see Figure 18 and Ref. [44]).

Figure 17. The same as in Figure 16 but for $Q^2<0.6 {\mathrm{GeV}}^2$.

Figure 18. The results for ${\Gamma }_1^{p-n}\left(Q^2\right)$ using the first four orders of APT, from fits of experimental data with $Q^2<0.6 {\mathrm{GeV}}^2$.

Figure 18 also shows contributions from conventional PT at the first two orders: the LO and NLO predictions, which have no resemblance to the experimental data. As mentioned above, higher orders lead to even worse agreement, and are, therefore, not shown. The purple curve emphasizes the key role of the twist-four contribution (see also [72,143,144]) and the discussions therein). Excluding this contribution, the value of ${\Gamma }_1^{p-n}\left(Q^2\right)$ is about 0.16, which is very far from the experimental data.

At $Q^2\le 0.3 {\mathrm{GeV}}^2$, we also see good agreement with the phenomenological models: LFHQCD [145] and the correct IR limit of the Burkert–Ioffe model [146,147]. For larger values of $Q^2$, our results are lower than the results of the phenomenological models, and for $Q^2\ge 0.5 {\mathrm{GeV}}^2$—below the experimental data.

Nevertheless, even in this case, where very good agreement with experimental data with $Q^2<0.6 {\mathrm{GeV}}^2$ is demonstrated, our results for ${\Gamma }_{\mathrm{MA},1}^{p-n}\left(Q^2\right)$ take negative unphysical values when $Q^2<0.02 {\mathrm{GeV}}^2$. The reason for this phenomenon can be seen by considering photoproduction within APT, which is the topic of the next subsection.

10.3. Photoproduction

To understand the problem demonstrated above, ${\Gamma }_{\mathrm{MA},1}^{p-n}(Q^2\to 0)<0$, we consider the photoproduction case. In the k-th order of MA QCD,

$$A_{\mathrm{MA}}^{\left(k\right)}(Q^2=0)\equiv {\tilde{A}}_{\mathrm{MA},m=1}^{\left(k\right)}(Q^2=0)=1,{\tilde{A}}_{\mathrm{MA},m}^{\left(k\right)}=0,\mathrm{when}m>1$$

(113)

and, so, we have

$$D_{\mathrm{MA},\mathrm{BS}}(Q^2=0)={\displaystyle \frac{4}{{\beta }_0}}\mathrm{and},\mathrm{hence},{\Gamma }_{\mathrm{MA},1}^{p-n}(Q^2=0)={\displaystyle \frac{g_A}{6}}\left(1-{\displaystyle \frac{4}{{\beta }_0}}\right)+{\widehat{\mu }}_{\mathrm{MA},4}.$$

(114)

The finiteness of the cross-section in the real photon limit leads to [107]

$${\Gamma }_{\mathrm{MA},1}^{p-n}(Q^2=0)=0\mathrm{and},\mathrm{thus},{\widehat{\mu }}_{\mathrm{MA},4}^{php}=-{\displaystyle \frac{g_A}{6}}\left(1-{\displaystyle \frac{4}{{\beta }_0}}\right).$$

(115)

For $f=3$, we have the following:

$${\widehat{\mu }}_{\mathrm{MA},4}^{php}=-0.118\mathrm{and},\mathrm{hence},|{\widehat{\mu }}_{\mathrm{MA},4}^{php}|<|{\widehat{\mu }}_{\mathrm{MA},4}|,$$

(116)

shown in (106) and in Table 1.

So, as can be seen from Table 1, the finiteness of the cross-section in the real photon limit is violated in our approaches. (Note that the results for ${\widehat{\mu }}_{\mathrm{MA},4}$ were obtained by only taking into statistical uncertainties. When adding systematic uncertainties, the results for ${\widehat{\mu }}_{\mathrm{MA},4}^{php}$ and ${\widehat{\mu }}_{\mathrm{MA},4}$ are completely consistent with each other, but the predictive power of such an analysis is small). This violation leads to negative values of ${\Gamma }_{\mathrm{MA},1}^{p-n}(Q^2\to 0)$. Note that this violation is less for experimental datasets with $Q^2\le 0.6 {\mathrm{GeV}}^2$, where the obtained values for $|{\widehat{\mu }}_{\mathrm{MA},4}|$ are essentially less than those obtained in the case of experimental data with $Q^2\le 5 {\mathrm{GeV}}^2$. Smaller values of $|{\widehat{\mu }}_{\mathrm{MA},4}|$ lead to negative values of ${\Gamma }_{\mathrm{MA},1}^{p-n}(Q^2\to 0)$, when $Q^2\le 0.02 {\mathrm{GeV}}^2$ (see Figure 4).

10.4. Gerasimov–Drell–Hearn and Burkhardt–Cottingham Sum Rules

Now, we plan to improve this analysis by involving the results from (110) at low $Q^2$ values and taking into account the “massive” twist-six term, similar to the twist-four shown in Equation (105).

Moreover, we take into account the GDH and BC sum rules, which lead to (see [104,107,108,109,148,149])

$${\displaystyle \frac{d}{d{Q}^2}}{\Gamma }_{\mathrm{MA},1}^{p-n}(Q^2=0)=G,G={\displaystyle \frac{{\mu }_n^2-{({\mu }_p-1)}^2}{8M_p^2}}=0.0631,$$

(117)

where ${\mu }_n=-1.91$ and ${\mu }_p=2.79$ are proton and neutron magnetic moments, respectively, and $M_p$ = 0.938 GeV is a nucleon mass. Note that the value of G is small.

In agreement with the definition (12), we have the following:

$$Q^2{\displaystyle \frac{d}{d{Q}^2}}{\tilde{A}}_n\left(Q^2\right)\sim {\tilde{A}}_{n+1}\left(Q^2\right).$$

(118)

Then, as $Q^2\to 0$, for any n value, we obtain the following:

$$Q^2{\displaystyle \frac{d}{d{Q}^2}}{\tilde{A}}_n\left(Q^2\right)\to 0,$$

(119)

but very slowly, so that the derivative behaves as follows:

$${\displaystyle \frac{d}{d{Q}^2}}{\tilde{A}}_n(Q^2\to 0)\to \infty .$$

(120)

Thus, after applying the derivative $d/d{Q}^2$ for ${\Gamma }_{\mathrm{MA},1}^{p-n}\left(Q^2\right)$, every term in $D_{\mathrm{MA},\mathrm{BS}}\left(Q^2\right)$ becomes divergent as $Q^2\to 0$. To produce finiteness at $Q^2\to 0$ for the l.h.s. of (117), we can assume the relation between twist-two and twist-four terms, which leads to the appearance of a new contribution:

$$-{\displaystyle \frac{g_A}{6}}{D}_{\mathrm{MA},\mathrm{BS}}\left(Q^2\right)+{\displaystyle \frac{{\widehat{\mu }}_{\mathrm{MA},4}{M}^2}{Q^2+M^2}}{D}_{\mathrm{MA},\mathrm{BS}}\left(Q^2\right),$$

(121)

which can be done to remain regular as $Q^2\to 0$.

The form (121) suggests the following idea about a modification of ${\Gamma }_{\mathrm{MA},1}^{p-n}\left(Q^2\right)$ in (110):

$${\Gamma }_{\mathrm{MA},1}^{p-n}\left(Q^2\right)={\displaystyle \frac{g_A}{6}}\left(1-D_{\mathrm{MA},\mathrm{BS}}\left(Q^2\right)·{\displaystyle \frac{Q^2}{Q^2+M^2}}\right)+{\displaystyle \frac{{\widehat{\mu }}_{\mathrm{MA},4}{M}^2}{Q^2+M^2}}+{\displaystyle \frac{{\widehat{\mu }}_{\mathrm{MA},6}{M}^4}{{(Q^2+M^2)}^2}},$$

(122)

where we add the “massive” twist-six term and introduce different masses in both higher-twist terms and into the modification factor $Q^2/(Q^2+M^2)$.

The finiteness of the cross-section in the real photon limit now leads to the following [107,108]:

$${\Gamma }_{\mathrm{MA},1}^{p-n}(Q^2=0)=0={\displaystyle \frac{g_A}{6}}+{\widehat{\mu }}_{\mathrm{MA},4}+{\widehat{\mu }}_{\mathrm{MA},6}$$

(123)

and, thus, we have the following:

$${\widehat{\mu }}_{\mathrm{MA},4}+{\widehat{\mu }}_{\mathrm{MA},6}=-{\displaystyle \frac{g_A}{6}}\approx -0.21205$$

(124)

From Equation (122) and condition (117), we obtain the following:

$$-{\displaystyle \frac{g_A}{6}}·D_{\mathrm{MA},\mathrm{BS}}(Q^2=0)-{\widehat{\mu }}_{\mathrm{MA},4}-2{\widehat{\mu }}_{\mathrm{MA},6}=G{M}^2,$$

(125)

where $D_{\mathrm{MA},\mathrm{BS}}(Q^2=0)=4/{\beta }_0$ (see Equation (114)).

Using $f=3$ (i.e., ${\beta }_0=9$), we have the following:

$${\widehat{\mu }}_{\mathrm{MA},4}+2{\widehat{\mu }}_{\mathrm{MA},6}=-G{M}^2-{\displaystyle \frac{2g_A}{3{\beta }_0}}=-G{M}^2-{\displaystyle \frac{2g_A}{27}}\approx -G{M}^2-0.0945.$$

(126)

Using the results from (123) and (126) together, we have the following:

$$\begin{array}{c}\hfill {\widehat{\mu }}_{\mathrm{MA},6}=-G{M}^2+{\displaystyle \frac{5g_A}{54}}=-G{M}^2+0.1182,\\ \hfill {\widehat{\mu }}_{\mathrm{MA},4}=-{\displaystyle \frac{g_A}{6}}-{\widehat{\mu }}_{\mathrm{MA},6}=G{M}^2-{\displaystyle \frac{7g_{A/V}}{27}}=G{M}^2-0.3309.\end{array}$$

(127)

Since the value of G is small, so ${\widehat{\mu }}_{\mathrm{MA},4}<0$ and ${\widehat{\mu }}_{\mathrm{MA},6}\approx -0.36{\widehat{\mu }}_{\mathrm{MA},4}>0$.

The fitting results of theoretical predictions based on Equation (122), with ${\widehat{\mu }}_{\mathrm{MA},4}$ and ${\widehat{\mu }}_{\mathrm{MA},6}$, as outlined in (127), are presented in Table 2 and Figure 19 and Figure 20.

Table 2. The values of the fit parameters.

	${\mathit{M}}^2$ [${\mathrm{GeV}}^2$] for ${\mathit{Q}}^2\le 5$ ${\mathrm{GeV}}^2$ (for ${\mathit{Q}}^{\mathbf{2}}\le \mathbf{0.6}$ GeV2)	${\mathit{\chi }}^2/\left(\mathbf{d}.\mathbf{o}.\mathbf{f}.\right)$ for ${\mathit{Q}}^2\le 5$ ${\mathrm{GeV}}^2$ (for ${\mathit{Q}}^{\mathbf{2}}\le \mathbf{0.6}{\mathbf{GeV}}^{\mathbf{2}}$)
LO	0.383 ± 0.014 (0.576 ± 0.046)	0.572 (0.575)
NLO	0.394 ± 0.013 (0.464 ± 0.039)	0.586 (0.590)
${\mathrm{N}}^2$LO	0.328 ± 0.014 (0.459 ± 0.038)	0.617 (0.584)
${\mathrm{N}}^3$LO	0.330 ± 0.014 (0.464 ± 0.039)	0.629 (0.582)
${\mathrm{N}}^4$LO	0.331 ± 0.013 (0.465 ± 0.039)	0.625 (0.584)

Figure 19. The results for ${\Gamma }_1^{p-n}\left(Q^2\right)$ (122) in the first four orders of APT.

Figure 20. As in Figure 19 but for $Q^2<0.6 {\mathrm{GeV}}^2$.

As one can see in Table 2, the obtained results for $M^2$ are different if we take the full dataset and the limited one with $Q^2<0.6 {\mathrm{GeV}}^2$. However, the difference is significantly less than in Table 1. Moreover, the results obtained in the fits using the full dataset and shown in Table 1 and Table 2 are quite similar.

We also see some similarities between the results shown in Figure 16 and Figure 19. The difference appears only at small $Q^2$ values, as can be seen in Figure 17 and Figure 20.

Figure 20 also shows that the results of fitting the full set of experimental data are in better agreement with the data at $Q^2\ge 0.55 {\mathrm{GeV}}^2$, as expected, since these data are involved in the analyses of the full set of experimental data.

The results shown in Table 1 and Table 2 are practically unchanged when heavy quark contributions [150] are taken into consideration (see [151,152,153]).

11. Conclusions

In this paper, we presented an overview of fractional analytic QCD and its application to the Higgs boson decay into a bottom–antibottom pair and for the polarized Bjorken sum rule.

We considered $1/L$-expansions of $\nu$-derivatives of the strong coupling $a_s$ expressed as combinations of operators ${\widehat{R}}_m$ (14) applied to the LO coupling $a_s^{\left(1\right)}$. Applying the same operators to the $\nu$-derivatives of the LO MA coupling $A_{\mathrm{MA}}^{\left(1\right)}$, we obtained four different representations for the $\nu$-derivatives of the MA couplings, i.e., ${\tilde{A}}_{\mathrm{MA},\nu }^{\left(i\right)}$, in each i-order of perturbation theory: one form contains a combination of polylogarithms; another contains an expansion of the generalized Euler $\zeta$-function; the third is based on dispersion integrals containing the LO spectral function; and the fourth representation is based on the dispersion integral containing the i-order spectral function. All results are presented up to the fifth order of the perturbation theory, where the corresponding coefficients of the QCD $\beta$-function are well known (see [2,3]).

The high-order corrections are negligible in the $Q^2\to 0$ and $Q^2\to \infty$ asymptotics and are nonzero in the vicinity of the point $Q^2={\Lambda }^2$. Thus, in fact, they are really only small corrections to the LO MA coupling $A_{\mathrm{MA},\nu }^{\left(1\right)}\left(Q^2\right)$. This proves the possibility of expansions of high-order couplings $A_{\mathrm{MA},\nu }^{\left(i\right)}\left(Q^2\right)$ via the LO couplings $A_{\mathrm{MA},\nu }^{\left(1\right)}\left(Q^2\right)$, as done in Ref. [12], as well as the possibility of various approximations used in [56,69,70,71,72,73,74].

As can be seen, all our results (up to the fifth order of perturbation theory) maintain a compact form and do not contain complicated special functions, such as the Lambert W-function [154,155,156], which already appears at the two-loop order as an exact solution to the usual coupling, and was used to evaluate MA couplings in [157,158].

Applying the same operators to the $\nu$-derivatives of the LO MA coupling $U_{\mathrm{MA}}^{\left(1\right)}$, we obtained two different representations (see Equations (29) and (71)) for the $\nu$-derivatives of the MA couplings (i.e., ${\tilde{U}}_{\mathrm{MA},\nu }^{\left(i\right)}$ introduced for time-like processes) in each i-order of perturbation theory: one form contains a combination of trigonometric functions, and the other is based on dispersion integrals containing the i-order spectral function. All results are presented up to the fifth order of perturbation theory.

As in the case of ${\tilde{A}}_{\mathrm{MA},\nu }^{\left(i\right)}$ [39] applied in the Euclidean space, high-order corrections for ${\tilde{U}}_{\mathrm{MA},\nu }^{\left(i\right)}$ are negligible in the $s\to 0$ and $s\to \infty$ limits and are nonzero in the vicinity of the point $s={\Lambda }^2$. Thus, there are only small corrections to the LO MA coupling $U_{\mathrm{MA},\nu }^{\left(1\right)}\left(s\right)$. In particular, this proves the possibility of expansions of high-order couplings $U_{\mathrm{MA},\nu }^{\left(i\right)}\left(s\right)$ via the LO couplings $U_{\mathrm{MA},\nu }^{\left(1\right)}\left(s\right)$, as done in Ref. [12].

As an example, we examined the Higgs boson decay into a $b\overline{b}$ pair and the obtained results are in good agreement with the standard model expectations [96] and with the experimental data [97,98]. Moreover, our results are also in good agreement with studies based on the principle maximum conformality [89,90,91,92,93].

As a second application, we considered the Bjorken sum rule ${\Gamma }_1^{p-n}\left(Q^2\right)$ in the framework of MA and perturbative QCD, and obtained results similar to those obtained in previous studies [44,69,70,72,77,79] for the first four orders of PT. The results based on conventional PT do not agree with the experimental data. For some $Q^2$ values, the PT results were negative, since the high-order corrections were large and entered the twist-two term with a minus sign. APT in the minimal version led to good agreement with experimental data when we used the “massive” version (110) for the twist-four contributions.

Examining low $Q^2$ behavior, we found that there was a disagreement between the results obtained in the fits and application of MA QCD to photoproduction. The fit results, when extended to low $Q^2$ values, led to negative values for the Bjorken sum rule ${\Gamma }_{\mathrm{MA},1}^{p-n}\left(Q^2\right)$: ${\Gamma }_{\mathrm{MA},1}^{p-n}(Q^2\to 0)<0$. This was contrary to the finiteness of the cross-section in the real photon limit leading to ${\Gamma }_{\mathrm{MA},1}^{p-n}(Q^2\to 0)=0$. Note that fits of experimental data at low $Q^2$ values (we used $Q^2<0.6 {\mathrm{GeV}}^2$) led to less magnitudes of negative values for ${\widehat{\mu }}_{\mathrm{MA},4}$ (see Table 1 and Table 2).

To solve the problem, we considered low $Q^2$ modifications of the OPE formula for ${\Gamma }_{\mathrm{MA},1}^{p-n}\left(Q^2\right)$. Carefully considering one of them, Equation (122), we find good agreement with full sets of experimental data for the Bjorken sum rule ${\Gamma }_{\mathrm{MA},1}^{p-n}\left(Q^2\right)$ and with its $Q^2\to 0$ limit, i.e., with photoproduction. We also see good agreement with phenomenological models [104,146,147,148,149], especially with LFHQCD [145].