Bjorken sum rule with analytic coupling

The polarized Bjorken sum rule (BSR) $\Gamma^{p-n}_{1}(Q^{2})$ Bjorken:1966jh , i.e. the difference in the first (Mellin) moments of spin-dependent structure functions (SFs) of proton and neutron, is a very important space-like QCD observable Deur:2018roz ; Kuhn:2008sy . Its isovector nature simplifies the theoretical description within the framework of perturbative QCD (pQCD) in comparison with the corresponding SF integrals for each nucleon. Experimental results for this quantity obtained in polarized deep inelastic scattering (DIS) are currently available in a ruther wide range of spacelike squared momenta $Q^{2}$: 0.021 GeV${}^{2}\leq Q^{2}<$5 GeV² Deur:2021klh ; E143:1998hbs ; SpinMuon:1993gcv ; COMPASS:2005xxc ; HERMES:1997hjr ; Deur:2004ti ; ResonanceSpinStructure:2008ceg . In particular, the most recent experimental results Deur:2021klh with significantly reduced statistical uncertainty make BSR an attractive value for testing various generalizations of pQCD at low $Q^{2}$ values: $Q^{2}\leq 1$GeV².

Theoretically, pQCD (with the operator product extension (OPE)) in the $\overline{MS}$ scheme, was a common approach to describing such quantities. This approach, however, has a theoretical disadvantage, which is that the strong coupling constant (couplant) $\alpha_{s}(Q^{2})$ has Landau singularities for small values of $Q^{2}$: $Q^{2}\leq 0.1$GeV², which makes it inconvenient to estimate spacelike observable quantities, such as BSR, at small values of $Q^{2}$. In recent years, the extension of the QCD couplant without Landau singularity for low $Q^{2}$ called (fractional) analytical perturbation theory [(F)APT)] ShS ; BMS1 ; Bakulev:2006ex (or the minimal analytic (MA) theory, as discussed in Cvetic:2008bn ), was applied to compare theoretical OPE expression and experimental BSR data Pasechnik:2008th ; Khandramai:2011zd ; Ayala:2017uzx ; Ayala:2018ulm ; Gabdrakhmanov:2023rjt (see also other recent BSR studies in Kotlorz:2018bxp ; Ayala:2023wpy ). Following Cvetic:2006mk , we introduce here the derivatives (in the $k$-order of perturbation theory (PT))

which play a key role for the construction of analytic QCD (but still have Landau pole). Hereafter $\beta_{0}$ is the first coefficient of the QCD $\beta$-function:

where $\beta_{i}$ are known up to $k=4$ Baikov:2008jh .

The series of derivatives $\tilde{a}_{n}(Q^{2})$ can be used as an analogue of $a_{s}$-powers series, as it was numerically tested in Kotikov:2022swl ). Although each derivative reduces the $a_{s}$ power, on the other hand it produces an additional $\beta$-function and, consequently, additional $a_{s}^{2}$ factor. According to the definition (1), in LO the expressions for $\tilde{a}_{n}(Q^{2})$ and $a_{s}^{n}$ exactly match. Beyond LO, there is one-to-one correspondence between $\tilde{a}_{n}(Q^{2})$ and $a_{s}^{n}$, established in Cvetic:2006mk ; Cvetic:2010di and extended to the fractional case in Ref. GCAK .

In this paper, we apply the inverse logarithmic expansion of the MA couplant, recently obtained in Kotikov:2022sos ; Kotikov:2023meh for any PT order (for a brief introduction, see Kotikov:2022vnx ). This approach is very convenient, since for LO the MA couplants have simple representations (see BMS1 ) and beyond LO the MA couplants are very close to LO ones, especially for $Q^{2}\to\infty$ and $Q^{2}\to 0$, where the differences between MA couplants of different PT orders become insignificant. Moreover, for $Q^{2}\to\infty$ and $Q^{2}\to 0$ the (fractional) derivatives of the MA couplants with $n\geq 2$ tend to zero, and therefore only the first term in perturbative expansions makes a valuable contribution. Along with that, the new modification of BSR allows us to make the derivative of its PT term finite in the IR limit and be in agreement with Gerasimov-Drell-Hearn and Burkhardt-Cottingham sum rules.

The polarized BSR is defined as the difference between the proton and neutron polarized SFs, integrated over the entire interval $x$

Theoretically, the quantity can be written in the OPE form (see Ref. Shuryak:1981pi ; Balitsky:1989jb )

where $g_{A}$=1.2762 $\pm$ 0.0005 PDG20 is the nucleon axial charge, $(1-D_{BS}(Q^{2}))$ is the leading-twist (or twist-two) contribution, and $\mu_{2i}/Q^{2i-2}$ $(i\geq 1)$ are the higher-twist (HT) contributions.¹¹1Below, in our analysis, the so-called elastic contribution will always be excluded.

Since we plan to consider in particular very small $Q^{2}$ values here, the representation (4) of the HT a number of infinite terms. To avoid that, it is preferable to use the so-called ”massive” twist-four representation, which includes a part of the HT contributions of (4) (see Refs. Teryaev:2013qba ; Gabdrakhmanov:2017dvg ): ²²2Note that Ref. Gabdrakhmanov:2017dvg also contains a more complicated form for the ”massive” twist-four part. It was included in our previous analysis in Gabdrakhmanov:2023rjt , but will not be considered here.

where the values of $\hat{\mu}_{4}$ and $M^{2}$ have been fitted in Refs. Ayala:2017uzx ; Ayala:2018ulm in the different analytic QCD models.

In the case of MA QCD, from Ayala:2018ulm one can see that in (5)

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

Up to the $k$-th PT order, the twist-two part has the form

where $d_{1}$, $d_{2}$ and $d_{3}$ are known from exact calculations (see, for example, Chen:2006tw ). The exact $d_{4}$ value is not known, but it was estimated in Ref. Ayala:2022mgz .

Converting the couplant powers into its derivatives, we have

where

and $b_{i}=\beta_{i}/\beta_{0}^{i+1}$.

For the case of 3 active quark flavors ($f=3$), which is accepted in this paper, we have ³³3 The coefficients $\beta_{i}$ $(i\geq 0)$ of the QCD $\beta$-function (2) and, consequently, the couplant $\alpha_{s}(Q^{2})$ itself depend on the number $f$ of active quark flavors, and each new quark enters/leaves the game at a certain threshold $Q^{2}_{f}$ according to Chetyrkin:2005ia . The corresponding QCD parameters $\Lambda^{(f)}$ in NⁱLO of PT can be found in Ref. Chen:2021tjz .

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

In MA QCD, the results (5) become as follows

where the perturbative part $D_{\rm{BS,MA}}(Q^{2})$ takes the same form, however, with analytic couplant $\tilde{A}^{(k)}_{\rm MA,\nu}$ (the corresponding expressions are taken from Kotikov:2022sos )

The fitting results of experimental data obtained only with statistical uncertainties are presented in Table 1 and shown in Figs. 1 and 2. For the fits we use $Q^{2}$-independent $M^{2}$ and $\hat{\mu}_{4}$ and the two-twist part shown in Eqs. (8), (12) for regular PT and APT, respectively.

As it can be seen in Fig. 1, with the exception of LO, the results obtained using conventional couplant are very poor. Moreover, the discrepancy in this case increases with the order of PT (see also Pasechnik:2008th ; Khandramai:2011zd ; Ayala:2017uzx ; Ayala:2018ulm for similar analyses). The LO results describe experimental points relatively well, since the value of $\Lambda_{\rm LO}$ is quite small compared to other $\Lambda_{i}$, and disagreement with the data begins at lower values of $Q^{2}$ (see Fig. 4 below). Thus, using the “massive” twist-four form (5) does not improve these results, since with $Q^{2}\to\Lambda_{i}^{2}$ conventional couplants become singular, which leads to large and negative results for the twist-two part (7). So, as the PT order increases, ordinary couplants become singular for ever larger $Q^{2}$ values, while BSR tends to negative values for ever 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 slightly different everywhere else. As can be seen in Fig. 2, the fit quality is pretty high, which is demonstrated by the values of the corresponding $\chi^{2}/({\rm d.o.f.})$ (see Table 1).

We have considered the Bjorken sum rule $\Gamma_{1}^{p-n}(Q^{2})$ in the framework of MA and perturbative QCD and obtained results similar to those obtained in previous studies Pasechnik:2008th ; Khandramai:2011zd ; Ayala:2017uzx ; Ayala:2018ulm ; Gabdrakhmanov:2023rjt for the first 4 orders of PT. The results based on the conventional PT do not agree with the experimental data. For some $Q^{2}$ values, the PT results become negative, since the high-order corrections are large and enter the twist-two term with a minus sign. APT in the minimal version leads to a good agreement with experimental data when we used the “massive” version (11) for the twist-four contributions.

Examining low $Q^{2}$ behaviour, we found that there is a disagreement between the results obtainded in the fits and application of MA QCD to photoproduction. The results of fits extented to low $Q^{2}$ lead to the negative values for Bjorken sum rule $\Gamma_{\rm{MA},1}^{p-n}(Q^{2})$: $\Gamma_{\rm{MA},1}^{p-n}(Q^{2}\to 0)<0$ that contrary to the finitness of cross-section in the real photon limit, which leads to $\Gamma_{\rm{MA},1}^{p-n}(Q^{2}\to 0)=0$. Note that fits of experimental data at low $Q^{2}$ values (we used $Q^{2}<$ 0.6 GeV²) lead to less magnitudes of negative values for $\Gamma_{\rm{MA},1}^{p-n}(Q^{2})$.

To solve the problem we considered low $Q^{2}$ modifications of OPE formula for $\Gamma_{\rm{MA},1}^{p-n}(Q^{2})$. Considering carefully one of them, Eq. (22), we find good agreemnet with full sets of experimental data for Bjorken sum rule $\Gamma_{\rm{MA},1}^{p-n}(Q^{2})$ and also with its $Q^{2}\to 0$ limit, i.e. with photoproduction. We see also good agreement with phenomenological modeles, especially with LFHQCD Brodsky:2014yha .

As the next step in our research, we plan to add to our analysis the heavy-quark contibution to $\Gamma_{\rm{MA},1}^{p-n}(Q^{2})$. It was calculated in a closed form in Ref. Blumlein:2016xcy . It is suppressed by the factor $a_{s}^{2}$, but contains a contribution of $\sim\ln(1/Q^{2})$ at low $Q^{2}$ values and should be important there.

Acknowledgments  Authors are grateful to Alexandre P. Deur for information about new experimental data in Ref. Deur:2021klh and discussions. Authors thank Andrei Kataev and Nikolai Nikolaev for careful discussions. This work was supported in part by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”.