Bjorken Sum Rule with Analytic Coupling at Low Q² Values

The experimental data obtained for the polarized Bjorken sum rule $\Gamma _{1}^{{p - n}}({{Q}^{2}})$ for small values of Q² are approximated by the predictions obtained in the framework of analytic QCD up to the fifth order perturbation theory, whose coupling constant does not contain the Landau pole. We found an excellent agreement between the experimental data and the predictions of analytic QCD, as well as a strong difference between these data and the results obtained in the framework of perturbative QCD.

INTRODUCTION

Polarized Bjorken sum rule (BSR) $\Gamma _{1}^{{p - n}}({{Q}^{2}})$ [1, 2], i.e., the difference between the first moments of the spin-dependent structure functions (SFs) of a proton and neutron, is a very important space-like QCD observable [3, 4]. Its isovector nature facilitates its theoretical description in perturbative QCD (pQCD) in terms of the operator product expansion (OPE), compared to the corresponding SF integrals for each nucleon. Experimental results for this quantity obtained in polarized deep inelastic scattering (DIS) are currently available in a wide range of the spacelike squared momenta Q²: 0.021 GeV² $\leqslant {{Q}^{2}} <$ 5 GeV² [5–19]. In particular, the most recent experimental results [5] with significantly reduced statistical uncertainties, derived mainly from the Jefferson Lab EG4 experiment on polarized protons and deuterons and E97110 one on polarized ³H, make BSR an attractive quantity for testing various pQCD generalizations at low Q² values: ${{Q}^{2}} \leqslant 1$ GeV².

Theoretically, pQCD (with OPE) in the $\overline {MS}$-scheme was the usual approach to describing such quantities. This approach, however, has the theoretical disadvantage that the running coupling constant (couplant) ${{\alpha }_{s}}({{Q}^{2}})$ has the Landau singularities for small Q² values: ${{Q}^{2}} \leqslant 0.1$ GeV², which makes it inconvenient for estimating spacelike observables at small Q², such as BSR. In the recent years, the extension of pQCD couplings for low Q² without Landau singularities called (fractional) analytic perturbation theory [(F)APT)] [30–37] (or the minimal analytic (MA) theory [38]), were applied to match the theoretical OPE expression with the experimental BSR data [39–46].

Following [47, 48], we introduce here the derivatives (in the kth order of perturbation theory (PT))

$$\begin{array}{*{20}{c}} {\tilde {a}_{{n + 1}}^{{(k)}}({{Q}^{2}}) = \frac{{{{{( - 1)}}^{n}}}}{{n!}}{\kern 1pt} \frac{{{{d}^{n}}a_{s}^{{(k)}}({{Q}^{2}})}}{{{{{(dL)}}^{n}}}},} \\ {a_{s}^{{(k)}}({{Q}^{2}}) = \frac{{{{\beta }_{0}}\alpha _{s}^{{(k)}}({{Q}^{2}})}}{{4\pi }} = {{\beta }_{0}}\bar {a}_{s}^{{(k)}}({{Q}^{2}}),} \end{array}$$

(1)

which are very convenient in the case of analytic QCD. Here, ${{\beta }_{0}}$ is the first coefficient of the QCD $\beta$‑function:

$$\beta (\bar {a}_{s}^{{(k)}}) = - {{(\bar {a}_{s}^{{(k)}})}^{2}}\left( {{{\beta }_{0}} + \sum\limits_{i = 1}^k {{\beta }_{i}}{{{(\bar {a}_{s}^{{(k)}})}}^{i}}} \right),$$

(2)

where ${{\beta }_{i}}$ are known up to k = 4 [49].

The series of derivatives ${{\tilde {a}}_{n}}({{Q}^{2}})$ can successfully replace the corresponding series of ${{a}_{s}}$-powers (see, e.g., [50]). Indeed, each derivative reduces the ${{a}_{s}}$ power 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 ${{\tilde {a}}_{n}}({{Q}^{2}})$ are exactly the same as $a_{s}^{n}$. Beyond LO, the relationship between ${{\tilde {a}}_{n}}({{Q}^{2}})$ and $a_{s}^{n}$ was established in [47, 51] and extended to the fractional case, where $n \to$ is a non-integer $\nu$, in [52].

In this short paper, we apply the inverse logarithmic expansion of the MA couplants, recently obtained in [53, 54] for any PT order (see [55] for a brief introduction). This approach is very convenient: for LO the MA couplants have simple representations (see [35]), while 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 various PT orders become insignificant. Moreover, for ${{Q}^{2}} \to \infty$ and ${{Q}^{2}} \to 0$ the (fractional) derivatives of the MA couplants with $n \geqslant 2$ tend to zero, and therefore only the first term in perturbative expansions makes a valuable contribution.

BJORKEN SUM RULE

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

$$\Gamma _{1}^{{p - n}}({{Q}^{2}}) = \int\limits_0^1 {dx{\kern 1pt} [g_{1}^{p}(x,{{Q}^{2}}) - g_{1}^{n}(x,{{Q}^{2}})]} .$$

(3)

Theoretically, the quantity can be written in the OPE form (see [57, 58])

$$\Gamma _{1}^{{p - n}}({{Q}^{2}}) = \frac{{{{g}_{A}}}}{6}{\kern 1pt} (1 - {{D}_{{{\text{BS}}}}}({{Q}^{2}})) + \sum\limits_{i = 2}^\infty \frac{{{{\mu }_{{2i}}}({{Q}^{2}})}}{{{{Q}^{{2i - 2}}}}},$$

(4)

where ${{g}_{A}} = 1.2762 \pm 0.0005$ [59] is the nucleon axial charge, $(1 - {{D}_{{{\text{BS}}}}}({{Q}^{2}}))$ is the leading-twist contribution, and ${{\mu }_{{2i}}}{\text{/}}{{Q}^{{2i - 2}}}$ $(i \geqslant 1)$ are the higher-twist (HT) contributions.¹

Since we include very small Q² values here, the representation (4) of the HT contributions is inconvenient. It is much better to use the so-called “massive” representation for the HT part (introduced in [60, 61]):

$$\Gamma _{1}^{{p - n}}({{Q}^{2}}) = \frac{{{{g}_{A}}}}{6}{\kern 1pt} (1 - {{D}_{{{\text{BS}}}}}({{Q}^{2}})) + \frac{{{{{\hat {\mu }}}_{4}}{{M}^{2}}}}{{{{Q}^{2}} + {{M}^{2}}}}{\kern 1pt} ,$$

(5)

where the values of ${{\hat {\mu }}_{4}}$ and M² have been fitted in [43, 45] in the different analytic QCD models.

In the case of MA QCD, from [45] one can see that in (5)

$${{M}^{2}} = 0.439 \pm 0.463,\quad {{\hat {\mu }}_{{{\text{MA}},4}}} = - 0.173 \pm 0.666,$$

(6)

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

Another form, which is correct at very small Q² values, has been proposed in [62]

$$\begin{gathered} \Gamma _{1}^{{p - n}}({{Q}^{2}}) = \frac{{{{g}_{A}}}}{6}{\kern 1pt} (1 - {{D}_{{{\text{BS}}}}}({{Q}^{2}})) \\ + \;\frac{{{{{\hat {\mu }}}_{4}}{{M}^{2}}({{Q}^{2}} + {{M}^{2}})}}{{{{{({{Q}^{2}} + {{M}^{2}})}}^{2}} + {{M}^{2}}{{\sigma }^{2}}}}, \\ \end{gathered} $$

(7)

where small value $\sigma \equiv {{\sigma }_{\rho }} = 145$ MeV (the $\rho$-meson decay width) has been used.

Up to the k-th PT order, the perturbative part has the form

$$\begin{array}{*{20}{c}} {D_{{{\text{BS}}}}^{{(1)}}({{Q}^{2}}) = \frac{4}{{{{\beta }_{0}}}}{\kern 1pt} a_{s}^{{(1)}},} \\ {D_{{{\text{BS}}}}^{{(k \geqslant 2)}}({{Q}^{2}}) = \frac{4}{{{{\beta }_{0}}}}{\kern 1pt} a_{s}^{{(k)}}\left( {1 + \sum\limits_{m = 1}^{k - 1} {{d}_{m}}{{{(a_{s}^{{(k)}})}}^{m}}} \right){\kern 1pt} ,} \end{array}$$

(8)

where ${{d}_{1}}$, ${{d}_{2}}$ and ${{d}_{3}}$ are known from exact calculations (see, e.g., [63, 64]). The exact ${{d}_{4}}$ value is not known, but it was recently estimated in [65].

Converting the powers of couplant into its derivatives, we have

$$\begin{array}{*{20}{c}} {D_{{{\text{BS}}}}^{{(1)}}({{Q}^{2}}) = \frac{4}{{{{\beta }_{0}}}}{\kern 1pt} \tilde {a}_{1}^{{(1)}},} \\ {D_{{{\text{BS}}}}^{{(k \geqslant 2)}}({{Q}^{2}}) = \frac{4}{{{{\beta }_{0}}}}{\kern 1pt} \left( {\tilde {a}_{1}^{{(k)}} + \sum\limits_{m = 2}^k {{{\tilde {d}}}_{{m - 1}}}\tilde {a}_{m}^{{(k)}}} \right),} \end{array}$$

(9)

where

$$\begin{gathered} {{{\tilde {d}}}_{1}} = {{d}_{1}},\quad {{{\tilde {d}}}_{2}} = {{d}_{2}} - {{b}_{1}}{{d}_{1}}, \\ {{{\tilde {d}}}_{3}} = {{d}_{3}} - \frac{5}{2}{{b}_{1}}{{d}_{2}} - \left( {{{b}_{2}} - \frac{5}{2}b_{1}^{2}} \right){\kern 1pt} {{d}_{1}}, \\ {{{\tilde {d}}}_{4}} = {{d}_{4}} - \frac{{13}}{3}{{b}_{1}}{{d}_{3}} - \left( {3{{b}_{2}} - \frac{{28}}{3}b_{1}^{2}} \right){\kern 1pt} {{d}_{2}} \\ - \left( {{{b}_{3}} - \frac{{22}}{3}{{b}_{1}}{{b}_{2}} + \frac{{28}}{3}b_{1}^{3}} \right){\kern 1pt} {{d}_{1}}, \\ \end{gathered} $$

(10)

and ${{b}_{i}} = {{\beta }_{i}}{\text{/}}\beta _{0}^{{i + 1}}$.

For the case of 3 active quark flavors (f = 3), we have²

$$\begin{gathered} {{d}_{1}} = 1.59,\;\;{{d}_{2}} = 3.99,\;\;{{d}_{3}} = 15.42,\;\;{{d}_{4}} = 63.76, \\ {{{\tilde {d}}}_{1}} = 1.59,\;\;{{{\tilde {d}}}_{2}} = 2.73,\;\;{{{\tilde {d}}}_{3}} = 8.61,\;\;{{{\tilde {d}}}_{4}} = 21.52, \\ \end{gathered} $$

(11)

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

In MA QCD, the results ((7)) become as follows

$$\begin{gathered} \Gamma _{{{\text{MA}},1}}^{{p - n}}({{Q}^{2}}) = \frac{{{{g}_{A}}}}{6}{\kern 1pt} (1 - {{D}_{{{\text{MA}}{\text{,BS}}}}}({{Q}^{2}})) \\ + \frac{{{{{\hat {\mu }}}_{{{\text{MA}},4}}}{{M}^{2}}({{Q}^{2}} + {{M}^{2}})}}{{{{{({{Q}^{2}} + {{M}^{2}})}}^{2}} + {{M}^{2}}{{\sigma }^{2}}}}, \\ \end{gathered} $$

(12)

where the perturbative part ${{D}_{{{\text{BS}}{\text{,MA}}}}}({{Q}^{2}})$ takes the form

$$D_{{{\text{MA}}{\text{,BS}}}}^{{(1)}}({{Q}^{2}}) = \frac{4}{{{{\beta }_{0}}}}{\kern 1pt} A_{{{\text{MA}}}}^{{(1)}},$$

(13)

$$D_{{{\text{MA}}{\text{,BS}}}}^{{k \geqslant 2}}({{Q}^{2}}) = \frac{4}{{{{\beta }_{0}}}}{\kern 1pt} \left( {A_{{{\text{MA}}}}^{{(k)}} + \sum\limits_{m = 2}^k {\kern 1pt} {{{\tilde {d}}}_{{m - 1}}}{\kern 1pt} \tilde {A}_{{{\text{MA}},\nu = m}}^{{(k)}}} \right).$$

RESULTS

The calculation results taking into account only statistical uncertainties are presented in Table 1 and in Fig. 1. Here we use the Q²-independent M and ${{\hat {\mu }}_{4}}$ values and the twist-two parts shown in Eqs. (9) and (13) for the cases of usual PT and APT, respectively.

Table 1. Fit parameters with $\sigma = {{\sigma }_{\rho }}$ ($\sigma = 0$)

Fig. 1.

(Color online) Results for $\Gamma _{1}^{{p - n}}({{Q}^{2}})$ in the first four orders of APT with $\sigma = {{\sigma }_{\rho }}$.

In the case of using MA couplants, we see in Table 1 that the cases $\sigma = 0$ and $\sigma = {{\sigma }_{\rho }}$ lead to very similar values for the fitting parameters and χ²-factor. So, in Fig. 1 we show only the case with $\sigma = {{\sigma }_{\rho }}$. The quality of the fits is very good, as evidenced quantitatively by the values of ${{\chi }^{2}}{\text{/}}({\text{d}}{\text{.o}}{\text{.f}}.)$. Moreover, our results obtained for different PT orders are very similar to each other: the corresponding curves in Fig. 1 are indistinguishable. One can also see the important role of the twist-four term (see also [42] and [70, 71] and discussions therein). Without it, the value of $\Gamma _{1}^{{p - n}}({{Q}^{2}})$ is about 0.16, which is very far from the experimental data.

At ${{Q}^{2}} \leqslant 0.3$ GeV² we also see good agreement with the phenomenological models: Burkert–Ioffe one [72, 73] and especially LFHQCD one [74]. For larger Q² values our results are below the results of the phenomenological models and at ${{Q}^{2}} \geqslant 0.5$ GeV² are below the experimental data. We hope to improve agreement with using massive forms of HT contributions ${{h}_{{2i}}}$ with $i \geqslant 3$. This is a subject of future investigations.

As seen in Fig. 1, the results obtained using conventional couplants are not good and worse for the NLO case to compare to the LO one. Indeed, the deterioration increases with the PT order in this case (see [39, 40, 43, 45, 53]). Thus, the use of the massive twist-four form (5) does not improve these results, since at ${{Q}^{2}} \to \Lambda _{i}^{2}$ conventional couplants become to be singular, that leads to large and negative results for the twist-two part (8). As the PT order increases, usual couplants become singular for ever larger Q² values, while BSR tends to negative values for ever larger Q² values (see, e.g., Fig. 15 in [53]). Thus, the discrepancy between theory and experiment increases with the PT order.

CONCLUSIONS

We have considered the Bjorken sum rule in the framework of MA and perturbative QCD and obtained results similar to those obtained in previous studies [39, 40, 43, 45, 53] for the first 4 orders of PT. The results based on the conventional PT do not agree with the experimental data. For some Q² 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 (12) for the twist-four contributions.

Now we would like to discuss the photoproduction (PhP) case, i.e., the ${{Q}^{2}} \to 0$ limit. In MA QCD, ${{A}_{{{\text{MA}}}}}({{Q}^{2}} = 0) = 1$ and $\tilde {A}_{{{\text{MA}},m}}^{{(k)}} = 0$ for $m > 1$ and we have

$$\begin{gathered} {{D}_{{{\text{MA}}{\text{,BS}}}}}({{Q}^{2}} = 0) = \frac{4}{{{{\beta }_{0}}}} , \\ \Gamma _{{{\text{MA}},1}}^{{p - n}}({{Q}^{2}} = 0,\sigma = 0) = \frac{{{{g}_{A}}}}{6}{\kern 1pt} \left( {1 - \frac{4}{{{{\beta }_{0}}}}} \right) + {{{\hat {\mu }}}_{{{\text{MA}},4}}}. \\ \end{gathered} $$

(14)

The finiteness of cross section in the real photon limit leads [60–62]

$$\begin{gathered} \Gamma _{{{\text{MA}},1}}^{{p - n}}({{Q}^{2}} = 0) = 0 {\kern 1pt} {\text{ and}}{\text{,}}\;{\text{hence}}{\text{,}} \\ \hat {\mu }_{{{\text{MA}},4}}^{{{\text{php}}}} = - \frac{{{{g}_{A}}}}{6}\left( {1 - \frac{4}{{{{\beta }_{0}}}}} \right).{\kern 1pt} \\ \end{gathered} $$

(15)

In the case of three active quarks, i.e., f = 3, we have

$$\hat {\mu }_{{{\text{MA}},4}}^{{{\text{php}}}} = - 0.118 {\kern 1pt} {\text{ and}}{\text{,}}\;{\text{hence}},{\kern 1pt} {\text{ |}}\hat {\mu }_{{{\text{MA}},4}}^{{{\text{php}}}}{\text{|}} < {\text{|}}{{\hat {\mu }}_{{{\text{MA}},4}}}{\text{|}},$$

(16)

shown in Table 1.

So, in our fits the finiteness of cross section in the real photon limit is violated.³

This is a common situation that appears as a consequence of the use of analytic versions of QCD for the Bjorken sum rule (see, e.g., [45]). Note that our results for ${{\hat {\mu }}_{4}}$ shown in Table 1 are smaller than in [45].

In our future investigations we plan to improve this analysis by taking several massive twists by analogy with twist-four one shown in Eq. (5). We hope that this will lead to better agreement with the real photon limit and with the studies in [75–77].