Determination of the effective strong coupling constant alpha_{s,g_1}(Q^2) from CLAS spin structure function data

We present a new extraction of the effective strong coupling constant \alpha_{s,g_1}(Q^2). The result agrees with a previous determination and extends the measurement of the low and high Q^2 behavior of \alpha_{s,g_1}(Q^2) that was previously deduced from sum rules. In particular, it experimentally verifies the lack of Q^2-dependence of \alpha_{s,g_1}(Q^2) in the low Q^2 limit. This fact is necessary for application of the AdS/CFT correspondence to QCD calculations. We provide a parameterization of \alpha_{s,g_1}(Q^2) thatcan equivalently be used to parameterize the Q^2-dependence of the generalized Gerasimov-Drell-Hearn and Bjorken sums.

In Quantum Chromodynamics (QCD), the gauge theory of the strong force, the magnitude of the coupling is given by the running coupling constant α s . While α s is well defined within perturbative QCD at large Q 2 [1], these calculations lead to an infinite coupling at large distances. However, many calculations, including lattice QCD or solving Dyson-Schwinger equations, indicate that α s remains finite, see e.g. [2] for a review. Most of these theoretical results also predict that α s loses its scale dependence at large distances ("freezing" of α s ) although there is still no firm consensus, see e.g. the new calculations [3] and [4] (the first result indicates that breaking of chiral symmetry causes α s to be divergent at large distances while the second confirms previous results of the freezing of α s ). Recently, an effective strong coupling constant, α s,g 1 , has been extracted [5] from experimental data on the Bjorken sum [6]. It is indicative of the freezing of α s,g 1 . When the data are complemented with the generalized Gerasimov-Drell-Hearn (GDH) sum rule [8] and the generalized Bjorken sum rule [7] predictions, the behavior of α s,g 1 can be established at any distance.
Although the connection among the various theoretical techniques used to compute α s at large distances is unclear, most results exhibit analog behavior and order of magnitude.
Likewise, the connection between the experimental results reported in [5] and theoretical techniques is not fully known but they display intriguing similarities. Remarkably, the Q 2 behavior of these calculations and of the data agree. In this paper, we present new results for α s,g 1 over an extended Q 2 range. We then propose a parameterization of α s,g 1 and finally discuss briefly the consequences of the behavior seen for α s,g 1 .
We have extracted new data points on α s,g 1 following the procedure described in [5] and based on the theoretical works of [9,10]. The advantages and limitations of such extraction are discussed in [5] and summarized here. Effective strong coupling constants are defined using first order pQCD equations. As a consequence, they are process dependent. However, because they can be related to each other using commensurate scale relations [10], they can meaningfully be used for QCD predictions. The definition of effective strong coupling constants using first order pQCD equations makes them renormalization scheme and gauge independent and free of divergence at low Q 2 . They also are analytic when crossing quark mass thresholds. Using the generalized Bjorken sum rule is particularly suited to define the effective strong coupling constant α s,g 1 . Its simplicity makes theoretical calculations easier and it is constrained at both large and low Q 2 by well established sum rules. Furthermore, there is a large amount of data available. A present limitation of the use of effective strong coupling constants is that the connection between various theoretical calculations of α s at low Q 2 is so far unclear due to the different approximations used. The relation between theoretical results and experimental extractions is also unclear. In ref. [5], we nonetheless compared all these quantities in order to shed light on possible similarities. We will repeat the comparison here keeping in mind that the same precautions apply.
The new data used to extract α s,g 1 were taken with the CLAS spectrometer [11] in Hall B at Jefferson Lab (JLab), using a polarized electron beam with energies ranging from 1 to 6 GeV. The data are reported in [13] and were used to form the Bjorken sum Γ p−n in a Q 2 -range from 0.06 to 2.92 GeV 2 [14]. Here, Q 2 is the square of the four-momentum transfered from the electron to the target. Apart from the extended Q 2 -coverage, one notable difference between these data and those of ref. [6] is that the neutron information originates from the longitudinally polarized deuteron target of CLAS while the previous data resulted from the longitudinally and transversally polarized 3 He target of JLab's Hall A [12]. The effective coupling α s,g 1 is defined by the Bjorken sum rule expressed at first order in pQCD and at leading twist. This leads to the relation: where g A is the nucleon axial charge. We used Eq. 1 to extract α s,g 1 /π. The results are shown in Fig. 1. The inner error bars represent the statistical uncertainties whereas the outer ones are the quadratic sum of the statistical and systematic uncertainties. Also plotted in the figure are the first data on α s,g 1 from [5] and from the world data of the Bjorken sum evaluated at < Q 2 >=5 GeV 2 , α s,F 3 from the Gross-Llewellyn Smith (GLS) sum rule [17] measured by the CCFR collaboration [18], and α s,τ [19]. See [5] for details.
The behavior of α s,g 1 is given near Q 2 = 0 by the generalized GDH sum rule and at large The values for α s,g 1 from the new data are in good agreement with the previous JLab data.
While the previous data were suggestive, the freezing of α s,g 1 at low Q 2 is now unambiguous and in good agreement with the GDH sum prediction. At larger Q 2 , the new data agree with the world data and the results from the Bjorken sum rule at leading twist. We fit the data using a functional form that resembles the pQCD evolution equation for α s , with an additional term m g (Q) that prevents α f it s,g 1 from diverging when Q 2 → Λ 2 and another term n(Q) that forces α f it s,g 1 to π when Q 2 → 0. Note that the latter constraint is a consequence of both the generalized GDH and Bjorken sum rules [5]. Our fit form is: Finally, we show the α s,g 1 formed using a phenomenological model of polarized lepton scattering off polarized nucleons (Burkert-Ioffe). These calculations are discussed in [5]. The magnitude of the Godfrey-Isgur and Cornwall results agrees with the estimate of the average value of α s using magnetic and color-magnetic spin-spin interactions [22]. We emphasize that the relation between these results is not fully known and that they should be considered as indications of the behavior of α s rather than strict predictions.
The data show that α s,g 1 loses its Q 2 -dependence both at large and small Q 2 . The Q 2scaling at large Q 2 is long known and is the manifestation of the asymptotic freedom of QCD [30]. The absence of Q 2 -dependence at low Q 2 has been conjectured and observed by many calculations but this is the first experimental evidence. This lack of scale dependence (conformal behavior) at low Q 2 shows that conformal field theories might be applicable to study the properties of hadrons. In particular the AdS/CFT correspondence [31] between strongly coupled gauge fields and weakly coupled string states can be used. This opens promising opportunities for calculations in the non-perturbative regime of QCD [32,33,34].
Finally, it is noteworthy that conformal behavior is broken in the Q 2 -range between ≈ 0.7 to a few GeV 2 . This domain is the transition region between the fundamental degrees of freedom of QCD (partons) to its effective ones (hadrons).
To summarize, we have used new JLab data on the Bjorken sum to form the effective strong coupling constant α s,g 1 . The Q 2 -range is extended by factors of 3 in both the small and large Q 2 sides compared to results previously reported. The results are in good agreement with sum rule predictions and show for the first time unambiguously that α s,g 1 loses its Q 2 dependence at low Q 2 . We provided an analytic form for α s,g 1 (or equivalently for the generalized GDH and Bjorken sums) based on the pQCD result for α s and including the sum rule constraints at small and large Q 2 . It appears that strong interaction is approximately conformal in both the large and small Q 2 limits. We remarked that conformal behavior breaks down when transiting from the fundamental degrees of freedom of QCD (quarks and gluons) to its effective ones (baryons and mesons). Establishing conformal behavior of strong interaction is a basic step in applying the AdS/CFT correspondence to the study of hadronic matter.
We are would like to thanks to S. Brodsky, S. Gardner. W. Melnitchouk, C. Roberts and P. Tandy for helpful discussions. We acknowledge S. Furui and P. Bowman for sending us their lattice results.