Experimental determination of the effective strong coupling constant

We present a first attempt to experimentally extract an effective strong coupling constant that we define to be a low Q2 extension of a previous definition by S. Brodsky et al. following an initial work of G. Grunberg. Using Jefferson Lab data and sum rules, we establish its Q2-behavior over the complete Q2-range. The result is compared to effective coupling constants inferred from different processes and to calculations based on Schwinger-Dyson equations, hadron spectroscopy or lattice QCD. Although the connection between the experimentally extracted effective coupling constants and the calculations is not established it is interesting to note that their behaviors are similar.

At experimentally accessible distances, the strong force remains the only interaction that resists satisfactory understanding. Quantum Chromodynamics (QCD), the gauge theory of the strong force, is well known at short distances ( 10 −16 m) where it is solvable perturbatively. QCD, however, is not perturbatively calculable at larger distances, typically the scale of the nucleon radius. Recent precision data on moments of nucleon structure functions [1,2,3,4] reveal a smooth transition from small to large scales, while in contrast, a feature of perturbative QCD (pQCD) is that at Λ QCD , the running strong coupling constant α s becomes infinite. An approach using an effective coupling constant could reconcile these two seemingly paradoxical aspects of strong interaction.
In lepton scattering a scale at which the target structure is probed is given by the inverse of Q 2 , the square of the four-momentum transfered to the target. One way to extract α s at large Q 2 is to fit the Q 2 -dependence of the moments of structure functions. Among all moments, the Bjorken sum rule [5] is a convenient relation for such an extraction [6].
Furthermore, as will be discussed, the Bjorken sum may offer unique advantages to define a effective coupling at low Q 2 .
In the limit where the energy transfer ν and Q 2 are infinite, while x ≡ Q 2 /(2Mν) remains finite (M is the nucleon mass), the Bjorken sum rule reads: g p 1 and g n 1 are spin structure functions for the proton and neutron. The axial charge of the nucleon, g A , is known from neutron β-decay. for finite Q 2 much greater than Λ 2 QCD , the Q 2 -dependence of the Bjorken sum rule is given by a double series in Q 2−t (t=2,4... being the twist) and in (α s /π) n . The (α s /π) n series is given by pQCD evolution equations. At leading twist (t = 2) and 3 rd order in α s , in the MS scheme and dimensional regularization, we have [7]: The validity of the sum rule is verified at Q 2 =5 GeV 2 to better than 10% [8].
The extraction of the Bjorken integral using data from the Thomas Jefferson National Accelerator Facility (JLab) in the Q 2 -range of 0.17-1.10 GeV 2 has been reported recently [9].
The use of Eq. 2 as an Ansatz for definition of an effective running coupling constant at low Q 2 has ambiguities and difficulties as well as some practical advantages. The advantages are the following: firstly, it was pointed out in Ref. [10] that the extraction of α s does not depend strongly on the low-x extrapolation. Secondly, this flavor non-singlet contribution does not mix quark and gluon operators when evolved. Hence the pQCD evolution is known to higher order. Thirdly, the JLab data are at constant Q 2 . This avoids a possible ambiguity encountered in previous experiments, namely that in order to combine neutron and proton data, structure functions must be evolved to a common Q 2 0 , which needs α s itself as input. This is especially important in our case since we cannot anticipate the value of α s at such low Q 2 , due to the breakdown of pQCD. Hence, no evolution of the data to Q 2 0 is possible. Difficulties within the pQCD approach are: firstly, at low Q 2 higher twist effects become important and are not well known [9]. Secondly, the pQCD expansion loses its meaning as Q approaches Λ QCD where, as a consequence of renormalization and regularization, α pQCD s itself is singular. It is necessary to use an appropriate theoretical framework to circumvent these difficulties. Such frameworks have been developed, see for example [11] and [12]. We use the method of "effective charges" of Grunberg [13], where the non-perturbative terms and higher order perturbative processes are absorbed in the definition of the coupling constant.
In our case, it obeys the following definition: Eq. 3 provides a definition of an effective QCD running coupling that we will explore here.
The inherent systematic uncertainties in this experimental Ansatz, and in those of the various theoretical approaches are unknown. Their comparison provides a framework for further analysis.
The coupling constant defined with Eq. 3 still obeys the renormalization group equation [13]. The first two terms in the Eq. 2 and 3 series are independent of the choice of gauge and renormalization scheme. Consequently, the effective coupling constant is renormalization scheme and gauge independent, but becomes process-dependent.
These process-dependent coupling constants can be related by using "commensurate scale relations" which connect observables without renormalization scheme or scale ambiguity [14].
In this topic, an important relation is the Crewther relation [14,15].
Considering an effective coupling constant yields other advantages beside renormalization scheme/gauge independence: such a procedure improves perturbative expansions [13,14], the effective charge is analytic when crossing quark mass thresholds, is non-singular at Q = Λ QCD , and is well defined at low Q 2 [16]. The choice of defining the effective charge with Eq. 3 has unique advantages: Low Q 2 data exist and near real photon data will be available soon from JLab [17,18]. Furthermore, sum rules constrain α s,g 1 at both low and high Q 2 limits, as will be discussed in the next paragraph. Another advantage is that Γ p−n 1 is a quantity well suited to be calculated at any Q 2 because of various cancellations that simplify calculations [9,19].
Using Eq. 3 and the JLab data, α s,g 1 can be formed. The elastic (x = 1) contribution is . The resulting α s,g 1 /π is shown in Fig. 1. Systematic effects dominate the uncertainties, see ref. [9] for details. The uncertainty from g A is small. We also used the world data of the Bjorken sum evaluated at < Q 2 >=5 GeV 2 to compute α s,g 1 /π. We can also use a model for Γ 1 and, using Eq. 3, form α s,g 1 . We chose the Burkert-Ioffe model [20] because of its good match with the experimental data on moments of spin structure functions [1]- [9]. It is interesting to note the behavior of α s,g 1 near Q 2 = 0 where it is constrained by the Gerasimov-Drell-Hearn (GDH) sum rule [21] that predicts the derivative of the Bjorken integral with respect to Q 2 at Q 2 = 0, [22]: where κ p (κ n ) is the proton (neutron) anomalous magnetic moment and α is the QED coupling constant. This, in combination with Eq. 3, yields: The constraint is shown by the dashed line. At Q 2 = 0 the Bjorken sum is zero and α s,g 1 = π, a particular property of the definition of α s,g 1 . At larger Q 2 , where higher twist effects are negligible, α s,g 1 can be evaluated by estimating the right hand side of Eq. 2 (using α pQCD s as predicted by pQCD) and equating it to g A [1 − α s,g 1 /π]/6. The resulting α s,g 1 is shown by the gray band [23]. The width of the band is due to the uncertainty in Λ QCD . Finally α s,F 3 can be computed using data on the Gross-Llewellyn Smith sum rule [24], which relates the number of valence quarks in the hadron, n v , to the structure function F 3 (Q 2 , x) measured in neutrino scattering. At leading twist, the GLS sum rule reads: Using the data taken by the CCFR collaboration [25], we can apply the same procedure as for the Bjorken sum rule to compute α s,F 3 . We expect α s,F 3 = α s,g 1 at large Q 2 , since the leading twist Q 2 -dependence of Eq. 2 and Eq. 6 is identical up to α 3 s, , up to a very small difference at order α 3 s, coming from the light-by-light contribution to the GLS sum. In principle, the commensurate scale relations derived by Brodsky and Lu in ref. [14] should be applied when comparing α s,g 1 to α s,F 3 (and α s,τ in Fig. 1). In practice, however, the resulting corrections are small; the same scale is used in the α s,g 1 and α s,F 3 comparison and the correction from the light by light contribution decreases the value of α s,F 3 by at most 1%. These corrections were neglected. The commensurate scale relations should be used when comparing α s,τ (Q) extracted from OPAL data on τ -decay [16] to α s,g 1 . At leading order (the only order available since only one Q value of α s,τ (Q) is available) the Q correction necessary to compare α s,g 1 to α s,τ leads to a 20% change on the effective charge. Such Q correction was applied.
In summary, assuming the validity of the GDH and Bjorken sum rules for low and large Q 2 respectively and using the JLab data at intermediate Q 2 , we can evaluate α s,g 1 at any value of Q 2 . The absence of divergence in α s,g 1 is obvious since it is defined from finite experimental data. It is interesting to notice that α s,g 1 loses its Q 2 -dependence at low Q 2 . This feature was suggested by the work of Brodsky et al. based on τ -decay data [16] and by other theoretical works, as will be discussed below.
Many theoretical or phenomenological studies of α s at low-Q 2 are available. See [26,27] for reviews. The theoretical studies comprise Schwinger-Dyson Equations (SDE), lattice QCD, non perturbative QCD vacuum, and analyticity arguments. Phenomenological studies are based on quark model spectroscopy, low-x-low-Q 2 reactions, parametrization of nucleon and pion form factors, heavy quarkonia decay, and ratio of hadron production cross section to µ + µ − production cross section in e + e − annihilation.
Similarly to experimental effective charges, different definitions of the strong coupling constant at low Q 2 [28] are possible in the theoretical calculations. How they are related is not fully known. Furthermore, these calculations should be viewed as indications of the behavior of α s rather than strict predictions. Although some theoretical uncertainties due to parameterizations are shown, the existence of unknown systematic/model uncertainties should be borne in mind. However, it is interesting to compare the various calculations to our result to see whether they show common features. In theory, solving the SDE equations can yield α s . In practice, approximations are necessary and choices of approximation lead to different values of α s . In the top panels and the bottom left panel of Fig. 2, we compare our data to different approaches [29,30,31,32,33]. The uncertainty in Cornwall's result is due to the uncertainties in parameters that enter the calculation. The uncertainty on the Bloch curve is due to Λ QCD . There is a good agreement between the absolute value obtained from the present Ansatz and the results of Bloch [31] and Fisher et al. [30], while the results from Maris-Tandy [32] and Bhagwat et al. [33] do not agree as well. The older calculation from Cornwall [29] disagree with the result of our Ansatz. The Godfrey and Isgur curve in the top right panel of Fig. 2 represents the coupling constant used in a quark model [34].
It is interesting to notice that the Q 2 -dependence of α s,g 1  for Q 2 > 0.6 GeV.
One Ansatz that has received recent theoretical attention for the QCD coupling at low Q 2 is related to the product of the gluon propagator dressing function with the ghost propagator dressing function squared [30]. Gluon and ghost propagators have been computed in lattice QCD by many groups. The lattice results offer a fairly consistent picture [27] and agree reasonably with the various SDE propagator results [35]. It is calculated in particular in refs. [27,36]. In the bottom right panel of Fig. 2, we compare α s,g 1 to the lattice result from ref. [36]. We plot the lattice result that is believed to be closer to the continuum limit [37].
There is a good agreement between the lattice QCD calculations and our data.
To conclude, we have formed an effective strong coupling constant α s,g 1 at low Q 2 . Data, together with sum rules, allow to obtain α s,g 1 at any Q 2 . The connection between the Bjorken and the GDH sum rules yields a value of α s,g 1 equal to π at Q 2 = 0. An important feature of α s,g 1 is its loss of Q 2 -dependence at low Q 2 . We compared our result to other coupling constants from different reactions. They agree with each other, although they were defined from different processes. We also compared α s,g 1 to SDE calculations, lattice QCD calculations and a coupling constant used in a quark model. Although the relation between the various calculations is not well understood, the data and calculations agree in most cases especially when only considering Q 2 -dependences. It will be interesting in the future to pursue the same analysis with lower Q 2 data that will be available both for the neutron [17] and proton [18].
We are grateful to S. Brodsky, S. Gardner. W. Melnitchouk and P. Tandy for helpful discussions. We thanks S. Furui  under contract DE-AC05-84ER40150.