The strong coupling constant at low
\abstractsWe extract an effective strong coupling constant using low- data and sum rules. Its behavior is established over the full -range and is compared to calculations based on lattice QCD, Schwinger-Dyson equations and a quark model. Although the connection between all these quantities is not known yet, the results are surprisingly alike. Such a similitude may be related to quark-hadron duality.
1 The strong coupling constant
A peculiar feature of strong interaction is asymptotic freedom: quark-quark interactions grow weaker with decreasing distances. Asymptotic freedom is expressed in the vanishing of the QCD coupling constant,, at large . Conversely, the fact that , as calculated in pQCD, becomes large when is often linked to quark confinement. Since it is not expected that pQCD holds at the confinement scale and since the condition when is far from necessary to assure confinement[1], it is interesting to study in the large distance domain.
Experimentally, moments of structure functions are convenient objects to extract . Among them, is the simplest to use. In pQCD, it is linked to the axial charge of the nucleon, , by the Bjorken sum rule:
(1) | |||
where () is the first spin structure function for the proton(neutron). The are higher twist corrections and become important at lower . This series, usually truncated to leading twist and to 3rd order, can be used to fit experimental data and to extract . The higher twists can be computed with non-perturbative models or can be extracted from data, although with limited precision at the moment[3]. This imprecise knowledge and the break down of pQCD at low prevent a priori the extraction of at low . However, an effective strong coupling constants was defined by Grunberg[4] in which higher twists and higher order QCD radiative corrections are incorporated. Eq. 1 becomes by definition:
(2) |
This definition yields many advantages: the coupling constant is extractable at any , is well-behaved when , is not renormalization scheme (RS) dependent and is analytic when crossing quark thresholds. The price to pay for such benefits is that it becomes process-dependent (hence the subscript in Eq. 2). However, as pointed out by Brodsky et al.[5], effective couplings can be related to each other, at least in the pQCD domain, by “commensurate scale equations”. These relate, using different scales, observables without RS or scale ambiguity. Thus, one effective coupling constant is enough to characterize the strong interaction.
Among the possible observables available to define an effective coupling constant, has unique advantages. The generalized Gerasimov-Drell-Hearn (GDH)[6, 7] and Bjorken sum rules predict at low and large , and is experimentally known between these two domains. Hence, can be extracted at any . In particular, it has a well defined value at =0. Furthermore, we will see that might best be suited to be compared to the predictions of theories and models.
2 Experimental determination of
A measurement of at intermediate was reported recently[8] and was used to extract [9]. The results are shown by the triangles in Fig. 1, together with extracted from SLAC data[10] at =5 GeV (open square). Note that the elastic contribution is not included in .
is related to the generalized GDH sums:
(3) |
where is the QED coupling constant. Hence, at =0, and
(4) |
At , the GDH sum rule implies:
(5) |
where () is the proton (neutron) anomalous magnetic moment. Combining Eq. 2 and 5, we get the derivative of at =0:
(6) |
Relations 4 and 6 constrain at low (dashed line in Fig. 1). At large , can be estimated using Eq. 1 at leading twist and calculated with pQCD. can be subsequently extracted (gray band).
These data and sum rules give at any . A similar result is obtained using a model of and Eq. 2 (dotted line). The Burkert-Ioffe[11] model is used because of its good match with data.
One can compare our result to effective coupling constants extracted using different processes. was extracted from -decay data[12] from the OPAL experiment (inverted triangle). It is compatible with . The Gross-Llewellyn Smith sum rule[13] (GLS) can be used to form . The sum rule relates the number of valence quarks in the hadron, , to the structure function . At leading twist, it reads:
(7) |
We expect at high , since the -dependence of Eq. 1 and 7 at leading twist are identical. The GLS sum was measured by the CCFR collaboration[14] and the resulting is shown by the star symbols.
3 Comparison with theory
Just like effective coupling constants extracted experimentally, there are also many possible theory definitions for the coupling constant and, contrarily to the experimental quantities, the relations between the various definitions are not well known. Furthermore, the connection between the experimental and the theoretical quantities is not clear. Hence, the remainder of this paper is to be understood as a candid comparison of quantities a priori defined differently, in order to see if they share common features.
Calculations of using Schwinger-Dyson equations (SDE), lattice QCD or quark models are available. Different SDE results are shown in Fig. 2. The pioneering result of Cornwall[15] is shown by the blue band in the top left panel. The more recent SDE results from Fisher et al., Bloch et al., Maris and Tandy, and Bhagwat et al. are shown in top left, top right, bottom left and bottom left panels respectively. There is a good match between the data and the result from Fisher et al. and a fair match with the curve from Bloch et al. The results from Maris-Tandy, Bhagwat et al. and Cornwall do not match the data. The Godfrey and Isgur curve in the top right panel of Fig. 2 is the coupling constant used in the framework of hadron spectroscopy[20]. -behavior of coupling constants can also be compared regardless of their absolute magnitudes by normalizing them to at (These curves are not shown here). The Godfrey-Isgur, Cornwall and Fisher et al. -behavior match well the data. The normalized curves from Maris-Tandy, Bloch et al. and Bhagwat et al. are slightly below the data (by typically one sigma) for GeV.
Gluon bremsstrahlung and vertex corrections contribute to the running of . Modern SDE calculations include those[21] but it is a priori not the case for the used in the one gluon exchange term of the Godfrey and Isgur quark model, or for older SDE works. If so, pQCD corrections should be added to these calculations. The effect of those corrections (on ) is given by the ratio of extracted using Eq. 2 to extracted using Eq. 1 at leading twist. For both Eq. 1 and 2, is given by a model[11]. Since model and data agree well, no strong model dependence is introduced. The difference between results using Eq. 1 up to 4 and 5 order is taken as the uncertainty due to the truncation of the pQCD series. The resulting are shown in the bottom right panel of Fig. 2.
Finally, we can compare lattice QCD data to our results. Many lattice results are available and are in general consistent. We chose to compare with the results of Furui and Nakajima[22], see bottom left panel in Fig. 2. They match well the data. The lowest point is afflicted by finite size effect and should be ignored.
The match between our data and the various calculations might be surprising since these quantities are defined differently. We can try to understand this fact. Choosing minimizes the rôle of resonances, in particular it fully cancels the contribution which usually dominates the moments at low . By furthermore excluding the elastic contribution, we obtain a quantity for which coherent reactions (elastic and resonances) are suppressed and we are back to a DIS-like case in which the interpretation is straightforward. One can also possibly invoke the phenomenon of quark-hadron duality to explain why the extraction of , using a formalism developed for DIS[12], seems to also work at lower .
4 Conclusion
We have extracted, using JLab data at low together with sum rules, an effective strong coupling constant at any . A striking feature is its loss of -dependence at low . We compared our result to SDE and lattice QCD calculations and to a coupling constant used in a quark model. Despite the unclear relation between these various coupling constants, data and calculations match in most cases, especially for relative -dependences. This could be linked to quark-hadron duality.
Acknowledgments
This work is supported by the U.S. Department of Energy (DOE). The Southeastern Universities Research Association (SURA) operates the Thomas Jefferson National Accelerator Facility for the DOE under contract DE-AC05-84ER40150.
References
- [1] See e.g. Y. L. Dokshitzer, hep-ph/9812252
- [2] J. D. Bjorken, Phys. Rev. 148, 1467 (1966)
- [3] J-P. Chen, A. Deur, Z-E Meziani, nucl-ex/0509007
- [4] G. Grunberg, Phys. Lett. B95 70 (1980); Phys. Rev. D29 2315 (1984); Phys. Rev. D40, 680 (1989)
- [5] S. J. Brodsky and H. J. Lu, Phys. Rev. D51 3652 (1995); S. J. Brodsky, G. T. Gabadadze, A. L. Kataev and H. J. Lu, Phys. Lett. B372 133 (1996); S. J. Brodsky, hep-ph/0310289
- [6] S. D. Drell and A. C. Hearn, Phys. Rev. Lett. 16, 908 (1966). S. Gerasimov, Sov. J. Nucl. Phys. 2, 430 (1966)
- [7] X. Ji and J. Osborne, J.Phys. G27 127 (2001)
- [8] A. Deur et al., Phys. Rev. Lett. 93 212001 (2004)
- [9] A. Deur et al., hep-ph/0509113
- [10] K. Abe et al., Phys. Rev. Lett. 79 26 (1997); P. L. Anthony et al., Phys. Lett. B493 19 (2000); Phys. Rev. D67 055008 (2003)
- [11] V. D. Burkert and B. L. Ioffe, Phys. Lett. B296, 223 (1992); J. Exp. Theor. Phys. 78, 619 (1994)
- [12] S. J. Brodsky et al., Phys. Rev. D67 055008 (2003)
- [13] D. J. Gross and C.H. Llewellyn Smith, Nucl. Phys B14 337 (1969)
- [14] J. H. Kim et al., Phys. Rev. Lett. 81 3595 (1998)
- [15] J. M. Cornwall, Phys. Rev. D26 1453 (1982)
- [16] C. S. Fischer and R. Alkofer, Phys. Lett. B536 177 (2002); C. S. Fischer, R. Alkofer and H. Reinhardt, Phys. Rev. D65 125006 (2002); R. Alkofer, C. S. Fischer and L. von Smekal, Acta Phys. Slov. 52 191 (2002)
- [17] J. C. R. Bloch, Phys. Rev. D66 034032 (2002)
- [18] P. Maris and P. C. Tandy, Phys. Rev. C60 055214 (1999)
- [19] Bhagwat et al., Phys. Rev. C68 015203 (2003)
- [20] S. Godfrey and N. Isgur, Phys, Rev. D32 189 (1985)
- [21] We thank P. Tandy for pointing it out to us.
- [22] S. Furui, H. Nakajima, hep-lat/ 0410038, S. Furui and H. Nakajima, Phys. Rev. D70 094504 (2004)