#
Experimental Determination of the QCD Effective Charge α_{g1}(Q)

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Extraction of ${\mathbf{\alpha}}_{{\mathbf{g}}_{\mathbf{1}}}$

**Figure 1.**Effective charge ${\alpha}_{{g}_{1}}\left(Q\right)/\pi $ obtained from JLab experiments E03006/E97110 [33] (solid stars), E03006/E05111 [33] (solid circles) and EG1dvcs [32] (solid triangles) and from COMPASS [31] (solid square). Inner error bars represent the statistical uncertainties, and outer ones represent the systematic and statistical uncertainties added quadratically. The open symbols show the older world data [27,28,29,30] with the error bars the quadratic sum of the systematic and statistical uncertainties. Also shown are the HLFQCD [24] (red line, using the HLFQCD scale $\kappa =0.534$ GeV [59]) and DSE [25] (magenta line and hatched band) parameter-free predictions of effective charges. The dashed line and hatched cyan band are ${\alpha}_{{g}_{1}}\left(Q\right)/\pi $ obtained from the GDH and Bjorken sum rules, respectively.

## 3. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- D’Enterria, D.; Kluth, S.; Zanderighi, G.; Ayala, C.; Benitez-Rathgeb, M.A.; Bluemlein, J.; Xie, K. The strong coupling constant: State of the art and the decade ahead. arXiv
**2022**, arXiv:2203.08271. [Google Scholar] - Yao, W.M.; Amsler, C.; Asner, D.; Barnett, R.M.; Beringer, J.; Burchat, P.R.; Staney, T. Review of Particle Physics. J. Phys. G Nucl. Part. Phys.
**2006**, 33, 1–1232. [Google Scholar] [CrossRef] - Brodsky, S.J.; de Téramond, G.F.; Dosch, H.G.; Erlich, J. Light-front holographic QCD and emerging confinement. Phys. Rep.
**2015**, 584, 1–105. [Google Scholar] [CrossRef] [Green Version] - Dobado, A.; Espriu, D. Strongly coupled theories beyond the Standard Model. Prog. Part. Nucl. Phys.
**2020**, 115, 103813. [Google Scholar] [CrossRef] - Dirac, P.A.M. Forms of relativistic dynamics. Rev. Mod. Phys.
**1949**, 21, 392. [Google Scholar] [CrossRef] [Green Version] - Brodsky, S.J.; de Téramond, G.F. Light-front hadron dynamics and AdS/CFT correspondence. Phys. Lett. B
**2004**, 582, 211–221. [Google Scholar] [CrossRef] [Green Version] - Maris, P.; Roberts, C.D. Dyson-Schwinger equations: A Tool for hadron physics. Int. J. Mod. Phys. E
**2003**, 12, 297–365. [Google Scholar] [CrossRef] [Green Version] - Deur, A.; Brodsky, S.J.; de Téramond, G.F. The QCD Running Coupling. Prog. Part. Nucl. Phys.
**2016**, 90, 1. [Google Scholar] [CrossRef] [Green Version] - Grunberg, G. Renormalization Group Improved Perturbative QCD. Phys. Lett.
**1980**, 95B, 70, Erratum in Phys. Lett.**1982**, 110B, 501. [Google Scholar] [CrossRef] - Gell-Mann, M.; Low, F.E. Quantum electrodynamics at small distances. Phys. Rev.
**1954**, 95, 1300. [Google Scholar] [CrossRef] - Brodsky, S.J.; Lu, H.J. Commensurate scale relations in quantum chromodynamics. Phys. Rev. D
**1995**, 51, 3652. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bjorken, J.D. Applications of the Chiral U(6)×U(6) Algebra of Current Densities. Phys. Rev.
**1966**, 148, 1467. [Google Scholar] [CrossRef] - Gross, D.J.; Smith, C.H.L. High-energy neutrino-nucleon scattering, current algebra and partons. Nucl. Phys. B
**1969**, 14, 337. [Google Scholar] [CrossRef] [Green Version] - Brodsky, S.J.; Menke, S.; Merino, C.; Rathsman, J. On the behavior of the effective QCD coupling alpha(tau)(s) at low scales. Phys. Rev. D
**2003**, 67, 055008. [Google Scholar] [CrossRef] [Green Version] - Gorishnii, S.G.; Kataev, A.L.; Larin, S.A. The O(αs3)-corrections to σ
_{tot}(e^{+}e^{-}→hadrons) and Γ(τ^{-}→ν_{τ}+hadrons) in QCD. Phys. Lett. B**1991**, 259, 144–150. [Google Scholar] [CrossRef] - Deur, A.; Brodsky, S.J.; de Téramond, G.F. Connecting the hadron mass scale to the fundamental mass scale of quantum chromodynamics. Phys. Lett. B
**2015**, 750, 528. [Google Scholar] [CrossRef] - Bjorken, J.D. Asymptotic Sum Rules at Infinite Momentum. Phys. Rev.
**1969**, 179, 1547–1553. [Google Scholar] [CrossRef] - Deur, A.; Brodsky, S.J.; de Téramond, G.F. The Spin Structure of the Nucleon. Rep. Prog. Phys.
**2019**, 82, 7. [Google Scholar] [CrossRef] [Green Version] - Kataev, A.L. The Ellis-Jaffe sum rule: The estimates of the next-to-next-to-leading order QCD corrections. Phys. Rev. D
**1994**, 50, 5469. [Google Scholar] [CrossRef] [Green Version] - Burkardt, M. Transverse force on quarks in deep-inelastic scattering. Phys. Rev. D
**2013**, 88, 114502. [Google Scholar] [CrossRef] - Deur, A. Spin Sum Rules and the Strong Coupling Constant at large distance. AIP Conf. Proc.
**2009**, 1155, 112–121. [Google Scholar] - Gerasimov, S.B. A Sum rule for magnetic moments and the damping of the nucleon magnetic moment in nuclei. Sov. J. Nucl. Phys.
**1966**, 2, 430. [Google Scholar] - Burkert, V.D. Comment on the generalized Gerasimov-Drell-Hearn sum rule in chiral perturbation theory. Phys. Rev. D
**2001**, 63, 097904. [Google Scholar] [CrossRef] [Green Version] - Brodsky, S.J.; de Téramond, G.F.; Deur, A. Nonperturbative QCD Coupling and its β-function from Light-Front Holography. Phys. Rev. D
**2010**, 81, 096010. [Google Scholar] [CrossRef] [Green Version] - Binosi, D.; Mezrag, C.; Papavassiliou, J.; Roberts, C.D.; Rodriguez-Quintero, J. Process-independent strong running coupling. Phys. Rev. D
**2017**, 96, 054026. [Google Scholar] [CrossRef] [Green Version] - Deur, A.; Burkert, V.; Chen, J.P.; Korsch, W. Experimental determination of the effective strong coupling constant. Phys. Lett. B
**2007**, 650, 244. [Google Scholar] [CrossRef] [Green Version] - Adeva, B.; Ahmad, S.; Arvidson, A.; Badelek, B.; Ballintijn, M.K.; Bardin, G.; Voss, R. Measurement of the spin dependent structure function g1(x) of the deuteron. Phys. Lett. B
**1993**, 302, 533. [Google Scholar] [CrossRef] [Green Version] - Airapetian, A.; Akopov, N.; Akushevich, I.; Amarian, M.; Arrington, J.; Aschenauer, E.C.; Schüler, K.P. The Q
^{2}dependence of the generalized Gerasimov-Drell-Hearn integral for the proton. Phys. Lett. B**2000**, 494, 1–8. [Google Scholar] [CrossRef] [Green Version] - Deur, A.; Bosted, P.; Burkert, V.; Cates, G.; Chen, J.P.; Choi, S.; Yun, J. Experimental determination of the evolution of the Bjorken integral at low Q
^{2}. Phys. Rev. Lett.**2004**, 93, 212001. [Google Scholar] [CrossRef] [Green Version] - Anthony, P.L.; Arnold, R.G.; Band, H.R.; Borel, H.; Bosted, P.E.; Breton, V.; Zapalac, G. Deep inelastic scattering of polarized electrons by polarized He-3 and the study of the neutron spin structure. Phys. Rev. D
**1996**, 54, 6620. [Google Scholar] [CrossRef] [Green Version] - Alekseev, M.G.; Alexakhin, V.Y.; Alexandrov, Y.; Alexeev, G.D.; Amoroso, A.; Austregesilo, A.; Padee, A. The Spin-dependent Structure Function of the Proton ${g}_{1}^{p}$ and a Test of the Bjorken Sum Rule. Phys. Lett. B
**2010**, 690, 466–472. [Google Scholar] [CrossRef] [Green Version] - Deur, A.; Prok, Y.; Burkert, V.; Crabb, D.; Girod, F.X.; Griffioen, K.A.; Kvaltine, N. High precision determination of the Q
^{2}evolution of the Bjorken Sum. Phys. Rev. D**2014**, 90, 012009. [Google Scholar] [CrossRef] [Green Version] - Deur, A.; Chen, J.P.; Kuhn, S.E.; Peng, C.; Ripani, M.; Sulkosky, V.; Zheng, X. Experimental study of the behavior of the Bjorken sum at very low Q2. Phys. Lett. B
**2022**, 825, 136878. [Google Scholar] [CrossRef] - Rodríguez-Quintero, J.; Binosi, D.; Mezrag, C.; Papavassiliou, J.; Roberts, C.D. Process-independent effective coupling. From QCD Green’s functions to phenomenology. Few Body Syst.
**2018**, 59, 121. [Google Scholar] [CrossRef] [Green Version] - Deur, A.; Shen, J.M.; Wu, X.G.; Brodsky, S.J.; de Téramond, G.F. Implications of the Principle of Maximum Conformality for the QCD Strong Coupling. Phys. Lett. B
**2017**, 773, 98. [Google Scholar] [CrossRef] - Deur, A.; Brodsky, S.J.; de Téramond, G.F. Determination of ${\Lambda}_{\overline{\mathrm{MS}}}$ at five loops from holographic QCD. J. Phys. G
**2017**, 44, 105005. [Google Scholar] [CrossRef] [Green Version] - Cui, Z.F.; Ding, M.; Gao, F.; Raya, K.; Binosi, D.; Chang, L.; Roberts, C.D.; Rodríguez-Quintero, J.; Schmidt, S.M. Kaon and pion parton distributions. Eur. Phys. J. C
**2020**, 80, 1064. [Google Scholar] [CrossRef] - Janik, R.A. The Dynamics of Quark-Gluon Plasma and AdS/CFT. Lect. Notes Phys.
**2011**, 828, 147–181. [Google Scholar] - Busza, W.; Rajagopal, K.; van der Schee, W. Heavy Ion Collisions: The Big Picture, and the Big Questions. Ann. Rev. Nucl. Part. Sci.
**2018**, 68, 339–376. [Google Scholar] [CrossRef] [Green Version] - Florkowski, W.; Heller, M.P.; Spalinski, M. New theories of relativistic hydrodynamics in the LHC era. Rept. Prog. Phys.
**2018**, 81, 046001. [Google Scholar] [CrossRef] [Green Version] - Jokela, N.; Järvinen, M.; Remes, J. Holographic QCD in the Veneziano limit and neutron stars. JHEP
**2019**, 3, 41. [Google Scholar] [CrossRef] [Green Version] - Sulkosky, V.; Singh, J.T.; Peng, C.; Chen, J.P.; Deur, A.; Abrahamyan, S.; Zhu, L. Measurement of the
^{3}He spin-structure functions and of neutron (^{3}He) spin-dependent sum rules at 0.035 ≤ Q^{2}≤ 0.24 GeV^{2}. Phys. Lett. B**2020**, 805, 135428. [Google Scholar] [CrossRef] - Alcorn, J.; Anderson, B.D.; Aniol, K.A.; Annand, J.R.M.; Auerbach, L.; Arrington, J.; McKeown, R.D. Basic Instrumentation for Hall A at Jefferson Lab. Nucl. Instrum. Meth. A
**2004**, 522, 294–346. [Google Scholar] [CrossRef] - Mecking, B.A.; Adams, G.; Ahmad, S.; Anciant, E.; Anghinolfi, M.; Asavapibhop, B.; Vlassov, A.V. The CEBAF Large Acceptance Spectrometer (CLAS). Nucl. Instrum. Meth. A
**2003**, 503, 513–553. [Google Scholar] [CrossRef] [Green Version] - Prok, Y.; Bosted, P.; Kvaltine, N.; Adhikari, K. Precision measurements of g
_{1}of the proton and the deuteron with 6 GeV electrons. Phys. Rev. C**2014**, 90, 025212. [Google Scholar] [CrossRef] [Green Version] - Zheng, X.; Deur, A.; Kang, H.; Kuhn, S.E.; Ripani, M.; Zhang, J.; Zachariou, N. Measurement of the proton spin structure at long distances. Nat. Phys.
**2021**, 17, 736. [Google Scholar] [CrossRef] - Adhikari, K.P.; Deur, A.; El Fassi, L.; Kang, H.; Kuhn, S.E.; Ripani, M.; CLAS Collaboration. Measurement of the Q2-dependence of the deuteron spin structure function g1 and its moments at low Q2 with CLAS. Phys. Rev. Lett.
**2018**, 120, 062501. [Google Scholar] [CrossRef] [Green Version] - Sulkosky, V. The Spin Structure of
^{3}He and the Neutron at Low Q^{2}: A Measurement of the Generalized Gdh Integrand. Ph.D. Thesis, College of William & Mary, Williamsburg, VA, USA, 2007. [Google Scholar] - Keith, C.D.; Anghinolfi, M.; Battaglieri, M.; Bosted, P.; Branford, D.; Bültmann, S.; Witherspoon, S. A polarized target for the CLAS detector. Nucl. Instrum. Meth. A
**2003**, 501, 327–339. [Google Scholar] [CrossRef] [Green Version] - Garibaldi, F.; Acha, A.; Ambrozewicz, P.; Aniol, K.A.; Baturin, P.; Benaoum, H.; Jefferson Lab Hall A Collaboration. High-resolution hypernuclear spectroscopy at Jefferson Lab, Hall A. Phys. Rev. C
**2019**, 99, 054309. [Google Scholar] [CrossRef] [Green Version] - Kim, J.H.; Harris, D.A.; Arroyo, C.G.; de Barbaro, L.; de Barbaro, P.; Bazarko, A.O.; Bernstein, R.H.; Bodek, A.; Bolton, T.; Budd, H.; et al. A measurement of α
_{s}(Q^{2}) from the Gross–Llewellyn Smith sum rule. Phys. Rev. Lett.**1998**, 81, 3595. [Google Scholar] [CrossRef] [Green Version] - De Alfaro, V.; Fubini, S.; Furlan, G. Conformal invariance in quantum mechanics. Nuovo Cim. A
**1976**, 34, 569. [Google Scholar] [CrossRef] [Green Version] - Trawinski, A.P.; Glazek, S.D.; Brodsky, S.J.; de Téramond, G.F.; Dosch, H.G. Effective confining potentials for QCD. Phys. Rev. D
**2014**, 90, 074017. [Google Scholar] [CrossRef] [Green Version] - Cornwall, J.M. Dynamical Mass Generation in Continuum QCD. Phys. Rev. D
**1982**, 26, 1453. [Google Scholar] [CrossRef] - Abbott, L.F. The Background Field Method Beyond One Loop. Nucl. Phys. B
**1981**, 185, 189–203. [Google Scholar] [CrossRef] [Green Version] - Binosi, D.; Chang, L.; Papavassiliou, J.; Roberts, C.D. Bridging a gap between continuum-QCD and ab initio predictions of hadron observables. Phys. Lett. B
**2015**, 742, 183–188. [Google Scholar] [CrossRef] [Green Version] - Aguilar, A.C.; Binosi, D.; Papavassiliou, J. Gluon and ghost propagators in the Landau gauge: Deriving lattice results from Schwinger-Dyson equations. Phys. Rev. D
**2008**, 78, 025010. [Google Scholar] [CrossRef] [Green Version] - Brodsky, S.J.; Shrock, R. Maximum wavelength of confined quarks and gluons and properties of quantum chromodynamics. Phys. Lett. B
**2008**, 666, 95. [Google Scholar] [CrossRef] [Green Version] - Sufian, R.S.; Liu, T.; de Téramond, G.F.; Dosch, H.G.; Brodsky, S.J.; Deur, A.; Hlfhs Collaboration. Nonperturbative strange-quark sea from lattice QCD, light-front holography, and meson-baryon fluctuation models. Phys. Rev. D
**2018**, 98, 114004. [Google Scholar] [CrossRef] [Green Version]

**Table 1.**Data on ${\alpha}_{{g}_{1}}\left(Q\right)$ from JLab experiments EG4 (top, from $Q=0.143$ GeV to 0.704 GeV), EG4/E97110 (middle, from $Q=0.187$ GeV to 0.490 GeV) and EG1dvcs (bottom, from $Q=0.775$ GeV to 2.177 GeV).

Q (GeV) | ${\mathit{\alpha}}_{{\mathit{g}}_{1}}\pm \mathbf{stat}.\pm \mathbf{syst}.$ |
---|---|

0.143 | 3.064 $\pm \phantom{\rule{3.33333pt}{0ex}}0.043\pm 0.018$ |

0.156 | 3.129 $\pm \phantom{\rule{3.33333pt}{0ex}}0.046\pm 0.019$ |

0.171 | 2.955 $\pm \phantom{\rule{3.33333pt}{0ex}}0.046\pm 0.023$ |

0.187 | 3.083 $\pm \phantom{\rule{3.33333pt}{0ex}}0.044\pm 0.024$ |

0.204 | 3.022 $\pm \phantom{\rule{3.33333pt}{0ex}}0.049\pm 0.024$ |

0.223 | 3.002 $\pm \phantom{\rule{3.33333pt}{0ex}}0.052\pm 0.027$ |

0.243 | 2.988 $\pm \phantom{\rule{3.33333pt}{0ex}}0.055\pm 0.031$ |

0.266 | 2.947 $\pm \phantom{\rule{3.33333pt}{0ex}}0.060\pm 0.035$ |

0.291 | 2.983 $\pm \phantom{\rule{3.33333pt}{0ex}}0.065\pm 0.035$ |

$\mathbf{Q}$(GeV) | ${\mathbf{\alpha}}_{{\mathbf{g}}_{\mathbf{1}}}\pm \mathbf{stat}\mathbf{.}\pm \mathbf{syst}\mathbf{.}$ |

0.317 | 2.961 $\pm \phantom{\rule{3.33333pt}{0ex}}0.062\pm 0.038$ |

0.347 | 2.730 $\pm \phantom{\rule{3.33333pt}{0ex}}0.070\pm 0.044$ |

0.379 | 2.853 $\pm \phantom{\rule{3.33333pt}{0ex}}0.077\pm 0.040$ |

0.414 | 2.745 $\pm \phantom{\rule{3.33333pt}{0ex}}0.076\pm 0.041$ |

0.452 | 2.779 $\pm \phantom{\rule{3.33333pt}{0ex}}0.090\pm 0.043$ |

0.494 | 2.451 $\pm \phantom{\rule{3.33333pt}{0ex}}0.094\pm 0.044$ |

0.540 | 2.397 $\pm \phantom{\rule{3.33333pt}{0ex}}0.092\pm 0.039$ |

0.590 | 2.349 $\pm \phantom{\rule{3.33333pt}{0ex}}0.101\pm 0.040$ |

0.645 | 2.431 $\pm \phantom{\rule{3.33333pt}{0ex}}0.109\pm 0.043$ |

0.704 | 1.996 $\pm \phantom{\rule{3.33333pt}{0ex}}0.131\pm 0.104$ |

$\mathbf{Q}$(GeV) | ${\mathbf{\alpha}}_{{\mathbf{g}}_{\mathbf{1}}}\pm \mathbf{stat}\mathbf{.}\pm \mathbf{syst}\mathbf{.}$ |

0.187 | 3.016 $\pm \phantom{\rule{3.33333pt}{0ex}}0.009\pm 0.027$ |

0.239 | 2.973 $\pm \phantom{\rule{3.33333pt}{0ex}}0.015\pm 0.035$ |

0.281 | 2.952 $\pm \phantom{\rule{3.33333pt}{0ex}}0.021\pm 0.041$ |

0.316 | 2.929 $\pm \phantom{\rule{3.33333pt}{0ex}}0.017\pm 0.048$ |

0.387 | 2.815 $\pm \phantom{\rule{3.33333pt}{0ex}}0.021\pm 0.076$ |

0.447 | 2.704 $\pm \phantom{\rule{3.33333pt}{0ex}}0.025\pm 0.086$ |

0.490 | 2.575 $\pm \phantom{\rule{3.33333pt}{0ex}}0.031\pm 0.053$ |

0.775 | 1.743 $\pm \phantom{\rule{3.33333pt}{0ex}}0.007\pm 0.071$ |

0.835 | 1.571 $\pm \phantom{\rule{3.33333pt}{0ex}}0.007\pm 0.101$ |

0.917 | 1.419 $\pm \phantom{\rule{3.33333pt}{0ex}}0.009\pm 0.132$ |

0.986 | 1.341 $\pm \phantom{\rule{3.33333pt}{0ex}}0.010\pm 0.147$ |

1.088 | 1.272 $\pm \phantom{\rule{3.33333pt}{0ex}}0.010\pm 0.156$ |

1.167 | 1.121 $\pm \phantom{\rule{3.33333pt}{0ex}}0.013\pm 0.153$ |

1.261 | 0.955 $\pm \phantom{\rule{3.33333pt}{0ex}}0.016\pm 0.146$ |

1.384 | 0.874 $\pm \phantom{\rule{3.33333pt}{0ex}}0.016\pm 0.269$ |

1.522 | 0.730 $\pm \phantom{\rule{3.33333pt}{0ex}}0.012\pm 0.280$ |

1.645 | 0.708 $\pm \phantom{\rule{3.33333pt}{0ex}}0.009\pm 0.257$ |

1.795 | 0.617 $\pm \phantom{\rule{3.33333pt}{0ex}}0.007\pm 0.254$ |

1.967 | 0.581 $\pm \phantom{\rule{3.33333pt}{0ex}}0.006\pm 0.223$ |

2.177 | 0.636 $\pm \phantom{\rule{3.33333pt}{0ex}}0.003\pm 0.187$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Deur, A.; Burkert, V.; Chen, J.-P.; Korsch, W.
Experimental Determination of the QCD Effective Charge *α*_{g1}(*Q*). *Particles* **2022**, *5*, 171-179.
https://doi.org/10.3390/particles5020015

**AMA Style**

Deur A, Burkert V, Chen J-P, Korsch W.
Experimental Determination of the QCD Effective Charge *α*_{g1}(*Q*). *Particles*. 2022; 5(2):171-179.
https://doi.org/10.3390/particles5020015

**Chicago/Turabian Style**

Deur, Alexandre, Volker Burkert, Jian-Ping Chen, and Wolfgang Korsch.
2022. "Experimental Determination of the QCD Effective Charge *α*_{g1}(*Q*)" *Particles* 5, no. 2: 171-179.
https://doi.org/10.3390/particles5020015