Abstract
We evaluate the Λ-parameter in the \( \overline{\mathrm{MS}} \) scheme for the pure SU(3) gauge theory with the twisted gradient flow (TGF) method. A running coupling constant g 2TGF (1/L) is defined in a finite volume box with size of L4 with the twisted boundary condition. This defines the TGF scheme. Using the step scaling method for the TGF coupling with lattice simulations, we can evaluate the Λ-parameter non-perturbatively in the TGF scheme. In this paper we determine the dimensionless ratios, \( {\varLambda}_{\mathrm{TGF}}/\sqrt{\sigma } \) and r0ΛTGF together with the Λ-parameter ratio ΛSF/ΛTGF on the lattices numerically. Combined with the known ratio \( {\varLambda}_{\overline{\mathrm{MS}}}/{\varLambda}_{\mathrm{SF}} \), we obtain \( {\varLambda}_{\overline{\mathrm{MS}}}/\sqrt{\sigma } = 0.5315(81)\left({}_{-48}^{+269}\right) \) and \( \mathrm{r}0{\varLambda}_{\overline{\mathrm{MS}}}=0.6062(92)\left({}_{-52}^{+309}\right) \), where the first error is statistical one and the second is our estimate of systematic uncertainty.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Allton, M. Teper and A. Trivini, On the running of the bare coupling in SU(N) lattice gauge theories, JHEP 07 (2008) 021 [arXiv:0803.1092] [INSPIRE].
G. Parisi, Recent progresses in gauge theories, World Sci. Lect. Notes Phys. 49 (1980) 349 [AIP Conf. Proc. 68 (1980) 1531] [LNF-80-52-P] [INSPIRE].
M. Lüscher, P. Weisz and U. Wolff, A numerical method to compute the running coupling in asymptotically free theories, Nucl. Phys. B 359 (1991) 221 [INSPIRE].
M. Lüscher, R. Narayanan, P. Weisz and U. Wolff, The Schrödinger functional: a renormalizable probe for non-Abelian gauge theories, Nucl. Phys. B 384 (1992) 168 [hep-lat/9207009] [INSPIRE].
M. Lüscher, R. Sommer, U. Wolff and P. Weisz, Computation of the running coupling in the SU(2) Yang-Mills theory, Nucl. Phys. B 389 (1993) 247 [hep-lat/9207010] [INSPIRE].
M. Lüscher, R. Sommer, P. Weisz and U. Wolff, A precise determination of the running coupling in the SU(3) Yang-Mills theory, Nucl. Phys. B 413 (1994) 481 [hep-lat/9309005] [INSPIRE].
ALPHA collaboration, A. Bode, P. Weisz and U. Wolff, Two loop computation of the Schrödinger functional in lattice QCD, Nucl. Phys. B 576 (2000) 517 [Erratum ibid. B 600 (2001) 453] [Erratum ibid. B 608 (2001) 481] [hep-lat/9911018] [INSPIRE].
ALPHA collaboration, M. Della Morte, R. Frezzotti, J. Heitger, J. Rolf, R. Sommer and U. Wolff, Computation of the strong coupling in QCD with two dynamical flavors, Nucl. Phys. B 713 (2005) 378 [hep-lat/0411025] [INSPIRE].
ALPHA collaboration, F. Tekin, R. Sommer and U. Wolff, The running coupling of QCD with four flavors, Nucl. Phys. B 840 (2010) 114 [arXiv:1006.0672] [INSPIRE].
S. Capitani, M. Lüscher, R. Sommer and H. Wittig, Non-perturbative quark mass renormalization in quenched lattice QCD, Nucl. Phys. B 544 (1999) 669 [Erratum ibid. B 582 (2000) 762] [hep-lat/9810063] [INSPIRE].
M. Dalla Brida and M. Lüscher, SMD-based numerical stochastic perturbation theory, Eur. Phys. J. C 77 (2017) 308 [arXiv:1703.04396] [INSPIRE].
S. Sint and R. Sommer, The running coupling from the QCD Schrödinger functional: a one loop analysis, Nucl. Phys. B 465 (1996) 71 [hep-lat/9508012] [INSPIRE].
G.M. de Divitiis, R. Frezzotti, M. Guagnelli and R. Petronzio, A definition of the running coupling constant in a twisted SU(2) lattice gauge theory, Nucl. Phys. B 422 (1994) 382 [hep-lat/9312085] [INSPIRE].
G.M. de Divitiis, R. Frezzotti, M. Guagnelli and R. Petronzio, Nonperturbative determination of the running coupling constant in quenched SU(2), Nucl. Phys. B 433 (1995)390 [hep-lat/9407028] [INSPIRE].
Alpha collaboration, G. de Divitiis et al., Universality and the approach to the continuum limit in lattice gauge theory, Nucl. Phys. B 437 (1995) 447 [hep-lat/9411017] [INSPIRE].
E. Bilgici et al., A new scheme for the running coupling constant in gauge theories using Wilson loops, Phys. Rev. D 80 (2009) 034507 [arXiv:0902.3768] [INSPIRE].
E. Itou, Properties of the twisted Polyakov loop coupling and the infrared fixed point in the SU(3) gauge theories, PTEP 2013 (2013) 083B01 [arXiv:1212.1353] [INSPIRE].
R. Narayanan and H. Neuberger, Infinite N phase transitions in continuum Wilson loop operators, JHEP 03 (2006) 064 [hep-th/0601210] [INSPIRE].
M. Lüscher, Properties and uses of the Wilson flow in lattice QCD, JHEP 08 (2010) 071 [Erratum ibid. 03 (2014) 092] [arXiv:1006.4518] [INSPIRE].
M. Lüscher and P. Weisz, Perturbative analysis of the gradient flow in non-Abelian gauge theories, JHEP 02 (2011) 051 [arXiv:1101.0963] [INSPIRE].
Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C.H. Wong, The Yang-Mills gradient flow in finite volume, JHEP 11 (2012) 007 [arXiv:1208.1051] [INSPIRE].
A. Ramos, The gradient flow running coupling with twisted boundary conditions, JHEP 11 (2014) 101 [arXiv:1409.1445] [INSPIRE].
V. Leino, J. Rantaharju, T. Rantalaiho, K. Rummukainen, J.M. Suorsa and K. Tuominen, The gradient flow running coupling in SU(2) gauge theory with N f = 8 fundamental flavors, Phys. Rev. D 95 (2017) 114516 [arXiv:1701.04666] [INSPIRE].
C.-J.D. Lin, K. Ogawa and A. Ramos, The Yang-Mills gradient flow and SU(3) gauge theory with 12 massless fundamental fermions in a colour-twisted box, JHEP 12 (2015) 103 [arXiv:1510.05755] [INSPIRE].
P. Fritzsch and A. Ramos, The gradient flow coupling in the Schrödinger functional, JHEP 10 (2013) 008 [arXiv:1301.4388] [INSPIRE].
ALPHA collaboration, M. Dalla Brida, P. Fritzsch, T. Korzec, A. Ramos, S. Sint and R. Sommer, Slow running of the gradient flow coupling from 200 MeV to 4 GeV in N f = 3 QCD, Phys. Rev. D 95 (2017) 014507 [arXiv:1607.06423] [INSPIRE].
ALPHA collaboration, M. Dalla Brida, P. Fritzsch, T. Korzec, A. Ramos, S. Sint and R. Sommer, A status update on the determination of \( {\Lambda}_{\overline{\mathrm{MS}}}^{N_f=3} \) by the ALPHA collaboration, MS PoS(LATTICE 2015)248 [arXiv:1511.05831] [INSPIRE].
E.I. Bribian and M. Garcia Perez, Perturbative running of the twisted Yang-Mills coupling in the gradient flow scheme, PoS(LATTICE2016)371 [arXiv:1611.07221] [INSPIRE].
M. Asakawa, T. Hatsuda, T. Iritani, E. Itou, M. Kitazawa and H. Suzuki, Determination of reference scales for Wilson gauge action from Yang-Mills gradient flow, arXiv:1503.06516 [INSPIRE].
A. González-Arroyo and M. Okawa, The string tension from smeared Wilson loops at large-N, Phys. Lett. B 718 (2013) 1524 [arXiv:1206.0049] [INSPIRE].
S. Necco and R. Sommer, The N f = 0 heavy quark potential from short to intermediate distances, Nucl. Phys. B 622 (2002) 328 [hep-lat/0108008] [INSPIRE].
K.-I. Ishikawa, I. Kanamori, Y. Murakami, A. Nakamura, M. Okawa and R. Ueno, Numerical determination of the Λ-parameter in SU(3) gauge theory from the twisted gradient flow coupling, PoS(LATTICE2016)185 [arXiv:1612.01676] [INSPIRE].
K. Fabricius and O. Haan, Heat bath method for the twisted Eguchi-Kawai model, Phys. Lett. B 143 (1984) 459 [INSPIRE].
ALPHA collaboration, U. Wolff, Monte Carlo errors with less errors, Comput. Phys. Commun. 156 (2004) 143 [Erratum ibid. 176 (2007) 383] [hep-lat/0306017] [INSPIRE].
Alpha collaboration, A. Bode, U. Wolff and P. Weisz, Two loop computation of the Schrödinger functional in pure SU(3) lattice gauge theory, Nucl. Phys. B 540 (1999) 491 [hep-lat/9809175] [INSPIRE].
T. Awaya, Two-dimensional curve fitting in counting experiments, Nucl. Instrum. Meth. 212 (1983) 311.
G.S. Bali and K. Schilling, Running coupling and the Λ parameter from SU(3) lattice simulations, Phys. Rev. D 47 (1993) 661 [hep-lat/9208028] [INSPIRE].
S. Aoki et al., Review of lattice results concerning low-energy particle physics, Eur. Phys. J. C 77 (2017) 112 [arXiv:1607.00299] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1702.06289
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Ishikawa, KI., Kanamori, I., Murakami, Y. et al. Non-perturbative determination of the Λ-parameter in the pure SU(3) gauge theory from the twisted gradient flow coupling. J. High Energ. Phys. 2017, 67 (2017). https://doi.org/10.1007/JHEP12(2017)067
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2017)067