Abstract
We investigate whether it is possible to extract the quark mass anomalous dimension and its scale dependence from the spectrum of the twisted mass Dirac operator in Lattice QCD. The answer to this question appears to be positive, provided that one goes to large enough eigenvalues, sufficiently above the non-perturbative regime. The obtained results are compared to continuum perturbation theory. By analyzing possible sources of systematic effects, we find the domain of applicability of the approach, extending from an energy scale of around 1.5 to 4 GeV. The lower limit is dictated by physics (non-perturbative effects at low energies), while the upper bound is set by the ultraviolet cut-off of present-day lattice simulations. The information about the scale dependence of the anomalous dimension allows also to extract the value of the \( {\varLambda}_{\overline{\mathrm{MS}}} \)-parameter of 2-flavour QCD, yielding the value 303(13)(25) MeV, where the first error is statistical and the second one systematic. We use gauge field configuration ensembles generated by the European Twisted Mass Collaboration (ETMC) with 2 flavours of dynamical twisted mass quarks, at 4 lattice spacings in the range between around 0.04 and 0.08 fm.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Banks and A. Casher, Chiral symmetry breaking in confining theories, Nucl. Phys. B 169 (1980) 103 [INSPIRE].
L. Giusti and M. Lüscher, Chiral symmetry breaking and the Banks-Casher relation in lattice QCD with Wilson quarks, JHEP 03 (2009) 013 [arXiv:0812.3638] [INSPIRE].
K. Cichy, E. Garcia-Ramos and K. Jansen, Chiral condensate from the twisted mass Dirac operator spectrum, JHEP 10 (2013) 175 [arXiv:1303.1954] [INSPIRE].
K. Cichy, E. Garcia-Ramos, K. Jansen and A. Shindler, Computation of the chiral condensate using N f = 2 and N f = 2 + 1 + 1 dynamical flavors of twisted mass fermions, PoS(LATTICE 2013)128 [arXiv:1312.3534] [INSPIRE].
G.P. Engel, L. Giusti, S. Lottini and R. Sommer, Chiral condensate from the Banks-Casher relation, arXiv:1309.4537 [INSPIRE].
E.V. Shuryak and J.J.M. Verbaarschot, Random matrix theory and spectral sum rules for the Dirac operator in QCD, Nucl. Phys. A 560 (1993) 306 [hep-th/9212088] [INSPIRE].
P.H. Damgaard, U.M. Heller, R. Niclasen and K. Rummukainen, Eigenvalue distributions of the QCD Dirac operator, Phys. Lett. B 495 (2000) 263 [hep-lat/0007041] [INSPIRE].
T.A. DeGrand and S. Schaefer, Chiral properties of two-flavor QCD in small volume and at large lattice spacing, Phys. Rev. D 72 (2005) 054503 [hep-lat/0506021] [INSPIRE].
C.B. Lang, P. Majumdar and W. Ortner, The condensate for two dynamical chirally improved quarks in QCD, Phys. Lett. B 649 (2007) 225 [hep-lat/0611010] [INSPIRE].
JLQCD collaboration, H. Fukaya et al., Two-flavor lattice QCD simulation in the ϵ-regime with exact chiral symmetry, Phys. Rev. Lett. 98 (2007) 172001 [hep-lat/0702003] [INSPIRE].
TWQCD collaboration, H. Fukaya et al., Two-flavor lattice QCD in the ϵ-regime and chiral random matrix theory, Phys. Rev. D 76 (2007) 054503 [arXiv:0705.3322] [INSPIRE].
JLQCD and TWQCD collaborations, H. Fukaya et al., Determination of the chiral condensate from QCD Dirac spectrum on the lattice, Phys. Rev. D 83 (2011) 074501 [arXiv:1012.4052] [INSPIRE].
F. Bernardoni, P. Hernández, N. Garron, S. Necco and C. Pena, Probing the chiral regime of N f = 2 QCD with mixed actions, Phys. Rev. D 83 (2011) 054503 [arXiv:1008.1870] [INSPIRE].
K. Splittorff and J.J.M. Verbaarschot, The microscopic twisted mass Dirac spectrum, Phys. Rev. D 85 (2012) 105008 [arXiv:1201.1361] [INSPIRE].
M. Lüscher and F. Palombi, Universality of the topological susceptibility in the SU(3) gauge theory, JHEP 09 (2010) 110 [arXiv:1008.0732] [INSPIRE].
K. Cichy, V. Drach, E. Garcia-Ramos and K. Jansen, Topological susceptibility and chiral condensate with N f = 2 + 1 + 1 dynamical flavors of maximally twisted mass fermions, PoS(Lattice 2011)102 [arXiv:1111.3322] [INSPIRE].
K. Cichy, E. Garcia-Ramos, K. Jansen and A. Shindler, Topological susceptibility from twisted mass fermions using spectral projectors, PoS(LATTICE 2013)129 [arXiv:1312.3535] [INSPIRE].
ETM collaboration, K. Cichy, E. Garcia-Ramos and K. Jansen, Topological susceptibility from the twisted mass Dirac operator spectrum, JHEP 02 (2014) 119 [arXiv:1312.5161] [INSPIRE].
T. DeGrand, Finite-size scaling tests for SU(3) lattice gauge theory with color sextet fermions, Phys. Rev. D 80 (2009) 114507 [arXiv:0910.3072] [INSPIRE].
L. Del Debbio and R. Zwicky, Hyperscaling relations in mass-deformed conformal gauge theories, Phys. Rev. D 82 (2010) 014502 [arXiv:1005.2371] [INSPIRE].
A. Cheng, A. Hasenfratz and D. Schaich, Novel phase in SU(3) lattice gauge theory with 12 light fermions, Phys. Rev. D 85 (2012) 094509 [arXiv:1111.2317] [INSPIRE].
A. Hasenfratz, A. Cheng, G. Petropoulos and D. Schaich, Mass anomalous dimension from Dirac eigenmode scaling in conformal and confining systems, PoS(Lattice 2012)034 [arXiv:1207.7162] [INSPIRE].
A. Cheng, A. Hasenfratz, G. Petropoulos and D. Schaich, Scale-dependent mass anomalous dimension from Dirac eigenmodes, JHEP 07 (2013) 061 [arXiv:1301.1355] [INSPIRE].
A. Cheng, A. Hasenfratz, G. Petropoulos and D. Schaich, Determining the mass anomalous dimension through the eigenmodes of Dirac operator, PoS(LATTICE 2013) 088 [arXiv:1311.1287] [INSPIRE].
A. Patella, GMOR-like relation in IR-conformal gauge theories, Phys. Rev. D 84 (2011) 125033 [arXiv:1106.3494] [INSPIRE].
A. Patella, A precise determination of the \( \overline{\psi} \) ψ anomalous dimension in conformal gauge theories, Phys. Rev. D 86 (2012) 025006 [arXiv:1204.4432] [INSPIRE].
L. Keegan, Mass anomalous dimension at large-N, PoS(Lattice 2012)044 [arXiv:1210.7247] [INSPIRE].
M.G. Perez, A. Gonzalez-Arroyo, L. Keegan and M. Okawa, Mass anomalous dimension from large-N twisted volume reduction, arXiv:1311.2395 [INSPIRE].
D. Landa-Marban, W. Bietenholz and I. Hip, Features of a 2d gauge theory with vanishing chiral condensate, arXiv:1307.0231 [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].
ALPHA collaboration, S. Capitani, M. Lüscher, R. Sommer and H. Wittig, Non-perturbative quark mass renormalization in quenched lattice QCD, Nucl. Phys. B 544 (1999) 669 [hep-lat/9810063] [INSPIRE].
ALPHA collaboration, M. Della Morte et al., Non-perturbative quark mass renormalization in two-flavor QCD, Nucl. Phys. B 729 (2005) 117 [hep-lat/0507035] [INSPIRE].
F. Bursa, L. Del Debbio, L. Keegan, C. Pica and T. Pickup, Mass anomalous dimension in SU(2) with two adjoint fermions, Phys. Rev. D 81 (2010) 014505 [arXiv:0910.4535] [INSPIRE].
P. Fritzsch, J. Heitger and N. Tantalo, Non-perturbative improvement of quark mass renormalization in two-flavour lattice QCD, JHEP 08 (2010) 074 [arXiv:1004.3978] [INSPIRE].
PACS-CS collaboration, S. Aoki et al., Non-perturbative renormalization of quark mass in N f = 2 + 1 QCD with the Schrödinger functional scheme, JHEP 08 (2010) 101 [arXiv:1006.1164] [INSPIRE].
T. DeGrand, Y. Shamir and B. Svetitsky, Running coupling and mass anomalous dimension of SU(3) gauge theory with two flavors of symmetric-representation fermions, Phys. Rev. D 82 (2010) 054503 [arXiv:1006.0707] [INSPIRE].
B. Svetitsky, Y. Shamir and T. DeGrand, Sextet QCD: slow running and the mass anomalous dimension, PoS(Lattice 2010)072 [arXiv:1010.3396] [INSPIRE].
T. DeGrand, Y. Shamir and B. Svetitsky, SU(4) lattice gauge theory with decuplet fermions: Schrödinger functional analysis, Phys. Rev. D 85 (2012) 074506 [arXiv:1202.2675] [INSPIRE].
J.G. López, K. Jansen, D.B. Renner and A. Shindler, A quenched study of the Schrödinger functional with chirally rotated boundary conditions: applications, Nucl. Phys. B 867 (2013) 609 [arXiv:1208.4661] [INSPIRE].
Y. Zhestkov, Domain wall fermion study of scaling in non-perturbative renormalization of quark bilinears and B K , hep-lat/0101008 [INSPIRE].
ETM collaboration, M. Constantinou et al., Non-perturbative renormalization of quark bilinear operators with N f = 2 (tmQCD) Wilson fermions and the tree-level improved gauge action, JHEP 08 (2010) 068 [arXiv:1004.1115] [INSPIRE].
RBC and UKQCD collaborations, R. Arthur and P.A. Boyle, Step scaling with off-shell renormalisation, Phys. Rev. D 83 (2011) 114511 [arXiv:1006.0422] [INSPIRE].
RBC and UKQCD collaborations, R. Arthur et al., Domain wall QCD with near-physical pions, Phys. Rev. D 87 (2013) 094514 [arXiv:1208.4412] [INSPIRE].
K.G. Chetyrkin, Quark mass anomalous dimension to O(α 4 S ), Phys. Lett. B 404 (1997) 161 [hep-ph/9703278] [INSPIRE].
J.A.M. Vermaseren, S.A. Larin and T. van Ritbergen, The four loop quark mass anomalous dimension and the invariant quark mass, Phys. Lett. B 405 (1997) 327 [hep-ph/9703284] [INSPIRE].
T.A. Ryttov and F. Sannino, Supersymmetry inspired QCD β-function, Phys. Rev. D 78 (2008) 065001 [arXiv:0711.3745] [INSPIRE].
K. Jansen and C. Urbach, tmLQCD: a program suite to simulate Wilson twisted mass lattice QCD, Comput. Phys. Commun. 180 (2009) 2717 [arXiv:0905.3331] [INSPIRE].
R. Frezzotti and G.C. Rossi, Chirally improving Wilson fermions 1. O(a) improvement, JHEP 08 (2004) 007 [hep-lat/0306014] [INSPIRE].
P. Weisz, Continuum limit improved lattice action for pure Yang-Mills theory (I), Nucl. Phys. B 212 (1983) 1 [INSPIRE].
Alpha collaboration, R. Frezzotti, P.A. Grassi, S. Sint and P. Weisz, Lattice QCD with a chirally twisted mass term, JHEP 08 (2001) 058 [hep-lat/0101001] [INSPIRE].
R. Frezzotti and G.C. Rossi, Chirally improving Wilson fermions 2. Four-quark operators, JHEP 10 (2004) 070 [hep-lat/0407002] [INSPIRE].
A. Shindler, Twisted mass lattice QCD, Phys. Rept. 461 (2008) 37 [arXiv:0707.4093] [INSPIRE].
F. Farchioni et al., Exploring the phase structure of lattice QCD with twisted mass quarks, Nucl. Phys. Proc. Suppl. 140 (2005) 240 [hep-lat/0409098] [INSPIRE].
F. Farchioni et al., The phase structure of lattice QCD with Wilson quarks and renormalization group improved gluons, Eur. Phys. J. C 42 (2005) 73 [hep-lat/0410031] [INSPIRE].
R. Frezzotti, G. Martinelli, M. Papinutto and G.C. Rossi, Reducing cutoff effects in maximally twisted lattice QCD close to the chiral limit, JHEP 04 (2006) 038 [hep-lat/0503034] [INSPIRE].
XLF collaboration, K. Jansen, M. Papinutto, A. Shindler, C. Urbach and I. Wetzorke, Quenched scaling of Wilson twisted mass fermions, JHEP 09 (2005) 071 [hep-lat/0507010] [INSPIRE].
ETM collaboration, P. Boucaud et al., Dynamical twisted mass fermions with light quarks, Phys. Lett. B 650 (2007) 304 [hep-lat/0701012] [INSPIRE].
ETM collaboration, P. Boucaud et al., Dynamical twisted mass fermions with light quarks: simulation and analysis details, Comput. Phys. Commun. 179 (2008) 695 [arXiv:0803.0224] [INSPIRE].
ETM collaboration, R. Baron et al., Light meson physics from maximally twisted mass lattice QCD, JHEP 08 (2010) 097 [arXiv:0911.5061] [INSPIRE].
ETM collaboration, B. Blossier et al., Average up/down, strange and charm quark masses with N f = 2 twisted mass lattice QCD, Phys. Rev. D 82 (2010) 114513 [arXiv:1010.3659] [INSPIRE].
ETM collaboration, K. Jansen, F. Karbstein, A. Nagy and M. Wagner, \( {\varLambda}_{\overline{\mathrm{MS}}} \) from the static potential for QCD with n f = 2 dynamical quark flavors, JHEP 01 (2012) 025 [arXiv:1110.6859] [INSPIRE].
C. Alexandrou, M. Constantinou, T. Korzec, H. Panagopoulos and F. Stylianou, Renormalization constants of local operators for Wilson type improved fermions, Phys. Rev. D 86 (2012) 014505 [arXiv:1201.5025] [INSPIRE].
K. Cichy, K. Jansen and P. Korcyl, Non-perturbative renormalization in coordinate space for N f = 2 maximally twisted mass fermions with tree-level Symanzik improved gauge action, Nucl. Phys. B 865 (2012) 268 [arXiv:1207.0628] [INSPIRE].
G. Martinelli, C. Pittori, C.T. Sachrajda, M. Testa and A. Vladikas, A general method for non-perturbative renormalization of lattice operators, Nucl. Phys. B 445 (1995) 81 [hep-lat/9411010] [INSPIRE].
G. Martinelli et al., Non-perturbative improvement of composite operators with Wilson fermions, Phys. Lett. B 411 (1997) 141 [hep-lat/9705018] [INSPIRE].
V. Giménez et al., Non-perturbative renormalization of lattice operators in coordinate space, Phys. Lett. B 598 (2004) 227 [hep-lat/0406019] [INSPIRE].
K.G. Chetyrkin and A. Retey, Renormalization and running of quark mass and field in the regularization invariant and MS schemes at three loops and four loops, Nucl. Phys. B 583 (2000) 3 [hep-ph/9910332] [INSPIRE].
K.G. Chetyrkin and A. Maier, Massless correlators of vector, scalar and tensor currents in position space at orders α 3 s and α 4 s : explicit analytical results, Nucl. Phys. B 844 (2011) 266 [arXiv:1010.1145] [INSPIRE].
F. Karbstein, A. Peters and M. Wagner, \( {\varLambda}_{\overline{\mathrm{MS}}}^{\left({n}_f=2\right)} \) MS from a momentum space analysis of the quark-antiquark static potential, arXiv:1407.7503 [INSPIRE].
P. Fritzsch et al., The strange quark mass and Lambda parameter of two flavor QCD, Nucl. Phys. B 865 (2012) 397 [arXiv:1205.5380] [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: 1311.3572
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
Cichy, K. Quark mass anomalous dimension and \( {\varLambda}_{\overline{\mathrm{MS}}} \) from the twisted mass Dirac operator spectrum. J. High Energ. Phys. 2014, 127 (2014). https://doi.org/10.1007/JHEP08(2014)127
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2014)127