Abstract
Off-lightcone Wilson-line operators are constructed using local operators connected by time-like or space-like Wilson lines, which ensure gauge invariance. Off-lightcone Wilson-line operators have broad applications in various contexts. For instance, space-like Wilson-line operators play a crucial role in determining quasi-distribution functions (quasi-PDFs), while time-like Wilson-line operators are essential for understanding quarkonium decay and production within the potential non-relativistic QCD (pNRQCD) framework. In this work, we establish a systematic approach for calculating the matching from the gradient-flow scheme to the \( \overline{\textrm{MS}} \) scheme in the limit of small flow time for off-lightcone Wilson-line operators. By employing the one-dimensional auxiliary-field formalism, we simplify the matching procedure, reducing it to the matching of local current operators. We provide one-loop level matching coefficients for these local current operators. For the case of hadronic matrix element related to the quark quasi-PDFs, we show at one-loop level that the finite flow time effect is very small as long as the flow radius is smaller than the physical distance z, which is usually satisfied in lattice gradient flow computations. Applications include lattice gradient flow computations of quark/gluon quasi-PDFs, gluonic correlators related to quarkonium decay and production in pNRQCD, and spin-dependent potentials in terms of chromoelectric and chromomagnetic field insertions into a Wilson loop.
Article PDF
Avoid common mistakes on your manuscript.
References
A.M. Polyakov, Gauge Fields as Rings of Glue, Nucl. Phys. B 164 (1980) 171 [INSPIRE].
J.-L. Gervais and A. Neveu, The Slope of the Leading Regge Trajectory in Quantum Chromodynamics, Nucl. Phys. B 163 (1980) 189 [INSPIRE].
V.S. Dotsenko and S.N. Vergeles, Renormalizability of Phase Factors in the Nonabelian Gauge Theory, Nucl. Phys. B 169 (1980) 527 [INSPIRE].
N.S. Craigie and H. Dorn, On the Renormalization and Short Distance Properties of Hadronic Operators in QCD, Nucl. Phys. B 185 (1981) 204 [INSPIRE].
H. Dorn, Renormalization of Path Ordered Phase Factors and Related Hadron Operators in Gauge Field Theories, Fortsch. Phys. 34 (1986) 11 [INSPIRE].
I.Y. Arefeva, Quantum contour field equations, Phys. Lett. B 93 (1980) 347 [INSPIRE].
I.Y. Arefeva, Elimination of Divergences in an Integral Formulation of Yang-Mills Theory, JETP Lett. 31 (1980) 393 [INSPIRE].
S. Samuel, Color zitterbewegung, Nucl. Phys. B 149 (1979) 517 [INSPIRE].
R.A. Brandt, F. Neri and D. Zwanziger, Lorentz Invariance From Classical Particle Paths in Quantum Field Theory of Electric and Magnetic Charge, Phys. Rev. D 19 (1979) 1153 [INSPIRE].
X. Ji, Parton Physics on a Euclidean Lattice, Phys. Rev. Lett. 110 (2013) 262002 [arXiv:1305.1539] [INSPIRE].
A.V. Radyushkin, Quasi-parton distribution functions, momentum distributions, and pseudo-parton distribution functions, Phys. Rev. D 96 (2017) 034025 [arXiv:1705.01488] [INSPIRE].
A. Pineda and J. Soto, Effective field theory for ultrasoft momenta in NRQCD and NRQED, Nucl. Phys. B Proc. Suppl. 64 (1998) 428 [hep-ph/9707481] [INSPIRE].
N. Brambilla, A. Pineda, J. Soto and A. Vairo, Potential NRQCD: An effective theory for heavy quarkonium, Nucl. Phys. B 566 (2000) 275 [hep-ph/9907240] [INSPIRE].
N. Brambilla, A. Pineda, J. Soto and A. Vairo, Effective Field Theories for Heavy Quarkonium, Rev. Mod. Phys. 77 (2005) 1423 [hep-ph/0410047] [INSPIRE].
N. Brambilla et al., New predictions for inclusive heavy quarkonium P wave decays, Phys. Rev. Lett. 88 (2002) 012003 [hep-ph/0109130] [INSPIRE].
N. Brambilla et al., Inclusive decays of heavy quarkonium to light particles, Phys. Rev. D 67 (2003) 034018 [hep-ph/0208019] [INSPIRE].
N. Brambilla, H.S. Chung and A. Vairo, Inclusive Hadroproduction of P-Wave Heavy Quarkonia in Potential Nonrelativistic QCD, Phys. Rev. Lett. 126 (2021) 082003 [arXiv:2007.07613] [INSPIRE].
N. Brambilla, H.S. Chung and A. Vairo, Inclusive production of heavy quarkonia in pNRQCD, JHEP 09 (2021) 032 [arXiv:2106.09417] [INSPIRE].
N. Brambilla, H.S. Chung, A. Vairo and X.-P. Wang, Production and polarization of S-wave quarkonia in potential nonrelativistic QCD, Phys. Rev. D 105 (2022) L111503 [arXiv:2203.07778] [INSPIRE].
N. Brambilla, H.S. Chung, A. Vairo and X.-P. Wang, Inclusive production of J/ψ, ψ(2S), and Υ states in pNRQCD, JHEP 03 (2023) 242 [arXiv:2210.17345] [INSPIRE].
V. Braun and D. Müller, Exclusive processes in position space and the pion distribution amplitude, Eur. Phys. J. C 55 (2008) 349 [arXiv:0709.1348] [INSPIRE].
X. Ji, Parton Physics from Large-Momentum Effective Field Theory, Sci. China Phys. Mech. Astron. 57 (2014) 1407 [arXiv:1404.6680] [INSPIRE].
T. Izubuchi et al., Factorization Theorem Relating Euclidean and Light-Cone Parton Distributions, Phys. Rev. D 98 (2018) 056004 [arXiv:1801.03917] [INSPIRE].
X. Ji et al., Large-momentum effective theory, Rev. Mod. Phys. 93 (2021) 035005 [arXiv:2004.03543] [INSPIRE].
A. Di Giacomo, H.G. Dosch, V.I. Shevchenko and Y.A. Simonov, Field correlators in QCD: Theory and applications, Phys. Rept. 372 (2002) 319 [hep-ph/0007223] [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, Trivializing maps, the Wilson flow and the HMC algorithm, Commun. Math. Phys. 293 (2010) 899 [arXiv:0907.5491] [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].
M. Lüscher, Chiral symmetry and the Yang-Mills gradient flow, JHEP 04 (2013) 123 [arXiv:1302.5246] [INSPIRE].
M. Lüscher, Future applications of the Yang-Mills gradient flow in lattice QCD, PoS LATTICE2013 (2014) 016 [arXiv:1308.5598] [INSPIRE].
BMW collaboration, High-precision scale setting in lattice QCD, JHEP 09 (2012) 010 [arXiv:1203.4469] [INSPIRE].
R. Sommer, Scale setting in lattice QCD, PoS LATTICE2013 (2014) 015 [arXiv:1401.3270] [INSPIRE].
H. Suzuki, Energy-momentum tensor from the Yang-Mills gradient flow, PTEP 2013 (2013) 083B03 [Erratum ibid. 2015 (2015) 079201] [arXiv:1304.0533] [INSPIRE].
H. Makino and H. Suzuki, Lattice energy-momentum tensor from the Yang-Mills gradient flow — inclusion of fermion fields, PTEP 2014 (2014) 063B02 [Erratum ibid. 2015 (2015) 079202] [arXiv:1403.4772] [INSPIRE].
R.V. Harlander, Y. Kluth and F. Lange, The two-loop energy-momentum tensor within the gradient-flow formalism, Eur. Phys. J. C 78 (2018) 944 [Erratum ibid. 79 (2019) 858] [arXiv:1808.09837] [INSPIRE].
Flavour Lattice Averaging Group (FLAG) collaboration, FLAG Review 2021, Eur. Phys. J. C 82 (2022) 869 [arXiv:2111.09849] [INSPIRE].
V. Leino, N. Brambilla, J. Mayer-Steudte and A. Vairo, The static force from generalized Wilson loops using gradient flow, EPJ Web Conf. 258 (2022) 04009 [arXiv:2111.10212] [INSPIRE].
J. Mayer-Steudte, N. Brambilla, V. Leino and A. Vairo, Implications of gradient flow on the static force, PoS LATTICE2022 (2023) 353 [arXiv:2212.12400] [INSPIRE].
V. Leino, N. Brambilla, J. Mayer-Steudte and P. Petreczky, Heavy quark diffusion coefficient with gradient flow, PoS LATTICE2022 (2023) 183 [arXiv:2212.10941] [INSPIRE].
C. Monahan and K. Orginos, Quasi parton distributions and the gradient flow, JHEP 03 (2017) 116 [arXiv:1612.01584] [INSPIRE].
C. Monahan, Smeared quasidistributions in perturbation theory, Phys. Rev. D 97 (2018) 054507 [arXiv:1710.04607] [INSPIRE].
J. Artz et al., Results and techniques for higher order calculations within the gradient-flow formalism, JHEP 06 (2019) 121 [Erratum ibid. 10 (2019) 032] [arXiv:1905.00882] [INSPIRE].
SymLat collaboration, Short flow-time coefficients of CP-violating operators, Phys. Rev. D 102 (2020) 034509 [arXiv:2005.04199] [INSPIRE].
A. Suzuki, Y. Taniguchi, H. Suzuki and K. Kanaya, Four quark operators for kaon bag parameter with gradient flow, Phys. Rev. D 102 (2020) 034508 [arXiv:2006.06999] [INSPIRE].
R.V. Harlander, F. Lange and T. Neumann, Hadronic vacuum polarization using gradient flow, JHEP 08 (2020) 109 [arXiv:2007.01057] [INSPIRE].
M. Boers and E. Pallante, Conserved vector current in QCD-like theories and the gradient flow, JHEP 10 (2020) 034 [arXiv:2007.02121] [INSPIRE].
A. Shindler, Flavor-diagonal CP violation: the electric dipole moment, Eur. Phys. J. A 57 (2021) 128 [INSPIRE].
R.V. Harlander and F. Lange, Effective electroweak Hamiltonian in the gradient-flow formalism, Phys. Rev. D 105 (2022) L071504 [arXiv:2201.08618] [INSPIRE].
E. Mereghetti et al., One-loop matching for quark dipole operators in a gradient-flow scheme, JHEP 04 (2022) 050 [arXiv:2111.11449] [INSPIRE].
R. Harlander, M.D. Rizik, J. Borgulat and A. Shindler, Two-loop matching of the chromo-magnetic dipole operator with the gradient flow, PoS LATTICE2022 (2023) 313 [arXiv:2212.09824] [INSPIRE].
N. Brambilla, H.S. Chung, A. Vairo and X.-P. Wang, QCD static force in gradient flow, JHEP 01 (2022) 184 [arXiv:2111.07811] [INSPIRE].
Ò.L. Crosas et al., One-loop matching of the CP-odd three-gluon operator to the gradient flow, Phys. Lett. B 847 (2023) 138301 [arXiv:2308.16221] [INSPIRE].
E. Eichten and B.R. Hill, An Effective Field Theory for the Calculation of Matrix Elements Involving Heavy Quarks, Phys. Lett. B 234 (1990) 511 [INSPIRE].
I.I.Y. Bigi, M.A. Shifman, N.G. Uraltsev and A.I. Vainshtein, The pole mass of the heavy quark. Perturbation theory and beyond, Phys. Rev. D 50 (1994) 2234 [hep-ph/9402360] [INSPIRE].
M. Beneke and V.M. Braun, Heavy quark effective theory beyond perturbation theory: Renormalons, the pole mass and the residual mass term, Nucl. Phys. B 426 (1994) 301 [hep-ph/9402364] [INSPIRE].
V.M. Braun, K.G. Chetyrkin and B.A. Kniehl, Renormalization of parton quasi-distributions beyond the leading order: spacelike vs. timelike, JHEP 07 (2020) 161 [arXiv:2004.01043] [INSPIRE].
H. Dorn, D. Robaschik and E. Wieczorek, Renormalization and Short Distance Properties of Gauge Invariant Gluonium and Hadron Operators, Annalen Phys. 40 (1983) 166 [INSPIRE].
J.-H. Zhang et al., Accessing Gluon Parton Distributions in Large Momentum Effective Theory, Phys. Rev. Lett. 122 (2019) 142001 [arXiv:1808.10824] [INSPIRE].
W. Wang, J.-H. Zhang, S. Zhao and R. Zhu, Complete matching for quasidistribution functions in large momentum effective theory, Phys. Rev. D 100 (2019) 074509 [arXiv:1904.00978] [INSPIRE].
J. Green, K. Jansen and F. Steffens, Nonperturbative Renormalization of Nonlocal Quark Bilinears for Parton Quasidistribution Functions on the Lattice Using an Auxiliary Field, Phys. Rev. Lett. 121 (2018) 022004 [arXiv:1707.07152] [INSPIRE].
X. Ji, J.-H. Zhang and Y. Zhao, Renormalization in Large Momentum Effective Theory of Parton Physics, Phys. Rev. Lett. 120 (2018) 112001 [arXiv:1706.08962] [INSPIRE].
T. Ishikawa, Y.-Q. Ma, J.-W. Qiu and S. Yoshida, Renormalizability of quasiparton distribution functions, Phys. Rev. D 96 (2017) 094019 [arXiv:1707.03107] [INSPIRE].
Z.-Y. Li, Y.-Q. Ma and J.-W. Qiu, Multiplicative Renormalizability of Operators defining Quasiparton Distributions, Phys. Rev. Lett. 122 (2019) 062002 [arXiv:1809.01836] [INSPIRE].
N.G. Stefanis, Gauge invariant quark two point Green’s function through connector insertion to O(αs), Nuovo Cim. A 83 (1984) 205 [INSPIRE].
N.G. Stefanis, Worldline techniques and QCD observables, Acta Phys. Polon. Supp. 6 (2013) 71 [arXiv:1211.7218] [INSPIRE].
N. Brambilla, A. Pineda, J. Soto and A. Vairo, The QCD potential at O(1/m), Phys. Rev. D 63 (2001) 014023 [hep-ph/0002250] [INSPIRE].
A. Pineda and A. Vairo, The QCD potential at O(1/m2): Complete spin dependent and spin independent result, Phys. Rev. D 63 (2001) 054007 [Erratum ibid. 64 (2001) 039902] [hep-ph/0009145] [INSPIRE].
A.M. Eller, The Color-Electric Field Correlator under Gradient Flow at next-to-leading Order in Quantum Chromodynamics, Ph.D. thesis, Technische Universität Darmstadt, 64289 Darmstadt, Germany (2021) [INSPIRE].
H. Dorn and E. Wieczorek, Renormalization and Short Distance Properties of String Type Equations in QCD, Z. Phys. C 9 (1981) 49 [Erratum ibid. 9 (1981) 274] [INSPIRE].
E. Eichten and B.R. Hill, Static effective field theory: 1/m corrections, Phys. Lett. B 243 (1990) 427 [INSPIRE].
A.F. Falk, B. Grinstein and M.E. Luke, Leading mass corrections to the heavy quark effective theory, Nucl. Phys. B 357 (1991) 185 [INSPIRE].
L.F. Abbott, Introduction to the Background Field Method, Acta Phys. Polon. B 13 (1982) 33 [INSPIRE].
G. Amoros, M. Beneke and M. Neubert, Two loop anomalous dimension of the chromomagnetic moment of a heavy quark, Phys. Lett. B 401 (1997) 81 [hep-ph/9701375] [INSPIRE].
A. Czarnecki and A.G. Grozin, HQET chromomagnetic interaction at two loops, Phys. Lett. B 405 (1997) 142 [Erratum ibid. 650 (2007) 447] [hep-ph/9701415] [INSPIRE].
D.J. Broadhurst and A.G. Grozin, Two loop renormalization of the effective field theory of a static quark, Phys. Lett. B 267 (1991) 105 [hep-ph/9908362] [INSPIRE].
X.-D. Ji and M.J. Musolf, Subleading logarithmic mass dependence in heavy meson form-factors, Phys. Lett. B 257 (1991) 409 [INSPIRE].
K.G. Chetyrkin and A.G. Grozin, Three loop anomalous dimension of the heavy light quark current in HQET, Nucl. Phys. B 666 (2003) 289 [hep-ph/0303113] [INSPIRE].
M. Constantinou and H. Panagopoulos, Perturbative renormalization of quasi-parton distribution functions, Phys. Rev. D 96 (2017) 054506 [arXiv:1705.11193] [INSPIRE].
HotQCD collaboration, Quark Mass Dependence of Heavy Quark Diffusion Coefficient from Lattice QCD, Phys. Rev. Lett. 132 (2024) 051902 [arXiv:2311.01525] [INSPIRE].
Acknowledgments
We would like to thank X. Ji, Y. Ji, C. Monahan, T. Neumann, A. Shindler, V. Harlander, F. Lange and P. Petreczky for useful discussions. We thank A. Vairo for reading the paper and giving very useful comments. The work of N. B. and X.-P. W. is supported by the DFG (Deutsche Forschungsgemeinschaft, German Research Foundation) Grant No. BR 4058/2-2. We acknowledge support from the DFG cluster of excellence “ORIGINS” under Germany’s Excellence Strategy - EXC-2094 - 390783311. The authors acknowledge support from STRONG-2020- European Union’s Horizon 2020 research and innovation program under grant agreement No. 824093.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2312.05032
Rights and permissions
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.
About this article
Cite this article
Brambilla, N., Wang, XP. Off-lightcone Wilson-line operators in gradient flow. J. High Energ. Phys. 2024, 210 (2024). https://doi.org/10.1007/JHEP06(2024)210
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2024)210