Non-perturbative tests of continuum HQET through small-volume two-flavour QCD

  • Patrick Fritzsch
  • Nicolas Garron
  • Jochen Heitger
Open Access
Regular Article - Theoretical Physics


We study the heavy quark mass dependence of selected observables constructed from heavy-light meson correlation functions in small-volume two-flavour lattice QCD after taking the continuum limit. The light quark mass is tuned to zero, whereas the range of available heavy quark masses m h covers a region extending from around the charm to beyond the bottom quark mass scale. This allows entering the asymptotic mass-scaling regime as 1/m h → 0 and performing well-controlled extrapolations to the infinite-mass limit. Our results are then compared to predictions obtained in the static limit of continuum Heavy Quark Effective Theory (HQET), in order to verify non-perturbatively that HQET is an effective theory of QCD. While in general we observe a nice agreement at the per cent level, we find it to be less convincing for the small-volume pseudoscalar decay constant when perturbative matching is involved.


Lattice QCD Nonperturbative Effects Heavy Quark Physics Effective field theories 


Open Access

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  1. [1]
    E. Eichten, Heavy quarks on the lattice, Nucl. Phys. Proc. Suppl. 4 (1988) 170 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    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].ADSCrossRefGoogle Scholar
  3. [3]
    H. Georgi, An effective field theory for heavy quarks at low-energies, Phys. Lett. B 240 (1990) 447 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    B. Grinstein, The static quark effective theory, Nucl. Phys. B 339 (1990) 253 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    ALPHA collaboration, J. Heitger and R. Sommer, Nonperturbative heavy quark effective theory, JHEP 02 (2004) 022 [hep-lat/0310035] [INSPIRE].
  6. [6]
    ALPHA collaboration, B. Blossier, M. Della Morte, N. Garron and R. Sommer, HQET at order 1/m: I. Non-perturbative parameters in the quenched approximation, JHEP 06 (2010) 002 [arXiv:1001.4783] [INSPIRE].
  7. [7]
    ALPHA collaboration, B. Blossier et al., HQET at order 1/m: II. Spectroscopy in the quenched approximation, JHEP 05 (2010) 074 [arXiv:1004.2661] [INSPIRE].
  8. [8]
    ALPHA collaboration, B. Blossier et al., HQET at order 1/m: III. Decay constants in the quenched approximation, JHEP 12 (2010) 039 [arXiv:1006.5816] [INSPIRE].
  9. [9]
    ALPHA collaboration, B. Blossier et al., Parameters of Heavy Quark Effective Theory from N f = 2 lattice QCD, JHEP 09 (2012) 132 [arXiv:1203.6516] [INSPIRE].
  10. [10]
    ALPHA collaboration, F. Bernardoni et al., The b-quark mass from non-perturbative N f = 2 Heavy Quark Effective Theory at O(1/m h), Phys. Lett. B 730 (2014) 171 [arXiv:1311.5498] [INSPIRE].
  11. [11]
    ALPHA collaboration, F. Bernardoni et al., Decay constants of B-mesons from non-perturbative HQET with two light dynamical quarks, Phys. Lett. B 735 (2014) 349 [arXiv:1404.3590] [INSPIRE].
  12. [12]
    ALPHA collaboration, F. Bernardoni et al., B-meson spectroscopy in HQET at order 1/m, Phys. Rev. D 92 (2015) 054509 [arXiv:1505.03360] [INSPIRE].
  13. [13]
    ALPHA collaboration, M. Della Morte, S. Dooling, J. Heitger, D. Hesse and H. Simma, Matching of heavy-light flavour currents between HQET at order 1/m and QCD: I. Strategy and tree-level study, JHEP 05 (2014) 060 [arXiv:1312.1566] [INSPIRE].
  14. [14]
    ALPHA collaboration, P. Korcyl, On one-loop corrections to matching conditions of Lattice HQET including 1/m b terms, PoS(LATTICE 2013)380 [arXiv:1312.2350] [INSPIRE].
  15. [15]
    ALPHA collaboration, J. Heitger, A. Jüttner, R. Sommer and J. Wennekers, Non-perturbative tests of heavy quark effective theory, JHEP 11 (2004) 048 [hep-ph/0407227] [INSPIRE].
  16. [16]
    ALPHA collaboration, D. Hesse and R. Sommer, A one-loop study of matching conditions for static-light flavor currents, JHEP 02 (2013) 115 [arXiv:1211.0866] [INSPIRE].
  17. [17]
    ALPHA collaboration, M. Della Morte, P. Fritzsch, J. Heitger, H.B. Meyer, H. Simma and R. Sommer, Towards a non-perturbative matching of HQET and QCD with dynamical light quarks, PoS(LATTICE 2007)246 [arXiv:0710.1188] [INSPIRE].
  18. [18]
    P. Fritzsch, B-meson properties from non-perturbative matching of HQET to finite-volume two-flavour QCD, Ph.D. Thesis, Westfälische Wilhelms-Universität, Münster Germany (2009).Google Scholar
  19. [19]
    ALPHA collaboration, 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].
  20. [20]
    ALPHA collaboration, 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].
  21. [21]
    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].ADSCrossRefGoogle Scholar
  22. [22]
    S. Sint, On the Schrödinger functional in QCD, Nucl. Phys. B 421 (1994) 135 [hep-lat/9312079] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    ALPHA collaboration, M. Della Morte, R. Hoffmann and R. Sommer, Non-perturbative improvement of the axial current for dynamical Wilson fermions, JHEP 03 (2005) 029 [hep-lat/0503003] [INSPIRE].
  24. [24]
    S. Sint and P. Weisz, Further results on O(a) improved lattice QCD to one loop order of perturbation theory, Nucl. Phys. B 502 (1997) 251 [hep-lat/9704001] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    ALPHA collaboration, M. Della Morte, R. Sommer and S. Takeda, On cutoff effects in lattice QCD from short to long distances, Phys. Lett. B 672 (2009) 407 [arXiv:0807.1120] [INSPIRE].
  26. [26]
    ALPHA collaboration, S. Schaefer, R. Sommer and F. Virotta, Critical slowing down and error analysis in lattice QCD simulations, Nucl. Phys. B 845 (2011) 93 [arXiv:1009.5228] [INSPIRE].
  27. [27]
    M. Lüscher, A Semiclassical Formula for the Topological Susceptibility in a Finite Space-time Volume, Nucl. Phys. B 205 (1982) 483 [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    M. Lüscher, Step scaling and the Yang-Mills gradient flow, JHEP 06 (2014) 105 [arXiv:1404.5930] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    ALPHA collaboration, M. Della Morte, A. Shindler and R. Sommer, On lattice actions for static quarks, JHEP 08 (2005) 051 [hep-lat/0506008] [INSPIRE].
  30. [30]
    ALPHA collaboration, M. Della Morte, P. Fritzsch and J. Heitger, Non-perturbative renormalization of the static axial current in two-flavour QCD, JHEP 02 (2007) 079 [hep-lat/0611036] [INSPIRE].
  31. [31]
    ALPHA collaboration, M. Kurth and R. Sommer, Heavy quark effective theory at one loop order: An Explicit example, Nucl. Phys. B 623 (2002) 271 [hep-lat/0108018] [INSPIRE].
  32. [32]
    R. Sommer, Introduction to Non-perturbative Heavy Quark Effective Theory, arXiv:1008.0710 [INSPIRE].
  33. [33]
    ALPHA collaboration, J. Heitger, G.M. von Hippel, S. Schaefer and F. Virotta, Charm quark mass and D-meson decay constants from two-flavour lattice QCD, PoS(LATTICE 2013)475 [arXiv:1312.7693] [INSPIRE].
  34. [34]
    S. Bekavac, A.G. Grozin, P. Marquard, J.H. Piclum, D. Seidel and M. Steinhauser, Matching QCD and HQET heavy-light currents at three loops, Nucl. Phys. B 833 (2010) 46 [arXiv:0911.3356] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    ALPHA collaboration, J. Heitger, M. Kurth and R. Sommer, Nonperturbative renormalization of the static axial current in quenched QCD, Nucl. Phys. B 669 (2003) 173 [hep-lat/0302019] [INSPIRE].
  36. [36]
    ALPHA collaboration, M. Della Morte, N. Garron, M. Papinutto and R. Sommer, Heavy quark effective theory computation of the mass of the bottom quark, JHEP 01 (2007) 007 [hep-ph/0609294] [INSPIRE].
  37. [37]
    ALPHA collaboration, M. Kurth and R. Sommer, Renormalization and O(a) improvement of the static axial current, Nucl. Phys. B 597 (2001) 488 [hep-lat/0007002] [INSPIRE].
  38. [38]
    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].
  39. [39]
    R. Tarrach, The pole mass in perturbative QCD, Nucl. Phys. B 183 (1981) 384 [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    M.A. Shifman and M.B. Voloshin, On annihilation of mesons built from heavy and light quark and \( {\overline{B}}_0\leftrightarrow {B}_0 \) oscillations, Sov. J. Nucl. Phys. 45 (1987) 292 [INSPIRE].Google Scholar
  41. [41]
    H.D. Politzer and M.B. Wise, Leading Logarithms of Heavy Quark Masses in Processes with Light and Heavy Quarks, Phys. Lett. B 206 (1988) 681 [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    N. Gray, D.J. Broadhurst, W. Grafe and K. Schilcher, Three loop relation of quark \( \overline{MS} \) and pole masses, Z. Phys. C 48 (1990) 673 [INSPIRE].ADSGoogle Scholar
  43. [43]
    E. Eichten and B.R. Hill, Static effective field theory: 1/m corrections, Phys. Lett. B 243 (1990) 427 [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    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].ADSCrossRefGoogle Scholar
  45. [45]
    X.-D. Ji and M.J. Musolf, Subleading logarithmic mass dependence in heavy meson form-factors, Phys. Lett. B 257 (1991) 409 [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    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].ADSCrossRefGoogle Scholar
  47. [47]
    V. Giménez, Two loop calculation of the anomalous dimension of the axial current with static heavy quarks, Nucl. Phys. B 375 (1992) 582 [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    G. Amorós, 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].ADSCrossRefGoogle Scholar
  49. [49]
    A. Czarnecki and A.G. Grozin, HQET chromomagnetic interaction at two loops, Phys. Lett. B 405 (1997) 142 [Erratum ibid. B 650 (2007) 447] [hep-ph/9701415] [INSPIRE].
  50. [50]
    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].ADSCrossRefGoogle Scholar
  51. [51]
    A.G. Grozin, P. Marquard, J.H. Piclum and M. Steinhauser, Three-Loop Chromomagnetic Interaction in HQET, Nucl. Phys. B 789 (2008) 277 [arXiv:0707.1388] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    E. Eichten and B.R. Hill, Renormalization of heavy-light bilinears and f B for Wilson fermions, Phys. Lett. B 240 (1990) 193 [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    D.J. Broadhurst and A.G. Grozin, Matching QCD and HQET heavy-light currents at two loops and beyond, Phys. Rev. D 52 (1995) 4082 [hep-ph/9410240] [INSPIRE].ADSGoogle Scholar
  54. [54]
    A.G. Grozin, Decoupling of heavy quark loops in light-light and heavy-light quark currents, Phys. Lett. B 445 (1998) 165 [hep-ph/9810358] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    G. ’t Hooft, Dimensional regularization and the renormalization group, Nucl. Phys. B 61 (1973) 455 [INSPIRE].
  56. [56]
    D.J. Gross and F. Wilczek, Ultraviolet Behavior of Nonabelian Gauge Theories, Phys. Rev. Lett. 30 (1973) 1343 [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    W.E. Caswell, Asymptotic Behavior of Nonabelian Gauge Theories to Two Loop Order, Phys. Rev. Lett. 33 (1974) 244 [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    O.V. Tarasov, A.A. Vladimirov and A. Yu. Zharkov, The Gell-Mann-Low Function of QCD in the Three Loop Approximation, Phys. Lett. B 93 (1980) 429 [INSPIRE].
  59. [59]
    T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin, The four-loop β-function in quantum chromodynamics, Phys. Lett. B 400 (1997) 379 [hep-ph/9701390] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    M. Czakon, The four-loop QCD β-function and anomalous dimensions, Nucl. Phys. B 710 (2005) 485 [hep-ph/0411261] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    K.G. Chetyrkin, Quark mass anomalous dimension to O(α s4), Phys. Lett. B 404 (1997) 161 [hep-ph/9703278] [INSPIRE].ADSCrossRefGoogle Scholar
  62. [62]
    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].ADSCrossRefGoogle Scholar
  63. [63]
    A.G. Grozin, Heavy quark effective theory, Springer Tracts Mod. Phys. 201 (2004) 1.CrossRefGoogle Scholar
  64. [64]
    M. Beneke, Renormalons, Phys. Rept. 317 (1999) 1 [hep-ph/9807443] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    I.I.Y. Bigi, M.A. Shifman and N. Uraltsev, Aspects of heavy quark theory, Ann. Rev. Nucl. Part. Sci. 47 (1997) 591 [hep-ph/9703290] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Patrick Fritzsch
    • 1
    • 2
  • Nicolas Garron
    • 3
  • Jochen Heitger
    • 4
  1. 1.Instituto de Física Teórica UAM/CSICUniversidad Autónoma de MadridMadridSpain
  2. 2.School of Physics & AstronomyUniversity of SouthamptonSouthamptonU.K.
  3. 3.School of Computing & MathematicsPlymouth UniversityPlymouthU.K.
  4. 4.Institut für Theoretische PhysikWestfälische Wilhelms-Universität MünsterMünsterGermany

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