Advertisement

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

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

Abstract

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.

Keywords

Lattice QCD Nonperturbative Effects Heavy Quark Physics Effective field theories 

Notes

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.

References

  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

Personalised recommendations