Journal of High Energy Physics

, 2013:15 | Cite as

Phase shifts in I = 2 ππ-scattering from two lattice approaches

  • T. KurthEmail author
  • N. Ishii
  • T. Doi
  • S. Aoki
  • T. Hatsuda


We present a lattice QCD study of the phase shift of I = 2 ππ scattering on the basis of two different approaches: the standard finite volume approach by Lüscher and the recently introduced HAL QCD potential method. Quenched QCD simulations are performed on lattices with extents N s  = 16, 24, 32, 48 and N t  = 128 as well as lattice spacing a ~ 0.115 fm and a pion mass of m π  ~ 940 MeV. The phase shift and the scattering length are calculated in these two methods. In the potential method, the error is dominated by the systematic uncertainty associated with the violation of rotational symmetry due to finite lattice spacing. In Lüscher’s approach, such systematic uncertainty is difficult to be evaluated and thus is not included in this work. A systematic uncertainty attributed to the quenched approximation, however, is not evaluated in both methods. In case of the potential method, the phase shift can be calculated for arbitrary energies below the inelastic threshold. The energy dependence of the phase shift is also obtained from Lüscher’s method using different volumes and/or nonrest-frame extension of it. The results are found to agree well with the potential method.


Lattice QCD Scattering Amplitudes QCD Lattice Quantum Field Theory 


  1. [1]
    V. Bernard, N. Kaiser and U.-G. Meissner, Chiral dynamics in nucleons and nuclei, Int. J. Mod. Phys. E 4 (1995) 193 [hep-ph/9501384] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    E. Epelbaum, H.-W. Hammer and U.-G. Meissner, Modern Theory of Nuclear Forces, Rev. Mod. Phys. 81 (2009) 1773 [arXiv:0811.1338] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    W. Detmold and M.J. Savage, A method to study complex systems of mesons in Lattice QCD, Phys. Rev. D 82 (2010) 014511 [arXiv:1001.2768] [INSPIRE].ADSGoogle Scholar
  4. [4]
    T. Doi and M.G. Endres, Unified contraction algorithm for multi-baryon correlators on the lattice, Comput. Phys. Commun. 184 (2013) 117 [arXiv:1205.0585] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    W. Detmold and K. Orginos, Nuclear correlation functions in lattice QCD, Phys. Rev. D 87 (2013)114512 [arXiv:1207.1452] [INSPIRE].ADSGoogle Scholar
  6. [6]
    J. Gunther, B.C. Toth and L. Varnhorst, A recursive approach to determine correlation functions in multi-baryon systems, Phys. Rev. D 87 (2013) 094513 [arXiv:1301.4895] [INSPIRE].ADSGoogle Scholar
  7. [7]
    N. Ishii, S. Aoki and T. Hatsuda, The Nuclear Force from Lattice QCD, Phys. Rev. Lett. 99 (2007)022001 [nucl-th/0611096] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    S. Aoki, T. Hatsuda and N. Ishii, Theoretical Foundation of the Nuclear Force in QCD and its applications to Central and Tensor Forces in Quenched Lattice QCD Simulations, Prog. Theor. Phys. 123 (2010) 89 [arXiv:0909.5585] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  9. [9]
    H. Nemura, N. Ishii, S. Aoki and T. Hatsuda, Hyperon-nucleon force from lattice QCD, Phys. Lett. B 673 (2009) 136 [arXiv:0806.1094] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    HAL QCD collaboration, T. Inoue et al., Bound H-dibaryon in Flavor SU(3) Limit of Lattice QCD, Phys. Rev. Lett. 106 (2011) 162002 [arXiv:1012.5928] [INSPIRE].CrossRefGoogle Scholar
  11. [11]
    HAL QCD collaboration, T. Doi et al., Exploring Three-Nucleon Forces in Lattice QCD, Prog. Theor. Phys. 127 (2012) 723 [arXiv:1106.2276] [INSPIRE].CrossRefGoogle Scholar
  12. [12]
    HAL QCD collaboration, T. Inoue et al., Two-Baryon Potentials and H-Dibaryon from 3-flavor Lattice QCD Simulations, Nucl. Phys. A 881 (2012) 28 [arXiv:1112.5926] [INSPIRE].ADSGoogle Scholar
  13. [13]
    HAL QCD collaboration, S. Aoki et al., Lattice QCD approach to Nuclear Physics, PTEP 2012 (2012)01A105 [arXiv:1206.5088] [INSPIRE].Google Scholar
  14. [14]
    M. Lüscher, Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. 2. Scattering States, Commun. Math. Phys. 105 (1986) 153.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    M. Lüscher, Two particle states on a torus and their relation to the scattering matrix, Nucl. Phys. B 354 (1991) 531 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    S.R. Sharpe, R. Gupta and G.W. Kilcup, Lattice calculation of I = 2 pion scattering length, Nucl. Phys. B 383 (1992) 309 [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    M. Fukugita, Y. Kuramashi, M. Okawa, H. Mino and A. Ukawa, Hadron scattering lengths in lattice QCD, Phys. Rev. D 52 (1995) 3003 [hep-lat/9501024] [INSPIRE].ADSGoogle Scholar
  18. [18]
    CP-PACS collaboration, S. Aoki et al., I = 2 pion scattering length with Wilson fermions, Phys. Rev. D 67 (2003) 014502 [hep-lat/0209124] [INSPIRE].Google Scholar
  19. [19]
    CP-PACS collaboration, T. Yamazaki et al., I = 2 pi pi scattering phase shift with two flavors of O(a) improved dynamical quarks, Phys. Rev. D 70 (2004) 074513 [hep-lat/0402025] [INSPIRE].Google Scholar
  20. [20]
    CP-PACS collaboration, S. Aoki et al., I=2 pion scattering length from two-pion wave functions, Phys. Rev. D 71 (2005) 094504 [hep-lat/0503025] [INSPIRE].Google Scholar
  21. [21]
    NPLQCD collaboration, S. Beane et al., The I=2 pipi S-wave Scattering Phase Shift from Lattice QCD, Phys. Rev. D 85 (2012) 034505 [arXiv:1107.5023] [INSPIRE].Google Scholar
  22. [22]
    J.J. Dudek, R.G. Edwards and C.E. Thomas, S and D-wave phase shifts in isospin-2 pi pi scattering from lattice QCD, Phys. Rev. D 86 (2012) 034031 [arXiv:1203.6041] [INSPIRE].ADSGoogle Scholar
  23. [23]
    J.E. Mandula, G. Zweig and J. Govaerts, Representations of the Rotation Reflection Symmetry Group of the Four-dimensional Cubic Lattice, Nucl. Phys. B 228 (1983) 91 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    S. Beane, P. Bedaque, A. Parreno and M. Savage, Two nucleons on a lattice, Phys. Lett. B 585 (2004)106 [hep-lat/0312004] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    C.W. Bernard and M.F. Golterman, Finite volume two pion energies and scattering in the quenched approximation, Phys. Rev. D 53 (1996) 476 [hep-lat/9507004] [INSPIRE].ADSGoogle Scholar
  26. [26]
    K. Rummukainen and S.A. Gottlieb, Resonance scattering phase shifts on a nonrest frame lattice, Nucl. Phys. B 450 (1995) 397 [hep-lat/9503028] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    W. Krolikowski and J. Rzewuski, Covariant One-time Formulation of the Many-body Problem in Quantum Theory, Nuovo Cim. 4 (1956) 1212.ADSCrossRefzbMATHGoogle Scholar
  28. [28]
    HAL QCD collaboration, N. Ishii et al., Hadron-Hadron Interactions from Imaginary-time Nambu- Bethe-Salpeter Wave Function on the Lattice, Phys. Lett. B 712 (2012) 437 [arXiv:1203.3642] [INSPIRE].ADSGoogle Scholar
  29. [29]
    M.S. Floater and K. Hormann, Barycentric rational interpolation with no poles and high rates of approximation, Num. Math. 107 (2007) 315.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    C.D. Lin, G. Martinelli, C.T. Sachrajda and M. Testa, K → ππ decays in a finite volume, Nucl. Phys. B 619 (2001) 467 [hep-lat/0104006] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    S. Capitani, S. Dürr and C. Hölbling, Rationale for UV-filtered clover fermions, JHEP 11 (2006)028 [hep-lat/0607006] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    S. Dürr et al., Lattice QCD at the physical point: light quark masses, Phys. Lett. B 701 (2011)265 [arXiv:1011.2403] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    S. Dürr et al., Lattice QCD at the physical point: Simulation and analysis details, JHEP 08 (2011)148 [arXiv:1011.2711] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  34. [34]
    S. Dürr et al., Precision computation of the kaon bag parameter, Phys. Lett. B 705 (2011) 477 [arXiv:1106.3230] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    S. Borsányi et al., High-precision scale setting in lattice QCD, JHEP 09 (2012) 010 [arXiv:1203.4469] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    S. Dürr et al., Scaling study of dynamical smeared-link clover fermions, Phys. Rev. D 79 (2009)014501 [arXiv:0802.2706] [INSPIRE].ADSGoogle Scholar
  37. [37]
    F. Csikor, Z. Fodor, S. Katz, T. Kovacs and B. Toth, A comprehensive lattice search for the Theta + pentaquark, Nucl. Phys. Proc. Suppl. 153 (2006) 49.ADSCrossRefGoogle Scholar
  38. [38]
    S. Dürr et al., Ab-Initio Determination of Light Hadron Masses, Science 322 (2008) 1224 [arXiv:0906.3599] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    S. Dürr et al., Sigma term and strangeness content of octet baryons, Phys. Rev. D 85 (2012) 014509 [arXiv:1109.4265] [INSPIRE].ADSGoogle Scholar
  40. [40]
    K. Murano, N. Ishii, S. Aoki and T. Hatsuda, Nucleon-Nucleon Potential and its Non-locality in Lattice QCD, Prog. Theor. Phys. 125 (2011) 1225 [arXiv:1103.0619] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  41. [41]
    K. Murano et al., Spin-Orbit Force from Lattice QCD, arXiv:1305.2293 [INSPIRE].
  42. [42]
    J. Gasser and H. Leutwyler, Low-Energy Theorems as Precision Tests of QCD, Phys. Lett. B 125 (1983)325 [INSPIRE].CrossRefGoogle Scholar
  43. [43]
    J. Bijnens, G. Colangelo, G. Ecker, J. Gasser and M. Sainio, Elastic pi pi scattering to two loops, Phys. Lett. B 374 (1996) 210 [hep-ph/9511397] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    G. Colangelo, J. Gasser and H. Leutwyler, ππ scattering, Nucl. Phys. B 603 (2001) 125 [hep-ph/0103088] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  • T. Kurth
    • 1
    Email author
  • N. Ishii
    • 2
  • T. Doi
    • 3
  • S. Aoki
    • 4
    • 5
  • T. Hatsuda
    • 3
    • 6
  1. 1.Bergische Universität WuppertalWuppertalGermany
  2. 2.Kobe Branch, Center for Computational SciencesUniversity of Tsukuba, in RIKEN Advanced Institute for Computational Science (AICS)PortIslandJapan
  3. 3.Theoretical Research DivisionNishina Center, RIKENWakoJapan
  4. 4.Yukawa Institute for Theoretical PhysicsKyoto UniversitySakyo-kuJapan
  5. 5.Center for Computational SciencesUniversity of TsukubaTsukubaJapan
  6. 6.Kavli IPMUThe University of TokyoKashiwaJapan

Personalised recommendations