Advertisement

The parton orbital angular momentum: Status and prospects

  • Keh-Fei LiuEmail author
  • Cédric Lorcé
Review
Part of the following topical collections:
  1. The 3-D Structure of the Nucleon

Abstract.

Theoretical progress on the formulation and classification of the quark and gluon orbital angular momenta (OAM) is reviewed. Their relation to parton distributions and open questions and puzzles are discussed. We give a status report on the lattice calculation of the parton kinetic and canonical OAM and point out several strategies to calculate the quark and gluon canonical OAM on the lattice.

References

  1. 1.
    M. Burkardt, A. Jarrah, arXiv:1011.1041 [hep-ph]
  2. 2.
    C.A. Aidala, S.D. Bass, D. Hasch, G.K. Mallot, Rev. Mod. Phys. 85, 655 (2013)ADSCrossRefGoogle Scholar
  3. 3.
    D. de Florian, R. Sassot, M. Stratmann, W. Vogelsang, Phys. Rev. Lett. 113, 012001 (2014)ADSCrossRefGoogle Scholar
  4. 4.
    NNPDF Collaboration (E.R. Nocera et al.), Nucl. Phys. B 887, 276 (2014)ADSCrossRefGoogle Scholar
  5. 5.
    E. Leader, C. Lorcé, Phys. Rev. Lett. 111, 039101 (2013)ADSCrossRefGoogle Scholar
  6. 6.
    M. Wakamatsu, Int. J. Mod. Phys. A 29, 1430012 (2014)ADSCrossRefGoogle Scholar
  7. 7.
    K.Y. Bliokh, F. Nori, Phys. Rep. 592, 1 (2015) arXiv:1504.03113 ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Wakamatsu, Phys. Rev. D 81, 114010 (2010)ADSCrossRefGoogle Scholar
  9. 9.
    X.-S. Chen, X.-F. Lu, W.-M. Sun, F. Wang, T. Goldman, Phys. Rev. Lett. 100, 232002 (2008)ADSCrossRefGoogle Scholar
  10. 10.
    X.-S. Chen, W.-M. Sun, X.-F. Lu, F. Wang, T. Goldman, Phys. Rev. Lett. 103, 062001 (2009)ADSCrossRefGoogle Scholar
  11. 11.
    C. Lorcé, Phys. Rev. D 88, 044037 (2013)ADSCrossRefGoogle Scholar
  12. 12.
    C. Lorcé, Nucl. Phys. A 925, 1 (2014)ADSCrossRefGoogle Scholar
  13. 13.
    M. Burkardt, Phys. Rev. D 88, 014014 (2013)ADSCrossRefGoogle Scholar
  14. 14.
    R.L. Jaffe, A. Manohar, Nucl. Phys. B 337, 509 (1990)ADSCrossRefGoogle Scholar
  15. 15.
    C. Lorcé, Int. J. Mod. Phys. Conf. Ser. 25, 1460030 (2014)CrossRefGoogle Scholar
  16. 16.
    M. Wakamatsu, Phys. Rev. D 83, 014012 (2011)ADSCrossRefGoogle Scholar
  17. 17.
    X.-D. Ji, Phys. Rev. Lett. 78, 610 (1997)ADSCrossRefGoogle Scholar
  18. 18.
    V. Berestetskii, E. Lifshitz, L. Pitaevskii, Quantum Electrodynamics (Pergamon, Oxford, 1982)Google Scholar
  19. 19.
    J. Jauch, F. Rohrlich, The Theory of Photons and Electrons (Springer-Verlag, Berlin, 1976)Google Scholar
  20. 20.
    C. Lorcé, Phys. Rev. D 87, 034031 (2013)ADSCrossRefGoogle Scholar
  21. 21.
    C. Lorcé, Phys. Lett. B 719, 185 (2013)ADSCrossRefGoogle Scholar
  22. 22.
    M.N. Stoilov, Bulg. J. Phys. 41, 251 (2014) arXiv:1011.5617 [hep-th]Google Scholar
  23. 23.
    J. François, S. Lazzarini, T. Masson, Phys. Rev. D 91, 045014 (2015)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    S. Bashinsky, R.L. Jaffe, Nucl. Phys. B 536, 303 (1998)ADSCrossRefGoogle Scholar
  25. 25.
    M. Wakamatsu, Eur. Phys. J. A 51, 52 (2015)ADSCrossRefGoogle Scholar
  26. 26.
    S.J. Van Enk, G. Nienhuis, J. Mod. Opt. 41, 963 (1994)ADSCrossRefGoogle Scholar
  27. 27.
    K.Y. Bliokh, J. Dressel, F. Nori, New J. Phys. 16, 093037 (2014)ADSCrossRefGoogle Scholar
  28. 28.
    J. Collins, Foundations of Perturbative QCD (Cambridge Univeristy Press, 2011)Google Scholar
  29. 29.
    S.K. Taneja, K. Kathuria, S. Liuti, G.R. Goldstein, Phys. Rev. D 86, 036008 (2012)ADSCrossRefGoogle Scholar
  30. 30.
    X.D. Ji, Phys. Rev. D 58, 056003 (1998)ADSCrossRefGoogle Scholar
  31. 31.
    E. Leader, Phys. Rev. D 85, 051501 (2012)ADSCrossRefGoogle Scholar
  32. 32.
    M. Burkardt, Phys. Rev. D 72, 094020 (2005)ADSCrossRefGoogle Scholar
  33. 33.
    X. Ji, X. Xiong, F. Yuan, Phys. Lett. B 717, 214 (2012)ADSCrossRefGoogle Scholar
  34. 34.
    E. Leader, Phys. Lett. B 720, 120 (2013)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    E. Leader, C. Lorcé, Phys. Rep. 541, 163 (2014)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Y. Hatta, K. Tanaka, S. Yoshida, JHEP 02, 003 (2013)ADSCrossRefGoogle Scholar
  37. 37.
    A. Harindranath, R. Kundu, A. Mukherjee, Phys. Lett. B 728, 63 (2014)ADSCrossRefGoogle Scholar
  38. 38.
    M. Penttinen, M.V. Polyakov, A.G. Shuvaev, M. Strikman, Phys. Lett. B 491, 96 (2000)ADSCrossRefGoogle Scholar
  39. 39.
    D.V. Kiptily, M.V. Polyakov, Eur. Phys. J. C 37, 105 (2004)ADSCrossRefGoogle Scholar
  40. 40.
    P. Hägler, A. Mukherjee, A. Schäfer, Phys. Lett. B 582, 55 (2004)CrossRefGoogle Scholar
  41. 41.
    Y. Hatta, S. Yoshida, JHEP 10, 080 (2012)ADSCrossRefGoogle Scholar
  42. 42.
    C. Lorcé, JHEP 08, 045 (2015)ADSCrossRefGoogle Scholar
  43. 43.
    A. Courtoy, G.R. Goldstein, J.O.G. Hernandez, S. Liuti, A. Rajan, Phys. Lett. B 731, 141 (2014)ADSCrossRefGoogle Scholar
  44. 44.
    X. Ji, X. Xiong, F. Yuan, Phys. Rev. D 88, 014041 (2013)ADSCrossRefGoogle Scholar
  45. 45.
    B.Q. Ma, I. Schmidt, Phys. Rev. D 58, 096008 (1998)ADSCrossRefGoogle Scholar
  46. 46.
    J. She, J. Zhu, B.Q. Ma, Phys. Rev. D 79, 054008 (2009)ADSCrossRefGoogle Scholar
  47. 47.
    H. Avakian, A.V. Efremov, P. Schweitzer, F. Yuan, Phys. Rev. D 81, 074035 (2010)ADSCrossRefGoogle Scholar
  48. 48.
    A.V. Efremov, P. Schweitzer, O.V. Teryaev, P. Zavada, PoS DIS 2010, 253 (2010)Google Scholar
  49. 49.
    C. Lorcé, B. Pasquini, M. Vanderhaeghen, JHEP 05, 041 (2011)ADSCrossRefGoogle Scholar
  50. 50.
    C. Lorcé, B. Pasquini, Phys. Lett. B 710, 486 (2012)ADSCrossRefGoogle Scholar
  51. 51.
    C. Lorcé, B. Pasquini, Phys. Rev. D 84, 034039 (2011)ADSCrossRefGoogle Scholar
  52. 52.
    M. Burkardt, Phys. Rev. D 66, 114005 (2002)ADSCrossRefGoogle Scholar
  53. 53.
    M. Burkardt, Nucl. Phys. A 735, 185 (2004)ADSCrossRefGoogle Scholar
  54. 54.
    M. Burkardt, D.S. Hwang, Phys. Rev. D 69, 074032 (2004)ADSCrossRefGoogle Scholar
  55. 55.
    S. Meissner, A. Metz, K. Goeke, Phys. Rev. D 76, 034002 (2007)ADSCrossRefGoogle Scholar
  56. 56.
    L. Gamberg, M. Schlegel, Phys. Lett. B 685, 95 (2010)ADSCrossRefGoogle Scholar
  57. 57.
    A. Bacchetta, M. Radici, Phys. Rev. Lett. 107, 212001 (2011)ADSCrossRefGoogle Scholar
  58. 58.
    M. Guidal, M.V. Polyakov, A.V. Radyushkin, M. Vanderhaeghen, Phys. Rev. D 72, 054013 (2005)ADSCrossRefGoogle Scholar
  59. 59.
    M. Diehl, T. Feldmann, R. Jakob, P. Kroll, Eur. Phys. J. C 39, 1 (2005)ADSCrossRefGoogle Scholar
  60. 60.
    S. Ahmad, H. Honkanen, S. Liuti, S.K. Taneja, Phys. Rev. D 75, 094003 (2007)ADSCrossRefGoogle Scholar
  61. 61.
    M. Diehl, P. Kroll, Eur. Phys. J. C 73, 2397 (2013)ADSCrossRefGoogle Scholar
  62. 62.
    P. Kroll, EPJ Web of Conferences 85, 01005 (2015)CrossRefGoogle Scholar
  63. 63.
    C. Lorcé, B. Pasquini, Phys. Rev. D 84, 014015 (2011)ADSCrossRefGoogle Scholar
  64. 64.
    C. Lorcé, B. Pasquini, X. Xiong, F. Yuan, Phys. Rev. D 85, 114006 (2012)ADSCrossRefGoogle Scholar
  65. 65.
    Y. Hagiwara, Y. Hatta, Nucl. Phys. A 940, 158 (2015)ADSCrossRefGoogle Scholar
  66. 66.
    X.d. Ji, Phys. Rev. Lett. 91, 062001 (2003)ADSCrossRefGoogle Scholar
  67. 67.
    A.V. Belitsky, X.d. Ji, F. Yuan, Phys. Rev. D 69, 074014 (2004)ADSCrossRefGoogle Scholar
  68. 68.
    D.E. Soper, Phys. Rev. D 15, 1141 (1977)ADSCrossRefGoogle Scholar
  69. 69.
    M. Burkardt, Phys. Rev. D 62, 071503 (2000) (66ADSCrossRefGoogle Scholar
  70. 70.
    S. Meissner, A. Metz, M. Schlegel, JHEP 08, 056 (2009)ADSCrossRefGoogle Scholar
  71. 71.
    C. Lorcé, B. Pasquini, JHEP 09, 138 (2013)ADSCrossRefGoogle Scholar
  72. 72.
    Y. Hatta, Phys. Lett. B 708, 186 (2012)ADSCrossRefGoogle Scholar
  73. 73.
    K. Kanazawa, C. Lorcé, A. Metz, B. Pasquini, M. Schlegel, Phys. Rev. D 90, 014028 (2014)ADSCrossRefGoogle Scholar
  74. 74.
    X. Ji, X. Xiong, F. Yuan, Phys. Rev. Lett. 109, 152005 (2012)ADSCrossRefGoogle Scholar
  75. 75.
    X. Ji, J.H. Zhang, Y. Zhao, Phys. Rev. Lett. 111, 112002 (2013)ADSCrossRefGoogle Scholar
  76. 76.
    X. Ji, J.H. Zhang, Y. Zhao, Phys. Lett. B 743, 180 (2015)ADSCrossRefGoogle Scholar
  77. 77.
    R.S. Sufian, M.J. Glatzmaier, Y.B. Yang, K.F. Liu, M. Sun, PoS LATTICE 2014, 166 (2015)Google Scholar
  78. 78.
    X. Ji, Phys. Rev. Lett. 110, 262002 (2013)ADSCrossRefGoogle Scholar
  79. 79.
    A. Mukherjee, S. Nair, V.K. Ojha, Phys. Rev. D 90, 014024 (2014)ADSCrossRefGoogle Scholar
  80. 80.
    A. Mukherjee, S. Nair, V.K. Ojha, Phys. Rev. D 91, 054018 (2015)ADSCrossRefGoogle Scholar
  81. 81.
    T. Liu, B.Q. Ma, Phys. Rev. D 91, 034019 (2015)ADSCrossRefGoogle Scholar
  82. 82.
    M. Wakamatsu, H. Tsujimoto, Phys. Rev. D 71, 074001 (2005)ADSCrossRefGoogle Scholar
  83. 83.
    T. Liu, B.Q. Ma, Phys. Lett. B 741, 256 (2014)ADSCrossRefGoogle Scholar
  84. 84.
    M. Burkardt, H. BC, Phys. Rev. D 79, 071501 (2009)ADSCrossRefGoogle Scholar
  85. 85.
    T. Liu, B.Q. Ma, Phys. Rev. D 91, 017501 (2015)ADSCrossRefGoogle Scholar
  86. 86.
    P. Lowdon, Nucl. Phys. B 889, 80 (2014)MathSciNetCrossRefGoogle Scholar
  87. 87.
    P. Hägler, A. Schäfer, Phys. Lett. B 430, 179 (1998)CrossRefGoogle Scholar
  88. 88.
    P. Hoodbhoy, X.D. Ji, W. Lu, Phys. Rev. D 59, 014013 (1999)ADSCrossRefGoogle Scholar
  89. 89.
    A. Harindranath, R. Kundu, A. Mukherjee, R. Ratabole, Phys. Rev. Lett. 111, 039102 (2013)ADSCrossRefGoogle Scholar
  90. 90.
    S.J. Brodsky, H.C. Pauli, S.S. Pinsky, Phys. Rep. 301, 299 (1998)ADSMathSciNetCrossRefGoogle Scholar
  91. 91.
    X. Ji, X. Xiong, F. Yuan, Phys. Rev. Lett. 111, 039103 (2013)ADSCrossRefGoogle Scholar
  92. 92.
    A. Harindranath, R. Kundu, Phys. Rev. D 59, 116013 (1999)ADSCrossRefGoogle Scholar
  93. 93.
    M. Burkardt, Int. J. Mod. Phys. A 18, 173 (2003)ADSCrossRefGoogle Scholar
  94. 94.
    M.V. Polyakov, Phys. Lett. B 555, 57 (2003)ADSMathSciNetCrossRefGoogle Scholar
  95. 95.
    K. Goeke, J. Grabis, J. Ossmann, M.V. Polyakov, P. Schweitzer, A. Silva, D. Urbano, Phys. Rev. D 75, 094021 (2007)ADSCrossRefGoogle Scholar
  96. 96.
    L. Adhikari, M. Burkardt, Nucl. Phys. Proc. Suppl. 251-252, 105 (2014)ADSCrossRefGoogle Scholar
  97. 97.
    STAR Collaboration (L. Adamczyk et al.), Phys. Rev. Lett. 115, 092002 (2015) arXiv:1405.5134 [hep-ex]ADSCrossRefGoogle Scholar
  98. 98.
    PHENIX Collaboration (A. Adare et al.), Phys. Rev. D 90, 012007 (2014)ADSCrossRefGoogle Scholar
  99. 99.
    See for example, S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972)Google Scholar
  100. 100.
    R. Jackiw, in Current Algebra and Anomalies (World Scientific, Singapore, 1985) p. 537Google Scholar
  101. 101.
    W. Wilcox, Phys. Rev. D 66, 017502 (2002)ADSMathSciNetCrossRefGoogle Scholar
  102. 102.
    B.L.G. Bakker, E. Leader, T.L. Trueman, Phys. Rev. D 70, 114001 (2004)ADSCrossRefGoogle Scholar
  103. 103.
    M. Deka, T. Streuer, T. Doi, S.J. Dong, T. Draper, K.F. Liu, N. Mathur, A.W. Thomas, Phys. Rev. D 79, 094502 (2009)ADSCrossRefGoogle Scholar
  104. 104.
    I. Montvay, G. Munster, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 1994) p. 491Google Scholar
  105. 105.
    C. Best et al., Phys. Rev. D 56, 2743 (1997)ADSCrossRefGoogle Scholar
  106. 106.
    A.S. Kronfeld, D.M. Photiadis, Phys. Rev. D 31, 2939 (1985)ADSCrossRefGoogle Scholar
  107. 107.
    H. Neuberger, Phys. Lett. B 417, 141 (1998)ADSMathSciNetCrossRefGoogle Scholar
  108. 108.
    M. Luscher, Phys. Lett. B 428, 342 (1998)ADSCrossRefGoogle Scholar
  109. 109.
    P. Hasenfratz, V. Laliena, F. Niedermayer, Phys. Lett. B 427, 125 (1998)ADSCrossRefGoogle Scholar
  110. 110.
    Y. Kikukawa, A. Yamada, Phys. Lett. B 448, 265 (1999)ADSCrossRefGoogle Scholar
  111. 111.
    D.H. Adams, Ann. Phys. (N.Y.) 296, 13 (2002)CrossRefGoogle Scholar
  112. 112.
    K. Fujikawa, Nucl. Phys. B 546, 480 (1999)ADSMathSciNetCrossRefGoogle Scholar
  113. 113.
    H. Suzuki, Prog. Theor. Phys. 102, 141 (1999)ADSCrossRefGoogle Scholar
  114. 114.
    I. Horvath, S.J. Dong, T. Draper, F.X. Lee, K.F. Liu, N. Mathur, H.B. Thacker, J.B. Zhang, Phys. Rev. D 68, 114505 (2003)ADSCrossRefGoogle Scholar
  115. 115.
    I. Horvath, A. Alexandru, J.B. Zhang, Y. Chen, S.J. Dong, T. Draper, K.F. Liu, N. Mathur, S. Tamhankar, H.B. Thacker, Phys. Lett. B 617, 49 (2005)ADSCrossRefGoogle Scholar
  116. 116.
    K.F. Liu, A. Alexandru, I. Horvath, Phys. Lett. B 659, 773 (2008)ADSMathSciNetCrossRefGoogle Scholar
  117. 117.
    A. Alexandru, I. Horvath, K.F. Liu, Phys. Rev. D 78, 085002 (2008)ADSMathSciNetCrossRefGoogle Scholar
  118. 118.
    M. Deka, T. Doi, Y.B. Yang, B. Chakraborty, S.J. Dong, T. Draper, M. Glatzmaier, M. Gong, H.W. Lin, K.F. Liu, D. Mankame, N. Mathur, T. Streuer, Phys. Rev. D 91, 014505 (2015)ADSCrossRefGoogle Scholar
  119. 119.
    J.E. Mandula, G. Zweig, J. Govaerts, Nucl. Phys. B 228, 109 (1983)ADSMathSciNetCrossRefGoogle Scholar
  120. 120.
    O.V. Teryaev, hep-ph/9904376Google Scholar
  121. 121.
    S.J. Brodsky, D.S. Hwang, B.-Q. Ma, I. Schmidt, Nucl. Phys. B 593, 311 (2001)ADSCrossRefGoogle Scholar
  122. 122.
    I.Y. Kobzarev, L.B. Okun, Zh. Eksp. Teor. Fiz. 43, 1904 (1962) (Sov. Phys. JETP 16Google Scholar
  123. 123.
    M. Glatzmaier, K.-F. Liu, arXiv:1403.7211 [hep-lat].
  124. 124.
    K.F. Liu, S.J. Dong, Phys. Rev. Lett. 72, 1790 (1994)ADSCrossRefGoogle Scholar
  125. 125.
    K.F. Liu, S.J. Dong, T. Draper, D. Leinweber, J.H. Sloan, W. Wilcox, R.M. Woloshyn, Phys. Rev. D 59, 112001 (1999)ADSCrossRefGoogle Scholar
  126. 126.
    K.F. Liu, Phys. Rev. D 62, 074501 (2000)ADSCrossRefGoogle Scholar
  127. 127.
    K.-F. Liu, W.-C. Chang, H.-Y. Cheng, J.-C. Peng, Phys. Rev. Lett. 109, 252002 (2012)ADSCrossRefGoogle Scholar
  128. 128.
    N. Mathur, S.J. Dong, K.F. Liu, L. Mankiewicz, N.C. Mukhopadhyay, Phys. Rev. D 62, 114504 (2000)ADSCrossRefGoogle Scholar
  129. 129.
    LHPC and SESAM Collaborations (P. Hägler et al.), Phys. Rev. D 68, 034505 (2003)CrossRefGoogle Scholar
  130. 130.
    QCDSF Collaboration (M. Gockeler et al.), Phys. Rev. Lett. 92, 042002 (2004)CrossRefGoogle Scholar
  131. 131.
    QCDSF-UKQCD Collaboration (D. Brommel et al.), PoS LATTICE 2007, 158 (2007)Google Scholar
  132. 132.
    LHPC Collaboration (J.D. Bratt et al.), Phys. Rev. D 82, 094502 (2010)Google Scholar
  133. 133.
    C. Alexandrou, J. Carbonell, M. Constantinou, P.A. Harraud, P. Guichon, K. Jansen, C. Kallidonis, T. Korzec et al., Phys. Rev. D 83, 114513 (2011)ADSCrossRefGoogle Scholar
  134. 134.
    S.N. Syritsyn, J.R. Green, J.W. Negele, A.V. Pochinsky, M. Engelhardt, P. Hagler, B. Musch, W. Schroers, PoS LATTICE 2011, 178 (2011)Google Scholar
  135. 135.
    C. Alexandrou, M. Constantinou, S. Dinter, V. Drach, K. Jansen, C. Kallidonis, G. Koutsou, Phys. Rev. D 88, 014509 (2013)ADSCrossRefGoogle Scholar
  136. 136.
    H.L. Lai, P. Nadolsky, J. Pumplin, D. Stump, W.K. Tung, C.-P. Yuan, JHEP 04, 089 (2007)ADSCrossRefGoogle Scholar
  137. 137.
    H.L. Lai, J. Huston, S. Kuhlmann, F.I. Olness, J.F. Owens, D.E. Soper, W.K. Tung, H. Weerts, Phys. Rev. D 55, 1280 (1997)ADSCrossRefGoogle Scholar
  138. 138.
    S.J. Dong, J.-F. Lagae, K.F. Liu, Phys. Rev. Lett. 75, 2096-2099 (1995)ADSCrossRefGoogle Scholar
  139. 139.
    G.S. Adkins, C.R. Nappi, E. Witten, Nucl. Phys. B 228, 552 (1983)ADSCrossRefGoogle Scholar
  140. 140.
    B.A. Li, AIP Conf. Proc. 343, 802 (1995)ADSCrossRefGoogle Scholar
  141. 141.
    X. Ji, Sci. China Phys. Mech. Astron. 57, 1407 (2014)ADSCrossRefGoogle Scholar
  142. 142.
    X. Xiong, X. Ji, J.H. Zhang, Y. Zhao, Phys. Rev. D 90, 014051 (2014)ADSCrossRefGoogle Scholar
  143. 143.
    Y.Q. Ma, J.W. Qiu, arXiv:1404.6860 [hep-ph]
  144. 144.
    A.V. Manohar, Phys. Rev. Lett. 65, 2511 (1990)ADSCrossRefGoogle Scholar
  145. 145.
    Y. Hatta, Phys. Rev. D 84, 041701 (2011)ADSCrossRefGoogle Scholar
  146. 146.
    Y. Zhao, K.F. Liu, Y. Yang, Phys. Rev. D 93, 054006 (2016) arXiv:1506.08832 [hep-ph]ADSCrossRefGoogle Scholar
  147. 147.
    Y. Yang, in preparationGoogle Scholar
  148. 148.
    K.F. Liu, Int. J. Mod. Phys. Conf. Ser. 40, 1660005 (2016) arXiv:1504.06601 [hep-ph]ADSCrossRefGoogle Scholar
  149. 149.
    C. Alexandrou, M. Constantinou, G. Koutsou, K. Ottnad, M. Petschlies, PoS LATTICE 2014, 144 (2015)Google Scholar
  150. 150.
    B.U. Musch, P. Hägler, J.W. Negele, A. Schäfer, Phys. Rev. D 83, 094507 (2011)ADSCrossRefGoogle Scholar
  151. 151.
    B.U. Musch, P. Hägler, M. Engelhardt, J.W. Negele, A. Schäfer, Phys. Rev. D 85, 094510 (2012)ADSCrossRefGoogle Scholar
  152. 152.
    M. Engelhardt et al., PoS LATTICE 2014, 167 (2014)Google Scholar
  153. 153.
    M. Engelhardt, P. Hägler, B. Musch, J. Negele, A. Schäfer, Phys. Rev. D 93, 054501 (2016) arXiv:1506.07826 [hep-lat]ADSCrossRefGoogle Scholar
  154. 154.
    Y.B. Yang, M. Gong, K.F. Liu, M. Sun, PoS LATTICE 2014, 138 (2014)Google Scholar
  155. 155.
    L.Y. Glozman, D.O. Riska, Phys. Rep. 268, 263 (1996)ADSCrossRefGoogle Scholar
  156. 156.
    K.F. Liu, Y. Chen, M. Gong, R. Sufian, M. Sun, A. Li, PoS LATTICE 2013, 507 (2014)Google Scholar
  157. 157.
    K.G. Wilson, J.B. Kogut, Phys. Rep. 12, 75 (1974)ADSCrossRefGoogle Scholar
  158. 158.
    J. Polchinski, Nucl. Phys. B 231, 269 (1984)ADSCrossRefGoogle Scholar
  159. 159.
    K.F. Liu, S.J. Dong, T. Draper, J.H. Sloan, W. Wilcox, R.M. Woloshyn, Phys. Rev. D 61, 118502 (2000)ADSCrossRefGoogle Scholar
  160. 160.
    A. Manohar, H. Georgi, Nucl. Phys. B 234, 189 (1984)ADSCrossRefGoogle Scholar
  161. 161.
    E. Witten, Nucl. Phys. B 160, 57 (1979)ADSCrossRefGoogle Scholar
  162. 162.
    B.A. Li, Phys. Rev. D 52, 5165 (1995)ADSCrossRefGoogle Scholar
  163. 163.
    B.A. Li, Phys. Rev. D 52, 5184 (1995)ADSCrossRefGoogle Scholar
  164. 164.
    B.J. Warr, Ann. Phys. (N.Y.) 183, 1 (1988)ADSCrossRefGoogle Scholar
  165. 165.
    Q. Wang, Y.P. Kuang, X.L. Wang, M. Xiao, hep-ph/9910289Google Scholar
  166. 166.
    G.E. Brown, M. Rho, Phys. Lett. B 82, 177 (1979)ADSCrossRefGoogle Scholar
  167. 167.
    A.W. Thomas, S. Theberge, G.A. Miller, Phys. Rev. D 24, 216 (1981)ADSCrossRefGoogle Scholar
  168. 168.
    M. Wakamatsu, Y. Nakakoji, Phys. Rev. D 74, 054006 (2006)ADSCrossRefGoogle Scholar
  169. 169.
    C. Lorcé, Phys. Lett. B 735, 344 (2014)ADSCrossRefGoogle Scholar
  170. 170.
    E. Leader, A.V. Sidorov, D.B. Stamenov, Phys. Rev. D 82, 114018 (2010)ADSCrossRefGoogle Scholar

Copyright information

© SIF, Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Physics and AstronomyUniversity of KentuckyLexingtonUSA
  2. 2.SLAC National Accelerator LaboratoryStanford UniversityMenlo ParkUSA
  3. 3.IFPA, AGO DepartmentUniversité de Liège, Sart-TilmanLiègeBelgium

Personalised recommendations