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An overview of transverse-momentum-dependent factorization and evolution

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Abstract.

I review TMD factorization and evolution theorems, with an emphasis on the treatment by Collins and originating in the Collins-Soper-Sterman (CSS) formalism. I summarize basic results while attempting to trace their development over that past several decades.

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References

  1. J.C. Collins, D.E. Soper, Nucl. Phys. B 194, 445 (1982)

    Article  ADS  Google Scholar 

  2. J.C. Collins, D.E. Soper, Nucl. Phys. B 193, 381 (1981) 213

    Article  ADS  Google Scholar 

  3. John C. Collins, Davison E. Soper, George Sterman, Phys. Lett. B 109, 388 (1982)

    Article  ADS  Google Scholar 

  4. John C. Collins, Davison E. Soper, George F. Sterman, Nucl. Phys. B 223, 381 (1983)

    Article  ADS  Google Scholar 

  5. J.C. Collins, D.E. Soper, G. Sterman, Nucl. Phys. B 250, 199 (1985)

    Article  ADS  Google Scholar 

  6. J.C. Collins, Foundations of Perturbative QCD (Cambridge University Press, Cambridge, 2011)

  7. R.P. Feynman, R.D. Field, G.C. Fox, Phys. Rev. D 18, 3320 (1978)

    Article  ADS  Google Scholar 

  8. D.C. Hom et al., Phys. Rev. Lett. 36, 1236 (1976)

    Article  ADS  Google Scholar 

  9. D.C. Hom et al., Phys. Rev. Lett. 37, 1374 (1976)

    Article  ADS  Google Scholar 

  10. S.W. Herb et al., Phys. Rev. Lett. 39, 252 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  11. Walter R. Innes et al., Phys. Rev. Lett. 39, 1240 (1977) 39

    Article  ADS  Google Scholar 

  12. C.S. Lam, Tung-Mow Yan, Phys. Lett. B 71, 173 (1977)

    Article  ADS  Google Scholar 

  13. Keh-Fei Liu, Cédric Lorcé, Eur. Phys. J. A 52, 160 (2016)

    Google Scholar 

  14. R.K. Ellis, W.J. Stirling, B.R. Webber, QCD and Collider Physics (Cambridge University Press, Cambridge, 1996)

  15. Robert N. Cahn, Phys. Lett. B 78, 269 (1978)

    Article  ADS  Google Scholar 

  16. D.W. Sivers, Phys. Rev. D 41, 83 (1990)

    Article  ADS  Google Scholar 

  17. R.D. Klem, J.E. Bowers, H.W. Courant, H. Kagan, M.L. Marshak et al., Phys. Rev. Lett. 36, 929 (1976)

    Article  ADS  Google Scholar 

  18. W.H. Dragoset, J.B. Roberts, J.E. Bowers, H.W. Courant, H. Kagan et al., Phys. Rev. D 18, 3939 (1978)

    Article  ADS  Google Scholar 

  19. J. Antille, L. Dick, L. Madansky, D. Perret-Gallix, M. Werlen et al., Phys. Lett. B 94, 523 (1980)

    Article  ADS  Google Scholar 

  20. V.D. Apokin, Yu.I. Arestov, O.V. Astafev, N.I. Belikov, B.V. Chuiko et al., Phys. Lett. B 243, 461 (1990)

    Article  ADS  Google Scholar 

  21. S. Saroff, B.R. Baller, G.C. Blazey, H. Courant, Kenneth J. Heller et al., Phys. Rev. Lett. 64, 995 (1990)

    Article  ADS  Google Scholar 

  22. D.L. Adams et al., Phys. Lett. B 261, 201 (1991)

    Article  ADS  Google Scholar 

  23. D.L. Adams et al., Phys. Lett. B 264, 462 (1991)

    Article  ADS  Google Scholar 

  24. Christine A. Aidala, Steven D. Bass, Delia Hasch, Gerhard K. Mallot, Rev. Mod. Phys. 85, 655 (2013)

    Article  ADS  Google Scholar 

  25. Jian-Ping Chen, Int. J. Mod. Phys. Conf. Ser. 20, 45 (2012)

    Article  Google Scholar 

  26. J.C. Collins, Nucl. Phys. B 396, 161 (1993)

    Article  ADS  Google Scholar 

  27. R.D. Tangerman, P.J. Mulders, Phys. Rev. D 51, 3357 (1995)

    Article  ADS  Google Scholar 

  28. P.J. Mulders, R.D. Tangerman, Nucl. Phys. B 461, 197 (1996)

    Article  ADS  Google Scholar 

  29. Daniel Boer, P.J. Mulders, Phys. Rev. D 57, 5780 (1998)

    Article  ADS  Google Scholar 

  30. Mauro Anselmino, Mariaelena Boglione, Francesco Murgia, Phys. Rev. D 60, 054027 (1999)

    Article  ADS  Google Scholar 

  31. June-gone Chay, Stephen D. Ellis, W. James Stirling, Phys. Rev. D 45, 46 (1992)

    Article  ADS  Google Scholar 

  32. Alessandro Bacchetta, Daniel Boer, Markus Diehl, Piet J. Mulders, JHEP 08, 023 (2008)

    Article  Google Scholar 

  33. Yuri L. Dokshitzer, Dmitri Diakonov, S.I. Troian, Phys. Rep. 58, 269 (1980)

    Article  ADS  Google Scholar 

  34. Yuri L. Dokshitzer, Dmitri Diakonov, S.I. Troian, Phys. Lett. B 78, 290 (1978)

    Article  ADS  Google Scholar 

  35. G. Parisi, R. Petronzio, Nucl. Phys. B 154, 427 (1979)

    Article  ADS  Google Scholar 

  36. W. Greiner, S. Schramm, E. Stein, Quantum Chromodynamics (Springer, 2007)

  37. D.E. Soper, Phys. Rev. Lett. 43, 1847 (1979)

    Article  ADS  Google Scholar 

  38. John C. Collins, Phys. Rev. D 21, 2962 (1980)

    Article  ADS  Google Scholar 

  39. Daniel Boer, Nucl. Phys. B 603, 195 (2001)

    Article  ADS  Google Scholar 

  40. Daniel Boer, Nucl. Phys. B 806, 23 (2009)

    Article  ADS  Google Scholar 

  41. Xiang-Dong Ji, Jian-Ping Ma, Feng Yuan, Phys. Rev. D 71, 034005 (2005)

    Article  ADS  Google Scholar 

  42. Xiang-Dong Ji, Jian-Ping Ma, Feng Yuan, Phys. Lett. B 597, 299 (2004)

    Article  ADS  Google Scholar 

  43. Ruibin Meng, Fredrick I. Olness, Davison E. Soper, Nucl. Phys. B 371, 79 (1992)

    Article  ADS  Google Scholar 

  44. Ruibin Meng, Fredrick I. Olness, Davison E. Soper, Phys. Rev. D 54, 1919 (1996)

    Article  ADS  Google Scholar 

  45. P. Nadolsky, D.R. Stump, C.P. Yuan, Phys. Rev. D 61, 014003 (1999)

    Article  ADS  Google Scholar 

  46. P.M. Nadolsky, D.R. Stump, C.P. Yuan, Phys. Rev. D 64, 114011 (2001)

    Article  ADS  Google Scholar 

  47. Yuji Koike, Junji Nagashima, Werner Vogelsang, Nucl. Phys. B 744, 59 (2006)

    Article  ADS  Google Scholar 

  48. John P. Ralston, Davison E. Soper, Nucl. Phys. B 172, 445 (1980)

    Article  ADS  Google Scholar 

  49. J.C. Collins, Phys. Rev. D 22, 1478 (1980)

    Article  ADS  Google Scholar 

  50. John P. Ralston, Davison E. Soper, Nucl. Phys. B 152, 109 (1979)

    Article  ADS  Google Scholar 

  51. G.T. Bodwin, S.J. Brodsky, G.P. Lepage, Phys. Rev. Lett. 47, 1799 (1981)

    Article  ADS  Google Scholar 

  52. J.C. Collins, D.E. Soper, G. Sterman, Nucl. Phys. B 261, 104 (1985)

    Article  ADS  Google Scholar 

  53. W.W. Lindsay, D.A. Ross, Christopher T. Sachrajda, Nucl. Phys. B 214, 61 (1983)

    Article  ADS  Google Scholar 

  54. Geoffrey T. Bodwin, Phys. Rev. D 31, 2616 (1985) 34

    Article  ADS  Google Scholar 

  55. John C. Collins, Davison E. Soper, George F. Sterman, Phys. Lett. B 438, 184 (1998)

    Article  ADS  Google Scholar 

  56. V.A. Abramovsky, V.N. Gribov, O.V. Kancheli, Yad. Fiz. 18, 595 (1973) Sov. J. Nucl. Phys. 18

    Google Scholar 

  57. John C. Collins, Andreas Metz, Phys. Rev. Lett. 93, 252001 (2004)

    Article  ADS  Google Scholar 

  58. G.A. Ladinsky, C.P. Yuan, Phys. Rev. D 50, R4239 (1994)

    Article  ADS  Google Scholar 

  59. Csaba Balázs, Jian-Wei Qiu, C.P. Yuan, Phys. Lett. B 355, 548 (1995)

    Article  ADS  Google Scholar 

  60. C. Balázs, C.P. Yuan, Phys. Rev. D 56, 5558 (1997)

    Article  ADS  Google Scholar 

  61. F. Landry, R. Brock, G. Ladinsky, C.P. Yuan, Phys. Rev. D 63, 013004 (2001)

    Article  ADS  Google Scholar 

  62. Jian-Wei Qiu, Xiao-Fei Zhang, Phys. Rev. D 63, 114011 (2001)

    Article  ADS  Google Scholar 

  63. Jian-Wei Qiu, Xiao-Fei Zhang, Role of nonperturbative input in QCD resummed heavy boson $Q_T$ distribution, arXiv:hep-ph/0205115 (2002)

  64. George I. Fai, Jian-Wei Qiu, Xiao-Fei Zhang, Phys. Lett. B 567, 243 (2003)

    Article  ADS  Google Scholar 

  65. F. Landry, R. Brock, P.M. Nadolsky, C.-P. Yuan, Phys. Rev. D 67, 073016 (2003)

    Article  ADS  Google Scholar 

  66. Anton V. Konychev, Pavel M. Nadolsky, Phys. Lett. B 633, 710 (2006)

    Article  ADS  Google Scholar 

  67. Pavel M. Nadolsky, AIP Conf. Proc. 753, 158 (2005)

    Article  ADS  Google Scholar 

  68. Marco Guzzi, Pavel M. Nadolsky, Bowen Wang, Phys. Rev. D 90, 014030 (2014)

    Article  ADS  Google Scholar 

  69. Rafael Lopes de Sa, Measurements of the $W$ Boson Mass with the D0 Detector, PhD Thesis, Stony Brook University, 2013

  70. S.J. Brodsky, D.-S. Hwang, I. Schmidt, Phys. Lett. B 530, 99 (2002)

    Article  ADS  Google Scholar 

  71. J.C. Collins, Phys. Lett. B 536, 43 (2002)

    Article  ADS  Google Scholar 

  72. Daniel de Florian, Massimiliano Grazzini, Phys. Rev. Lett. 85, 4678 (2000)

    Article  ADS  Google Scholar 

  73. Daniel de Florian, Massimiliano Grazzini, Nucl. Phys. B 616, 247 (2001)

    Article  ADS  Google Scholar 

  74. Stefano Catani, Daniel de Florian, Massimiliano Grazzini, Nucl. Phys. B 596, 299 (2001)

    Article  ADS  Google Scholar 

  75. Stefano Catani, Leandro Cieri, Giancarlo Ferrera, Daniel de Florian, Massimiliano Grazzini, Phys. Rev. Lett. 103, 082001 (2009)

    Article  ADS  Google Scholar 

  76. Stefano Catani, Leandro Cieri, Daniel de Florian, Giancarlo Ferrera, Massimiliano Grazzini, Eur. Phys. J. C 72, 2195 (2012)

    Article  ADS  Google Scholar 

  77. Giuseppe Bozzi, Stefano Catani, Giancarlo Ferrera, Daniel de Florian, Massimiliano Grazzini, Phys. Lett. B 696, 207 (2011)

    Article  Google Scholar 

  78. Stefano Catani, Leandro Cieri, Daniel de Florian, Giancarlo Ferrera, Massimiliano Grazzini, Nucl. Phys. B 881, 414 (2014)

    Article  ADS  Google Scholar 

  79. J.C. Collins, F. Hautmann, Phys. Lett. B 472, 129 (2000)

    Article  ADS  Google Scholar 

  80. A.V. Belitsky, X. Ji, F. Yuan, Nucl. Phys. B 656, 165 (2003)

    Article  ADS  Google Scholar 

  81. J.C. Collins, Acta Phys. Pol. B 34, 3103 (2003)

    ADS  Google Scholar 

  82. Daniel Boer, P.J. Mulders, F. Pijlman, Nucl. Phys. B 667, 201 (2003)

    Article  ADS  Google Scholar 

  83. C.J. Bomhof, P.J. Mulders, F. Pijlman, Phys. Lett. B 596, 277 (2004)

    Article  ADS  Google Scholar 

  84. C.J. Bomhof, P.J. Mulders, F. Pijlman, Eur. Phys. J. C 47, 147 (2006)

    Article  ADS  Google Scholar 

  85. I.O. Cherednikov, N.G. Stefanis, New results on gauge-invariant TMD PDFs in QCD, arXiv:0809.1315 (2008)

  86. I.O. Cherednikov, N.G. Stefanis, Phys. Rev. D 77, 094001 (2008)

    Article  ADS  Google Scholar 

  87. I.O. Cherednikov, N.G. Stefanis, Nucl. Phys. B 802, 146 (2008)

    Article  ADS  Google Scholar 

  88. F. Hautmann, Phys. Lett. B 655, 26 (2007)

    Article  ADS  Google Scholar 

  89. John Collins, PoS LC, 2008 (2008)

    Google Scholar 

  90. F. Hautmann, Acta Phys. Pol. B 40, 2139 (2009)

    ADS  Google Scholar 

  91. John Collins, CSS equation, etc., follow from structure of TMD factorization, arXiv:1212.5974 (2012)

  92. Miguel G. Echevarría, Ahmad Idilbi, Ignazio Scimemi, JHEP 07, 002 (2012)

    Article  ADS  Google Scholar 

  93. Thomas Becher, Matthias Neubert, Eur. Phys. J. C 71, 1665 (2011)

    Article  Google Scholar 

  94. John C. Collins, Davison E. Soper, Nucl. Phys. B 197, 446 (1982)

    Article  ADS  Google Scholar 

  95. S. Mert Aybat, Ted C. Rogers, Phys. Rev. D 83, 114042 (2011)

    Article  ADS  Google Scholar 

  96. John Collins, Ted Rogers, Phys. Rev. D 91, 074020 (2015)

    Article  ADS  Google Scholar 

  97. S. Mert Aybat, John C. Collins, Jian-Wei Qiu, Ted C. Rogers, Phys. Rev. D 85, 034043 (2012)

    Article  ADS  Google Scholar 

  98. Alessandro Bacchetta, Alexei Prokudin, Nucl. Phys. B 875, 536 (2013)

    Article  Google Scholar 

  99. M.A.G. Aivazis, John C. Collins, Fredrick I. Olness, Wu-Ki Tung, Phys. Rev. D 50, 3102 (1994)

    Article  ADS  Google Scholar 

  100. John C. Collins, Phys. Rev. D 58, 094002 (1998)

    Article  ADS  Google Scholar 

  101. M. Boglione, J.O. Gonzalez Hernandez, S. Melis, A. Prokudin, JHEP 02, 095 (2015)

    Article  ADS  Google Scholar 

  102. X. Artru, J. Czyzewski, H. Yabuki, Z. Phys. C 73, 527 (1997)

    Article  Google Scholar 

  103. Sonny Mantry, Frank Petriello, Phys. Rev. D 81, 093007 (2010)

    Article  ADS  Google Scholar 

  104. Sonny Mantry, Frank Petriello, Phys. Rev. D 84, 014030 (2011)

    Article  ADS  Google Scholar 

  105. John C. Collins, Ted C. Rogers, Phys. Rev. D 87, 034018 (2013)

    Article  ADS  Google Scholar 

  106. R. Angeles-Martinez, Transverse momentum dependent (TMD) parton distribution functions: status and prospects, arXiv:1507.05267 (2015)

  107. Kai-bao Chen, Shu-yi Wei, Zuo-tang Liang, Three Dimensional Imaging of the Nucleon and Semi-Inclusive High Energy Reactions, arXiv:1506.07302 (2015)

  108. John Collins, Jian-Wei Qiu, Phys. Rev. D 75, 114014 (2007)

    Article  ADS  Google Scholar 

  109. John Collins, 2-soft-gluon exchange and factorization breaking, arXiv:0708.4410 (2007)

  110. Ted C. Rogers, Piet J. Mulders, Phys. Rev. D 81, 094006 (2010)

    Article  ADS  Google Scholar 

  111. Stefano Catani, Daniel de Florian, German Rodrigo, JHEP 07, 026 (2012)

    Article  Google Scholar 

  112. Jeffrey R. Forshaw, Michael H. Seymour, Andrzej Siodmok, JHEP 11, 066 (2012)

    Article  ADS  Google Scholar 

  113. Ted C. Rogers, Phys. Rev. D 88(1), 014002 (2013)

    Article  ADS  Google Scholar 

  114. Christian W. Bauer, Frank J. Tackmann, Phys. Rev. D 76, 114017 (2007)

    Article  ADS  Google Scholar 

  115. J.C. Collins, T.C. Rogers, A.M. Staśto, Phys. Rev. D 77, 085009 (2008)

    Article  ADS  Google Scholar 

  116. Ted C. Rogers, Phys. Rev. D 78, 074018 (2008)

    Article  ADS  Google Scholar 

  117. S. Jadach, M. Skrzypek, Acta Phys. Pol. B 40, 2071 (2009)

    ADS  Google Scholar 

  118. Ambar Jain, Massimiliano Procura, Wouter J. Waalewijn, JHEP 04, 132 (2012)

    Article  ADS  Google Scholar 

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Communicated by M. Anselmino

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Rogers, T.C. An overview of transverse-momentum-dependent factorization and evolution. Eur. Phys. J. A 52, 153 (2016). https://doi.org/10.1140/epja/i2016-16153-7

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