Skip to Main Content

Access Provided by Xinjiang University

Physical Review D

Access by Xinjiang University

Next-to-leading-order correction to pion form factor in $k_{T}$ factorization

Hsiang-nan Li^1,2,3, Yue-Long Shen^1,4, Yu-Ming Wang^5,6, and Hao Zou⁶

¹Institute of Physics, Academia Sinica, Taipei, Taiwan 115, Republic of China
²Department of Physics, National Cheng-Kung University, Tainan, Taiwan 701, Republic of China
³Department of Physics, National Tsing-Hua University, Hsinchu, Taiwan 300, Republic of China
⁴College of Information Science and Engineering, Ocean University of China, Qingdao, Shandong 266100, People’s Republic of China
⁵Theoretische Physik 1, Fachbereich Physik, Universität Siegen, D-57068 Siegen, Germany
⁶Institute of High Energy Physics and Theoretical Physics Center for Science Facilities, Post Office Box 918(4) Beijing 100049, China

Phys. Rev. D 83, 054029 – Published 31 March, 2011

DOI: https://doi.org/10.1103/PhysRevD.83.054029

Abstract

We calculate the next-to-leading-order (NLO) correction to the pion electromagnetic form factor at leading twist in the $k_{T}$ factorization theorem. Partons off-shell by $k_{T}^{2}$ are considered in both quark diagrams and effective diagrams for the transverse-momentum-dependent pion wave function. The light-cone singularities in the transverse-momentum-dependent pion wave function are regularized by rotating the Wilson lines away from the light cone. The soft divergences from gluon exchanges among initial- and fal-state partons cancel exactly. We derive the infrared-finite $k_{T}$ -dependent NLO hard kernel for the pion electromagnetic form factor by taking the difference of the above two sets of diagrams. Varying the renormalization and factorization scales, we find that the NLO correction is smaller, when both the scales are set to the invariant masses of internal particles: it becomes lower than 40% of the leading-order contribution for momentum transfer squared $Q^{2} > 7 {GeV}^{2}$ . It is observed that the NLO leading-twist correction does not play an essential role in explaining the experimental data, but the leading-order higher-twist contribution does.

Article Text

References (45)

S. Catani, M. Ciafaloni, and F. Hautmann, Phys. Lett. B 242, 97 (1990); Nucl. Phys. B366, 135 (1991).
J. C. Collins and R. K. Ellis, Nucl. Phys. B360, 3 (1991).
E. M. Levin, M. G. Ryskin, Yu. M. Shabelskii, and A. G. Shuvaev, Sov. J. Nucl. Phys. 53, 657 (1991).
J. Botts and G. Sterman, Nucl. Phys. B325, 62 (1989).
H-n. Li and G. Sterman, Nucl. Phys. B381, 129 (1992).
T. Huang and Q. X. Shen, Z. Phys. C 50, 139 (1991); J. P. Ralston and B. Pire, Phys. Rev. Lett. 65, 2343 (1990).
F. Dominguez, B. W. Xiao, and F. Yuan, Phys. Rev. Lett. 106, 022301 (2011).
S. P. Baranov, A. V. Lipatov, and N. P. Zotov, arXiv:1012.3022; A. V. Lipatov, M. A. Malyshev, and N. P. Zotov, arXiv:1102.1134.
M. Nagashima and H-n. Li, Phys. Rev. D 67, 034001 (2003).
F. Feng, J. P. Ma, and Q. Wang, Phys. Lett. B 674, 176 (2009).
S. Nandi and H-n. Li, Phys. Rev. D 76, 034008 (2007).
H-n. Li and S. Mishima, Phys. Lett. B 674, 182 (2009).
J. C. Collins, Acta Phys. Polon. B 34, 3103 (2003).
J. P. Ma and Q. Wang, J. High Energy Phys. 01 (2006) 067; Phys. Lett. B 642, 232 (2006).
H-n. Li and H. S. Liao, Phys. Rev. D 70, 074030 (2004).
R. D. Field, R. Gupta, S. Otto, and L. Chang, Nucl. Phys. B186, 429 (1981).
F. M. Dittes and A. V. Radyushkin, Yad. Fiz. 34, 529 (1981) [Sov. J. Nucl. Phys. 34, 293 (1981)].
M. H. Sarmadi, Ph.D. thesis, University of Pittsburgh, 1982.
R. S. Khalmuradov and A. V. Radyushkin, Yad. Fiz. 42, 458 (1985) [Sov. J. Nucl. Phys. 42, 289 (1985)].
E. Braaten and S. M. Tse, Phys. Rev. D 35, 2255 (1987).
E. P. Kadantseva, S. V. Mikhailov, and A. V. Radyushkin, Yad. Fiz. 44, 507 (1986) [Sov. J. Nucl. Phys. 44, 326 (1986)].
B. Melic, B. Nizic, and K. Passek, Phys. Rev. D 60, 074004 (1999).
V. M. Braun, A. Khodjamirian, and M. Maul, Phys. Rev. D 61, 073004 (2000).
J. Bijnens and A. Khodjamirian, Eur. Phys. J. C 26, 67 (2002).
Y. Y. Keum, H-n. Li, and A. I. Sanda, Phys. Lett. B 504, 6 (2001); Phys. Rev. D 63, 054008 (2001); Y. Y. Keum and H-n. Li, 63, 074006 (2001); C. D. Lu, K. Ukai, and M. Z. Yang, 63, 074009 (2001).
H-n. Li and S. Mishima, Phys. Rev. D 80, 074024 (2009).
B. Aubert et al. (BABAR Collaboration), Phys. Rev. D 80, 052002 (2009).
Z. T. Wei and M. Z. Yang, Phys. Rev. D 67, 094013 (2003); J. W. Chen, H. Kohyama, K. Ohnishi, U. Raha, and Y. L. Shen, Phys. Lett. B 693, 102 (2010).
H-n. Li, Phys. Rev. D 64, 014019 (2001); M. Nagashima and H-n. Li, Eur. Phys. J. C 40, 395 (2005).
X. Ji and F. Yuan, Phys. Lett. B 543, 66 (2002); A. V. Belitsky, X. Ji, and F. Yuan, Nucl. Phys. B656, 165 (2003).
I. O. Cherednikov and N. G. Stefanis, Nucl. Phys. B802, 146 (2008).
W. Siegel, Phys. Lett. 84B, 193 (1979).
J. C. Collins and D. E. Soper, Nucl. Phys. B193, 381 (1981).
G. P. Korchemsky, D. Pirjol, and T. M. Yan, Phys. Rev. D 61, 114510 (2000), and references therein.
J. P. Ma and Q. Wang, Phys. Rev. D 75, 014014 (2007), and references therein.
R. Akhoury, G. F. Sterman, and Y. P. Yao, Phys. Rev. D 50, 358 (1994).
H-n. Li, Phys. Rev. D 66, 094010 (2002); K. Ukai and H-n. Li, Phys. Lett. B 555, 197 (2003).
H-n. Li and H. L. Yu, Phys. Rev. Lett. 74, 4388 (1995); Phys. Lett. B 353, 301 (1995); Phys. Rev. D 53, 2480 (1996).
H-n. Li, Phys. Rev. D 55, 105 (1997).
R. Jakob and P. Kroll, Phys. Lett. B 315, 463 (1993); 319, 545(E) (1993).
A. P. Bakulev, K. Passek-Kumericki, W. Schroers, and N. G. Stefanis, Phys. Rev. D 70, 033014 (2004); 70, 079906(E) (2004).
H. P. Blok, G. M. Huber, and D. J. Mack, arXiv:nucl-ex/0208011.
C. J. Bebek et al., Phys. Rev. D 17, 1693 (1978).
G. Duplancic, A. Khodjamirian, T. Mannel, B. Melic, and N. Offen, J. High Energy Phys. 04 (2008) 014.
G. M. Huber et al. (Jefferson Lab Collaboration), Phys. Rev. C 78, 045203 (2008).