1. 中国物理学会高能物理分会第九届全国会员代表大会暨学术年会
CCNU, WuHan,
NLO twist-3 corrections to π electro-magenatic form factors in factorization
程山，樊莹莹，肖振军
南京师范大学
Based on PRD (2014), S.Cheng, Y.Y.Fan, and Z.J.Xiao

2. Outline Background and motivations; Quark and effective dias;
Cancellation of the IR divs;
NLO hard kernel;
Numerical results and discusses;
Summary;
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3. I: Backgrounds
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4. pQCD calculatons for the πγ*→π form factors:
LO calculations:
G.P.Lepage, S.J.Brodsky, PLB87-359(1979) ;
PRD (1980) ;
N.Isgur, C.H.Llewllyn Smith, NPB (1989) ;
T.H, Q.Z.Shen, ZPC50-139(1991);
H.N.Li, G.Sterman, NPB (1992);
Completeness and accuracy for NLO contributions:
NLO factorization proofs at TW-2 & TW-3 ;
M.Nagashima, H.N.Li, PRD (2003) ;
M.Nagashima, H.N.Li, EPJC40-395(2005) ;
NLO corrections of form factor at TW-2;
H.N.Li, Y.L.Shen, Y.M.Wang, PRD (2012);
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5. I: Motivations Numerical at LO ( ): current algebra OPE
Analytical at LO :
TW-2 :
TW-3 :
Numerical at LO ( ):
TW-2 : ~45%
TW-3 : ~55%
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6. LO Form Factors
Dirac equation;
Perturbative vacuum
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7. The basic idea of k_T factorization
No end-point singularity;
No soft form factor, no zero-bin substrction;
The calculable form factors;
IR divs in the high orders:
Light partons are off mass shell by ;
IR divs are regurlated by ;
IR divs are absorbed into the high order WFs;
Cancellation of the IR divs;
IR finite dependent high order hard kernels.
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9. a. Self-Energy Corrections Diagrams :
II: Quark and Effective Dias
a. Self-Energy Corrections Diagrams :
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10. self-energy to the external quark line -- field renarmalization;
i.e. subdiagram (a):
self-energy to the external quark line
-- field renarmalization;
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11. b. Vertex Corrections Diagrams :
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12. sub-dia(a): coll div at the first sight,
but supressed by the numerator;
sub-dia(b): coll div l//p_2;
sub-dia(c): coll divs l//p_1,p_2
& soft div;
sub-dia(d): coll divs l//p_1,p_2;
sub-dia(e): coll div l//p_2;
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14. c. Box and Pentagon Corrections Diagrams :
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15. sub-dia(a): coll div from l//k_2; sub-dia(b): 5-point integral,
reducible dia,
(would cancelled);
sub-dia(c): 5-point integral,
soft & collinear divs;
sub-dia(d): coll divs & soft divs,
reducible diagram;
sub-dia(e): coll & soft div;
sub-dia(f): coll & soft div;
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17. Effective Diagrams For pion Meson Wave Functions
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18. One loop corrections of the initial and final meson collect the order matrix element of the leading Fock states:
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21. III: IR Cancellation
Consider the IR contributions that don't cancelled directely by their counterparts in the effective diagrams for the quark diagrams.
i: The soft divs are cancelled exactly ;
ii: The remaining collinear divs can be cancelled
by the IR divs from the ffective diagrams;
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22. set-a:
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23. set-b:
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25. IV: NLO Hard Kernel
After UV renomalization,
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26. After subtract the Jet function arise from the tiny x_1
(double counting)
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27. V: Numerical Results and discussion
Inputs Pion DAs
General resolution for the evolution equation of the pion DAs:
a: , asymptotic behaviour(SET-I);
b: non- asymptotic behaviour(SET-II):
with the Gegenbauer Polynomial(low-energy, mass);
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28. Inputs
SET-I:
All the Gegenbauer moments are zero -- sympototic;
SET-II:
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29. Results
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30. SET-I
SET-II
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31. low energy: TW-3 dominate; high energy: TW-2 dominate;
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32. VI:Conclutions Up to TW-3 NLO level:
The UV divergences are renomalizated;
The soft divs are cancelled;
The collinear divs are absorbed;
The scalesμ and t are chosen as the hard sclae;
The in the NLO hard kernel are suppressed;
The fraction NLO/LO ~20% when 20Gev^2<Q^2,
0<NLO/LO <20% when 5<Q^2<20,
NLO/LO <0 when Q^2<5;
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33. Thank you !
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