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, 0418-0423 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 PRD89-054015(2014), S.Cheng, Y.Y.Fan, and Z.J.Xiao

Outline Background and motivations; Quark and effective dias; Cancellation of the IR divs; NLO hard kernel; Numerical results and discusses; Summary; 2018/8/12018/8/1

I: Backgrounds 2018/8/1

pQCD calculatons for the πγ*→π form factors: LO calculations: G.P.Lepage, S.J.Brodsky, PLB87-359(1979) ; PRD22-2157(1980) ; N.Isgur, C.H.Llewllyn Smith, NPB317-526(1989) ; T.H, Q.Z.Shen, ZPC50-139(1991); H.N.Li, G.Sterman, NPB325-129(1992); Completeness and accuracy for NLO contributions: NLO factorization proofs at TW-2 & TW-3 ; M.Nagashima, H.N.Li, PRD67-034001(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, PRD83-054029(2012); 2018/8/1

I: Motivations Numerical at LO ( ): current algebra OPE Analytical at LO : TW-2 : TW-3 : Numerical at LO ( ): TW-2 : ~45% TW-3 : ~55% 2018/8/1

LO Form Factors Dirac equation; Perturbative vacuum 2018/8/1

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. 2018/8/1

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a. Self-Energy Corrections Diagrams : II: Quark and Effective Dias a. Self-Energy Corrections Diagrams : 2018/8/1

self-energy to the external quark line -- field renarmalization; i.e. subdiagram (a): self-energy to the external quark line -- field renarmalization; 2018/8/1

b. Vertex Corrections Diagrams : 2018/8/1

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; 2018/8/1

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c. Box and Pentagon Corrections Diagrams : 2018/8/1

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; 2018/8/1

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Effective Diagrams For pion Meson Wave Functions 2018/8/1

One loop corrections of the initial and final meson collect the order matrix element of the leading Fock states: 2018/8/1

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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; 2018/8/1

set-a: 2018/8/1

set-b: 2018/8/1

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IV: NLO Hard Kernel After UV renomalization, 2018/8/1

After subtract the Jet function arise from the tiny x_1 (double counting) 2018/8/1

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); 2018/8/1

Inputs SET-I: All the Gegenbauer moments are zero -- sympototic; SET-II: 2018/8/1

Results 2018/8/1

SET-I SET-II 2018/8/1

low energy: TW-3 dominate; high energy: TW-2 dominate; 2018/8/1

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; 2018/8/1

Thank you ! 2018/8/1