中科院理論物理研究所

1、中科院理論物理研究所中科院理論物理研究所HFCPV-2006 ，April，Nanjing* Introduction* Transverse Momentum Dependent(TMD) wave function of pion.* TMD factorization of pion transition form factor at one-loop level.* Conclusion. * + +* * Simplest process for for exclusive processes.* Easy to * Experiment measurement: CELLO (19

2、91), CLEO (1996).* Data analysis: assume (higher-loop)/(1-loop) = 20%, (higher-twist)/(twist-4) = 20%. A.Schmedding et al. PRD62: 116002(2000), A.P.Bakulev et al. PRD67: 074012(2003); PLB508:279(2001).Q2=1.5GeV2 9.2GeV2. Light-cone formalism: Lepage & Brodsky, PRD22, 2157 (1980) Collinear factor

3、ization to one-loop:Aguila & Chase, NPB193:517 (1981), Braaten, PRD28, 524 (1983) MFA and intrinsic transverse momentum: Saro Ong, PRD52, 3111 (1995). Kroll et al. hep-ph/9306259, 9410304, 9605264 Collinear factorization formula: Hard kernel Hc(x,): B.Melic et al, hep-ph/0107295, 0212346 TMD e+e

4、- back-to-back jets, resum nln2n-1(QT2/Q2). Sudakov factor in hadron-hadron elastic scattering, Modified Factorization Approach(MFA) in pion form factor, Three options for unintegrated PDF definition. TMD PDF and factorization for SIDIS and Drell-Yan.* Definition:* Gauge link:* Direction u : u+u- Tr

5、ee level wave function : Real correction: Virtual correction: Regulators: gluon mass , quark mass . Parameters: . Renormalization equation: Can not obtain DA directly with kt integral. Change to b space: Relation in b space: evolution in b space: Agree with Botts & Sterman NPB325325, 62(1989). S

6、olution will resum sudakov logarithms. We consider: Coordinate frame: Form factor F(Q2): TMD factorization formula: Tree level hard kernel: Other persons choice: One-loop hard kernel:Agree with Braatens result. UV divergence appear when convolute wave function with hard kernel. We can not use dimens

7、ional regularization here. In other kt dependent cases, the convolution may be UV finite. UV div will cancel with soft factor. Soft factor in B case, S=S2/S4: v1-v1+, v2-v2+. We take v1-v1+ first. Problem: end-point singularity. Real corrections: Virtual corrections: TMD wave fucntion has charge sym

8、metry, it is symmetric under kP-k. Real correction (a) is antisymmetric under kP-k, and v1v2. Real correction (a) is UV finite when convoluted with hard kernel. But wave function is UV divergent.S1 is designed to subtract divs from real corrections (b-d). (a) is zero in S1. All virtual corrections a

9、re unchanged.S2 is designed to subtract divs from real correction (a). All other diagrams are zero. Box diagram:+=+x1 = Vertex correction, quark side:=+x1 Vertex correction, anti-quark side:=+x1 Vertex correction: Box diagram:Quark side:Anti-quark side:We give a gauge invariant definition of TMD pion wave function, derive Collins-Soper equation, our result is the same as that obtained by Botts & Sterman. We design a soft factor which is the soft approximation of wave function and has no end-point singula