1. On the Drell-Levy-Yan Relation for Fragmentation Functions T. Ito, W.Bentz(Tokai Univ.) K.Yazaki (Tokyo Woman’s University, and RIKEN) A.W.Thomas (JLab)

2. DLY Relation: Case of nucleon Using spectral representations (or the reduction formula), one can show the following relations (“Crossing of nucleon line”): local field operator(fermion type: + sign, boson type: - sign); Choose (i)(current operator) in hadronic tensor, (ii)(quark field) in operator definition of quark distribution → DLY relation can be shown in 2 ways:

3. DLY Relation: (i) hadronic tensor Hadronic tensors for e p → e’ X, and e + e - → p X Crossing gives: for bosons (fermions). Then the DLY relation follows (0 < z < 1): distribution of quark (q) in hadron (h);fragmentation of q into h

4. DLY Relation: (ii) operator definition Crossing gives again the DLY relation for bosons (+) and fermions (-). (Here 0 < z < 1) origin of factor 1/6: Average over quark spin and color. Important result: For boson case, at x = 1 ! The “generalized distribution” must be must not vanish positive everywhere:

5. Numerical calculations: valence quarks Use the NJL model and a simple valence quark picture to calculate the generalized distribution function from the Feynman diagrams (nucleon case) Nucleon ≡ quark + scalar diquark Use pole approximation for diquark propagator. Regularization: Transverse cut-off. Parameters: Constituent quark mass M = 0.4 GeV, cut-off Λ Tr = 0.407 GeV.

6. Numerical calculations: sea quarks Include sea quark distributions (or: “unfavored” fragmentation process of sea quarks) as an effect of pion cloud around constituent quarks. For the generalized distribution function, we use the convolution formalism to evaluate diagrams like …… Use pole approximation for diquark and pion propagators; on-shell (parent quark) approximation in the convolution integral.

7. Generalized u V distribution in proton Results at the NJL model scale Red line: without pion cloud Black line:with pion cloud

8. Generalized u V distribution in π + Results at the NJL model scale Red line: without pion cloud Black line:with pion cloud

9. u V distribution in proton Results at Red line: without pion cloud Black line:with pion cloud Blue line:Empirical (MRST)

10. u V fragmentation into proton Results at Red line: without pion cloud Black line:with pion cloud Blue line:Empirical (M.Hirai et al).

11. u V distribution in π + Results at Red line: without pion cloud Black line:with pion cloud Blue line:Empirical (P.J.Sutton et al) At x = 1: Input artificially set to zero, in order to use the Q 2 evolution program.

12. u V fragmentation into π + Results at Red line: without pion cloud Black line:with pion cloud Blue line:Empirical (M.Hirai et al). At z = 1: Input artificially set to zero, in order to use the Q 2 evolution program.

13. Conclusions DLY relation is based on crossing symmetry, and is very general. It expresses the fragmentation function by the distribution function in the “unphysical” region x > 1. Simple chiral quark model describes the distributions (x 1) by factors of 10 ～ 100. Possible reasons for failure: (i)Point NJL vertex functions (ii) On-shell approximation in convolution integrals (iii) What else ??? Thanks to: S.Kumano, M.Miyama, M.Hirai for Q 2 evolution; M.Strattman for discussions.