On the DrellLevyYan Relation for Fragmentation Functions

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On the Drell-Levy-Yan Relationfor Fragmentation
Functions

• T. Ito, W.Bentz(Tokai Univ.)
• K.Yazaki (Tokyo Womans University, and RIKEN)
• A.W.Thomas (JLab)

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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
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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 lt z
lt 1)
distribution of quark (q) in hadron (h)
fragmentation of q into h
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DLY Relation (ii) operator definition
Crossing gives again the DLY relation
for bosons () and fermions (-). (Here 0 lt z lt 1)
origin of factor 1/6 Average over quark spin and
color.
Important result For boson case,
must not vanish
at x 1 ! The generalized distribution must be
positive everywhere
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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 ?Tr
0.407 GeV.
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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.
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Generalized uV distribution in proton
Results at the NJL model scale
Red line without pion cloud Black linewith pion
cloud
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Generalized uV distribution in p
Results at the NJL model scale
Red line without pion cloud Black linewith pion
cloud
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uV distribution in proton
Results at
Red line without pion cloud Black linewith pion
cloud Blue lineEmpirical (MRST)
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uV fragmentation into proton
Results at
Red line without pion cloud Black linewith pion
cloud Blue lineEmpirical (M.Hirai et al).
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uV distribution in p
Results at
Red line without pion cloud Black linewith pion
cloud Blue lineEmpirical (P.J.Sutton et al)
At x 1 Input artificially set to zero, in
order to use the Q2 evolution program.
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uV fragmentation into p
Results at
Red line without pion cloud Black linewith pion
cloud Blue lineEmpirical (M.Hirai et al).
At z 1 Input artificially set to zero, in
order to use the Q2 evolution program.
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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 gt 1.
Simple chiral quark model describes the
distributions (x lt 1) very well, but fails for
the fragmentation functions (x gt 1) by factors of
10100.

• Possible reasons for failure
• Point NJL vertex functions
• On-shell approximation in convolution integrals
• What else ???

Thanks to S.Kumano, M.Miyama, M.Hirai for Q2
evolution M.Strattman for discussions.