Factorization,

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Factorization, Chiral Symmetry, and Large Nc
Xiangdong Ji University of Maryland
Workshop on Soft Pions in Hard Processes,
Regensburg, Aug. 4, 2006
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Outline

• Introduction the Sullivan process
• Factorization and chiral non-analytical behavior
  of parton distributions
• Chiral contribution to the Gottfried sum
• Role of Delta resonance at large Nc
• Spin structure of the nucleon
• Conclusion

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Introduction

• The picture of the physical nucleon with a pion
  cloud has been around for a long time
  (electromagnetic form factors)
• The first example of the role of pion cloud in
  hard process is the so-called Sullivan process

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Sullivan process

• Sullivan process has been used to explain sea
  quark distribution in the nucleon.
• A certain level of success has been achieved
• Studies by Strikman et al, and many others.
• It was claimed that the Sullivan process is the
  key to understand the Gottfried sum (Thomas et
  al.)
• How do we understand the Sullivan process from
  the point view of chiral perturbation theory?

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Factorization

• Chiral dynamics is manifest only when
  low-momentum scales are involved. On the other
  hand, DIS or other hard processes involve
  typically large momentum transfers.
• To make use of chiral symmetry, one must first
  have scale factorization (effective theory)
• Separate physics at different momentum scales and
  treat each scale separately.
• The chiral dynamics associated with soft scales
  can be studied in the usual way
• Parton distributions,
• Distributions amplitudes
• Generalized parton distributions

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DIS factorization

• DIS factorization is a standard textbook exercise
  in which one separates physics at scale Q from
  that at low-momentum scale ?
• Therefore, the question becomes how the chiral
  dynamics contribute to the the parton
  distributions?

f is the parton distribution function,
nonperturbative C is the coefficient function, a
power series in coupling as
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Refactorization of parton distributions

• Parton distribution contains physics at hard
  scale ?QCD and soft scale m?. One can separate
  out these physics through additional
  factorization.
• This factorization is achieved by matching
  twist-2 quark-gluon operator to hadron operators

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Chiral correction to the moments of PDF

• One can now use the standard ChiPT to calculate
  the leading non-analytical behavior of the parton
  distributions

Chen Ji Arndt Savage
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Convolution

• The matching condition
• has an alternative interpretation
• as a hadron dis. in the physical nucleon
• and parton distribution in this intermediate
  hadron
• For example, part of the chiral contribution can
  be expressed in terms of the quark distributions
  in the pion and pion distribution in the nucleon

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An example contribution to Gottfried sum

• Consider the difference between the up and down
• sea quark distributions in the nucleon,

• Use the above, one finds

• Additional chiral contribution from bare nucleon

Chen Ji
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Large Nc limit

• The chiral contribution from the nucleon alone
  becomes large in the large Nc limit,
• f? goes like ?Nc
• gA goes like Nc
• Therefore the chiral logs go like Nc, divergent
  pert.!
• This has the same origin as the pi-N scattering
  violates the unitary in the large-Nc limit.
• The solution is that there is a tower of hadron
  states degenerate with the nucleon Contracted
  SU(4) symmetry in the 2-flavor case.
• Dashen, Jenkins and Manohar

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Delta resonance

• In the large Nc world, the ? resonance is
  degenerate with the nucleon
• It is expected that the chiral contribution is
  strongly affected by the presence of ?.
• In the real world, the mass difference is about
  300 MeV, and the importance of the chiral
  contribution depends on the specific spin-isospin
  channels.
• Must make combined expansions in m? and ?

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Non-commutivity of large Nc and chiral limits

• Cohen and Broniowski

• Neither limit is good in reality. Ratio m?/?
  shall be kept to all orders.

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Delta structure functions

• In the large Nc limit, the ?structure function
  can be obtained from that of the nucleon (Chen
  Ji)

There relations are good up to order 1/Nc2!
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Chiral corrections including Delta
Scalar-isoscalar (TS0) case

• Moments of u(x)d(x) get no chiral corrections
  from the nucleon state along and go like Nc1-n
  in large Nc limit.
• The non-analytical chiral correction to u(x)
  d(x) goes like
• Therefore the chiral correction from ? alone is
  subleading in 1/Nc

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Scalar-isovector (S0,T1)

• The chiral correction to u(x)-d(x) from the
  nucleon intermediate state alone goes like Nc in
  large Nc limit
• The contribution from ? is
• Add two contributions together, and using the
  relation

Then the total chiral correction is
which goes like 1/Nc
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Vector-isoscalar (S1,T0)

• ?u ?d gets a chiral contribution from nucleon
  intermediate state.
• Contribution from delta is
• Using the relation between N-? structure
  function, Combine chiral log is
• which is again 1/Nc suppressed.

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Vector-isovector (S1, T1)

• The leading chiral correction to the moments of
  ?u?d is of order Nc in the large Nc limit.
• Adding the ? contribution, one finds

The correction now goes like 1/Nc, For realistic
? mass, cancellation happen at 60 level.
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Spin structure of the nucleon

• The spin of the nucleon has contributions from
  quarks and gluons
• These contributions can be calculated from the
  form factor of the energy-momentum tensor

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Matching to hadronic operators

• Expand the quark-gluon energy-momentum tensor
• The pion operator goes like
• Where aq,g are the fractions of pion momentum
  carried by quarks and gluons

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Nucleon operator

• Nucleon operator
• Constraints in chiral limit

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Feynman diagrams result
It grows like large Nc as Nc goes to infinity!
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Delta contribution

• Energy-momentum tensor
• Feynman diagram

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Large Nc behavior

• Total contribution
• Large Nc relation
• Therefore the order Nc contribution cancels.

Chen Ji

• Chiral behavior of tensor form factors was
  studied by Belitsky
• Ji.

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Conclusion

• After factorization of soft and hard scales, the
  chiral perturbation theory can be
  straightforwardly implemented for any low-energy
  observables.
• The chiral expansion can be strongly affected by
  the delta resonance in the large Nc limit. A
  combined chiral and large Nc expansion shall be
  made.
• For realistic case, the importance of the ?
  resonance depends strongly on the spin-isospin
  channels.