Nucleon Spin Structure and Gauge Invariance

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Nucleon Spin StructureandGauge Invariance

• X.S.Chen, X.F.Lu
• Dept. of Phys., Sichuan Univ.
• W.M.Sun, Fan Wang
• Dept. of Phys. Nanjing Univ.

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Outline

• Introduction
• Gauge invariance and canonical angular momentum
  commutation relation of nucleon spin
• Energy, momentum, orbital angular momentum of
  hydrogen atom and the em multipole radiation
• IV. There is no proton spin crisis but quark
  spin confusion
• V. Summary

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I. Introduction

• 1.It is still a quite popular idea that the quark
  spin measured in polarized
• deep inelastic lepton-nucleon scattering (DIS)
  invalidates the constituent
• quark model (CQM).
• I will show that this is not true. After
  introducing minimum relativistic
• modification, as usual as in other cases where
  the relativistic effects are
• introduced to the non-relativistic models, the
  quark spin measured in DIS
• can be accomodated in CQM.
• 2.One has either gauge invariant or non-invariant
  decomposition of the total
• angular momentum operator of nucleon and atom,
  but up to now one has
• no decomposition which satisfies both gauge
  invariance and canonical
• angular momentum commutation relation.
• I will show the third decomposition where the
  gauge invariance and
• canonical angular momentum commutation relation
  are both satisfied.
• To use canonical spin and orbital angular
  momentum operators is
• important for the consistency between hadron
  spectroscopy and internal
• structure studies.

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• The question is
• whether the two fundamental requirements
• gauge invariance,
• canonical commutation relation for J,
• i.e., angular momentum algebra
• for the individual component of the nucleon
  spin,
• can both be satisfied or can only keep one,
• such as gauge invariance, but the other one, the
  
• canonical commutation relation should be
• given up?

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• Our old suggestion
• keep both requirements,
• the canonical commutation relation is intact
• and the gauge invariance is kept for the
• matrix elements, but not for the operator itself.
• Other suggestion
• keep gauge invariance only and give up
• canonical commutation relation.
• This is dangerous! One can not stay with it!

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New solution

• We found a new decomposition of the
• angular momentum operator for atom (QED)
• and nucleon (QCD), both the gauge
• Invariance and angular momentum algebra
• are satisfied for individual components.
• In passing, the energy and momentum of
• hydrogen atom can also be gauge invariant.
• The key point is to separate the transverse
• and longitudinal components of the gauge
• field.

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IV.Gauge Invariance and canonical Commutation
relation of nucleon spin operators

• From QCD Lagrangian, one can get the total
  angular momentum by Noether theorem

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• Each term in this decomposition satisfies the
  canonical angular momentum algebra, so they are
  qualified to be called quark spin, orbital
  angular momentum, gluon spin and orbital angular
  momentum operators.
• However they are not gauge invariant except the
  quark spin. Therefore the physical meaning is
  obscure.

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• One can have the gauge invariant decomposition,

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• However each term no longer satisfies the
  canonical angular momentum algebra except the
  quark spin, in this sense the second and third
  term is not the quark orbital and gluon angular
  momentum operator.
• The physical meaning of these operators is
  obscure too.
• One can not have gauge invariant gluon spin and
  orbital angular momentum operator separately, the
  only gauge invariant one is the total angular
  momentum of gluon.
• The photon is the same, but we have the
  polarized photon beam already.

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• How to reconcile these two fundamental
  requirements, the gauge invariance and canonical
  angular momentum algebra?
• One choice is to keep gauge invariance and give
  up canonical commutation relation.

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Dangerous suggestion

• It will ruin the multipole radiation analysis
• used from atom to hadron spectroscopy.
• Where the canonical spin and orbital angular
• momentum of photon have been used.
• Even the hydrogen energy is not an observable,
• neither the orbital angular momentum of
• electron nor the polarization (spin) of photon
• is observable either.
• It is totally unphysical!

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New Solution

• A new decomposition
• Gauge invariance and angular momentum
• algebra both satisfied for individual terms.
• Key point to separate the transverse and
• Longitudinal part of gauge field.

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New decomposition
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Esential taskto define properly the pure gauge
field and physical one
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V.Hydrogen atom and em multipole radiation have
the same problem

• Hydrogen atom is a U(1) gauge field system, where
  the canonical momentum, orbital angular momentum
  had been used about one century, but they are not
  the gauge invariant ones. Even the Hamiltonian of
  the hydrogen atom used in Schroedinger equation
  is not a gauge invariant one. After a time
  dependent gauge transformation, the energy of
  hydrogen will be changed.
• Totally unphysical and absurd!

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Momentum operator inquantum mechanics

• Generalized momentum for a charged particle
• moving in em field
• It is not gauge invariant, but satisfies the
  canonical
• momentum commutation relation.
• It is both gauge invariant and canonical momentum
• commutation relation satisfied.

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• We call
• physical momentum.
• It is neither the canonical momentum
• nor the mechanical momentum

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• Gauge transformation
• only affects the longitudinal part of the vector
  potential
• and time component
• it does not affect the transverse part,
• so is physical and which is used in Coulomb
  gauge.

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Hamiltonian of hydrogen atom

• Coulomb gauge
• Hamiltonian of a nonrelativistic particle
• Gauge transformed one

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• Follow the same recipe, we introduce a new
• Hamiltonian,
• which is gauge invariant, i.e.,
• This means the hydrogen energy calculated in
• Coulomb gauge is gauge invariant and physical.

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Multipole radiation

• Multipole radiation analysis is based on the
• decomposition of em vector potential in
• Coulomb gauge. The results are physical
• and gauge invariant, i.e.,
• gauge transformed to other gauges one will
• obtain the same results.

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III.There is no proton spin crisis but quark spin
confusion

• The DIS measured quark spin contributions are

While the pure valence q3 S-wave quark model
calculated ones are
.
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• It seems there are two contradictions between
  these two results
• 1.The DIS measured total quark spin contribution
  to nucleon spin is about one third while the
  quark model one is 1
• 2.The DIS measured strange quark contribution is
  nonzero while the quark model one is zero. New
  measurement
• gave smaller strange contribution.

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• To clarify the confusion, first let me emphasize
  that the DIS measured one is the matrix element
  of the quark axial vector current operator in a
  nucleon state,

Here a0 ?u?d?s which is not the quark spin
contributions calculated in CQM. The CQM
calculated one is the matrix element of the Pauli
spin part only.
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The axial vector current operator can be expanded
as
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• Only the first term of the axial vector current
  operator, which is the Pauli spin part, has been
  calculated in the non-relativistic quark models.
• The second term, the relativistic correction, has
  not been included in the non-relativistic quark
  model calculations. The relativistic quark model
  does include this correction and it reduces the
  quark spin contribution about 25.
• The third term, creation and annihilation,
  will not contribute in a model with only valence
  quark configuration and so it has never been
  calculated in any quark model as we know.

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An Extended CQM with Sea Quark Components

• To understand the nucleon spin structure
  quantitatively within CQM and to clarify the
  quark spin confusion further we developed a CQM
  with sea quark components,

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Where does the nucleon get its Spin

• As a QCD system the nucleon spin consists of the
  following four terms,

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• In the CQM, the gluon field is assumed to be
  frozen in the ground state and will not
  contribute to the nucleon spin.
• The only other contribution is the quark orbital
  angular momentum .
• One would wonder how can quark orbital angular
  momentum contribute for a pure S-wave
  configuration?

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• The quark orbital angular momentum operator can
  be expanded as,

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• The first term is the nonrelativistic quark
  orbital angular momentum operator used in CQM,
  which does not contribute to nucleon spin in a
  pure valence S-wave configuration.
• The second term is again the relativistic
  correction, which takes back the relativistic
  spin reduction.
• The third term is again the creation and
  annihilation contribution, which also takes back
  the missing spin.

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• It is most interesting to note that the
  relativistic correction and the creation
  and annihilation terms of the quark spin and the
  orbital angular momentum operator are exact the
  same but with opposite sign. Therefore if we add
  them together we will have
• where the , are the non-relativistic
  part of
• the quark spin and angular momentum operator.

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• The above relation tell us that the nucleon spin
  can be either solely attributed to the quark
  Pauli spin, as did in the last thirty years in
  CQM, and the nonrelativistic quark orbital
  angular momentum does not contribute to the
  nucleon spin or
• part of the nucleon spin is attributed to the
  relativistic quark spin, it is measured in DIS
  and better to call it axial charge to distinguish
  it from the Pauli spin which has been used in
  quantum mechanics over seventy years, part of the
  nucleon spin is attributed to the relativistic
  quark orbital angular momentum, it will provide
  the
• exact compensation missing in the
  relativistic quark spin no matter what quark
  model is used.
• one must use the right combination otherwise will
  misunderstand the nucleon spin structure.

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VI. Summary

• 1.The DIS measured quark spin is better to be
  called quark axial charge, it is not the quark
  spin calculated in CQM.
• 2.One can either attribute the nucleon spin
• solely to the quark Pauli spin, or partly
  attribute to the quark axial charge partly to the
  relativistic quark orbital angular momentum. The
  following relation should be kept in mind,

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• 3.We suggest to use the physical momentum,
  angular momentum, etc.
• in hadron physics as well as in atomic
  physics, which is both gauge invariant and
  canonical commutation relation satisfied, and had
  been measured in atomic physics with well
  established physical meaning.

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• Thanks