Nucleon Internal Structure the Quark Gluon Momentum and Angular by nyut545e2


									 Nucleon Internal Structure:
the Quark Gluon Momentum
    Angular Momentum
            X.S.Chen, X.F.Lu
       Dept. of Phys., Sichuan Univ.
         W.M.Sun, Fan Wang
               NJU and PMO
Joint Center for particle nuclear physics and
I.     Introduction
II.    Conflicts between Gauge invariance
       and Canonical Quantization
III.   Quantum mechanics
IV.    QED
V.     QCD
VI.    Nucleon internal structure
VII.   Summary
• Quark gluon contribution to the nucleon
  observables, mass, momentum, spin,
  magnetic moment, etc. are unavoidable in
  the study of nucleon internal structure.
• We never have the quark gluon
  momentum and angular momentum
  operators which satisfy both gauge
  invariance and canonical commutation
II. Conflicts between
   gauge invariance
canonical quantization
          Quantum Mechanics
   Even though the Schroedinger equation is
gauge invariant, the matrix elements of the
canonical momentum, orbital angular momentum,
and Hamiltonian of a charged particle moving in
eletromagnetic field are gauge dependent,
            orbital angular momentum
especially the
and Hamiltonian of the hydrogen atom
 are “not the measurable ones” !?
            It is absurd!
The canonical momentum and orbital
angular momentum of electron are gauge
dependent and so their physical meaning is
  The canonical photon spin and orbital
angular momentum operators are also
gauge dependent. Their physical meaning is
obscure too. Even it has been claimed that
it is impossible to have photon spin and
orbital angular momentum operators.
             Multipole radiation
 The multipole radiation theory is based on the
decomposition of a polarized em wave into multipole
radiation field with definite photon spin and orbital
angular momentum coupled to a total angular
momentum quantum number LM,

A = ξ peik⋅r = 2π Σ∞=1i L 2L +1DMp (ϕ,θ ,0)[ALM (m) + ip ALM (e)]

                        L                     L +1
         A LM (e) = −       ς L +1T LL +1M +       ς L −1T LL −1M
                      2L + 1                 2L + 1
                      A LM (m) = ς L T LLM
  Multipole radiation measurement and
analysis are the basis of atomic, molecular,
nuclear and hadron spectroscopy. If the
orbital angular momentum of photon is
gauge dependent and not measurable, then all
determinations of the
of these microscopic systems would be
• Because the canonical parton (quark and
  gluon) momentum is “gauge dependent”,
  so the present analysis of parton
  distribution of nucleon uses the covariant
  derivative operator instead of the
  canonical momentum operator; uses the
  Poynting vector as the gluon momentum
 They are not the right momentum
• The quark spin contribution to nucleon
  spin has been measured, the further study
  is hindered by the lack of gauge invariant
  quark orbital angular momentum, gluon
  spin and orbital angular momentum
  operators. The present gluon spin
  measurement is even under the condition
“there is not a gluon spin
               can be measured”.
    III. Quantum Mechanics
Gauge is an internal degree of freedom, no
matter what gauge used, the canonical
coordinate and momentum of a charged
particle is r and p = − i ∇ , the orbital
angular momentum is
                             ∇ ,
               L = r× p = r×
the Hamiltonian is           i

                ( p − e A)2
            H =             + eϕ
 Gauge transformation
                  ψ → ψ ' = eieω ( x )ψ ,
   A → A = A + ∇ω,
         '                                  ϕ → ϕ ' = ϕ − ∂ tω ,
The matrix elements transformed as
         ψ | p |ψ → ψ | p |ψ + ψ | e∇ω |ψ ,

         ψ | L|ψ → ψ | L|ψ + ψ | er×∇ω|ψ ,
         ψ | H |ψ → ψ | H |ψ − ψ | e∂tω |ψ ,
They are not gauge invariant,
      even though the Schroedinger equation is.
      New momentum operator in
         quantum mechanics
Canonical momentum for a charged particle
moving in em field:

      p = m r + q A = m r + q A ⊥ + q A //
Its ME is not gauge invariant, but satisfies the canonical
momentum commutation relation.
             p − q A // = m r + q A ⊥
           ∇ ⋅ A⊥ = 0,          ∇× A// = 0
It is both gauge invariant and satisfies
       canonical momentum commutation relation.
We call
            Dphy             1
                 = p − qA// = ∇− qA//
             i               i
physical momentum.
It is neither the canonical momentum
             p = mr + q A = ∇
nor the mechanical momentum
             p − q A = mr = D
Gauge transformation
    ψ ' = e iq ω ( x )ψ ,     Aμ = Aμ + ∂ μ ω ( x),

only affects the longitudinal part of the vector potential
      A// = A// + ∇ ω ( x ),

and time component
       ϕ ' = ϕ − ∂ tω ( x),
it does not affect the transverse part,
          A = A⊥,
so A⊥ is physical and which is used in Coulomb gauge.
A // is unphysical, it is caused by gauge transformation.
 Hamiltonian of hydrogen atom
Coulomb gauge:
                                                A = ϕ ≠ 0.
          c                      c
                    = 0,
                                                    c   c
      A   //                   A ≠ 0,
                                 ⊥                  0
Hamiltonian of a nonrelativistic particle
                           (p − qA ) ⊥
                      Hc =           + qϕ c .
Gauge transformed one
               c                                c
  A // = A + ∇ω ( x ) = ∇ω ( x ), A ⊥ = A , ϕ = ϕ c − ∂ t ω ( x )
               //                               ⊥

       ( p − q A)        ( p − q∇ω − q A )
    H=            + qϕ =                   + qϕ c − q∂ tω.
           2m                    2m
Follow the same recipe, we introduce a new Hamiltonian,
                                   ( p − q A// − q A )
      H phy   = H + q∂ t ω ( x ) =                     + qϕ c

                          ω = ∇ −2∇ ⋅ A//

which is gauge invariant, i.e.,

                 ψ | H phy | ψ = ψ c | H c | ψ c
This means the hydrogen energy calculated in
Coulomb gauge is gauge invariant and physical.
       A rigorous derivation
Start from a ED Lagrangian including
electron, proton and em field, under the
heavy proton approximation, one can derive
a Dirac equation and a Hamiltonian for
electron and proved that the time evolution
operator is different from the Hamiltonian
exactly as we obtained phenomenologically.
The nonrelativistic one is the above
      Schroedinger or Pauli equation.
Different approach will obtain different energy-momentum
tensor and four momentum, they are not unique:
Noether theorem                    ∇
                    P = ∫ d 3 x{ψ + ψ + E i ∇Ai }
Gravitational theory (Weinberg) or Belinfante tensor
                   P = ∫ d 3 x{ψ + ψ + E × B}
It appears to be perfect , but individual part does not
                            satisfy the momentum algebra.
Usually one supposes these two expressions are
equivalent, because the integral is the same.
We are experienced in quantum mechanics, so we
introduce               D
              P = ∫ d x{ψ   +
                                        ψ + E i ∇A⊥ }
                     3            phy             i

                A = A       //   + A     ⊥

                 D phy = ∇ − ieA//

They are both gauge invariant and momentum
algebra satisfied. They return to the canonical
expressions in Coulomb gauge.
We proved the renowned Poynting vector is not the
 correct momentum of em field

J γ = ∫ d xr ×(E × B) = ∫ d xE × A⊥ + ∫ d xr × E ∇A ⊥
        3                 3             3      i   i

It includes photon spin and
       orbital angular momentum
Electric dipole radiation field
 Blm = a h (kr) LYlm ,......E lm = ik Alm = ∇ × Blm
        lm l

  1     ∗          | a11 |2 3 1+ cos2 θ      sinθ
    Re[E11 ×B11] =          ⋅ ⋅[        nr +      nϕ ]
  2                 (kr) 16π
                                  2           kr
  1     i∗        | a11 |2 3 1 + cos2 θ      sinθ
    Re[E11∇A11] =
                           ⋅ ⋅[         nr +      nϕ ]
  2                (kr) 16π
                                  2           2kr

     dP | a11 |2 3 1 + cos 2 θ    dJ z
        =       ⋅              =k
     dΩ    k 2
                  16 π  2         dΩ

         dJ z | a11 |2 3
             =        ⋅  ⋅ sin 2 θ
         dΩ      k 3 16π
J   QED   = S e + L e + S γ + Lγ
• Each term in this decomposition satisfies
  the canonical angular momentum algebra,
  so they are qualified to be called electron
  spin, orbital angular momentum, photon
  spin and orbital angular momentum
• However they are not gauge invariant
  except the electron spin. Therefore the
  physical meaning is obscure.
• 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.
J QED = S e + L ' e + J 'γ
• However each term no longer satisfies the
  canonical angular momentum algebra except
  the electron spin, in this sense the second and
  third term is not the electron orbital and photon
  angular momentum operator.
  The physical meaning of these operators is
  obscure too.
• One can not have gauge invariant photon spin
  and orbital angular momentum operator
  separately, the only gauge invariant one is the
  total angular momentum of photon.
 The photon spin and orbital angular
 momentum had been measured!
       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.

      It is totally unphysical!
J   QED   = S   e   + L e ' '+ S   γ   ' '+ L γ ' '
         Multipole radiation
Multipole radiation analysis is based on the
decomposition of em vector potential in
Coulomb gauge. The results are physical
and gauge invariant.
                                V. QCD
             P = ∫ d x{ψ   ψ + E i ∇ Ai }
                            3         +


             P = ∫ d x{ψ   ψ + E × B}
                            3        +

                            D phy
       P = ∫ d xψ
                3       +
                                    ψ + ∫ d 3 xE i a D phy Aiphys

D phy = ∇ − ig A pure                         a   D phy = ∇ − ig[ A pure, ]
• From QCD Lagrangian, one can get the total
  angular momentum by Noether theorem:
• One can have the gauge invariant decomposition,
New decomposition
                                          ''            ''        ''
 J QCD = S q + L + S + L                  q             g         g

                         ∫            xψ                 ψ
          S   q     =        d    3

                                  D phy
                   L = ∫ d xψ r ×
                              3       +

          S   g     =        d 3 x E × A phy

 L   g    =         d 3 xEi r ×                a   D p h y Ai   phy
  Esential task:to define properly the pure gauge
            field Apure and physical one A phy
Phys.Rev.Lett.100,232002(2008), arXiv:0904.0321[hep-ph]

            D phy = ∇ − ig A pure          A pure = T A

                        A = A pure + A phy

          D phy × A pure = ∇ × A pure − ig A pure × A pure = 0

    a   D phy ⋅ Aphy = ∇ ⋅ Aphy − ig[ Apure , Aphy ] = 0
     VI. Nucleon internal structure
       it should be reexamined!
• The present parton distribution is not the
right quark and gluon momentum distribution.
In the asymptotic limit, the gluon only
contributes ~1/5 nucleon momentum, not 1/2 !
• The nucleon spin structure study should
be based on the new decomposition and
new operators.
• One has to be careful when one compares
  experimental measured quark gluon
  momentum and angular momentum to the
  theoretical ones.
  The proton spin crisis is mainly due to
  misidentification of the measured quark
  axial charge to the nonrelativistic Pauli
  Phys. Rev. D58,114032 (1998)
               VII. Summary
• The renowned Poynting vector is not the right
  momentum operator of photon and gluon field.
• The space time translation operators of the Fermion
  part are not observables.
• The gauge invariant and canonical quantization rule
  satisfied momentum, spin and orbital angular
  momentum operators of the individual part do exist.
• The Coulomb gauge is physical, expressions in
  Coulomb gauge, even with vector potential, are
  gauge invariant, including the hydrogen atomic
  Hamiltonian and multipole radiation.
       Nucleon Internal Structure
• 1. Nucleon anomalous magnetic moment
     Stern’s measurement in 1933;
     first indication of nucleon internal structure.
• 2. Nucleon rms radius
     Hofstader’s measurement of the charge
     and magnetic rms radius of p and n in 1956;
     Yukawa’s meson cloud picture of nucleon,
              p->p+ π 0 ; n+ π + ;
              n->n+ π 0 ; p+ π − .
• 3. Gell-mann and Zweig’s quark model
    SU(3) symmetry:
    baryon qqq; meson q q .
    SU(6) symmetry:
    B(qqq)=     [χms(q3)ηms(q3)+χma(q3)ηma(q3)] .
    color degree of freedom.
    quark spin contribution to nucleon spin,
                      4         1
               Δu =     ; Δd = − ; Δs = 0.
                      3         3

    nucleon magnetic moments.
  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:
• 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.
• 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= Δu+Δd+Δs which is not the quark spin
  contributions calculated in CQM. The CQM
  calculated one is the matrix element of the Pauli spin
  part only.
The axial vector current operator can
be expanded as
• 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, q 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.
         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,
   Where does the nucleon get its
• As a QCD system the nucleon spin consists of
  the following four terms,
• 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 Lq .
• One would wonder how can quark orbital
  angular momentum contribute for a pure
  S-wave configuration?
• The quark orbital angular momentum operator
  can be expanded as,
• 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
• The third term is again the qq creation and
  annihilation contribution, which also takes back
  the missing spin.
• It is most interesting to note that the relativistic
  correction and the qq 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.
• 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.
              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,
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.

To top