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Review on Nucleon Spin Structure

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Review on Nucleon Spin Structure Powered By Docstoc
					   Nucleon Internal Structure:
  the Quark Gluon Momentum,
Angular Momentum operators and
       parton distributions
             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
                  cosmology
                  (J-CPNPC)
        T.Goldman T.D., LANL, USA
                   Outline
I.      Introduction
II.     Conflicts between Gauge invariance and
        Canonical Quantization
III. A new set of quark, gluon momentum,
        angular momentum, and spin operators
   III.0 Decomposing the gauge field into pure
        gauge and physical parts
   III.1 Quantum mechanics
   III.2 QED
   III.3 QCD
IV. Nucleon internal structure
V. Summary
               I.Introduction
 Fundamental principles of quantum physics:

1.Quantization rule: operators corresponding to
  observables satisfy definite quantization rule;
2.Gauge invariance: operators corresponding to
  observables must be gauge invariant;
3.Lorentz covariance: operators in quantum
  field theory must be Lorentz covariant.
       Apply fundamental principles
           to Internal structure
• Quark gluon (electron) contribution to the
  nucleon (atom) observables, mass, momentum,
  spin, magnetic moment, etc. are unavoidable in
  the study of nucleon (atom) internal structure.
• We never have the quark gluon (electron)
  momentum and angular momentum operators
  which satisfy both gauge invariance and
  canonical commutation relations or Lie algebra.
   Pauli-Landau-Feynman-…….
II. Conflicts between
   gauge invariance
          and
canonical quantization
          Quantum Mechanics
  Even though the Schroedinger equation is
gauge invariant, the matrix elements (ME) of the
canonical momentum, orbital angular momentum,
and Hamiltonian of a charged particle moving in
an eletromagnetic field are gauge dependent,
            orbital angular momentum
especially the
and Hamiltonian of the hydrogen atom
 are “not the measurable ones” !?
            It is absurd!
                            QED
• The canonical momentum and orbital angular
momentum of electron are gauge dependent and
so their physical meaning is obscure.
• The canonical photon spin and orbital angular
momentum operators are also gauge dependent.
Their physical meaning is obscure too.
• Even it has been claimed in some textbooks that
it is impossible to have photon spin and orbital
angular momentum operators.
V.B. Berestetskii, A.M. Lifshitz and L.P. Pitaevskii, Quantum
electrodynamics, Pergamon, Oxford, 1982.
C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Photons and
atoms, Wiley, New York,1987.
           Multipole radiation
  Multipole radiation measurement and
analysis are the basis of atomic, molecular,
nuclear and hadron spectroscopy. If the spin and
orbital angular momentum of photon is
gauge dependent and not measurable or even
meaningless , then all determinations of the
                  parity
of these microscopic systems would be
               meaningless!
                   Multipole field
 The multipole radiation theory is based on the
decomposition of an em field into multipole
radiation field with definite photon spin and orbital
angular momentum quantum numbers coupled to a
total angular momentum quantum number LM,


A   p eik r  2 1i L 2L  1DMp ( , ,0)[ ALM (m)  ip ALM (e)]
                     L
                                   L



                           L                     L 1
          A LM (e)            L 1T LL 1M         L 1T LL 1M
                         2L  1                 2L  1

                         ALM (m)   L T LLM
                   QCD
• 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
  operator.
 They are not the right momentum
         operators!
• 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 that
 “there is not a gluon spin
              can be measured”.
             III.
          A New set of
quark, gluon (electron, photon)
          momentum,
   orbital angular momentum
              and
         spin operators
 III.0 Decomposing the gauge field
into pure gauge and physical parts
• There were gauge field decompositions
  discussed before, mainly mathematical.
 Y.S.Duan and M.L.Ge, Sinica Sci. 11(1979)1072;
 L.Fadeev and A.J.Niemi, Nucl. Phys. B464(1999)90; B776(2007)38.


• We suggest a new decomposition based
  on the requirement: to separate the gauge
  field into pure gauge and physical parts.
 X.S. Chen, X.F.Lu, W.M.Sun, F.Wang and T.Goldman, Phys. Rev.
 Lett. 100(2008)232012.
U(1) Abelian gauge field
                                   
             A  Apure  Aphys
                                
   Fpure   Apure   Apure  0
 Apure  0   Aphys   A
                            Aphys  0
                              A phys ( x  )  0
                ' A( x ') 3
Aphys                  d x'
               4 x  x '
      i A0
          phys    i A   t ( A  A
                         0           i      i
                                            phys   )
        Aphys    dxi (i A0  t Ai  t Aiphys )
         0        x


Other solution
                      Apure   ( x)
                           ' A( x ') 3
              ( x)                d x '  0 ( x)
                          4 x  x '
                        A 0
                          pure    t ( x)
                          0 ( x )  0
                             2
Under a gauge transformation,
        A  A    ( x)
          '         

The physical and pure gauge parts will be
transformed as
                  A  '
                     phys    Aphys
              '
              A
              pure     Apure  ( x)

    A 0'
      phys   A
              0
              phys   ,       A0'
                              pure   A
                                      0
                                      pure     ( x)
                                                0
SU(3) non-Abelian gauge field
                        a                     
            A  A T  Apure  Aphys
                              a


                               
 Fpure    A   Apure  ig[ Apure , A ]  0
              pure                        pure


                  Dpure    igApure

      Dpure  Apure   Apure  igApure  Apure  0

                  a   Dpure    ig[ Apure ,]

     a Dpure  Aphys   Aphys  ig[ Apure , Aphys ]  0
                                       i       i
       The above equations can be
              rewritten as
                     Aphys  ig[ Ai , Aphys ]
                                         i




            Aphys   A  ig ( A  Aphys )  ( A  Aphys )


i A
   0
   phys    i A   t ( A  A
                0          i    i
                                phys   )  ig[ A  A
                                              i    i
                                                   phys   ,A A
                                                           0    0
                                                                phys   ]


 a perturbative solution in power of g
           through iteration can be obtained
Under a gauge transformation,

          U ,'
                                         U e    ig ( x )

           '     i    1    1
        A  UA U  U  U
                  g
                                          1
            A       '
                    phys    UAphysU

                               i     1
                                    1
       A '
         pure        UApureU  U U
                               g
      III.1 Quantum mechanics
The classical canonical momentum of a
charged particle moving in an electromagnetic
field, an U(1) gauge field, is
             p  m dr dt  eA
       It is not gauge invariant!
The gauge invariant one is

         p  eApure  mdr dt  eAphys
 Gauge is an internal degree of freedom,
no matter what gauge is used, the canonical
momentum of a charged particle is quantized
as
                 p  i
 The orbital angular momentum is
                             
               L  r p  r
                             i
The Hamiltonian is
                  ( p  e A) 2
              H                eA0
                      2m
 Under a gauge transformation,
                      '  eie ( x ) ,
     A  A  A  ,
          '                               A0  A0'  A0   t ,
the matrix elements transformed as
          | p |   | p |   | e | ,

          | L |   | L |   | er  | ,
          | H |   | H |   | et | .

They are not gauge invariant,
      even though the Schroedinger equation is.
       New momentum operator
• The canonical momentum is,
                                      
                   p  m dr dt  eA 
                                      i
It satisfies the canonical momentum
commutation relation, but its matrix element is
not gauge invariant.
The new momentum operator is,
                                       
                p pure  p  eApure     eApure
                                       i
It satisfies the canonical momentum
commutation relation and its matrix element is
gauge invariant.
We call
            D pure                 1
                    p  e A pure    e A pure
              i                    i
The physical momentum.
It is neither the canonical momentum
                          1
             p  mr  eA  
                          i
nor the mechanical momentum
                          1
             p  eA  mr  D
                          i
     Hamiltonian of hydrogen atom
    Coulomb gauge
        Apure  0,       Aphys  A , c
                                                A0   c
                     1
                H   c
                       ( p  eA )  e
                               c 2     c

                    2m
    Gauge transformed one
'
A
pure    Apure   , A   '
                              phys        Aphys ,  '   c   t
                   1
              H '
                     ( p  eA' )2  e( c  t )
                  2m
Follow the same recipe, we introduce a new Hamiltonian,
                                                   c
                                   ( p  eApure  eA
                                                   phys)2

        H phy  H  e t ( x) 
                  '
                                                             e c
                                            2m

                            2  A

which is gauge invariant, i.e.,

                ' | H phy | '   c | H c | c
This means the hydrogen energy calculated in
Coulomb gauge is physical.
       A rigorous derivation
Start from a QED 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
      Schroedinger or Pauli equation.
                     III.2 QED
Different approach will obtain different energy-momentum
tensor and four momentum, they are not unique:
Noether theorem
                                  
                     P   d x{
                             3
                                        E i Ai }
                                    i
They are not gauge invariant.
Gravitational theory (Weinberg) or Belinfante tensor
                                 D
                  P   d 3 x{    E  B}
                                  i
It appears to be perfect , but individual part does not
                            satisfy the momentum algebra.
   New momentum for QED system

We are experienced in quantum mechanics, so we
introduce              D pure
              P   d 3 x{          E i Aiphys }
                                i

                  A  A pure  A phys

                D pure    ieApure

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    d xx  ( E  B)   d x( x  E A
         3                  3       i   i
                                        phys    E  Aphys )


It includesphoton spin and
      orbital angular momentum
Electric dipole radiation field
                                           i
 B lm  a h (kr) LYlm ,...... lm  ik Alm    B lm
         lm l
             (1)
                            E
                                           k
  1                  | a11 |2 3 1  cos 2       sin 
    Re[ E 11 B11 ]            [          nr        n ]
  2                    (kr) 16
                             2
                                      2            kr
  1      i         | a11 |2 3 1  cos 2       sin 
    Re[ E11A11 ] 
             i
                              [          nr        n ]
  2                  (kr) 16
                           2
                                    2            2kr

     dP | a11 |2 3 1  cos 2     dJ z
                             k
     d    k 2
                  16   2         d

          dJ z | a11 |2 3
                         sin 2 
          d      k 3 16
Usual  Spin  decomposition
                    
J QED Se  Le  S  L
• 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
  operators.
• However they are not gauge invariant
  except the electron spin. Therefore the
  physical meaning is obscure.
Gauge  in var iant  spin  decomposition
                      
      J QED  Se  L'e  J '
• 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 unphysical!
New spin decomposition for QED
            system
  J QED  Se  L  S  L''
                          e
                                         ''        ''


                   
       Se   d x   3        †

                   2
                                   D pure
       L   d x x 
         ''
         e
                  3       †
                                              
                                     i
       S   d xE  Aphys
             ''       3



       L   d xE x A
        ''            3        i                  i
                                                  phys
         Multipole radiation
• Photon spin and orbital angular momentum
are well defined now and they will take the
canonical form in Coulomb gauge.
• Multipole radiation analysis is based on the
decomposition of em vector potential in
Coulomb gauge. The results are physical and
these multipole field operators are in fact
gauge invariant.
              III.3 QCD
       three decompositions of
t
h            momentum
r
e
                      
e
          P   d x{
                    3
                          E A }
                             i  i 

                      i
                       D
           P   d x{ 3
                           E  B}
                                  

                       i
                         D pure
      P   d x
               3   
                                     d 3 xE i a D pure Aphys
                                                           i

                           i

D pure    ig A pure                      a   D pure    ig[ A pure , ]
 Three decompositions of angular
           momentum
1. From QCD Lagrangian, one can get the total
  angular momentum by Noether theorem:
2. One can have the gauge invariant decomposition,
3.New decomposition
                                  ''        ''     ''
 J QCD  S q  L  S  L          q         g      g


                                           
   Sq              d
                          3
                              x       
                                             
                                           2

                             D pure
              L   d x x 
               ''
               q
                      3       
                                    
                               i


                    
         ''
    S    g           d 3 xE  A phy


   L   d 3 xEi x  a Dpure Ai
    ''
    g                                            phy
     IV. Nucleon internal structure
         it should be reexamined!
• The present parton distribution is not the
real quark and gluon momentum distribution.
In the asymptotic limit, the gluon only
contributes ~1/5 nucleon momentum, not 1/2 !
   arXiv:0904.0321[hep-ph],Phys.Rev.Lett. in press.
• The nucleon spin structure should be
reexamined based on the new decomposition
                    and new operators.
    Consistent separation of nucleon
         momentum and spin




Standard construction of orbital angular momentum L   d 3 x x  P
           Quantitative example:
Old quark/gluon momentum in the nucleon

                    1
         Pq   d x i D
                   3

    If: 
         Pq  d 3 xE  B
                
                              2 ng    nf 
           2 d 
                  Pq   s   9            
                                        3   Pq 
    Then Q    2                            
            dQ  Pg  2  2ng          n f   Pg 
                                      
                              9         3 
                         2 ng         1
    Q   : Pg 
     2
                                 PN     PN (n f  5)
                      2ng  3n f      2
Real quark/gluon momentum in
           nucleon
      C           1
      Pq   d x i Dpure
                3

 if: 
      Pq C  d 3 xE ai Aai
                        phys


                               ng      nf 
                                
         2 d
                Pq C   s  18        3  q
                                              P C 
             2 
                             
                P C  2  n                   C
 Then: Q
          dQ  g                                 
                                         n f   Pg 
                                        
                                   g
                              
                               18        3 
                          1
                            ng
                          2             1
 Q   : Pg 
  2            C
                                    PN    PN (n f  5)
                       1                5
                         ng  3n f
                       2
• 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
  spin matrix elements.
  D. Qing, X.S. Chen and F. Wang, Phys. Rev. D58,114032 (1998)
Conventional and new construction of
 parton distribution functions (PDFs)
                          
The pure  gauge term Apure can be used instead of
   the full gauge field A to construct the gauge link
    Wilson line  to accomplish gauge invariance

                      
 The physical term Aphys can be used instead of
                         
  the field strength F        as the gauge  covariant
  canonical variable
 The conventional gauge-invariant “quark” PDF




The gauge link (Wilson line) restores gauge invariance,
       but also brings quark-gluon interaction,
        as also seen in the moment relation:
The new quark PDF




With a second moment:
The conventional gluon PDF




  Relates to the Poynting vector:
   The new gluon PDF




Relates to the new gauge-invariant
         gluon momentum
Gauge-invariant polarized gluon PDF
  and gauge-invariant gluon spin




  Its first moment gives the gauge-invariant local operator:
                     M g ij  F  i ij  Aphys ,
                                           j


 which is the + component of the gauge-invariant gluon spin
                         S g  E  Aphys
 To measure the new quantities
The same experiments as to measure the
 conventional PDFs

New factorization formulae and extraction
 of the new PDFs needed

New quark and gluon orbital angular
 momentum can in principle be measured
 through generalized (off-forward) PDFs
         VII. Summary: general
• The gauge field can be separated into pure gauge
  and physical parts.
• The renowned Poynting vector is not the right
  momentum operator of photon and gluon field.
• The canonical momentum, angular momentum
  operators of the Fermion part are not observables.
• The gauge invariant and canonical quantization rule
  both satisfied momentum, spin and orbital angular
  momentum operators of the individual part do exist.
  They had been confirmed in QM and QED.
• The Coulomb gauge is physical, operators used in
  Coulomb gauge, even with vector potential, are
  gauge invariant, including the hydrogen atom
  Hamiltonian and multipole radiation field operators.
 special to nucleon internal structure
• The nucleon internal structure should be reanalyzed
  and our picture of it might be modified

• A new set of quark, gluon momentum, orbital
  angular momentum and spin operators for the study
  of nucleon internal structure is provided

• Gluon spin is indeed meaningful and measurable

• Gluons carry not much of the nucleon momentum,
  not ½ but 1/5
                Prospect
• Computation of asymptotic partition of
  nucleon spin
• Reanalysis of the measurements of
  unpolarized quark and gluon PDFs
  New factorization formulas are needed
• Reanalysis and further measurements of
  polarized gluon distributions. A lattice QCD
  calculation of gluon spin contribution to
  nucleon spin.
• The Lorentz covariance can be kept to
  what extent, the meaning of non Lorentz
  covariance.
• The possibility of the gauge non-invariant
  color dependent part of an operator might
  have zero matrix element in a color singlet
  nucleon state should be studied further.
Thanks
       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+  0 ; n+   ;
              n->n+  0 ; p+   .
• 3. Gell-mann and Zweig’s quark model
    SU(3) symmetry:
    baryon qqq; meson q q .
    SU(6) symmetry:
               1
    B(qqq)=      [  ms (q3 )ms (q3 )   ma (q3 )ma (q3 )] .
               2
    color degree of freedom.
    quark spin contribution to nucleon spin,
                          4         1
                   u      ; d   ; s  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, qq 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
               Spin
• 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  the quark
                              is
  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
  reduction.
• 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.
Thanks

				
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posted:10/9/2012
language:Latin
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