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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.
II.
Introduction
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,
especially the
orbital angular momentum
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,
L
A   p eikr  2  L 1i L 2 L  1DMp
( , ,0)[ ALM (m)  ip ALM (e)]
ALM (e)  
L
L 1
 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
Aphys
 ' A( x ') 3
  
d x'
4 x  x '
i A
0
phys
 i A   t ( A  A
0
i
i
phys
)
0
x
i
Aphys
  
dxi ( i A0   t Ai   t Aphys
)
Other solution
Apure   ( x)
 ' A( x ') 3
 ( x)   
d x '  0 ( x)
4 x  x '
0
pure
A
 t ( x)
 0 ( x )  0
2
Under a gauge transformation,
'


A  A    ( x)
The physical and pure gauge parts will be
transformed as
'
Aphys  Aphys
'
pure
A
 Apure   ( x)
0'
0
Aphys
 Aphys
,
0'
pure
A
A
0
pure
  ( x)
0
SU(3) non-Abelian gauge field

a


A  A T  Apure  Aphys
a



Fpure
   Apure   Apure
 ig[ Apure
, Apure ]  0
Dpure    igApure
Dpure  Apure   Apure  igApure  Apure  0
a
Dpure    ig[ Apure ,]
i
i
D

A



A

ig
[
A
,
A
a
pure
phys
phys
pure
phys ]  0
The above equations can be
rewritten as
  Aphys  ig[ Ai , Aiphys ]
 Aphys   A  ig ( A  Aphys )  ( A  Aphys )
0
0
i Aphys
 i A0  t ( Ai  Aiphys )  ig[ Ai  Aiphys , A0  Aphys
]
a perturbative solution in power of g
through iteration can be obtained
Under a gauge transformation,
  U ,
'
U e
 ig ( x )
i
 1
A  UA U  U  U
g
'

'
phys
A
'
pure
A
1
 UAphysU
1
i
1
 UApureU  U U
g
1
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  m dr 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  eA pure    eA 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
Aphys  A ,
Apure  0,
c
A 
0
c
1
H 
( p  eAc )2  e c
2m
c
Gauge transformed one
'
pure
A
 Apure    , A
'
phys
'
c



  t
 Aphys ,
1
H 
( p  eA' ) 2  e( c   t )
2m
'
Follow the same recipe, we introduce a new Hamiltonian,
H phy  H  et ( x) 
'
c
2
phys
( p  eApure  eA
)
2m
 e c
   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
3
 
P   d x{
  E i A i }
i
They are not gauge invariant.
Gravitational theory (Weinberg) or Belinfante tensor
3
 D
P   d 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{ 
i
  E i Aiphys }
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 )
photon spin and
orbital angular momentum
It includes
Electric dipole radiation field
i
B lm  a h (kr) LYlm ,......E lm  ik Alm    B lm
k
(1)
lm l

1
| a11 |2 3 1  cos 2 
sin 
Re[ E11 B11 ] 

[
nr 
n ]
2
2
(kr) 16
2
kr
2
2
1
|
a
|
3
1

cos

sin 
i
i
11
Re[ E11A11 ] 

[
nr 
n ]
2
2
(kr) 16
2
2kr
dP | a11 |2 3 1  cos 2 
dJ z


k
2
d
k
16
2
d
dJ z | a11 |2 3
2



sin

3
d
k
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

2
3
†
L   d x x 
''
e
3
†
Dpure
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.
t
h
r
e
e
III.3 QCD
three decompositions of
momentum

P   d x{
  E i A i }
i

3
D
P   d x{
  E  B}
i

3
P   d x
3

Dpure
D pure    ig A pure
i
i
   d 3 xE i a Dpure Aphys
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
''
q
''
g
J QCD  S q  L  S  L
Sq 
d
3
x

''
g


2
D pure
L   d x x 

i
''
q
S
''
g

3

3
d
 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

3
1
 Pq   d x i D
If: 
 Pq  d 3 xE  B


 2 ng


P
s
9
2 d  q 

Then Q

2 
dQ  Pg  2  2ng

 9
2 ng
2
Q   : Pg 
PN
2ng  3n f
nf 

3   Pq 
 
n f   Pg 
 
3 
1
PN (n f  5)
2
Real quark/gluon momentum in
nucleon
 C
3
1
 Pq   d x i Dpure
if: 
 Pq C  d 3 xE ai Aai

phys

nf 
 ng

C
C






P
P
s
18
3
q
q
2 d

 C 
Then: Q


C
2 
n f   Pg 
dQ  Pg  2  ng
 

3 
 18
1
ng
1
2
C
2
Q   : Pg 
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:
j
M g ij  F  i ij  Aphys
,
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,
u 
4
1
; 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 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
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