Transcript PPT

Nucleon Internal Structure:
the Quark Gluon Momentum
and
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
cosmology
(J-CPNPC)
Outline
I.
II.
III.
IV.
V.
VI.
VII.
Introduction
Conflicts between Gauge invariance
and Canonical Quantization
Quantum mechanics
QED
QCD
Nucleon internal structure
Summary
I.Introduction
• Hadron spctroscopy->hadron structure
• Three Thematic Melodies of Twentieth
Century Physics :
Symmetry, Quantization, Phase (C.N.Yang)
• The combination of symmetry and phase
lead to Gauge Invariance Principle and
gauge field theory (C.N.Yang)
• There are conflicts between these three
thematic melodies when one wants to
apply them to study the internal structure.
Pauli-Landau-Feynman-…….
II. Conflicts between
gauge invariance
and
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,
especially the
orbital angular momentum
and hamiltonian of the hydrogen atom
are “not the measurable ones” !?
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 that
it is impossible to have photon spin an
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,
L
A   p eikr  2  L 1i L 2 L  1DMp
( , ,0)[ A LM (m)  ip A LM (e)]
ALM (e)  
L
L 1
 L1T LL 1M 
 L1T LL 1M
2L  1
2L  1
ALM (m)   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
parity
of these microscopic systems would be
meaningless!
QCD
• Because the 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
“they are measuring a not measurable
quantity”.
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
i
the Hamiltonian is
( p  e A) 2
H 
 e
2m
Gauge transformation
  '  eie ( x ) ,
A  A  A   ,
'
   '     t ,
The matrix elements transformed as
 | p |   | p |   | e | ,
 | L |   | L |   | er   | ,
 | H |   | H |   | et | ,
even though the Schroedinger equation is
gauge invariant.
New momentum operator in
quantum mechanics
Generalized momentum for a charged particle
moving in em field:
p  mr  q A  mr  q A  q A//
It is not gauge invariant, but satisfies the canonical
momentum commutation relation.
p  q A//  mr  q A
  A  0,
  A//  0
It is both gauge invariant and canonical momentum
commutation relation satisfied.
We call
D phy
1
 p  q A//    q A//
i
i
physical momentum.
It is neither the canonical momentum
1
p  mr  q A  
i
nor the mechanical momentum
1
p  q A  m r  D
i
Gauge transformation
 '  eiq ( x ) ,
A'  A   ( x),
only affects the longitudinal part of the vector potential
A//'  A//   ( x),
and time component
 '     t ( 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:
c
//
c

A  0,
A  0,
A    0.
c
0
c
Hamiltonian of a nonrelativistic particle
c
2

(p  qA )
Hc 
 q c .
2m
Gauge transformed one
c
//
c

A//  A   ( x)   ( x), A  A ,    c   t ( x)
c
2

( p  q A)
( p  q  q A )
H
 q 
 q c  q t.
2m
2m
2
Follow the same recipe, we introduce a new Hamiltonian,
H phy
c
2

( p  q A//  q A )
 H  q t ( x) 
 q 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 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.
IV.QED
Different approach will obtain different energy-momentum
tensor and four momentum, they are not unique:
Noether theorem

P   d 3 x{    E i Ai }
i
Gravitational theory (Weinberg) or Belinfante tensor
D
P   d 3 x{    E  B}
i
It appears to be perfect and has been used in parton
distribution analysis of nucleon, 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 3 x{ 
phy
i
  E i A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    d xr  ( E  B)   d x E  A   d xr  E A 
3
3
3
photon spin and
orbital angular momentum
It includes
i
i
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





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.
• 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  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.
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!





J QED  S e  Le ' ' S ' ' L ' '
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.
V. QCD

P   d x{
  E i  Ai }
i

3
D
P   d x{
  E  B}
i

3
P   d x
3

D pure    ig A pure
Dpure
i
i
   d 3 xE i a Dpure Aphys
a
D pure    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
''
q
''
g
J QCD  S q  L  S  L
Sq 
d
3
x

''
g


2
D pure
L   d x r 

i
''
q
S
''
g
L 
''
g

3

3
d
 xE  A phy
3
d
 xEi r  a Dpure Ai
phy
Esential task:to define properly the pure gauge
field A pure and physical one A phy
Phys.Rev.Lett.100,232002(2008), arXiv:0904.0321[hep-ph]
D pure    ig A pure
A pure  T A
a
a
pure
A  A pure  A phy
D pure  A pure    A pure  ig A pure  A pure  0
a
Dpure  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
spin.
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.
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