Transcript Review on Nucleon Spin Structure

Review on Nucleon Spin
Structure
X.S.Chen, Dept. of Phys., Sichuan Univ.
X.F.Lu, Dept. of Phys., Sichuan Univ.
W.M.Sun, Dept. of Phys., Nanjing Univ.
Fan Wang, Dept. of Phys. Nanjing Univ.
Outline
I. Introduction
II. Nucleon internal structure
III. There is no proton spin crisis but quark
spin confusion
IV. Gauge invariance and canonical
commutation relation of nucleon spin
operators
V. Summary
I. Introduction
•
•
There are various reviews on the nucleon spin structure, such as B.W.
Filippone & X.D. Ji, Adv. Nucl. Phys. 26(2001)1.
We will not repeat those but discuss two problems related to nucleon spin
which we believe where confusions remain.
1.It is still a quite popular idea that the polarized deep inelastic lepton-nucleon
scattering (DIS) measured quark spin 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 DIS measured quark spin can
be accomodated in CQM.
2.One has either gauge invariant or non-invariant decomposition of the total
angular momentum operator of nucleon, a quantum gauge field system, but
one has no gauge invariant and canonical commutation relation of the
angular momentum operator both satisfied decomposition.
Unharmony within three thematic
melodies of 20th century physics
• Three thematic melodies:
Symmetry, quantization, phase
Combined the phase and symmetry leads
to gauge invariance.
• When applied these three thematic
melodies to atom and nucleon internal
structures, one meets unharmony.
II. 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.
• SLAC-MIT e-p deep inelastic scattering
Bjorken scaling.
quark discovered.
there are really spin one half, fractional
charge, colorful quarks within nucleon.
c,b,t quark discovered in 1974, 1977,1997
complete the history of quark discovery.
there are only three quark generations.
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:
.
• 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.
IV.Gauge Invariance and canonical
Commutation relation of nucleon
spin operator
• Up to now we use the following decomposition,
• Each term in this decomposition satisfies
the canonical commutation relation of
angular momentum operator, 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.
• We can have the gauge invariant decomposition,
• However each term no longer satisfies the
canonical commutation relation of angular
momentum operator except the quark spin, in
this sense the second and third term is not the
quark orbital and gluon angular momentum
operator.
• 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.
• How to reconcile these two fundamental
requirements, the gauge invariance and
canonical commutation relation?
• One choice is to keep gauge invariance
and give up canonical commutation
relation.
• This is a dangerous suggestion. Is this the
unavoidable choice?
Our solution
• We found a third decomposition, where
each term both the gauge invariance and
the canonical commutation relation are
satisfied.
PRL, 100(2008) 232002
arXiv:0806.3166[hep-ph];0709.1284[hep-ph];
0709.3649[hep-ph];0710.1427[hep-ph];
0802.2891[hep-ph].





J QED  S e  Le ' ' S ' ' L ' '
QCD
''
q
''
g
J QCD  S q  L  S  L
Sq 
''
q
L 
S
L
''
g
''
g
x

3

d
x

r



d
3
d
3
a


2
D phy

i
xE  A
a
phy
3
a
a
d
xE
r


A
i
i phy

''
g
Non Abelian complication
D phy    ig A pure
A pure  T A
a
a
pure
A  A pure  A phy
D phy  A pure    A pure  ig A pure  A pure  0
A phy  E  E  A phy  0
• It is not a special problem for angular
momentum;
• But all of the fundamental operators of QM
and QFT:
Four momentum;
Hamiltonian.
Quantum Mechanics
• The fundamental operators in QM
r
p  i

L  r p  r
i
( 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  m r  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.
• Few days ago, we derived the Dirac
equation and the Hamiltonian of electron
in the presence of a massive proton from a
ED Lagrangian with electron and proton
and found that indeed the time translation
operator and the Hamiltonian are different,
exactly as we obtained
phenomenologically before.
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)
P   d 3 x{ 
D
  E  B}
i
It appears to be perfect and has been used in parton
distribution analysis of nucleon, but do 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
VI. Summary
•
•
•
There is no proton spin crisis but quark spin
confusion.
The DIS measured quark spin is better to be
called quark axial charge, it is not the quark
spin calculated in CQM.
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,
• The renowned Poynting vector is not the right
momentum operator of em field.
• The space-time translation and space rotation
generators are not the observable momentum,
Hamiltonian and angular momentum operator.
• The gauge invariance and canonical quantization
rule for momentum, spin and orbital angular
momentum can be satisfied simultaneously.
• The Coulomb gauge is physical, expressions in
Coulomb gauge, even with vector potential, are
gauge invariant, including the hydrogen atomic
Hamiltonian and multipole radiation.
• We suggest to use the physical
momentum, angular momentum, etc.
in hadron physics as have been used
in atomic, nuclear physics so long a time,
Thanks
谢谢
xie xie