Download Review on Nucleon Spin Structure

Problems
in nucleon structure study
X.S.Chen, Dept. of Phys., HUST.
W.M.Sun, Dept. of Phys., Nanjing Univ.
Fan Wang, Dept. of Phys. Nanjing Univ
C.W. Wong, Dept. of Phys., UCLA
[email protected]
Outline
I.
II.
III.
Introduction
The first proton spin crisis and quark spin confusion
The second proton spin crisis and the quark orbital
angular momentum confusion
IV. A consistent decomposition of the momentum and
angular momentum operators
V. Debate on the decomposition
V.1, Renormalization ruins the canonical commutation
relation
V.2, Non-local operator
V.3, Poincare covariance
V.4, Uniqueness of gauge invariant extension
VI. Summary
I. Introduction
• It is still a popular idea that the polarized deep Inelastic
scattering (DIS) measured quark spin invalidates the constituent
quark model (CQM), lead to the so-called proton spin crisis.
I will show that there is no proton spin crisis but has a
misidentifying of the relativistic quark spin to the non-relativistic
quark spin.
• There are different quark momentum and orbital angular
Momentum operators used in the literature. This will cause further
confusion in the nucleon spin structure study and might have
already caused it, such as the second proton spin crisis.
• For a long time it is a widely accepted idea that there is no spin
operator for massless particles. I will show it is not true!
• There is a new debate on what are the right quark, gluon
spin and orbital angular momentum operators. I will review this
Debate.
II.The first proton spin crisis and
quark spin confusion
quark spin contribution to nucleon spin in
naïve non-relativistic quark model
4
1
u  , d   , s  0.
3
3
Lq  0, G  0, LG  0.
consistent with nucleon magnetic moments.
 p / n  3 / 2
DIS measured quark spin
The DIS measured quark spin contributions are:
  u  d  s
  0.800  0.467  0.126  0.207(Q  1GeV , G  0)
2
2
  0.812  0.455  0.114  0.243(Q  1GeV , G  0)
2
2
(E.Leader, A.V.Sidorov and D.B.Stamenov, PRD75,074027(2007);
hep-ph/0612360)
  0.813  0.458  0.114  0.242(Q2  10GeV 2 , G  0.084)
(D.de Florian, R.Sassot, M.Statmann and W.Vogelsang, PRL101,
072001(2008); 0804.0422[hep-ph])
.
Proton spin crisis!?
It seems there are serious contradiction
between our understanding of nucleon spin
structure and measurements:
The DIS measured total quark spin
contribution to nucleon spin is about 25%,
while we believe that the nucleon spin is
solely coming from quark spin;
Quark spin confusion
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.
Quark axial vector current operator
The quark axial vector current operator can be
expanded instantaneously as
Contents of axial vector current operator
• Only the first term of the axial vector current
operator, which is the Pauli spin, 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, has
not been calculated in models with only valence
quark configuration. Various meson cloud
models have not calculated this term either.
Extending the valence CQM to
Fock states expansion quark model
(D.Qing, X.S.Chen and F.Wang,PRD58,114032(1998))
• 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,
N  c0 q  c (q ) (qq )
3
3
Model prediction of quark spin
contribution to nucleon spin
Main spin reduction mechanism
III.The second proton spin crisis
and quark orbital angular
momentum confusion
Quark orbital angular momentum
confusion
• The quark “orbital
angular momentum”
Lq   d 3 xx  q ( p  gA) q
calculated in LQCD and measured in DVCS is not the
real orbital angular momentum used in quantum
mechanics. It does not satisfy the Angular Momentum
Algebra,
L  L  iL
and the gluon contribution is ENTANGLED in it.
Where does the nucleon get spin?
Real quark orbital angular momentum
• As a QCD system the nucleon spin consists of
the following four terms (in Coulomb gauge),
The Real quark orbital angular momentum
operator
The real quark orbital angular momentum operator
can be expanded instantaneously as
Quark orbital angular momentum will
compensate the quark spin reduction
• The first term is the non-relativistic quark orbital
angular momentum operator used in CQM, which
does not contribute to nucleon spin in the naïve CQM.
• The second term is again the relativistic correction,
which will compensate the relativistic quark spin
reduction.
• The third term is again the qq creation and annihilation
contribution, which will compensate the quark spin
reduction due to qq creation and annihilation.
Relativistic versus non-relativistic
spin-orbital angular momentum sum
• 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 operators are exact the same but
with opposite sign. Therefore if we add them
together we will have
where the
,
are the non-relativistic quark
spin and orbital angular momentum operator
used in quantum mechanics.
• The above relation tells us that the quark contribution to nucleon spin can be
either attributed to the quark Pauli spin, as done in the last thirty years in
CQM, and the non-relativistic quark orbital angular momentum which does
not contribute to the nucleon spin in naïve CQM; or
• part of the quark contribution is attributed to the relativistic quark spin as
measured in DIS, the other part is attributed to the relativistic quark orbital
angular momentum which will provide
the exact compensation of the missing part in t
he relativistic “quark spin”
Prediction
• I have mentioned, in the introduction, a
widely accepted idea that there is no spin
operator for massless particles.
• These confusions made up our mind to
check the ideas of momentum, spin and
orbital angular momentum of a gauge field
system critically!
IV.A consistent decomposition of
the momentum and angular
momentum of a gauge system
Jaffe-Manohar decomposition:
R.L.Jaffe and A. Manohar,Nucl.Phys.B337,509(1990).
• 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.
Gauge invariant decomposition
X.S.Chen and F.Wang, Commun.Theor.Phys. 27,212(1997).
X.Ji, Phys.Rev.Lett.,78,610(1997).
• 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 terms are not the
real quark orbital and gluon angular momentum
operators.
• 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.
• In QED this means there is no photon spin and
orbital angular momentum! This contradicts the
well established multipole radiation analysis.
Standard definition of momentum and
orbital angular momentum
Momentum and angular momentum
operators for charged particle moving
in electro-magnetic field
For a charged particle moving in em field,
the canonical momentum is,
p  mr  qA
• It is gauge dependent, so classically it is
Not measurable.

p  , no
i
• In QM, we quantize it as
matter what gauge is.
• It appears to be gauge invariant, but in fact
Not!
Under a gauge transformation
  '  eiq ( x ) ,
A  A  A  ,
'
   '     t ,
The matrix elements transform as
 | p |   | p |   | q | ,
 | L |   | L |   | qr  | ,
 | H |   | H |   | qt | ,
New momentum operator
 A   A.
We call
D pure
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 or the kinematical momentum
1
p  q A  mr  D
i
Gauge invariance and canonical
quantization both satisfied decomposition
• Gauge invariance is not sufficient to fix the
decomposition of the angular momentum of a gauge
system.
• Canonical quantization rule of the angular momentum
operator must be respected. It is also an additional
condition to fix the decomposition.
• Measurable one must be physical.
X.S.Chen, X.F.Lu, W.M.Sun, F.Wang and T.Goldman, Phys.Rev.Lett. 100(2008)
232002.
arXiv:0806.3166; 0807.3083; 0812.4366[hep-ph];
0909.0798[hep-ph]
J QED  Se  L  S  L
"
e
"
"

Se   d x

2
Dpure
''
3

Le   d x x 

i
3

S"   d 3 xE  Aphy
L"   d 3 xE i x  D pure Aiphys
A  Aphys  Apure ,
Aphy
  Aphys  0,  Aphys   A.
,
1
B
(
x
)
3 ,
 
d x
,

,
4
xx
Dpure    ieApure .
It provides the theoretical basis of the multipole radiation analysis
QCD
''
q
''
g
J QCD  S q  L  S  L
Sq 
d
3
x



2
D pure
L   d x x 

i
''
q
S
''
g

3
''
g
L 

d
3
a
a
phys
xE  A
3
ia
adj
ia
d
xE
x

D
A
pure
phys

''
g
Non Abelian complication
A  A pure  A phy
A pure  T A
a
a
pure
D pure  A pure    A pure  ig A pure  A pure  0
D pure    ig A pure
adj
Dpure
 Aphys    Aphys  ig[ Aipure, Aiphys ]  0
adj
Dpure
   ig[ Apure , ]
Consistent separation of nucleon
momentum and angular momentum
Standard construction of orbital angular momentum L   d 3 x x  P
• Each term is gauge invariant and so in
principle measurable.
• Each term satisfies angular momentum
commutation relation and so can be
compared to quark model ones.
• In Coulomb gauge it reduces to JaffeManohar decomposition, the usual one
used in QM and QFT.
• Jaffe-Manohar’s quark, gluon orbital
angular momentum and gluon spin are
gauge dependent. Ours are gauge
invariant.
V.Debate on the decomposition
V.1, Renormalization ruins the canonical
commutation relations of the bare field
operators, do we have to insist on the
commutation relations of bare field operators?
No way to avoid it!
V.2, New decomposition uses non-local
operators? Yes, it did! non-local operator is
popular in nucleon structure study. Parton
distribution operators are all non-local.
Moreover the new one reduces to local one in
Coulomb gauge!
V.3,Poincare covariance
No full Poincare covariance for
the individual operators
3-dimensional translation and rotation
for individual part can be retained
General Lorentz covariance
for individual part are retained
QED
' phys


phys
  ( A
  
' pure


pure
  
A
A
  ( A
  
phys


phys

),
pure

),
pure

,
Two special cases
• Case 1

phys

 
pure

,   0
In this case, the vector potential A transforms with the
usual homogeneous Lorentz transformation law.
• Case 2
  
phys

,
pure

0
In this case, the vector potential A transforms
inhomogeneously as the physical part.
QCD
Remarks about Lorentz Covariance
• The Lorentz transformation law of the gauge
potential is gauge fixing dependent.
• Both the kinematical 4-momentum operator
and the new physical 4-momentum operator
are Lorentz covariant, but the later transforms
with the
inhomogeneous Lorentz transformation law.
V.4, Gauge Invariant extension
• Among the infinite possible gauge invariant
extensions, only one might be used to construct
the observable which is the physical part of the
gauge potential.
• For QED, it is the Coulomb gauge fixing parts
which are physical because there are only two
transverse components or helicity components
retained. They are observed in Compton
scattering.
• For QCD a natural choice is to choose the two
transverse or helicity components of the gluon
field as physical parts which might be observed
in gluon jet.
• The light cone gauge invariant extension is not
physical one, because the gauge potential is
entangled in already.
• Only in light cone gauge, the Jaffe-Bashinsky
gluon spin operator can be related to gluon spin
and so can the measured gluon helicity
distribution be related to the matrix element of
J-B operator, to the Collins-Soper-Manohar
polarized gluon parton distribution function
should be studied further.
• The three components of the kinematic
momentum p  g A do not commute and
cannot be measured simultaneously and so
cannot be used to describe the 3-dimensional
quark parton momentum distribution.
• The kinematical momentum can only be
measured classically but cannot be measured in
quantum physics.
• DIS “measured” quark parton momentum
distribution is only one component of the
kinematical momentum-the light-cone
component.
•
Experimental Check
Energy-momentum (E-M) tensor T  is the
starting point of various gauge invariant
decomposition and extension.
• The popular idea is that one has the freedom
to choose the form of the E-M tensor because
by adding a surface term the conservation law
satisfied by the E-M tensor is unchanged.
• The gauge invariant symmetric E-M tensor is
prefered in gauge field theory and the
kinematical momentum is derived from the
symmetric E-M tensor.
• In fact the classic em E-M tensor is
measurable and there is already measurement.
Optical evidence
• For symmetric em E-M tensor, there should be
no difference of the diffraction pattern for an
orbital or spin polarized light beam if the total
J z  lz ~ or ~ sz  1
• For asymmetric em E-M tensor, there should be
difference of the diffraction pattern between
orbital and spin polarized beams, because only
for orbital polarized beam there is momentum
density circular flow in the transverse plane.
A detailed analysis had been given in
arXiv:1211.4407[physics.class-ph]
Spin is different from orbital angular
momentum
• In 1920’s one already knew that the fundamental spin can not
be related to the orbital motion. So to use the symmetric E-M
tensor to express the angular momentum operator as
physically is misleading even though mathematically correct. It
also leads to the wrong idea that the Poynting vector is not only
the energy flow but also the momentum density flow of em field.
The Belinfante derivation of the symmetric E-M tensor and in
turn the derived momentum and angular momentum density
operators physically is misleading too.
Spin half electron field needs
asymmetric energy-momentum tensor
• Symmetric E-M tensor for a spin half electron field will lead to
contradiction between angular momentum and magnetic
momentum measurement. Suppose we have a spin polarized
electron beam moving along the z direction with momentum
density flow K ,
1
 ( x K ) z dV   J z dV  N 2
K
j
  n,
E e
E
E
 ( x  K ) z dV  e  ( x  j ) z dV  e  2 z dV  N 1
• For symmetric E-M tensor, the momentum density flow is the
same as the energy density flow. For spin s=1/2 electron this
will lead to a contradiction. The energy flow and momentum
density flow should be different.
A detaild analysis had been given in arXiv:1211.2360[gr-qc].
To meet the requirement of a gauge
invariant but asymmetric E-M tensor we
derived the following one based on the
decomposition of gauge potential.
Hope experimental colleagues to check
further which one is the correct one.

cw
T
i  
  phys

   D pure  h.c.  F  A  g L(x)
2
Remarks about various proposals
• We appreciate E. Leader’s effort, especially
on the full quantum version in the covariant
gauge.
• E. Leader’s proposal: gauge non-invariant
operators might have gauge invariant matrix
elements for physical states.
We studied this approach in 1998.
X.S. Chen and Fan Wang, Gauge invariance and hadron
structure, hep-ph/9802346.
At most, it might be true for gauge transformations
within the covariant or the extended Lorentz gauge.
Hamiltonian of hydrogen atom
Coulomb gauge:
c
//
c

A  0,
A  0,
A    0.
c
0
c
Hamiltonian of a non-relativistic charged 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
H phy
c
2

( p  q A//  q A )
 H  q t ( x) 
 q c
2m
  2 A
 | H phy |    c | H c |  c
A check
• We derived the Dirac equation and the
Hamiltonian of electron in the presence of a
massive proton from a em Lagrangian with
electron and proton and found that indeed
the time translation operator and the
Hamiltonian are different, exactly as we
obtained phenomenologically before.
W.M. Sun, X.S. Chen, X.F. Lu and F. Wang, arXiv:1002.3421[hep-ph]
• We appreciate Hatta’s effort .
But just say you use another gauge invariant
one, it is not physical, because it includes
unphysical gluon field.
• We also appreciate P.M. Zhang, D. Pak and
Y.M. Cho’s effort. They claim their approach
can avoid the Gribov ambiguity.
• We appreciate X. Ji’s criticism which pushed us
to understand more of this topic.
VI. Summary
• There are different quark and gluon momentum and
orbital angular momentum operators. Confusions
disturbing or even misleading the nucleon spin
structure studies.
• Quark spin missing can be understood with the Fock
space extension of the CQM.
• It is quite possible that the real relativistic quark
orbital angular momentum will compensate the
missing quark spin.
• A LQCD calculation of the matrix elements of u,d
quark physical orbital angular momentum might
illuminate the nucleon spin structure study.
• For a gauge system, the momentum and
angular momentum operators of the individual
part (quark~gluon, electron~photon), the
existing ones are either gauge invariant or
satisfy the canonical commutation relation only
but not both.
• We suggest a decomposition which satisfies
both the gauge invariance and canonical
commutation relations. It might be useful and
modify our picture of nucleon internal structure.
• It is not a special problem for quark and gluon angular
momentum operators, but a fundamental problem of gauge
field systems.
Gauge potential can be separated into gauge invariant
and pure gauge two parts. The pure gauge part should be
subtracted from the operators of the individual momentum
and angular momentum of quark and gluon to make them
gauge invariant.
The spin and orbital angular momentum of gluon or
photon can be separated gauge invariantly.
X.S.Chen, X.F.Lu, W.M.Sun, F.Wang and T.Goldman, Phys.Rev.Lett.
100,232002(2008), arXiv:0806.3166; 0807.3083; 0812.4336[hep-ph];
0909.0798[hep-ph]
Thanks
for the invitation.