do gluons carry half of the nucleon momentum? x.s.chen, x.f.lu dept. of phys., sichuan univ....
TRANSCRIPT
Do gluons carry half of the nucleon 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)
T.Goldman T.D., LANL, USA
Outline
I. IntroductionII. Conflicts between Gauge invariance and Can
onical Quantization III. A new set of quark, gluon momentum, angul
ar momentum, and spin operators III.0 A lemma:Decomposing the gauge field into
pure gauge and physical parts III.1 Quantum mechanics III.2 QED III.3 QCDIV. Nucleon internal structureV. Summary
I.Introduction
• A widely accepted nucleon internal momentum structure: quark and gluon carry half the nucleon momentum in the asymptotic limit.
D.J.Gross and F. Wilczek, PRD9,980(1974); H.Georgi and H.D. Politzer, PRD9,416
(1974); H.D.Politzer, Phys.Rep. 14,129(1974).
• This is a distorted picture, because it is obtained based on a momentum decomposition where both quark and gluon momentum operators are not proper.
• A proper momentum operator must be gauge invariant and satisfy canonical momentum algebra.
• The nucleon internal spin structure has the same problem. The quark gluon angular momentum operators used in the nucleon spin sum rule is not proper either.
• The nucleon internal momentum and spin structure should be reexamined based on the proper quark and gluon momentum, orbital angular momentum and spin operators.
II. Conflicts between
gauge invariance
and
canonical quantization
Quantum mechanics
The classical canonical momentum of a
charged particle moving in an electromagnetic
field, an U(1) gauge field, is
It is not gauge invariant!
The gauge invariant one is
drp m eA
dt
pure phys
drp eA m eA
dt
Gauge is an internal degree of freedom,
no matter what gauge is used, the canonical
momentum of a charged particle is quantized
as
The orbital angular momentum is
The Hamiltonian is
ip
irprL
20( )
2
p eAH eA
m
����������������������������
Under a gauge transformation,
the matrix elements transformed as
They are not gauge invariant,
even though the Schroedinger equation is.
' ( ) ,ie xe ' ,A A A
������������������������������������������ 0 0' 0 ,tA A A
| | | | | | ,p p e
| | | | | | ,L L er
| | | | | | .tH H e
We call
The physical momentum.
It is neither the canonical momentum
nor the mechanical momentum
1 purepure purephys
Dp p eA eA
i i
��������������������������������������������������������
1p mr eA
i
��������������������������������������������������������
1p eA mr D
i
��������������������������������������������������������
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 andatoms, 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,
LLMLLM TmA )(
)]()()[0,,(122 1 eAipmADLieA LMLMLMp
LL
rikp
MLLLMLLLLM TL
LT
L
LeA 1111 12
1
12)(
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 proper momentum
operators!
D��������������
p
E B
• 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
pure physA A A
0pure pure pureF A A
0pure physA A A
0physA
( ) 0physA x
3' ( ')'
4 'phys
A xA d x
x x
Other solution
0 0 ( )i ii phys i t physA A A A
( )pureA x
30
' ( ')( ) ' ( )
4 '
A xx d x x
x x
0 ( )pure tA x2
0 ( ) 0x
0 0( )x i i iphys i t t physA dx A A A
( )pureA x
Under a gauge transformation,
The physical and pure gauge parts will be
transformed as
' ( )A A x
'phys physA A
' ( )pure pureA A x
0' 0 ,phys physA A 0' 0 0 ( )pure pureA A x
SU(3) non-Abelian gauge fielda a
pure physA A T A A
[ , ] 0pure pure pure pure pureF A A ig A A
pure pureD igA
0pure pure pure pure pureD A A igA A
[ , ]a pure pureD ig A
[ , ] 0i ia pure phys phys pure physD A A ig A A
The above equations can be rewritten as
a perturbative solution in power of g
through iteration can be obtained
[ , ]i iphys physA ig A A
( ) ( )phys phys physA A ig A A A A
0 0 0 0( ) [ , ]i i i ii phys i t phys phys physA A A A ig A A A A
Under a gauge transformation,
' ,U ' 1 1i
A UA U U Ug
' 1phys physA UA U
' 1 1pure pure
iA UA U U U
g
( )ig xU e
III.1 Quantum mechanics
The classical canonical momentum of a
charged particle moving in an electromagnetic
field, an U(1) gauge field, is
It is not gauge invariant!
The gauge invariant one is
drp m eA
dt
pure phys
drp eA m eA
dt
Gauge is an internal degree of freedom,
no matter what gauge is used, the canonical
momentum of a charged particle is quantized
as
The orbital angular momentum is
The Hamiltonian is
ip
irprL
20( )
2
p eAH eA
m
����������������������������
New momentum operator• The canonical momentum is,
It satisfies the canonical momentum commutation relation, but its matrix element is not gauge invariant.The new momentum operator is,
It satisfies the canonical momentum commutation relation and its matrix element is gauge invariant.
p mdr dt eAi
pure pure purep p eA eAi
We call
The physical momentum.
It is neither the canonical momentum
nor the mechanical momentum
1 purepure purephys
Dp p eA eA
i i
��������������������������������������������������������
1p mr eA
i
��������������������������������������������������������
1p eA mr D
i
��������������������������������������������������������
Hamiltonian of hydrogen atom
Coulomb gauge
Gauge transformed one
0,pureA
,cphysA A
0 cA
21( )
2c c cH p eA e
m
' ,pure pureA A
' ,phys physA A ' c
t
' ' 21( ) ( )
2c
tH p eA em
Follow the same recipe, we introduce a new Hamiltonian,
which is gauge invariant, i.e.,
This means the hydrogen energy calculated in
Coulomb gauge is physical.
2' ( )
( )2
cphyspure c
phy t
p eA eAH H e x e
m
����������������������������
' '| | | |c c cphyH H
2 A
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
They are not gauge invariant.
Gravitational theory (Weinberg) or Belinfante tensor
It appears to be perfect , but individual part does not
satisfy the momentum algebra.
}{3 ii AEi
xdP
}{3 BEi
DxdP
New momentum for QED system
We are experienced in quantum mechanics, so we
introduce
They are both gauge invariant and momentum
algebra satisfied. They return to the canonical
expressions in Coulomb gauge.
3 { }pure i i
phys
DP d x E A
i
����������������������������
pure physA A A ������������������������������������������
pure pureD ieA��������������
We proved the renowned Poynting vector is not the proper momentum of em field
It includes photon spin and
orbital angular momentum
3 3( ) ( )i iphys physJ d xx E B d x x E A E A
��������������
Electric dipole radiation field
lmlmlmlmllmlm Bk
iAikEYLkrhaB ,......)()1(
]sin
2
cos1[
16
3
)(
||]Re[
2
1 2
2
211
1111
n
krn
kr
aBE r
]2
sin
2
cos1[
16
3
)(
||]Re[
2
1 2
2
211
1111
n
krn
kr
aAE r
ii
d
dJk
k
a
d
dP z
2
cos1
16
3|| 2
2
211
23
211 sin
16
3||
k
a
d
dJ z
LSLSJ eeQED
Usual Spin decomposition
• 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.
'' JLSJ eeQED
varGauge in iant spin decomposition
• 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
'' '' ''QED e eJ S L S L
3 † ,2eS d x
'' 3 † puree
DL d x x
i
'' 3physS d xE A
'' 3 i iphysL d xE x A
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 QCDthree decompositions of
momentum
}{3 ii AEi
xdP
3 3pure i ia pure phys
DP d x d xE D A
i
}{3 BEi
DxdP
pure pureD ig A������������������������������������������
,[ ]a pure pureD ig A������������������������������������������
three
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''''''
ggqqQCD LSLSJ
2
3xdS q
'' 3 pureq
DL d x x
i
����������������������������
'' 3phygS d xE A
������������������������������������������
'' 3g i a pure i phyL d xE x D A
��������������
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
3Standard construction of orbital angular momentum xL d x P
Quantitative example:Old quark/gluon momentum in the nucleon
3
22
2
3
2:
1
If:
1( 5)
2
9 32
2 3
Then
2
2
9 3
g f
q qs
g gg f
N f
q
g
g
gN
g f
n n
P d x
P n
Di
P d
nQ P
P PdQ
P Pn n
P
B
n
E
Q
n
x
d
Proper quark/gluon momentum in nucleon
phys
3pure
3
2
22
1
if:
Then: 318
18
( 5
112
1 53
2
: )
3
2
fC C
q qsC
Cg
Cg
g
g
g
g
N f
Cq
C ai a
f
f
g
ig
N
n nP Pd
QP Pn
Q P P
P d x D
n
i
P d xE A
n
nP n
dQ
n
• 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)
• 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
• The quark orbital angular momentum operator can be expanded as,
• It is most interesting to note that the relativistic correction and the 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.
Conventional and new construction ofparton distribution functions (PDFs)
pure
phys
The pure gauge term can be used instead of
the full gauge field to construct the gauge link
Wilson line to accomplish gauge invariance
The physical term can be used
inst
A
A
A
ead of
the field strength as the gauge covariant
canonical variable
F
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
phys
phys
Its first moment gives the gauge-invariant local operator:
which is the + component of the gauge-invariant gluon spi
,
n
ij i jg ij
g
M F A
S E A
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 a
nd physical parts.• The renowned Poynting vector is not the proper mo
mentum operator of photon and gluon field.• The canonical momentum, angular momentum oper
ators 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 measured in QM and QED.• The Coulomb gauge is physical, operators used in
Coulomb gauge, even with gauge 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.
• For the quark (electron), gluon(photon) momentum and angular momentum operators the Lorentz covariance can be kept to what extent, the meaning of non Lorentz covariance.
• The possibility of the gauge non-invariant operator might have gauge invariant matrix element for special states 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+ ; n+ ;
n->n+ ; p+ .
0
0
• 3. Gell-mann and Zweig’s quark model
SU(3) symmetry:
baryon qqq; meson q .
SU(6) symmetry:
B(qqq)= .
color degree of freedom.
quark spin contribution to nucleon spin,
nucleon magnetic moments.
q
4 1; ; 0.
3 3u d s
3 3 3 31[ ( ) ( ) ( ) ( )]
2 ms ms ma maq q q q
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, 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 .
• One would wonder how can quark orbital angular momentum contribute for a pure S-wave configuration?
qL
• 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 creation and annihilation contribution, which also takes back the missing spin.
• It is most interesting to note that the relativistic correction and the 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