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Quark Correlations and Single Quark Correlations and Single Spin Asymmetry Spin Asymmetry
G. Musulmanbekov JINR, Dubna, Russiae-mail:[email protected]
Contents
•Introduction•Strongly Correlated Quark Model (SCQM)•Spin in SCQM•Single Spin Asymmetry•Conclusions
SPIN’05
Introduction
)]([)(2
15
3 BExAEixxd
JJJ gq
Where does the Proton Spin come from?
Spin "Crisis“:DIS experiments: ΔΣ=Δu+Δd+Δs 1≪SU(6) 1
QCD sum rule for the nucleon spin:1/2 =(1/2)ΔΣ(Q²)+Lq(Q²)+Δg(Q²)+Lg(Q²)
QCD angular momentum operator (X. Ji, PRL 1997):
Introduction
)]([)(2
15
3 BExAEixxd
JJJ gq
Where does the Proton Spin come from?
Spin "Crisis“:DIS experiments: ΔΣ=Δu+Δd+Δs 1≪SU(6) 1
QCD sum rule for the nucleon spin:1/2 =(1/2)ΔΣ(Q²)+Lq(Q²)+Δg(Q²)+Lg(Q²)
QCD angular momentum operator (X. Ji, PRL 1997):
SCQM All nucleon spin comes from circulating around each valence quarks gluon and quark-antiquark condensate (term 4)
What is Chiral Symmetry and its Breaking?
• Chiral Symmetry
SU(2)L × SU(2)R for ψL,R = u, d
• The order parameter for symmetry breaking is quark or chiral condensate:
<ψψ> - (250 MeV)³, ψ = ≃ u,d
• As a consequence massless valence quarks (u, d) acquire dynamical masses which we call constituent quarks
MC ≈ 350 – 400 MeV
Strongly Correlated Quark Model
(SCQM)
Attractive Force
Attractive Force
Vacuum polarization around single quark
Quark and Gluon Condensate
Vacuum fluctuations(radiation) pressure
Vacuum fluctuations(radiation) pressure
(x)
Strongly Correlated Quark Model
1. Constituent Quarks – Solitons
0),(sin),( txtx
2
21
1/cosh
1/sinhtan4),(
uxu
uuttx ass
x
txtx ass
ass
),(),(
Sine- Gordon equation
Breather – oscillating soliton-antisoliton pair, the periodic solution of SG:
The density profile of the soliton-antisoliton pair (breather)
)(tanh2)( 2 mxMxU
Effective soliton – antisoliton potential
Interplay Between Current and Constituent Quarks
Chiral Symmetry Breaking and Restoration Dynamical Constituent Mass Generation
2 0 21
0
1
x, fermi
Po
lari
zati
on
Fie
ld
2 0 20
0.5
1
x, fermi
Had
ron
ic M
att
er
Dis
trib
uti
on
d=0.64
t = 0
2 0 21
0
1
x, fermi
Po
lari
zati
on
Fie
ld
2 0 20
0.5
1
x, fermi
Had
ron
ic M
att
er
Dis
trib
uti
on
d=0.05
2 0 21
0
1
x, fermi
Po
lari
zati
on
Fie
ld
2 0 20
0.5
1
x, fermi
Had
ron
ic M
att
er
Dis
trib
uti
on
d=0.05
t = T/4
2 0 21
0
1
x, fermi
Po
lari
zati
on
Fie
ld
2 0 20
0.5
1
x, fermi
Had
ron
ic M
att
er
Dis
trib
uti
on
d=0.64 t = T/2
The Strongly Correlated Quark Model
Hamiltonian of the Quark – AntiQuark System
)2()1()1( 2/122/12 xV
mmH qq
q
q
q
q
, are the current masses of quarks, = (x) – the velocity of the quark (antiquark), is the quark–antiquark potential.
qm qm
qqV
)(
)1()(
)1( 2/122/12 xUm
xUm
Hq
q
q
q
)2(2
1)( xVxU
qq is the potential energy of the
quark.
Conjecture:
),(2),()(2 xMrxxdydzdxU Q
where is the dynamical mass of the constituent quark and
)()(
xMQQ
),,(),,(
),(),(
zyxxzyxxC
rxCrx
For simplicity
XAXA
rT
exp)(det
)(2/3
2/1
I
II
U(x) > I – constituent quarksU(x) < II – current(relativistic) quarks
Quark Potential and “Confining Force” inside Light Hadons
Generalization to the 3 – quark system (baryons)
ColorSU )3(
3 RGB,
_ 3 CMY
qq 1 33-
qqq 3 3
3
3
3
31- -
-
_ ( 3)Color
The Proton
3
1
)()(i
iiColor cxax
One–Quark color wave function
ic
ijji cc
Where are orthonormal states with i = R,G,B
Nucleon color wave function
kjijk
iijkColor cccex 6
1)(
Considering each quark separately
SU(3)Color U(1)
Destructive Interference of color fields Phase rotation of the quark w.f. in color space:
Colorxig
Color xex )()( )(
Phase rotation in color space dressing (undressing) of the quark the gauge transformation chiral symmetry breaking (restoration)
);()()( xxAxA
here
)0,0,0,( A
Chiral Symmerty Breaking and its Restoration
Consituent Current Quarks Consituent Quarks Asymptotic Freedom Quarks
t = 0x = xmax
t = T/4x = 0
t = T/2x = xmax
During the valence quarks oscillations:
...321132113211 gqqqcqqqqqcqqqcB
Parameters of SCQM
in pp_
2.Maximal Displacement of Quarks: xmax=0.64 fm,
3.Constituent quark sizes (parameters of gaussian distribution): x,y=0.24 fm, z =0.12 fm
,36023
1)( max)( MeV
mmxM N
Parameters 2 and 3 are derived from the calculations of Inelastic Overlap Function (IOF) and in and pp – collisions.
1.Mass of Consituent Quark
Summary on SCQM
• Quarks and gluons inside hadrons are strongly correlated;
• Constituent quarks are identical to solitons.• Hadronic matter distribution inside hadrons is
fluctuating quantity resulting in interplay between constituent and current quarks.
• Strong interactions between quarks are nonlocal: they emerge as the vacuum response (radiation field) on violation of vacuum homogeneity by embedded quarks.
• Parameters of SCQM:1. Maximal displacement of quarks in hadrons x 0.64f
2. Sizes of the constituent quark: x,y 0.24f, z 0.12f
• Hadronic matter distributions inside nucleons are deformed (oblate).
Inelastic Overlap Function
sxxM 21
+ energy – momentum conservation
Monte-Carlo Simulation of Inelastic Events
Spin in SCQM
a
rd )(3 BErJJ gQ
Our conjecture: spin of consituent quark is entirely analogous to the angular momentum carried by classical circularly polarized wave:
Classical analog of electron spin – F.Belinfante 1939; R. Feynman 1964; H.Ohanian 1986; J. Higbie 1988.
Electron surrounded by proper electric E and B fields creates circulating flow of energy:
S=ɛ₀c²E×B.
Total angular momentum created by this Pointing’s vector
is associated with the entire spin angular momentum of the electron. Here if a = 2/3 r0.entiire mass of electron is contained in its field.
a
rd )((...) 3 BErLs
Spin in SCQM
a
rd )((...) 3chchgQ BErLs
1. Now we accept that
Sch = c²Ech× Bch .
3. Total angular momentum created by this Pointing’s vector
is associated with the entire spin angular momentum of the constituent quark.
A},{ A
and intersecting Ech and Bch create around VQ color
analog of Pointing’s vector
4. Quark oscillations lead to changing of the values of Ech and Bch : at the origin of oscillations they are concentrated
in a small space region around VQ. As a result hadronic current is concentrated on a narrow shell with small
radius.
2. Circulating flow of energy carrying along with it hadronic matter is associated with hadronic matter current.
7. At small displacements spins of both u – quarks inside the proton are predominantly parallel.
8. Velocity field is irrotational:
Analogue from hydrodynamics
((∂ξ)/(∂t))+ ×(ξ×v)=0,∇
ξ= ×v,∇
∇⋅v=0,
const.srv d
r/1v
This means that sea quarks are not polarized
5. Quark spins are perpendicular to the plane of oscillation.
6. Quark spin module is conserved during oscillation:
Single Spin Asymmetry in proton – proton collisions
• In the factorized parton model
'' // qqqqp Df
)(),,(
),,(cos
),(),(
3/
2
/2
/2
3
hkzphzPDhddz
d
Pddd
ryfrddykxfkddxpd
d
qqh
q
prpq
Xpp
0
pq
pqpq
pq
pqq
f
ff
f
fP
/
//
/
/
where
Single Spin Asymmetry in proton – proton collisions
• In the factorized parton model
'' // qqqqp Df
)(),,(
),,(cos
),(),(
3/
2
/2
/2
3
hkzphzPDhddz
d
Pddd
ryfrddykxfkddxpd
d
qqh
q
prpq
Xpp
0
pq
pqpq
pq
pqq
f
ff
f
fP
/
//
/
/
where
)()()()1(4
1
)(),(
2
/2
,
2
33
kkpxzxkfPP
zDkxfkdkddzdx
pd
d
pd
d
Fqqq
qqduq
qp leading
Lund model
Numerical Calculationsof Collins Effect
)exp(1
)(
kkf
q
Chqh zCczD )1)(1()(/
22
)(2
k
kzkPq
Single Spin Asymmetry in
proton – proton collisions
)(),,(
),,(cos
),(),(
3/
2
/2
/2
3
hkzphzPDhddz
d
Pddd
ryfrddykxfkddxpd
d
qqh
q
prpq
Xpp
0
Collision of Vorticing Quarks
Anti-parallel Spins Parallel Spins
Single Spin Asymmetries
Double Spin Asymmetries