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Single Spin Asymmetry in Single Spin Asymmetry in Correlated Quark Model Correlated Quark Model
G. Musulmanbekov JINR, Dubna
e-mail:[email protected]
Contents
•Introduction•Strongly Correlated Quark Model (SCQM)•Single Spin Asymmetry in hadronic reactions
–Collins Effect–Sivers Effect
•Conclusions
SPIN2006
IntroductionWhere does the Proton Spin come from?
Spin "Crisis“:DIS experiments: ΔΣ=Δu+Δd+Δs 1≪
SU(6) 1
Sum rule for the nucleon spin:1/2 =(1/2)ΔΣ(Q²)+Δg(Q²)+L(Q2)q+g
gqqLs
SCQM: Total nucleon spin comes from circulating around each of three valence quarks gluon and quark-antiquark condensate.
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)
Constituent Quarks – Solitons
0),(sin),( txtx
2
21
1/cosh
1/sinhtan4),(
uxu
uuttx ass
x
txtx ass
ass
),(),(
Sine- Gordon (SG) equation
Breather – oscillating soliton-antisoliton pair, the periodic solution of SG:
The density profile of the breather:
)(tanh2)( 2 mxMxU
Breather solution of SG is Lorenz – invariant.
Effective soliton – antisoliton potential
Breather (soliton –antisoliton) solution of SG equation
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 inside Light Hadons
Quark Potential inside Light Quark Potential inside Light HadronsHadrons
Uq = 0.36tanh2(m0x) Uq x
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
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 total spin angular momentum of the constituent quark.
A},{ A
and intersecting Ech and Bch create around VQ
circulating flow of energy, color analog of the Pointing’s vector
Classical analog of electron spin – F.Belinfante 1939; R. Feynman 1964; H.Ohanian 1986; J. Higbie 1988.
2. Circulating flow of energy carrying along with it hadronic matter is associated with hadronic matter current.
Analogue from hydrodynamicsHelmholtz laws for velocity field
((∂ξ)/(∂t))+ ×(∇ ξ×v)=0,
ξ= ×∇ v,∇⋅v=0,
const.s rv d
σξ
r/1v
5. Quark spins are perpendicular to the plane of oscillation.
6. Quark spin module is conserved during oscillation:
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.
lead to
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
Structure Function of Valence Quarks in Proton
Summary on SCQM
• Quarks and gluons inside hadrons are strongly correlated;
• Constituent quarks are identical to vortical solitons.
• Hadronic matter distribution inside hadrons is fluctuating quantity resulting in interplay between constituent and current quarks.
• Hadronic matter distribution inside the nucleon is deformed;
it is oblate in relation to the spin direction.
Single Spin Asymmetry in proton – proton collisions
• In the factorized parton model
'' // qqqqp Df
)(),,(
),,(cos
)(),(
3/
2
//2
3
hkzphzPDhddz
d
Pddd
yfdykxfkddxpd
d
qqh
q
prpq
Xpp
0
pq
pqpq
pq
pqq
f
ff
f
fP
/
//
/
/
where
Determination of PDFs
'' // qqqqp Df
)(),,(
),,(cos
)(),(
3/
2
//2
3
hkzphzPDhddz
d
Pddd
yfdykxfkddxpd
d
qqh
q
prpq
Geometrical view of possible quark
configurations inside colliding protons at the instant of collision
Geometrical view of possible quark
configurations inside colliding protons at the instant of collision
Determination of cross sections
'' // qqqqp Df
)(),,(
),,(cos
),(),(
3/
2
/2
/2
3
hkzphzPDhddz
d
Pddd
ryfrddykxfkddxpd
d
qqh
q
prpq
Calculation of cross section with the use of Inelastic Overlap Fuction
23
21
)()(4
mdMM
sxxM
jijjii pqppqq
X
rrrr '
+ energy – momentum conservation
Monte-Carlo simulation of inelastic eventsusing modified Heisenberg picture:
)()()()1(4
1
)(),(
2
/2
,
2
33
kkpxzxkfPP
zDkxfkdkddzdx
pd
d
pd
d
Fqqq
qqduq
qp leading
Calculationsof Collins Effect
Pure Collins Effect: leading quark is a spectator
)exp(1
)(
k
kfq
Chqh zCczD )1)(1()(/
22
)(2
k
kzkPq
Collins Effect in SSA
Inclusion of “Sivers” Effect
'' // qqqqp Df
)(),,(
),,(cos
),(),(
3/
2
/2
/2
3
hkzphzPDhddz
d
Pddd
ryfrddykxfkddxpd
d
qqh
q
prpq
“Sivers” Effect in SSA
Single Spin Asymmetries
Spin-up polarized quark - vortex
Chou & Yang 1976Hadronic matter current distributions insidepolarized hadrons and nuclei
)1(
),,(
zineff
zyxeff
vPP
bbbPOpaqueness
Collins & “Sivers” Effect in SSA
Single Spin Asymmetries
Experiments with Polarized Protons
Anti-parallel Spins
Double Spin Asymmetries
Parallel Spins
Experiments with Polarized Protons
