Contents Introduction Strongly Correlated Quark Model (SCQM) Spin in SCQM Single Spin Asymmetry

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

Breather (soliton –antisoliton) solution of SG equation

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

QQ

QQ

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

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

qq

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

QQ

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 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

Collins Effect in SSA

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

Experiments with Polarized Protons