Modification of scalar field inside

dense nuclear matter

Hit quark has momentum j + = x p +

ExperimentalyExperimentaly x =x = Q2/2Mand is iterpreted as fraction of longitudinal nucleon momentum carried by parton(quark) for 2Q2jorken jorken lim)lim)

D I S

remnant

e

p

Q2

, j

On light cone Bjorken x is defined as x = j+ /p+ where p+ =p0 + pz

d~l W WW1 , W2)

Bjorken Scaling

F2(x)=lim[(W2(q2,Bjo

In Nuclear Matter due to final state NN interaction, nucleon mass M(x) depends on x , and consequently from energy and density .

for large x (no NN int.) the nucleon mass has limit

Due to renomalization of the nucleon mass in medium we have enhancement of the pion cloud from momentum sum rule

Fermie NB MM

F2A(x)= F2[xM/M(x)] + F2

(x)

Rescaling inside nucleus

Relativistic Mean Field Problemsconnected with Helmholz-van Hove theorem - e(pF)=M-

In standard RMF electrons will be scattered on nucleons in average scalar and vector potential:

p +M+US) - (e -UV

where US=-gS /mSSUV =-gV

/mV

US = -400MeV

UV = 300MeV

Gives the nuclear distribution f(y) of longitudinal nucleon momenta p+=yAMA

SN() - spectral fun. - nucleon chemical pot.

)(

)(1)p,(

)2(

4)( 3030

4

4 ppy

pE

ppS

pdy NA

A

Strong vector-scalar cancelation

*

3

22

/,)1(

4

3)( FFA

A

AAA

A Epvv

yvy

Nuclear Final State Interaction in the Deep Inelastic Scatt.

•

z(x)

Effective nucleon Mass M(x)=M( z(x) , rC ,rN ) renomalization of the nucleon mass in medium with the enhancement of the pion cloud

Relativistic MF rN - av. NN distance

rC - nucleon radius

if z(x) > rN

M(x) = MN

if z(x) < rC

M(x) = MB 1/Mx = z distance how far can propagate the quark in the medium. (Final state quark interaction - not known)

Fermie NB MM

Kazimierz 2009

M(x) & in RMF the nuclear pions almost

disappear

Nuclear sea is slightly enhanced in nuclear medium - pions have bigger mass according to chiral restoration scenario BUT also change sea quark contribution to nucleon SF

rather then additional (nuclear) pions appears

Because of Momentum Sum Rule in DIS

The pions play role rather on large distances?

SF - Evolution in Density

Fe eq. density fm-3

Soft EOS (density - 4)

Non Linear RMF Models

Pions take 5% of longitudinal momenta

Good compressibility Chiral instability

(phase transition)

correction to effective NNinteraction for high density?

Stiff EOS (density - 4

Walecka RMF Model

No enhancement of pion cloud

Bad compressibilty K>300MeV

R(x) = F2NM(x)/ F2

N(x) “no” free parameters

with G. Wilk Phys.Rev. C71 (2005)

J Rozynek Int. Journal of Mod. Phys. In print

Correction to Equation Of State for Nuclear Matter - ap. in Astrophysics

Results“no” free paramerers

Fermi Smearing

Constant effective nucleon mass

x dependent effective nucleon mass

with G. Wilk Phys.Rev. C71 (2005)

Nuclear deep inelastic limit revisitedx dependent nucleon „rest” mass in NM

• Momentum Sum Rule violation

NNx

NNN

Vxf

MM

FxfxFxfx

2

)(1

))(1()()()( 22

22F

)1(]1)(

)(1

2

2

M

V

dxxF

dxxFA N

N

AAA

C[f

f(x) - probability that struck quark originated from correlated nucleon

But in the medium we have correction to the Hugenholtz-van Hove theorem:

On the other hand we have nuclear energy the energy of quarks (plus gluons) as the integral over the structure function F2(x) and is given by EF. Therefore in this model we have to scale Bjorken x=-q/2M the ratio of old and new nucleon mass.

)1(]1)(

)(1

2

2

M

V

dxxF

dxxFA N

N

AAA

C[fEF/M > (E/A)/M

Nuclear energy per nucleon for Walecka abd nonlinear models

The density dependent energy carried by meson field

Now the new nucleon nass will dependent on the nucleon energy in pressure but will remain constant below saturation point.

Mm=M/(1+(dE/d

and we have new equation for the relativistic (Walecka type) effective mass which now include the pressure correction.

Density correction started from ρ=.16fm-3

Density correction started from ρ=.19fm-3

Dash curve include the reduction of the field in medium.

Pre

ssur

e(M

eV/f

m3 )

Wal

ecka

Flow Constrain

P.Danielewicz Science(2002)

Results

Maxwell construction

Spinodal phase transition

EOS in NJL

• pion mass in the medium in chiral symmetry restoration

• Nucleon mass in the medium ?

Bernard,Meissner,Zahed PRC (1987)

EMC effect

Condensates and quark masses

The pressure at critcal temperature

CEP and Statistics

Physically

CONCLUSIONS

Presented model correspond to the scenario where the part   of nuclear momentum carried by meson field and coming from the strongly correlation region,  reduce the nucleon mass  by corrections proportional to the pressure. In the same time in the low density limit the spin-orbit splitting of single particle levels remains in agreement with experiments, like in the classical Walecka Relativistic Mean Field Approach, but the equation of state  for nuclear matter is softer from the classical scalar-vector Walecka model and now the compressibility K-1=9(r2d2/dr2)E/A = 230MeV, closed to experimental estimate. Pressure dependent meson contributions when added to EOS and to the nuclear structure function improve the EOS and give well satisfied Momentum Sum Rule for the parton constituents.

New EOS is enough soft to be is in agreement with estimates from compact stars [3] for higher densities.

Finally we conclude that we found the corrections to Relativistic Mean Field Approach from parton structure measured in DIS, which improve the mean field description of nuclear matter from saturation density to 3. 

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