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Modification of scalar field inside
dense nuclear matter
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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
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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
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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
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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?
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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
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Results“no” free paramerers
Fermi Smearing
Constant effective nucleon mass
x dependent effective nucleon mass
with G. Wilk Phys.Rev. C71 (2005)
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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
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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
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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.
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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
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Maxwell construction
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Spinodal phase transition
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EOS in NJL
• pion mass in the medium in chiral symmetry restoration
• Nucleon mass in the medium ?
Bernard,Meissner,Zahed PRC (1987)
EMC effect
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Condensates and quark masses
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The pressure at critcal temperature
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CEP and Statistics
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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.
0