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Bjorken Scaling & modification of nucleon mass
inside dense nuclear matter
Jacek Rozynek INS Warsaw
Nuclear Physics Workshop
HCBM 2010
1. Nuclear structure function in Deep
Inelastic Scattering (DIS) - review.
2. Finite pressure corrections.
3. Implication to the EOS for Nuclear matter.
4. Conclusion.
Plan
• EMC effect• Relativistic Mean Field Problems• Hadron with quark primodial distributions• Pion contributions
• Nuclear Bjorken Limit - MN(x)
EMC effect
Historically ratio
R(x) = F2A(x)/ AF2
N(x)
x
Pion excess
Three approaches to EMC effect in term of nucleon degrees of freedom through the nuclear spectral function. (nonrelativistic off shell effects) G.A.Miller&J. Smith, O. Benhar, I. Sick, Pandaripande,E Oset
in terms of quark meson coupling model modification of quark propagation by direct coupling of quarks to nuclear envirovment A.Thomas+Adelaide/Japan group, Mineo, Bentz, Ishii, Thomas, Yazaki (2004)
by the direct change of the partonic primodial distribution. S.Kinm, R.Close Sea quarks from pion cloud. G.Wilk+J.R.,
The graphical representation of the convolution modelfor the deep inelastic electron nucleus scattering with: (a) theactive nucleon N (with hit quark q) including exchange finalstate interaction with nucleon spectators and (b) virtual pion
Hit quark has momentum j + = x p +
ExperimentalyExperimentaly x =x = and is iterpreted as fraction of longitudinal nucleon momentum carried by parton(quark) for 2Q2
On light cone Bjorken x is defined as x = j+ /p+
where p+ =p0 + pz
e
p r(emnant)
Q2
,
Q2/2M
D I Sj
Light cone coordinates
fixedMQxMxq
QfixedxwithQ
qQQvq
pJJpedW iq
2/),0,0,(
0)/(
),,0,0,(
)0()(
2
222
2222
4
MxMx
Mx
qqq
Mxqbutq
qqq
/1||and/1||so
/2||but0
thenif
2/
limit Bjorken in)(2/1
30
30
Relativistic Mean Field Problems
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 = 300MeV
UV = 300MeV
Relativistic Mean Field Problems
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
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
Today - Convolution modelToday - Convolution model for x <0.15
• We We willwill show that in deep inelastic show that in deep inelastic scattering the magnitude of the scattering the magnitude of the nuclear Fermi motion is sensitive nuclear Fermi motion is sensitive to residual interaction between to residual interaction between partons influencing both the partons influencing both the Nucleon Structure Function Nucleon Structure Function
• and nucleon mass in th and nucleon mass in th NMNM
• MMBB (x) (x)
• We We willwill show that in deep inelastic show that in deep inelastic scattering the magnitude of the scattering the magnitude of the nuclear Fermi motion is sensitive nuclear Fermi motion is sensitive to residual interaction between to residual interaction between partons influencing both the partons influencing both the Nucleon Structure Function Nucleon Structure Function
• and nucleon mass in th and nucleon mass in th NMNM
• MMBB (x) (x)
• Relativistic Mean Field problems
• Primodial parton distributions
• Bjorken x scaling in nuclear medium
F2N(x)
)()()()(1
22 xFyxyxx
dxdyAxF
xN
AA
AAAAA
A
N O S H A D O W I N G
RMF failure &Where the nuclear pions are
• M Birse PLB 299(1985), JR IJMP(2000), G Miller J Smith PR (2001)• GE Brown, M Buballa, Li, Wambach , Bertsch, Frankfurt, Strikman
M(x) & in RMF solution 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?
z=9fm
TTwo resolutions scales in deep inelastic scattering
1 1/ Q 2 connected with virtuality of probe . (A-P evolution equation - well known)
1/Mx = z distance how far can propagate the quark in the medium. (Final state quark interaction - not known)
For x=0.05 z=4fm
•
Nuclear final state interaction
z(x)
Effective nucleon Mass M(x)=M( z(x) , rC ,rN )
J.R. Nucl.Phys.A
rN - av. NN distance
rC - nucleon radius
if z(x) > rN
M(x) = MN
if z(x) < rC
M(x) = MB
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
Drell Yan Calculations
Good description due to the x dependence of nucleon mass
(no nuclear pions in Sum Rules)
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
Results
Fermi Smearing
Results
Fermi Smearing
Constant effective nucleon mass
Results“no” free paramerers
Fermi Smearing
Constant effective nucleon mass
x dependent effective nucleon mass
with G. Wilk Phys.Rev. C71 (2005)
Conclusions part1
• Good fit to data for Bjorken x>0.1 by modfying the nucleon mass in the medium (~24 MeV depletion) in x dependent way with
• Unchanged nucleon M for medium and small x.
• Although such subtle changes of nucleons mass is difficult to measure inside nuclear medium due to final state interaction this reduction of nucleon mass is comp atible with recent observation of similar reduction in Delta invariant mass in the decay spectrum to (N+Pion)T.Matulewicz Eur. Phys. J A9 (2000)
• (~ 1% only) of nuclear momentum is carried by sea quarks nuclear pions) due to x dependent effective nucleon mass supported by Drell-Yan nuclear experiments for higher densities increase for soft EOS towards chiral phase transition.
• Increase of the „additional nuclear pion mass” 5% compatible with chiral restoration.
• x – dependent correction to the distribution kT2
In vacuum
In nuclear medium
Phys.Rev.C45 1881
The QCD vacuum
is the vaccum state of quark & gluon system. It is an example of a non-perturbative vacuum state, characterized by many non-vanishing condensates such as
the gluon <gg> & quark <qq> condensates.
These condensates characterize the normal phase or the confined phase of quark matter.
Unsolved problems in physics: QCD in the non-perturbative regime: confinement The equations of QCD remain unsolved at energy scale relevant for describing atomic nuclei. How does QCD give rise to the physics of nuclei and nuclear constituents?
• pion mass in the medium in chiral symmetry restoration
• Nucleon mass in the medium ?
EOS in NJL
Bernard,Meissner,Zahed PRC (1987)
EMC effect
R. Rapp and J.Wambach, Adv. Nucl. Phys. 25, 1 (2000)
Brown- Rho scaling
Derivative Coupling for scalars RMF Models ZMA. Delfino, CT Coelho and M. Malheiro, Phys. Rev. C51, 2188 (1995).
{Tensor coupling vector (Bender, Rufa)} Review J. R. Stone, P.-G. Reinhard nucl-th/0607002 (2006).
M. Baldo, Nuclear Methods and the Nuclear Equation of State (World Scientific, 1999)
Effective Mass in RMF
• W - Nucleon bare mass in the Walecka mean field approach
• ZM - constructed by changing of covariant derivative in W model. Langrangian describes the motion of baryons with effective mass and the density dependent scalar (vector) coupling constant.
ZM - Zimanyi Moszkowski
Relativistic Mean Field & EOSquark condensate < qq>m in the medium 0
• Delfino, Coelho, Malheiro
22
2
22
2
2
2
22)()1(
)(
1 *
*
m
g
MMMg
m
MMgm
fm NNN
NN
qqNm
fmeff
qq22
1
for models)
<qq>m
Condensate Ratios in RMF
Positive Pressure in NM
A
O
U
R
M
O
D
E
L
But in the medium we have correction to the Hugenholtz-van Hove theorem:
On the other hand we have momentum sum rule of quarks (plus gluons) as the integral over the structure function FN
2(x) shold compensate factor EF/(E/A). Therefore in this model we have to scale Bjorken x=q/2M .
)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.
Mmod=M/(1+(d(E/A)/d
and we have new equation for the relativistic (Walecka type) effective mass which now include the pressure correction.
Basic Equations
o
Masses - solution of modified RMF equation with pressure corrections compare to ZM
Partially published in IJMP & Acta Phys Pol
EOS results
P.Danielewicz et al Science(2002)
David Blaschke
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 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
Dependence from initial in p-A collision
2kT
X-N Wang Phys. Rev.C (2000)
Spinodal phase transition
Deep inelastic scattering
fixedMQxMxq
qQQvq
ScalingBjorkenxFqWM
qWqqMpqqMvp
MqWqqqgW
pJJpedW
pJXXJprqpW
Wld
T
T
v
iq
x
2/),0,0,(
),,0,0,(
)(),(lim)/(
),())/()()/((
/1),()/(
)0()(
)0()0()(
2
2222
22
2
22
22
221
2
4
Quark inside nucleus
)()()())(( 0 nnn qVEqrmi
0,0
2
22/
~
1
|)()0(
)0()(|)(
PPQe
PQePedxF
iV
iViMx
QMC model
0,00
2
0
0
2
02/
~
1
|)()0(
)0()(|)(
PPQe
PQePedxF
iV
iViMx
Condensates and quark masses
Maxwell construction
Chiral solitons in nuclei
Miller, Smith, Phys. Rev. Lett. 2003
0)()0(
( )()(_
5
rnriMeiL
EENM vCN
)(
)(arctan)(
r
rr
qs
qps
)'()'()()( '30 rrrrdrr v
svs
qs
)(
)(4
2
3
vsN
vsN
kNs
qMk
Mkd
F
)()()2()(
42
222
3
3
FBv
vFN
k
FB
km
gkMk
kd
kA
E F
nNnn
nNC xMpEMNxq |)()1(|)( 330
_
Chiral Quark Soliton Model Petrov- Diakonov So far effect to strong
x dependent nucleon effective mass
• it is possible to show that in DIS <kT2>
M2
22 / TMediumT kk
In the x>0.6 limit
(no NN interaction)
<kT2> Nuclear= <kT
2> Nukleon
Bartelski Acta Phys.Pol.B9 (1978)
