Bjorken Scaling modification of nucleon mass inside dense

Bjorken Scaling & modification of nucleon mass inside dense nuclear matter Jacek Rozynek INS Warsaw Nuclear Physics Workshop HCBM 2010

Plan 1. Nuclear structure function in Deep Inelastic Scattering (DIS) - review. • • • EMC effect Relativistic Mean Field Problems Hadron with quark primodial distributions Pion contributions Nuclear Bjorken Limit - MN(x) 2. Finite pressure corrections. 3. Implication to the EOS for Nuclear matter. 4. Conclusion.

EMC effect Historically ratio Pion excess R(x) = F 2 A(x)/ AF 2 N(x) x

Three approaches to EMC effect ª 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 model for the deep inelastic electron nucleus scattering with: (a) the active nucleon N (with hit quark q) including exchange final state interaction with nucleon spectators and (b) virtual pion

D I S e Q , n 2 p j r(emnant) Hit quark has momentum j + = x p + Experimentaly x = Q /2 M n 2 and is iterpreted as fraction of longitudinal nucleon momentum carried by parton(quark) for n 2 >Q 2 -> oo On light cone Bjorken x is defined as x = j+ /p+ where p+ =p 0 + pz

Light cone coordinates

Relativistic Mean Field Problems In standard RMF electrons will be scattered on nucleons in average scalar and vector potential: [ap + b(M+US) - (e -UV)]y=0 where US=-g. S /m. Sr. S UV =-g. V /m. Vr US = 300 Me. Vr/ r 0 UV = 300 Me. Vr/r 0

Relativistic Mean Field Problems In standard RMF electrons will be scattered on nucleons in average scalar and vector potential: Gives the nuclear distribution f(y) of longitudinal nucleon momenta p+=y. AMA [ap + b(M+US) - (e -UV)]y=0 where US=-g. S /m. Sr. S UV =-g. V /m. Vr US = -400 Me. Vr/ r 0 UV = 300 Me. Vr/r 0 SN() - spectral fun. chemical pot. m - nucleon

Relativistic Mean Field Problems connected with Helmholz-van Hove theorem - e(p. F)=M-e In standard RMF electrons will be scattered on nucleons in average scalar and vector potential: Gives the nuclear distribution f(y) of longitudinal nucleon momenta p+=y. AMA [ap + b(M+US) - (e -UV)]y=0 where US=-g. S /m. Sr. S UV =-g. V /m. Vr US = -400 Me. Vr/ r 0 UV = 300 Me. Vr/r 0 Strong vector-scalar cancelation SN() - spectral fun. chemical pot. m - nucleon

N O S H A D O W I N G Today - Convolution model for x <0. 15 • Relativistic Mean • We will show that in deep inelastic Field problems scattering the magnitude of the nuclear Fermi motion is sensitive to residual interaction between • Primodial partons influencing both the distributions Nucleon Structure Function F 2 N(x) • and nucleon mass in th NM • MB (x) • Bjorken x scaling in nuclear medium

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 Because of Momentum Sum Rule in DIS 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 The pions play role rather on large distances?

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) z=9 fm For x=0. 05 z=4 fm

Nuclear final state interaction r. N - av. NN distance r. C - nucleon radius if z(x) > r. N • M(x) = MN if z(x) < r. C M(x) = MB z(x) Effective nucleon Mass M(x)=M( z(x) , r. C , r. N ) J. R. Nucl. Phys. A

Nuclear deep inelastic limit revisited x dependent nucleon „rest” mass in NM f(x) - probability that struck quark originated from correlated nucleon • Momentum Sum Rule violation

Drell Yan Calculations Good description due to the x dependence of nucleon mass (no nuclear pions in Sum Rules)

D I S Hit quark has momentum j + = x p + e p 2 Q , n j remnant Experimentaly x = Q 2/2 Mn and is iterpreted as fraction of longitudinal nucleon momentum carried by parton(quark) for n 2 >Q 2 -> oo (Bjorken lim) ds ~ l mn Wmn(W 1 , W 2) Bjorken Scaling F 2(x)=lim[(n/M)W 2(q 2, n)]Bjo Rescaling inside nucleus F 2 A(x)= F 2[x. M/M(x)] + F 2 p(x) On light cone Bjorken x is defined as x = j+ /p+ where p+ =p 0 + pz In Nuclear Matter due to final state NN interaction, nucleon mass M(x) depends on x , and consequently from energy e and density r. 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

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. C 71 (2005)

Conclusions part 1 • Good fit to data for Bjorken x>0. 1 by modfying the nucleon mass in the medium (~24 Me. V 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 A 9 (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

In vacuum In nuclear medium Phys. Rev. C 45 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 nonperturbative 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?

EOS in NJL EMC effect • pion mass in the medium in chiral symmetry restoration • Nucleon mass in the medium ? Bernard, Meissner, Zahed PRC (1987) R. Rapp and J. Wambach, Adv. Nucl. Phys. 25, 1 (2000) Brown- Rho scaling

Derivative Coupling for scalars RMF Models ZM A. Delfino, CT Coelho and M. Malheiro, Phys. Rev. C 51, 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 & EOS quark condensate < qq>m in the medium • Delfino, Coelho, Malheiro <qq>m 0 0 for a=1 (ZM models)

Condensate Ratios in RMF

Positive Pressure in NM A

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/2 Mn. EF/M > (E/A)/M

Now the new nucleon nass will dependent on the nucleon energy in pressure but will remain constant below saturation point. Mmod=M/(1+(rd(E/A)/dr)/(E/A)) and we have new equation for the relativistic (Walecka type) effective mass which now include the pressure correction. The density dependent energy carried by meson field Nuclear energy per nucleon for Walecka abd nonlinear models

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 scalarvector Walecka model and now the compressibility K-1=9(r 2 d 2/dr 2)E/A = 230 Me. V, 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 r 0 to 3 r 0.

X-N Wang Phys. Rev. C (2000) Dependence from initial in p-A collision

Spinodal phase transition

Deep inelastic scattering

Quark inside nucleus QMC model

Condensates and quark masses

Maxwell construction

Miller, Smith, Phys. Rev. Lett. 2003 Chiral solitons in nuclei Chiral Quark Soliton Model Petrov- Diakonov So far effect to strong

x dependent nucleon effective mass • it is possible to show that in DIS Bartelski Acta Phys. Pol. B 9 (1978) M 2 <k. T 2> In the x>0. 6 limit (no NN interaction) <k. T 2> Nuclear= <k. T 2> Nukleon