JLAB-THY-03-03, ADP-03-107/T545 Bound Nucleon Form Factors, Quark-Hadron Duality, and Nuclear EMC Effect K. Tsushima1∗, D.H. Lu2, W. Melnitchouk3, K. Saito4, and A.W. Thomas5 1Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602, USA 2Department of Physics and Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310027, China 3Jefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606, USA 4Physics Division, Tohoku College of Pharmacy, Sendai 981-8558, Japan 3 5Department of Physics and Mathematical Physics, and Special Research Centre for the 0 0 Subatomic Structure of Matter, Adelaide University, SA 5005, Australia 2 n a J Wediscusstheelectromagneticformfactors,axialformfactors,andstruc- 4 turefunctionsofaboundnucleon inthequark-mesoncoupling(QMC)model. 2 Free space nucleon form factors are calculated using the improved cloudy bag 1 model(ICBM). After describingfinitenucleiandnuclear matter inthequark- v based QMC model, we compute the in-medium modification of the bound 8 7 nucleon form factors in the same framework. Finally, limits on the medium 0 modification of the bound nucleon F structure function are obtained using 2 1 thecalculatedin-mediumelectromagneticformfactorsandlocalquark-hadron 0 3 duality. 0 / h t I. INTRODUCTION - l c u Partial restoration of chiral symmetry in a nuclear medium, to which the reduction in n the mass of a bound nucleon is sometimes ascribed, plays a key role in understanding the : v medium modification of bound nucleon (hadron) properties. At relatively high energies i X and/or temperature and/or densities, quark and gluon degrees of freedom are expected to r a be efficient in describing physical phenomena according to perturbative QCD. However, it is not at all obvious whether such degrees of freedom are indeed necessary or efficient in describing low energy nuclear phenomena, such as the static properties of finite nuclei. In thisarticle,wedemonstratethatthequarkdegreesoffreedomdoindeedseemtobenecessary to understand recent polarization transfer measurements in the 4He(~e,e′~p)3H reaction [1,2], which cannot be explained within the best existing treatments of traditional physics (solely based on hadronic degrees of freedom). Over the past few years there has been considerable interest in possible changes in bound nucleon properties in a nuclear medium. There is a significant constraint on the possible ∗Talk presented by K. Tsushima at the Joint JLab-UGA Workshop on “Modern Sub-Nuclear PhysicsandJLabExperiments”,tohonortheoccasionofDr. JoeHamilton(VanderbiltUniversity), September 13, 2002, University of Georgia, Athens, Georgia, USA, and to be published in the proceedings. 1 change in the radius of a bound nucleon based on y-scaling — especially in 3He [3]. On the other hand, the space (time) component of the effective one-body axial coupling constant is known to be quenched [4] (enhanced [5]) in Gamow-Teller (first-forbidden) nuclear β decay, and a change in the charge radius of a bound proton provides a natural suppression of the Coulomb sum rule [6]. One of the most famous nuclear medium effects — the nuclear EMC effect [7], or the change in the inclusive deep-inelastic structure function of a nucleus relative to that of a free nucleon — has stimulated theoretical and experimental efforts for almost two decades now which seek to understand the dynamics responsible for the change in the quark-gluon structure of the nucleon in the nuclear environment [8]. Recently the search for evidence for some modification of nucleon properties in medium has been extended to electromagnetic form factors, in polarized (~e,e′~p) scattering exper- iments on 16O [9] and 4He [1,2]. These experiments measured the ratio of transverse to longitudinal polarization of the ejected protons, which for a free nucleon is proportional to the ratio of electric to magnetic elastic form factors [10], Gp P′ E +E′ E = − x e e tan(θ /2) . (1) Gp P′ 2M e M z N Here P′ and P′ are the transverse and longitudinal polarization transfer observables, E x z e and E′ the incident and recoil electron energies, θ the electron scattering angle, and M e e N the nucleon mass. Compared with the traditional cross section measurements, polarization transfer experiments provide more sensitive tests of dynamics, especially of any in-medium changes in the form factor ratios. The feasibility of this technique was first demonstrated in the commissioning experiment at Jefferson Lab on 16O [9] at Q2 = 0.8 GeV2. In the subsequent experiment at MAMI on 4He [1] at Q2 ≈ 0.4 GeV2, and at Jefferson Lab at Q2 = 0.5,1.0,1.6 and 2.6 GeV2, which had much higher statistics, the polarization ratio in 4He was found to differ by ≈ 10% from that in 1H. Conventional models using free nucleon form factors and the best phenomenologically determined optical potentials and bound state wave functions, as well as relativistic cor- rections, meson exchange currents, isobar contributions and final state interactions [11–14], fail to account for the observed effect in 4He [1,2]. Indeed, full agreement with the data was only obtained when, in addition to these standard nuclear corrections, a small change in the structure of the bound nucleon, which had been estimated within the quark-meson coupling (QMC) model [15–21], was taken into account. The final analysis [2] seems to favor this scenario even more, although the error bars may still be too large to draw a definite conclusion. On the other hand, there has recently been considerable interest in the interplay between form factors and structure functions in the context of quark-hadron duality. As observed originally by Bloom and Gilman [22], the F structure function measured in inclusive lepton 2 scattering at low W (where W is the mass of the hadronic final state) generally follows a global scaling curve which describes high W data, to which the resonance structure function averages. Furthermore, the equivalence of the averaged resonance and scaling structure functions appears to hold for each resonance region, over restricted intervals of W, so that the resonance–scaling duality also exists locally. These findings were dramatically confirmed in recent high-precision measurements of the proton and deuteron F structure function at 2 Jefferson Lab [23,24], which demonstrated that local duality works remarkably well for each of the low-lying resonances, including surprisingly the elastic, to rather low values of Q2. 2 Inthisarticlewefirstbrieflyreviewhowfinitenucleiandnuclearmatteraretreatedinthe quark-based QMC model [20,21]. We then discuss the modification of the electromagnetic and axial form factors of a bound nucleon in the same model. Finally, using the concept of quark-hadron duality and the calculated bound nucleon electromagnetic form factors, we extract the F structure function of the bound nucleon [25]. To the extent that local 2 duality is a good approximation, the relations among the nucleon form factors and structure functions are model independent, and can in fact be used to test the self-consistency of the models. We findthat the recent formfactor datafor a protonbound in4He[1,2]place strong constraints on the medium modification of inclusive structure functions at large Bjorken-x. In particular, they appear to disfavor models in which the bulk of the nuclear EMC effect is attributed to deformation of the intrinsic nucleon structure off-shell – see e.g. Ref. [26]. This article is organized as follows. In Section II we briefly review the treatment of finite nuclei in the QMC model [20,21]. Then, in Section III, we discuss the in-medium modification of bound nucleon electromagnetic form factors in the QMC model [15–18] as inferred from the recent polarization transfer experiments, as well as that of the axial form factor G (Q2) [27]. In Section IV quark-hadron duality is used to relate the observed A form factor modification to that which would be expected in the deep-inelastic F structure 2 function [25]. Finally, we summarize our findings in Section V. II. FINITE NUCLEI AND NUCLEAR MATTER IN THE QMC MODEL In this Section we briefly review the treatment of finite nuclei and symmetric nuclear matter in the QMC model [19–21]. We consider static, spherically symmetric nuclei, and adopt the Hartree, mean-field approximation, ignoring the ρNN tensor coupling as usually done in the Hartree treatment of quantum hadrodynamics (QHD) [28] (see Refs. [20,21] for discussions about the ρNN tensor coupling). Using the Born-Oppenheimer approximation, mean-field equations of motion are derived for a nucleus in which the quasi-particles moving in single-particle orbits are three-quark clusters with the quantum numbers of a nucleon. A relativistic Lagrangian density at the hadronic level can then be constructed [20,21], similar to that obtained in QHD [28], which produces the same equations of motion when expanded to the same order in velocity: τN e L = ψ (~r) iγ ·∂ −M∗(σ)−(g ω(~r)+g 3 b(~r)+ (1+τN)A(~r))γ ψ (~r) QMC N " N ω ρ 2 2 3 0# N 1 1 − [(∇σ(~r))2 +m2σ(~r)2]+ [(∇ω(~r))2 +m2ω(~r)2] 2 σ 2 ω 1 1 + [(∇b(~r))2 +m2b(~r)2]+ (∇A(~r))2, (2) 2 ρ 2 where ψ (~r) and b(~r) are respectively the nucleon and ρ meson (the time component in the N third direction of isospin) fields, while m , m and m are the masses of the σ, ω and ρ σ ω ρ meson fields. g and g are the ω-N and ρ-N coupling constants which are related to the ω ρ corresponding (u,d)-quark-ω, gq, and (u,d)-quark-ρ, gq, coupling constants by g = 3gq ω ρ ω ω and g = gq [20,21]. (Hereafter, we will denote the light quark flavors by q ≡ u,d.) The field ρ ρ dependent σ-N coupling strength predicted by the QMC model, g (σ), which is related to σ the Lagrangian density of Eq. (2) at the hadronic level, is defined by: 3 M∗(σ) ≡ M −g (σ)σ(~r) . (3) N N σ Note that the dependence of these coupling strengths on the applied scalar field must be calculatedself-consistently atthequarklevel[20,21]. FromtheLagrangiandensityinEq.(2), a set of equations of motion for the nuclear system can be obtained: τN e [iγ ·∂ −M∗(σ)−(g ω(~r)+g 3 b(~r)+ (1+τN)A(~r))γ ]ψ (~r) = 0, (4) N ω ρ 2 2 3 0 N ∂M∗(σ) (−∇2 +m2)σ(~r) = −[ N ]ρ (~r) ≡ g C (σ)ρ (~r), (5) r σ ∂σ s σ N s (−∇2 +m2)ω(~r) = g ρ (~r), (6) r ω ω B g (−∇2 +m2)b(~r) = ρρ (~r), (7) r ρ 2 3 (−∇2)A(~r) = eρ (~r), (8) r p where, ρ (~r), ρ (~r), ρ (~r) and ρ (~r) are the scalar, baryon, third component of isovector, s B 3 p and proton densities at position ~r in the nucleus [20,21]. On the right hand side of Eq. (5), −[∂M∗(σ)/∂σ] = g C (σ), where g ≡ g (σ = 0), is a new, and characteristic feature of N σ N σ σ QMC beyond QHD [28]. The effective mass for the nucleon, M∗, is defined by: N ∂M∗(σ) ∂ N = −n gq d3x ψ (~x)ψ (~x) ≡ −n gqS (σ) = − [g (σ)σ], (9) ∂σ q σ q q q σ N ∂σ σ Zbag where n is the number of light quarks (u and d), and the MIT bag model quantities and q the in-medium bag radius satisfying the mass stability condition are given by [19–21]: n Ω∗ −z 4 M∗(σ) = q q N + π(R∗ )3B , (10) N R∗ 3 N q=u,d N X S (σ) = Ω∗/2+m∗R∗ (Ω∗ −1) / Ω∗(Ω∗ −1)+m∗R∗ /2 , (11) N q q N q q q q N h i h i Ω∗ = x2 +(R∗ m∗)2, m∗ = m −gqσ(~r) , (12) q q N q q q σ dM∗/qdR | = 0 , (13) N N RN=R∗N where gq is the quark-σ meson coupling constant. Here, the MIT bag model quantities are σ calculated in a local density approximation using the spin and spatial part of the wave func- tions, ψq(x) = Nqe−iǫqt/R∗Nψq(~x), where Nq is the normalization factor. The wave functions ψ (x) satisfy the Dirac equations for the quarks in the nucleon bag centered at position ~r in q the nucleus (|~x−~r| ≤ R∗ [20,21]): N 1 ψ (x) iγ ·∂ −(m −Vq(~r))∓γ0 Vq(~r)± Vq(~r) u = 0 , (14) (cid:20) x q σ (cid:18) ω 2 ρ (cid:19)(cid:21) ψd(x) ! where we approximate the constant, mean meson fields within the bag and neglect the Coulomb force. The constant, mean-field potentials within the bag centered at~r are defined by Vq(~r) ≡ gqσ(~r), Vq(~r) ≡ gqω(~r) and Vq(~r) ≡ gqb(~r). The eigenenergies in units of 1/R∗ σ σ ω ω ρ ρ N are given by: ǫ 1 u = Ω∗ ±R∗ Vq(~r)± Vq(~r) . (15) ǫd ! q N (cid:18) ω 2 ρ (cid:19) 4 In Eqs. (10) - (13), z , B, x , and m are the parameters for the sum of the c.m. and gluon N q q fluctuation effects, bag pressure, lowest eigenvalues for the quark q, and the corresponding current quark masses, respectively. z and B are fixed by fitting the nucleon mass in free N space. We use the current quark masses m = 5 MeV, and obtained z = 3.295 and q=u,d N B = (170.0MeV)4 bychoosing thebagradiusforthenucleon infreespaceR = 0.8fm. The N parameters at the hadronic level, which are already fixed by the study of nuclear matter and finite nuclei [21], are as follows: m = 783 MeV, m = 770 MeV, m = 418 MeV, ω ρ σ e2/4π = 1/137.036, g2/4π = 3.12, g2/4π = 5.31 and g2/4π = 6.93. The sign of m∗ in σ ω ρ q the nucleus in Eq. (12) reflects nothing but the strength of the attractive, negative scalar potential, and thus the naive interpretation of the mass for a physical particle, which is positive, should not be applied. At the hadronic level, the entire information on the quark dynamics is condensed into the effective coupling C (σ) of Eq. (5). Furthermore, when C (σ) = 1, which corresponds N N to a structureless nucleon, the equations of motion given by Eqs. (4)-(8) can be identified with those derived from QHD [28], except for the terms arising from the tensor coupling and the non-linear scalar field interaction introduced beyond naive QHD. FIG. 1. Charge density distributions for 40Ca and 208Pb calculated in the QHD [28] and QMC [21] models. As examples, we show in Fig. 1 the charge density distributions calculated for 40Ca and 208Pb, and also the energy spectra obtained for 40Ca and 208Pb in Figs. 2 and 3 [21], respectively. Next, we consider the limit of infinite symmetric nuclear matter [19–21]. In this limit all meson fields become constant, and we denote the mean-values of the ω and σ fields by ω and σ. Then, equations for the ω and self-consistency condition for the σ are given by [19–21], 4 g 2k3 g ω = d3kθ(k −k) = ω F = ω ρ , (16) (2π)3 F m2 3π2 m2 B Z ω ω g 4 M∗(σ) g σ = σ C (σ) d3kθ(k −k) N = σ C (σ)ρ , (17) m2σ N (2π)3 Z F M∗2(σ)+~k2 m2σ N s N q 5 where g = 3gqS (0) (see Eq. (11)), k is the Fermi momentum, ρ and ρ are the baryon σ σ N F B s and scalar densities, respectively. Note that M∗(σ) in Eq. (17) must be calculated self- N consistently in the MIT bag model through Eqs. (9)–(14) for a given baryon density. This self-consistency equation for the σ is the same as that in QHD, except that in the latter model one has C (σ) = 1 [28]. N FIG. 2. Energy spectrum for 40Ca [21] in the QMC model compared with experiment (Exp.), and that of QHD [28]. FIG. 3. Same as Fig. 2 (for 209Pb). 6 III. NUCLEAR MEDIUM MODIFICATION OF FORM FACTORS In this Section we outline the medium modification of the electromagnetic form fac- tors of the nucleon, as suggested in the recent polarization transfer measurements in the 4He(~e,e′p~)3H reaction [1,2]. The first data were analyzed in Ref. [1] using a variety of mod- els, nonrelativistic and relativistic, based on conventional nucleon-nucleon potentials and well-established bound state wave functions, including corrections from meson exchange currents, final state interaction and other effects [11–14]. The observed deviation, which was of order 10%, could only be explained by supplementing the conventional nuclear de- scription with the effects associated with the medium modification of the nucleon internal structure calculated by the QMC model [15–18]. In Fig. 4 we show the “super ratio”, R/R , which was made for the final analysis of PWIA the polarization transfer measurements on 4He [2]. Here, R stands for the prediction PWIA based on therelativistic plane-wave impulse approximation (PWIA), andthe measured ratio R is defined by: R = (Px′/Pz′)4He . (18) (Px′/Pz′)1H FIG. 4. Super ratio R/R , as a function of Q2, taken from Ref. [2]. See caption of Fig. 1 in PWIA Ref. [2] for detailed explanations. In Fig. 4, the modification of electromagnetic form factors of the bound nucleon cal- culated in the QMC model [15–18] (the solid line denoted by “Udias RDWIA + QMC”) uses the improved cloudy bag model (ICBM) [29,30] for the free nucleon form factors. The ICBM [30] includes a Peierls-Thouless projection to account for center of mass and recoil corrections, and a Lorentz contraction of the internal quark wave functions. The electromagnetic current is given by the sum of the contributions from the quark core and the pion cloud, jµ(x) = Q eψ (x)γµψ (x)−ie[π†(x)∂µπ(x)−π(x)∂µπ†(x)] , (19) q q q q X 7 where Q is the charge operator for a quark flavor q, and π(x) destroys a negatively charged q (or creates a positively charged) pion. Relevant diagrams included in the calculation of free space electromagnetic form factors are depicted in Fig. 5 [30]. N N N B C N N N B (a) (b) (c) FIG. 5. Diagrams included in the calculation of free space electromagnetic form factors in the ICBM [30]. The intermediate baryon states B and C are restricted to the N and ∆. In the Breit frame the quark core contribution to the electromagnetic form factors of the bound nucleon is given by [15–18]: G∗(Q2) = η2 Gsph∗(η2Q2) , (20a) E E G∗ (Q2) = η2 Gsph∗(η2Q2) , (20b) M M where Q2 ≡ −q2 = ~q2, and the scaling factor η = (M∗/E∗ ), with E∗ = M∗2 +Q2/4 the N N N N energy and M∗ the mass of the nucleon in medium. Gsph∗ are the formqfactors calculated N E,M with the static spherical bag wave function, 1 Gsph∗(Q2) = d3r j (Qr) f (r) K(r) , (21a) E D 0 q Z 1 2M∗ Gsph∗(Q2) = N d3r j (Qr) β∗ j (x r/R∗ ) j (x r/R∗ ) K(r) . (21b) M D Q 1 q 0 q N 1 q N Z Here f (r) = j2(x r/R∗ ) + β∗2 j2(x r/R∗ ), where R∗ is the nucleon bag radius in q 0 q N q 1 q N N medium, x the lowest eigenfrequency, and β∗2 = (Ω∗ − m∗R∗ )/(Ω∗ + m∗R∗ ), with q q q q N q q N Ω∗ = x2 +(m∗R∗ )2 and m∗ = m − gqσ (see also Section II). The recoil function q q q N q q σ K(r) =q d3xf (~x)f (−~x − ~r) accounts for the correlation of the two spectator quarks, q q and D = d3r f (r) K(r) is the normalization factor. The scaling factor η in the argument R q of Gsph∗ arises from the coordinate transformation of the struck quark, and the prefactor in E,M R Eqs. (20) comes from the reduction of the integral measure of the two spectator quarks in the Breit frame. The contribution from the pionic cloud is calculated along the lines of Ref. [15–18]. Although the pion mass would be slightly smaller in the medium than in free space, we use m∗ = m , which is consistent with chiral expectations and phenomenological constraints. π π Furthermore, since the ∆ isobar is treated on the same footing as the nucleon in the CBM, and because it contains three ground state light quarks, its mass should vary in a similar manner tothat ofthe nucleon intheQMC model. As afirst approximationwe therefore take the in-medium and free space N–∆ mass splittings to be approximately equal, M∗ −M∗ ≃ ∆ N M −M . ∆ N 8 The change in the ratio of the electric to magnetic form factors of the proton from free to bound, Gp∗(Q2) Gp(Q2) Rp∗ (Q2) Rp (Q2) = E E , (22) EM EM Gp∗(Q2)!, Gp (Q2)! . M M is illustrated in Fig. 6 for 4He, 16O (left panel) and for nuclear matter densities, ρ = ρ and 0 ρ = 1ρ (right panel) with ρ = 0.15 fm−3. 2 0 0 1.0 1 ρ= 1 ρ − 0.9 2 0 M )Mfree pRE G G/E0.8 */M0.8 ρ=ρ0 G)/(M 1s ,4He pRE (G/E0.7 111sps113///222,,,111666OOO(B) 1p1/2,16O(B) 3/2 0.6 0.6 0.0 0.5 1.0 1.5 2.0 2.5 0 0.5 1 1.5 2 Q2 (GeV2) 2 2 Q (GeV ) FIG. 6. The change in the ratio of the electric to magnetic form factors from free to bound protons,Rp∗ /Rp = (Gp∗/Gp∗)/(Gp /Gp ),for4He(1s state)and16O(1s and1s states) EM EM E M E M 1/2 1/2 3/2 (left panel) [18] and nuclear matter (right panel) [25]. 16O(B) stands for the results which allow changes of the bag constant according to the nuclear density (ρ = 0.15 fm−3 [17]). 0 Because theaverage nuclear densities for allexisting stable nuclei heavier thandeuterium lie in the range 21ρ0 ∼< ρ ∼< ρ0, we consider these two specific nuclear densities to give the upper and lower bounds for the change of the electromagnetic form factors (and structure functions at large x) of the bound nucleon. We emphasize that in the present analysis the absolute value of the proton magnetic form factor at Q2 = 0 (the magnetic moment), which is enhanced in medium, plays animportant role – as it did in the analysis of polarized (~e,e′~p) scattering experiments. Because of charge conservation, the value of Gp at Q2 = 0 remains unity for any ρ. On E the other hand, the proton magnetic moment is enhanced in the nuclear medium, increasing with ρ, so that Rp∗ < Rp at Q2 = 0. In fact, the electric to magnetic ratio is ∼ 5% EM EM smaller in medium than in free space for ρ = 1ρ , and ∼ 10% smaller for ρ = ρ . The effect 2 0 0 increases with Q2 out to ∼ 2 GeV2, where the (bound/free) ratio deviates by ∼ 20% from unity. The extension to the in-medium modification of the bound nucleon axial form factor G∗(Q2) can be made in a straightforward manner [27]. Since the induced pseudoscalar A form factor, G (Q2), is dominated by the pion pole, and can be derived using the PCAC P relation [29], we do not discuss it here. The relevant axial current operator is then simply given by 9 τ Aµ(x) = ψ (x)γµγ aψ (x)θ(R−r), (23) a q 5 2 q q X where ψ (x) is the quark field operator for flavor q. q Similarly to the case of electromagnetic form factors, in the preferred Breit frame the resulting bound nucleon axial form factor is given by [27]: G∗(Q2) = η2Gsph∗(η2Q2), (24) A A 5 Gsph∗(Q2) = d3r{ j2(x r/R∗ )−β∗2j2(x r/R∗ ) j (Qr) A 3 0 q N q 1 q N 0 Z h +2β∗2j2(x r/R∗ )[ij (Qr)/Qr]} K(r)/D. (25) q 1 q N 1 In Fig. 7 we show the (normalized) free space axial form factor G (Q2) calculated in the A ICBM [27] together with the experimental data (left panel), and (the space component of) that calculated at nuclear densities ρ = (0.5,0.7,1.0,1.5)ρ with ρ = 0.15 fm−3. 0 0 1 1 Kitagaki83 0.8 R=0.90 fm =0) in free space 0.6 RR==01..9050 ffmm 2ρρQ,=0) for fixed 00..89 22Q)/G(QA0.4 2ρQ,)/G(A 001...570 ρρρ00 G(A0.2 G(A0.7 1.5 ρ00 0 0.6 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Q2 (GeV2) Q2 (GeV2) FIG. 7. Free space (normalized) axial form factor G (Q2) calculated in the ICBM A (left panel) [27] together with experimental data [31] summarized by a dipole form: G (Q2) = g /(1 +Q2/m2)2 with m = (1.03 ±0.04) GeV, and the ratio of in-medium to free A A A A axial form factors [27] (right panel), where g = 1.14 is used in the ICBM calculation. A At Q2 = 0 the space component G∗(Q2 = 0) ≡ g∗ is quenched [4] by about 10 % at A A normal nuclear matter density. The modification calculated here may correspond to the “model independent part” in meson exchange language, where the axial current attaches itself to one of the two nucleon legs, but not to the exchanged meson [4]. This is because the axial current operator in Eq. (23) is a one-body operator which operates on the quarks and pions belonging to a bound nucleon. The medium modification of the bound nucleon axial form factor G∗(Q2) may be observed for instance in neutrino-nucleus scattering, similar to A that observed in the “EMC-type” experiments, or ina similar experiment to the polarization transfer measurements performed on 4He [1,2,9]. However, at present the experimental uncertainties seem to be too large to detect such medium effect directly. We should also note that the medium modification of the parity-violating F structure functions of a bound 3 10