Density Dependence of Parity Violation in Electron Quasi-elastic Scattering

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1 Journal of the Korean Physical Society, Vol. 66, No. 12, June 2015, pp Brief Reports Density Dependence of Parity Violation in Electron Quasi-elastic Scattering K. S. Kim School of Liberal Arts and Science, Korea Aerospace University, Goyang , Korea Myung-Ki Cheoun Department of Physics, Soongsil University, Seoul , Korea (Received 14 April 2015, in final form 18 May 2015) Within the framework of a relativistic single-particle model, we investigate the effect of the density dependent form factor on parity violating electron scattering from a heavy nucleus in the quasielastic region through the leading order of the interference of the electromagnetic current and the electro-weak neutral current. 208 Pb is used as a target nucleus in the incident electron energy range up to 1 GeV. The density-dependent form factors are obtained from a quark-meson coupling model. The sensitivities of the density-dependence are studied separately for each Dirac (F 1), Pauli (F 2), and axial (G A) form factors. The density enhances the parity violation asymmetry, in particular, in the forward angle region, and the density dependence turns out to be most sensitive to the Pauli form factor. PACS numbers: Fj, Jv Keywords: Parity violation, Density-dependent form factor, Quasi-elastic region DOI: /jkps Parity violation (PV) in the electron scattering has been acknowledged as a useful tool for understanding the weak structure of nucleons and nuclei in the framework of the standard model by the Z 0 probe, such as weak vector and axial vector form factors, the strangeness content of the nucleon, and so on. Since the first experiment on PV at SLAC in the late 1970 s [1], many experiments have been performed at MIT-Bates [2], G0 [3], HAPPEX [4, 5], and etc. Furthermore, at Jlab [6], the Pb (Lead) Radius Experiment (PREX) was performed to measure the neutron radius of 208 Pb and recently, 48 Ca Radius EXperiment (CREX) [6] using PV electron elastic scattering was also proposed. The PV electro-weak asymmetry at Bates [2] yielded the isoscalar vector hadronic coupling constant, which agreed with the value predicted by using the standard model. The combination of strange electric and magnetic form factors was extracted from measuring the PV asymmetry at forward angles [3 5]. In particular, the strange electric form factor of the nucleon was isolated and is reported in Ref. [5]. Recent experiment at Jlab [7] foundtheelectricandthemagneticstrangeformfactors to be insensitive to variations in axial form factor. Since Kaplan and Manohar s paper [8], many theoretical works have focused on the form factors associated with the strange quark components of the hadronic neu- kyungsik@kau.ac.kr cheoun@ssu.ac.kr tral current. Recently, the electro-weak hadronic structure for PV in elastic electron-nucleon scattering was extensively reviewed within the framework of the standard model; then, a possible sensitivity to the strangeness was investigated with available data [9]. The authors in Ref. [10] investigated the influence of nuclear isospin mixing onpv elastic electron scattering for even-even nuclei and then observed the effects from isospin mixing to be larger than those reported in an older work [11]. In particular, we investigated the possible uncertainties of PV in the quasi-elastic (QE) electron scattering, which are the sensitivities of electromagnetic (EM) form factors, strange form factors, final state interaction, and electron Coulomb distortion [12]. On the other hand, since the first discovery of the EMC effect [13], medium modifications of nucleon form factors have been studied for over thirty years. Among various approaches, two theoretical models have exploited the modifications of the electric and the magnetic form factors of nucleons in a nuclear medium: the quark-meson coupling (QMC) model proposed by Thomas and collaborators [14], and the cloudy bag model (CBM) proposed by Cheon and Jeong [15]. In particular, by using the QMC model, Thomas and collaborators [16] calculated the density dependence of weak form factors. That model successfully described various properties of finite nuclei [17]. The results in Ref. [17] showed a simple scaling relation for the change of the hadron masses whichwasdescribedintermsofthenumbersofnon

2 Density Dependence of Parity Violation in Electron Quasi-elastic Scattering K. S. Kim and Myung-Ki Cheoun strange quarks in a hadron and the strength of the scalar mean field in a nucleus. In Ref. [18], the 12 C(ν μ,μ )X reaction was calculated by using the density-dependent weak form factors within a relativistic Fermi gas model; then, the bound nucleon form factors were shown to reduce the total cross sections by 8% (or higher for a heavier nucleus). In our previous papers [19,20], we studied the effects of the density dependence of the weak form factors on the charged-current (CC) and neutral-current (NC) reactions of the neutrino-nucleus scattering within the framework of the quasi-particle random phase approximation (QRPA) in the low energy range (E ν( ν) 80 MeV). As aresult,theeffectsreducethecrosssectionmostlyby 5 % at normal nuclear density, ρ 0, but in antineutrinonucleus scattering, the cross sections were found to be decreased by up to 35 % by comparison with the calculations using the free form factors. In the QE region, the effect of the density-dependent weak form factor on the CC neutrino-nucleus scattering from 12 C, 40 Ca, and 208 Pb was also estimated, and the cross section was reduced maximally by 60 % at ρ =2.0ρ 0 for the antineutrino around the peak position [21]. In particular, the effect on NC neutrino-nucleus scattering [22] was separately studied for protons and neutrons participated in the reactions, each response cross section, and the asymmetry. The density effects reduce the cross section at high densities and show different behaviors in the asymmetry. In the present work, we investigate the medium effect by the density-dependent form factors for PV asymmetry from 208 Pb. The sensitivity of the density dependence is studied separately on each Dirac, Pauli, and axial form factor. As a nuclear model, we use the σ ω model [23] to obtain the bound state wave functions that are generated by solving the Dirac equation in the presence of strong vector and scalar potentials, that is, called TIMORA code. The scattered state wave functions of the knockedout nucleons are also generated under the same potential of the bound nucleons, called the relativistic mean field (RMF), in order to include the final state interaction. This RMF model guarantees current conservation and gauge invariance and provides very good agreement with the Bates (e, e ) experimental data with the inclusion of the Coulomb distortion of the electrons [24]. However, we do not include the Coulomb distortion of the incident and the outgoing electrons because the distortion effect was calculated for PV asymmetry [12]. Unfortunately, in this work, only theoretical results are presented because there are no proper experimental data on QE electron scattering. Only one data point for the asymmetry from 208 Pb was reported in Ref. [25]. To calculate the QE electron scattering, we choose the nucleus-fixed frame in which the target nucleus is seated at the origin of the coordinate system. The four-momenta of the incident and outgoing electron and of the knocked-out nucleon are labeled p μ i = (E i, p i ), p μ f =(E f, p f ), and p μ =(E,p), respectively. The fourmomentum transfer is given by q μ =(p i p f ) μ =(ω, q), where q is along the ẑ-axis. In the laboratory frame, the inclusive cross section, which does not detect the knocked-out nucleons, is given by the contraction between the lepton and the hadron tensors. The asymmetry of PV electron scattering is written as the ratio of the contribution by the EM current to that by the weak current as follows: dσ + A = ( dσ dσ + )/( + dσ ) dω f de f dω f de f dω f de f dωde f W PV = A 0, (1) W EM where the constant A 0 is given by A 0 = G F Q 2 2πα 2, (2) with the Fermi constant G F and the fine structure constant α. The squared four-momentum transfer is given by Q 2 = q 2 ω 2. The EM total response function is decomposed into the longitudinal and the transverse response functions as follows: W EM = v L R L EM + v T R T EM, (3) where R L EM = J 0 EM 2 is the longitudinal response function and R T EM = J x EM 2 + J y EM 2 is the transverse response function. The weak current is the summation of the neutral vector current and the neutral axial vector current through Z 0 boson exchange: W PV = W V PV + W A PV. (4) The weak vector current is given by W V PV = v L R L PV + v T R T PV, (5) where the longitudinal and the transverse response functions are given by RPV L = JEM 0 J NC 0 and RT PV = JEM x J NC x y +JEM J y NC, respectively. The weak axial vector current is written as W A PV = v T R T PV, (6) where the transverse response function is contributed in the way R T PV = J EM J AV. The factors of the electron kinematics are given by v L = Q4 q 4, v T =tan θ e 2 v T =tan 2 θ e 2 + Q2 2q 2, tan 2 θ e 2 + Q2 q 2, (7) where θ e denotes the scattering angle. The nuclear current is calculated by using the Fourier transform of the nucleon current operator: J μ = ψ p Ĵ μ ψ b e iq r d 3 r, (8)

3 Journal of the Korean Physical Society, Vol. 66, No. 12, June 2015 Fig. 1. (Color online) PV asymmetry of the density dependence in terms of the energy transfer at incident energies of 500 MeV and 1 GeV for scattering angles of 10 0 and The solid (red) lines are the results for no density dependence, the dashed (black) lines are for ρ =1.0ρ 0, and the dotted (blue) lines are for ρ =2.0ρ 0. where Ĵμ is a free weak nucleon current operator, and ψ p and ψ b are the wave functions of the knocked-out and the bound state nucleons, respectively. For a free nucleon, the current operator for the EM interaction is composed of the Dirac and the Pauli form factors and is given by Ĵ μ EM = F 1(Q 2 )γ μ + F 2 (Q 2 ) iκ σ μν q ν, (9) 2M N where κ represents the nucleon magnetic moment. The Dirac form factor F 1 and the Pauli form factor F 2 are related to the electric and the magnetic Sachs form factors given by G E = F 1 +(κ/qμ/4m 2 2 )F 2 and G M = F 1 +κf 2, which are assumed to take the standard form 1 G M G E = ( ) 2 = (κ +1), (10) 1+ Q2 Λ 2 where the standard value for Λ 2 is 0.71 (GeV/c) 2 and M N denotes the mass of the nucleon. The current operator of the neutral current reaction in the response functions of Eqs. (5) and (6) consists of the weak vector and the axial vector form factors and is given by Ĵ μ NC = F 1 V (Q 2 )γ μ + F2 V (Q 2 ) iκ σ μν q ν. (11) 2M N By the conservation of the vector current (CVC) hypothesis, the vector form factors for the proton (neutron), F V, p(n) i (Q 2 ), are expressed as V, p(n) Fi (Q 2 ) = ( ) 1 2 2sin2 θ W F p(n) i (Q 2 ) 1 2 F n(p) i (Q 2 ) 1 2 F s i (Q 2 ), (12) where θ W is the Weinberg angle given by sin 2 θ W = The strange vector form factors are usually given on a dipole form, independently of the nucleon isospin, F1 s (Q 2 F1 s (0)Q 2 )= (1 + τ)(1 + Q 2 /MV 2, )2 F2 s (Q 2 F2 s (0) )= (1 + τ)(1 + Q 2 /MV 2, (13) )2 where τ = Q 2 /(4MN 2 ) and M V = GeV is the cut off mass parameter usually adopted for nucleon EM form factors. F1 s (0)isdefinedasthesquaredstrange radius of the nucleus, F1 s (0) = < r 2 > /6 = dg s E (Q2 )/dq 2 Q 2 =0 =0.53 GeV 2,andF2 s (0) = μ s = 0.04 is an anomalous strange magnetic moment in Eq. (4) [26]. The axial vector current operator is given by Ĵ μ AV = G A(Q 2 )γ μ γ 5. (14)

4 Density Dependence of Parity Violation in Electron Quasi-elastic Scattering K. S. Kim and Myung-Ki Cheoun Fig. 2. (Color online) PV asymmetry where only the Dirac form factor (F 1) depends on the density. The kinematics and the explanations of the curves are the same as they are for Fig. 1. The axial form factor for the neutral current reaction is given by G A (Q 2 )= 1 2 ( g A + g s A)/(1 + Q 2 /M 2 A) 2, (15) where g A =1.262, M A =1.032 GeV, and g s A = 0.19, which represent the strange quark contents on the nucleon. (+) coming from the isospin dependence denotes the knocked-out proton (neutron), respectively. In the following, we present the density dependence of the form factors by multiplying the ratio, R(F V 1,2) = F V 1,2(ρ, Q 2 )/F V 1,2(ρ =0,Q 2 ) generated by using the QMC model [16], on the free form factors. Detailed features of the form factors and their modifications in nuclear matter used in this study are found at Refs. [19, 20]. The weak form factors including the axial form factor, are shown to be recovered to the standard form factors in the ρ = ρ 0 limit in the appendix of Ref. [20]. As shown in Fig. 1, we calculate the PV asymmetry in terms of the energy transfer at incident energies of E = 500 MeV and 1 GeV for two scattering angles of 10 0 and The solid (red) lines are the result for no density dependence, the dashed (black) lines are for ρ =1.0ρ 0, and the dotted (blue) lines are for ρ =2.0ρ 0. Here we show results for ρ =0 2.0ρ for pedagogical reasons. Note that ρ fm 3 is the normal density, and that ρ for 208 Pb is about ρ 0.6ρ 0. At forward angles, the effect of the density dependence changes even the sign of the asymmetry, but at backward angles, the effect reduces the magnitude. According to our previous paper [22], the effect enhances the cross section for QE electron scattering, but reduces the cross section for NC neutrino-nucleus scattering. From our results in Fig. 1, we learn that the effect on the electro-weak reaction is larger than it is the EM reaction because the PV asymmetry is proportional to the ratio of the electro-weak PV reaction to the EM reaction. In order to study the roles of each form factor on the density-dependent effects, in Fig. 2, we show the sensitivity of the density dependence to only the Dirac form factor (F 1 ).Thekinematicsandtheexplanationsofthe curves are the same as they are for Fig. 1. The effect of the density dependence on the Dirac form factor is very small, almost indiscernible, which is the same as our previous result [22]. According to the results in Ref. [22], the effect of the density dependence on the F 2 enhances the EM cross section, but reduces the cross section of electro-weak interaction. In Fig. 3, the sensitivity of the density dependence to only Pauli form factor (F 2 ) is shown. The kinematics and the explanations of the curves are the same as they are for Fig. 1. The effect reduces the PV asymmetry, but at forward angles the effect appears to be different. If we compare the contribution by the F 2 form factor in Fig. 3 to the total results in Fig. 1, the effect of the density dependence stems mainly from the

5 Journal of the Korean Physical Society, Vol. 66, No. 12, June 2015 Fig. 3. (Color online) PV asymmetry where only the Pauli form factor (F 2) depends on the density. The kinematics and the explanations of the curves are the same as they are for Fig. 1. Fig. 4. (Color online) PV asymmetry where only the axial form factor (G A) depends on the density. The kinematics and the explanations of the curves are the same as they are for Fig. 1.

6 Density Dependence of Parity Violation in Electron Quasi-elastic Scattering K. S. Kim and Myung-Ki Cheoun Pauli form factor. Finally, we study the sensitivity of the density dependence to only the axial form factor (G A ). The kinematics and the explanations of the curves are the same as they are for Fig. 1. The effect of the density dependence is within a few percent because the contribution of the transverse response function is much smaller than those by other response functions, as shown at Fig. 8 in Ref. [12]. However, we should note that by comparing with other two form factors, in neutrino-nucleus scattering, theeffectreducesthecrosssectionandislargest[22]. In the present work, we calculate the PV asymmetry of electron scattering off 208 Pb within the framework of a relativistic single-particle model, called σ-ω model. In the QE region, where the inelastic processes, such as Δ resonance, are excluded, the density dependence reduces thepvasymmetryandkeepsthesameshapeatbackward angles, but at forward angles, the behavior appears to be totally different, i.e., it may change the asymmetry sign. The effect on the Dirac and the axial form factors is small, but that on the Pauli form factor is relatively large. In particular, the density-dependent Pauli form factor changes not only the magnitude but also the shape of the PV asymmetry. On the other hand, although we study the sensitivity of density-dependence to the form factor, the current calculation is not fully consistent. In conclusion, for more physical quantities for 208 Pb to be used for determining the nuclear matter properties, such as neutron skin thickness of finite heavy nuclei, experiments for precisely probing the PV asymmetry should be improved although that is a very challenging task experimentally. ACKNOWLEDGMENTS This work was supported by the National Research Foundation of Korea (Grant Nos. 2012R1A1A , 2012M7A1A and 2014R1A2A2A ). REFERENCES [1]C.Y.Prescottet al., Phys. Lett. B 77, 347 (1978). [2]P.A.Souderet al.,phys.rev.lett.65, 694 (1990) [3] D. S. Armstrong et al., Phys. Rev. Lett. 95, (2005). [4] K. A. Anio et al., Phys. Rev. C69, (2004). [5] K. A. Anio et al.,phys.rev.lett.96, (2006). [6] [7] A. Acha et al.,phys.rev.lett.98, (2007). [8] D. B. Kaplan and A Manohar, Nucl. Phys. B 310, 527 (1988). [9] R. Gonzalez-Jimenez, J. A. Caballero and T. W. Donnelly, Phys. Rep. 524, 1 (2013). [10] O. Moreno, P. Sarriguren, E. Moya de Guerra, J. M. Udias, T. W. Donnelly and I. Sick, Nucl. Phys. A 828, 306 (2009). [11] T. W. Donnelly, J. Dubach and I. Sick, Nucl. Phys. A 503, 589 (1989). [12] K. Kim and M.-K. Cheoun, J. Phys. Soc. Jpn. 82, (2013). [13] J. Aubert et al., Phys. Lett. B 123, 275 (1983); J. Ashman et al., Phys. Lett B 206, 364 (1988). [14] D. H. Lu et al., Nucl. Phys. A 634, 443 (1998): K. Saito, K. Tsushima and A. W. Thomas, Phys. Lett. B 465, 27 (1999). [15] I.-T. Cheon and M. T. Jeong, J. Phys. Soc. Jpn, 61, 2726 (1992); M. T. Jeong and I.-T. Cheon, Phys. Rev. D 43, 3725 (1991). [16] D. H. Lu, K. Tsushima, A. W. Thomas, A. G. Williams and K. Saito, Phys. Lett. B 441, 27 (1998); D. H. Lu, K. Tsushima, A. W. Thomas, A. G. Williams and K. Saito, Phys. Rev. C 60, (1999). [17] K. Saito, K. Tsushima and A. W. Thomas, Nucl. Phys. A 609, 339 (1996); K. Saito, K. Tsushima and A. W. Thomas, Phys. Rev. C 55, 2637 (1997). [18] K. Tsushima, H. Kim and K. Saito, Phys. Rev. C 70, (2004). [19] M.-K. Cheoun et al., Phys. Lett. B 723, 464 (2013). [20] M.-K. Cheoun et al.,phys.rev.c87, (2013). [21] K. S. Kim, M.-K. Cheoun and W. Y. So, Phys. Rev. C 90, (2014). [22] K. S. Kim, M.-K. Cheoun, W. Y. So and H. Kim, Phys. Rev. C 91, (2015). [23] C. J. Horowitz and B. D. Serot, Nucl. Phys. A 368, 503 (1981). [24] K. S. Kim, L. E. Wright, Y. Jin and D. W. Kosik, Phys. Rev. C 54, 2515 (1996); K. S. Kim, L. E. Wright, Phys. Rev. C (1997); K. S. Kim, B. G. Yu, and M. K. Cheoun,Phys.Rev.C.74, (2006). [25] S. Abrahamyan et al., Phys. Rev. Lett. 108, (2012). [26] M.-K. Cheoun and K. S. Kim, J. Phys. G 35, (2008).