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Volume 197, Number 4 PHYSICS LETTERS B 5 November 1987 VACUUM POLARIZATION EFFECTS in ELASTIC SCATTERING of PROTONS by NUCLEI N

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Volume 197, Number 4 PHYSICS LETTERS B 5 November 1987 VACUUM POLARIZATION EFFECTS in ELASTIC SCATTERING of PROTONS by NUCLEI N Volume 197, number 4 PHYSICS LETTERSB 5 November 1987 VACUUM POLARIZATION EFFECTS IN ELASTIC SCATTERING OF PROTONS BY NUCLEI N. OTTENSTEIN, S.J. WALLACE Department of Physics and Astronomy, University of Maryland, College Park, MD 20742, USA and J.A. TJON Institute of Theoretical Physics, University of Utrecht, 3508 TA Utrecht, The Netherlands Received 24 May 1987; revised manuscript received 29 July 1987 Sensitivity of spin observables in elastic proton scattering by nuclei to the vacuum polarization correction 8PvAcis explored. The Dirac optical potential is calculated from a complete set of Lorentz invariant amplitudes, Dirac-Hartree densities and the existing estimate of SPvAcbased on quantum hadrodynamics. Spin observables in elastic scattering of protons by principle may affect significantly the scattering nuclei have been interpreted successfully in a rela- observables, it is clearly of interest to study their ef- tivistic approach based on the Dirac equation. Large fect. In this letter we examine for various nuclei the scalar and vector potentials in the Dirac equation sensitivity of elastic proton nucleus scattering to these yield a simple description of the analyzing power, Av, vacuum polarization corrections. and the spin rotation function, Q, over a wide energy The Dirac optical potential is determined in a fac- range. Calculations of the optical potential in the torized form by the NN amplitude and the nuclear Dirac equation are based on the nucleon-nucleon density according to the relation [9 ] (NN) interaction and the nuclear density [1-5]. Recently this parameter-free approach has been gen- (7(p', p) = - 1Tr2 {/I)/(p, - ½q ~p', + ½q)~(q) }, eralized by developing a complete set of Lorentz (1) invariant NN amplitudes from a relativistic meson exchange description of NN scattering [6-8]. The where M is the Feynman amplitude for NN scatter- point of this generalization is to use a dynamical ing and/~ is the relativistic nuclear density. The latter model which specifies the coupling between positive is characterized by scalar, vector and tensor terms as energy and negative energy states, the + to - cou- follows. pling. In the original form of the impulse approxi- mation, the + to - coupling was not tied to ~2 'q . dynamics. Fermi covariants were assumed to pro- P(q) =Ps(q) +7°Pv(q) - -~-m-mPTtq) • (2) vide a useful extension of the NN data to the full Dirac space of two nucleons [4]. Relativistic Hartree wave functions of the nucleus Another immediate consequence of a relativistic [ 10] are used to calculate the densities ps(r), pv(r) meson theoretical description of the nuclear inter- and px(r) in coordinate space and these are Fourier action is the existence of vacuum polarization con- transformed to obtain the densities in (2) as func- tributions to the optical potential. Since they in tions of the momentum transfer, q. Neutron-proton 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. 493 (North-Holland Physics Publishing Division) Volume 197, number 4 PHYSICS LETTERS B 5 November 1987 differences are incorporated by evaluating eq. (1) 5PvAC(r) = -- (1/Tt2)[ M*( r) 3 ln( M*( r)/M) + I M 3 once with the proton-proton amplitude, Mpp, times the proton density, /Sp, and once with the pro- -~M2M*(r)+3MM*(r)Z-~M*(r) 3 ] , (4) ton-neutron amplitude, ~pn, times the neutron den- sity,/5,, and then adding the two results. Several papers have noted that spin observables in where M*(r) is calculated from the total scalar den- proton scattering are quite sensitive to the difference sity (proton plus neutron), between scalar and vector terms in the Dirac optical M*( r) _~M- (g2 /M~s)Ps( r) , (5) potential [ 11,12 ]. Owing to the large scalar and vec- tor components of the NN amplitudes, which have and M is the nucleon mass, gs is the scalar meson opposite signs, there is sensitivity to the difference, coupling constant and ms is the scalar meson mass. 5p(r)-pv(r)-ps(r), between scalar and vector Eq. (4) is derived for infinite nuclear matter and then densities. For Dirac-Hartree wave functions, 5p(r) applied to finite nuclei by using the local density to is determined by the lower components of the single determine M*(r) in (5). Finite range effects are particle wave functions, F~(r), as follows, neglected in eq. (5). Regarding the vector density as fixed, the total 2j+ 1 r , "2 8PLc(r) =2 ~ -~---~tr) , (3) scalar density is found by subtracting the Hartree lower component density, 6PLc(r), of eq. (3), and where c~ is summed over the occupied states. The then adding the vacuum polarization correction of proton vector density is constrained by electron scat- eq. (4), tering data and is relatively well known. Although there are uncertainties in neutron densities, the total Ps (r) = Pv (r) - 6PLc (r) + 6PVAC(r) . ( 6 ) vector density, Pv, is fairly well known. On the other It is necessary to determine 8PvAc self consistently hand, the scalar density is not constrained by elec- from eqs. (4)-(6) due to the highly nonlinear tron scattering but one expects 8PLc(r) to be small dependence of 6PvAc on M*. A straightforward cal- since it vanishes for nonrelativistic wave functions. culation shows that gPvAc is about 5% of the vector Employing scattering data to constrain pv(r) and density. Perry [16] has considered the vacuum using Dirac-Hartree wave functions to determine polarization for a finite nucleus within QHD. Extra 8PLc(r), one may calculate the scalar density, terms are obtained in that case and the vacuum ps(r) =pv(r) -SPec(r), which is needed for the Dirac polarization effect causes changes in both the scalar optical potential of (1). However, there is another and vector densities. However, the difference contribution to the scalar density due to vacuum 8p=pv-Ps is quite similar to the mean field result polarization. of eq. (4). In the quantum hadrodynamics (QHD) model of In this paper, the effect of the vacuum polarization Serot and Walecka [ 13 ], filled negative-energy states on spin observables in proton scattering is consid- of the Dirac sea are shifted in energy due to the sca- ered. The vacuum correction ~PvAc is estimated lar and vector potentials. The resulting change of the based on mean field theory using eqs. (4)-(6). Har- total energy of the system, ZXEvAc, due to this vac- tree vector and lower-component densities, Pv and uum polarization is a function of the mean scalar field 6PLc, are kept fixed and scalar meson parameters 00 (r). A functional derivative of ZXEvAcwith respect gg=54.289 and ms=458 MeV are used [ 15]. Fig. to the mean scalar field determines the vacuum 1 shows 5PvAc for Ca and Pb nuclei. Somewhat more polarization correction to the scalar density, consistent values of fiPvAc can be obtained by refit- 8PvAc(r)=S&EvAc/8Oo(r). The energy correction ting the Hartree density to obtain an improved esti- &EvAc has been calculated in a mean field approx- mate of 8PLc(r) including the vacuum polarization imation to quantum hadrodynamics after introduc- corrections to the nuclear binding energy. However, ing suitable counter terms for renormalization the result is quite similar to that given by eq. (4). We [ 14,15 ]. The result is that 8PvAc is determined from do not consider such refinements of the calculation the effective nucleon mass, M*, as follows. because dynamical uncertainties exist in the deter- 494 Volume 197, number 4 PHYSICS LETTERS B 5 November 1987 104 0.04 I I I I I I I I I ) t I I P I I CA 500 MeV 103 'E / 0.02 102 O-- \\\\ i0 ~ 60 I ,.Q 0.0 I I \q, I I i0 ° l 2 3 4 5 6 7 8 9 10 r (fm) 10 -I Fig. 1. Negative of vacuum corrections, 8PVAC, for 4°Ca (solid 10 -2 line) and 2°SPb (dashed line). b i0 -3 mination of realistic vacuum polarization correc- tions. For example, the mean field theory does not i0 -4 I I I I I I I include pion exchange dynamics. The values in fig. 8 12 16 20 2q 28 32 1 represent the simplest model. It is conceivable that 9- (deg) mean field theory based on scalar and vector mesons overestimates the effect because it yields a rather low l.O I I I I I " ~ ~~~I\r p~"\I ~++'+ I I .8 value of 34* in the nuclear interior. The optical .6 potential based on complete sets of NN amplitudes .q .2 )1 \\ as in eq. (1) predicts smaller scalar and vector 0 potentials. This implies a larger value for 34* and <:~ -.2 therefore a smaller vacuum correction than mean field theory. -.8 -1.0. I I I I In order to explore the sensitivity of proton scat- 2 4 6 8 tO 12 1't t6 18 20 22 24 26 28 30 tering observables in a phenomenological way, cal- CA 500 MeV 1.0 It I le I culations are given for three cases: (1) not including .8 ~. " \ .6 6PvAc(dashed lines); (2) including 50% of .4 6PvAc(Solid lines); and (3) including 100% of .2 5pvgc(dotted lines). Figs. 2 and 3 show the proton CY_. ° scattering observables for 4°Ca and 2°SPb at 500 MeV. -,4 Without the vacuum polarization correction, the spin -,8 I I observables are in reasonable agreement with exper- -1.0 I I I I I I I J f I I l 4 6 8 tO 12 14 t6 18 20 22 2~ 26 28 30 imental data for 4°Ca but the agreement is less good for 2°Spb.
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