Manifestation of one- and two-body currents in longitudinal and transverse response functions of the 12C nucleus at q = 300 MeV/c
aa r X i v : . [ nu c l - e x ] S e p MANIFESTATION OF ONE- AND TWO-BODY CURRENTS IN LONGITUDINALAND TRANSVERSE RESPONSE FUNCTIONS OF THE C NUCLEUS AT q =
300 MeV / c A.Yu. Buki ∗ , I.S. Timchenko National Science Center ”Kharkov Institute of Physics and Technology”, 1, Akademicheskaya St., Kharkov, 61108, Ukraine
Abstract
The experimental values of longitudinal and transverse response functions of the C nucleus have been obtained at the 3-momentumtransfer q =
300 MeV / c. The data are compared with the calculations made with due regard to the dynamics of all the nucleonsconstituting the C nucleus, and also, to the contributions of both the one-body currents only, and their combination with two-bodycurrents.
Keywords: electron scattering, C nucleus, longitudinal and transverse response functions, currents
PACS: + n The exact calculation of the longitudinal R L ( q , ω ) and trans-verse R T ( q , ω ) response functions of nuclei with full consider-ation of the dynamics of all their constituent nucleons is oneof the challenges in quantum many-body physics. So far, theresponse function calculations, which are in fairly good agree-ment with the experiment, have been performed only for thenuclei with A ≤ R L ( q , ω ) and R T ( q , ω ) of the C nucleus were performedon the basis of the AV18 + IL7 combination of two and three-nucleon potentials and accompanying set of two-body electro-magnetic currents. The Green’s function Monte Carlo methodsand maximum-entropy techniques were used in the calculation.In case of the longitudinal response function, the considera-tion of contributions from one-body currents only, or from acombination of one- and two-body currents, causes an insignif-icant change in R L ( q , ω ) only in the vicinity of the threshold.However, since the two-body currents generate a large excessof strength in R T ( q , ω ) over the whole ω − spectrum, the com-parison with the experimental data could be a good test of thecalculations.The calculations of response functions in ref. [3] were com-pared with the experimental response functions of C, deter-mined from the world data analysis of J. Jourdan [4, 5] and,for q =
300 MeV / c, from the Saclay data [6, 7]. The dataof the mentioned works di ff er widely. In view of this, itshould be noted that the experimental data on the functions R L ( q , ω ) and R T ( q , ω ) of the C nucleus were obtained inSaclay [6, 7] at constant momentum transfers q ranging from ∗ Corresponding author
Email address: [email protected] (A.Yu. Buki )
200 to 550 MeV / c. In his papers [4, 5], J. Jourdan has rean-alyzed the primary data from refs. [6, 7] and the measureddata obtained at SLAC [8, 9, 10], which were then used fordetermining the ”world” response function values of the Cnucleus. However, not all researchers were content with the re-sults of the reanalysis [4, 5]. For example, J. Morgenstern andZ.-E. Meziani have carried out their own reanalysis of the ex-perimental data for a variety of nuclei, and demonstrated [11]that the results changed only insignificantly with the combina-tion of the SLAC and Saclay data.It follows from the above that for testing the calculations ofref. [3], there is a need to use other experimental data on theresponse functions of the C nucleus, which would be inde-pendent of the ones in refs. [4, 5, 6, 7]. These data are derivedin the present work and are used for comparison with the calcu-lations [3]. The present experimental response functions were obtainedfrom processing the spectra of electrons scattered by C nu-cleus, which were measured at the NSC KIPT LUE-300 linacat initial electron energies E , ranging from 149 to 208 MeV,through the scattering angle θ = ◦ , and at E =
200 MeVand θ = ◦ to 90 ◦ .Below we give a short description of the measurementand data processing procedures for obtaining the experimental R L ( q , ω ) and R T ( q , ω ) values (a more detailed information onthe topic can be found, e.g., in ref. [12]).The electron beam from the accelerator (current being up to0.2 µ A ) is incident on the target. The scattered electrons aremomentum analyzed by the spectrometer having the solid an-gle of 2 . × − sr, and the dispersion of 13.7 mm / percent. In Preprint submitted to Physics Letters B September 7, 2020 he focal plane of the spectrometer, the electrons are detectedby 8 scintillators, each with an energy acceptance of 0.31%.After that, the electrons come to organic-glass Cherenkov ra-diators. The pulses from the photomultipliers of scintillationand Cherenkov detectors are registered by a coincidence circuitwith a time resolution of 9 nsec.The spectral measurements of electrons scattering by nu-clei involved the measurement of the contributions that cometo the data from the background processes, viz., the detectorregistration of radiation background in the experimental hall(physical background), and also, of random coincidences as thepulses from scintillation and Cherenkov detectors arrive simul-taneously at the coincidence circuit (random coincidence back-ground). The electron scattering by the target is accompaniedby photoproduction of e + , e − -pairs in the target substance. Theelectrons of the pairs form one more background. This back-ground is measured through reversing the polarity of the spec-trometer magnet, and registering the positron spectrum, whichis identical to the electron spectrum from the e + , e − -pairs. Themeasurement of this sort was performed in our experiment, butno positrons were observed. Perhaps, that was due to theirlow yield under those experimental conditions. To check theconclusion, the positron yield was numerically estimated un-der the conditions of the described measurements. The estima-tions were performed using the calculation methods from [13].As result was found that the manifestation of electron-positronbackground in our measurements was well below the measure-ment error.After taking into account the contributions from di ff erentbackgrounds, the spectra were corrected for the radiation-ionization e ff ects by equations of refs. [14, 15]. The mea-surement data were normalized with the coe ffi cient k = F ( q ) / F ( q ), where F ( q ) represents the nuclear ground-stateform factor values obtained in our measurements, and F ( q )stands for the data taken from work [16]. At that, the 3% cor-rection (see ref. [17]) to the data of [16] was considered.The experimental values of the longitudinal R L ( q , ω ) andtransverse R T ( q , ω ) response functions of the nucleus are deter-mined from the analysis of the inclusive electron-nucleus scat-tering cross-sections measured at large and small scattering an-gles θ . In this case, the equation from ref. [18] is used, whichconnects the response functions with the twice di ff erential elec-tron scattering cross-section d σ / d Ω d ω , by the relationship R θ ( q , ω ) = d σ d Ω d ω ( θ, E , ω ) /σ M ( θ, E ) = q µ q R L ( q , ω ) + q µ q + tan θ R T ( q , ω ) . (1)Here R θ ( q , ω ) is the angular response function, E is the initialenergy of electron scattered through the angle θ with the trans-fer of energy ω , the e ff ective 3-momentum q = { E e f f [ E e f f − ω ]sin ( θ/ + ω } / and 4-momentum q µ = ( q − ω ) / to the nu-cleus studied; σ M ( E , θ ) = e cos ( θ/ / [4 E sin ( θ/ e is the electron charge. The term E e f f inthe definition of the e ff ective 3-momentum is the e ff ective en-ergy, which is the sum of the initial energy E and the correction E C that takes into account the action of the electrostatic field ofthe nucleus on the incoming electron. According to [19], thiscorrection is written as E C = Ze < r > − / , where Z and < r > are, respectively, the charge and r.m.s. radius of thenucleus.To obtain the experimental values of the longitudinal andtransverse response functions, it is essential that the set of equa-tions (1) should be solved for two angular response functions R θ ( q , ω ) measured at large and small electron scattering angles,but at the same ω and q . However, in the plane of arguments q and ω , the functions R θ ( q , ω ) can have only one point in com-mon. Therefore, for obtaining the R L ( q , ω ) and R T ( q , ω ) values,the set of cross-sections for electrons scattered by the nucleus ismeasured in experiment, from which, after transformation intothe function R θ ( q , ω ) by means of certain interpolations and ex-trapolations with respect to q and ω , the sought-for values areobtained (for more details, see, e.g., ref. [20]).The described processing of the measured data has re-sulted in the experimental values of the functions R L ( q , ω ) and R T ( q , ω ) of the C nucleus at a constant momentum transfer q =
300 MeV / c. The data are illustrated in Figs. 1a and 1b, di-vided by the square of the proton charge form factor [ G pE ( q µ )] from ref. [21]. Figure 1 shows calculations from work [3] for the longi-tudinal and transverse response functions of the C nucleus.The dash-and-dot line represents the plane-wave impulse-approximation (PWIA) calculation using the single-nucleonmomentum distribution [22]. The other calculations are basedon the realistic dynamic pattern of the description of nucleus forthe cases with consideration of only one-body (O1b) currents inthe electromagnetic operator, and also, with the combination ofone- and two-body currents (O1b-2b). In the last calculationsthe AV18 + IL7 combination of two- and three-nucleon poten-tials is used.The comparison of the calculation data for R L ( q , ω ) with theexperimental information points to the fact that the PWIA cal-culation overestimates the response value in the longitudinalcomponent of the quasielastic-scattering peak maximum. Inview of the smallness of the two-body current e ff ect on the lon-gitudinal response function, the curves for the O1b and O1b-2bcalculations di ff er only insignificantly. Therefore, none of thecalculation variants can be singled out.Unlike the R L ( q , ω ) case, in the R T ( q , ω ) case, the calcula-tions with the contribution of only one-body currents or with thecontribution from combination of one- and two-body currentsshow quite a di ff erence, thereby making possible the test of thecalculations. As is seen from Fig. 1b, our data on the function R T ( q , ω ) are in excellent agreement with the O1b-2b calculationat all ω values under study, except in the near-threshold region,where the contributions of C low-lying levels were excludedin the O1b and O1b-2b calculations.Thus, in the present study, we have determined the experi-mental functions R L ( q , ω ) and R T ( q , ω ) of the C nucleus at2
50 100 150 2000.010.020.030.040.050.060.07 R L ( q , ) / [ G p E ( q )] , M e V - , MeV (a) , MeV R T ( q , ) / [ G p E ( q )] , M e V - (b) Figure 1: C response functions at constant q =
300 MeV / c: (a) longitudinalfunction R L ( q , ω ); (b) transverse function R T ( q , ω ). The lines show the cal-culations of work [3]: the dash-and-dot line shows the PWIA calculation; thedashed line - the calculation with due regard to one-body currents only; thesolid line - with due regard to a combination of one- and two-body currents; thepoints show the experimental data of the present work. q =
300 MeV / c. The results are independent of the data ofrefs. [4, 5] and [6, 7], which were earlier used for testing thecalculations of ref. [3]. Our present experimental values of theresponse functions under consideration correspond to the cal-culation variant of ref. [3], in which the combination of one-and two-body currents was taken into account. References [1] S. Bacca, N. Barnea, W. Leidemann, G. Orlandini, Role of theFinal-State Interaction and Three-Body Force on the LongitudinalResponse Function of He, Phys. Rev. Lett. 102 (2009) 162501,https: // doi.org / / PhysRevLett.102.162501.[2] W. Leidemann, V.D. Efros, G. Orlandini, E.L. Tomusiak, Threshold Heand H Transverse Electron Scattering Response Functions, Few-BodySystems 47 (2010) 157, https: // doi.org / / s00601-009-0078-8.[3] A. Lovato, S. Gandolfi, J. Carlson, S.C. Pieper, R. Schiavilla, Electromag-netic Response of C: A First-Principles Calculation, Phys. Rev. Lett.117 (2016) 082501, https: // doi.org / / PhysRevLett.117.082501.[4] J. Jourdan, Quasi-elastic response functions. The Coulomb sum re-visited, Nucl. Phys. A 603 (1996) 117, https: // doi.org / / // doi.org / / // doi.org / / // doi.org / / ∆ Electroproduc-tion and Inelastic Charge Scattering from Carbon and Iron, Phys. Rev.Lett. 61 (1988) 400, https: // doi.org / / PhysRevLett.61.400.[9] D.T. Baran, The Electroproduction of the Delta Isobar in Nuclei, PhD the-sis, Northwestern University, 1989.[10] D.B. Day, J.S. McCarthy, Z.E. Meziani, R. Minehart, R. Sealock,S.T. Thornton, J. Jourdan, I. Sick, B.W. Filippone, R.D. McKeown,R.G. Milner, D.H. Potterveld, Z. Szalata, Inclusive electron-nucleusscattering at high momentum transfer, Phys. Rev. C 48 (1993) 1849,https: // doi.org / / PhysRevC.48.1849.[11] J. Morgenstern and Z.-E. Meziani, Is the Coulomb sum rule violated innuclei? Phys. Lett. B 515 (2001) 269, https: // doi.org / / S0370-2693(01)00873-5.[12] A.Yu. Buki, N.G. Shevchenko, I.A. Nenko, V.N. Polishchuk,A.A. Khomich, S.V. Dmytriyeva, Moments of the response function forthe H nucleus at q = − , Phys. of Atom. Nucl. 65 (2002) 753,https: // doi.org / / e + , e − )-pair contribution tothe spectra of electrons scattered by nuclei, East European Journal ofPhysics, 2(2) (2015) 38, https: // doi.org / / ep and up Scattering, Rev. Mod. Phys. 41 (1969) 205,https: // doi.org / / RevModPhys.41.205.[15] Y.S. Tsai, Radiative corrections to electron scatterings: Report SLAC-PUB-848. Stanford Linear Accelerator Center, 1971, 66 p.[16] W. Reuter, G. Fricke, K. Merle, H. Miska, Nuclear charge distribution andrms radius of C from absolute electron scattering measurements, Phys.Rev. C 26 (1982) 806, https: // doi.org / / PhysRevC.26.806.[17] A.Yu. Buki, I.S. Timchenko, About absoluteness of data on elastic elec-tron scattering with C nucleus, Problems of atomic science and technol-ogy. 5(48) (2007) 45, arXiv:1012.2691.[18] T. de Forest Jr., J.D. Walecka, Electron scattering and nuclear structure,Adv. Phys. 15 (1966) 1, https: // doi.org / /