The ratio of p and n yields in NC v(v̄) nucleus scattering and strange form factors of the nucleon

15 October 1998 Ž. Physics Letters B 438 1998 9–13 ž/ The ratio of p and n yields in NC nn nucleus scattering and strange form factors of the nucleon W.M. Alberico a, M.B. Barbaro a, S.M. Bilenky b, J.A. Caballero c,1, C. Giunti a, C. Maieron a, E. Moya de Guerra c, J.M. Udıas c,2 ´ aINFN, Sezione di Torino and Dipartimento di Fisica Teorica, UniÕersita di Torino, Via P. Giuria 1, 10125 Torino, Italy ` bJoint Institute for Nuclear Research, Dubna, Russia cInstituto de Estructura de la Materia, CSIC, Serrano 123, E-28006 Madrid, Spain Received 5 January 1998; revised 8 July 1998 Editor: J.-P. Blaizot Abstract Ž. We calculate the ratio of proton and neutron yields in NC induced nn –nucleus inelastic scattering at neutrino energies of about 1 GeV. We show that this ratio depends very weakly on the nuclear models employed and that in n and n cases the ratios have different sensitivity to the axial and vector strange form factors; moreover the ratio of n –nucleus cross sections turns out to be rather sensitive to the electric strange form factor. We demonstrate that measurements of these ratios will allow to get information on the strange form factors of the nucleon in the region Q 2 G0.4 GeV 2 .q1998 Published by Elsevier Science B.V. All rights reserved. PACS: 12.15.mn; 25.30.Pt; 13.60.Hb; 14.20.Dh; 14.65.Bt Keywords: Neutrino–nucleus scattering; Strange form factors; Nuclear model effects The determination of the one–nucleon matrix eleŽ. ments of the axial and vector weak strange currents has become an important challenge both for theory and experiment: after the measurements of the polarized structure function of the proton gin deep 1 wx inelastic scattering 1,2 , the value of the axial strange 1Permanent address: Dpto. de Fısica Atomica, Molecular y ´´ Nuclear, Universidad de Sevilla, Apdo. 1065, E-41080 Sevilla, Spain. 2Present address: Dpto. de Fısica Atomica, Molecular y Nu- ´´ clear, Fac. de CC. Fısicas, Univ. Complutense de Madrid, Ciudad ´ Universitaria, E-28040 Madrid, Spain. ss wx constant ghas been set to gsy0.10"0.03 3 , AA while the value of the strange magnetic form factor of the nucleon has been recently determined at Bates wx 4 via measurements of the P–odd asymmetry in electron–proton scattering, with the result sŽ2. G0.1GeV s0.23"0.37"0.15"0.19. The latMŽ ter is still affected by large experimental and theo- . retical uncertainties, which are compatible with vanishing magnetic strange form factor; the former seems to indicate a non–zero value of the strange axial constant, but the theoretical analysis of the data leading to the above mentioned result still suffers from some uncertainties and model dependence. Fur0370-2693r98r$ - see front matter q1998 Published by Elsevier Science B.V. All rights reserved. Ž. PII: S0370-2693 98 01043-0 () W.M. Alberico et al.rPhysics Letters B 438 1998 9–1310 ther progress is thus needed in order to assign a reliable quantitative estimate of the strange form factors of the nucleon. wx In previous works 5,6 , we have shown that an investigation of elastic and inelastic neutral current Ž. Ž . NC scattering of neutrinos and antineutrinos on nucleons and nuclei is an important tool to disentangle the isoscalar strange components of the nucleonic current. In this letter we focus on the ratio between the cross sections of the inelastic production of Ž. protons and neutrons in neutrino antineutrino processes: nn qA,Z™ nn qpqAy1,Zy1, Ž. Ž . Ž. Ž. m mmm 1 Ž. nn qA,Z™ nn qnqAy1,Z,2 Ž. Ž .Ž. Ž. Ž. m mmm Ž. where A,Zis a nucleus with Anucleons and atomic number Z. This ratio has been first suggested as a probe for strange form factors by Garvey et al. wx Ž 7,8 , at rather low incident neutrino energies E, n . 200 MeV , a kinematical condition which is appropriate for LAMPF. The influence of the nuclear dynamics on this wx ratio, has been thoroughly discussed in Ref. 9 and wx 6 . It has been found that at Eof the order of 200 n MeV the theoretical uncertainties associated, e.g., Ž. with the final states interaction FSI of the ejected nucleon with the residual nucleus could introduce ambiguities in the determination of the strange axial wx and magnetic form factors 6 . In our opinion, incident neutrino energies of the order of 1 GeV appear interesting, from the point of view of the determination of the strange form factors of the nucleon, since the nuclear model effects are within percentage range and are well under control. Neutrinos with such energies are available at Brookhaven, KEK, Protvino Ž and probably will be available at Fermilab see wx. BOONE proposal 10 . In this letter we calculate the contributions of the axial and vector strange form factors to the ratio of Ž. Ž. the cross sections of the processes 1 and 2 , d s rdT Ž. Ž. nn ,p N n Ž n . R Rs,3 Ž. prnd s rdT Ž. Ž. nn ,n N for incident neutrino energies Es1 GeV and for n Ž n . 12 C. In the above Tis the kinetic energy of the N outgoing nucleon. We present here calculations in Ž. plane wave impulse approximation PWIA , within Ž. two nuclear models: the relativistic Fermi gas RFG Ž. and a relativistic shell model RSM . Calculations in Ž. distorted wave impulse approximation DWIA are also included for the RSM, with FSI taken into Ž. account through a relativistic optical potential ROP . wx For details of these models see Refs. 6,11 , and references therein. We also consider the ratio of integrated cross sections, dT d s rdT Ž. HŽ. nn ,p NN n Ž n . Rs.4 Ž. prndT d s rdT Ž. HŽ. nn ,n NN n Ž. In Fig. 1a,b we present the ratio R Ra and prn n Ž. R Rb for incident neutrino energy Es1 GeV as prn n a function of T, at different values of the parameN ters that characterize the strange form factors. The solid lines correspond to the pure RSM, the dot– Ž. dashed lines to the DWIA RSMqROP and the dotted lines to the RFG. The latter almost coincide with the solid ones in Fig. 1a, while small differŽ ences are seen in the ratio of n –cross sections Fig. . 1b . Also the effect of FSI appears to be somewhat more relevant in the n processes, while it is fairly negligible in R R n . prnwx As already noticed in Ref. 6 , at Es1 GeV the n ratio R R n is substantially unaffected by the nuclear prn model description, even by including the distortion Ž of the knocked out nucleon in spite of the fact that the FSI sizably reduce the separated cross section . n with respect to the PWIA ; moreover R Ris fairly prn constant as a function of the ejected nucleon energy over the whole interval of kinematically allowed TN values, thus providing a wide range of energy for testing the effects of the strange form factors. n On the contrary R Rshows a more pronounced prn dependence upon the energy of the emitted nucleon, Ž. stemming from the fact that the n ,ncross sections Ž. decrease faster than the n ,pones. As a consequence the range of Twhere the ratio increases N appears to be more sensitive to the nuclear model Ž and to FSI we have partially cut the curves in the large Tregion, the latter being uninteresting for the N () W.M. Alberico et al.rPhysics Letters B 438 1998 9–13 11 n n Ž. Ž. Fig. 1. The ratio R Ra and R Rb for NC neutrino prnprn processes, versus the kinetic energy of the final nucleon TsT Np sT, at incident energy Es1 GeV. The dotted lines corren n Ž n . spond to the RFG model, the solid lines to the RSM calculation, the dot–dashed lines include the effect of FSI accounted for by the ROP model. Four different choices of the strangeness parameters are shown, as indicated in the figure. . discussion . If we further restrict to the region where n R Rremains fairly constant, the sensitivity of the prn ratio to the nucleonic strangeness is comparable to the one of R R n . prn Models for the strange form factors of the nucleon 2wx exist in the low Qlimit 13 ; a soliton model has wx been recently employed by Kolbe et al. 12 in a Ž. study of the ratio 4 under the LAMPF kinematical wx conditions. It was shown in Ref. 5 that information 2Ž on the Qdependence of the strange axial and .22 magnetic form factors in the region QR0.5GeV can be obtained from the measurement of the asymŽ. metry of elastic nn –proton scattering. To illustrate the size of the effects of strangeness we have adopted sŽ2. here the standard dipole behaviour, both for GQ M sŽ2.sŽ. sŽ. s and FQ, with G0s m and F0sg, usAMsAA ing the same cutoff masses of the non–strange vector Ž. s axial form factors. A stronger decrease of Gand M s2Ž Fat high Qas suggested by the asymptotic quark A. counting rule would indeed reduce the global effects of strangeness, the size of this reduction and the scale where it becomes important being determined by the specific form assumed for the Q2dependence: for example a ‘‘Galster-like’’ parameterizawx tion as the one used in Ref. 5 would reduce the effects we are considering of about 25%. The comparison of Figs. 1a and 1b indicates that the interplay between axial and magnetic strangeness is opposite for the n and n ratios. For instance, if sŽ. gand m are assumed to have the same negative As Žs sign e.g. in our calculation gsy0.15, m s As . n y0.3 , their effects on R Rhave a constructive prn interference, which enhances the global effect of strangeness, while the opposite occurs for anti–neun trinos. On the contrary, R Ris more sensitive than prn R R n to the strange form factors when, e.g., gss prnA y0.15 but m sq0.3. s The interest of considering positive m values s stems from the recent measurement of this quantity performed at Bates in parity violating electron scatwx tering on the proton 4 . Though affected by large errors, which give a result still compatible with zero magnetic strangeness, a positive strange magnetic moment of the nucleon is allowed. The value of Gssq0.23 "0.37 "0.15 "0.19 at Q2s0.1 M GeV2corresponds to a m s0.30"0.48"0.20" s 0.25 if extrapolated down to the origin by using form Ž factors of dipole type the quoted uncertainties are, respectively, the statistical and systematic errors together with the theoretically estimated radiative corwx. rections 14 . Thus far we have discussed results obtained for the ratio R Rby setting to zero the electric strange prn form factor, Gs: we have included the latter in our EsŽ2.VŽ2. calculations, using the form GQs rt GQ, EsD VŽ2. r being a constant and GQ the usual dipole sD form factor of the vector currents. We have found Ž that, for rather large values of r of the order of s . "2 the ratio R Ris appreciably modified, in prn particular it is enhanced by a negative r and res () W.M. Alberico et al.rPhysics Letters B 438 1998 9–1312 duced by a positive one 3. Moreover we have found that the electric strangeness has a quite different n n impact on R Rand on R R. In the first case prnprn Ž n .s R Rthe effect of Gdoes not exceed 25% of the prnE correction associated to the axial strange form factor, which remains the dominant one, while it can be of the order of 50% of the correction associated with a strange magnetic moment m sy0.3. s Instead, for the ratio obtained with antineutrino n Ž. beams R R, the interference between the electric prn and magnetic strange form factors appears to be n much more important: it turns out that R Ris even prn more sensitive to Gsthan to Gs, although, again, EM the axial strange form factor plays the major role. This introduces a third unknown in the analysis of n n R Rand R R. However it is worth reminding that prnprn it is quite difficult to determine the electric strange form factor in parity violating electron scattering: this component can affect the PV asymmetry by at wx most 20% at very small scattering angles 16 , while it is possible to measure Gs, as shown by the M SAMPLE experiment and more precise measurements are indeed under way. Thus one can exploit n s the sensitivity of R Rto Gprecisely to extract prnE the relevant information on the electric strange form factor. In order to illustrate this point, we present in Fig. Ž. 2a,b the ratio 4 , where the cross sections have been integrated in the interval 100 MeV FTF400 MeV N Žthe maximum reliable interval for which the n ratio . is fairly stable versus T. The ratio is displayed as a N function of m , fixing gss0 and gssy0.15 and sA A showing, around this last value, the ‘‘band’’ associated with a variation of r between y2 and q2. s This band is rather narrow in Fig. 2a, referring to the ratio measurable with neutrino beams, while it is larger in Fig. 2b, referring to the n case: yet, in this last instance, room enough is left to appreciate different values of gs. Concerning the sensitivity of the A integrated ratio to the magnetic strange form factor, one can see that the n case shows a perceptible slope with increasing m , whereas the n case appears to be s almost independent upon the value of m : this fact s 3The value r s2 is compatible with the vector strange form s wx factors employed in fit IV of Garvey et al. 15 in the analysis of Ž. nn –p cross sections. n n Ž. Ž. Fig. 2. The ratio Ra and Rb of the integrated NC prnprn neutrino–nucleus cross sections, as a function of m : all curves s are evaluated in the RFG. The incident energy is Es1 GeV n Ž n . and the integration limits for the cross sections are 100FT'T pn F400 MeV. The solid line corresponds to gss r s0, in the As wŽ. Ž.xs other three curves both in a and in b we have fixed gs A Ž. y0.15 and chosen r to be: r s0 dashed line , r sy2 ss s Ž.Ž. dot–dashed line and r sq2 dotted line . s favours the extraction of the electric strange form factor 4. We also recall that the most recent data on the electromagnetic form factors have shown a signifi4Fig. 2a shows that without strangeness R n ,0.73. Considprn wx ering Ref. 8 , only the dominant pure axial–vector contribution to the cross sections, one should expect this value to be 1. However, under the kinematical conditions considered here, also the pure vector and especially the vectorraxial interference contributions Ž can be important the pure axial term contributes only about 60% . to the n pand 45% to the n ncross sections , giving rise to the above deviation from 1. () W.M. Alberico et al.rPhysics Letters B 438 1998 9–13 13 cant deviation from the dipole behaviour at Q2G1 GeV2. We have investigated the sensitivity of the n Ž n . ratios R Rto different parameterizations of the prnwx Sach’s form factors 17,18 , and found that the effect of different forms for Gand Gdoes not exceed EM 1%2%. We have also investigated the sensitivity of the ratios considered here to the axial cutoff M. For A Ž. neutrinos the effect is small less than 3% but for antineutrinos a 6%7% variation in Mcan induce A n an effect as large as 7%8% in R. prn In conclusion we have focussed our analysis on n Ž n . the interplay, in the ratio R R, between axial, prn magnetic and electric strange form factors. The largest effect is associated with the axial strange form factor: the interplay between gsand m cruAs cially depends on their relative sign and turns out to n n act in opposite ways on R Rand R R. Moreover prnprn n we have found a strong sensitivity of R Rto the prn electric strange form factor. Thus, by assuming that PV electron scattering experiment will support a Ž. more precise than the present one determination of the magnetic strange form factor, the combined mean n surement of R Rand R Rcould allow a determiprnprn nation of all three strange form factors of the nucleon. References wx Ž. 1 D. Adams et al., Phys. Lett. B 329 1994 399. wx Ž. 2 K. Abe et al., Phys. Rev. Lett. 74 1995 346. wx Ž. 3 J. Ellis, M. Karliner, Phys. Lett. B 341 1995 397. wx 4 B. Mueller et al., SAMPLE Collaboration, Phys. Rev. Lett. Ž. 78 1997 3824. wx 5 W.M. Alberico, S.M. Bilenky, C. Giunti, C. Maieron, Z. fur ¨ Ž. Physik C 70 1996 463. wx 6 W.M. Alberico, M.B. Barbaro, S.M. Bilenky, J.A. Caballero, C. Giunti, C. Maieron, E. Moya de Guerra, J.M. Udias, Nucl. Ž. Phys. A 623 1997 471. wx 7 G.T. Garvey, S. Krewald, E. Kolbe, K. Langanke, Phys. Lett. Ž. B 289 1992 249. wx 8 G.T. Garvey, E. Kolbe, K. Langanke, S. Krewald, Phys. Rev. Ž. C 48 1993 1919. wx 9 M.B. Barbaro, A. De Pace, T.W. Donnelly, A. Molinari, M.J. Ž. Musolf, Phys. Rev. C 54 1996 1954. wx 10 E. Church et al., nucl-exr9706011. wx 11 J.M. Udıas, P. Sarriguren, E. Moya de Guerra, J.A. Ca- ´Ž. Ž. ballero, Phys. Rev. C 53 1996 R1488; C 48 1993 2731. wx Ž. 12 E. Kolbe, S. Krewald, H. Weigel, Z. fur Physik A 358 1997 ¨ 445. wx Ž. 13 See for example H. Forkel et al., Phys. Rev. C 50 1994 3108; H.C. Kim, T. Watabe, K. Goeke, Nucl. Phys. A 616 Ž. 1997 606; M. Kirchbach, D. Arenhovel, in Juelich 1994, in: ¨ Proceedings, Physics with GeV–particle beams, p. 414, hepphr9409293. wx Ž. 14 M.J. Musolf, B.R. Holstein, Phys. Lett. B 242 1990 461. wx 15 G.T. Garvey, W.C. Louis, D.H. White, Phys. Rev. C 48 Ž. 1993 761. wx 16 T.W. Donnelly, M.J. Musolf, W.M. Alberico, M.B. Barbaro, Ž. A. De Pace, A. Molinari, Nucl. Phys. A 541 1992 525. wx Ž. 17 K. Watanabe, H. Takahashi, Phys. Rev. D 51 1995 1423. wx Ž. 18 P.E. Bosted, Phys. Rev. C 51 1995 509.