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Nuclear models and neutrino cross sections 6 0 Giampaolo Co’ a 0 2 aDip. di Fisica Università di Lecce and INFN sez. di Lecce, I-73100 Lecce, Italy n Meritsand faults of theeffectivetheory RandomPhase Approximationsarediscussed in theperspectiveof its a J usein theprediction of neutrino-nucleuscross sections. 1 1 1 The Random Phase Approximation (RPA) is v 40.0 an effective theory aiming to describe the exci- 4 3 tation of many-body systems. In nuclear physics 30.0 (cid:10)16O(e,e’)16O(cid:10) (cid:10)ε i = 1 GeV θ=30 o(cid:10) 0 the RPA has been applied with success over a b](cid:10) n 1 wide range of excitation energies. In Fig. 1 we Ω [ 0 show inclusive (e,e’) and (ν,ν’) cross sections on ω d 20.0 06 16O target nucleus, as a function of the nuclear 2σ / d / excitationenergy. Inbothcasesthe incomingen- (cid:10)d 10.0 (cid:10)x 100(cid:10) h ergy of the lepton has been fixed at 1 GeV and t - the scattering angle at 30o. In the figure, three 0.0 l 100 101 102 103 c different excitation regions are emphasized. At 400.0 u few MeV of excitation energy there are discrete :n states, from 15 up to 30 MeV there is the giant 2cm](cid:10) 300.0 (cid:10)16O(ν,ν’)16O(cid:10) Xiv rqeusTaoshni-eaenlRacesPtsAiecxdpceeitsaacktr.iiobnes,atnhdeantuhculenadrreedxcoitfeMdesVtatthees - 42Ω [10 200.0 (cid:10) (cid:10) ar as a linear combination of particle-hole (ph) and ωd d hole-particle (hp) excitations σ / 100.0 2 d (cid:10) |Ψn >= (Xpha+pah+Ypha+hap)|Ψo > . (1) 0.0100 101 102 103 Xph (cid:10)ω [MeV](cid:10) Aim of the theory is the evaluation of the X ph Figure1. Electron(upperpanel)andneutrino(lower andY amplitudes for eachexcited state |Ψ >. ph n panel) cross sections. This is done by solving the secular equations A B X X (cid:18) B∗ A∗ (cid:19)(cid:18) Y (cid:19)=(En−Eo)(cid:18) −Y (cid:19) , wherethe coefficientsofthe matrixareexpressed intermsofsingleparticleenergiesandwavefunc- Single particle wave functions and energies are tions as: input of the theory. In our calculations they Aph,p′h′ =(ǫp−ǫh)δph,p′h′ + havebeengeneratedbyaWoods-Saxonpotential <ph′|Veff|hp′ >−<ph′|Veff|p′h>, whose parameters have been fixed to reproduce the energies of the levels close to the Fermi sur- and face and the rms charge radii [1]. Bph,p′h′ =<pp′|Veff|hh′ >−<pp′|Veff|h′h> . The other input of theory is the effective interaction Veff. This is not the vacuum nucleon- 1 2 (cid:10)12C(e,e’)12C(cid:10) 12C 12N 12B 10-2 LM1 17.2 20.2 14.3 (cid:10)LM1(cid:10) LM2 18.8 21.7 15.9 10-3 (cid:10)LM2(cid:10) (cid:10)PP(cid:10) PNPuInt05 1156..17 1189..064 1123..28 -1V](cid:10) 10-4 (cid:10)NuInt05(cid:10) e M exp 15.1 17.3 13.4 [T 10-5 R (cid:10) Table 1 10-6 Energies, in MeV, of the isospin triplet 1+ excited states referred to the12C ground state. 10-7 0.0 1.0 2.0 3.0 10-2 (cid:10)LM1 (0.46)(cid:10) 10-3 (cid:10)LM2 (0.60)(cid:10) (cid:10)PP (0.40)(cid:10) nucleon interaction, but it is an effective interac- -1V](cid:10) 10-4 (cid:10)NuInt05 (0.30)(cid:10) tion in the medium, behaving well at short in- e M ternucleon distances. We would like to point out [T 10-5 R the sensitivity of the various RPA results on the (cid:10) choice of the effective interaction. For this rea- 10-6 son we used various effective interactions which 10-7 have the same dignity from first principles point 0.0 1.0 2.0 3.0 (cid:10)q [fm -1](cid:10) of view. We used twozero-rangeinteractionswhose pa- Figure 2. Magnetic form factors of the 1+ state in rameters have been fixed to reproduce muonic 12Ccalculatedwiththevariousinteractionsandcom- atom polarization isotope shifts in the 208Pb re- pared with the experimental data. In the panel (b) gion (LM1) [2], and spin responses in 12C [3]. theresultsof(a)havebeenmultipliedbythequench- Wealsousedthepolarizationpotential(PP)[4] ing factors given in the figure in order to reproduce which is a finite-range interaction whose param- the data in thepeak. eters have been tuned to reproduce some prop- erties of nuclear matter. In addition we defined a new interaction of zero-range type (NuInt05) whose parameters have been fixed to reproduce excitation. This indicates the limits of the pre- theenergyoftheisovector1+statein12Cat15.11 diction power of the RPA theory. In the eval- MeV. uation of some observable, the use of effective, In Tab. 1 we compare the energies of the and phenomenological, nucleon-nucleon interac- isovector1+ statesinbothchargeconservingand tions cannot substitute the explicit treatment of charge exchange reactions with the experimental many-particle many-hole excitations. values. The two main issues we want to discuss Theabovediscussedissuesareshownevenbet- in the present report are already emerging from ter in Fig. 2 where the magnetic form factors of these results: the sensitivity of the RPA results the1+statein12C,calculatedwiththevariousin- to the residualinteractionand the need ofan ex- teractions, are comparedto the inelastic electron plicittreatmentofdegreesoffreedombeyondone- scattering data [5]. There is a large spreading of particle one-hole excitations. The uncertainty on the theoretical curves. A common feature is that the energies calculatedwith the differentinterac- all the results overestimate the data. This is a tionsisofabout2MeV.EventhoughtheNuInt05 well known problem of the RPA in the descrip- interaction reproduces the excitation energy of tion of the magnetic form factors of medium and the charge conserving 1+ state, it is unable to heavynuclei[6,7]. Ithasbeenarguedthattheex- reproduce the energies of the charge exchange plicit inclusion of many-particles many-holes ex- 3 4.00 300.0 (cid:10)(ν,ν’)(cid:10) (cid:10)(e,e ’) (cid:10) 2m](cid:10) 200.0 V sr](cid:10)3.00 -4210 c 2m / Me2.00 σ(cid:10) [ 100.0 -610 f1.00 (cid:10)(cid:10) (cid:10)[ 0.0 20 40 60 80 100 0.00 0 50 100 150 200 250 300 400.0 600 ](cid:10) 300.0 (cid:10)(ν,e+)(cid:10) -4220 cm 200.0 MeV sr](cid:10) 450000 (cid:10)(ν,ν ’)(cid:10) σ [1 100.0 2m / 300 (cid:10) c -42 200 0.020 40 60 80 100 (cid:10)[10 100 600.0 0 0 50 100 150 200 250 300 (cid:10)(ν,e -)(cid:10) ](cid:10) (cid:10)ω [MeV](cid:10) 2m 400.0 c -420 Fquigausir-eela4s.tiEclercetgrioonn.anTdhneeuentreirngoycorofstshseeclteipotnosnisnitsh1e 1 σ [ 200.0 GeVandthescatteringangle30o. Thefulllineshave (cid:10) been obtained with RPA calculations using the LM1 0.0 interaction, thedottedlineswith thePPinteraction. 20 40 60 80 100 The dashed lines show the mean-field results. (cid:10)(cid:10)εεii [[MMeeVV]](cid:10)(cid:10) Figure 3. Total neutrino cross sections exciting the 1+ states of Tab. 1 as a function of the neutrino energy. RPA results, while the lower curves, the thicker ones,havebeenobtainedbymultiplyingtheRPA resultswiththequenchingfactorsfixedinFig. 2. While in the electron excitation the use of citations can solve the problem [8]. In a very quenching factors reduced the spreading of the crude and phenomenological approach the vari- results, in the neutrino case this spreading has ous curves are multiplied by a factor to repro- increased. This is a further indication of the ducethedatainthemaximumoftheformfactor. fact that electrons and neutrinos excite the same Thevaluesofthesequenchingfactorsaregivenin states in different manners. In the case of elec- the panel (b) of Fig. 2, where the renormalized tronstheexcitationisinducedbyavectorcurrent results are shown. Obviously, the spreading be- while neutrinos excitations are dominated by the tween the various curves is reduced. axial vector current [1]. The consequences of these theoretical uncer- The roleofthe effectiveinteractionsandofthe tainties on the neutrino cross sections are pre- many-particles many-holes degrees of freedom in sented in Fig. 3 where the total neutrino cross the giant resonance region, has been thoroughly sections for the excitation of the three 1+ states investigatedin[1]andwereportherethemainre- ofTab. 1are shownas a function ofthe neutrino sults. The uncertainty on the total cross section energies. The thinner upper lines are the bare is large for neutrinos of 20-40 MeV. These un- 4 used. In the quasi-elastic peak the the probe can (cid:10)16O(cid:10) (cid:10) 12C(cid:10) resolve distances of about 0.5 fm, therefore zero- 100 40 80 (cid:10)εi=(cid:10)θ7=0302 M o(cid:10)eV(cid:10) 30 (cid:10)εi=(cid:10)θ6=8306 M o(cid:10)eV(cid:10) ranOguerincatelcrualcattioionnssasrheonwotthraetl,iaibnleth.equasi-elastic 60 20 region, the RPA effects are rather small. How- 40 ever, many-body effects beyond RPA are not 10 20 negligible, as it is shown in Fig. 5 where elec- 0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 tron scattering cross sections calculated within V sr](cid:10) 50 15 the mean-field model are compared to data. In e nb / M 3400 (cid:10)εi=(cid:10)θ8=8302 M o(cid:10)eV(cid:10) 10 (cid:10)εi=(cid:10)θ4=4600 M o(cid:10)eV(cid:10) tbhoedyqueaffsei-cetlsa,swtichircehgiionnt,hteheRsePAcomlapnlgiucaatgeedamreandye-- ω [ 20 scribedasmany-particlesmany-holesexcitations, d 5 Ω are usually called Final State Interactions (FSI). d 10 σ / 0 0 OurtreatmentoftheFSIassumesthattheydo 2 0 50 100 150 200 250 300 0 50 100 150 200 250 300 not dependent on the angular momentum ad the d (cid:10) 20 3.0 parity of the excitation. Under this assumption 16 (cid:10)εi=(cid:10)1θ0=8302 Mo(cid:10)eV(cid:10) 22..05 (cid:10)εi=(cid:10)θ6=8600 M o(cid:10)eV(cid:10) it is possible to correct RPA, or mean-field, re- 12 sponses for the presence of FSI by folding them 1.5 8 with a Lorentz function [11]. The parameters of 1.0 4 0.5 this function are fixed by hadron scattering data 0 0.0 [12]. We have shown that the inclusion of the 0 50 100 150 200 250 300 0 50 100 150 200 250 300 (cid:10)ω [MeV](cid:10) (cid:10)ω [MeV](cid:10) FSI reduces the quasi-elastic total neutrino cross sections by a 10-15%factor [13]. Insummary,theeffectivetheoryRPAallowsus Figure 5. Electron scattering cross sections in the to investigate spectroscopic and dynamical prop- quasi-elasticregion. Themean-fieldresultsareshown bythedashedlines. TheinclusionoftheFSIproduces erties of the nuclear excitations. RPA calcula- the full lines. tions are necessary to produce giant resonances and collective low-lying states. The RPA results are however strongly dependent on the effective nucleon-nucleon interaction used. Interactions certainties have heavy consequences on the cross equivalent from the spectroscopic point of view, sections of low energy neutrinos such as super- canhoweverproduceverydifferentexcitedstates. nova neutrinos and neutrinos coming from muon The interactionswe haveusedgiverathersimilar decayatrest. Whentheneutrinoenergyisabove results for charge conserving natural parity exci- the 50 MeV, the results are rather independent tations. The situation for unnatural parity and fromthenucleon-nucleoninteractionandindicate charge-exchangeexcitation is quite uncertain. In that the inclusion of many-particles many-holes these cases it emerges the necessity of including excitations reduces the RPA cross sections by a degrees of freedom beyond the RPA assumptions 10-15% factor. since their effects cannot be simulated by read- We show in Fig. 4 an example of RPA ef- justingtheparametersoftheeffectiveinteraction. fects in the quasi-elastic region. Full and dotted For neutrinos of energy smaller than 100 MeV lines havebeencalculatedinthe continuumRPA the neutrino-nucleus cross sections are still very framework by using the LM1 and the PP inter- model dependent. action respectively. These results are compared In the quasi-elastic region, which is dominated to those obtained with a mean-field model, i.e. by single particle dynamics, our results indicate by setting Veff to zero. The use of zero-range that RPA effects are not relevant. However the interaction overestimates the RPA effects. They comparison with electron scattering data shows are negligible when a finite-range interaction is the need of considering FSI, whose effects low- 5 ers the total neutrino cross sections by a 10-15% factor. REFERENCES 1. A. Botrugno, G. Co’, Nucl. Phys. A 761 (2005) 200. 2. G.Rinker,J.Speth,Nucl.Phys.A306(1978) 360. 3. F.T.Baker,etal.,Phys.Rep.289(1997)235. 4. D. Pines, K.Q. Quader, J. Wambach, Nucl. Phys. A 477 (1988) 365. 5. T.W. 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