Nuclear models and neutrino cross sections

0.12 MB·English

by  Giampaolo Co'

Nuclear models and neutrino cross sections
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0 Giampaolo Co’ a
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2 aDip. di Fisica Universit`a 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.

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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 in-
teraction 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.
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