Cross Sections for Neutral-Current Neutrino-Nucleus Interactions: Applications for 12C and 16O

PHYSICAL REVIEW C VOLUME 59, NUMBER 6 JUNE 1999

Cross sections for neutral-current neutrino-nucleus interactions: Applications for 12C and 16O

N. Jachowicz,* S. Rombouts, K. Heyde,† and J. Ryckebusch Institute for Theoretical Physics, Vakgroep Subatomaire en Stralingsfysica, Proeftuinstraat 86, B-9000 Gent, Belgium ͑Received 29 September 1998; revised manuscipt received 4 February 1999͒ We study quasielastic neutrino-nucleus scattering on 12C and 16O within a continuum random phase approximation ͑RPA͒ model. The RPA equations are solved using a Green’s function approach with an effective Skyrme ͑SkE2͒ two-body interaction. Besides a comparison with existing calculations, we carry out a detailed study using various methods ͑Tamm-Dancoff approximation and RPA͒ and different forces ͑SkE2 and Landau- Migdal͒. We evaluate cross sections that may be relevant for neutrino nucleosynthesis. ͓S0556-2813͑99͒01606-4͔ PACS number͑s͒: 25.30.Pt, 24.10.Eq, 26.30.ϩk

I. INTRODUCTION function approach in which the polarization propagator is approximated by an iteration of the first-order contribution Neutrinos are extremely well-suited probes to provide de- ͓14͔. The unperturbed wave functions are generated using tailed information about the structure and properties of the either a Woods-Saxon potential or a HF calculation using a weak interaction, as they are only interacting via the weak Skyrme force. The latter approach makes self-consistent HF- forces. Moreover, being intrinsically polarized and coupling RPA calculations possible. Calculations using either a to the axial vector as well as to the vector part of the had- Skyrme or a Landau-Migdal force must give indications ronic current, neutrinos are able to reveal other and more about possible differences and sensitivity in methods to the precise nuclear structure information than, e.g., electrons do. residual two-body interaction used. Finally, we discuss ap- The most important problem in extracting information from plications for neutral-current reactions on 16O and 12C, two neutrino-nucleus scattering experiments remains the very nuclei that are very important from an astrophysical point of small interaction cross sections. view. In Sec. VI, we discuss some aspects of neutrino nu- These restrictions become rather unimportant when con- cleosynthesis and in Sec. VII, the stability of the RPA and sidering astrophysical processes. The amount of neutrinos Tamm-Dancoff approximation ͑TDA͒ methods, using differ- produced at the end of the lifetime of a massive star during ent two-body forces in calculating the neutrino scattering the neutronization of the collapsing core of a star and its cross section, is discussed. subsequent cooling, is as large as ϳ1058, representing approximately 99% of the total released energy ͓1,2͔. These II. QUASIELASTIC NEUTRINO SCATTERING supernova neutrinos will play an important role in explosive nucleosynthesis processes, causing a considerable transfor- In the neutral-current neutrino scattering reactions, a neu- ជ mation of the material synthesized during the hydrostatic trino with four-momentum (⑀i ,ki) is scattered inelastically burning phases in the life of the star. from a nucleus with initial energy and momentum Ei ,Pជ i . During the last few years, a large number of quasielastic The nucleus is supposed to be spherically symmetric and in neutrino-nucleus scattering studies have been carried out, in- its ground state, J␲iϭ0ϩ. After the reaction, the nucleus is cluding shell model ͓3͔, relativistic fermi gas ͓3,4͔, and ran- i left in an excited state with final parity and angular momen- dom phase approximation ͑RPA͒ calculations using various tum J ,␲ . The energy transferred from neutrino to nucleus forces ͓4–8͔. Since both the LSND and the KARMEN Col- f f equals ␻ϭ⑀ Ϫ⑀ ϭE ϪE ,⑀ and E being the final neu- laborations have been measuring neutrino-nucleus scattering i f f i f f trino and nuclear energy respectively. The transferred mo- cross sections on 12C ͓9,10͔, special attention was paid to the 12 Ϫ mentum is denoted by qជ ϭkជ iϪkជ f , ␬ϭ͉qជ͉. calculation of cross sections for the reactions C(␯e ,e )X 12 Ϫ As the interaction between the neutrino and a nucleus is and C(␯␮ ,␮ )X. Despite the theoretical and experimental efforts, no clear agreement between theory and experiment, mediated by the weak interaction, the cross section can be and between the different theoretical results for derived using Fermi’s golden rule. In Born approximation, 12 Ϫ 12 the outgoing neutrino is described by a plane wave. Using the C(␯␮ ,␮ ) X reaction could be reached ͓8,9,11–13͔. Next to these problems, we observe that relatively few natural units (បϭcϭ1), the differential cross section is then neutral-current calculations have been performed ͓3,5͔. given by In the present work, we examine quasielastic neutral- d2␴ 1 current neutrino-nucleus reactions in an energy region rel- ϭ͑2␲͒4k ␧ ͉͗ f͉Hˆ ͉i͉͘2. ͑1͒ d⍀d␻ f f ͚ 2J ϩ1 ͚ W evant to explosive nucleosynthesis processes, using a con- s f ,si i M f ,M i tinuum random phase approximation ͑CRPA͒ formalism ˆ ͑Sec. III͒. The CRPA equations are solved using a Green’s The weak interaction Hamiltonian HW has the current- current structure G *Corresponding author. ˆ ˆ ˆ ␮,hadronic HWϭ dxជ j␮,leptonic͑xជ ͒J ͑xជ ͒, ͑2͒ †Present address: CERN CH-1211, Geneva 23, Switzerland. ͱ2 ͵

Ϫ11 Ϫ2 with Gϭ1.16639ϫ10 MeV the weak coupling con- ˆ and ˆ denote the Coulomb and longitudinal opera- stant. The neutral leptonic current reads MJM LJM ˆ el ˆ mag tor respectively, whereas JM and JM are the transverse ˆj0 ͑x͒ϭ⌿¯ ͑x͒␥ ͑1Ϫ␥ ͒⌿ ͑x͒ J J ␮ ␯e ␮ 5 ␯e electric and magnetic operators:

ϩ⌿¯ ͑x͒␥ ͑1Ϫ␥ ͒⌿ ͑x͒ ␯␮ ␮ 5 ␯␮ ¯ ˆ ͑␬͒ϭ dxជ͓j ͑␬r͒YM͑⍀ ͔͒Jˆ ͑xជ͒, ϩ⌿␯ ͑x͒␥␮͑1Ϫ␥5͒⌿␯ ͑x͒. ͑3͒ JM ͵ J J x o ␶ ␶ M

The weak neutral hadronic current is given by ͓15͔

ˆ 0 ˆ vector ˆ axial vector i M ˆ ,J␮ϭJ␮ ϩJ␮ ˆ ͑␬͒ϭ dxជٌ͕ជ ͓j ͑␬r͒Y ͑⍀ ͔͖͒Jជ͑xជ͒ LJM ␬͵ J J x 2 ˆ V3 2 ˆ S ˆ A3 ϭ͑1Ϫ2sin ␪W͒J␮ Ϫ2sin ␪WJ␮ϩJ␮ 1 ¯ V 2 V 2 ␯ 1 ˆ ϭ⌿N GE͑q ͒␥␮ϩ GM͑q ͒␴␮␯q ˆ el ជ ជ ជ M ជ ជ ,2M JM͑␬͒ϭ dxٌ͕ϫ͓jJ͑␬r͒ J,J͑⍀x͔͖͒J͑x͒ ͭ J ␬ ͵ Y

A 2 ϩG ͑q ͒␥␮␥5ͮ . ͑4͒ ˆ ˆ mag͑␬͒ϭ dxជ͓j ͑␬r͒ជ M ͑⍀ ͔͒ Jជ͑xជ͒. ͑9͒ 2 JM ͵ J J,J x • Here sin ␪Wϭ0.2325, where ␪W is the Weinberg angle. For J Y the isospin operators, the convention ␶3͉p͘ϭϩ1,␶3͉n͘ϭ ˆ V3 Ϫ1 is adopted. J␮ denotes the ␶3 component of the electro- S III. THE CONTINUUM RANDOM PHASE magnetic current and J␮ is its scalar counterpart. The vector form factors can be obtained directly from those in electro- APPROXIMATION magnetic interactions applying the CVC theorem: The transition densities necessary to calculate the cross 1 V 2 2 section ͑8͒ are determined within a continuum random phase GEϭ ͑1Ϫ2sin ␪W͒␶3Ϫsin ␪W , ͑5͒ 2 approximation. The unperturbed single-particle wave- functions are generated by a Hartree-Fock calculation, using 1 V 2 2 GMϭ ͑1Ϫ2sin ␪W͒͑␮ pϪ␮n͒␶3Ϫsin ␪W͑␮ pϩ␮n͒. a Skyrme potential ͓16–19͔. The CRPA formalism is based 2 on a Green’s function approach ͓14,16,20–22͔. The RPA 6 ͑ ͒ polarization propagator is obtained by the iteration to all orders of the ﬁrst-order contribution to the particle-hole GA is given by Green’s function A G ϭϪga␶3ϭϪ1.262 ␶3 . ͑7͒

(RPA) (0) For low momentum transfers (␬Ͻ400 MeV/c), only ⌸ ͑x1 ,x2 ;␻͒ϭ⌸ ͑x1 ,x2 ;␻͒ lowest-order contributions to the hadronic current have to be retained ͓15͔. The differential cross section for neutral- 1 ϩ dx dxЈ⌸(0)͑x ,x;␻͒ current neutrino scattering ﬁnally becomes ប͵ ͵ 1 ␪ 2 cos2 ϫ˜V͑x,xЈ͒⌸(RPA)͑xЈ,x ;␻͒, ͑10͒ d2␴ G2␧2 ͩ 2ͪ ϱ ϱ 2 i f f → J J ϭ ͚ ␴CLϩ͚ ␴T , ͑8͒ ͩ d⍀d␻ ͪ ␯ ␲ 2Jiϩ1 ͫ Jϭ0 Jϭ1 ͬ ¯␯ where ⌸(0) denotes the zeroth-order contribution to the po- where larization propagator. ˜V is the antisymmetrized form of the 2 residual interaction which in the following will be supposed ␻ ˜ J ˆ ˆ to be rotationally invariant, allowing us to write V(x1 ,x2)as ␴CLϭ Jf J͑␬͒ϩ J͑␬͒ Ji , ͯͳ ͯͯM ͉qជ͉L ͯͯ ʹͯ

q2 ␪ ˜ J JM† ˆ JM ˆ J ␮ 2 ˆ mag 2 V͑x1,x2͒ϭ ͚ U␣␤͑r1 ,r2͒X␣ ͑x1͒X␤ ͑x2͒, ͑11͒ ␴Tϭ Ϫ ϩtan ͓͉͗J f͉͉ J ͑␬͉͉͒Ji͉͘ ␣␤,JM ͩ 2͉qជ͉2 ͩ 2ͪ ͪ J

2 ␪ q␮ ␪ ϩ J ˆ el͑␬͒ J 2͔ϯtan Ϫ ϩtan2 where the XJM(xˆ) represent spherical tensor operators of ͉͗ f͉͉ J ͉͉ i͉͘ 2 J ͩ 2ͪͱ ͉qជ͉ ͩ 2ͪ rank J,M. A solution to the RPA Bethe-Salpeter equation ͑10͒ is ϫ ͓2Re͑͗J ͉͉ ˆ mag͑␬͉͉͒J ͗͘J ͉͉ ˆ el͑␬͉͉͒J ͘*͔͒. f J J i f J J i given by the set of wave functions implicitly deﬁned by 3248 JACHOWICZ, ROMBOUTS, HEYDE, AND RYCKEBUSCH PRC 59

␺ ͑x ͒␺† ͑x ,⑀ ͒ Ϫ1 ˜ hЈ 1 pЈ 1 pЈ Ϫ1 ͉⌿C͑E͒͘ϭ͉ph ͑E͒͘ϩ͵ dx1 ͵ dx2 V͑x1 ,x2͚͒ ͵ d⑀pЈ ͉pЈhЈ ͑⑀pЈhЈ͒͘ c P ͫ EϪ⑀ Ј pЈhЈ ␺† ͑x ͒␺ ͑x ,⑀ ͒ hЈ 1 pЈ 1 pЈ Ϫ1 ˆ † ˆ Ϫ ͉hЈpЈ ͑Ϫ⑀p h ͒͘ ͗⌿0͉␺ ͑x2͒␺͑x2͉͒⌿C͑E͒͘, ͑12͒ Eϩ⑀ Ј Ј ͬ pЈhЈ

where denotes the Cauchy principal value, and the Ϫ1 P Ϫ1 dr dr R(0) r,r ;E ͉ph (E)͘ and ͉hp (ϪE)͘ are the unperturbed particle- ͵ ͵ Ј ␩␮;JM͑ Ј ͒ hole and backward hole-particle solutions of the mean-field problem. The summation index C stands for all quantum 1 ϭ dx dxЈ X ͑x͒ ⌸(0)͑x,xЈ;␻͒X† ͑xЈ͒, numbers defining a reaction channel unambiguously: C ប͵ ͵ ␩JM ␮JM ϭ͕nh ,lh , jh ,mh ,⑀h ;l p , j p ,m p ,␶z͖. The first term in expres- sion ͑12͒ corresponds to the Tamm-Dancoff approximation ͑15͒ and the second one represents the negative energy RPA con- the RPA transition densities are determined by the set of tribution. From the energy dependence of the denominators, coupled integral equations it is clear that the backward RPA contribution becomes only important for states whose energy eigenvalue deviates sub- ͗⌿0͉͉X␩J͉͉⌿C͑J;E͒͘r stantially from the unperturbed value ⑀ ph . This makes RPA a well-suited tool for describing collective excitations in nu- ϭϪ͗h͉͉X␩J͉͉p͑⑀ph͒͘r clei. From the wave functions ͑12͒ the correctly normalized J (0) ϩ dr1 dr2 U␮␯͑r1 ,r2͒ Re͑R␩␮;J͑r,r1 ;E͒͒ solutions of the scattering problem can be obtained by taking ͚␮,␯ ͵ ͵ suitable linear combinations. Defining the K matrix by ϫ ⌿ X ⌿ ͑J;E͒ . ͑16͒ ͗ 0͉͉ ␯J͉͉ C ͘r2 J Ϫ1 J KC,C ϭ dr1 dr2 U␣␤͑r1 ,r2͒ Discretizing these equations on a mesh in the radial coordi- Ј 2Jϩ1 ͚␣,␤ ͵ ͵ nate, the transition densities for each reaction channel ͑16͒ ϫ͗hЈ͉͉XJ ͑x ͉͉͒pЈ͑Eϩ⑀ ͒͘* are obtained as the solutions of the matrix equation ␣ 1 hЈ r1 1 ϫ͗⌿ ͉͉XJ ͑x ͉͉͒⌿ ͑J;E͒͘ , ͑13͒ ␳RPAϭϪ ␳HF . ͑17͒ 0 ␤ 2 C r2 C 1ϪRU C where the subscript r of the transition densities denotes that Here ␳RPA and ␳HF represent column vectors containing the all coordinates except the radial one have been integrated, it RPA and the Hartree-Fock transition densities for all in- can be shown that the wave functions constructed by putting cluded interaction channels ␩ and for a number of mesh ϩ points in coordinate space. R and U are block matrices con- ͉⌿C ͑JM;E͒͘ taining the unperturbed response function ͑15͒ and the radial part of the interaction ͑11͒, evaluated at the appropriate chan- ϭ 1ϩi KJ Ϫ1 JM;E 14 ͚ ͓ ␲ ͔C,CЈ͉⌿CЈ͑ ͒͘ ͑ ͒ nels and r values. The discretization in coordinate space is CopenЈ : ⑀pϭ⑀hϩEϾ0 well under control. It does not demand large numbers of mesh points for the calculated transition densities to become contain asymptotically only one incoming wave. They allow mesh independent, thus keeping the dimension of matrix in- us to describe systems where one particle is excited to an versions to be performed sufficiently small. From the form of unbound state with ⑀ pϾ0, and is able to escape from the Eq. ͑17͒ it is clear that the wave functions ͑12͒ can be con- nuclear potential. Furthermore they obey the same normal- sidered as the solution to the RPA equivalent of the ization conditions as the unperturbed ͉phϪ1͘ wave functions. Lippmann-Schwinger integral scattering equations. In Eq. From the above, if follows that the wave functions ͑14͒ are ͑17͒ as well as in Eq. ͑16͒, the minus sign arises from the suited to evaluate the transition densities in the cross section phase convention adopted in the definition of the K matrix ͑8͒. ͑13͒. This formalism has the interesting feature that treating Defining the unperturbed radial response functions as the RPA equations in coordinate space allows us to deal with

TABLE I. Parameter set for the SkE2-interaction.

t0 t1 t2 t3 t4 x0 W0 x3 Ϫ1299.30 802.41 Ϫ67.89 19558.96 Ϫ15808.79 0.270 120 0.43 MeV fm3 MeV fm5 MeV fm5 MeV fm6 MeV fm8 MeV fm5 PRC 59 CROSS SECTIONS FOR NEUTRAL-CURRENT . . . 3249

FIG. 1. Cross section for the reaction 16O ϩ␯ 16O*ϩ␯Ј ͑full line͒ and its domi- 50 MeV→ nant multipole contributions. J␲ϭ1Ϫ ͑dashed line͒, J␲ϭ1ϩ ͑small dashes below͒, J␲ϭ2Ϫ ͑dotted line͒, and J␲ϭ0Ϫ ͑dashed-dotted͒. The total cross section includes multipoles up to J ϭ4.

the energy continuum in an exact way, without cutoff or hole channels and this over the whole mass table. This is discretization of the excitation energies ͓14͔. done by replacing part of the three-particle contribution to the original Skyrme versions by a momentum dependent two-particle term. The extra free parameter thus obtained IV. THE SkE2 INTERACTION is used to guarantee correct two-body characteristics in The Hartree-Fock and RPA calculations were performed nuclei containing few valence nucleons outside of the closed with an extended Skyrme force. The parameter values used shells. Furthermore, the SkE2 parameter set allows a good to obtain the presented results are those of the SkE2 param- reproduction of the experimental single-particle energies etrization ͓16–19͔. This parameter set was designed to yield ͓16–19͔. a realistic description of nuclear structure properties in both In coordinate space, the antisymmetrized residual interac- the particle-particle ͑pairing properties͒ and in the particle- tion takes the form

1 1 ͒ V͑rជ ,rជ ͒ϭt ͑1ϩx Pˆ ͒␦͑rជ Ϫrជ ͒Ϫ t ٌ͓͑ឈ Ϫٌឈ ͒2␦͑rជ Ϫrជ ͒ϩ␦͑rជ Ϫrជ ٌ͒͑ជ Ϫٌជ ͒2͔ϩ t ٌ͑ឈ Ϫٌឈ ͒␦͑rជ Ϫrជ 1 2 0 0 ␴ 1 2 8 1 1 2 1 2 1 2 1 2 4 2 1 2 1 2

e2 1 ͒ ˆϫٌ͑ជ Ϫٌជ ͒ϩ ϩiW ͑␴ជ ϩ␴ជ ٌ͒͑ឈ Ϫٌឈ ͒ϫ␦͑rជ Ϫrជ ٌ͒͑ជ Ϫٌជ ͒ϩ t ͑1Ϫx ͒͑1ϩP 1 2 0 1 2 1 2 1 2 1 2 6 3 3 ␴ ͉rជ1Ϫrជ2͉

rជ1ϩrជ2 1 ϫ␳ ␦͑rជ Ϫrជ ͒ϩx t ␦͑rជ Ϫrជ ͒␦͑rជ Ϫrជ ͒Ϫ t ٌ͕͓͑ឈ Ϫٌឈ ͒2ϩٌ͑ឈ Ϫٌឈ ͒2ϩٌ͑ឈ Ϫٌឈ ͒2͔ ͩ 2 ͪ 1 2 3 3 1 2 1 3 24 4 1 2 2 3 3 1

2 2 2 ϫ␦͑rជ1Ϫrជ2͒␦͑rជ1Ϫrជ3͒ϩ␦͑rជ1Ϫrជ2͒␦͑rជ1Ϫrជ3ٌ͓͒͑ជ 1Ϫٌជ 2͒ ϩٌ͑ជ 2Ϫٌជ 3͒ ϩٌ͑ជ 3Ϫٌជ 1͒ ͔͖, ͑18͒

with P␴ the spin exchange operator. Table I illustrates the difﬁculties, makes it a good test for the reliability of the parameter values for the SkE2 set. As the same interaction formalism. Therefore, the study of neutrino-nucleus interac- with the same parameter values is adopted for the calculation tions with the CRPA formalism was started with cross- of the unperturbed as well as the RPA wave functions, the section calculations for the neutral-current reaction formalism is self-consistent with respect to the residual in- 16Oϩ␯ 16O*ϩ␯Ј. ͑19͒ teraction used. →

V. APPLICATIONS TO 16O AND 12C In all of the following results, calculations were per-

16 formed with an incoming neutrino energy ⑀iϭ50 MeV. A. The nucleus O Multipoles up to Jϭ4 were taken into account. Contribu- As one of the major products of the thermonuclear burn- tions of higher-order multipole excitations were found to be ing processes in massive stars, 16O plays an important role in negligibly small. The differential neutrino scattering cross supernova nucleosynthesis ͓1,23͔. Moreover, having closed sections are of the order of 10Ϫ42 cm2 per MeV. In Fig. 1, proton and neutron shells, the lack of major nuclear structure we show the total cross section and some important multi- 3250 JACHOWICZ, ROMBOUTS, HEYDE, AND RYCKEBUSCH PRC 59

FIG. 2. Comparison between the vector ͑dot- ted͒ and the axial vector ͑dashed line͒ contribution to the reaction 16O(␯,␯Ј)16O*.

pole contributions. At excitation energies between 20 and 25 tations will dominate isoscalar ones. Isoscalar excitations are MeV, the broad resonance structure of the giant dipole reso- even further suppressed: not only is the axial vector current 2 nance shows up. At energies below 20 MeV, smaller peaks, not contributing to isoscalar transitions, but due to the sin ␪W related to excitations with a stronger single-particle charac- factor, the isoscalar form factors are considerably smaller ter, are present. In RPA results, resonances are pushed to than the vector ones as well. somewhat higher energies than those in mean-ﬁeld calcula- In Fig. 3 we then compare the contribution of the different tions due to the repulsive character of the residual interaction operators to the cross section. Transverse transitions are in the isovector channels. For excitation energies above clearly prominent. The difference between the sum of trans- ϳ30 MeV, the cross section decreases almost purely expo- verse and CL terms and the total cross section is again due to nentially, according to the energy dependence of Eq. ͑8͒. The the transverse interference term. According to Eq. ͑8͒ it is Jϭ1 excitations are clearly prominent. Jϭ0 excitations are this interference contribution that is responsible for the dif- suppressed due to the fact that only Coulomb and longitudi- ference in the nuclear response to neutrino and antineutrino nal terms contribute to these channels. But still, some clear perturbations. The sign of the interference term determines 0Ϫ resonances show up in the differential cross sections. which cross section will be dominant. Generally neutrino In Fig. 2, we make a comparison between the contribution cross sections are slightly larger than antineutrino cross sec- of the vector and the axial vector part of the hadronic current tions. Only at around 23 MeV does the interference term to the total cross section. The axial vector current is clearly change sign and antineutrino excitations become more im- more sensitive to the weak neutrino probes. The vector con- portant. tribution is suppressed by almost two orders of magnitude. Our self-consistent calculations substantiate the results of The splitting of the cross section into a vector and axial Kolbe et al. ͓5,6͔ on neutral-current reactions. Contrary to vector part excludes the interference contribution. This ex- the present work, there the CRPA equations are solved using plains the discrepancy between the sum of both curves in the techniques proposed by Ref. ͓24͔. The standard RPA Fig. 2 and the total cross section. Due to the fact that the equations are extended to the continuum by the introduction axial vector current is completely isovector, isovector exci- of an integration over the excitation energies. The resulting

FIG. 3. Coulomb ͑dotted line͒, longitudinal ͑dashed-dotted͒, transverse electric ͑short dashed͒, and transverse magnetic ͑dashed͒ contributions to the reaction 16O(␯,␯Ј)16O*. The full line gives the total differential cross section. PRC 59 CROSS SECTIONS FOR NEUTRAL-CURRENT . . . 3251

FIG. 4. Cross section for the reaction 12C ϩ␯ 12C*ϩ␯Ј ͑full line͒ and its domi- 50 MeV→ nant multipole contributions. J␲ϭ1Ϫ ͑dashed line͒, J␲ϭ1ϩ ͑small dashes below͒, J␲ϭ0Ϫ ͑dotted line͒, and J␲ϭ2Ϫ ͑dashed-dotted͒. The total cross section includes multipoles up to J ϭ4.

integrodifferential equations are then solved by an expansion single-particle energy levels. The Woods-Saxon parameters in Weinberg states, already having the correct asymptotic were taken from Ref. ͓25͔. Thereby, the full self-consistency behavior and thus making a fast convergence possible. As is spoiled but it is clearly an approach that allows us to take residual interaction an effective force derived from the Bonn into account some effects of the deformed ground state, al- meson exchange potential is used. The single-particle wave beit in a semiphenomenological way. functions are generated using a Woods-Saxon potential. A Cross sections for neutral-current neutrino scattering on comparison of results obtained with the different methods 12C show mainly the same behavior as those for the makes clear that total cross sections as well as the cross 16 16 * ␲ reaction O(␯,␯Ј) O . In Fig. 4 we show the differential section for transitions to speciﬁc J ﬁnal states show remark- cross section and the principal multipole contributions J␲ able agreement in overall strength and in the position of the ϭ1ϩ,1Ϫ, and 2 Ϫ. Again axial vector and isovector excita- resonances. Moreover, also the more detailed conclusions tions are prominent. Coulomb and longitudinal contributions concerning relative strength of vector and axial vector, neu- are suppressed. Neutrino cross sections dominate an- trino and antineutrino, isovector and isoscalar, and transverse 16 and Coulomb-longitudinal contributions are in excellent tineutrino excitations. As was the case for O, our results agreement. are in good agreement with those of Kolbe et al. ͓5,6͔.

VI. NEUTRINO NUCLEOSYNTHESIS B. The nucleus 12C As 12C is known to be deformed in its ground state, As neutrinos only participate in weak interaction pro- whereas the Hartree-Fock calculation assumes the nucleus to cesses and cross sections for scattering reactions involving have a spherical ground state, the single-particle wave func- neutrinos are very small, the importance of neutrinos to as- tions for 12C were determined in a different way: the unper- trophysical processes has long been underestimated. How- turbed wave functions were generated with a Woods-Saxon ever, models describing the explosion mechanism of type II potential, yielding a more reliable splitting between the supernovae provide an important role for neutrinos in these

FIG. 5. Differential cross section for the 16 16 reaction Oϩ␯FD O*ϩ␯Ј, averaged over neutrinos and antineutrinos→ and over a Fermi- Dirac distribution with temperature T. T ϭ12 MeV ͑full line͒, Tϭ10 MeV ͑dashed͒, T ϭ8 MeV ͑short dashed͒, Tϭ6 MeV ͑dotted͒, and Tϭ4 MeV ͑dashed-dotted͒. 3252 JACHOWICZ, ROMBOUTS, HEYDE, AND RYCKEBUSCH PRC 59

TABLE II. Cross section per nucleon for the reaction 16O TABLE III. Cross section per nucleon for the reaction 12C 16 12 ϩ␯FD O*ϩ␯Ј, averaged over neutrinos and antineutrinos and ϩ␯FD C*ϩ␯Ј, averaged over neutrinos and antineutrinos and over a→ Fermi-Dirac distribution with temperature T. over a→ Fermi-Dirac distribution with temperature T.

T ͑MeV͒ 4681012 T ͑MeV͒ 4681012

MF 0.0033 0.045 0.22 0.71 1.7 MF 0.0018 0.024 0.12 0.41 1.0 16O d␴ 12C d␴ (10Ϫ42cm2) (10Ϫ42cm2) d␻ d␻ CRPA 0.0056 0.070 0.32 1.0 2.2 CRPA 0.0016 0.026 0.14 0.45 1.1 processes ͓1,2͔. The cooling of a newly formed neutron star study are in good agreement with those of Ref. ͓5͔ and ͓1͔ by the production and subsequent emission of neutrino pairs and thus substantiate the reliability ͑cross section magnitude causes a ﬂux of ϳ1058 neutrinos, representing an energy of and overall shape͒ of CRPA calculations determining 1053 erg. Although the neutrinos are only weakly interacting, neutrino-nucleus scattering processes. this enormous amount of particles and energy traveling The sensitivity of the cross sections to the temperature is through the different layers of the star is able to cause a due to the strong energy dependence of the nuclear response considerable transformation of the elements synthesized dur- in the considered energy region. The temperatures studied ing the thermonuclear burning processes in the life of the correspond to average neutrino energies between 12 and 38 star. MeV. The large differences are then easily explained by not- The energy distribution of the supernova neutrinos is de- ing that the continuum in 16O only opens at ϳ11 MeV. scribed by a Fermi-Dirac spectrum ͓1͔ Next to a bare energy effect, the number of particles with energies above the 16O particle threshold increases consider- N E2 ably with the temperature of the neutrino distribution, en- n ͑E,T͒ϭ , ͑20͒ ␯ 3 E/T hancing the cross sections accordingly. As a consequence, it T 1ϩe is mainly the neutrinos in the high-energy tail of the spectrum that are responsible for the excitation of nuclei. It is where the normalization factor N equals 0.533. The tempera- therefore important to have a good description of this part of ture of the spectra amounts to approximately 3.5 MeV for the energy spectrum for supernova neutrinos. Monte Carlo electron neutrinos and 5 MeV for electron antineutrinos ͓26͔. simulations of neutrino transport in supernovae indicate that As typical supernova energies are not high enough to pro- in this energy region, the spectrum is somewhat depleted duce heavy leptons, heavy ﬂavor neutrinos do not participate compared to the distribution ͑20͒. The tail of the Fermi-Dirac in charged current reactions. They escape from regions distribution can be adjusted to a more accurate reproduction closer to the center of the star with temperatures of 8–10 of the actual supernova spectrum by bringing in a chemical MeV. As the cross sections are roughly proportional to the potential ͓27,28͔. The new neutrino energy spectrum then square of the incoming neutrino energy, the higher tempera- reads ture ␮ and ␶ neutrinos and neutral-current reactions will dominate nucleosynthesis processes. N E2 In Fig. 5 we show the results for calculations where ␣ n␯͑E,T,␣͒ϭ , ͑21͒ the 16O cross sections have been folded with a Fermi-Dirac T3 1ϩe(E/Tϩ␣) spectrum with temperatures between 4 and 12 MeV. Table II summarizes the total cross sections for the Hartree-Fock and where ␣ is the parameter associated with the nonzero chemi- the CRPA calculation. The results obtained in the present cal potential and N␣ is a normalization factor depending on

FIG. 6. Differential cross section for the 16 16 reaction Oϩ␯FD O*ϩ␯Ј, averaged over neutrinos and antineutrinos→ and over a Fermi- Dirac distribution with temperature Tϭ6 MeV, ␣ϭ0 ͑full line͒ and ␣ϭ5 ͑dashed͒. PRC 59 CROSS SECTIONS FOR NEUTRAL-CURRENT . . . 3253

FIG. 7. Comparison of RPA ͑full line͒ to TDA results ͑dashed͒ for the reaction 16O(␯,␯Ј)16O*. The discrepancy is most pro- nounced in the J␲ϭ1ϩ,2Ϫ, and 0ϩ channels. The incoming neutrino energy is 50 MeV. All multipoles up to Jϭ4 are taken into account.

␣. In Fig. 6 we show how the introduction of a nonzero ␣ Fig. 7 shows some important differences. The contribution of parameter affects the differential cross sections. The influ- 1ϩ,2Ϫ, and 0ϩ excitations are unproportionally large in ence of such an adjustment on the total cross sections is TDA, compared to those in RPA. relatively small. Only in the giant dipole resonance region Moreover, the TDA solutions are unstable against small the adjustment of the neutrino spectrum causes slight changes in the residual interaction. This can easily be illus- changes in the results. Further investigations showed that trated by using different two-body forces. In Fig. 8 we com- differences caused by the variation of the parameter ␣ in the pare the RPA and TDA results using the SkE2-Skyrme and range ␣ϭ3to␣ϭ6 have a negligible influence on the re- Landau-Migdal forces. The Landau-Migdal parameters were sults. taken from Ref. ͓29͔. Apart from slight differences in the In Table III we show a similar picture for neutrino nucleo- relative weight of the resonances, the RPA cross sections synthesis reactions on 12C. Here, similar conclusions as in 16 obtained using different residual interactions are in good the case of O can be drawn in comparison with other cal- agreement. The outcome of the TDA calculations, however, culations. is extremely sensitive to the force used, as is illustrated in Fig. 8. VII. ON THE RELIABILITY OF THE TDA The anomalous behavior of the TDA solution may well be AND RPA CALCULATIONS caused by the intrinsic asymmetry underlying the TDA equa- Starting from the CRPA calculations, as discussed in Sec. tions ͓30͔. The TDA accepts the shell-model closed-shell V and VI, one can easily evaluate TDA results by switching configuration as the ground state, whereas excited states are off the backward-going term in the wave functions ͓see ex- described by a coherent superposition of particle-hole exci- pression ͑12͔͒. Due to the energy factors appearing in the tations. The random phase approximation considers particle- denominators of Eq. ͑12͒, the latter terms are not expected to hole and negative energy hole-particle configurations out of have a major influence on the resulting transition densities. a correlated ground state. Whereas in the RPA formalism the Nevertheless, the comparison of the TDA and RPA results in ground state is treated on an equal basis with all excited

FIG. 8. Comparison of the dependence of RPA and TDA results on the residual interaction used. Whereas RPA calculations are stable against changes in the interaction, TDA cross sections are not. RPA-SkE2 ͑full line͒, RPA-LM ͑short dashed͒, TDA-SkE2 ͑dashed͒, TDA-LM ͑dotted͒. 3254 JACHOWICZ, ROMBOUTS, HEYDE, AND RYCKEBUSCH PRC 59

FIG. 9. Comparison of RPA ͑full line͒ to Hartree-Fock ͑dashed͒ results. The incoming neutrino energy is 50 MeV. All multipoles up to J ϭ4 are taken into account. The large difference between the random phase and the mean-ﬁeld response at low excitation energies indicates that RPA calculations in this energy region may not give as accurate a description as one could ex- pect. For excitation energies above ϳ27 MeV, RPA calculations provide a reasonable correction to the mean-ﬁeld ones.

states, the lack of backward going terms in the TDA equa- results do not seem to be very sensitive to the particular tions implies that the TDA vacuum is a static reference state. choice of the residual two-body force. We also noted that the Furthermore it should be stressed that even the RPA re- TDA results differ substantially from RPA results and thus sults have to be considered with extreme care, especially for much caution has to be taken when using the former method. excitation energies below ϳ25 MeV. The large contribu- As expected there also appears a clear sensitivity to the use tions introduced by the RPA correlations compared to the of the particular unperturbed single-particle energy spectrum Hartree-Fock results, as is illustrated in Fig. 9, make it dif- used as input. This can cause shifts in the resonance struc- ficult to see the RPA as a correction that can be handled ture. These latter conclusions also point towards the need to within perturbation theory. These differences are indications incorporate more complex configurations in order to produce that more complex nuclear configurations should be incorpo- both the correct excitation energies and widths of the various rated in order to give a more reliable description of both the resonances. resonance energies and widths in the giant resonance region. In 16O as well as in 12C the dominant multipole transi- A related problem is the sensitivity of the results to the tions are J␲ϭ1Ϫ,1ϩ,2Ϫ. The Jϭ0 exitations are suppressed. choice of single-particle energies and wave functions, which Due to the dominance of the axial vector contribution, is- makes it hard to obtain unambiguous cross section results. ovector excitations are clearly prominent. In both nuclei, neutrino cross sections are slightly larger than cross sections VIII. CONCLUSION for excitations induced by antineutrinos. Both nuclei studied play an important role in explosive In the present paper, we have carried out RPA studies of nucleosynthesis. We made an estimate of the importance of neutrino-nucleus scattering reactions and corresponding nu- the reactions studied to neutrino nucleosynthesis by folding cleosynthesis processes. The RPA equations have been stud- the neutrino-nucleus scattering cross sections with a Fermi- ied using a Green’s function approach with an effective Dirac energy spectrum. The resulting cross sections depend Skyrme force ͑SkE2͒ and a Landau-Migdal force as residual sensitively on the temperature of the neutrino spectrum. The two-body interactions. The SkE2-force already showed its influence of the introduction of a chemical potential is rather strength in evaluating various electromagnetic processes. We small. now have extended applications into the weak interaction sector. ACKNOWLEDGMENTS Besides performing detailed calculations for 16O and 12C, we have carefully studied the consequences of us- One of the authors ͑N.J.͒ thanks the University Research ing TDA instead of RPA, and the effects of using different Board ͑BOZF͒, S.R., K.H., and J.R. are grateful to the Fund residual interactions in order to obtain conclusions on the for Scientific Research ͑FWO͒ Flanders for financial support. reliability of present-day RPA calculations. Moreover, K.H. thanks CERN for financial support in the A first important conclusion is that our results substantiate later stages of this work. The authors are grateful to M. Ar- earlier ones, obtained by Kolbe et al. ͓5,6͔ using a different nould, S. Goriely, E. Kolbe, K. Langanke, and F. Thieleman technique to solve the RPA equations. Moreover, the RPA for many fruitful discussions.

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