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 1999! We study quasielastic neutrino-nucleus scattering on 12C and 16O within a continuum random phase ap- proximation ~RPA! model. The RPA equations are solved using a Green’s function approach with an effective Skyrme ~SkE2! two-body interaction. Besides a comparison with existing calculations, we carry out a detailed study using various methods ~Tamm-Dancoff approximation and RPA! and different forces ~SkE2 and Landau- Migdal!. We evaluate cross sections that may be relevant for neutrino nucleosynthesis. @S0556-2813~99!01606-4# PACS number~s!: 25.30.Pt, 24.10.Eq, 26.30.1k 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- @14#. 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 ~TDA! 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 ;1058, representing ap- proximately 99% of the total released energy @1,2#. 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- W mation of the material synthesized during the hydrostatic trino with four-momentum (ei ,ki) is scattered inelastically burning phases in the life of the star. from a nucleus with initial energy and momentum Ei ,PW 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, Jpi501. After the reaction, the nucleus is cluding shell model @3#, relativistic fermi gas @3,4#, and ran- i left in an excited state with final parity and angular momen- dom phase approximation ~RPA! calculations using various tum J ,p . The energy transferred from neutrino to nucleus forces @4–8#. Since both the LSND and the KARMEN Col- f f equals v5e 2e 5E 2E ,e 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 @9,10#, special attention was paid to the 12 2 mentum is denoted by qW 5kW i2kW f , k5uqWu. calculation of cross sections for the reactions C(ne ,e )X 12 2 As the interaction between the neutrino and a nucleus is and C(nm ,m )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 2 12 the outgoing neutrino is described by a plane wave. Using the C(nm ,m ) X reaction could be reached @8,9,11–13#. Next to these problems, we observe that relatively few natural units (\5c51), the differential cross section is then neutral-current calculations have been performed @3,5#. given by In the present work, we examine quasielastic neutral- d2s 1 current neutrino-nucleus reactions in an energy region rel- 5~2p!4k « u^ fuHˆ ui&u2. ~1! dVdv f f ( 2J 11 ( W evant to explosive nucleosynthesis processes, using a con- s f ,si i M f ,M i tinuum random phase approximation ~CRPA! formalism ˆ ~Sec. III!. 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 m,hadronic HW5 dxW jm,leptonic~xW !J ~xW !, ~2! †Present address: CERN CH-1211, Geneva 23, Switzerland. A2 E 0556-2813/99/59~6!/3246~10!/$15.00PRC 59 3246 ©1999 The American Physical Society PRC 59 CROSS SECTIONS FOR NEUTRAL-CURRENT . 3247 211 22 with G51.16639310 MeV the weak coupling con- ˆ and ˆ denote the Coulomb and longitudinal opera- stant. The neutral leptonic current reads MJM LJM ˆ el ˆ mag tor respectively, whereas JM and JM are the transverse ˆj0 ~x!5C¯ ~x!g ~12g !C ~x! J J m ne m 5 ne electric and magnetic operators: 1C¯ ~x!g ~12g !C ~x! nm m 5 nm ¯ ˆ ~k!5 dxW@j ~kr!YM~V !#Jˆ ~xW!, 1Cn ~x!gm~12g5!Cn ~x!. ~3! JM E J J x o t t M The weak neutral hadronic current is given by @15# ˆ 0 ˆ vector ˆ axial vector i M ˆ Jm5Jm 1Jm ˆ ~k!5 dxW$¹W @j ~kr!Y ~V !#%JW~xW!, LJM kE J J x 2 ˆ V3 2 ˆ S ˆ A3 5~122sin uW!Jm 22sin uWJm1Jm 1 ¯ V 2 V 2 n 1 ˆ 5CN GE~q !gm1 GM~q !smnq ˆ el W W W M W W 2M JM~k!5 dx$¹3@jJ~kr! J,J~Vx!#%J~x!, H J k E Y A 2 1G ~q !gmg5J . ~4! ˆ ˆ mag~k!5 dxW@j ~kr!W M ~V !# JW~xW!. ~9! 2 JM E J J,J x • Here sin uW50.2325, where uW is the Weinberg angle. For J Y the isospin operators, the convention t3up&511,t3un&5 ˆ V3 21 is adopted. Jm denotes the t3 component of the electro- S III. THE CONTINUUM RANDOM PHASE magnetic current and Jm 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 ~8! are determined within a continuum random phase GE5 ~122sin uW!t32sin uW , ~5! 2 approximation. The unperturbed single-particle wave- functions are generated by a Hartree-Fock calculation, using 1 V 2 2 GM5 ~122sin uW!~m p2mn!t32sin uW~m p1mn!. a Skyrme potential @16–19#. The CRPA formalism is based 2 on a Green’s function approach @14,16,20–22#. The RPA 6 ~ ! polarization propagator is obtained by the iteration to all or- ders of the first-order contribution to the particle-hole GA is given by Green’s function A G 52gat3521.262 t3 . ~7! (RPA) (0) For low momentum transfers (k,400 MeV/c), only P ~x1 ,x2 ;v!5P ~x1 ,x2 ;v! lowest-order contributions to the hadronic current have to be retained @15#. The differential cross section for neutral- 1 1 dx dx8P(0)~x ,x;v! current neutrino scattering finally becomes \E E 1 u 2 cos2 3˜V~x,x8!P(RPA)~x8,x ;v!, ~10! d2s G2«2 S 2D ` ` 2 i f f → J J 5 ( sCL1( sT , ~8! S dVdv D n p 2Ji11 F J50 J51 G ¯n where P(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 v ˜ J ˆ ˆ to be rotationally invariant, allowing us to write V(x1 ,x2)as sCL5 Jf J~k!1 J~k! Ji , UK UUM uqWuL UU LU q2 u ˜ J JM† ˆ JM ˆ J m 2 ˆ mag 2 V~x1,x2!5 ( Uab~r1 ,r2!Xa ~x1!Xb ~x2!, ~11! sT5 2 1tan @u^J fuu J ~k!uuJi&u ab,JM S 2uqWu2 S 2D D J 2 u qm u 1 J ˆ el~k! J 2#7tan 2 1tan2 where the XJM(xˆ) represent spherical tensor operators of u^ fuu J uu i&u 2 J S 2DA uqWu S 2D rank J,M. A solution to the RPA Bethe-Salpeter equation ~10! is 3 @2Re~^J uu ˆ mag~k!uuJ &^J uu ˆ el~k!uuJ &*!#. 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 c ~x !c† ~x ,e ! 21 ˜ h8 1 p8 1 p8 21 uCC~E!&5uph ~E!&1E dx1 E dx2 V~x1 ,x2!( E dep8 up8h8 ~ep8h8!& c P F E2e 8 p8h8 c† ~x !c ~x ,e ! h8 1 p8 1 p8 21 ˆ † ˆ 2 uh8p8 ~2ep h !& ^C0uc ~x2!c~x2!uCC~E!&, ~12! E1e 8 8 G p8h8 where denotes the Cauchy principal value, and the 21 P 21 dr dr R(0) r,r ;E uph (E)& and uhp (2E)& are the unperturbed particle- E E 8 hm;JM~ 8 ! hole and backward hole-particle solutions of the mean-field problem.