Study on ν-A Reaction Cross Sections within CRPA Jeong-Yeon LEE and Yeong-Duk KIM Sejong University, KOREA

I. Why ν-A interactions?  ν : play an important role in various astrophysical processes, dynamics of core-collapse supernovae and supernova-nucleosynthesis, with the detection of neutrinos from SN1987A.  ν : interesting tools to study weak interaction, the limits of the standard model, and nuclear structure.  Though neutral-current ν scattering is important in astrophysics, experimental works concentrate on charged-current ν reactions, since outgoing charged leptons are more easily detected.  Scattering off nuclei like 12 C and 16 O (i.e., the main constituents of scintillator and water Cerenkov detectors) has been the object of many investigations.  Longstanding problems concerning discrepancy between theoretical and experimental results for 12 C(ν µ, µ - ) 12 N * ⇒ could not be solved satisfactorily. ⇒ Motivation of a new study of charged-current ν-A reactions, including calculations of cross sections for nuclei of experimental interests.

II. Continuum RPA Random phase approximation (RPA) - A nucleus is excited primarily through ph excitation. - Interaction between p and h produces correlations between ph pairs, which play an important role in determining characteristics of energy spectrum.  Giant resonance (GR) states : Collective states described as superpositions of many 1p-1h configurations. ⇒ well describe the positions & strengths, but not the widths of GR. ( ∵ states are treated as discrete ones even in continuum.)

Continuum RPA Nuclear response in continuum - ph correlations. - damping (absorption) effects. - continuum boundary condition. ⇒ Quite successful in explaining  Giant resonances : Hadronic inelastic scattering (Lee et al., JNST S2, 770(2002), JKPS 36, 323 (2000), 36, 13 (2000) & 33, 388 (1998)) Electronic inelastic scattering ( Kyum, Ph. D. thesis, 1996.)  Δ-excitations by charge exchange reactions : (p,n), ( 3 He,t) ( Udagawa et al., PLB 245, 1 (1990), PRC 49, 3162 (1994) )

III. Purposes  Calculations of cross sections for ν-A reactions, 12 C(ν l, l - ) 12 N *, 16 O(ν l, l - ) 16 F *, using the continuum RPA.  Comparison with the experimental data.  Comparison with cross sections for reactions by other probes.

Charged-current reactions ν l + X(Z,A) ⇒ l - + X(Z+1,A), l = e, µ ν e : coming from decay-at-rest of µ +. ν µ : coming from decay-in-flight of π +. Assumption : Target is a spherical nucleus with J π = 0 +. ν-A reaction cross section G : Weak interaction coupling constant. θ c : Cabibbo angle. F(Z’,E) : Fermi function. IV. Charge-exchange ν-A scattering εiεi εf εf θ νlνl l - X

M J (κ), L J (κ), J J mag (κ), J J el (κ) : Coulomb, longitudinal, transverse magnetic, transverse electric multipole operators.

V. Nuclear strength function in Continuum RPA Strength function i, f : Quantum numbers of initial and final states. : Generic many-body operator. Assumption :Target is a spherical nucleus with J π = 0 +.

Source function Y p : Spin-angle wave function of particle p. Φ h : Hole wave function of h (=time reversal state of h). ( | | › : Integrals are carried over only spin-angle variables. ~ ~

Green’s function ω : Excitation energy of system (ω >0 for forward amplitude, ω <0 for backward amplitude) H h : Hamiltonian of hole nucleus H p (=T p +U p ): Hamiltonian of excited particle p (U p =V p +iW p, W p : deals with damping effects of particle going into more complicated nuclear states) V ph : Residual interaction (responsible for ph correlations)

How can we obtain ?? ⇒ Introduce !!! Λ ph : Correlated source function. G 0 : Free Green’s function without V ph. ⇒ Inhomogeneous coupled-channels integral equation. 

⇒ Use Lanczos method to solve ! (Whitehead et al., Adv. Nucl. Phys. 9, 123 (1977). S ↓ : Damping(spreading) process. S ↑ : Direct knockout (particle emission) process. (χ p : Distorted wave function of knocked-out p against residual nucleus)

VI. Applications Apply to 12 C(ν l, l - ) 12 N *, 16 O(ν l, l - ) 16 F * H h = T h +U h (U h : Mean field real potential of Woods-Saxon type) H p = T p +U p (U p : Optical potential of Woods-Saxon type ) V ph (r 1,r 2 )= V ph (|r 1 -r 2 |) [W + BP σ +M P x + H P σ P x ] (P σ,P x : Spin, coordinate exchange operators) Assume : - V ph is a local 2-body operator. - We use δ-interaction approximation. ⇒ V ph (r 1,r 2 )= V p δ(r 1 -r 2 ) [a +bP σ ], a= W+M, b=B+H

Cross section for 12 C(ν µ, µ - ) 12 N *

VII. Summary  We study charge exchange ν-A scattering reactions, 12 C(ν l, l - ) 12 N *, 16 O(ν l, l - ) 16 F *, in a self-consistent manner within Continuum RPA.  Numerical calculations for cross sections will be done up to higher order of multipole transitions and be compared with the experimental data.  Further works - Comparison of ν-A and ν-N scattering cross sections. - Comparison with cross sections for the reactions by other probes.