1. Realistic Calculations of Neutrino-Nucleus Reaction Cross sections T.S. Kosmas Realistic Calculations of Neutrino-Nucleus Reaction Cross sections T.S. Kosmas Division of Theoretical Physics, University of Ioannina, Greece Collaborators: P. Divari, V. Chasioti, K. Balasi, V. Tsakstara, G. Karathanou, K. Kosta MEDEX’07 International workshop on DBD and Neutrino Physics Prague, Czech Republic, June 11 – 14, 2007

2. Outline Introduction Cross Section Formalism 1. Multipole operators ( Donnelly-Walecka method ) 2. Compact expressions for all basic reduced matrix elements Applications – Results 1. Exclusive and inclusive neutrino-nucleus reactions 2. Differential, integrated, and total cross sections for the nuclei : 40 Ar, 56 Fe, 98 Mo, 16 O 40 Ar, 56 Fe, 98 Mo, 16 O 3. Dominance of specific multipole states – channels 4. Nuclear response to SN ν (flux averaged cross sections) Summary and ConclusionsSummary and Conclusions

3.  Charged-current reactions (l= electron, muon, tau)  Neutral-current reactions Introduction There are four types of neutrino-nucleus reactions to be studied :

4. 1-body semi-leptonic electroweak processes in nuclei Donnely-Walecka method provides a unified description of semi- leptonic 1-body processes in nuclei

5. The Effective Interaction Hamiltonian ( leptonic current ME ) Matrix Elements between initial and final Nuclear states are needed for obtaining a partial transition rate : The effective interaction Hamiltonian reads (momentum transfer)

6. One-nucleon matrix elements (hadronic current) Polar-Vector current: 2). Assuming CVC theory Axial-Vector current: 1). Neglecting second class currents : 3). Use of dipole-type q-dependent form factors 4. Static parameters, q=0, for nucleon form factors (i) Polar-Vector (i) Axial-Vector

7. Non-relativistic reduction of Hadronic Currents The nuclear current is obtained from that of free nucleons, i.e. The free nucleon currents, in non-relativistic reduction, are written α = +, -, charged-current processes, 0, neutral-current processes

8. Multipole Expansion – Tensor Operators The ME of the Effective Hamiltonian reads Apply multipole expansion of Donnely-Walecka in the quantities : The result is (for J-projected nuclear states) :

9. The basic multipole operators are defined as (V – A Theory) The multipole operators, which contain Polar Vector + Axial Vector part, The multipole operators are : Coulomb, Longitudinal, Tranverse-Electric, Transverse-Magnetic for Polar-Vector and Axial-Vector components

10. Nucleon-level hadronic current for neutrino processes For charged-current ν -nucleus processes For neutral-current ν -nucleus processes The form factors, for neutral-current processes, are given by The effective nucleon level Hamiltonian takes the form

11. Kinematical factors for neutrino currents Summing over final and averaging over initial spin states gives

12. Neutral-Current ν–Nucleus Cross sections In Donnely-Walecka method [PRC 6 (1972)719, NPA 201(1973)81] ============================================================================================================== where The Coulomb-Longitudinal (1 st sum), and Transverse (2 nd sum) are:

13. The seven basic single-particle operators Normal Parity Operators Abnormal Parity Operators

14. Compact expressions for the basic reduced ME For H.O. bases w-fs, all basic reduced ME take the compact forms The Polynomials of even terms in q have constant coefficients as Advantages of the above Formalism : (i)The coefficients P are calculated once (reduction of computer time) (ii)They can be used for phenomenological description of ME (iii)They are useful for other bases sets (expansion in H.O. wavefunctions) Chasioti, Kosmas, Czec.J. Phys.

15. Polynomial Coefficients of all basic reduced ME

16. Nuclear Matrix Elements - The Nuclear Model Q RPA (QRPA) The initial and final states, |J i >, |J f >, in the ME 2 are determined by using the Quasi-particle RPA (QRPA) 1). Interactions: Woods Saxon+Coulomb correction (Field) Bonn-C Potential (two-body residual interaction) 2). Parameters: 2). Parameters: In the BCS level: the pairing parameters g n pair, g p pair In the QRPA level: the strength parameters g pp, g ph j1, j2 run over single-particle levels of the model space (coupled to J) D(j1, j2; J) one-body transition densities determined by our model 3). Testing the reliability of the Method : Low-lying nuclear excitations (up to about 5 MeVLow-lying nuclear excitations (up to about 5 MeV) magnetic momentsmagnetic moments (separate spin, orbital contributions)

17. Particle-hole, g ph, and particle-particle g pp parameters for 16 O, 40 Ar, 56 Fe, 98 Mo H.O. size-parameter, b, model space and pairing parameters, n, p pairs for 16 O, 40 Ar, 56 Fe, 98 Mo

18. experimentaltheoretical Low-lying Nuclear Spectra (up to about 5 MeV) 98 Mo

19. experimentaltheoretical Low-lying Nuclear Spectra (up to about 5 MeV) 40 Ar

20. State-by-state calculations of multipole contributions to dσ/dΩ 56 Fe

21. Angular dependence of the differential cross-section 56 Fe

22. Total Cross section: Coherent & Incoherent contributions g. s. g. s. f_ exc 56 Fe

23. Dominance of Axial-Vector contributions in σ 56 Fe

24. Dominance of Axial-Vector contributions in σ _tot 40 Ar

25. Dominance of Axial-Vector contributions in σ 16 O

26. Dominance of Axial-Vector contributions in σ 98 Mo

27. State-by-state calculations of dσ/dΩ 40 Ar

28. Total Cross section: Coherent + Incoherent contributions 40 Ar

29. State-by-state calculations of dσ/dΩ 16 O

30. Coherent and Incoherent

31. State-by-state calculations of dσ/dΩ 98 Mo

32. Angular dependence of the differential cross-section 98 Mo

33. Angular dependence of the differential cross section for the excited states J=2+, J=3 -

34. Coherent and Incoherent 98 Mo

35. Nuclear response to the SN- ν for various targets Assuming Fermi-Dirac distribution for the SN- ν spectra Using our results, we calculated for various ν –nucleus reaction channels normalized to unity as α = 0, 3 2.5 < Τ < 8 Results of Toivanen-Kolbe-Langanke-Pinedo-Vogel, NPA 694(01)395 56Fe ===========================================================

36. Flux averaged Cross Sections for SN- ν α = 0, 3 2.5 < Τ < 8 (in MeV) A= _A V= _V 56 Fe

37. Flux averaged Cross Sections for SN- ν A= V= α = 0, 3 16O16O16O16O 2.5 < Τ < 8 (in MeV)

38. SUMMARY-CONCLUSIONS Using H.O. wave-functions, we have improved the Donnelly-Walecka formalism : compact analytic expressions for all one-particle reduced ME as products (Polynomial) x (Exponential) both functions of q. Using QRPA, we performed state-by-state calculations for inelastic ν –nucleus neutral-current processes (J-projected states) for currently interesting nuclei. The QRPA method has been tested on the reproducibility of : a) the low-lying nuclear spectrum (up to about 5 MeV) b) the nuclear magnetic moments Total differential cross sections are evaluated by summing-over-partial-rates. For integrated-total cross-sections we used numerical integration. 56 Fe, 16 O Our results are in good agreement with previous calculations (Kolbe-Langanke, case of 56 Fe, and Gent-group, 16 O ). We have studied the response of the nuclei in SN- ν spectra for Temperatures in the range : 2.5 < T < 8 and degeneracy-parameter α values : α = 0, 3 Acknowledgments: Acknowledgments: I wish to acknowledge financial support from the ΠΕΝΕΔ-03/807, Hellenic G.S.R.T. project to participate and speak in the present workshop.