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MEDEX’07 International workshop on DBD and Neutrino Physics Prague, Czech Republic, June 11 – 14, 2007. Realistic Calculations of Neutrino-Nucleus Reaction Cross sections T.S. Kosmas Division of Theoretical Physics, University of Ioannina , Greece. Collaborators:

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

  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: 40Ar, 56Fe, 98Mo, 16O 3. Dominance of specific multipole states – channels 4. Nuclear response to SN ν (flux averaged cross sections) • Summary 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 The effective interaction Hamiltonian reads Matrix Elements between initial and final Nuclear states are needed for obtaining a partial transition rate : (leptonic current ME) (momentum transfer)

  6. One-nucleon matrix elements (hadronic current) 1). Neglecting second class currents : Polar-Vector current: Axial-Vector current: 2). Assuming CVC theory 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 The multipole operators, which contain Polar Vector + Axial Vector part, (V – A Theory) are defined as 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 The effective nucleon level Hamiltonian takes the form For charged-current ν-nucleus processes For neutral-current ν-nucleus processes The form factors, for neutral-current processes, are given by

  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 (1st sum), and Transverse (2nd 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 Chasioti, Kosmas, Czec.J. Phys. Advantages of the above Formalism : • The coefficients P are calculated once (reduction of computer time) • They can be used for phenomenological description of ME • They are useful for other bases sets (expansion in H.O. wavefunctions)

  15. Polynomial Coefficients of all basic reduced ME

  16. Nuclear Matrix Elements - The Nuclear Model The initial and final states, |Ji>, |Jf>, in the ME <Jf ||T(qr)||Ji>2 are determined by using the Quasi-particle RPA (QRPA) j1, j2run over single-particle levelsof the model space (coupled to J) D(j1, j2; J)one-body transition densitiesdetermined by our model • 1). Interactions: • Woods Saxon+Coulomb correction (Field) • Bonn-C Potential (two-body residual interaction) • 2). Parameters: • In the BCS level: the pairing parameters gnpair , gppair • In the QRPA level: the strength parameters gpp,gph • 3). Testing the reliability of the Method: • Low-lying nuclear excitations (up to about 5 MeV) • magnetic moments(separate spin, orbital contributions)

  17. H.O.size-parameter, b, model space and pairing parameters, n, p pairs for 16O ,40Ar,56Fe,98Mo Particle-hole, gph, and particle-particle gppparameters for16O ,40Ar,56Fe,98Mo

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

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

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

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

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

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

  24. Dominance of Axial-Vector contributions in σ_tot 40Ar

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

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

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

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

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

  30. Coherent and Incoherent 16O

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

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

  33. 98Mo Angular dependence of the differential cross section for the excited states J=2+, J=3-

  34. Coherent and Incoherent 98Mo

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

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

  37. Flux averaged Cross Sectionsfor SN-ν α = 0, 3 2.5 < Τ < 8 (in MeV) A= <σ> V= <σ> 16O

  38. SUMMARY-CONCLUSIONS • Using H.O. wave-functions, we have improved the Donnelly-Walecka formalism: compact analytic expressions for all one-particle reducedMEas products (Polynomial) x(Exponential) bothfunctions of q. • UsingQRPA, 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. • Our results are in good agreement with previous calculations (Kolbe-Langanke, case of 56Fe, and Gent-group, 16O). •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: 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.

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