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ECT* Workshop 20007 ‘‘Fundamental Symmetries : From Nuclei and Neutrinos to the Universe’’ ECT*, Trento, Italy, June 24 – 29, 2007.

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  1. ECT* Workshop 20007 ‘‘Fundamental Symmetries : From Nuclei and Neutrinos to the Universe’’ ECT*, Trento, Italy, June 24 – 29, 2007 ‘‘Inelastic Neutrino-Nucleus Reaction Cross sections at low and intermediate energies’’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= e, μ, τ) • 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. Exotic Semi-leptonic Nuclear Processes 1). LF violating process : Conversion of a bound μ-b toe-in nuclei μ-b+ (Α, Ζ) e- + (Α,Ζ)* a) Coherent (g.s => g.s.) and Incoherent i> => f> Transitions exist: b) Both Fermi and Gammow-Teller like contributions occur c) Dominance of Coherent channel, ‘measured’ by experiments : (i) TRIUMF : 48Ti, 208Pb (ii) PSI : 48Ti, 208Pb, 197Au Best limit Rμe < 10-13 A. van derShaafJ.Phys.G 29 (2003)1503 (iii) MECO at Brookhaven on 27Al (Cancelled, planned limit Rμe < 2x 10-17) W,Molzon, Springer Tracts in Mod. Phys., (iV) PRIME at PRISM on 48Ti planned limit Rμe < 10-18) Y.Kuno, AIP Conf.Proc. 542(2000)220 d) Theoretically QRPA: TSK, NPA 683(01)443, E.Deppisch, TSK, JWF.Walle, NPB 752(06)80 2). LF and L violating process: Conversion of a μ-b toe+ in nuclei μ-b+ (Α, Ζ) e+ + (Α,Ζ-2)* • DCEx process like 0νββ-decayF.Simkovic, A.Faessler • b) 2-body (very complicated operator), P.Divari,T.S.K.,Vergados, NPA

  6. LSP-nucleus elastic (+ inelestic) scattering The Content of the universe: Dark Energy ≈ 74%,Cold Dark Matter ≈ 22%( Atoms ≈ 4% Χ + (Α, Ζ) χ’ + (Α,Ζ)* • Coherent - Incoherent event rates : Vector & Axial-Vector part • Dominance of Axial-Vector contributions • Odd-A nuclear targets : 73Ge, 127I, 115In, 129,131Xe • C) Theoretically: MQPM, SM for : 73Ge, 127I, 115In, 81Ga • TSK, J.Vergados, PRD 55(97)1752, Korteleinen, TSK, Suhonen, Toivanen, PLB 632(2006)226,

  7. Detection of WIMPs Prominent Odd-A Nuclear Targets : 73Ge, 115In, 127I

  8. Conclusions: Experimental ambitions for Recoils

  9. Semi-leptonic 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)

  10. 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

  11. 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

  12. Multipole Expansion – Tensor Operators The ME of the Effective Hamiltonian reads Apply multipole expansion of Donnely-Walecka[PRC 6 (1972)719, NPA 201(1973)81] in the quantities : For J-projected nuclear states the result is written:

  13. 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

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

  15. 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)

  16. Polynomial Coefficients of all basic reduced ME

  17. 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: ==============================================================================================================

  18. 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 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)

  19. 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

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

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

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

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

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

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

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

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

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

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

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

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

  32. Coherent and Incoherent 16O

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

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

  35. 98Mo Angular dependence of the differential cross section for the excited states J=2+, J=3-

  36. Coherent and Incoherent 98Mo

  37. 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

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

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

  40. 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.

  41. 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

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

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