‘‘Inelastic Neutrino-Nucleus Reaction Cross sections at low and intermediate energies’’ T.S. Kosmas ‘‘Inelastic Neutrino-Nucleus Reaction Cross sections at low and intermediate energies’’ T.S. Kosmas Division of Theoretical Physics, University of Ioannina, Greece Collaborators: P. Divari, V. Chasioti, K. Balasi, V. Tsakstara, G. Karathanou, K. Kosta ECT* Workshop ‘‘Fundamental Symmetries : From Nuclei and Neutrinos to the Universe’’ ECT*, Trento, Italy, June 24 – 29, 2007

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

 Charged-current reactions (l= e, μ, τ )  Neutral-current reactions Introduction There are four types of neutrino-nucleus reactions to be studied :

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

Exotic Semi-leptonic Nuclear Processes 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 : 48 Ti, 208 Pb (ii) PSI : 48 Ti, 208 Pb, 197 Au Best limit R μe < A. van der Shaaf J.Phys.G 29 (2003)1503 (iii) MECO at Brookhaven on 27 Al (Cancelled, planned limit R μe < 2x ) W,Molzon, Springer Tracts in Mod. Phys., (iV) PRIME at PRISM on 48 Ti planned limit R μe < ) 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 μ - b + (Α, Ζ) e + + (Α,Ζ-2)* μ - b + (Α, Ζ) e - + (Α,Ζ)* 1). LF violating process : Conversion of a bound μ - b to e - in nuclei 2). LF and L violating process: Conversion of a μ - b to e + in nuclei a)DCEx process like 0νββ-decay F.Simkovic, A.Faessler b) 2-body (very complicated operator), P.Divari,T.S.K.,Vergados, NPA

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

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

Conclusions: Experimental ambitions for Recoils

Semi-leptonic 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)

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

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

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:

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

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

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.

Polynomial Coefficients of all basic reduced ME

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:

Nuclear Matrix Elements - The Nuclear Model QRPA The initial and final states, |J i >, |J f >, in the ME 2 are determined by using 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 j 1, j 2 run over single-particle levels of the model space (coupled to J) D(j 1, j 2 ; 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)

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

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

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

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

Angular dependence of the differential cross-section 56 Fe

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

Dominance of Axial-Vector contributions in σ 56 Fe

Dominance of Axial-Vector contributions in σ _tot 40 Ar

Dominance of Axial-Vector contributions in σ 16 O

Dominance of Axial-Vector contributions in σ 98 Mo

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

Total Cross section: Coherent + Incoherent contributions 40 Ar

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

Coherent and Incoherent

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

Angular dependence of the differential cross-section 98 Mo

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

Coherent and Incoherent 98 Mo

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, < Τ < 8 Results of Toivanen-Kolbe-Langanke-Pinedo-Vogel, NPA 694(01)395 56Fe ===========================================================

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

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

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.

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

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