collaborators:
DESCRIPTION
ECT* Workshop 20007 ‘‘Fundamental Symmetries : From Nuclei and Neutrinos to the Universe’’ ECT*, Trento, Italy, June 24 – 29, 2007. - PowerPoint PPT PresentationTRANSCRIPT
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‘‘‘‘Inelastic Neutrino-Nucleus Reaction Inelastic Neutrino-Nucleus Reaction Cross sections at low and Cross sections at low and
intermediate energies’’intermediate energies’’
T.S. KosmasT.S. Kosmas
Division of Theoretical Physics, University of Ioannina,
Greece
Collaborators: Collaborators:
P. Divari, V. Chasioti, K. Balasi, V. Tsakstara, G. Karathanou, P. Divari, V. Chasioti, K. Balasi, V. Tsakstara, G. Karathanou, K. Kosta K. Kosta
ECT* Workshop 20007‘‘Fundamental Symmetries : From Nuclei
and Neutrinos to the Universe’’ECT*, Trento, Italy, June 24 – 29, 2007
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OutlineOutline
• 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:
4040Ar, Ar, 5656Fe, Fe, 9898Mo, Mo, 1616OO 3. Dominance of specific multipole states – channels 4. Nuclear response to SN ν (flux averaged cross sections)
• Summary and ConclusionsSummary and Conclusions
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Charged-current reactions (l= e, μ, τ)
Neutral-current reactions
Introduction
There are four types of neutrino-nucleus reactions to be studied :
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1-body semi-leptonic electroweak processes in nuclei
Donnely-Walecka method provides a unified description of semi-leptonic 1-body processes in nuclei
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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 occurc) Dominance of Coherent channel, ‘measured’ by experiments :
(i) TRIUMF : 48Ti, 208Pb
(ii) PSI : 48Ti, 208Pb, 197Au Best limit Rμe < 10-13 A. van der Shaaf J.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
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
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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 : 73Ge, 127I, 115In, 129,131XeC) Theoretically: MQPM, SM for : 73Ge, 127I, 115In, 81Ga
TSK, J.Vergados, PRD 55(97)1752, Korteleinen, TSK, Suhonen, Toivanen, PLB 632(2006)226,
Χ + (Α, Ζ) χ’ + (Α,Ζ)*
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Detection of WIMPs
Prominent Odd-A Nuclear Targets : 73Ge, 115In, 127I
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Conclusions: Experimental ambitions for Recoils
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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)
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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
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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
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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:
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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
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The seven basic single-particle operators
Normal Parity Operators
Abnormal Parity Operators
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Compact expressions for the basic reduced MEFor 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.
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Polynomial Coefficients of all basic reduced ME
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Neutral-Current ν–Nucleus Cross sectionsIn Donnely-Walecka method [PRC 6 (1972)719, NPA 201(1973)81]
==============================================================================================================
where
The Coulomb-Longitudinal (1st sum), and Transverse (2nd sum) are:
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Nuclear Matrix Elements - The Nuclear Model 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 QRPA
1). Interactions: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 gn
pair , gppair
• In the QRPA level: the strength parameters gpp , gph
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). 3). Testing the reliability of the MethodTesting the reliability of the Method::• Low-lying nuclear excitations Low-lying nuclear excitations (up to about 5 MeV(up to about 5 MeV)• magnetic momentsmagnetic moments (separate spin, orbital contributions)
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Particle-hole, gph, and particle-particle gpp parameters for 16O ,40Ar, 56Fe, 98Mo
H.O. size-parameter, b, model space and pairing parameters, n, p pairs for 16O ,40Ar, 56Fe, 98Mo
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experimental theoretical
Low-lying Nuclear Spectra (up to about 5 MeV)
98Mo
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experimental theoretical
Low-lying Nuclear Spectra (up to about 5 MeV)
40Ar
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State-by-state calculations of multipole contributions to dσ/dΩ
56Fe
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Angular dependence of the differential cross-section
56Fe
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Total Cross section: Coherent & Incoherent contributions
g.s. g.s.
g.s. f_exc
56Fe
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Dominance of Axial-Vector contributions in σ
56Fe
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Dominance of Axial-Vector contributions in σ_tot
40Ar
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Dominance of Axial-Vector contributions in σ
16O
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Dominance of Axial-Vector contributions in σ
98Mo
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State-by-state calculations of dσ/dΩ
40Ar
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Total Cross section: Coherent + Incoherent contributions
40Ar
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State-by-state calculations of dσ/dΩ
16O
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16O
Coherent and Incoherent
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State-by-state calculations of dσ/dΩ
98Mo
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Angular dependence of the differential cross-section
98Mo
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98Mo
Angular dependence of the differential cross section for the excited states J=2+, J=3-
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Coherent and Incoherent
98Mo
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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
===========================================================
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Flux averaged Cross Sections for SN-ν
α = 0, 3
2.5 < Τ < 8 (in MeV)
A= <σ>_A
V= <σ>_V
5656Fe Fe
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Flux averaged Cross Sections for SN-ν
A= <σ>
V= <σ>
α = 0, 3
1616O O
2.5 < Τ < 8 (in MeV)
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SUMMARY-CSUMMARY-CONCLUSIONSONCLUSIONS• 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.
• Our results are in good agreement with previous calculations (Kolbe-Langanke, case of 5656Fe,Fe, and Gent-group, 1616OO).
• 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.
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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
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Kinematical factors for neutrino currents
Summing over final and averaging over initial spin states gives