Realistic Calculations of Neutrino-Realistic Calculations of Neutrino-Nucleus Reaction Cross sections Nucleus Reaction Cross sections

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

MEDEX’07International workshop on DBD and Neutrino Physics Prague, Czech Republic, June 11 – 14, 2007

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

Charged-current reactions (l= electron, muon, tau)

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

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)

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 in the quantities :

The result is (for J-projected nuclear states) :

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

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

Neutral-Current ν–Nucleus Cross sectionsIn Donnely-Walecka method [PRC 6 (1972)719, NPA 201(1973)81]

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where

The Coulomb-Longitudinal (1st sum), and Transverse (2nd sum) are:

The seven basic single-particle operators

Normal Parity Operators

Abnormal Parity Operators

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.

Polynomial Coefficients of all basic reduced ME

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 the QQuasi-particle RPA (QRPA)RPA (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)

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

experimental theoretical

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

98Mo

experimental theoretical

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

40Ar

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

56Fe

Angular dependence of the differential cross-section

56Fe

Total Cross section: Coherent & Incoherent contributions

g.s. g.s.

g.s. f_exc

56Fe

Dominance of Axial-Vector contributions in σ

56Fe

Dominance of Axial-Vector contributions in σ_tot

40Ar

Dominance of Axial-Vector contributions in σ

16O

Dominance of Axial-Vector contributions in σ

98Mo

State-by-state calculations of dσ/dΩ

40Ar

Total Cross section: Coherent + Incoherent contributions

40Ar

State-by-state calculations of dσ/dΩ

16O

16O

Coherent and Incoherent

State-by-state calculations of dσ/dΩ

98Mo

Angular dependence of the differential cross-section

98Mo

98Mo

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

Coherent and Incoherent

98Mo

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

Flux averaged Cross Sections for SN-ν

A= <σ>

V= <σ>

α = 0, 3

1616O O

2.5 < Τ < 8 (in MeV)

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.