1
Inelastic Neutrino-Nucleus Reaction Cross
sections at low and intermediate energiesT.S.
Kosmas Division of Theoretical Physics,
University of Ioannina, Greece
ECT Workshop 20007 Fundamental Symmetries
From Nuclei and Neutrinos to the
Universe ECT, Trento, Italy, June 24 29,
2007
Collaborators P. Divari, V. Chasioti, K.
Balasi, V. Tsakstara, G. Karathanou, K. Kosta
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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 ? (flux averaged
  cross sections)
• Summary and Conclusions

3
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
1). LF violating process Conversion of a bound
µ-b to e- in nuclei
µ-b (?, ?) e- (?,?)
a) Coherent (g.s gt g.s.) and Incoherent igt gt
fgt 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 lt 10-13 A. van der Shaaf J.Phys.G 29
(2003)1503 (iii) MECO at Brookhaven on 27Al
(Cancelled, planned limit Rµe lt 2x
10-17) W,Molzon, Springer Tracts in Mod. Phys.,
(iV) PRIME at PRISM on 48Ti planned limit
Rµe lt 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 to e in nuclei
µ-b (?, ?) e (?,?-2)

• DCEx process like 0?ßß-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
? (?, ?) ? (?,?)

• 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,

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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
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)
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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, q0, 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
a , -, 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
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
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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 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

1. The coefficients P are calculated once
   (reduction of computer time)
2. They can be used for phenomenological description
   of ME
3. They are useful for other bases sets (expansion
   in H.O. wavefunctions)

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Polynomial Coefficients of all basic reduced ME
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Neutral-Current ?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



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Nuclear Matrix Elements - The Nuclear Model
The initial and final states, Jigt, Jfgt, in
the ME ltJf T(qr)Jigt2 are determined by
using QRPA
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

• 1). Interactions
• Woods SaxonCoulomb 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)

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H.O. size-parameter, b, model space and pairing
parameters, n, p pairs for 16O
,40Ar, 56Fe, 98Mo
Particle-hole, gph, and particle-particle gpp
parameters for 16O ,40Ar, 56Fe, 98Mo
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Low-lying Nuclear Spectra (up to about 5 MeV)
98Mo
experimental
theoretical
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Low-lying Nuclear Spectra (up to about 5 MeV)
40Ar
experimental
theoretical
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State-by-state calculations of multipole
contributions to ds/dO
56Fe
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Angular dependence of the differential
cross-section
56Fe
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Total Cross section Coherent Incoherent
contributions
56Fe
g.s. g.s.

g.s. f_exc

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Dominance of Axial-Vector contributions in s
56Fe
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Dominance of Axial-Vector contributions in s_tot
40Ar
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Dominance of Axial-Vector contributions in s
16O
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Dominance of Axial-Vector contributions in s
98Mo
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State-by-state calculations of ds/dO
40Ar
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Total Cross section Coherent Incoherent
contributions
40Ar
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State-by-state calculations of ds/dO
16O
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Coherent and Incoherent
16O
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State-by-state calculations of ds/dO
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 J2, J3-
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Coherent and Incoherent
98Mo
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Nuclear response to the SN-? for various targets
Assuming Fermi-Dirac distribution for the SN-?
spectra
normalized to unity as
Using our results, we calculated for various
?nucleus reaction channels


Results of Toivanen-Kolbe-Langanke-Pinedo-Vogel,
NPA 694(01)395
a 0, 3
2.5 lt ? lt 8
56Fe
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Flux averaged Cross Sections for SN-?
a 0, 3
2.5 lt ? lt 8 (in MeV)
A ltsgt_A
V ltsgt_V
56Fe
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Flux averaged Cross Sections for SN-?
a 0, 3
2.5 lt ? lt 8 (in MeV)
A ltsgt
V ltsgt
16O
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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 ?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-? spectra for
Temperatures in the range 2.5 lt T lt 8
and degeneracy-parameter a values a 0, 3
Acknowledgments I wish to acknowledge
financial support from the ?????-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
The effective nucleon level Hamiltonian takes the
form
For charged-current ?-nucleus processes
For neutral-current ?-nucleus processes
The form factors, for neutral-current processes,
are given by
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Kinematical factors for neutrino currents
Summing over final and averaging over initial
spin states gives