Electroweak Interactions with nuclei: The Relativistic Mean Field Approach

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1 Electroweak Interactions with nuclei: The Relativistic Mean Field Approach Juan Antonio Caballero Departamento de Física Atómica, Molecular y Nuclear Universidad de Sevilla INT Workshop on neutrino-nucleus interactions INT, Seattle, December 5th 1th, 16 Seattle, 1/6/16 p. 1

2 SOME QUESTIONS OF INTEREST Importance of relativistic effects: kinematics & dynamics Evidence for such effects in electron scattering? Inclusive & semi-inclusive reactions Natural outcomes. Transverse enhancement Model extended to neutrino reactions Parity-violating electron scattering MEC effects handled within RMF? Seattle, 1/6/16 p.

3 The model: Relativistic Impulse Approximation (RIA) Nuclear Current = One-body operator J µ N (ω, q) = d p Ψ F ( p+ q)ĵµ N Ψ B( p) Scattering off a nucleus= incoherent sum of single nucleon scattering processes Seattle, 1/6/16 p. 3

4 Basic ingredients of RIA Nucleon Wave Functions = Solutions of Dirac equation with phenomenological relativistic potentials (scalar and vector terms): Seattle, 1/6/16 p. 4

5 Basic ingredients of RIA Nucleon Wave Functions = Solutions of Dirac equation with phenomenological relativistic potentials (scalar and vector terms): Ψ B : Bound nucleon wave function= Relativistic Mean Field (RMF) Approach Local potentials obtained from a Lagrangian fitted to properties of nuclear matter, radii and nuclear masses. The non-relativistic reduction of the RMF formalism leads to a Schrödinger-like equation but with presence of non-local terms. ] [ V DEB φ nr (r) = E nr φ nr (r); Ψ up (r) = K(r)φ nr (r) m N K(r).8 in the nuclear interior going to unity asymptotically Seattle, 1/6/16 p. 4

6 Basic ingredients of RIA Nucleon Wave Functions = Solutions of Dirac equation with phenomenological relativistic potentials (scalar and vector terms): Ψ B : Bound nucleon wave function= Relativistic Mean Field (RMF) Approach Local potentials obtained from a Lagrangian fitted to properties of nuclear matter, radii and nuclear masses. The non-relativistic reduction of the RMF formalism leads to a Schrödinger-like equation but with presence of non-local terms. ] [ V DEB φ nr (r) = E nr φ nr (r); Ψ up (r) = K(r)φ nr (r) m N K(r).8 in the nuclear interior going to unity asymptotically Ψ F : Ejected nucleon wave function= Dependence with final state interactions (FSI) Seattle, 1/6/16 p. 4

7 The Relativistic Mean Field Approach (RMF) Large scalar (attractive) and vector (repulsive) potentials that lead to saturation. Nonlocalities & correlation effects accounted for by the RMF? Important difference with non-relativistic models 4 C-1 O-16 Repulsive Vector Potential V [MeV] - Attractive Scalar Potential -4 O-16 C r [fm] Seattle, 1/6/16 p. 5

8 Final State Interactions (FSI) Final State Interactions between the ejected nucleon and the residual nucleus described by the Optical Model: complex energy-dependent relativistic potentials fitted to elastic nucleon-nucleus scattering data 4 Parte real Parte imaginaria Vector potenciales (MeV) Escalar 3 MeV 5 MeV 1 MeV 1 1 Escalar Vector r(fm) r(fm) 4 Parte real Parte imaginaria potentiales (MeV) Escalar Vector EDAIO EDAD1 EDAD EDAD3 1 1 Escalar Vector r(fm) r(fm) Seattle, 1/6/16 p. 6

9 Nucleon current operator and uncertainties Electromagnetic current: (e,e ) Ĵ µ cc1 = (F 1 +F )γ µ F m N (P +P N ) µ Ĵ µ cc = F 1 γ µ + if m N σ µν Q ν Nucleon (bound and ejected) off-shell = dependence with the current operator: off-shell effects or Gordon ambiguities RIA model = Gauge invariance violation (Q µ J µ ): Lorentz Gauge: use ofĵ, Ĵ 3 Coulomb Gauge: Ĵ 3 ω qĵ Weyl Gauge: Ĵ q ωĵ3 Seattle, 1/6/16 p. 7

10 Important terminology RELATIVISTIC DISTORTED WAVE IMPULSE APPROXIMATION (RDWIA) Relativistic distorted (Dirac) wave functions,ψ B,Ψ F and the relativistic nucleon current operatorj µ p. RELATIVISTIC PLANE WAVE IMPULSE APPROXIMATION (RPWIA) Final State Interactions neglected= Ψ F -relativistic plane wave (u-dirac spinor) Bound Wave Function: Ψ B = φup φ = up φ down σ p E+M+S V φup i.e. Ψ B includes negative energy components= coupling to Dirac sea = αu+βv PLANE WAVE IMPULSE APPROXIMATION (PWIA) Negative Energy Components inψ B are projected out e' µ P N = Nuclear dynamics and electron-proton interaction are decoupled. The cross section factorizes: γ Q µ ~ σ en PWIA dσ dω e dε f dω p = Kf 1 rec σep N(p) e ~ S P A-1 µ P A µ withn(p)-single-particle momentum distribution andσ ep -single-proton cross section: σ ep η µν W µν = η µν { s i s f [ u(pf,s f )J µ pu(p i,s i ) ] [ u(p f,s f )J ν pu(p i,s i ) ]} Seattle, 1/6/16 p. 8

11 Non-relativistic approaches Relativistic vs non-relativistic approaches KINEMATICAL EFFECTS: Relativistic (4x4) current operator vs non-relativistic (x) reductions. Non-relativistic reduction in powers of: DYNAMICAL EFFECTS: q m N, ω m N, p m N? Relativistic vs non-relativistic wave functions: Upper components of Ψ vs Schrödinger solutions Darwin Factor Lower components of Ψ = Dynamical enhancement due to (S-V) relativistic potentials Seattle, 1/6/16 p. 9

12 THE RIA MODEL APPLIED TO(e,e N) REACTIONS Seattle, 1/6/16 p. 1

13 Analysis of reduced cross sections p 3 MeV/c: Darwin term. Larger spectroscopic factors p > 3 MeV/c: Dynamical enhancement of lower components Seattle, 1/6/16 p. 11

14 Cross Section for 16 O at low- Q TL response &A TL asymmetry at high- Q Kinematics: ε i = 445 MeV, q = 1 GeV/c, ω 439 MeV, Q =.8 (GeV/c) Observables: A TL = σ(o ) σ(18 o ) σ( o )+σ(18 o ) R TL = σ(o ) σ(18 o ) Kv TL Solid: CC Dashed: PCC Long-dashed: NR+SO Dotted: NR (no SO) Seattle, 1/6/16 p. 1

15 n(p) [fm 3 ] Reduced Cross Section and the RMF RFG HO no FSI, RMF no FSI, projected FSI, RMF n LFD Jastrow CDFM p [fm -1 ] Strong scalar & vector potentials: Significant enhancement of the reduced cross section at high momentum. Role of lower components in the nucleon wave functions (mainly in the final state). Explicit correlations? Seattle, 1/6/16 p. 13

16 INCLUSIVE(e,e ) PROCESSES & SCALING Seattle, 1/6/16 p. 14

17 The SuperScaling Approach (SuSA) Scaling of the first kind below the QE peak (ψ ) Excellent scaling of the second kind in the same region Breaking of scaling above the QE peak (ψ > )= Effects beyond the IA (mainly located in the T channel) LONGITUDINAL RESPONSE SUPERSCALES Seattle, 1/6/16 p. 15

18 The SuperScaling Approach (SuSA).8 Scaling of the first kind below the QE peak (ψ ) RFG.7 exp fit Breaking of scaling above the QE.6peak (ψ > )= Effects beyond the IA PRC6 Excellent (1999) scaling 655 of the second kind in the same region PRL8 (1999) 31 PRC65 () 55 (mainly located in the T channel) LONGITUDINAL RESPONSE SUPERSCALES Experimental superscaling function: asymmetric shape with a long tail extended to positive ψ-values strong constraints to models f(ψ ) ψ Seattle, 1/6/16 p. 15

19 Nucleon wave functions in RIA Solutions of Dirac equation with phenomenological relativistic potentials Ψ B : Bound nucleon wave function= Relativistic Mean Field (RMF) Approach Local potentials from a Lagrangian fitted to properties of nuclear matter and nuclei Seattle, 1/6/16 p. 16

20 Nucleon wave functions in RIA Solutions of Dirac equation with phenomenological relativistic potentials Ψ B : Bound nucleon wave function= Relativistic Mean Field (RMF) Approach Local potentials from a Lagrangian fitted to properties of nuclear matter and nuclei Ψ F : Ejected nucleon wave function= Dependence with final state interactions (FSI) Retain inelastic channels = Use of pure real potentials (no absortion) RMF rrop RPWIA Description of FSI with the RMF approach: Preserves orthogonalization property. Maintains the continuity equation (aprox.). Seattle, 1/6/16 p. 16

21 How Scaling of the 1 er kind behaves (RMF) Scaling of first kind. Results for 1 C(e, e ) and ǫ e = 1 GeV. CC current prescription fl(ψ ) RMF q=1 GeV/c exp ψ Seattle, 1/6/16 p. 17

22 Scaling of the second kind in RIA f(ψ ) CC RMF ψ 1 1 C 16 O 4 Ca CC1 RMF ψ 1 1 C 16 O 4 Ca Scaling of a kind: excellent with the CC current operator Seattle, 1/6/16 p. 18

23 RMF: Comparison with (e,e ) data C CC f L (RMF) f T (RMF) rrop RPWIA exp fit f(ψ ).4.3 CC..1 q=.7 GeV/c ψ Only the description of FSI provided by RMF leads to an asymmetric function f(ψ ) in accordance with the behavior shown by data. Moreover,f T > f L Seattle, 1/6/16 p. 19

24 Asymmetry in the RMF approach The RMF approach produces a shift of strength to higher missing momenta. E eff = E V, M eff = M S; p eff = p +(V S )+M(S V) T N V When increasing T N, the value of p eff decreases due to the strong relativistic potentials.5 RMF-3 RMF-5 TOTAL.5 pm < 3 pm < 5 TOTAL sigma RMF sigma PW sigma CC1, Landau Gauge (nbarns/mev/sr) w (MeV) w (MeV) RMF-3 PW-3 ROP-3 RMF-5 ROP w (MeV) w (MeV) pm<3 pm<5 total Seattle, 1/6/16 p.

25 Other models: th kind scaling & asymmetry RFG: by construction f L (ψ) = f T (ψ) = f(ψ) RPWIA: f L (ψ) = f T (ψ) = f(ψ) symmetric Semi-relativistic (SR) and/or NR approaches with FSI: Woods-Saxon potential: symmetric scaling functions. Dirac Equation-Based (DEB) potential: leads to an asymmetric function f(ψ). fl GeV 1.5 GeV GeV.5 1 ψ ft GeV 1.5 GeV GeV.5 1 ψ In all cases: f L (ψ) = f T (ψ) = f(ψ) Seattle, 1/6/16 p. 1

26 Scaling of the th kind in RMF:T enhancement f(ψ ) ε e = 1 GeV q = 1 GeV/c f(ψ ) RMF CC EMA CC ψ (e, e ) f L f T (e, e ) f L f T RMF CC EMA CC ψ (e, e ) f L f T (e, e ) f L f T Seattle, 1/6/16 p.

27 Scaling in QEL/T -channels Seattle, 1/6/16 p. 3

28 Scaling in QEL/T -channels Seattle, 1/6/16 p. 3

29 INTERACTION OF NEUTRINOS with NUCLEI Seattle, 1/6/16 p. 4

30 RELATIVISTIC IMPULSE APPROXIMATION Incident neutrino interacts with only one nucleon which is then emitted, while the(a 1) remaining nucleons are simply spectators. The weak nuclear current is the sum of single nucleon currents. Target and residual nuclei can be adequately described within an independent particle model. Weak single-nucleon current operator Ĵ µ wsn = Ĵµ V Ĵµ A = F 1 γ µ + i F m N σ µν Q ν +G A γ µ γ 5 + G P m N Q µ γ 5 Neutral Currents: Strangeness content in G E,M,A and dependence with Weinberg angle (nog P ) Charge-changing Currents: Pure isovector form factors F V i = (F p i Fn i ) Seattle, 1/6/16 p. 5

31 RIA & SR approximations. FSI effects dσ (1 39 cm /GeV/sr) deµdωµ RPWIA RMF SR-PWIA SR-FSI 1 C E µ [MeV] Seattle, 1/6/16 p. 6

32 RMF applied to (ν,µ) & Scaling PROCEDURE Evaluate the inclusive(ν,µ) cross section with a specific RIA model and divide it by the corresponding single-nucleon cross section [weighted by the appropriate proton (Z) and neutron (N ) numbers] = THEORETICAL SCALING FUNCTION Does the theoretical RIA scaling function satisfy scaling properties? Scaling of the first kind: f(q,ψ) q f(ψ) Scaling of the second kind: f(ψ) independent on the nucleus Seattle, 1/6/16 p. 7

33 RMF applied to (ν,µ) & Scaling f(ψ ) f(ψ ) f(ψ ).8.7 RPWIA GeV 1.5 GeV GeV PROCEDURE Evaluate the inclusive(ν,µ) cross section with a specific RIA.4 model and divide it by the corresponding single-nucleon cross section [weighted by the appropriate proton (Z) and neutron (N ) numbers] = THEORETICAL SCALING FUNCTION Does rrop the theoretical 1 GeV RIA scaling function satisfy scaling properties? RMF GeV GeV Scaling of the first kind: f(q,ψ) q f(ψ) Scaling of the second kind: ψ GeV 1.5 GeV GeV f(ψ ) f(ψ ) RPWIA rrop.3 f(ψ) independent on the nucleus f(ψ ) RMF ψ 1 C 16 O 4 Ca C 16 O 4 Ca C 16 O 4 Ca Seattle, 1/6/16 p. 7

34 RMF applied to (ν,µ) & Scaling PROCEDURE Evaluate the inclusive(ν,µ) cross section with a specific RIA model and divide it by the corresponding single-nucleon cross section [weighted by the appropriate proton (Z) and neutron (N ) numbers] = THEORETICAL SCALING FUNCTION Does the theoretical RIA scaling function satisfy scaling properties? Scaling of the first kind: f(q,ψ) q f(ψ) Scaling of the second kind: f(ψ) independent on the nucleus Is the function f(ψ) obtained from(ν, µ) cross sections evaluated within RIA consistent with the functionf(ψ) obtained from(e,e ) calculations (with the same model)?, and withf exp (ψ)? Similar scaling function f(ψ) for (e,e ) and (ν,µ) processes? Seattle, 1/6/16 p. 7

35 (e,e ) vs (ν,µ). SuSA vs RMF f(ψ ) RMF CC f L (e, e ) f T (e, e ) (ν µ, µ ) (ν µ, µ + ) exp q =.7 GeV/c ψ Basic result: the function f(ψ) evaluated for(ν, µ) processes agrees better with the contributionf L (ψ) [corresponding to(e,e )] than withf T (ψ). Seattle, 1/6/16 p. 8

36 BASIC CONCLUSIONS: (e,e ) &(ν µ,µ) ELECTRON SCATTERING Superscaling shows up in RIA, even with FSI: rrop & RMF. RMF-FSI description leads to an asymmetric superscaling function which fits data. Some first-kind scaling violation with FSI switched on. Contrary to most non-relativistic models,f L (ψ) f T (ψ) within RMF Seattle, 1/6/16 p. 9

37 BASIC CONCLUSIONS: (e,e ) &(ν µ,µ) ELECTRON SCATTERING Superscaling shows up in RIA, even with FSI: rrop & RMF. RMF-FSI description leads to an asymmetric superscaling function which fits data. Some first-kind scaling violation with FSI switched on. Contrary to most non-relativistic models,f L (ψ) f T (ψ) within RMF Superscaling fulfilled by RIA calculations. NEUTRINO SCATTERING Scaling functions from QE(e,e ) and(ν,µ) cross sections follow similar trends. Differences between(e,e ) and(ν,µ) RIA+RMF results are consistent with the isoscalar/isovector nucleon f.f. contributions in the two processes. Seattle, 1/6/16 p. 9

38 COMPARISON WITH DATA: MiniBooNE, MinerνA, NOMAD & TK Seattle, 1/6/16 p. 3

39 Flux-averaged double-differential CCQE: SuSA & RMF Seattle, 1/6/16 p. 31

40 Antineutrinos: strong enhancement of ph effects? 9.8 < cosθ <.9 d σ/dcosθ/dt µ (1-39 cm ) SuSA SuSAv RMF MiniBooNE 1,5 1 1,5 T µ (GeV) Seattle, 1/6/16 p. 3

41 MiniBooNE & NOMAD: SuSA vs RMF 14 1 σ (1-39 cm ) RFG RMF SuSA SuSAv MiniBooNE NOMAD, E ν (GeV) Seattle, 1/6/16 p. 33

42 MiniBooNE & NOMAD: SuSA vs RMF σ (1-39 cm ) σ (1-39 cm ) RFG RMF SuSA SuSAv MiniBooNE NOMAD RFG RMF SuSA SuSAv MiniBooNE NOMAD,1, E ν (GeV) E ν (GeV) Seattle, 1/6/16 p. 33

43 NOMAD: spinor distortion in RMF σ [1-39 cm ] neutrino 6 4 MiniBooNE NOMAD RMF EMA E ν [GeV] Seattle, 1/6/16 p. 34

44 NOMAD: spinor distortion in RMF σ [1-39 cm ] σ [1-39 cm ] neutrino antineutrino MiniBooNE NOMAD 4 RMF EMA MiniBooNE NOMAD RMF EMA E ν [GeV] E ν [GeV] Seattle, 1/6/16 p. 34

45 PARITY-VIOLATION in ELECTRON SCATTERING Feynman diagrams in first Born approximation: Seattle, 1/6/16 p. 35

46 Scaling in PV (e,e ) responses Phys. Rev. C91, 455 (15) Seattle, 1/6/16 p. 36

47 APPLICATION TO NEUTRAL CURRENT NEUTRINO PROCESSES:(ν,N)ν Seattle, 1/6/16 p. 37

48 NCQE & axial strangeness dσ/dt N (1-4 cm /MeV) CH ; s = -.3 CH ; s =.3 1 C; s = C; s =.3 ε ν = 1 MeV dσ/dt N (1-4 cm /MeV) p; 1 C; s =.3 n; 1 C; s =.3 p; 1 C; s = -.3 n; 1 C; s = T N (MeV) T N (MeV) Seattle, 1/6/16 p. 38

49 NCQE & axial strangeness dσ/dt N (1-4 cm /MeV) CH ; s = -.3 CH ; s =.3 1 C; s = C; s =.3 dσ/dt N (1-4 cm /MeV) ε ν = 1 MeV dσ/dt N (1-4 cm /MeV) CH ; s = -.3 CH ; s =.3 1 C; s = C; s =.3 ε _ ν = 1 MeV dσ/dt N (1-4 cm /MeV) 3 p; 1 C; s =.3 n; 1 C; s =.3 p; 1 C; s = -.3 n; 1 C; s = p; 1 C; s =.3 n; 1 C; s =.3 p; 1 C; s = -.3 n; 1 C; s = T N (MeV) T N (MeV) T N (MeV) T N (MeV) Seattle, 1/6/16 p. 38

50 P/N RATIO: axial strangeness vs axial mass.65 R [p/(p+n)] s = -.3; M A = 1.45 GeV s = -.3; M A = 1.3 GeV s =.; M A = 1.45 GeV s =.; M A = 1.3 GeV s =.3; M A = 1.45 GeV s =.3; M A = 1.3 GeV Correlation between axial strangeness & axial mass: for s-positive (grey band) NCQE is almost insenstive to M A, whereas for s-negative (green) uncertainty due tom A is 1 1% T N (MeV) Seattle, 1/6/16 p. 39

51 Comparison with MiniBooNE data T N (GeV),1,,3,4,5,6,7,8,9 dσ/dq (1-39 cm /GeV ) 1,1 MiniBooNE - CH RFG - CH SUSA - CH RMF - CH RPWIA - CH,1 Galster / M A = 1.3 GeV / g A (s) = / ρs = / µ s =,,4,6,8 1 1, 1,4 1,6 Q QE (GeV ) Seattle, 1/6/16 p. 4

52 È ½ Ã ¾ È ¼ ¾ È ¾ È ¾ È ½ Ã ½ È ¼ ¾ È ¾ È ¾ È ¼ ½ È ½ È ¼ ¾ È ¾ Ã ½ Ã ¾ È ¼ ½ È ¾ È ¼ ¾ È ¾ SuSAv-MEC: QE + p-h MEC + Inelastic È ¼ ½ È ¼ ¾ È ¼ ½ È ¼ ¾ È ¼ ½ È ¼ ¾ MEC É É É µ µ µ É È ¼ ½ È ½ Ã ¾ È ¼ ½ È ½ Ã ½ É É È ½ Ã ¾ È ½ Ã ½ É correlations and -MEC µ µ µ µ Application to CC neutrino scattering processes Seattle, 1/6/16 p. 41

53 Integrated cross section: p-h effects σ ν (1-39 cm ) QE + MEC (L+T) QE (SuSAv) MEC(L+T) MEC (T) MEC (T VA ) MEC (T AA ) MEC (T VV ) NOMAD MiniBooNE E ν (GeV) Seattle, 1/6/16 p. 4

54 Integrated cross section: p-h effects σ ν (1-39 cm ) QE + MEC (L+T) QE (SuSAv) MEC(L+T) MEC (T) MEC (T VA ) MEC (T AA ) MEC (T VV ) NOMAD MiniBooNE σ ν (1-39 cm ) MiniBooNE NOMAD MEC (T VV ) MEC (T AA ) MEC (T VA ) MEC (T) MEC (L+T) QE (SuSAv) QE + MEC (L+T) E ν (GeV) E ν (GeV) Seattle, 1/6/16 p. 4

55 Flux-averaged double-differential CCQE-ν µ & ν µ < cosθ µ <.9 p-h MEC (T) MiniBooNE p-h MEC (L+T) QE QE+MEC(L+T) T µ (GeV) Seattle, 1/6/16 p. 43

56 Flux-averaged double-differential CCQE-ν µ & ν µ < cosθ µ <.9 p-h MEC (T) MiniBooNE p-h MEC (L+T) QE QE+MEC(L+T) < cosθ µ < T µ (GeV).5 1 T µ (GeV) Seattle, 1/6/16 p. 43

57 MINERνA: case of ν e.5 dσ/dθ e (1-39 cm /degree/nucleon) MINERνA MEC QE QE+MEC θ e (deg) Seattle, 1/6/16 p. 44

58 TK experiment: -contribution dσ/dp e (1-39 cm /nucleon/(gev/c)) TK QE MEC 1π QE+MEC+1π p e (GeV/c) Seattle, 1/6/16 p. 45

59 SUMMARY The RIA/RMF describes in a reasonable way QE(e,e ) data, satisfying scaling behavior and providing an asymmetric superscaling L function in accordance with data. Contrary to most NR/SR models (likewise RFG), RMF violates scaling of zeroth order, i.e., f T > f L. This seems to be consistent with(e,e ) data analysis. RMF applied to neutrino scattering also satisfies scaling/superscaling properties. RMF provides results in excellent agreement with SuSA/SuSAv approaches. Significant discrepancy with MiniBooNE data: Important enhancement of p-h effects. RMF results (likewise SuSA/SuSAv) in accordance with NOMAD data. Correlation between axial strangeness and axial mass in NC processes. Some discrepancy with MiniBooNE data: role of p-h?, strangeness?... RMF leads to scaling for the PV (e,e ) responses. Similar to the EM ones. SuSAv-MEC applied to neutrino reactions describes properly MiniBooNE, MinerVa and TK data. Significant enhancement due to ph-mec. Seattle, 1/6/16 p. 46

60 COLLABORATION R. González-Jiménez, G. Megías Universidad de Sevilla J.E. Amaro, I. Ruíz Simo Universidad de Granada M.B. Barbaro Universitá di Torino T.W. Donnelly, C.F. Williamson, Massachusetts Institute of Technology A. Meucci, C. Giusti, Universitá di Pavia M.C. Martínez, E. Moya, J.M. Udías, Universidad Complutense de Madrid A.N. Antonov, M.V. Ivanov INRNE, Sofia C. Maieron Juan Antonio Caballero (6/1/16) Seattle, 1/6/16 p. 47

arxiv: v1 [nucl-th] 10 Apr 2018

arxiv: v1 [nucl-th] 10 Apr 2018 Meson-exchange currents and quasielastic predictions for neutrino-nucleus scattering M.B. Barbaro Dipartimento di Fisica, Universita di Torino and INFN, Torino, Italy E-mail: barbaro@to.infn.it arxiv:.33v