Using Electron Scattering Superscaling to predict Charge-changing Neutrino Cross Sections in Nuclei Maria B. Barbaro University of Torino (Italy) and INFN Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 1
Collaboration J.E. Amaro (Granada) M.B. B. (Torino) J.A. Caballero (Sevilla) T.W. Donnelly (MIT) C. Maieron (Catania) A. Molinari (Torino) I. Sick (Basel) Phys. Rev. C 71 (2005) 015501 [arxiv:nucl-th/0409078] Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 2
Motivation The interpretation of neutrino experiments requires good control of nuclear physics From (e, e ) we know that nuclear effects can be large Can we extract predictions for ν-a in the QEP and regions from e-a experimental cross sections? General strategy [1.] Extract scaling function from (e, e ) data, embodying the nuclear physics content of the problem [2.] Invert the scaling approach to predict (ν l, l) cross sections in the few-gev region Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 3
Summary General formalism for charge-changing ν-nucleus scattering y-scaling in inclusive electron scattering: the quasi-elastic peak (QEP) the -resonance region Predictions for (ν, µ) and ( ν, µ) cross sections Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 4
Charge-changing neutrino cross section Cross section in lab. frame [ d 2 σ ] dωdk χ σ 0 F 2 χ χ = + χ = for neutrinos for antineutrinos σ 0 (G cos θ c) 2 [ 2π 2 k cos θ/2 ] 2 Fermi G = 1.16639x10 5 GeV 2 Cabibbo cos θ c = 0.9741 θ=generalized scattering angle tan 2 θ/2 Q 2 v 0, v 0 (ɛ + ɛ ) 2 q 2 = 4ɛɛ Q 2 Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 5
Weak responses Generalized Rosenbluth decomposition (q along z) ] Fχ 2 = [ V CC R CC + 2 V CL R CL + V LL R LL + V T R T [ + χ 2 V ] T R T Hadronic tensor: W µν = fi δ(e f E i ω) f J µ (Q) i i J ν (Q) f R CC = W 00, R CL = W 03, R LL = W 33 R T = W 11 + W 22, R T = iw 12 Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 6
Nuclear weak current: J µ = J µ V J µ A Vector and axial-vector contributions R i = Ri V V R T = RT V A + R AA i, i = CC, CL, LL, T R V V CL = ω q RV V CC V CC RCC V V + 2 V CL RCL V V + V LL RLL V V = V L RL V V Full response: F χ 2 = X V V L R V V LL = ω2 q 2 RV V CC CVC X V V L V CC R AA CC + 2 V CL R AA CL + V LL R AA LL X AA V T [ R V V T + R AA T C/L Longitudinal Collapse does not occur for the AA terms. ] XT Transverse 2 V T RT V A X T V/A interference + XC/L AA + X T + χx T Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 7
Relativistic Fermi Gas On-shell nucleons move freely (except for Pauli blocking) inside the nucleus with E p = p 2 + m 2 N and are described by free Dirac spinors u(p, s) One free parameter: k F Lorentz covariance: W µν is a Lorentz tensor Gauge invariance: Q µ J µ = 0 vector current conservation Analytic expressions for response functions Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 8
Analytic integration yields W µν X(RF G) (q, ω) = N m NT F Q 2 k 3 F q U µν X (q, ω)f RF G(ψ X ) X = QE, ; N = Z, N; T F = ɛ F m N U µν X single nucleon tensor RFG superscaling function f RF G (ψ) = 3 4 (1 ψ2 )θ(1 ψ 2 ) is universal: when plotted against ψ no dependence is left either on momentum transfer (scaling of the I kind) or on nuclear species via k F (scaling of the II kind) Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 9
I + II kind scaling superscaling RFG scaling variables (X = QE, ) T X P +1 ω ω 0 X ψ X (q, ω) = T F P 1 ω ω X ωx P = q 2 + m 2 X m N, T0 X = q ρ 2 2 X + Q2 ωρ 4m 2 N 2 X m N, RFG response region 1 ψ X 1 ρ X = 1 + m2 X m2 N Q 2 QE and peaks: ψ X = 0 Energy shift: ω ω = ω E shift ψ X ψ X Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 10
QE CC single nucleon responses Vector and axial-vector currents: j µ = j µ V jµ A [ j µ V = ū(p ) F (1) 1 γ µ + i ] F (1) 2 σ µν Q ν u(p ), 2m N [ j µ A = ū(p ) G A γ µ + 1 ] G P Q µ γ 5 u(p ), 2m N Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 11
s.n. response functions (at leading order in k F /m N ) RL V V = q2 Q 2 R AA CC = ω2 Q 2 R AA LL = q2 Q 2 [ [ [ G (1) E G (1) A G (1) A ( RT V A = 1 m N Q 2 1 + Q2 4m 2 N ] 2 R V V T = Q2 2m 2 N ] 2 RCL AA = ωq Q 2 ] 2 R AA T = 2 ) G (1) M G(1) A G A G A Q2 4m 2 N [ G (1) M [ ] 2 G (1) A ( 1 + Q2 4m 2 N G P ] 2 ) [ ] G (1) 2 A Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 12
single-nucleon responses Elementary reactions ν µ p µ ++, ν µ n µ +, ν µ p µ + 0, ν µ n µ + Associated currents J µ N (q) = T ū( ) α (p, s )Γ αµ u(p, s) T = 3 for ++ and production and = 1 for + and 0 production, u ( ) α (p, s ): Rarita-Schwinger spinor Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 13
Vertex tensor [Alvarez-Ruso, Singh, Vicente Vacas, PRC 59, 3368 (1999)] Γ αµ = [ C V 3 m N (g αµ q/ q α γ µ ) + CV 4 + CV 5 m 2 N + ] (g αµ q p q α p µ ) m 2 N γ 5 [ C A 3 (g αµ q/ q α γ µ ) + CA 4 m N m 2 N ] q α q µ + C A 5 g αµ + CA 6 m 2 N ( g αµ q p q α p µ ) ( g αµ q p q α p µ) CVC C V 6 = 0 PCAC C A 6 = CA 5 m2 N /(m2 π + Q 2 ) Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 14
hadronic tensor w µν = m m N Tr { J µ N (q)j ν N (q) = w µν V V + wµν AA + wµν V A w µν V V = w 1V (g µν + qµ q ν Q 2 ) + w 2V (k µ + 2ρ q µ )(k ν + 2ρ q ν ) w µν AA = w 1A(g µν + qµ q ν Q 2 ) + w 2A(k µ + 2ρ q µ )(k ν + 2ρ q ν ) u 1A q µ q ν Q 2 + u 2A(2q µ k ν + 2k µ q ν ) w µν V A = 4iw 3ɛ αβµν k α q β w i w i (C V,A K ) are obtained by performing the traces } Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 15
Scaling in quasielastic (e, e ) Basic assumption: the QEP is dominated by free ejection of one nucleon e' e γ Q µ µ A-1 P µ P A P µ N e' e γ Q µ ~ σ en S µ P N P A-1 µ P A ~ PWIA µ Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 16
Guided from the RFG result, define an experimental QE scaling function as F (q, ψ QE) [d2 σ/dω e dω] QE S QE Dividing function S QE σ M [ v L G QE L ] + v T G QE T G QE L = q Q 2 G QE T = Q2 2q [ ] ZG 2 Ep + NG 2 En + O(η 2 F ) [ ] ZG 2 Mp + NG 2 Mn + O(η 2 F ) Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 17
Scaling of first kind F (q, ψ ) q F (ψ ) F (, ψ ) Superscaling function f QE (q, ψ ) k F F QE (q, ψ ) Extensively studied in Day et al., Phys.Rev.Lett.59(1987)427; Day, McCarthy, Donnelly & Sick, Ann.Rev.Part.Sci.40 (1990) 357; Donnelly & Sick, Phys.Rev.C60,065502 (1999); Phys.Rev.Lett.82, 3212 (1999); Maieron, Donnelly & Sick, Phys.Rev.C65,025502 (2002). Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 18
Donnelly & Sick, Phys.Rev.C60,065502 (1999); Phys.Rev.Lett.82, 3212 (1999)
Scaling og I kind Donnelly & Sick, Phys.Rev.C60,065502 (1999); Phys.Rev.Lett.82, 3212 (1999)
Scaling of II kind (E e = 3.6 GeV, θ e =16 deg.) - linear scale Donnelly & Sick, Phys.Rev.C60,065502 (1999); Phys.Rev.Lett.82, 3212 (1999)
Scaling of II kind (E e = 3.6 GeV, θ e =16 deg.)-semilog scale Donnelly & Sick, Phys.Rev.C60,065502 (1999); Phys.Rev.Lett.82, 3212 (1999)
Donnelly & Sick, Phys.Rev.C60,065502 (1999); Phys.Rev.Lett.82, 3212 (1999)
Longitudinal superscaling function Donnelly & Sick, Phys.Rev.C60,065502 (1999); Phys.Rev.Lett.82, 3212 (1999)
Transverse superscaling function Donnelly & Sick, Phys.Rev.C60,065502 (1999); Phys.Rev.Lett.82, 3212 (1999)
A transverse scaling function can be constructed where no LT separation is available. Assumption: f L is universal. Then 1. Σ L = 1 k F f L σ M v L G L, 2. Σ T = d2 σ dω e dω Σ L 3. f T (ψ ) = Σ T σ M v T G T Maieron, Donnelly & Sick, Phys.Rev.C 65, 025502 (2002)
8 f T 7 6 5 10 0 10-1 10-2 10-3 4 10-4 -2-1.5-1 -0.5 0 0.5 1 3 2 E e θ e = 3.6 GeV = 16 o 1 C Al Fe Au 0-2 -1 0 1 2 3 4 5 Maieron, Donnelly & Sick, Phys.Rev.C 65, 025502 (2002) ψ'
10 f T 8 6 C Fe Au θ e =15 23 30 37 45 55 74 4 2 0-0.5 0 0.5 1 1.5 2 2.5 3 Maieron, Donnelly & Sick, Phys.Rev.C 65, 025502 (2002) ψ'
the longitudinal part of the cross section superscales the transverse part does not : reasonably good scaling of the second kind, but something is breaking the scaling of the first kind, especially above the QE peak: inelastic scattering from the nucleons MEC and their associated correlations (required by gauge invariance) in both 1p-1h and 2p-2h channels initial and final state interaction effects
Fit of QE Superscaling Function
Scaling in the region The scaling analysis performed in the QEP can be extended to the -resonance peak To isolate the contributions in the region, we use the following procedure: [ ] d 1. 2 σ dω e = 1 dω k QE F f QE (ψ QE )σ M(v L G L + v T G T ) [ ] [ ] [ ] 2. = d 2 σ dω e dω d 2 σ dω e dω tot 3. superscaling function d 2 σ dω e dω QE
f (q, ψ ) k F [d 2 σ/dω e dω] σ M[v L G L +v T G T ] Dividing factors G L = κ 2τ [ N {( 1 + τρ 2 ) w 2 (τ) w 1 (τ) }] + O(η 2 F ) G T = 1 κ [ N { w 1 (τ) }] + O(η 2 F )
N single-baryon responses: w 1 (τ) = ν 1 w 2 (τ) = ν 2 [ G 2 M, (τ) + 3G 2 E, [ (τ)] G 2 M, (τ) + 3G 2 E, (τ) + 4τ ] G 2 µ C, (τ) 2 Electric, magnetic and Coulomb N form factors G M, (τ) = 2.97g (Q 2 ) G E, (τ) = 0.03g (Q 2 ) G C, (τ) = 0.15G M, (τ) g (Q 2 ) G Ep(τ) 1 + τ ; G Ep (τ) = 1 [1 + 4.97τ] 2
By doing this subtraction, we remove the impulsive L and T contributions that arise from elastic en scattering, leaving (at least) MEC effects with their associated correlations ( 10-20%) impulsive contributions arising from inelastic en scattering This approach can work only at ψ < 0, since at ψ > 0 other resonances and the tail of DIS start contributing Plot f vs ψ
Amaro et al., Phys.Rev.C71, 015501 (2005) [ 12 C and 16 O data, energies spanned from 0.30 to 4 GeV, angles from 12 to 145 deg.]
Superscaling Function Amaro et al., Phys.Rev.C71, 015501 (2005) [ 12 C and 16 O data, energies spanned from 0.30 to 4 GeV, angles from 12 to 145 deg.]
Amaro et al., Phys.Rev.C71, 015501 (2005)
Test of Superscaling Functions Amaro et al., Phys.Rev.C71, 015501 (2005)
Test of Superscaling Functions Amaro et al., Phys.Rev.C71, 015501 (2005)
Test of Superscaling Functions
Predictions for (ν, µ) f RF G (ψ QE ) f QE (ψ QE ) f RF G (ψ ) f (ψ )
(ν, µ) cross sections; 12 C
( ν, µ) cross sections; 12 C
Angular dependence of ν cross section at the peak
Angular dependence of ν cross section at the peak
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Comparison of superscaling with RFG
Results Dramatic difference between ν and ν at backward angles, since X T X T (constructive interference in ν scattering, cancellation in ν): small changes in the model could have large effects on the predictions for ν In the ν case the T and T responses dominate (CC, CL, LL < 5%) in both QEand regions The QE and contributions are comparable for θ 90 o RFG (and mean field) predictions differ significantly from the scaling predictions Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 49
Conclusions (e, e ) Scaling is observed in inclusive electron scattering for high energies (several hundred MeV to a few GeV) from the scaling region up to the peak. For the longitudinal contribution it is possible to find a scaling function f QE which, when plotted versus ψ QE, is seen to superscale. This universal scaling function embodies the basic nuclear dynamics in the problem. The scaling function f QE is used to obtain the impulsive part of the transverse response Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 50
The residual is analyzed in terms of a scaling function f by dividing by the elementary N cross section and plotted versus a scaling variable ψ Very good representation of inclusive electron scattering at relatively high energies up to the peak. (ν, l) Invert the scaling procedure: use the empirical scaling functions and the N N and N weak amplitudes to predict charge-changing νa and νa cross sections for the same range of kinematics. Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 51
Future work (in progress): 1. extend the scaling approach to neutral current neutrino and antineutrino scattering 2. microscopic model: role of MEC and associated correlations in ph and 2p2h channels 3. explaining the specific nature of the scaling functions (in particular the asymmetric shape) Maria B. Barbaro NUFACT05, Frascati, June 21-26, 2005 Page 52
J.A. Caballero et al., nucl-th/0504040 Page 53
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