Confronting electron and neutrino scattering Can the axial mass controversy be resolved?

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1 Confronting electron and neutrino scattering Can the axial mass controversy be resolved? Omar Benhar Center for Neutrino Physics, Virginia Tech, Blacksburg, VA INFN and Department of Physics, Sapienza University, I Roma, Italy UVA, Charlottesville February 11, 2014

2 Outline Preamble & motivation Lessons from electron scattering the e + A e + X cross section impulse approximation and beyond Neutrino-nucleus scattering at beam energy < 1 GeV: what do accelerator based neutrino oscillation experiments actually measure? Interpretation and theoretical description of the detected signal in the charged current quasi elastic (CCQE) channel Summary & Outlook Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

3 Preamble & motivation Vast supply of precise e A data available Q 2 = 4E e E e sin 2 θ e 2, x = Q2 2Mω Carbon target Different rection mechanisms contributimg to the mesured cross sections can be readily identified e + A e + X Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

4 Ab initio calculations of the electron scattering cross section (mostly in the quasielastic channel) carried out for lightest nuclei in the low-energy region provide a remarkably good description of the data Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

5 Models based on the same dynamical input and the impulse approximation (IA) scheme have been remarkably successful in describing electron scattering data at larger beam energies, up to few GeV, for a variety of targets The extension of these models to the case of neutrino scattering is needed to reduce the systematic uncertainty of LBL neutrino oscillation experiments Oscillation probablity (two flavors, for simplicity) ( m P να ν β = sin 2 2θ sin 2 2 ) L 4E ν Sensitivity reaches its maximum at [ L ] E ν 0.5 GeV 250 km Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

6 Theory of electron-nucleus scattering Double differential electron-nucleus cross section dσ ea dω e de e = α2 E e Q 4 L µν W µν E A e L µν is fully specified by the measured electron kinematical variables the determination of the target response tensor = 0 J µ A n n Jν A 0 δ(4) (P 0 + k e P n k e ) W µν A n requires the description of the target internal dynamics and electromagnetic current approximations needed to go beyond the non relativistic regime Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

7 Why worry about relativity dσ/d q (10 38 cm 2 /GeV 2 ) MINERνA q (GeV) 2 ν µ ν µ 3 dσ/d q (10 38 cm 2 /GeV 2 ) MiniBooNE 1 q (GeV) 2 ν µ ν µ 3 q -dependence of the CCQE cross section averaged with the Minerνa (left) and MiniBooNE (right) fluxes Unlike the initial state, the nuclear current and the final hadronic state can not be described using non relativistic many-body theory Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

8 The impulse approximation The IA amounts to the replacement 2 2 q,ω q,ω x i Σi J µ A = j µ i, X p X (A 1) i Nuclear dynamics and electromagnetic interactions are decoupled dσ A = d 3 kde dσ N P h (k, E) The electron-nucleon cross section dσ N can be written in terms of stucture functions extracted from electron-proton and electron-deuteron scattering data The hole spectral function P h (k, E), momentum and energy distribution of the knocked out nucleon, can be obtained from ab initio many-body calculations Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

9 Hole state spectral function Definition P h (k, E) = n a k 0 2 δ(e 0 + E E n ) n Exact calculations have been carried out for A = 2, 3. Accurate results, obtained using correlated wave functions, are also available for nuclear matter. For medium-haevy nuclei, approximated spectral functions have been constructed combining nuclear matter results and experimental information from (e, e p) experiments in the local density approximation (LDA) P h (k, E) = P exp (k, E) + P corr (k, E) P exp (k, E) = Z n φ n (k) 2 F n (E E n ) P corr (k, E) = n d 3 r ρ A (r) P NM corr(k, E; ρ = ρ A (r)) Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

10 Spectral function and momentum distribution of Oxygen shell model states account for 80% of the strenght the remaining 20%, arising from NN correlations, is located at high momentum and large removal energy (k k F, E ɛ) Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

11 Measured correlation strength the correlation strength in the 2p2h sector has been measured by the JLAB E Collaboration using a carbon target strong energy-momentum correlation: E E thr + A 2 A 1 k 2 2m n(p m ) [fm 3 sr -1 ] p m [GeV/c] Measured correlation strength 0.61 ± 0.06, to be compared with the theoretical predictions 0.64 (CBF) and 0.56 (G-Matrix) Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

12 Beyond IA: final state interactions (FSI) The measured (e, e p) x-sections provide overwhelming evidence of the importance of FSI q,ν + q,ν dσ A = d 3 kde dσ N P h (k, E)P p ( k + q, ω E) the particle-state spectral function P p ( k + q, ω E) describes the propagation of the struck particle in the final state the IA is recovered replacing P p ( k + q, ω E) with the particle spectral function of the non interacting system Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

13 FSI (continued) effects of FSI on the inclusive cross section (A) shift in energy transfer mean field of the spectators (B) redistributions of the strenght coupling of 1p 1h final state to np nh final states high energy approximation (A) the struck nucleon moves along a straight trajectory with constant velocity (B) the fast struck nucleon sees the spectator system as a collection of fixed scattering centers. δ(ω E k + q 2 + m 2 ) Tδ( ω E k + q 2 + m 2 ) +(1 T)f ( ω E k + q 2 + m 2 )) the nuclear transparency T and the folding function f can be computed within nuclear many-body theory using the measured nucleon-nucleon scattering amplitude Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

14 Theory vs data Recall: theoretical calculations involve no adjustable parameters Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

15 The quasi elastic (QE) sector Elementary interaction vertex described in terms of the vector form factors, F (p,n) 1 and F (p,n) 2, precisely measured over a broad range of Q 2 Position and width of the peak are determined by P h (k, E) The tail extending to the region of high energy loss is due to nucleon-nucleon correlations in the initial state, leading to the occurrence of two particle-two hole (2p2h) final states Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

16 Extension to CCQE neutrino nucleus scattering Consider the data sample of charged current QE data collected by the MiniBooNE Collaboration using a CH 2 target The measured double differential cross section is averaged over the energy of the incoming neutrino, distributed according to the flux Φ dσ A = 1 dσ A de ν Φ(E ν ) dt µ d cos θ µ N Φ de ν dt µ d cos θ µ In addition to F 1 and F 2, the QE electron-nucleon cross section is determined by the axial form factor F A, assumed to be of dipole form and parametrized in terms of the axial mass M A According to the paradigm succesfully employed to describe electron scattering data, in order to minimize the bias associated with nuclear effetcs, M A must be determined from measurement carried out using a deuterium target. The resulting value is M A = 1.03 GeV Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

17 Analysis of CCQE data MiniBooNe CCQE data MiniBooNE flux Theoretical calculations carried out setting M A = 1.03 GeV. A fit to the data within the Relativistic Fermi Gas Model yields M A = 1.25 GeV. Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

18 Contribution of different reaction mechanisms In neutrino interactions the lepton kinematics is not determined. The flux-averaged cross sections at fixed T µ and cos θ µ picks up contributions at different beam energies, corresponding to a variety of kinematical regimes in which different reaction mechanisms dominate x = 1 E ν = GeV, x = 0.5 E ν = GeV Φ(0.975)/Φ(0.788) = 0.83 Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

19 Flux averaged electron-nucleus x-section The electron scattering x-section off Carbon at θ e = 37 has been measured for a number of beam energies MIT-Bates data compared to theoretical calculations including QE scattering only Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

20 Where does the excess strength come from? It has been suggested that 2p2h (CCQE like) final states provide a large contribution to the measured neutrino cross section Two particle-two hole final states may be produced through different mechanisms Initial state correlations : lead to the tail extending to large energy loss, clearly visible in the calculated QE cross section. The corresponding strength is consistent with the measurements of the coincidence (e, e p) x-section carried out by the JLAB E Collaboration. Final state interactions : lead to a redistribution of the inclusive strength, mainly affecting the region of x > 1, i.e. low energy loss, where the cross section is small Coupling to the two-body current : leads to the appearance of strength at x < 1, mainly in the dip region between the QE and -excitation peaks the description of the measured neutrino cross sections requires that all the above mechanism be taken into account in a consistent fashion Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

21 The nuclear current operator The nuclear current includes one- and two-nucleon contributions A A J µ A = j µ µ i + j ij i=1 j>i=1 j i µ defined in terms of nucleon structure functions j ij µ takes into acount interactions in which the momentum transfer is shared between two nucleons Recall: unlike the ground state, the nuclear current operator depends on momentum trasfer. Hence, the relativistic description may become inadeqaute Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

22 this framework. Now in a realistic approach of the nuclear dynamics correlations the nuclear region of response is not MEC restrictedwithin to the Fermi motion bandindependent around the quasielastic lineparticle model (IPM) (as in Fig. 1)butitcoversthewholeω and q plane from multinucleon emission. As a consequence, for a given set of values of E J. µ and Nieves θ, allvaluesoftheenergytransferω, et, Phys. Lett. B 707, henceofthe 72 neutrino energy, E ν = E µ + ω, contributeandoneexplores the full (2012) energy spectrum of neutrinos above the muon energy. 5 The results of our present evaluation with the relativistic corrections of the double differential cross section are Full Model displayed in Fig. 2, withandwithout Full QE the (with inclusion RPA) ofthe np-nh 2 component and compared to Multinucleon the experimental data. No RPA, No Multinuc. This evaluation, like all those in this No article, RPA,No is Multin., donemwith A =1.32 the free value of the axial mass. The agreement is quite good in all 1.5the measured ranges once the multinucleon M A =1.049 component GeV is incorporated. Similar conclusions have been recently reported in Ref. [9]. The relativistic corrections 0.80 are < cos significant, " 1 µ < 0.90 as illustrated in Fig. 3 which compares the two approaches for the genuine quasielastic contributions. The relativistic treatment, which suppresses the kinematical pathologies, improves the 0.5 description, in particular, in the backward direction. This is illustrated in Fig. 4 in the case 0.4 GeV<T µ < 0.5 GeVin which 0 the 2p-2h component was added for comparison with data. The good 0.5agreement with1 data of Fig is absent in the T µ (GeV) nonrelativistic case. FIG. 3: MuonOur angle and responses energy distribution ared 2 σ/d described, cos θµdtµ for 0.80 as< in cos θµ our < previous Experimentalworks data from Ref. [3,4], [5] and calculation with MA =1.32 GeV are multiplied by 0.9. Axial mass for the other curves is MA =1.049 GeV. in the framework of random phase approximation. Its role with electron, is shown photon and pion in probes Figs. and contains 5 andno additional 6 where free parameters. the double RPA and multinucleon differential knockout have been found to be essential for the description of the data. Our main conclusion is that MiniBooNE data are fully compatible cross with former sections determinations asofathe function nucleon axial mass, ofboth cosθ using neutrino or T and µ electron are displayed beams in contrast with several previous analyses. The results also suggest that the neutrino flux could have been underestimated. Besides, with we have found andthat without the procedure RPA. commonly The usedrpa to reconstruct produces the neutrinoa energy quenching for quasielasticand events from the muon angle and energy could be unreliable for a wide region of the phase space, due to the large importance of multinucleon somevents. shift toward larger angles or larger T µ.infig.6 we It is clear that experiments on neutrino reactions on complex nuclei have reached a precision level that requires for a quantitative present descriptionthe of sophisticated comparison theoretical approaches. with data Apartadding from being important the np-nh in the studyto of neutrino the physics, these experiments are starting to provide very valuable information on the axial structure of hadrons. genuine QE with or without RPA. The fit is significantly d 2!/dT µ d cos" µ (10-38 cm 2 /GeV) Acknowledgments We thank L. Alvarez Ruso, R. Tayloe and G. Zeller for useful discussions. This research was supported by DGI and FEDER funds, under contracts FIS /FIS, FIS , and the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD ), by Generalitat Valenciana contract PROMETEO/2009/0090 and by the EU HadronPhysics2 project, grant agreement n I.R.S. acknowledges support from the Ministerio de Educación. embodied in the Landau-Migdal parameter g.alargepart of this quenching arises from the mixing of the p-h states with -hole ones. This is the Lorentz-Lorenz effect, which concerns exclusively the spin isospin response, hence the axial or magnetic matrix elements. In the graphical illustration of the response, the Lorentz-Lorenz effect on the quasielastic one is illustrated in Fig. 7. Figure6 shows the dominance of d 2 σ/(dcos θ dt µ ) (10-39 cm 2 /GeV) M. Martini et al, Phys. Rev.C 80, (2009); MiniBooNE QE RPA +np-nh QE bare + np-nh QE bare QE RPA QE RPA without LL quenching 0.8 < cos θ < T µ (GeV) After the inclusion of fully relativistic MEC, calculations based on the IPM provide a quantitative description of the MiniBooNE data FIG. 6. (Color online) MiniBooNE flux-averaged CC quasielastic νµ- 12 C double differential cross section per neutron for 0.8 < cosθ < 0.9 as a function of the muon kinetic energy. (Dashed curve) Pure quasielastic calculated in RPA; (solid curve) RPA quasielastic with the inclusion of np-nh component; (dot-dot-dashed) bare quasielastic with the inclusion of np-nh component; (dot-dashed curve) bare quasielastic; (dot-dashed-dashed) RPA quasielastic without the Lorentz-Lorenz (LL) quenching. However, within these approaches NN correlations are neglected altogether, or included using oversimplified ad hoc procedures Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

23 MEC within realistic nuclear models the interplay of MEC and correlation effects can be studied within nonrelativistic models, that can be solved exactly using stochastic methods for nuclei as heavy as as 12 C inclusion of the two-body current leads to an enhancement of the integrated electromagnetic response of light nuclei in the transverse channel S T (q) = dωs T (q, ω), where and S T (q, ω) = S xx (q, ω) + S yy (q, ω), S αβ = 0 JA α N N Jβ A 0 δ(e 0 + ω E N ). N Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

24 . In order current weak interactions is discussed in Ref. [15]. rator betate to be down the contribution of the terms arising from inter- We have employed the approach of Ref. [14] to pin n-nucleon ference between correlations and MEC to the transverse picture, Sum rule sumof rule, the which electromagnetic is long known response to be strongly of 12 C inaffected the transverse by lear wave channel processes involving two-nucleon currents. Interference between MEC and correlation amplitudes ween tararticipattribution which the omentum and, origtruck nulso result Sxx yy q J T b J b T J T b J T b J b T J b T J T b J b T J T b J b T J T b J b T J T b J b T moderate in which be applidescribed on realnstrained FIG. 1: terms Sum rule areof large. the electromagnetic MEC and correlations response of must carbon bein treated Interference the transverse channel. The line marked with crosses shows nd threevistic cal- consistently the results obtained including the one-nucleon current only, while the one marked with squares corresponds to the full Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

25 Generalization of IA factorization Using relativistic MEC and a realistic description of the nuclear ground state requires the extension of the IA scheme to two-nucleon emission amplitudes Rewrite the hadronic final state n in the factorized form X p, p X (A 2) X j µ ij 0 d 3 kd 3 k M n (k, k ) pp j µ ij kk The amplitude M n (k, k ) = n (A 2), k, k 0 is independent of q and can be obtained from non relativistic many-body theory Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

26 Two-nucleon spectral function Calculations have been carried out for uniform isospin-symmetric nuclear matter P(k 1, k 2, E) = M n (k 1, k 2 ) 2 δ(e + E 0 E n ) n n(k 1, k 2 ) = de P(k 1, k 2, E) Relative momentum distribution n(q) = 4π Q 2 ( Q d 3 q n 2 + q, Q ) 2 q q = k 1 + k 2, Q = k 1 k 2 2 Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

27 Summary & Outlook Ab initio calculations based on nuclear many-body theory and the available experimantal information on electron-nucleon interactions provide a remarkably accurate description of the nuclear cross sections in the impulse approximation regime The generalization to neutrino-nucleus scattering, needed to reduce the systematic uncertainty of LBL neutrino oscillation experiments, involves additional difficulties, arising from the flux average, and requires a consistent description of all relevant reaction mechanisms Calculations carried out using realistic models of nuclear effects suggest that the excess cross section observed in the quasi elastic sector can be explained without advocating medium modifications of the axial form factor. The implementation of realistic nuclear models in event generation codes employed for data analysis of neutrino oscillation searches is under way Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

28 Background slides Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

29 ab initio many-body approach The nucleus is seen as a collection of pointlike protons and neutrons interacting through the hamiltonian p 2 i H = 2m + v ij + i j>i k>j>i U ijk the potentials are determined by a fit to the properties of the exactly solvable two- and three-nucleon systems once the eigenstates of H are known from the solution of the Schrödinger equation H n = E n n calculations of nuclear observables do not involve any additional parameters Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

30 Correlated basis function formalism The eigenstates of the nuclear hamiltonian are approximated by the set of correlated states, obtained from independent particle model states [e.g. Fermi Gas (FG) states for nuclear matter] n = F n MF n MF F F n MF 1/2 = 1 F n MF, Nn F = S the structure of the two-nucleon correlation operator reflects the complexity of nuclear dynamics f ij = [f TS (r ij ) + δ S1 f tt (r ij )S ij ]P ST S,T=0,1 P ST spin isospin projector operator, S ij = σ α i σβ j j>i r α ij rβ ij shapes of f TS (r ij ) and f tt (r ij ) determined form minimization of the ground-state energy r 2 ij f ij δ αβ Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27

31 Nucleon-nucleon potential and correlation functions Omar Benhar (Virginia Tech & INFN) UVA, Charlottesville February 11, / 27