Understanging ν–nucleus interactions.Where we are, and why it matters

Omar Benhar

INFN and Department of Physics, “Sapienza” UniversityI-00185 Roma, Italy

IPPP/NuSTEC Topical Meeting on Neutrino-Nucleus ScatteringIPPP, Durham, April 18-20, 2017

OUTLINE

? Varius, Multiplex, Multiformis1

? Where we are: the issue of degeneracy

? Unraveling the nuclear cross section

? Impact on neutrino energy reconstruction

? Summary & Outlook

1Varius, Complex, Manifold. M. Yourcenar, Memoirs of Hadrian (Modern Library,New York, 1984)p.27

1 / 16

VARIUS, MULTIPLEX, MULTIFORMIS

? eA vs νA x-section: the issue of flux average

I electron-nucleus scatteringcross section at beam energyaround 1GeV

I electron-carbon cross sectionat beam energy between ∼ 0.7and ∼1.3 GeV.

? The flux integrated cross section at fixed emission angle andkinetic energy of the outgoing lepton picks up contributionsfrom different reaction mechanisms

2 / 16

THE LEPTON-NUCLEUS X-NECTION

? Double differential cross section of the process `+A→ `′ +Xat fixed beam energy

dσAdΩk′dk′0

∝ LµνWµνA

I Lµν is fully specified by the lepton kinematical variablesI The determination of the nuclear response tensor

WµνA =

∑N

〈0|JµA†|N〉〈N |JνA|0〉δ(4)(P0 + k − PN − k′)

requires the description of the target initial and final states, and ofthe nuclear current operator

I In the non-relativistic regime all elements can be consistentlyobtained from a realistic microscopic Hamiltonian, stronglyconstrained by phenomenology

I Accurate calculations of the cross section at momentum transfer<∼ 400 MeV can be carried out using Quantum Monte Carlotechniques for targets of mass number A ≤ 12

3 / 16

MODELLING dσA AT LARGE MOMENTUM TRANSFER

? In the kinematical regime in which the non-relativisticdescription breaks down, one has to resort to approximationschemes

I independent particle model for the target ground state,supplemented with diagrammatic expansions based on somewhatsimplified models of nuclear dynamics (Valencia model, see Juan’stalk; Martini et al model)

I y-scaling and superscaling analysis of electron scattering data,allowing to obtain the νA cross section from a universal scalingfunction (originally developed by Donnelly, McCarthy and Sick,recently extended by Megias et al)

I factorization ansatz and spectral function formalism, allowing todecouple ground state dynamics—which can be accurately treatedwithin nuclear many-body theory —from the elementaryelectroweak interaction vertex (OB et al)

I Giessen Boltzmann-Uehling-Uhlenbeck (GiBUU) formalism (seeUlrich’s talk)

4 / 16

WHERE WE ARE

? Over the ∼ 15 years since the first NuINT Workshop—the postFermi gas age—a number of models taking into account both thedynamcs of strong interactions and the variety of reactionmechanisms contributing to the nuclear cross section have beendeveloped

? Electron scattering data, mainly inclusive cross sections, havebeen exploited to assess the validity of the some of proposedmodels

? Several models have achieved the degree of maturity requiredfor a meaningful comparison between their predictions and themeasured neutrino-nucleus cross sections

5 / 16

12C(e, e′): FACTORIZATION vs SUPERSCALING

I N. Rocco et al

where dσ denotes the cross section in the absence of FSI,the effects of which are accounted for by the foldingfunction

fqðωÞ ¼ffiffiffiffiffiffiTA

pδðωÞ þ ð1 −

ffiffiffiffiffiffiTA

pÞFqðωÞ: ð10Þ

The above equations show that inclusion of FSI involvesthree elements: (i) the real part of the optical potential UVextracted from proton-carbon scattering data [28], respon-sible for the shift in ω, (ii) the nuclear transparency TAmeasured in coincidence ðe; e0pÞ reactions [29], and (iii) afunction FqðωÞ, sharply peaked at ω ¼ 0, whose width isdictated by the in-mediumNN scattering cross section [27].A comprehensive analysis of FSI effects on the electron-

carbon cross sections has been recently carried out by theauthors of Ref. [15]. In this work we have followed closelytheir approach, using the same input.Figure 3 illustrates the effects of FSI on the electron-

carbon cross section in the kinematical setups of Fig. 2. InFig. 3(a), both the pronounced shift of the quasielastic peakand the redistribution of the strength are clearly visible, andsignificantly improve the agreement between theory anddata. For larger values of Q2, however, FSI play a lessrelevant, in fact almost negligible, role. This feature isillustrated in Fig. 3(b), showing that at beam energy Ee ¼1.3 GeV and scattering angle θe ¼ 37.5 deg, correspond-ing toQ2 ∼ 0.5 GeV2, the results of calculations carried out

with and without inclusion of FSI give very similar results,yielding a good description of the data.Note that, being transverse in nature, the calculated two-

nucleon current contributions to the cross sections exhibit astrong angular dependence. At Ee ¼ 1.3 GeV, we find thatthe ratio between the integrated strengths in the 1p1h and2p2h sectors grows from 4% at electron scattering angleθe ¼ 10 deg to 46% at θe ¼ 60 deg.The results of our work show that the approach based on

the generalized factorization ansatz and the spectral func-tion formalism provides a consistent framework for aunified description of the electron-nucleus cross section,applicable in the kinematical regime in which relativisticeffects are known to be important.The extension of our approach to neutrino-nucleus

scattering, which does not involve further conceptualdifficulties, may offer new insight into the interpretationof the cross section measured by the MiniBooNECollaboration in the quasielastic channel [30,31]. Theexcess strength in the region of the quasielastic peak isin fact believed to originate from processes involvingtwo-nucleon currents [32–34], whose contributions areobserved at lower muon kinetic energy as a result of theaverage over the neutrino flux [35]. The strong angulardependence of the two-nucleon current contribution mayalso provide a clue for the understanding of the differencesbetween the quasielastic cross sections reported by theMiniBooNE Collaboration and the NOMAD Collaboration[36], which collected data using neutrino fluxes withvery different mean energies: 880 MeV and 25 GeV,respectively [35].As a final remark, it has to be pointed out that a clear-cut

identification of the variety of reaction mechanisms con-tributing to the neutrino-nucleus cross section will require acareful analysis of the assumptions underlying differentmodels of nuclear dynamics. All approaches based on theindependent particle model fail to properly take intoaccount correlation effects, leading to a significant reduc-tion of the normalization of the shell-model states [37], aswell as to the appearance of sizable interference terms inthe 2p2h sector. However, in some instances these twodeficiencies may largely compensate one another, leadingto accidental agreement between theory and data. Forexample, the two-body current contributions computedwithin our approach turn out to be close to those obtainedwithin the Fermi gas model.The development of a nuclear model having the pre-

dictive power needed for applications to the analysis offuture experiments—most notably the Deep UndergroundNeutrino Experiment (DUNE) [38]—will require that thedegeneracy between different approaches be resolved. Asystematic comparison between the results of theoreticalcalculations and the large body of electron scattering data,including both inclusive and exclusive cross sections, willgreatly help to achieve this goal.

(a)

(b)

FIG. 3. (a) Double differential electron-carbon cross section atbeam energy Ee ¼ 680 MeV and scattering angle θe ¼ 36 deg.The dashed line corresponds to the result obtained neglecting FSI,while the solid line has been obtained within the approach ofRef. [15]. The experimental data are taken from Ref. [24].(b) Same as (a) but for Ee ¼ 1300 MeV and θe ¼ 37.5 deg.The experimental data are taken from Ref. [25].

PRL 116, 192501 (2016) P HY S I CA L R EV I EW LE T T ER Sweek ending13 MAY 2016

192501-4

I Megias et al

0 0.05 0.1 0.150

1e+05

2e+05

0 0.05 0.1 0.15 0.23

0.23

0.24

0.24

0.25E=400 MeV, θ=36

o, qQE=239 MeV/c

0 0.05 0.10

10000

20000

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50000

0.24

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0.26

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o, qQE=263.739 MeV/c

0 0.1 0.2 0.30

25000

50000

75000

1e+05

0.28

0.3

0.32

0.34

E=480 MeV, θ=36o, qQE=286.4 MeV/c

0 0.05 0.1 0.150

10000

20000

30000

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0.28

0.29

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0.31

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E=320 MeV, θ=60o, qQE=299.4 MeV/c

0 0.1 0.2 0.3 0.40

2e+05

4e+05

6e+05

8e+05

0.3

0.35

0.4

0.45

0.5E=1500 MeV, θ=11.95

o, qQE=311 MeV/c

0 0.1 0.2 0.3 0.40

10000

20000

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40000

50000

60000

70000

0.32

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0.36

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E=560 MeV, θ=36o, qQE=332.9 MeV/c

0 0.05 0.1 0.15 0.20

5000

10000

15000

20000

25000

0.31

0.32

0.33

0.34

0.35

0.36E=361 MeV, θ=60

o, qQE=335.7 MeV/c

0 0.1 0.2 0.3 0.4 0.50

1e+05

2e+05

3e+05

4e+05

5e+05

6e+05

0.50.2

0.3

0.4

0.5

0.6

0.7

0.8E=1650 MeV, θ=11.95

o, qQE=343 MeV/c

0 0.1 0.2 0.3 0.40

1e+05

2e+05

3e+05

4e+05

0.3

0.35

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0.45

0.5

0.55

E=1500 MeV, θ=13.5o, qQE=352 MeV/c

0 0.05 0.1 0.150

2000

4000

6000

8000

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0.32

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0.37

E=280 MeV, θ=90o, qQE=352 MeV/c

0 0.1 0.2 0.3 0.40

10000

20000

30000

40000

50000

0.37

0.4

0.42

0.45

0.47

E=620 MeV, θ=36o, qQE=367 MeV/c

0 0.05 0.1 0.15 0.2 0.250

5000

10000

15000

0.34

0.36

0.38

0.4E=401 MeV, θ=60

o, qQE=370.8 MeV/c

0 0.1 0.2 0.3 0.4 0.50

1e+05

2e+05

3e+05

ω (GeV)

0.4

0.5

0.6

E=1650 MeV, θ=13.5o, qQE=388 MeV/c

0 0.1 0.2 0.3 0.4 0.5ω (GeV)

0

10000

20000

30000

40000

0.35

0.4

0.45

0.5

0.55E=680 MeV, θ=36

o, qQE=402.5 MeV/c

0 0.05 0.1 0.15 0.2 0.250

5000

10000

15000

0 0.05 0.1 0.15 0.2 0.25ω (GeV)

0.38

0.39

0.4

0.41

0.42

0.43

0.44E=440 MeV, θ=60

o, qQE=404.7 MeV/c

FIG. 5. Comparison of inclusive 12Cðe; e0Þ cross sections and predictions of the QE-SuSAv2 model (long-dashed red line), 2p-2hMEC model (dot-dashed brown line) and inelastic-SuSAv2 model (long dot-dashed orange line). The sum of the three contributions isrepresented with a solid blue line. The q dependence with ω is also shown (short-dashed black line). The y axis on the left representsd2σ=dΩ=dω in nb=GeV=sr, whereas the one on the right represents the q value in GeV=c.

G. D. MEGIAS et al. PHYSICAL REVIEW D 94, 013012 (2016)

013012-6

0 0.1 0.2 0.3 0.4 0.50

500

1000

1500

2000

2500

0.58

0.6

0.62

0.64

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E=680 MeV, θ=60o, qQE=610 MeV/c

0 0.1 0.2 0.3 0.4 0.50

5000

10000

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20000

25000

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0.75

0.8

E=2130 MeV, θ=18o, qQE=640 MeV/c

0.1 0.15 0.2 0.25 0.3 0.350

100

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0.45

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0.55

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o, qQE=650 MeV/c

0 0.2 0.4 0.6 0.80

5000

10000

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20000

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0.2 0.4 0.60

1000

2000

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4000

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0.8

E=1108 MeV, θ=37.5o, qQE=674.6 MeV/c

0.1 0.2 0.3 0.40

2000

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10000

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14000

16000

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0.72

0.74

0.76

E=2020 MeV, θ=20o, qQE=700.2 MeV/c

0.2 0.4 0.6 0.80

500

1000

1500

2000

0.8

0.85

0.9

0.95

E=1299 MeV, θ=37.5o, qQE=792 MeV/c

0.2 0.3 0.4 0.50

100

200

300

400

500

0.65

0.7

0.75

0.8

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E=560 MeV, θ=145o, qQE=795 MeV/c

0.2 0.3 0.4 0.5 0.6 0.7 0.80

250

500

750

1000

0.9

0.95

1

1.05

1.1

E=1501 MeV, θ=37.5o, qQE=917 MeV/c

0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

1000

2000

3000

1

1.25

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1.75

E=3595 MeV, θ=16o, qQE=1044.3 MeV/c

0 0.2 0.4 0.6 0.8 10

500

1000

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2500

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1.1

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E=4045 MeV, θ=15o, qQE=1113.8 MeV/c

0.2 0.4 0.6 0.80

100

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1.25

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E=3595 MeV, θ=20o, qQE=1316 MeV/c

0.4 0.8 10

50

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150

0.6ω (GeV)

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1.65

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1.8E=3595 MeV, θ=25

o, qQE=1640 MeV/c

0.6 0.8 1 1.2 1.4 1.6 1.8 2ω (GeV)

0

10

20

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50

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2.5E=4045 MeV, θ=30

o, qQE=2247 MeV/c

2.4 2.6 2.8 3ω (GeV)

0

1

2

3

4

5

3.35

3.4

3.45

3.5

3.55

3.6E=4045 MeV, θ=55

o, qQE=3432.2 MeV/c

FIG. 7. As for Fig. 5, but now for kinematics corresponding to the highest qQE values considered.

G. D. MEGIAS et al. PHYSICAL REVIEW D 94, 013012 (2016)

013012-8

? Mechanisms other than single nucleon knock-out—leading tothe appearance of 2p2h final states—play a significant role

6 / 16

12C(νµ, µ−): VALENCIA MODEL vs SUPERSCALING

? Comparison to the flux-integrated cross section measured by theMiniBooNE collaboration

I Nieves et al5

0.5 1 1.5Tµ (GeV)

0

0.5

1

1.5

2

d2 !/d

T µ d

cos" µ

(10-3

8 cm

2 /GeV

) Full ModelFull QE (with RPA)MultinucleonNo RPA, No Multinuc.No RPA,No Multin., MA=1.32

0.80 < cos "µ< 0.90

MA=1.049 GeV

FIG. 3: Muon angle and energy distribution d2σ/d cos θµdTµ for 0.80 < cos θµ < 0.90. Experimental data from Ref. [5] andcalculation with MA = 1.32 GeV are multiplied by 0.9. Axial mass for the other curves is MA = 1.049 GeV.

with electron, photon and pion probes and contains no additional free parameters. RPA and multinucleon knockouthave been found to be essential for the description of the data. Our main conclusion is that MiniBooNE data are fullycompatible with former determinations of the nucleon axial mass, both using neutrino and electron beams in contrastwith several previous analyses. The results also suggest that the neutrino flux could have been underestimated.Besides, we have found that the procedure commonly used to reconstruct the neutrino energy for quasielastic eventsfrom the muon angle and energy could be unreliable for a wide region of the phase space, due to the large importanceof multinucleon events.

It is clear that experiments on neutrino reactions on complex nuclei have reached a precision level that requires for aquantitative description of sophisticated theoretical approaches. Apart from being important in the study of neutrinophysics, these experiments are starting to provide very valuable information on the axial structure of hadrons.

Acknowledgments

We thank L. Alvarez Ruso, R. Tayloe and G. Zeller for useful discussions. This research was supported by DGIand FEDER funds, under contracts FIS2008-01143/FIS, FIS2006-03438, and the Spanish Consolider-Ingenio 2010Programme CPAN (CSD2007-00042), by Generalitat Valenciana contract PROMETEO/2009/0090 and by the EUHadronPhysics2 project, grant agreement n. 227431. I.R.S. acknowledges support from the Ministerio de Educacion.

[1] C. F. Perdrisat, V. Punjabi and M. Vanderhaeghen, Prog. Part. Nucl. Phys. 59, 694 (2007).[2] V. Bernard, L. Elouadrhiri and U. G. Meissner, J. Phys. G 28 (2002) R1.

I Megias et al

13

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

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dσ/d

T µ/d

cosθ

µ (1

0-39 cm

2 /GeV

)

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

5

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MiniBooNEQE+MECQEMEC

0.8 < cosθµ < 0.9

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µ < 0.8

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T µ/d

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µ (1

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)

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µ < 0.5

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T µ/d

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µ (1

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)

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µ < 0.2

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T µ/d

cosθ

µ (1

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)

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µ < 0.0

0 0.1 0.2 0.3 0.4 0.5Tµ (GeV)

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10-0.2 < cosθ

µ < -0.1

FIG. 6. (Color online) MiniBoone flux-folded double differen-tial cross section per target nucleon for the νµ CCQE processon 12C displayed versus the µ− kinetic energy Tµ for variousbins of cos θµ obtained within the SuSAv2+MEC approach.QE and 2p-2h MEC results are also shown separately. Dataare from [1].

? The degeneracy issue: calculation based on differentapproximations and including different reaction mechanismsyield similar results

7 / 16

UNRAVELING THE NUCLEAR CROSS SECTION

? An accurate description of the 2p2h sector, providing ∼ 20% ofthe nuclear cross section is only relevant to the extent to whichthe remaining ∼ 80%, arising from processes involving 1p1hfinal states, is fully understood

? (e, e′p) experiments, in which the scattered electron and theoutgoing proton are detected in coincidence, have provided awealth of information on single nucleon knock-out processes,associated with 1p1h final states, as well as clear-cut evidence ofthe coupling between the 1p1h and 2p2h sectors

? The large database of (e, e′p) cross sections—measured mainly atSaclay, NIKHEF-K and Jefferson Lab—can be exploited to testthe theoretical approaches employed to study neutrino-nucleusinteraction, and assess their predictive power

8 / 16

THE (e, e′p) REACTION

I Consider the process

e+A→ e′ + p+ (A− 1)

in which both the outgoing electronand the proton, carrying momentump′, are detected in coincidence, andthe recoiling nucleus can be left in aany (bound or continuum) state |n〉with energy En

e e′

p′

q,ω

I In the absence of final state interactions (FSI)—which can betaken into acount as corrections—the the measured missingmomentum and missing energy can be identified with themomentum of the knocked out nucleon and the excitationenergy of the recoiling nucleus, En − E0

pm = p′ − q , Em = ω − Tp′ − TA−1 ≈ ω − Tp′

9 / 16

(e, e′p) CROSS SECTION AND NUCLEAR SPECTRAL FUNCTION

I In the absence of FSI (to be discussed at a later stage)

dσAdEe′dΩe′dEpdΩp

∝ σepP (pm, Em)

I Kallen-Lehman representation of the spectral function

P(pm, Em) = PMF(pm, Em) + Pcorr(pm, Em)

I In the kinematical region corresponding to knock-out from theshell-model states (Em <∼ 50 MeV and |pm| <∼ 250 MeV)

PMF(pm, Em) =∑α∈F

Zα |φα(pm)|2 Fα(Em − εα)

I According to the nuclear shell model

Zα → 2jα + 1 , Fα(Em − εα)→ δ(Em − εα)

10 / 16

PINNING DOWN THE 1P1H SECTOR

I At moderate missing energy—typically Em <∼ 50 MeV—therecoiling nucleus is left in a bound state

I The final state is a 1p1h state of the A-nucleon systemI The missing energy spectrum exhibits spectroscopic lines,

corresponding to knock out from the shell model states.However the normalization of the shell model states issuppressed with respect to the predictions of the independentparticle model.

I The momentum distributions of nucleons in the shell modelstates can be obtained measuring the missing momentumspectra at fixed missing energy

I Consider 12C(e, e′p)11B, as an example. The expected 1p1h finalstates are

|11B(1/2−

), p〉 , |11B(3/2−

), p〉 , . . .

11 / 16

C(e, e′p) AT MODERATE MISSING ENERGY

I Missing energy spectrum of12C measured at Saclay(Mougey et al, 1976)

QUASI-FREE (e, e’p) 473

8% PG 180 M&J/”

MISSIffi ENERGY (McV)

Fig. 9. Missing energy spectra from “C(e, e’p), (a) 0 S P 5 36 MeV/c, (b) SO $ P 5 180 MeV/c and (c) 0 s P s 60 MeV/c for 20 5 E 5 60 MeV.

3OG E< 50 MeV

0 50 la, ls0 2co 250 300 RECOIL MOMENTUM (M&/c)

Fig. 10. Momentum ~s~ibution from “C(e, e’p); (a) I5 s E 4 21.5 MeV and (b) 30 5 E s 50 MeV. The solid and dashed lines represent DWIA and PWIA ~lcula~ons respectively, with nonfiction

obtained by a fit to the data.

shells of “C. The lp, shell, at a separation energy of 16 MeV (fig. 9), exhibits the expected I = 1 distribution having a zero at P = 0 and a single maximum at PW 100 MeVJc. The two lines occurring in S(E, P) at 18 and 21 MeV correspond

I P - state momentumdistribution. Solid line: LDAspectral function (OB et al,1994)

? The description of the nuclear ground state based on theindependent particle model fails to account for the measuredspectroscopic factors

12 / 16

THE ISSUE OF NEUTRINO ENERGY RECONSTRUCTION

I Oscillation probability after traveling a distance L (two neutrinoflavors, for simplicity)

Pα→β = sin2 2θ sin2

(∆m2L

4Eν

)

Observable'Oscilla9on'Parameters'

Elba,'June'26,'2014'ElectronNNucleus'ScaPering'XIII' 4'

I The energy of the incoming neutrino, Eν is not precisely known,and must be reconstructed on a event-by-event basis using theinformation carried by the observed final state particles

13 / 16

KINEMATIC NEUTRINO ENERGY RECONSTRUCTION

I In the charged current quasi elastic (CCQE) channel, assumingsingle nucleon single knock out, the relevant elementary processis

ν` + n→ `− + p

I The reconstructed neutrino energy is

Eν =m2p −m2

µ − En2 + 2EµEn − 2kµ · pn + |pn2|2(En − Eµ + |kµ| cos θµ − |pn| cos θn)

,

where |kµ| and θµ are measured, while pn and En are theunknown momentum and energy of the interacting neutron

I Existing simulation codes routinely use |pn| = 0 , En = mn − ε ,with ε ∼ 20 MeV for carbon and oxygen, or the predictions of theFermi gas model

14 / 16

RECONSTRUCTED NEUTRINO ENERGY IN THE CCQE CHANNEL

I Neutrino energyreconstructed using 2×104 pairs of (|p|, E)values sampled from LDA(SF) and Fermi gas oxygenspectral functions

I The average value 〈Eν〉obtained from the realisticspectral function turns outto be shifted towardslarger energy by∼ 70 MeV

15 / 16

SUMMARY & OUTLOOK

? A number of advanced models of the electroweak nuclear crosssection are being developed and tested

? The degeneracy between models based on different physics mustbe resolved. The availablity of electron scattering data inexclusive channels will play a critical role in this context.

? A necessary next step will be the consistent extension to theinelastic sector, required for the analysis of experiments usinghigher energy neutrino beams

? Breaking News: Jlab experiment E12-12-14-012 just measuredthe Ar,Ti(e, e′p) cross section. These data will allow thedetrmination of the spectral function needed for the analysis ofboth ν and ν interactions in liquid argon detectors

16 / 16

Backup slides

17 / 16

SPECTRAL FUNCTION OF 16O? The spectral function of medium-mass nuclei has obtained

combining (e, e′p) data and results of accurate nuclear mattercalculations within the Local Density Approximation (LDA)

? 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 kF , E ε )18 / 16

SPECTRAL FUNCTION FORMALISMI e+ 12C→ e′ +X cross section computed within the impulse

approximation and including final state interactions.

(a)

240 MeV, 36 deg∼ 143 MeV, 0.02 GeV2

1007550250

900

600

300

0

(b)

200 MeV, 60 deg∼ 186 MeV, 0.03 GeV2

1007550250

200

150

100

50

0

(c)

240.4 MeV, 60 deg∼ 224 MeV, 0.05 GeV2

150100500

120

90

60

30

0dσ/d

ωdΩ

(nb/M

eVsr

)

(d)

280.3 MeV, 60 deg∼ 259 MeV, 0.06 GeV2

150100500

60

40

20

0

(e)

320.3 MeV, 60 deg∼ 295 MeV, 0.08 GeV2

200150100500

45

30

15

0

(f)

560 MeV, 36 deg∼ 331 MeV, 0.10 GeV2

3002001000

80

60

40

20

0

(g)

620 MeV, 36 deg∼ 366 MeV, 0.13 GeV2

3002001000

60

40

20

0

(h)

1650 MeV, 13.5 deg∼ 390 MeV, 0.14 GeV2

ω (MeV)

3002001000

300

200

100

0

(i)

500 MeV, 60 deg∼ 450 MeV, 0.19 GeV2

3002001000

9

6

3

0

A. Ankowski et al, PRD 91 033005, (2015)19 / 16

EXTENSION TO THE INELASTIC SECTOR

? Factorization ansatz and LDA spectral function (Vagnoni et al)

20 / 16

QUENCHING OF THE 1P1H STRENGTH

? Nucleon-nucleon correlations move strength from the 1p1hsector to the 2p2h sector

I Spectroscopic factors ofvalence states (Lapikas, 1993)

Nuclear Structure: a wide angle view 8

Removal probability forRemoval probability forvalence protonsvalence protons

fromfromNIKHEF dataNIKHEF data

L. L. LapikLapikááss, , NuclNucl. Phys. A553,297c (1993). Phys. A553,297c (1993)

Note:

We have seen mostly

data for removal of

valence protons

S ≈ 0.65 for valence protonsReduction ⇒ both SRC and LRC

I Spectroscopic factors of theshell model states of 208Pb(OB et al, 1991)

-60 -40 -20 00.0

0.2

0.4

0.6

0.8

1.0

1.2

nNM

ZNM

Z(208Pb)

nNMc

-Em [MeV]

stre

ngth

z α

/(2j+

1)

NIKHEF results: 208Pb(e,e"p)207Tl

for nucleons at surface:binding energy ≈ excitation energy for nuclear vibrations fragmentation especially at Fermi edge (surface)

nucleons in the interior: deep hole stateslarger binding energies more difficult to excite zα approaches occupation number n of nuclear matter

theoretical curves:nuclear matter calculation: Correlated Basis Function TheoryBenhar, Fabrocini, Fantoni: NPA 505 (1985) 267modified for finite nuclei:PRC 41(1990) R24Modification of Im Σ to reproduce exp. width of the hole states

n = Σ zα + ncα

If fragmentation occurs spectroscopic factors of different states have to be summed up:

SRC

LRC

? Short range correlations account for a large fraction of theobserved quenching

21 / 16

MEASURED CORRELATION STRENGTH

? The correlation strength in the 2p2h sector has been investigatedby the JLAB E97-006 Collaboration using a carbon target

0.2 0.3 0.4 0.5 0.6pm [GeV/c]

10-3

10-2

10-1

n(p m

) [fm

3 sr-1

]? Measured correlation strength (Rohe et al, 2005)

6

0.2 0.3 0.4 0.5 0.6pm [GeV/c]

10-3

10-2

10-1

n(p m

) [fm

3 sr-1

]Figure 6. Momentum distribution of the data(circles) compared to the theory of refs. [3] (dots),[4] (solid) and [24] (dashed). The lower integra-tion limit is chosen as 40 MeV, the upper one toexclude the ∆ resonance.

Experiment 0.61 ±0.06Greens function theory [3] 0.46CBF theory [2] 0.64SCGF theory [4] 0.61

Table 1Correlated strength (quoted in terms of the num-ber of protons in 12C.)

shape of the spectral function for C, Al, and Feist quite similar. For Au a larger contributionfrom the broader resonance region is obvious andthe maximum of the spectral function is shiftedto higher Em. The correlated strength for Al, Feand Au is 1.05, 1.12 and 1.7 times the strengthfor C normalized to the same number of pro-tons. This increase cannot be solely explainedby rescattering but MEC’s have probably takeninto account. Another contribution may be com-ing from the stronger tensor correlations in asym-metric nuclear matter [26,27].

REFERENCES

1. P.K.A. deWitt Huberts, J. Phys. G 16 (1990)507.

2. O. Benhar, A. Fabrocini, and S. Fantoni,Nucl. Phys. A 505 (1989) 267.

3. H. Muther, G. Knehr, and A. Polls, Phys.Rev. C 52 (1995) 2955.

4. T. Frick and H. Muther, Phys. Rev. C 68(2003) 034310.

5. D. Rohe, Habilitationsschrift, UniversityBasel, 2004.

6. R. Ent et al., Phys. Rev. C 64 (2001) 054610.7. C. Barbieri, this proceeding8. C. Barbieri, D. Rohe, I. Sick and L. Lapikas,

Phys. Lett. B 608 (2005) 47.9. C. Barbieri, private communication.10. T. de Forest, Nucl. Phys. A 392 (1983) 232.11. D. Rohe, O. Benhar et al., Phys. Rev. C 72

(2005) 054602.12. T.G. O′Neill et al., Phys. Lett. B 351 (1995)

87.13. D. Abbott et al., Phys. Rev. Lett. 80 (1998)

5072.14. K. Garrow et al., Phys. Rev. C 66 (2002)

044613.15. G. Garino et al., Phys. Rev. C 45 (1992) 780.16. O. Benhar, A. Fabrocini, S. Fantoni, and I.

Sick, Nucl. Phys. A 579 (1994) 493.17. J. Mougey et al., Nucl. Phys. A 262 (1976)

461.18. I. Sick, private communication, 1993.19. J.W. Van Orden, W. Truex, and M.K. Baner-

jee, Phys. Rev. C 21 (1980) 2628.20. O. Benhar, A. Fabrocini, and S. Fantoni,

Phys. Rev. C 41 (1990) R24.21. H. Muther and I. Sick, Phys. Rev. C 70 (2004)

041301(R).22. D. Rohe et al., Phys. Rev. Lett. 93 (2004)

182501.23. T. Frick, Kh.S.A. Hassaneen, D. Rohe,

H. Muther, Phys. Rev. C 70 (2004) 024309.24. O. Benhar, A. Fabrocini, S. Fantoni and I.

Sick, Nucl. Phys. A 579 (1994) 493.25. C. Barbieri and L. Lapikas, Phys. Rev. C 70

(2004) 054612.26. P. Konrad, H. Lenske, U. Mosel, Nucl. Phys.

A 756 (2005) 192.27. Kh. S. A. Hassaneen and H. Muther, Phys.

Rev. C 70, 054308 (2004).

22 / 16

CALORIMETRIC NEUTRINO ENERGY RECONSTRUCTIONI The effects of final state interactions (FSI), whose importance can

be gauged studying the nuclear transparency to protons,significantly affect neutrino energy reconstruction based oncalorimetric analyses (Ankowski)

I Rohe et alNUCLEAR TRANSPARENCY FROM QUASIELASTIC 12C(e, e′p) PHYSICAL REVIEW C 72, 054602 (2005)

0 1 2 3 4 5proton kinetic energy Tp (GeV)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

nucl

ear t

rans

pare

ncy

TA

C

Fe

Au

FIG. 4. Nuclear transparency TA for C, Fe, and Au as a functionof the proton kinetic energy Tp compared to the correlated Glaubercalculations (solid lines). The data indicated by circles are from theNE18 experiment at SLAC [22], squares and diamonds are Jlab dataof Refs. [23] and [1] and from Bates [3] (triangle down). The resultindicated by stars is obtained with the correlated spectral function ofRef. [8].

(circles) and Jlab [1,23] (squares and diamonds). The errorbars shown in the figure contain the statistical and systematicuncertainty but not the model-dependent error. This appliesalso to the data points of the previous works. Since the previousexperiments were analyzed using the same assumption andingredients the model-dependent error is the same for them,

while it is somewhat lower in the case of using the CBF spectralfunction.

The solid lines drawn in Fig. 4 are the result of the theorypresented in this paper. For comparision also results fromprevious experiments [1,22,23] for iron and gold are shown.For all three nuclei and large proton kinetic energy (>1.5 GeV)the theory describes the data well within the error bars. Atlow energy there is remarkable agreement between theoryand the experimental results obtained using the CBF spectralfunction. The two data points at the lowest Tp for 12C couldindicate a deviation from the prediction, but considering themodel-dependent error bar no firm conclusion can be drawn.With the standard analysis the experimental results are ≈5%too low but in agreement with previous analyses using the sameingredients. On the other hand the data points for gold seemto exceed the theory. For these analyses a correction factor1/ϵSRC = 0.78 was used [22,23]. If one would have used theCBF spectral function the results would be lowered by ≈7%and thus closer to the theory.

ACKNOWLEDGMENTS

One of the authors (O.B.) wishes to express his gratitudeto S. C. Pieper for a very useful correspondence and forproviding tables of the medium modified NN cross sections.Many illuminating discussions with V. R. Pandharipande arealso gratefully acknowledged.

This work was supported by the Schweizerische Nation-alfonds (SNF), the U.S. Department of Energy and the U.S.National Science Foundation.

[1] K. Garrow et al., Phys. Rev. C 66, 044613 (2002).[2] V. R. Pandharipande and S. C. Pieper, Phys. Rev. C 45, 791

(1992).[3] G. Garino, M. Saber, R. E. Segel, D. F. Geesaman, R. Gilman,

M. C. Green, R. J. Holt, J. P. Schiffer, B. Zeidman, E. J. Beise,G. W. Dodson, S. Hoibraten, L. D. Pham, R. P. Redwine, W. W.Sapp, C. F. Williamson, S. A. Wood, N. S. Chant, P. G. Roos,J. D. Silk, M. Deady, and X. K. Maruyama, Phys. Rev. C 45,780 (1992).

[4] O. Benhar, A. Fabrocini, and S. Fantoni, Nucl. Phys. A505, 267(1989).

[5] O. Benhar, N. N. Nikolaev, J. Speth, A. A. Usmani, and B. G.Zakharov, Nucl. Phys. A673, 241 (2000).

[6] M. L. Goldberger and K. M. Watson, Collision Theory (J. Wiley& Sons, New York, 1964).

[7] R. J. Glauber, in Lectures in Theoretical Physics edited byW. E. Brittin et al. (Interscience Publishers Inc., New York,1959).

[8] O. Benhar, A. Fabrocini, S. Fantoni, and I. Sick, Nucl. Phys.A579, 493 (1994).

[9] A. V. Dobrovolsky et al., Nucl. Phys. B214, 1 (1983); B. H.Silverman et al., ibid. A499, 763 (1989); W. Grein, ibid. B131,255 (1977).

[10] O. Benhar, in Probing Nucleons and Nuclei via the (e, e′p)Reaction, edited by E. Voutier, J.-M. Laget, and D. W. Higin-botham (World Scientific, Singapore, 2004), p. 110.

[11] R. Schiavilla et al., Nucl. Phys. A473, 267 (1987).

[12] T. G. O’Neill (private communication).[13] A. Baldini et al., in Landoldt-Bornstein, New Series, Vol. I/12b,

edited by H. Schopper (Springer, Berlin, 1987).[14] E. Christy (private communication).[15] D. Mack (private communication).[16] R. Ent, B. W. Filippone, N. C. R. Makins, R. G. Milner, T. G.

ONeill, and D. A. Wasson, Phys. Rev. C 64, 054610 (2001).[17] T. de Forest, Nucl. Phys. A392, 232 (1983).[18] P. Mergell et al., Nucl. Phys. A596, 367 (1996).[19] M. K. Jones et al., Phys. Rev. Lett. 84, 1398 (2000).[20] O. Gayou et al., Phys. Rev. Lett. 88, 092301 (2002).[21] D. Rohe, Habilitationsschrift, University of Basel, 2004.[22] T. G. O’Neill et al., Phys. Lett. B351, 87 (1995).[23] D. Abbott et al., Phys. Rev. Lett. 80, 5072 (1998).[24] J. Mougey et al., Nucl. Phys. A262, 461 (1976).[25] I. Sick (private communication), 1993.[26] J. W. Van Orden, W. Truex, and M. K. Banerjee, Phys. Rev. C

21, 2628 (1980).[27] H. Muther, G. Knehr, and A. Polls, Phys. Rev. C 52, 2955 (1995).[28] O. Benhar, A. Fabrocini, and S. Fantoni, Phys. Rev. C 41, R24

(1990).[29] H. Muther and I. Sick, Phys. Rev. C 70, 041301(R) (2004).[30] S. Frullani and J. Mougey, Advances in Nuclear Physics, edited

by E. W. Vogt and J. W. Negele (Springer, 1984), Vol. 14, p. 1.[31] L. Lapikas, Nucl. Phys. A553, 297c (1993).[32] D. Rohe, C. S. Armstrong, R. Asaturyan, O. K. Baker,

S. Bueltmann, C. Carasco, D. Day, R. Ent, H. C. Fenker,

054602-7

23 / 16

THE E12-14-012 EXPERIMENT AT JEFFERSON LAB

? The reconstruction of neutrino and antineutrino energy in liquidargon detectors will require the understanding of the spectralfunctions describing both protons and neutrons

? The Ar(e, e′p) cross section only provides information on protoninteractions. The information on neutrons can be obtained fromthe Ti(e, e′p), exploiting the pattern of shell model levels

16

Physics MotivationExperimental Goals

Experimental conditionsTitanium idea

Physics motivation

Use few hours of beam time investigating the feasibility of runningon a titanium target, as suggested by the PAC.The neutron spectral function of argon is needed to modelquasielastic neutrino scattering. In pion production both neutronsand protons take part in charged-current interactions.

4018Ar

p’s n’s

4822Ti

p’s n’s

C. Mariani for E12-14-012 Collaboration Spectral function of 40Ar through the (e, e0p) reaction24 / 16