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FROM (e,e’p) TO NEUTRINO SCATTERING
Carlotta Giusti
Università and INFN Pavia
International Workshop on (e,e’p) Processes, 2-6 July 2017, Bled
(e,e’p)
1pKO
both e’ and p detected
(A-1) in a bound or continuum state
Em missing energy (W*A-1 – WA)
discrete spectrum (low Em) peaks
continuum spectrum (larger Em)
for Em corresponding to a peak (A-1) in a discrete eigenstate
(e,e’p) discrete low-lying states
(e,e’p) in the continuum
EXCLUSIVE
QE
(e,e’p) (e,e’) QE
1pKO
both e’ and p detected
(A-1) in a bound or continuum state
Em missing energy (W*A-1 – WA)
discrete spectrum (low Em) peaks
continuum spectrum (larger Em)
for Em corresponding to a peak (A-1) in a discrete eigenstate
(e,e’p) discrete low-lying states
(e,e’p) in the continuum
EXCLUSIVE
(e,e’p)
only e’ detected
all final states included
discrete and continuum spectrum
INCLUSIVE
(e,e’) QE
1pKO
both e’ and p detected
(A-1) in a bound or continuum state
Em missing energy (W*A-1 – WA)
discrete spectrum (low Em) peaks
continuum spectrum (larger Em)
for Em corresponding to a peak (A-1) in a discrete eigenstate
(e,e’p) discrete low-lying states
(e,e’p) in the continuum
EXCLUSIVE
(e,e’p)
EXCLUSIVE
INCLUSIVE
proton-hole states
properties of bound protons
s.p. aspects of nuclear structure
validity and limitation of IPSM, MFA
nuclear correlations SRC
only e’ detected
all final states included
discrete and continuum spectrum
INCLUSIVE
(e,e’) QE
(e,e’p)
EXCLUSIVE
INCLUSIVE
proton-hole states
properties of bound protons
s.p. aspects of nuclear structure
validity and limitation of IPSM, MFA
nuclear correlations SRC short-range correlations
SRC
high Em beyond threshold for
the emission of a second N
two-nucleon knockout
only e’ detected
all final states included
discrete and continuum spectrum
INCLUSIVE
(e,e’) QE
(e,e’p)
EXCLUSIVE
INCLUSIVE
proton-hole states
properties of bound protons
s.p. aspects of nuclear structure
validity and limitation of IPSM, MFA
nuclear correlations SRC short-range correlations
SRC
high Em beyond threshold for
the emission of a second N
two-nucleon knockout
only e’ detected
all final states included
discrete and continuum spectrum
INCLUSIVE
(e,e’) QE
NEUTRINO-NUCLEUS SCATTERING
(e,e’p)
EXCLUSIVE
INCLUSIVE
proton-hole states
properties of bound protons
s.p. aspects of nuclear structure
validity and limitation of IPSM, MFA
nuclear correlations SRC short-range correlations
SRC
high Em beyond threshold for
the emission of a second N
two-nucleon knockout
only e’ detected
all final states included
discrete and continuum spectrum
INCLUSIVE
(e,e’) QE
The work done for electron scattering can be exploited to describe nuclear effects in
neutrino-nucleus scattering This is very important for neutrino oscillation experiments where nuclei are used as neutrino detectors.
Experimental data: Em and pm distributions
For Em corresponding to a peak we assume that the residual nucleus is in a discrete eigenstate
ONE-HOLE SPECTRAL FUNCTION
exclusive reaction
joint probability of removing from the target a nucleon p1 leaving the residual nucleus in a state with energy Em
ONE-HOLE SPECTRAL FUNCTION
exclusive reaction
joint probability of removing from the target a nucleon p1 leaving the residual nucleus in a state with energy Em
inclusive reaction : one-body density
MOMENTUM DISTRIBUTION
probability of finding in the target a nucleon with momentum p1
E0’
E0
,q
p n
0
(e,e’p)
exclusive reaction n
DKO mechanism: the probe interacts through a one-body current with one nucleon which is then emitted the remaining nucleons are spectators
impulse approximation IA |i >
|f >
,q
p n
0
(e,e’p)
|i >
|f >
FSI = 0
exclusive reaction n
DKO mechanism: the probe interacts through a one-body current with one nucleon which is then emitted the remaining nucleons are spectators
impulse approximation IA
E0’
E0
PW
factorized cross section
PW FSI=0 PLANE-WAVE IMPULSE
APPROXIMATION
PWIA
spectral function
spectroscopic factor
overlap function
For each Em the mom. dependence of the SF is given by the mom. distr. of the quasi-hole states n produced in the target nucleus at that energy and described by the normalized OVF
The norm of the OVF, the spectroscopic factor gives the probability that n is a pure hole state in the target.
IPSM s.p. SM state
1 occupied SM states
0 empty SM states
There are correlations and the strength of the quasi-hole state is fragmented over a set of s.p. states
spectroscopic factor overlap function
,q
p n
0
DWIA (e,e’p)
|i >
|f >
FSI
exclusive reaction n
DKO IA
FSI DWIA
unfactorized c.s.
non diagonal SF
DW
j one-body nuclear current
(-) s.p. scattering w.f. H+(+Em)
n s.p. bound state overlap function H(-Em)
n spectroscopic factor
(-) and consistently derived as eigenfunctions of a Feshbach optical model Hamiltonian
Direct knockout DWIA (e,e’p)
phenomenological ingredients usually adopted
(-) phenomenological optical potential
n phenomenological s.p. wave functions WS, HF MF (some calculations including correlations are available)
n extracted in comparison with data: reduction factor applied to the calculated c.s. to reproduce the magnitude of the experimental c.s.
DWIA-RDWIA calculations
phenomenological ingredients usually adopted
(-) phenomenological optical potential
n phenomenological s.p. wave functions WS, HF MF (some calculations including correlations are available)
n extracted in comparison with data: reduction factor applied to the calculated c.s. to reproduce the magnitude of the experimental c.s.
DWIA-RDWIA calculations
DWIA and RDWIA: excellent description of (e,e’p) data
NIKHEF data & CDWIA calculations
reduction factors applied: spectroscopic factors
0.6 - 0.7
Experimental data: pm distribution
depletion due to NN correlations
different independent investigations:
SRC (Short-Range Correlations) account for only a few % of the depletion, up to 10-15 % with TC
LRC (Long-Range Correlations) main contribution to the depletion
LRC collective excitations of nucleons at the nuclear surface
SPECTROSCOPIC FACTORS and NN CORRELATIONS
account for only a small part of the depletion
depletion due to SRC compensated by the admixture of high-momentum components in the s.p. w.f.
SRC effects on (e,e’p) cross sections at high pm are small for low-lying states
calculations of the 1BDM and of the momentum distribution indicate that the missing strength due to SRC is found at large pm and Em, beyond the continuum threshold, where various processes involving more nucleons can be present
SRC
account for only a small part of the depletion
depletion due to SRC compensated by the admixture of high-momentum components in the s.p. w.f.
SRC effects on (e,e’p) cross sections at high pm are small for low-lying states
calculations of the 1BDM and of the momentum distribution indicate that the missing strength due to SRC is found at large pm and Em, beyond the continuum threshold, where various processes involving more nucleons can be present
(e,e’p) at high values of Em
SRC
16O(e,e’p) at high Em
N. Liyanage et al. PRL 86 (2001) 5670
<pm> = 50 MeV/c <pm> = 145 MeV/c <pm> = 280 MeV/c <pm> = 340 MeV/c
16O(e,e’p) at high Em
N. Liyanage et al. PRL 86 (2001) 5670
1p1/2 1p3/2
<pm> = 50 MeV/c <pm> = 145 MeV/c <pm> = 280 MeV/c <pm> = 340 MeV/c
16O(e,e’p) RDWIA
JLab (,q) const kin E0=2445 MeV =439 MeV Tp= 435 MeV
diff opt.pot.
n = 0.7
A. Meucci, C. Giusti, F.D. Pacati PRC 64 (2001) 014604
16O(e,e’p) at high Em
N. Liyanage et al. PRL 86 (2001) 5670
1p1/2 1p3/2 1s1/2
<pm> = 50 MeV/c <pm> = 145 MeV/c <pm> = 280 MeV/c <pm> = 340 MeV/c
RDWIA (J. Kelly).
2NKO (Ghent).
16O(e,e’p) at high Em
N. Liyanage et al. PRL 86 (2001) 5670
1p1/2 1p3/2 1s1/2
<pm> = 50 MeV/c <pm> = 145 MeV/c <pm> = 280 MeV/c <pm> = 340 MeV/c
RDWIA (J. Kelly).
2NKO (Ghent).
16O(e,e’p) at high Em
N. Liyanage et al. PRL 86 (2001) 5670
1p1/2 1p3/2 1s1/2
<pm> = 50 MeV/c <pm> = 145 MeV/c <pm> = 280 MeV/c <pm> = 340 MeV/c
RDWIA (J. Kelly).
2NKO (Ghent).
16O(e,e’p) at high Em
N. Liyanage et al. PRL 86 (2001) 5670
1p1/2 1p3/2 1s1/2
<pm> = 50 MeV/c <pm> = 145 MeV/c <pm> = 280 MeV/c <pm> = 340 MeV/c
RDWIA (J. Kelly).
2NKO (Ghent).
the exp. c.s contains other contributions beyond 1NKO+2NKO.
DWIA
FSI
FSI described by a complex OP, the imaginary part gives a reduction of the calculated c.s. which is essential to reproduce (e,e’p) data for low Em (low-lying states)
DWIA
DWIA
FSI
FSI described by a complex OP, the imaginary part gives a reduction of the calculated c.s. which is essential to reproduce (e,e’p) data for low Em (low-lying states)
not suited to address the continuum
the OP describes elastic nucleon-nucleus scattering. The Im part accounts for the flux lost in the elastically scattered beam toward the inelastic channels that are open
DWIA: the Im part removes the contribution of inelastic channels
(e,e’p) at low Em : only the final-state channel n is considered, it can be correct to account for the flux lost in the considered channel. The main contribution comes from the 1pKO process where the outgoing proton scatters elastically with the residual nucleus in the state n.
inclusive (e,e’) all the final-state channels are included, the flux must be redistributed and conserved in the sum over all the channels. In every channel flux is lost toward the other channels and flux is gained due to the flux lost in the other channels toward the considered channel
inclusive scattering: GREEN’S FUNCTION MODEL
FSI and OPTICAL POTENTIAL
GREEN’S FUNCTION MODEL
Based on the IA: one-body current interacts with a nucleon that is then emitted. Sum over all the nucleons of the target
FSI are accounted for by the complex energy dependent OP: the formalism translates the flux lost toward inelastic channels, represented by the Im part of the OP, into the strength observed in inclusive reactions.
The OP is responsible for the redistribution of the flux in all the final-state channels and in the sum over all the channels the flux is conserved.
The OP becomes a powerful tool to include important inelastic contributions beyond 1NKO (multi-nucleon, rescattering, non-nucleonic…) not included in usual models based on the IA
GREEN’S FUNCTION MODEL
Based on the IA: one-body current interacts with a nucleon that is then emitted. Sum over all the nucleons of the target
FSI are accounted for by the complex energy dependent OP: the formalism translates the flux lost toward inelastic channels, represented by the Im part of the OP, into the strength observed in inclusive reactions.
The OP is responsible for the redistribution of the flux in all the final-state channels and in the sum over all the channels the flux is conserved.
The OP becomes a powerful tool to include important inelastic contributions beyond 1NKO (multi-nucleon, rescattering, non-nucleonic…) not included in usual models based on the IA
MODEL SUCCESSFULLY APPLIED TO QE (e,e’) and CCQE NCE
16O(e,e’)
data from Frascati NPA 602 405 (1996)
RPWIA
RGF
RDWIA
A. Meucci, F. Capuzzi, C. Giusti, F.D. Pacati PRC 67 (2003) 054601
(e,e’)
E0 = 1080 MeV # = 32o
E0 = 841 MeV # = 45.5o
E0 = 2020 MeV # = 20o
RGF
A. Meucci et al. PRC 87 (2013) 054620
FSI in (e,e’p)
DWIA not suited to address the continuum
at high Em and pm : contributions beyond 1pKO
removed in DWIA by the Im part of the OP and included by the Im part of the OP in the GF model
e.g. multi-scattering effects: the ejectile proton in its way through the nucleus undergoes secondary collisions where it can change direction and lose energy before being emitted and detected
FSI in (e,e’p)
DWIA not suited to address the continuum
at high Em and pm : contributions beyond 1pKO
removed in DWIA by the Im part of the OP and included by the Im part of the OP in the GF model
e.g. multi-scattering effects: the ejected proton in its way through the nucleus undergoes secondary collisions where it can change direction and lose energy before being emitted and detected
Multi-scattering effects in (e,e’p) knockout: P.Demetriou, S. Boffi, C. Giusti, F.D. Pacati, Nucl. Phys. A 625 (1997) 513
Multi-Scattering Effects in (e,e’p)
Improve the treatment of FSI The method follows the lines of the multi-step direct (MSD) scattering theory of Feshbach. Kerman, and Koonin which allows one to trace the secondary collisions of the emitted nucleon. The proton, following the initial elm int., is excited to the continuum and subsequently undergoes a series of 2-b interactions with the residual nucleus before being emitted. Thus there are a series of collisions leading to the excitation of intermediate states of increasing complexity. At each step a N can be emitted. The theory combines a QM treatment of multiple scattering with statistical assumptions that lead to the convolution nature of the multistep c.s. and enables the calculation of higher-order contributions which would be otherwise impracticable
(e,e’p) c.s. = 1-step + n-step terms
DWIA
n-step term convolution of 1NKO and 1-step MSD c.s. over all intermediate energies E1, E2 … and angles 1, 2…..
DWIA DWBA
Multi-Scattering Effects in (e,e’p)
Multi-Scattering Effects in (e,e’p)
The method follows the trace of the fast emitted proton but does not follow the fate of secondary nucleons
2NKO not included Pion contribution and reabsorption not included
Suited to study the continuum spectrum with not too high Em
P . Demetriou et al Nucl. Pys. A 625 (1997) 513
E0=497 MeV, µ=52.9o, != 140 MeV , Tp= 87 10 MeV
Em =28 MeV Em =43 MeV
Em =58 MeV Em =73 MeV
tot
1-s
1-s
1-s 1-s
2-s
2-s
tot
3-s
tot tot
2-s
2-s
3-s
3-s
3-s 4-s
4-s
P . Demetriou et al Nucl. Phys. A 625 (1997) 513
E0=497 MeV, µ=52.9o, != 140 MeV , Tp= 87 10 MeV
Em =28 MeV Em =43 MeV
Em =58 MeV Em =73 MeV
tot
1-s
1-s
1-s 1-s
2-s
2-s
tot
3-s
tot tot
2-s
2-s
3-s
3-s
3-s 4-s
4-s
increasing Em 2-s and 3-s become gradually more important. The domination of multi-step processes at large scatt. angles is expected since the proton gradually loses memory of its initial direction yielding thus increasingly symm. ang. distribution
In the continuum effects and mechanisms beyond direct 1NKO may be important or even dominant Strong sensitivity to kinematic conditions. In electron scattering suitable kinematic conditions can be selected able to minimize or emphasize and therefore study specific contributions ELECTRON SCATTERING IS SELECTIVE
In the continuum effects and mechanisms beyond direct 1NKO may be important or even dominant Strong sensitivity to kinematic conditions. In electron scattering suitable kinematic conditions can be selected able to minimize or emphasize and therefore study specific contributions ELECTRON SCATTERING IS SELECTIVE
from e-nucleus to n-nucleus scattering
CCQE
from e-nucleus to n-nucleus scattering
only final lepton detected inclusive CC
same model as for inclusive (e,e’)
CCQE
from e-nucleus to n-nucleus scattering
only final lepton detected inclusive CC
same model as for inclusive (e,e’)
both final lepton and nucleon …. exclusive (e,e’N)
CCQE
from e-nucleus to n-nucleus scattering
only final lepton detected inclusive CC
same model as for inclusive (e,e’)
both final lepton and nucleon …. exclusive (e,e’N)……
BUT
different exp. situation in electron and º scattering
electron scattering :
beam energy known, ! and q determined
neutrino scattering:
beam energy not known, ! and q not determined, flux averaged c.s.
calculations over the energy range relevant for the neutrino flux,
broader kinematic region
different exp. situation in electron and º scattering
electron scattering :
beam energy known, ! and q determined
neutrino scattering:
beam energy not known ! and q not determined, flux averaged c.s.
calculations over the energy range relevant for the neutrino flux,
broader kinematic region
different exp. situation in electron and º scattering
the flux-average procedure picks up contributions from different kinematic regions, not only QE, contributions other than direct 1-nucleon emission are more important than in (e,e’)
RGF
RPWIA
rROP
RGF EDAI
RGF-EDAD1
A. Meucci and C. Giusti PRD 85 (2012) 093002
Comparison with MiniBooNe CCQE data
CCQE
from e-nucleus to n-nucleus scattering
both final lepton and nucleon …. (e,e’p) where the incident electron energy is not know
calculations for different values of Eº , !, and in a certain range of Em
Experimental and theoretical work already done and that can be done for electron scattering extremely useful to provide reliable º-nucleus cross sections
Comparison with electron-scattering data test of the validity and predictive power of a nuclear model Electron scattering experiments in suitable kinematics can give information on the relevant reaction mechanisms, nuclear dynamics FSI, nuclear current (1-body, 2-body)…. Information on nuclear structure and correlations, contained in the spectral function, can be obtained from dedicated (e,e’p) experiments (e.g. JLab experiment on 40Ar)
from e-nucleus to n-nucleus scattering