[PPT] - Lepton-nucleus interactions within many-body approaches: from the PowerPoint Presentation

SLIDE 1

Lepton-nucleus interactions within many-body approaches: from the quasi-elastic to the DIS region

Noemi Rocco

In Collaboration with: O.Benhar (La Sapienza University of Rome), H.Lee (Argonne National Laboratory), A.Lovato (Argonne National Laboratory, TIFPA), S. Nakamura (University of Tokyo), R.Schiavilla (Old Dominion University, JLab)

HEP Division Seminar, Argonne National Laboratory

October 30, 2019

SLIDE 2

Motivations: (e,e’p) scattering experiments

VOX.UME 13, NUMSER 10

PHYSICAL

REVIK%' LETTERS

7 SEPTEMBER 1964

4

eQUNTlNG

RATE

I I I i

E =406 MeY

T=110 MeV

/

510

520 530 540 550 560 570 580 590 600

Eo {MeV)

Q

1Q

20

30 40

50

60

SINONG ENERGY

(MeV)

COUNTING

RATE

1.0
Q5

500

510

I I I

Al

E =406 MeV

T = 100 MeV

t

2s 1d

j~[

/

l~t

520

530

54Q

550

56Q

570

5&0

590

Eo (MeV) 10

20

30 45

50 60

BNOING

ENERGY

(MeY)

FIG. 2.

Electron-proton

coincidence

counting

rate per 10~' equivalent

quanta

at 550 MeV as a function

f

the incident

energy. The dashed lines indicate the

contributions

f the various

shells

and the background

as explained

in the text.

which is naturally

very narrow, has a width here

f about 12 MeV, only slightly

larger

than the

calculated resolution. The contribution

f the

two 1s protons is not clearly separated with such

a resolution.

Our results are, however,

fully

consistent

with its presence

at the binding ener-

gy and with the width observed

in the (p, 2p) ex-

periments

and a relative

height calculated

by a

Monte Carlo program

n our IBM-7040 computer.

The calculation

is based on the impulse

approxi-

mation assuming

momentum

distributions

for the s and p protons

fitting the (p, 2p) results'

and

integrating

ver the energies

and angles fixed by

ur apparatus.

The counting rate on the C"P peak was about two counts per minute per elec-

tron momentum

channel

and agrees within a fac-

tor two with that calculated.

An assumed

back-

ground is shown in Fig. 2. The origin of this

background

is not yet clear,

but it comes at

least partly from the multiple scattering

f pro-

tons before leaving the original

nucleus.

This

effect is enhanced

with respect to existing (P, 2P)

results

because of the large solid angle of our

proton detector,

since the multiply

scattered

protons

have a wider angular

distribution.

For Al ' the spectrum

shows one clear peak,

and bumps

near 30- and 60-MeV binding energy.

%e assign the peak to the five protons

which, ac-

cording to the shell model,

are in the outermost 2s-1d shell,

and the bumps

to the six 1P protons

and the two 1s protons,

respectively.

The position and width of the 2s-1d peak agrees with

those observed

in (P, 2P) experiments;

the 1s and

1P have not been seen with that reaction.

After subtracting

an estimated

background,

we obtain

a good fit to the data with peaks at 14.5-, 32-,

and 59-MeV binding

energy,

with total natural widths

f 7, 17, and 21 MeV, respectively,

and

areas in the ratio of 1:0.

9:0.

4. The ratio of the

number

f protons

in the shells is 1:1.

2:0.4, in

reasonable

agreement

taking into account absorp- tion in the nucleus. Aside from the rough agreement

f the ratios

and absolute areas of the C" and Al" peaks with

the expected values, the most interesting

new

results are the binding

energies of the 1s and 1P

peaks in Al". The position of the P peak falls

roughly

where expected extrapolating

in Z from

nearby

nuclei,

in which it has been measured

through

(p, 2p) reactions,

and it is broadened

as expected from the P„,-P3» separation

and the

fact that the nucleus is heavily distorted.

It is

worthwhile noting that the P and s peaks are not

resolved because of their natural

width and not

for experimental

reasons.

The fact that the s peak seems to fall nearly

n a linear extrapola-

tion of the (P, 2P) results

from He4 to 0", how-

ever, is much more informative. Its observed

binding

energy

f -60 MeV is already

consider-

ably greater

than the -45-MeV well depth usual- ly assigned

to the shell-model.

potential,

pre-

sumably

indicating

an effective proton mass of

less than W.6 free masses

in the s shell of Al '.

The curve representing

the 1s binding

energy as

a function

f Z must level off eventually,

and it

will be most interesting

to follow it to heavier

nuclei. The width of the observed s peak of

roughly

20 MeV (compared

with 14 MeV in 0",

for instance)

gives some hope that the lifetime

f

the 1s hole is becoming short sufficiently

slowly

as to permit observation

f this shell to consid-

erably higher Z.

%e acknowledge

the help given to the experiment by the staff of the Frascati synchrotron

in running

the machine

according to strict stability requirements.

One of us (P.H. ) wishes to express his grati- tude to Comitato

Nazionale

per 1'Energia Nucle-

342 1s1/2 1p3/2

e e’ p

(e,e’p) experiments are important tools to

investigate the internal structure of the nucleus

The peak coming from four 1p protons is visible
The contribution of the two 1s protons is not

clearly separated with this resolution ✐ U.Amaldi et al, Phys. Rev. Lett. 13, 10 (1964) 12C

Assuming NO FSI, the energy and momentum of

the initial nucleon can be identified with the measured pmiss and Emiss

SLIDE 3

Electron and proton experiments have exposed the role of nuclear correlations
The large-momentum (short-range) component of the wave

function is dominated by the presence of Short Range Correlated (SRC) pairs of nucleons

kF (Fermi-gas like)

136 Many-body theory exposed!

Fig. 7.6

Spectroscopic factors from the (e, e'p) reaction as a function of target mass. The dotted line with a height of 1, illustrates the prediction of the independent-particle

model. Data have been obtained at the NIKHEF accelerator in Amsterdam [Lapikas

(1993)].

momentum can also have negative values when it is directed opposite to the momentum transferred to the target. A correct description of the reaction requires a good fit at all values of this quantity. Figure 7.5 demonstrates that the shapes of the valence nucleon wave functions accurately describe the observed cross sections. Such wave functions have been employed for years in nuclear-structure calculations, which have relied on the independent-particle model. The description of the data in Fig. 7.5, however, requires a significant departure of the independent- particle model, with regard to the integral of the square of these wave

functions. Indeed, the spectroscopic factors, necessary to obtain the solid

curves, are substantially less than 1. Similar spectroscopic factors are extracted for nuclei all over the periodic table4. A compilation for the spectroscopic factor of the last valence orbit for different nuclei, adapted from [Lapikas (1993)], is shown in Fig. 7.6. The results in Fig. 7.6 indicate that there is an essentially global reduction of the sp strength of about 35% for these valence holes in most nuclei. Such a substantial deviation from the prediction of the independent-particle model, requires a detailed

4Most experiments have been performed on closed-shell nuclei.

Quenching of the spectroscopic factors of valence

states has been confirmed by a number of high resolution (e,e’p) experiments

Semi-exclusive 2N-SRC

experiments at x>1 allows to detect both nucleons and reconstruct the initial state

The high momentum tail of

the nuclear wave function consists mainly of 2N-SRC ✐ Subedi et al., Science 320, 1476 (2008)

Motivations: Short Range Correlations

Figure by Or Hen

SLIDE 4

Schematic representation of the inclusive cross section as a function of the energy loss.

Broad peak due to quasi-

elastic electron-nucleon scattering.

Excitation of the nucleon to

distinct resonances (like the Δ) and pion production.

O. Benhar, et al. RMP 80, 189 (2008)
Deep Inelastic Scattering

region, productions of hadrons

ther than protons and

neutrons The different reaction mechanisms can be clearly identified

Inclusive electron-nucleus scattering

SLIDE 5

(GeV)

ν

E

1 2 3 4 5 6 7 8

Arbitrary

T2K/Hyper-K MicroBooNE/SBND MINERvA (ME) NOvA DUNE

Energy distribution of neutrino fluxes of different

experiments

Many-different reaction mechanisms come into play
Neutrino beams are a secondary decay product (π

and K decay): their energy it is not sharply defined (as in electron scattering experiments) but broadly distributed

Addressing Neutrino-Oscillation Physics

✐ T. Katori and M. Martini, J.Phys. G45 (2018) no.1, 013001 ✐ https://www.particlezoo.net

Neutrinos are extremely elusive particles
To increase their interaction rate and obtain

information on their nature medium- and large-mass nuclei are used as detectors in modern experiments Number of Interactions =

σ × Φ × N

# Targets Cross Section

? Where does Nuclear Physics come into play

Neutrino Flux

SLIDE 6

Ab-initio calculations (GFMC,

LIT) provide an accurate description of the QE region including one- and two-body currents

O. Benhar, et al. RMP 80, 189 (2008)

1st Part of the Presentation

Outline of the talk

SLIDE 7

More approximate approach:

Extended Factorization scheme + Semi-phenomenological SF can be currently used to also tackle QE, dip and π- production regions.

O. Benhar, et al. RMP 80, 189 (2008)

2nd Part of the Presentation

Outline of the talk

New results for medium-mass

nuclei obtained within the Self Consistent Green’s function and factorization scheme. Uncertainty estimate

SLIDE 8

The inclusive cross section of the process in which a lepton scatters off a nucleus and the hadronic final state is undetected can be written as

The Hadronic tensor contains all the information on target response

d2σ dΩ`dE`0 = Lµ⌫W µ⌫

W µν = X

f

h0|Jµ†(q)|fihf|Jν(q)|0iδ(4)(p0 + q pf) |0i = |ΨA

0 i , |fi = |ΨA f i, |ψN p , ΨA−1 f

i, |ψπ

k , ψN p , ΨA−1 f

i . . .

The initial and final state

dσ dE0dΩ0 = σMott h⇣ q2 q2 ⌘2 RL + ⇣−q2 2q2 + tan2 θ 2 ⌘ RT i

The cross section can be separated in the longitudinal and transverse channel

where RL = W 00

RT = W xx + W yy

Electron-nucleus scattering

SLIDE 9

Within NMBT the nucleus is described as a collection of A point-like nucleons, the dynamics of

which are described by the non relativistic Hamiltonian

H = X

i

p2

i

2m + X

i<j

vij + X

i<j<k

Vijk + . . .

π π ∆

Some of the diagrams effectively included in this potential are

N N N N N N N N N N N N

Argonne v18 is a finite, local, configuration-space potential which has been fit to ~4300 np and pp scattering data below 350 MeV of the Nijmegen database, low-energy nn scattering parameters, and deuteron binding energy.

Non relativistic Nuclear Many Body Theory

SLIDE 10

Within NMBT the nucleus is described as a collection of A point-like nucleons, the dynamics of

which are described by the non relativistic Hamiltonian The nuclear electromagnetic current is constrained through the continuity equation

r · JEM + i[H, J0

EM] = 0

The above equation implies that JEM involves two-

nucleon contributions. DISCLAIMER: all the results presented in this presentation have been obtained with semi- phenomenological interactions and currents

π ∆ π π π π ρ, ω

H = X

i

p2

i

2m + X

i<j

vij + X

i<j<k

Vijk + . . .

Chiral-EFT allows one to derive electromagnetic

currents consistent with the nuclear potentials

Non relativistic Nuclear Many Body Theory

SLIDE 11

The Green’s Function Monte Carlo approach

Suitable to solve A ≤ 12 nuclei with ~1% accuracy

100
90
80
70
60
50
40
30
20

Energy (MeV)

AV18 AV18 +IL7 Expt.

0+

4He

0+ 2+

6He

1+ 3+ 2+ 1+

6Li

3/2− 1/2− 7/2− 5/2− 5/2− 7/2−

7Li

0+ 2+

8He

2+ 2+ 2+ 1+ 0+ 3+ 1+ 4+

8Li

1+ 0+ 2+ 4+ 2+ 1+ 3+ 4+ 0+

8Be

3/2− 1/2− 5/2−

9Li

3/2− 1/2+ 5/2− 1/2− 5/2+ 3/2+ 7/2− 3/2− 7/2− 5/2+ 7/2+

9Be

1+ 0+ 2+ 2+ 0+ 3,2+

10Be

3+ 1+ 2+ 4+ 1+ 3+ 2+ 3+

10B

3+ 1+ 2+ 4+ 1+ 3+ 2+ 0+ 0+

12C

Argonne v18 with Illinois-7 GFMC Calculations

24 November 2012

J. Carlson et al. RMP 87, 1067 (2015)

GFMC algorithms use imaginary-time projection technique to enhance the ground-state component of a starting (correlated) trial wave function.

|Ψ0i = lim

τ→∞ e−(H−E0)τ|ΨT i

Quantum Monte Carlo techniques are successfully used to describe nuclear structure All the nucleon spin and isospin degrees of freedom are retained, the computational cost grows exponentially with A.

|Ψ3H⟩ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ a ↑↑↑ a ↑↑↓ a ↑↓↑ a ↑↓↓ a ↓↑↑ a ↓↑↓ a ↓↓↑ a ↓↓↓ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

le

⊗

  apnn anpn annp   Spin-isospin component of the 3H wave function

SLIDE 12

Using the completeness relation for the final states, we are left with a ground-state expectation value

✤ Accurate GFMC calculations of the electroweak responses of 4He and 12C have been recently

performed:

Rαβ(ω, q) = X

f

h0|J†

α(q)|fihf|Jβ(q)|0iδ(ω Ef + E0)

Eαβ(σ, q) = Z dωK(σ, ω)Rαβ(ω, q) = hψ0|J†

α(q)K(σ, H E0)Jβ(q)|ψ0i

K

Valuable information can be obtained from the integral transform of the response function

Integral Transform Techniques

E(σ, q) R(ω, q)

ill-posed problem; current solution is Maximum Entropy

A. Lovato et al, Phys.Rev.Lett. 117 (2016), 082501, Phys.Rev. C97

(2018), 022502 We are working on using Machine Learning techniques to invert

?

SLIDE 13

Limitations of the original method:

★ The quantum mechanical approach (e.g. the kinematics) is non relativistic—relativistic correction up

to order q2/m2 are included in the currents

★ The computational effort required by the inversion of makes the direct calculation of inclusive

cross sections unfeasible use smart interpolation procedure

Eαβ

4He

GFMC electromagnetic responses

SLIDE 14

Relativistic effects in a correlated system

The importance of relativity emerges in the frame dependence of non relativistic calculations at high

values of q

In a generic reference frame the longitudinal non relativistic response reads

Rfr

L =

X

f

hψi|

X

j

ρj(qfr, ωfr)|ψfi

2

δ(Efr

f

Efr

i

ωfr)

δ(Efr

f

− Efr

i

− ωfr) ≈ δ[efr

f + (P fr f )2/(2MT ) − efr i

− (P fr

i )2/(2MT ) − ωfr]

The response in the LAB frame is given by the Lorentz transformation

where

RL(q, ω) = q2 (qfr)2 Efr

i

M0 Rfr

L (qfr, ωfr)

qfr = γ(q − βω), ωfr = γ(ω − βq), P fr

i

= −βγM0, Efr

i

= γM0

We extend the applicability of GFMC in the quasielastic region to intermediate momentum transfers

by performing the calculation in a reference frame that minimizes the maximum nucleon momenta in the final state

SLIDE 15

Longitudinal responses of 4He for |q|=700 MeV in the four different reference frames.

The curves show differences in both peak positions and heights.

He

4

Relativistic effects in a correlated system

SLIDE 16

pfr = µ ⇣ pfr

N

mN − pfr

X

MX ⌘ P fr

f

= pfr

N + pfr X

The relative momentum is derived in a relativistic fashion

ωfr = Efr

f

− Efr

i

Efr

f

= q m2

N + [pfr + µ/MXPfr f ]2 +

q M 2

X + [pfr − µ/mNPfr f ]2

And it is used as input in the non relativistic kinetic energy

efr

f = (pfr)2/(2µ)

Analogy with NN potential model where the NN relative scattering momentum p12 is determined in a

relativistically correct fashion and used E12=p122/2μ

The frame dependence can be drastically reduced if one assumes a two-body breakup model with

relativistic kinematics to determine the input to the non relativistic dynamics calculation

Relativistic effects in a correlated system

N,pN X,pX

SLIDE 17

Longitudinal responses of 4He for |q|=700 MeV in the four different reference frames.

The different curves are almost identical.

He

4

Relativistic effects in a correlated system

SLIDE 18

Relativistic effects are much smaller in the ANB frame where the final nucleon momentum

is ∝q/2, the position of the peak remains almost unchanged

4He

Relativistic effects in a correlated system

We use the two-fragment model to account for relativistic corrections in the kinematics

SLIDE 19

GFMC electron 4He-cross sections

4He
✐ N.R, W. Leidemann, et al PRC 97 (2018) no.5, 055501

SLIDE 20

GFMC results for muon capture in 4He

Negative muons can be captured by the nucleus in a weak-

interaction process resulting in the change of one of the protons into a neutron and a neutrino emission: inverse process of charge current neutrino scattering

n

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µ−

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p

<latexit sha1_base64="Iet7OPTkf1gkRuQASJltGdM+Yxc=">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</latexit><latexit sha1_base64="Iet7OPTkf1gkRuQASJltGdM+Yxc=">ACb3icdVFdSxtBFJ1sa9VY69eDwUZGgr6EnajaHwL9MVHhUYD2UVmZ2+SwflYZu4qYdlf4Kv9cf0Z/Qed3aRgpF4YOJx7DufOvWkuhcMw/N0KPnxc+7S+sdne+rz9ZWd3b/WmcJyGHIjR2lzIEUGoYoUMIot8BUKuEufhR9+8ewTph9E+c5AoNtViIjhDT93k97udsBudn/Yv+7QGvX64AGcX0SWNumFTHbKs6/u91ijODC8UaOSOTeOwhyTklkUXELVjgsHOeMPbApjDzVT4JKymbSi3z2T0Ymx/mkDfvaUTLl3FylXqkYztzbXk3+rzcucNJPSqHzAkHzRdCkBQNrb9NM2GBo5x7wLgVflbKZ8wyjn45KylT0M0EK2QdiMZIV7XbsYnbpRiOivj9BF4NY4Sj4zMaqORtOxEVfVGN2PY6FaFsacXcm/wt/i3cPo+uO1I49vzjqDwfIqG+Qr+UaOSUQuyIBckWsyJwAeSYv5FfrT3AYHAV0IQ1aS8BWang5C9T0L+v</latexit><latexit sha1_base64="Iet7OPTkf1gkRuQASJltGdM+Yxc=">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</latexit><latexit sha1_base64="Iet7OPTkf1gkRuQASJltGdM+Yxc=">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</latexit>

νµ

<latexit sha1_base64="P9M/dpXaXxO4WUjCfpE48kLDhyk=">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</latexit><latexit sha1_base64="P9M/dpXaXxO4WUjCfpE48kLDhyk=">ACdXicdVFNa9tAEF0rTZO4afN1DIGlbkNOrpQEmtwMueSYQBwbLGFWq7G9eD/E7ijFCP2IXNf1l/Sa1eyC3XaDiw83rzHm51JcykchuGPVrDxZvPt1vZO+93u+w97+weHj84UlkOfG2nsMGUOpNDQR4EShrkFplIJg3R+U/cHT2CdMPoBFzkik21mAjO0FODWBfjWBXj/U7YPb+Ci+u6d8g6oZNdciq7sYHrWGcGV4o0Mglc24UhTkmJbMouISqHRcOcsbnbAojDzVT4JKymbeinz2T0Ymx/mkDfuno2TKuYVKvVIxnLnXvZr8V29U4OQqKYXOCwTNl0GTQlI0tP48zYQFjnLhAeNW+FkpnzHLOPoVraVMQTcTrJF1IBojXdVuxq+caMU01kZp0/Aq1GUeGRkVhuNpGUnqpXuhnDRrcujD29lHuDv8XvhdP/g8fzbuTx/Wn1tdZsck4/kjETkK+mRW3JH+oSTOXkmL+R762dwEnwKTpfSoLXyHJG1Cr78AjmXwlw=</latexit><latexit sha1_base64="P9M/dpXaXxO4WUjCfpE48kLDhyk=">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</latexit><latexit sha1_base64="P9M/dpXaXxO4WUjCfpE48kLDhyk=">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</latexit>

dΓ dEν = G2

V

2π |ψ(0)|2 E2

ν

⇥ R00(Eν) + Rzz(Eν) + R0z(Eν) + Rxx(Eν) − Rxy(Eν) ⇤

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Atomic wave function of the muon approximated as

ψ(x) ' ψ(0) = (Zαµ)3 /π

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A calculation of the total inclusive rate requires requires knowledge of both the low-lying discrete states and higher-energy continuum spectrum of the final nucleus; it is given in terms of five response functions

CC-1B CC-12B exp Γ(s-1) 265±9 306±9 336±75

Two-body currents increase the capture rate by ~15% More data for muon capture rates on light and medium mass nuclei would be helpful to constrain the axial two- body currents ✐ A.Lovato, N.R, R.Schiavilla Phys.Rev. C100 (2019) no.3, 035502

SLIDE 21

(GeV)

E
1

10 1 10

2

10

/ GeV)

2

cm

38

(10

cross section / E
0.2

0.4 0.6 0.8 1 1.2 1.4 (GeV)

E
1

10 1 10

2

10

/ GeV)

2

cm

38

(10

cross section / E
0.2

0.4 0.6 0.8 1 1.2 1.4

TOTAL QE DIS RES

/ GeV)
/ GeV)
Addressing DUNE’s Physics
Liquid Argon TPC Technology

✐ J.A. Formaggio and G.P. Zeller, Rev. Mod. Phys. 84 (2012)

The dominant reaction mechanism changes dramatically over the region of interest to oscillation

experiment OSCILLATION ZONE Ar p Ar n

X µ

𝜉

N N µ

𝜉

N N µ π ∆

𝜉

N

CCQE
CC 1π
DIS

Ar has a complicated structure,

ut of the

reach of most of the ab initio methods

SLIDE 22

Impulse Approximation and Spectral Function

The matrix element of the current can be written in the factorized form
For sufficiently large values of |q|, the IA can be applied under the assumptions
The nuclear cross section is given in terms of the one describing the interaction with individual bound

nucleons

|fi ! |pi ⌦ |fiA−1

Jα = X

i

ji

α

h0|Jα|fi ! X

k

h0|[|ki ⌦ |fiA−1]hk| X

i

ji

α|pi

dσA = Z dE d3kdσNP(k, E)

The intrinsic properties of the nucleus are described by the hole spectral function

SLIDE 23

The CBF Spectral Function of finite nuclei

16O Spectral Function obtained within CBF

and using the Local Density Approximation

136 Many-body theory exposed!

Fig. 7.6

Spectroscopic factors from the (e, e'p) reaction as a function of target mass. The dotted line with a height of 1, illustrates the prediction of the independent-particle

model. Data have been obtained at the NIKHEF accelerator in Amsterdam [Lapikas

(1993)].

momentum can also have negative values when it is directed opposite to the momentum transferred to the target. A correct description of the reaction requires a good fit at all values of this quantity. Figure 7.5 demonstrates that the shapes of the valence nucleon wave functions accurately describe the observed cross sections. Such wave functions have been employed for years in nuclear-structure calculations, which have relied on the independent-particle model. The description of the data in Fig. 7.5, however, requires a significant departure of the independent- particle model, with regard to the integral of the square of these wave

functions. Indeed, the spectroscopic factors, necessary to obtain the solid

curves, are substantially less than 1. Similar spectroscopic factors are extracted for nuclei all over the periodic table4. A compilation for the spectroscopic factor of the last valence orbit for different nuclei, adapted from [Lapikas (1993)], is shown in Fig. 7.6. The results in Fig. 7.6 indicate that there is an essentially global reduction of the sp strength of about 35% for these valence holes in most nuclei. Such a substantial deviation from the prediction of the independent-particle model, requires a detailed

4Most experiments have been performed on closed-shell nuclei.

PLDA(k, E) = PMF (k, E) + Pcorr(k, E)

X

n

Zn|φn(k)|2Fn(E − En)

✐ O. Benhar, A. Fabrocini, and S. Fantoni, Nucl. Phys. A505, 267 (1989). ✐ O. Benhar, A. Fabrocini, S. Fantoni, and I. Sick, Nucl. Phys. A579, 493 (1994)

SLIDE 24

The CBF Spectral Function of finite nuclei

16O Spectral Function obtained within CBF

and using the Local Density Approximation

Z d3rP NM

corr (k, E; ρ = ρA(r))

The Correlated Basis Function approach accounts for

correlations induced by the nuclear interactions

Φn(x1 . . . xA) F Φn(x1 . . . xA)

H = X

i

p2

i

2m + X

i<j

vij + X

i<j<k

Vijk + . . .

Argonne v18 UIX, IL7

The one-body Spectral function of nuclear matter:

PLDA(k, E) = PMF (k, E) + Pcorr(k, E)

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

SLIDE 25

Two-nucleon emission

Meson Exchange currents
Pcorr(k,E) accounts for the presence of strongly

correlated pairs: appearance of a tail in the large energy transfer region in the cross section

+

The Impulse Approximation has been generalized:

W µν

2p2h = W µν ISC + W µν MEC + W µν int

)

2

(GeV

QE 2

Q 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 )

2

/GeV

2

(cm

QE 2

/dQ σ d 2 4 6 8 10 12 14 16 18

39

10 ×

MiniBooNE data with shape error =1.000) κ =1.03 GeV,

eff A

RFG model (M =1.007) κ =1.35 GeV,

eff A

RFG model (M 1.08 × =1.007) κ =1.35 GeV,

eff A

RFG model (M

Initial State Correlations

The axial mass extracted assuming a Fermi Gas description of the nucleus (MA ~1.35 GeV) is incompatible with former measurements: MA~1.03 GeV The observed excess of CCQE cross-section may be traced back to the occurrence of events with two nucleon emission, which are often referred to CCQE-like.

SLIDE 26

Results for 12C(e,e’) cross sections

Separate contributions: IA
Including FSI in the QE region

NR, A. Lovato, O.Benhar, PRL 116, 192501 (2016)

Purple curve: Quasi-elastic, RES and DIS obtained using Bodek-Ritchie parametrization
Green curve: pure two-body current contribution
Red curve: interference one-and two-body current contribution

SLIDE 27

We extend the calculation to include vector and axial vector relativistic two-body currents

(Anti)neutrino -12C scattering cross sections

The calculation of the MEC current matrix is carried out automatically
9d-integral + use of realistic SFs implies dealing with a broader phase space: we developed an

highly parallel Monte Carlo code with importance sampling

π

∆

π π π π π π

∆

We generalized the SF formalism to include vector and axial vector relativistic two-body currents
Nuclear correlations are accounted for by the product of two one-body spectral functions

W µν

2b (q, ω) /

Z d ˜ E d3k (2π)3 d ˜ E0 d3k0 (2π)3 Ph(k, ˜ E)Ph(k0, ˜ E0) X

ij

hk k0|jµ

ij †|p p0ihp p0|jν ij|k k0i

NR et al, Phys.Rev. C99 (2019) no.2, 025502

SLIDE 28

CCQE neutrino -12C cross sections

ν

µ− W +

The 2b contribution mostly affects the ‘dip’ region, in analogy with

the electromagnetic case

Meson exchange currents strongly enhance the cross section for large

values of the scattering angle

NR et al, Phys.Rev. C99 (2019) no.2, 025502

SLIDE 29

NCQE neutrino -12C cross sections

ν ν

The 2b contribution mostly affects the ‘dip’ region, in analogy with

the electromagnetic case

Meson exchange currents strongly enhance the cross section for large

values of the scattering angle

Z0

NR et al, Phys.Rev. C99 (2019) no.2, 025502

SLIDE 30

Pion-production mechanisms

The pion-production mechanism can be described within the IA as
For the description of the pion production reaction mechanisms we adopted the DCC (dynamical

couple-channel) model; H. Kamano, S.X. Nakamura, T.-S.H. Lee, and T. Sato, Phys.Rev. C 88, 035209 (2013),Phys. Rev. C 94, 015201 (2016)

The diagrams considered are resonant and non resonant

hpkπ|jµ

1 |ki = hpkπ|[jµ 1 ]Born|ki + hpkπ|[jµ 1 ]∆|ki

One-body current operator for pion production is defined by its matrix element in momentum space

+ + ∆

π π π

dσA = Z dEd3kd3kπ P(k, E)dσγN→Nπ

The parameters entering the Hamiltonian have been obtained by fitting about 26,000 data points
f the πN and γN reactions up to the invariant mass W ≃ 2 GeV;

SLIDE 31

12C(e,e’)

The interference between one- and two-body currents is missing
The contribution of higher resonances has not yet been included
Electron -12C cross sections

✐ NR, S.X. Nakamura, T.S.H.Lee, A.Lovato, PRC100 (2019) no.4, 045503

SLIDE 32

CCQE neutrino -12C cross sections

12C(νµ, µ−)

✐ NR, S.X. Nakamura, T.S.H.Lee, A.Lovato, PRC100 (2019) no.4, 045503

SLIDE 33

Quasi-elastic and Deep Inelastic Scattering

Within the Spectral Function and Impulse Approximation Formalism

dσA = Z dE d3kdσNP(k, E)

dσN ∝ W2(ω, q2)cos2 θ 2 + 2W1(ω, q2)sin2 θ 2 W p

1 = − q2

4m2 G2

Mpδ

ω + q2

2m

W p

2 =

G2

Ep − q2/(4m2)G2 Mp

1 − q2/(4m2) δ

ω + q2

2m

ωW2(ω, q2) → F2(x) =

X

i

e2

i xfi(x)

mW1(ω, q2) → F1(x) = 1 2xF2(x)

Q2 → ∞, ω → ∞

SLIDE 34

An estimate of the uncertainty of the Spectral Function calculation can be assessed: work in this

direction is currently being carried out

Self Consistent Green’s Function approach

Σ∗ = Σ∗[G(E)]

, an iterative procedure is required to solve the Dyson equation self-consistently

✤ V. Somà et al, Phys.Rev. C87 (2013) no.1, 011303 : generalization of this formalism within Gorkov

theory allows to describe open-shell nuclei such as 12C, 40Ar, 48Ti…

SCGF is a polynomially-scaling many-body method that allows to efficiently describe nuclei with mass

number A up to ∼100

+

Σ∗(E)

G0(E) G(E)

Ab-initio method accurately solves the Dyson equation

=

initial reference state, HF Correlated propagator Self energy: encoding nuclear medium effects on the particle propagation

SLIDE 35

✐ C. Barbieri, NR, and V. Somà, arXiv:1907.01122 The experimental data are from the E12-14-012 experiment at Jefferson Lab Hall A: electron scattering on 40Ar and natural Ti targets

40Ar(e,e’) and 48Ti(e,e’) cross sections w-w/o FSI

Predicting Ar cross sections within SCGF

16O(e,e’) obtained SCGF spectral function + FSI

✐ NR, and C. Barbieri, Phys.Rev. C98 (2018) no.2, 025501

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 20 40 60 80 100 120 exp. SF IA SF IA+FSI

ω [GeV] d휎/dΩeʹ dEeʹ [nb/st MeV]

Ar(e,eʹ)

20 40 60 80 100 120 140 160

Ee = 2.2 GeV, 휃e = 15.5°

Ti(e,eʹ)

SLIDE 36

✐ C. Barbieri, NR, and V. Somà, arXiv:1907.01122

The band comes from a first estimate of the uncertainty on the spectral function calculation obtained

by varying the model-space and the harmonic oscillator frequency

The neutron spectrum of 40Ar is mirrored by the proton spectrum of Ti isotope

To gain indirect information on the neutron spectral function of 40Ar measuring the proton spectrum of Ti

Ti and Ar cross section within SCGF

Ti p Ar n

50 100 150 200 250 300 350 400 450 5 10 15 20 25 30 35 40 45 50

ω [MeV] d휎/dΩℓʹ dEℓʹ [10-8 nb/st MeV]

12C(휈휇,휇-) 40Ar(휈휇,휇-) 48Ti[p](휈휇,휇-)

휃휇 = 30°

2 4 6 8 10 12 14 16

휃휈 = 30°

12C(휈휇,휈휇) 40Ar(휈휇,휈휇) 40Ar[p]+48Ti[p](휈휇,휈휇)

150 300 450

휔

0.8
0.4

0.4 0.8

d휎(mix) - d휎(Ar)

150 300 450

휔

0.1

0.1

d휎(mix) - d휎(Ar)

SLIDE 37

Summary & Conclusions

Neutrino physics is entering a new precision era; realistic models of nuclear dynamics are fundamental for an accurate analysis of neutrino oscillation data

❖ Relativistic effects in the kinematics can be accounted for choosing a convenient reference

frame + two-fragment model

❖ We include relativistic two-body currents, mostly effective in the dip region ❖ Substantial progress in medium-mass nuclei predictions using the SCGF method. Obtain

results for asymmetric nuclei in the high energy region within the EFS. Refine the uncertainty estimate

GFMC calculations:

❖ Two-body currents enhance the cross-sections for transverse kinematics

Extended factorization scheme:

❖ We account for π-production mechanisms, extending the predictive power of the model to

higher energy transfers

❖ Predictions of muon capture rates on 4He (and 3H)

SLIDE 38

Back up slides

SLIDE 39

We start by defining the nuclear response functions, for a given value of q and ω
Electron case we write the double differential cross section as:

dσ dE0dΩ = σMott "✓ q2 q2 ◆2 RL + ✓−q2 2q2 + tan2 θ 2 ◆ RT #

where:

RL = W00 , RT = Wxx + Wyy

Neutrino case:

⇣ dσ dE0dΩ ⌘

ν/¯ ν = G2

4π2 k0 2Eν h ˆ LCCRCC + 2ˆ LCLRCL + ˆ LLLRLL + ˆ LT RT ± 2ˆ LT 0RT 0 i ,

Where the nuclear responses are given by

RCC = W 00 RCL = −1 2(W 03 + W 30)

RT 0 = − i 2(W 12 − W 21)

, ,

RLL = W 33 RT = W 11 + W 22

Electron- and neutrino-nucleus scattering

W µν(q, ω) = X

f

h0|(Jµ)†(q, ω)|fihf|Jν(q, ω)|0iδ(4)(p0 + ω pf)

SLIDE 40

0.05

0.1 0.15 0.2 0.1 0.2 0.3 0.4 0.5 0.6 E [GeV] k [GeV] SF FG

High energy and momentum correlated pairs

P FG(k, E) = (E − ✏B)✓(pF − |k|)

Fermi gas contribution Realistic SF: 80% shell model picture, 20% SRC

Within the Fermi Gas model we can define the SF as:
VMC: exact calculation of the momentum

distribution including SRC pairs

CBF: calculation
1h corresponds to the MF

, rapidly drops

FG: unrealistic momentum distribution, totally

missing the high momentum component

The CBF Spectral Function of finite nuclei

SLIDE 41

The VMC Spectral Function of 4He

P MF

p

(k, E) = nMF

p

(k)δ ⇣ E − B4He + B3H − k2 2m3H ⌘

|hΨ

4He

|[|ki ⌦ |Ψ

3H

0 i]|2

Pτk(k, E) = X

n

|hΨA

0 |[|ki |ΨA−1 n

i]|2 ⇥ δ(E + EA

0 EA−1 n

)

The single-nucleon overlap has been computed within

VMC ( center of mass motion fully accounted for)

SLIDE 42

p

X

τk0=p,n

Z d3k0 (2π)3 np,τk0 (k, k0)δ ⇣ E + EA

0 − e(k0) − ¯

BA2 + (k + k0)2 2mH2 ⌘

1 2 3 4 5 10-2 100 102 104 Q=0.00 4He 1 2 3 4 5 10-2 100 102 104 0.25 1 2 3 4 5 10-2 100 102 104 0.50 1 2 3 4 5 10-2 100 102 104 0.75 1 2 3 4 5 10-2 100 102 104 1.00 1 2 3 4 5 10-2 100 102 104 q (fm-1) ρpp(q,Q) (fm3) 1.25

P corr

p

(k, E) = X

n

Z d3k0 (2π)3 |hΨA

0 |[|ki |k0i |ΨA2 n

i]|2δ(E + EA

0 e(k0) EA2 n

)

The VMC Spectral Function of 4He

R.B. Wiringa, et al, PHYSICAL REVIEW C 89, 024305 (2014)

SLIDE 43

The VMC Spectral Function of 4He

Only SRC pairs should be considered: and be orthogonalized

|ψ

3H

0 i

|k0i|ψA2

n

i

We put cuts on the relative distance between the particles

In collaboration with R.B. Wiringa

SLIDE 44

The VMC Spectral Function of 4He

We put cuts on the relative distance between the particles

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Pp(k, E) = P MF

p

(k, E) + P corr

p

(k, E)

SLIDE 45

How do neutrino oscillate?

Simple case of two-flavor neutrino oscillation
The neutrino mass eigenstates are eigenstates of the free-particle Hamiltonian

|νii = e−Eit|νii

The neutrino flavor eigenstates can be written as a combination of the mass ones

The 𝜉e propagation in vacuum is written as

νe νe

νµ

The oscillation probability in the ultra relativistic limit (t ≃ L, Ei-|pi|2≃mi2/2Ei)

Dependence on the energy distribution of the neutrino source

SLIDE 46

Cross section using the VMC spectral function

4He(e,e’)

4He(e,e’) 3He(e,e’) 3He(e,e’)

Preliminary

SLIDE 47

Pion production mechanisms

10 20 1110 1130 1150 1170 1190 1210 10 1230 1250 1270 1290 1310 1330 5 10 1350 1370 1390 1410 1430 1 cos θπ* 1450 5 10

1

1 cos θπ* 1470 1 cos θπ* 1490 1 cos θπ* 1510 1 cos θπ* 1530 1 cos θπ* 1550

0.2 0.4 0.6 0.8 0.5 1 1.5 2 (a) σ (10-38cm2) Eν (GeV) BNL(corrected) ANL(corrected) 0.5 1 1.5 2 (b) Eν (GeV) 0.5 1 1.5 2 (c) Eν (GeV)

To fix the Q2-dependence of the

vector form factors: analyze data for electron-induced reactions off the proton and neutron targets

For the axial-current matrix

element at Q2=0 the PCAC relation is used. νµ + p → µ−π+p νµ + n → µ−π0p

νµ + n → µ−π+n

Virtual photon cross section at

Q2=0.4 GeV2 for p(e,e’π+)n S.X. Nakamura et al, Phys. Rev. D 92, 074024 (2015)

SLIDE 48

12C charge-current response

Preliminary

W −

µ+

¯ ν

12C, q=700 MeV

We computed the charged-current response function of 12C
Two-body currents have little effect in the vector term, but enhance the axial contribution at

energy larger than quasi-elastic kinematics

A. Lovato et al. in preparation

SLIDE 49

12C charge-current response

Preliminary

W −

µ+

¯ ν

12C, q=700 MeV
We computed the charged-current response function of 12C
Two-body currents have a sizable effect in the transverse response, both in the vector and in

the axial contributions

A. Lovato et al. in preparation

SLIDE 50

12C charge-current response

Preliminary

W −

µ+

¯ ν

We computed the charged-current response function of 12C
12C, q=700 MeV
Two-body currents have a sizable effect in the interference between the axial and vector

current contributions, important to asses neutrino/antineutrino event rates

A. Lovato et al. in preparation

SLIDE 51

The Impulse Approximation and convolution scheme

In the kinematical region in which the interactions between the struck

particle and the spectator system can not be neglected, the IA results have to be modified to include the effect of final state interactions (FSI).

The theoretical approach to calculate the folding function consists on a generalization of Glauber theory
f high energy proton-nucleus scattering

O.Benhar, Phys. Rev. C87, 024606 (2013)

Optical Potential

fq(ω) = δ(ω) p Tq + Z dt 2π eiωt h ¯ U F SI

q

(t) − p Tq i = δ(ω) p Tq + (1 − p Tq)Fq(ω),

Glauber Factor Nuclear Transparency

A.Ankowski et al,Phys. Rev. D91, 033005 (2015)

dσF SI = Z dω0fq(ω − ω0)d˜ σIA , ˜ e(p) = ˜ e(p) + U(tkin(p))