Modeling NuWro–Rwik Interactions

Rwik Dharmapal Banerjee (supervisor: Jan T. Sobczyk)
"Will-be-looking-for-PostDoc-soon" PhD scholar, University of Wrocław, Poland
NuWro workshop 2026
Bronze NuWro gnome mascot holding the Wrocław Neutrino Event Generator medallion Uniwersytet Wrocławski

NuWro team welcomes you in the beautiful city of Wrocław!

Line drawing of the NuWro team in front of Wrocław landmarks, under the NuWro logo

In this talk I will talk about

  • Spectral-function framework (work with Artur Ankowski)
  • Wigner-distribution implementation (collaboration with Alexis Nikolakopoulos)

Introduction

SKIP!

Impulse approximation

  • Dominant process of lepton-nucleus interaction is scattering off a single nucleon, with the remaining nucleons acting as a spectator system. Valid when momentum transfer is high enough
Hadronic current is decomposed
$$J^\mu = \sum_i j_i^\mu$$
The exchanged boson strikes a single nucleon, which leaves the nucleus, while the remaining A minus 1 nucleons stay behind as spectators

Impulse approximation

$$\frac{d\sigma_{lA}}{d\omega\,d\Omega} = \sum_N \int d\omega'\,d^3p\,dE\; P^N_{\mathrm{hole}}(p,E)\; \frac{M}{E_p}\, \frac{d\sigma^{\mathrm{elem}}_{lN}}{d\omega'\,d\Omega}\; P^N_{\mathrm{part}}(p',\mathcal{T}',\omega')$$
Hole spectral function
Elementary
cross section
Particle
spectral
function

Impulse approximation

$$\frac{d\sigma_{lA}}{d\omega\,d\Omega} = \sum_N \int d\omega'\,d^3p\,dE\; P^N_{\mathrm{hole}}(p,E)\; \frac{M}{E_p}\, \frac{d\sigma^{\mathrm{elem}}_{lN}}{d\omega'\,d\Omega}\; P^N_{\mathrm{part}}(p',\mathcal{T}',\omega')$$
Describes ground state
nuclear properties
No nuclear effects
describes propagation
and available final-state phase space
of struck nucleon

Spectral-function framework

  • Nucleon spectral function P(p, E) describes the probability distribution of removing a nucleon of momentum p from the target nucleus, leaving the residual system with excitation energy \(E - E_{thr}\)
  • Fundamental property of nucleus, independent of interaction
$$P(p,E) = P_{\mathrm{MF}}(p,E) + P_{\mathrm{CORR}}(p,E)$$
  • Mean-field part describes shell structure. Can be determined from experimental data. 70-80% contribution
  • Correlated part describes correlated high energetic nucleons. Easier to determine from theoretical calculation
An electron of energy E and momentum k scatters off a nucleus by exchanging a photon, knocking out a proton

Global Fermi-gas

$$P_{\mathrm{GFG}}(p,E) \;=\; \frac{3}{4\pi p_F^3}\; {\theta\!\left(p_F - |\mathbf{p}|\right)}\; {\delta\!\left(E - E_B\right)}$$
  • Nucleus treated as a fragment of non-interacting infinite nuclear matter of constant density
  • Nucleons move freely inside nuclear volume and occupy states up to the maximal momentum
  • In a constant potential every nucleon shares a single removal energy \(E_B\)

Local Fermi-gas

$$P_{\mathrm{LFG}}(p,E) \;=\; \frac{1}{A}\int d^3r\; \rho(r)\; \frac{3}{4\pi p_F^3(r)}\; {\theta\!\left(p_F(r) - |\mathbf{p}|\right)}\; {\delta\!\left(E - E_B(r)\right)}$$
$$p_F(r) \;=\; \left(\tfrac{3}{2}\pi^2 \rho(r)\right)^{1/3}$$
  • Nucleus treated locally as non-interacting nuclear matter, with a density \(\rho(r)\) that varies with position
  • Nucleons occupy states up to a local Fermi momentum, set by the density
  • The potential varies with position, so the removal energy becomes local

Fermi-gas Results for electron-carbon scattering

Electron-carbon double differential cross-section at 620 MeV, 36 degrees Electron-carbon double differential cross-section at 961 MeV, 37.5 degrees Electron-carbon double differential cross-section at 1299 MeV, 37.5 degrees
data from Barreau et. al. and Sealock et. al.

Nuclear shell-model

Measured spectral function of oxygen-16 versus removal energy, showing discrete shell peaks labelled by spin and parity
MF spectral-function describes nuclear orbital structures (Example: Oxygen nucleus)
Leuschner et al., PRC 49, 955 (1994)
Spectroscopic factors for valence protons across target masses, all near 0.65 rather than the mean-field-theory value of 1
Shell-model state-depletion due to short-range correlations explained by CORR spectral-function
credit: Zhihong Ye

Liquid argon detector

MicroBooNE event display: a neutrino enters from the left and produces a muon and several protons in liquid argon, with an inset showing the initial and final state

JLab spectral-function of argon and titanium

JLab spectral-function of argon and titanium

Momentum distribution decomposed shell by shell, from 1s one-half up to 1d three-halves, plus the correlated contribution Removal-energy distribution decomposed shell by shell, plus the correlated contribution
Phys. Rev. D105, 112002

Why titanium? Why?

  1. Neutron SF of Ar cannot be directly obtained due to charge-exchange background
  2. Proton shell structure of Ti mirrors neutron shell structure of Ar
  3. Proton SF of Ti is used as neutron SF of Ar, with necessary corrections
Shell filling of argon-40 and titanium-48: protons to the left of each energy axis, neutrons to the right. Titanium's proton levels are filled exactly as argon's neutron levels are.

Short-range NN correlation

Cartoon of a two-nucleon knockout: a muon neutrino strikes a neutron inside the nucleus, the neutron is tied by a spring to a correlated partner, and a proton leaves via W-plus exchange

Signatures for SRC

  • A high momentum nucleon whose momentum is almost balanced by one other nucleon
  • NN Pair with
    • Large Relative Momentum
    • Small Centre-of-mass Momentum
A correlated nucleon pair at the centre of a nucleus carrying equal and opposite momenta p1 and p2, with the other nucleons dim in the background. Two labelled distances compare the typical 1.7 fm spacing between background nucleons against the pair's separation of under 1.0 fm

Correlations and High Momentum

Momentum density against momentum for helium-3 and oxygen-16 on a logarithmic scale: the mean-field calculation falls off steeply, while the full calculation develops a high-momentum tail above about 1.5 inverse femtometres; the shaded area between them is the contribution of correlations
Ciofi degli Atti, PRC 53 (1996) 1689

Quasideuteron pairs are overwhelming

So, we disregard the nn and pp contributions

Pie chart of nucleons in carbon-12: 80 percent single nucleons, 18 percent in np short-range pairs, and 1 percent each in pp and nn pairs
Measured SRC pair fraction against missing momentum in carbon-12: np pairs sit near 90 to 100 percent across 0.3 to 0.6 GeV per c, while pp pairs sit near 5 percent
R. Subedi et al., Science 320, 1476 (2008)
SRC pair fraction against mass number for carbon, aluminium, iron and lead: the np fraction stays near 95 percent and the pp fraction near 5 percent across the whole range
O. Hen et al, Science 346 614 (2014)

simple correlation model

The pair alone carries the missing momentum: \(\vec{k}_2=-\vec{k_1},\ \vec{k}_{3}=0\). That fixes the removal energy:

$$ P^{\mathrm{}}_{\mathrm{CORR}}(\vec{k_1},E) = n_{\mathrm{CORR}}(\vec{k_1})\, \delta\!\left[E - E^{(2)}_{\mathrm{thr}} - \underbrace{\frac{A-2}{A-1}\frac{k_{1}^{2}}{2M}} _{\substack{\text{intrinsic excitation} \\ \text{energy of residual}}}\right] $$
The nucleus A: two nucleons near the top are joined by a short-range link and carry momenta k1 and k2, while the A minus 2 remnant below is left at rest, k3 = 0 The CORR removal-energy profile for argon: a single delta-function spike at fixed E for each of k=300, 400 and 500 MeV per c

Realistic correlation model

\(\vec{k}_3 \neq 0\): \((A-2)\) remnant recoils. The partner and remnant together form a residual \((A-1)\)-body system.

$$ \mu = \frac{M\,(A-2)M}{M+(A-2)M} = \frac{A-2}{A-1}\,M $$

partner's momentum relative to remnant

$$ \vec{t} = \frac{(A-2)\vec{k}_2 - \vec{k}_3}{A-1} $$

which gives the residual system's intrinsic kinetic energy,

$$ T^{\mathrm{int}}_{A-1} = \frac{t^2}{2\mu} $$
$$ E = E^{(2)}_{\mathrm{thr}} + T^{\mathrm{int}}_{A-1} $$

\(\vec{k}_3\) now fluctuates — a fixed \(k_1\) no longer pins down a single \(E\).

Correlated spectral-function

$$ \begin{aligned} P_{\mathrm{CORR}}(k_1,E) = \int d\vec{k}_3 \; &\delta\Big( E - \underbrace{E^{(2)}_{\mathrm{thr}}} _{\substack{\text{2N threshold} \\ \text{energy}}} - \underbrace{\frac{A-2}{2M(A-1)} \Big[ \vec{k_1} + \frac{A-1}{A-2}\vec{k}_3 \Big]^{2}} _{\substack{\text{kinetic energy of} \\ \text{the } (A-1) \text{ system}}} \Big) \\[10pt] &\times\; \underbrace{n_{\mathrm{rel}} \Big( \Big| \vec{k_1} + \frac{\vec{k}_3}{2} \Big| \Big)} _{\substack{\text{relative momentum} \\ \text{distribution}}} \;\; \underbrace{n_{\mathrm{CM}} \big( |\vec{k}_3| \big)} _{\substack{\text{centre-of-mass} \\ \text{momentum distribution}}} \end{aligned} $$
$$ n_{\mathrm{rel}}(k) = {C^{A}}\, n_D(k) $$ $$ n_{\mathrm{CM}}(k) = \left( \frac{\alpha_{\mathrm{CM}}}{\pi} \right)^{3/2} \exp\left( - {\alpha_{\mathrm{CM}}}\, k_{\mathrm{}}^{2} \right) $$

\(n_{\mathrm{rel}}\) is scaled from deuteron momentum distribution, since SRC pairs are dominantly quasi-deuteron at short range; \(n_{\mathrm{CM}}\) is taken Gaussian since the pair's CM motion is a slow, mean-field-like, s-wave degree of freedom.

A nucleus A drawn as a circle: two nucleons near the top are joined by a short-range link and carry momenta k1 and k2, while the A minus 2 remnant below carries k3 The boson q strikes one nucleon of the pair, which leaves with momentum p1 equal to k1 plus q, while its partner keeps k2 and the A minus 2 remnant keeps k3 inside the A minus 1 system

SRC in NuWro

  1. Argon MF and CORR tables stored separately in NuWro.
  2. Correlated nucleon added to the primary vertex, if struck nucleon belongs to CORR contribution
  3. Relative and centre-of-mass motion of the pair generates complex dynamics. No back-to-back approximation.
The correlated part of the argon spectral function: p squared times P_corr against removal energy, for nucleon momenta of 300, 400 and 500 MeV; the peak falls and moves to higher energy as the momentum rises

SRC in NuWro

Two panels for nucleon momenta of 300, 400 and 500 MeV: (a) the distribution of the pair relative momentum h, peaking between 0.2 and 0.5 GeV, and (b) the distribution of cos theta between the struck nucleon and its partner, rising sharply towards minus one, with an inset covering the full range

Final-state interactions

  1. FSI affects BOTH the outgoing lepton and outgoing nucleons.
  2. FSI is modeled using a convolution scheme, hadronic reinteractions are modeled through intranuclear cascade.
The Change My Mind meme: a man sitting at a table outdoors behind a sign reading FSI is not just cascade!

convolution scheme

$$ \frac{d^{2}\sigma^{\mathrm{FSI}}}{d\omega\, d\Omega} = \int d\omega'\; f_q(\omega - \omega' - U_V)\; \frac{d^{2}\sigma^{\mathrm{PWIA}}}{d\omega'\, d\Omega} $$ $$ f_{q}(\omega) = \delta(\omega)\sqrt{T_A} + F_{\mathbf{q}}(\omega)\left( 1 - \sqrt{T_A} \right) $$
  1. FSI modifies struck nucleon energy spectrum using real part of the optical potential
  2. Folding function describes broadening of inclusive cross-section.
Phys. Rev. D 91, 033005

transparency and optical potential

Left: nuclear transparency T_A against nucleon kinetic energy for carbon, oxygen, argon and iron, with world data points overlaid; the curves drop between 0.25 and 1 GeV and then flatten. Right: the real part of the optical potential U_V against kinetic energy for the same four nuclei, rising from between minus 55 and minus 35 MeV towards about minus 15 MeV at 120 MeV

Event categories

$$ f_{q}(\omega) = \delta(\omega)\sqrt{T_A} + F_{q}(\omega)\left( 1 - \sqrt{T_A} \right) $$
Transparent
Non-transparent
Correlated
Uncorrelated
Correlated
Uncorrelated
Two cartoon panels. Left, a transparent event: a muon neutrino strikes a nucleon which leaves the nucleus undisturbed. Right, a nontransparent event: the struck nucleon collides again on its way out, so at least one intranuclear collision occurs

Cascade algorithm

\(T_A\) enters the inclusive cross section through the convolution. The cascade is then told to produce a final state matching the class the cross section has already assigned.

CategoryStruck nucleonCORR partner
T–MF leaves directly with no INC interaction —
T–CORR leaves directly with no INC interaction standard cascade
NT–MF at least one interaction; redraw if none —
NT–CORR at least one interaction; redraw if none standard cascade

redraw: the mean free path is shortened so the required collision happens sooner

$$ \tilde{\lambda}^{(n)} = \alpha^{n} \tilde{\lambda}, \qquad \alpha = 0.5 $$
  • Redraw changes only the sampled cascade history — not the event weight — and \(\tilde{\lambda}\) is restored once it succeeds.

Result

Stacked histogram of the number of QE events against removal energy of carbon, on a log scale from 17 to 60 MeV, split into four categories: transparent mean-field, transparent correlated, non-transparent mean-field and non-transparent correlated. The transparent mean-field part dominates the sharp peak below 22 MeV, while the non-transparent and correlated parts build up above 30 MeV

Results: electron-carbon scattering

Three panels of the inclusive electron-carbon cross section against energy transfer, at 620 MeV and 36 degrees, 961 MeV and 37.5 degrees, and 1299 MeV and 37.5 degrees. In each, the calculation without FSI peaks higher and further right than the data, while the calculation with FSI is shifted and broadened onto the measured quasi-elastic peak
data from Barreau et al. and Sealock et al.

Why this kinematics?

Energy transfer against three-momentum transfer. An orange contour encloses 68 percent of MicroBooNE CC events, and three purple lines mark the kinematics of the electron-scattering measurements at 620 MeV and 36 degrees, 961 MeV and 37.5 degrees, and 1299 MeV and 37.5 degrees; all three cross the MicroBooNE region

MicroBooNE exclusive observable

  • We test our model with transverse-kinematic variables reported by MicroBooNE.
  • The event topology we choose is CC1p0π as it is QE-dominated. The selection criteria are:
    • Exactly 1 proton with momentum (0.3, 1.0) GeV/c
    • 1 muon with momentum (0.1, 1.2) GeV/c
    • No neutral pions. Any number of neutrons
    • No charged pion over 70 MeV/c momentum
A muon neutrino strikes a neutron inside an argon-40 nucleus; a muon leaves upwards to the right and a proton leaves to the right

Transverse-kinematic imbalance variable

An event in perspective: the neutrino beam runs up to the right, the outgoing muon and proton leave the nucleus, and both are projected back along the beam onto the transverse plane, where the angle between q T and the proton transverse momentum is delta phi T and the vector closing them is delta p T
\(\vec{p}_\nu\)
\(\vec{p}_{\mu}\)
\(\vec{p}_{p}\)
\(\vec{p}^{\;T}_{\mu}\)
\(\vec{p}^{\;T}_{p}\)
\(\vec{q}_T = -\vec{p}^{\;T}_{\mu}\)
\(\delta\phi_T\)
\(\delta\vec{p}_T\)
\(\delta\alpha_T\)

For this presentation, lets choose the variable \(\delta\phi_T\) ("acoplanarity"). \(p^T_\mu\) and \(p^T_p\) are transverse components of the outgoing muon and proton momenta

$$ \delta\phi_T = \angle\left(-\vec{p}^{\;T}_{\mu},\; \vec{p}^{\;T}_{p} \right) $$

zero if the proton balances the muon exactly.

$$ \delta\vec{p}_T = \vec{p}^{\;T}_{\mu} + \vec{p}^{\;T}_{p} $$

Results: neutrino-argon scattering

Four panels of the CC1p0pi cross section against delta phi T, compared with MicroBooNE data: all events, and three bins of delta p T below 0.2, between 0.2 and 0.4, and above 0.4 GeV per c. Each prediction is stacked into RES, MEC and the four QE categories — transparent and non-transparent, mean-field and correlated. The mean-field transparent part dominates at low delta p T, while MEC and RES take over in the highest bin
\(\chi^2/\mathrm{ndf} = 11.8/12\)
\(\chi^2/\mathrm{ndf} = 4.8/4\)
\(\chi^2/\mathrm{ndf} = 15.1/9\)
\(\chi^2/\mathrm{ndf} = 7.0/6\)
data from Abratenko et al.

Why a Wigner distribution?

  • Scattering calculations need momentum and position together.
  • Spectral-function: momentum and removal energy, but not where the interaction happened.
  • That position starts cascade, and sets how much nuclear matter the hadrons cross. Also matters for Pauli-blocking.
  • Wigner distribution gives the joint \((r, p)\) distribution, shell by shell.
Left: with a spectral function the struck nucleon's momentum is known but its position is not, drawn as three faded candidate vertices. Right: the Wigner distribution fixes one vertex at a definite radius r together with the same momentum

Wigner distribution

$$ W_\tau(\vec{r}, \vec{p}) = \int d^3s \; e^{-i\vec{p}\cdot\vec{s}} \; \rho_\tau\!\left( \vec{r} - \frac{\vec{s}}{2},\; \vec{r} + \frac{\vec{s}}{2} \right) $$

built from the one-body density matrix, with the position and momentum densities as its marginals

$$ \rho_\tau(r) = \int \frac{d^3p}{(2\pi)^3}\, W_\tau(\vec{r}, \vec{p}), \qquad n_\tau(p) = \int d^3r \; W_\tau(\vec{r}, \vec{p}) $$

and angle-averaged over \(\cos\theta = \hat{r}\cdot\hat{p}\) for use in NuWro

$$ w_\tau(r, p) \equiv \frac{1}{2} \int d\cos\theta \; W_\tau(\vec{r}, \vec{p}) $$

It is a quasi-probability — in a finite nucleus it must go negative somewhere.

Fermi-gas as Wigner distribution

The Fermi-gas models can be expressed as Wigner distributions.

$$ W^{\mathrm{FG}}_\tau(\vec{r}, \vec{p}) \propto \rho^{\mathrm{FG}}_\tau(\vec{r}) \; \Theta\!\left( k_{F,\tau}(\vec{r}) - |\vec{p}| \right) $$
  • Non-negative everywhere, \(W_\tau \ge 0\) — can be sampled directly.
  • GFG has no \(\vec{r}\)–\(\vec{p}\) correlation; LFG only through the sharp local Fermi sphere.

Carbon Wigner distributions

Proton shell-resolved Wigner distributions for carbon-12 in the r-p plane. Top row, the 1s shell; bottom row, the 1p shell. Left, the signed distribution; middle, its positive component; right, the magnitude of its negative component, which sits around 400 to 600 MeV at small radius
1s
1s (+)
1s (−)
1p
1p (+)
1p (−)
work-in-progress

Wigner distributions in NuWro

  • Wigner tables for each shell \(w_\alpha\) of \(^{12}\mathrm{C}\), \(^{16}\mathrm{O}\), \(^{40}\mathrm{Ar}\), \(^{56}\mathrm{Fe}\) are stored in NuWro.
$$ f^{\tau}_{-} = \frac{\sum_{\alpha \in \tau} \int dr\, r^2 \int dp\, p^2 \; \max\!\left[ -w_\alpha(r,p),\, 0 \right]} {\sum_{\alpha \in \tau} \int dr\, r^2 \int dp\, p^2 \; \left| w_\alpha(r,p) \right|} $$
Target\(f^p_-\) [%]\(f^n_-\) [%]
\(^{12}\mathrm{C}\)3.73.7
\(^{16}\mathrm{O}\)4.34.3
\(^{40}\mathrm{Ar}\)8.89.3
\(^{56}\mathrm{Fe}\)10.010.7

Marginal distributions

Shell-resolved radial and momentum marginals of the Wigner distributions. Top row, argon-40; bottom row, iron-56. Left, the radial density against r; right, the momentum distribution against p. In each the black total is decomposed into the individual orbitals, with 1s-one-half and 2s-one-half dominating at small r and small p
work-in-progress

Cross section formula

$$ \begin{aligned} \frac{d^2\sigma_{\mathrm{WD}}}{dE_\ell d\Omega_\ell} &= \mathcal{N} \sum_{\tau = p,n} \int d^3r \int \frac{d^3p}{(2\pi)^3} \; W_\tau(\mathbf{r}, \mathbf{p}) \, \mathcal{K}_\tau(\mathbf{p}, \mathbf{q}) \; \Theta^{\mathrm{PB}}\!\left( |\mathbf{p} + \mathbf{q}| - k_F^{\tau'}(\mathbf{r}) \right) \end{aligned} $$

with

$$ \mathcal{N} = \frac{G_F^2 \cos^2\theta_C}{(2\pi)^2} \, \frac{|\mathbf{k}'| E_\ell}{E_\nu}, \qquad \mathcal{K}_\tau(p, q) = \frac{M}{E_p} \, \mathcal{L}_{\mu\nu} \, \mathcal{W}^{\mu\nu}_\tau(p, q) $$

The Fermi-gas result is recovered by putting the GFG or LFG distribution in place of \(W_\tau\).

Nuclear binding

  • WD keeps the identity of the shell the nucleon is removed from, so the binding follows from it.
$$ E_\tau + E_\nu - B = E_\ell + E_{\tau'} $$
$$ B_{\alpha,\tau}(p) = \underbrace{S^{\tau}_{\alpha}} _{\substack{\text{separation energy} \\ \text{of shell } \alpha}} + \underbrace{T_{A-1}} _{\substack{\text{recoil of the} \\ (A-1) \text{ system}}} + \underbrace{T_{\tau}} _{\substack{\text{kinetic energy of} \\ \text{the struck nucleon}}} $$
Binding energy distribution P(B) for carbon-12. The GFG is a single dashed line at 34 MeV, the LFG is one band peaking sharply near 46 MeV, and the WD is a broad curve with two maxima, near 28 and 47 MeV, corresponding to the 1p and 1s shells
1p
1s
work-in-progress

By contrast the GFG uses a single constant, \(B = 34\) MeV, and the LFG uses \(E_F(r) + 7\) MeV.

Weighted mode sampling

  • The WD is not positive definite, so it cannot be sampled directly. In the weighted mode we keep the signed distribution and split each shell into two non-negative branches.
$$ w^{\pm}_{\alpha}(r,p) = \tfrac{1}{2} \left[ \, \left| w_\alpha(r,p) \right| \pm w_\alpha(r,p) \, \right], \qquad w_\alpha = w^{+}_{\alpha} - w^{-}_{\alpha} $$

integrated strength of each branch

$$ A^{\pm}_{\alpha} = \int dr\, r^2 \int dp\, p^2 \; w^{\pm}_{\alpha}(r,p), \qquad A^{\pm}_{\tau} = \sum_{\alpha \in \tau} A^{\pm}_{\alpha} $$

Both branches are non-negative, so both can be used as Monte Carlo sampling distributions.

Weighted mode sampling

shell and branch, once the struck nucleon type is known

$$ P(\alpha, \pm \,|\, \tau) = \frac{A^{\pm}_{\alpha}}{A^{\mathrm{abs}}_{\tau}}, \qquad A^{\mathrm{abs}}_{\tau} = A^{+}_{\tau} + A^{-}_{\tau}, \qquad s_\alpha = \pm 1 $$

radial position

$$ P^{\pm}_{\alpha}(r) = \frac{r^2 \int dp\, p^2 \, w^{\pm}_{\alpha}(r,p)} {A^{\pm}_{\alpha}} $$

momentum at that radius

$$ P^{\pm}_{\alpha}(p \,|\, r) = \frac{p^2 \, w^{\pm}_{\alpha}(r,p)} {\int dp' \, p'^2 \, w^{\pm}_{\alpha}(r,p')} $$

The binding energy then follows from \(B_{\alpha,\tau}(p)\).

Weighted mode sampling

every accepted event carries a sign and a correction

$$ \omega_j = s_{\alpha_j} \, C_{\tau_j} \, \omega^{(0)}_j $$

fixed by the negative fraction of the target

$$ C_\tau = \frac{A^{\mathrm{abs}}_{\tau}}{A^{\mathrm{signed}}_{\tau}} , \qquad A^{\mathrm{signed}}_{\tau} = A^{+}_{\tau} - A^{-}_{\tau} $$

the physical prediction is the difference of the two branches

$$ \sigma = \sigma^{+} - \sigma^{-} $$
Target\(C_p\)\(C_n\)
\(^{12}\mathrm{C}\)1.081.08
\(^{16}\mathrm{O}\)1.091.09
\(^{40}\mathrm{Ar}\)1.211.23
\(^{56}\mathrm{Fe}\)1.251.27

A weighted run has \(N_{\mathrm{eff}} \simeq N_{\mathrm{gen}} / C^2_\tau\): reaching a given precision costs 17% more events for carbon, and up to 62% for iron neutrons.

Result

Differential cross section against Q squared for 500 MeV muon-neutrino quasielastic scattering on argon. The positive-weight contribution peaks near 1.15, the negative-weight contribution is a small negative band around minus 0.12, and their difference is the total signed Wigner result
work-in-progress

500 MeV \(\nu_\mu\) QE scattering on argon — the signed result is the difference of the two branches

Unweighted mode sampling

  • NuWro generally generates unweighted events — equal statistical weight — and the rest of the chain expects that.
  • The signed WD is not positive definite, so it cannot be a sampling density there.
  • We build a non-negative coarse-grained \(w^{\mathrm{cg}}_\alpha(r,p) \geq 0\) by averaging the signed WD locally - the negative strength is absorbed, not discarded.
  • We keep the momentum marginal exactly, at the cost of distorting the radial density.

Results: electron scattering

Six panels of the inclusive electron-carbon cross section against energy transfer, comparing the Wigner distribution with the local Fermi gas and with data. In every panel the LFG peak is narrower and higher, while the WD is broader and lower; the WD total is decomposed into its 1s-one-half and 1p-three-halves shell contributions
1.930 GeV, 16.0°
0.961 GeV, 37.5°
1.108 GeV, 37.5°
2.020 GeV, 20.0°
1.299 GeV, 37.5°
1.501 GeV, 37.5°
work-in-progress

LFG: narrower, higher QE peak. WD: broader in \(\omega\).

data from Bagdasaryan et. al., Sealock et. al., and Day et. al.

Sensitivity to \(\vec{r}\)–\(\vec{p}\) correlations

  • We define a factorized WD (FACT) which keeps both marginals of original (CORR) distribution exactly and removes the joint dependence of position and momentum.
$$ w^{\mathrm{FACT}}_{\alpha}(r,p) = \frac{\left[ \int dp'\, p'^2 \, w_\alpha(r,p') \right] \left[ \int dr'\, r'^2 \, w_\alpha(r',p) \right]} {\int dr\, r^2 \int dp\, p^2 \, w_\alpha(r,p)} = \frac{\rho_\tau \times n_\tau}{A_\tau} $$
  • To test, we take the same CC1p0π topology on argon, and \(\delta\phi_T\) with \(\delta p_T > 0.4 \ \mathrm{GeV/c}\) imbalance

Results: neutrino–argon

Two panels of the differential cross section against delta phi T for CC1p0pi events on argon with delta p T above 0.4 GeV. Left: the CORR Wigner distribution in solid green, the factorized FACT Wigner distribution in dashed pink and the local Fermi gas in dashed blue; CORR sits above FACT across the whole range. Right: the same CORR prediction broken into the individual neutron shells as stacked histograms
work-in-progress

FACT is ~14% smaller than CORR over the selected region — primarily change in normalisation.

QE CC1p0π on argon, \(E_\nu = 300\) MeV, \(\delta p_T > 0.4\) GeV

Where the difference comes from

mean initial-nucleon radius of the NT component

\(\alpha\) \(\langle r \rangle_{\mathrm{CORR}}\) [fm] \(\langle r \rangle_{\mathrm{FACT}}\) [fm]
\(2s_{1/2}\)2.042.51
\(1p_{1/2}\)2.182.48
\(1p_{3/2}\)2.142.56
\(1d_{3/2}\)2.543.00
\(1d_{5/2}\)2.553.01
\(1f_{7/2}\)2.943.39
  • The four least-bound orbitals — \(1f_{7/2}, 1d_{3/2}, 2s_{1/2}, 1d_{5/2}\) — give most of the difference: the outermost and most extended ones, whose joint \((r,p)\) structure departs furthest from a product of its marginals.
  • Removing the correlation pushes the selected NT events outward.
  • In CORR they start deeper in, cross more nuclear matter on the way out, and are more likely to pick up the rescattering that makes the large \(\delta p_T\).

I am at the end of my talk

  • Argon SF from the JLab E12-14-012 measurement, with SRC as a correlated quasideuteron pair.
  • FSI consistently in two places: the optical potential folded into the inclusive spectrum, controlled cascade for the hadrons.
  • Tested against inclusive \(^{12}\mathrm{C}(e,e')\) and MicroBooNE CC1p0π on argon.
  • Wigner distributions shell by shell — \(^{12}\mathrm{C}\), \(^{16}\mathrm{O}\), \(^{40}\mathrm{Ar}\) and \(^{56}\mathrm{Fe}\) tabulated in NuWro. (Work-in-progress)
  • Many possibilities come with it. Few will be explored in our upcoming paper.
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JUST-IN-CASE SLIDES

Low energy corrections

Proton-neutron energy level difference:

$$ P_{(n)}(p, E) = P_{(p)}(p, E - \delta) $$

Nuclear recoil:

$$ E_\nu + M_A = E_l + E_p + \sqrt{(M_A - M + E)^2 + |\vec{p}|^2} $$

Coulomb distortion:

Alter momenta of charged particles
Introduce focusing of wave functions

$$ E_l \to E_l^{\mathrm{eff}} = E_l + \htmlClass{vc}{V_C} $$
$$ k_l \to k_l^{\mathrm{eff}} = \sqrt{(E_l + V_C)^2 - m_l^2} $$
$$ \sigma \to \sigma^{\mathrm{eff}} = \left( \frac{k_l^{\mathrm{max}}}{k_l^{\mathrm{eff}}} \right)^{2} \sigma $$
Average Coulomb potential
Differential cross section against energy transfer for muon neutrinos on argon-40 at 0.2 GeV. Without Coulomb distortion the curve peaks near 45 MeV; with distortion it is pushed to higher energy transfer, with maxima near 52 and 71 MeV

Folding function

The folding function F_q against energy transfer, peaking at about 5.4 times ten to the minus four per MeV at zero and falling to nearly zero by plus or minus 250 MeV

Mean-field spectral function

$$ P_{\mathrm{MF}}(p, E) = \sum_{\alpha} \underbrace{S_\alpha} _{\substack{\text{spectroscopic} \\ \text{factor}}} \quad \underbrace{\left| \phi_\alpha(p) \right|^{2}} _{\substack{\text{momentum-space} \\ \text{wave function}}} \quad \underbrace{f_\alpha(E - E_\alpha)} _{\substack{\text{missing-energy} \\ \text{distribution}}} $$
$$ f_\alpha(E - E_\alpha) = \frac{1}{\sqrt{2\pi}\,\sigma_\alpha} \exp\left[ -\frac{(E - E_\alpha)^{2}}{2\sigma_\alpha^{2}} \right] $$

Test spectral-function parameters

protons in argon

\(\alpha\) \(N_\alpha\) \(S_\alpha\) \(E_\alpha\) [MeV] \(\sigma_\alpha\) [MeV]
\(1d_{3/2}\)21.612.532
\(2s_{1/2}\)21.612.932
\(1d_{5/2}\)64.818.234
\(1p_{1/2}\)21.628.06
\(1p_{3/2}\)43.233.06
\(1s_{1/2}\)21.652.010
corr.—3.620.60—

protons in titanium

\(\alpha\) \(N_\alpha\) \(S_\alpha\) \(E_\alpha\) [MeV] \(\sigma_\alpha\) [MeV]
\(1f_{7/2}\)21.611.452
\(1d_{3/2}\)43.212.212
\(2s_{1/2}\)21.612.842
\(1d_{5/2}\)64.815.464
\(1p_{1/2}\)21.635.06
\(1p_{3/2}\)43.240.06
\(1s_{1/2}\)21.662.010
corr.—4.422.09—

\(N_\alpha\): occupation number in the independent-particle shell model  ·  for the correlated part \(S_\alpha\) is the total normalisation and the fourth column is the two-nucleon knockout threshold \(E_{\mathrm{thr}}\) (Phys. Rev. D105, 112002, Phys. Rev. D107, 012007)

Electron-Argon scattering

Inclusive cross section against energy transfer for 2.222 GeV electrons on argon-40 at 15.541 degrees. Both NuWro curves, with and without FSI, reproduce the quasielastic peak position but sit below the data on the low-energy side and miss the flat tail above 0.3 GeV

Missing energy and missing momentum

An electron of energy E and momentum k scatters off a nucleus by exchanging a photon, knocking out a proton
$$ E_e + M_A = E'_e + E'_p + \htmlClass{det}{E^{*}_{A-1}} $$
$$ \vec{k}_e + \vec{0} = \vec{k}\,'_e + \vec{p}\,' + \htmlClass{det}{\vec{p}_{A-1}} $$

known  ·  determined from the rest

In general,

$$ E^{*}_{A-1} = \sqrt{(M_A - M + E)^2 + \mathbf{p}^{2}_{A-1}} $$

Without final state interactions,

$$ -\vec{p}_{A-1} = \vec{p} $$

is the initial proton momentum

Missing energy and missing momentum

An electron of energy E and momentum k scatters off a nucleus by exchanging a photon, knocking out a proton
$$ E_e + M - \htmlClass{det}{E} = E'_e + E'_p $$
$$ \vec{k}_e + \htmlClass{det}{\vec{p}} = \vec{k}\,'_e + \vec{p}\,' $$

known  ·  determined from the rest

For negligible recoil energy,

$$ E^{*}_{A-1} = M_A - M + E $$

Without final state interactions,

$$ -\vec{p}_{A-1} = \vec{p} $$

is the initial proton momentum

argon test → exp spectral-function

Two panels of the argon spectral function S(E_m) against missing energy from 0 to 100 MeV, with an arrow from the left panel to the right. The strength is stacked shell by shell - 2s one-half, 1d three-halves, 1d five-halves, 1p one-half, 1p three-halves, 1s one-half - plus the correlated contribution. On the left the shells overlap into one broad shoulder between 10 and 40 MeV; on the right the same shells are resolved into separate peaks, with 2s one-half sharp near 12 MeV, the 1p pair near 32 MeV and 1s one-half spread around 55 MeV
credit: Artur Ankowski

titanium test → exp spectral-function

Two panels of the titanium spectral function S(E_m) against missing energy from 0 to 100 MeV, with an arrow from the left panel to the right. The strength is stacked shell by shell - 1f seven-halves, 2s one-half, 1d three-halves, 1d five-halves, 1p one-half, 1p three-halves, 1s one-half - plus the correlated contribution. On the left the 1d peak reaches 1.58 near 12 MeV and the 1p strength is one broad bump near 38 MeV; on the right the peaks are narrower and separated, with a gap around 25 MeV and 1s one-half resolved near 52 MeV
credit: Artur Ankowski

Wigner Coarse-graining algorithm

How one coarse-graining block works: a binned slice of the phase-space weighted Wigner distribution with four negative bins, and the reconstruction in which those bins are gone and the positive bins inside the block are reduced, shown against dashed outlines of their original height, keeping the block integral unchanged
  1. At each fixed \(r\), coarse-grain the phase-space weighted profile \(G_{\alpha,r}(p) = p^2 w_\alpha(r,p)\).
  2. Grow a block \(B\) outward from the negative bins, adding neighbours until \(I_{B,\alpha} = \int_B dp\, G_{\alpha,r} \ge 0\).
  3. Keep only its positive part, rescaled to that same integral: \(G^{\mathrm{cg}}_{B,\alpha} = (I_{B,\alpha}/I^{+}_{B,\alpha}) \max[G_{\alpha,r},0]\) — the negative strength is absorbed locally, not discarded. Bins in no block are left unchanged.
  4. Assemble the slices, then restore the momentum marginal exactly: \(w^{\mathrm{cg}}_\alpha = \eta_\alpha \tilde{w}^{\mathrm{cg}}_\alpha \ge 0\) with \(\eta_\alpha(p) = n_\alpha(p)/\tilde{n}^{\mathrm{cg}}_\alpha(p)\).

Generating the partner momentum

once \(\vec{k}_1\) and \(E\) are sampled from \(P_{\mathrm{CORR}}\), energy conservation fixes

$$ E' = E - E^{(2)}_{\mathrm{thr}}, \qquad \kappa^2 = 2\mu E' $$

\(\vec{k}_2\) is confined to a sphere of radius \(\kappa\) centred at \(-\vec{k}_1/(A-1)\) — sample the opening angle \(x=\cos\theta_{12}\), weighted by \(n_{\mathrm{rel}}\) and \(n_{\mathrm{CM}}\) along it

$$ \Big|\vec{k}_2 + \frac{\vec{k}_1}{A-1}\Big|^2 = \kappa^2 $$

for that \(x\), the radius has up to two roots — choose between them by their weight

$$ k_{2,\pm} = -\frac{k_1 x}{A-1} \pm \sqrt{\Delta}, \qquad \Delta = \kappa^2 - \frac{k_1^2}{(A-1)^2}\big(1-x^2\big) $$

finally the azimuth \(\varphi\) is uniform — \((k_2,x,\varphi)\) fixes \(\vec{k}_2\).

Wigner binding energy

remove a nucleon \(\tau\) of momentum \(\vec p\) from a target at rest: the \((A-1)\) system recoils with \(-\vec p\),

$$ E_{A-1}(p) = M^{*}_{A-1,\alpha} + T_{A-1}(p), \qquad T_{A-1}(p) = \frac{p^2}{2M_{A-1}} $$

while the struck nucleon's own on-shell energy is

$$ E_\tau(p) = m_\tau + T_\tau(p), \qquad T_\tau(p) = \frac{p^2}{2m_\tau} $$

energy conservation lets it appear off-shell by \(B_{\alpha,\tau}(p)\),

$$ \big[E_\tau(p) - B_{\alpha,\tau}(p)\big] + E_{A-1}(p) = M_A $$

and its \(p \to 0\) limit is exactly the shell's separation energy,

$$ S^{\tau}_{\alpha} \equiv M^{*}_{A-1,\alpha} + m_\tau - M_A $$

Rearranging:

$$ B_{\alpha,\tau}(p) = S^{\tau}_{\alpha} + T_{A-1} + T_{\tau} $$