---
title: Antineutron Pseudopotential Overview
url: https://www.emergentmind.com/topics/antineutron-pseudopotential
type: topic
---

# Antineutron Pseudopotential Overview

The **antineutron pseudopotential** is the effective complex one-body interaction used to describe an antineutron in matter, either in a nucleus or in a material reflector. In nuclear applications it is usually identified with a complex optical potential \(U_{\bar n}(r)\) or \(V_{\rm opt}(r)\), whose real part governs elastic phase shifts and whose imaginary part encodes annihilation and other inelastic loss. In material-reflection problems relevant to neutron–antineutron oscillation searches, the same concept appears as a Fermi or optical potential \(U_{\bar n}=V_{\bar n}-iW_{\bar n}\) derived from the complex antineutron–nucleus scattering length and the number density of nuclei in the wall medium [1402.3968; 1810.04988; 2508.17725]. Across these contexts, the term “pseudopotential” denotes a reduced effective interaction replacing the many-body antineutron–matter problem by a single-channel Schrödinger-, Klein–Gordon-, or Schrödinger-like equation with a complex potential.

## 1. Definitions and physical meaning

In the most general low-energy usage, the antineutron pseudopotential is written as
\[
U_{\bar n}(E;r)=V_{\bar n}(E;r)+i\,W_{\bar n}(E;r),
\]
with \(W_{\bar n}<0\) representing loss of flux from the elastic channel through annihilation [2205.02529]. For antineutron–nucleus scattering, this effective potential is inserted into a single-channel wave equation,
\[
\Bigl[-(\hbar^2/2\mu)\nabla^2+U_{\bar n}(E;r)-E\Bigr]\psi(E;r)=0,
\]
where \(\mu\) is the reduced mass and \(E\) the center-of-mass kinetic energy [2205.02529]. In the folded \(t\rho\) optical-model formulation used for low-energy antinucleon–nucleus interactions, the full potential is written as
\[
V_{\rm opt}(r)=V_N(r)+V_C(r),
\]
with \(V_C=0\) for \(\bar n\), so that only the strong-interaction nuclear term survives [1402.3968].

A central distinction in the literature is between **nuclear pseudopotentials** and **material pseudopotentials**. In the first case, the target is a finite nucleus and the effective interaction is radial, density-dependent, and strongly absorptive. In the second, the target is a condensed-matter wall medium and the effective interaction is often approximated by a step potential with constant complex value inside the material, as in semi-infinite mirror or one-dimensional bottle models [1810.04988; 2508.17725].

The physical interpretation is correspondingly bifurcated. In nuclei, the pseudopotential represents coherent elastic antineutron propagation plus annihilation in a many-body medium. In mirror-reflection problems, the real part determines the index-of-refraction-like reflection behavior, while the imaginary part determines the annihilation probability per wall collision [1810.04988; 2508.17725]. This suggests that the same formal object links low-energy antineutron nuclear phenomenology to the systematics of neutron–antineutron oscillation searches.

## 2. Standard optical-model constructions for antineutron–nucleus systems

A widely used starting point is the isoscalar folded \(t\rho\) form. Friedman writes the nuclear part as
\[
2\mu\,V_N(r)= -4\pi\Bigl(1+\frac{\mu}{M}\frac{A-1}{A}\Bigr)\,[b_0\,\rho(r)]_{\rm folded},
\qquad \rho(r)=\rho_n(r)+\rho_p(r),
\]
with finite-range Gaussian folding and \(b_1=0\) to very good approximation [1402.3968]. Explicitly,
\[
V_N(r)= -\,\frac{2\pi}{\mu} \Bigl(1+\tfrac{\mu}{M}\tfrac{A-1}{A}\Bigr)
\int d^3r'\; b_0\,\bigl[\rho_n(r')+\rho_p(r')\bigr]\;
\frac{1}{(\sqrt{\pi}\,a_G)^3}e^{-(r-r')^2/a_G^2}.
\]
The potential is separated as \(V_N(r)=U(r)-\tfrac{i}{2}W(r)\), with \(U(r),W(r)\propto \Re b_0,\Im b_0\) respectively [1402.3968].

An analogous formulation is given in the 2015 reanalysis of low-energy antinucleon–nucleus interactions:
\[
2\mu\,V_{\rm opt}(r)= -4\pi\Bigl(1+\frac{\mu}{m}\frac{A-1}{A}\Bigr)b_0\,[\rho_p^F(r)+\rho_n^F(r)],
\]
where the folded densities use either a Gaussian form factor of range \(a_G\) or a Yukawa form factor of range \(a_Y\) [1502.07127]. In that treatment, \(b_1\) is again found to be consistent with zero and is set to zero [1502.07127].

The densities entering these potentials are usually taken from two-parameter Fermi forms. For example,
\[
\rho_i(r)=\frac{\rho_{0,i}}{1+\exp[(r-R_i)/a_i]}, \qquad (i=n,p),
\]
with \(R_i=c_i=A^{1/3}r_{0,i}\), proton densities taken from measured charge distributions, and neutron densities constrained through a neutron-skin shift
\[
r_n-r_p=\gamma\,\frac{N-Z}{A}+\delta,
\qquad \gamma\approx1.0\,{\rm fm},\;\delta\approx0
\]
[1402.3968; 1502.07127].

Once \(U(r)\) is specified, the reaction or annihilation cross section is obtained either through a partial-wave sum or through the optical-theorem form. Friedman gives, above threshold,
\[
\sigma_R=\frac{\pi}{k^2}\sum_{l=0}^\infty(2l+1)\Bigl[1-\exp\{-4\,\Im\delta_l(k)\}\Bigr],
\]
where the complex phase shifts are generated by inserting the same \(V_{\rm opt}\) into a Klein–Gordon equation [1402.3968]. The revisit paper gives the equivalent reaction formula in terms of \(S_\ell\),
\[
\sigma_{\rm ann}=\frac{\pi}{k^2}\sum_{\ell=0}^{\infty}(2\ell+1)\bigl(1-|S_\ell|^2\bigr),
\]
and emphasizes that no further “local equivalent” pseudopotential is needed beyond \(V_{\rm opt}(r)\) [1502.07127].

## 3. Global parameterizations and empirical performance

Friedman’s global analysis constructs a single isoscalar optical-model potential that describes very low-energy \(\bar p\)-nucleus data both below and above threshold, and then tests the same potential against available \(\bar n\)-nucleus annihilation data [1402.3968]. In the sub-threshold antiprotonic-atom fit, ninety level-shift and width data points from \(^{16}\)O through \(^{208}\)Pb were fitted with three free parameters \(\Re b_0,\Im b_0,a_G\), yielding
\[
\Re b_0\approx0.40\pm0.04\ {\rm fm},\qquad
\Im b_0\approx1.25\pm0.05\ {\rm fm},\qquad
a_G\approx1.34\pm0.05\ {\rm fm},
\]
with \(\chi^2/{\rm dof}\simeq2.2\) and rms folding radius \(\sqrt{3/2}\,a_G\approx1.64\) fm [1402.3968].

The same parameter set was then tested against low-energy elastic scattering and above-threshold annihilation without refitting. For elastic differential cross sections at \(p_{\rm lab}=48\) MeV/\(c\) on \(^{12}\)C and \(^{40}\)Ca, later also \(^{208}\)Pb, the outcome agreed within errors with the atom-derived values and yielded \(\chi^2/{\rm dof}\approx2.2\) for 88 data points [1402.3968]. For measured \(\bar p\)-nucleus annihilation cross sections on Ne, Ni, Sn, and Pt, the same potential gave overall agreement at the 10–20% level [1402.3968].

When transferred unchanged to antineutrons by setting \(V_C=0\), the potential reproduces \(\bar n\) annihilation cross sections above about 250 MeV/\(c\) within \(\sim15\%\), but below 100 MeV/\(c\) the calculated cross sections fall below the OBELIX data by factors of 3–4 and fail to reproduce the steep rise observed experimentally [1402.3968]. Attempts to restore agreement by refitting \(b_0\) or \(a_G\) require unphysically large ranges, specifically \(a_G\sim3.3\) fm, and are rejected [1402.3968]. The 2015 reanalysis reaches the same broad conclusion: a common nuclear optical potential interpolating the \(\bar p\)-nucleus data reveals unexpected features in the \(\bar n\)-nucleus annihilation data at very low energy [1502.07127].

The following summary collects representative global optical-model numbers already used in the literature.

| Framework | Parameters | Stated outcome |
|---|---|---|
| Folded \(t\rho\), Gaussian [1402.3968] | \(\Re b_0\approx0.40\pm0.04\) fm, \(\Im b_0\approx1.25\pm0.05\) fm, \(a_G\approx1.34\pm0.05\) fm | Coherent description of \(\bar p\)-nucleus data; underpredicts low-energy \(\bar n\) OBELIX data |
| Folded \(t\rho\), Yukawa/Gaussian [1502.07127] | \(\Re b_0\approx0\pm0.1\) fm, \(\Im b_0\approx1.3\pm0.05\) fm, \(a_G\approx1.30\pm0.05\) fm, \(a_Y\approx1.45\pm0.05\) fm | Excellent agreement with available low-energy \(\bar p\) data |
| Momentum-dependent Woods–Saxon [1803.01820] | Target-dependent \(V'_o,b_0,b_1,W_o,W_{oD},r_V,a_V,r_{W_D},a_{W_D}\) | Good description of \(\bar nA\) annihilation cross sections for \(p_{\rm lab}\lesssim500\) MeV/\(c\) |

These results frame the main empirical issue. For \(\bar p\)-nucleus observables, a simple isoscalar folded potential with no isovector term is broadly successful over a large momentum interval. For \(\bar n\)-nucleus annihilation at the lowest momenta, however, the same ansatz fails in a way that the papers describe as an unresolved “antineutron puzzle” [1402.3968].

## 4. Alternative phenomenological and microscopic pseudopotentials

Beyond the global folded \(t\rho\) form, the literature contains both phenomenological and microscopic alternatives. One phenomenological approach uses a momentum-dependent local Woods–Saxon potential,
\[
U_{\bar n}(r,p)= -V_o(p)\,f(r,R_V,a_V)-i\Bigl[W_o\,f(r,R_V,a_V)-4a_{W_D}W_{oD}\frac{d}{dr}f(r,R_{W_D},a_{W_D})\Bigr],
\]
with
\[
f(r,R,a)=\frac{1}{1+\exp[(r-R)/a]},
\]
and a real-volume depth
\[
V_o(p)=V'_o\,
\frac{\cosh\!\bigl(\sqrt{b_0+p}-\sqrt{b_0}\bigr)}
{\cosh\!\bigl(\sqrt{b_1+p}-\sqrt{b_1}\bigr)}.
\]
Here \(V_C(r)=0\) for the neutral antineutron, only the real volume term carries momentum dependence, and the imaginary part combines volume absorption with a derivative surface term [1803.01820]. For \(^{12}\)C and \(^{206}\)Pb, explicit fitted parameter sets are given, for example \(^{12}\)C with \(V'_o=52\) MeV, \(W_o=12\) MeV, \(W_{oD}=5.98\) MeV, \(b_0=14.04\), \(b_1=7.92\), and \(^{206}\)Pb with \(V'_o=110\) MeV, \(W_o=2.80\) MeV, \(W_{oD}=5.98\) MeV, \(b_0=243.08\), \(b_1=106.60\) [1803.01820].

A more microscopic construction derives the optical potential from the Paris \(\bar NN\) interaction. In one formulation the antineutron–nucleus potential is decomposed into local \(S\)-wave and gradient \(P\)-wave terms,
\[
U(r)=U_S(r)+U_P(r),
\]
with
\[
U_S(r)=-(2\pi/\mu)(1+\mu/M_N)\,f_S(\sqrt s,\rho(r))\,\rho(r),
\]
\[
U_P(r)=-4\pi(\mu/M_N)\,\nabla\!\cdot\![f_P(\sqrt s,\rho(r))\,\rho(r)\,\nabla].
\]
The \(S\)-wave amplitude is dressed in medium by the Waas–Rho–Weise prescription, while no further medium renormalization is applied to the \(P\)-wave [1506.06965]. The local invariant energy is written as \(\sqrt s(r)=E_{\rm th}+\delta\sqrt s(r)\), with
\[
\delta\sqrt s(r)= -B_N[\rho(r)/\bar\rho]
-\xi_N[T_N(\rho(r)/\bar\rho)^{2/3}-E_{\rm lab}]
+\xi_{\bar n}\Re U(r),
\]
and one solves for \(U(r)\) and \(\sqrt s(r)\) iteratively until self-consistency is reached [1506.06965].

A parallel Paris-based derivation writes the optical potential directly in isospin components and includes a \(P\)-wave Kisslinger-type term. The consolidated working form is
\[
\begin{aligned}
2E_{\bar n}V_{\rm opt}(r)
&= -4\pi\Bigl[F_{\bar np}(r,\sqrt s(r))\rho_p(r)+F_{\bar nn}(r,\sqrt s(r))\rho_n(r)\Bigr] \\
&\quad +3\nabla\!\cdot\!\Bigl\{4\pi\frac{m_N}{\sqrt s(r)}
\bigl[f^P_{\bar np}(\delta\sqrt s)\rho_p(r)+f^P_{\bar nn}(\delta\sqrt s)\rho_n(r)\bigr]\Bigr\}\nabla ,
\end{aligned}
\]
with self-consistent \(\sqrt s(r)\) and in-medium amplitudes \(F_{\bar np},F_{\bar nn}\) [1708.06204].

A more schematic review-level representation describes the antineutron pseudopotential as a “G-parity + core” form,
\[
U_{\bar n}(E;r)=\sum_{i=\pi,\sigma,\rho,\omega}(-1)^{G_i}g_i f_i(r)
-(V_0+iW_0)e^{-(r/r_c)^2},
\]
or, for medium and heavy nuclei, as a \(t\rho\) potential
\[
U_{\bar n}(E;r)=-(2\pi/\mu)(1+\mu/m_N)\,[\bar a(E)]\,\rho(r)
\]
with
\[
\bar a \simeq (1.0\pm0.2)+i(1.0\pm0.2)\ {\rm fm},
\]
leading at \(\rho_0=0.17\ {\rm fm}^{-3}\) to \(U_{\bar n}(0;0)\approx-(50\text{–}70)-i(50\text{–}70)\) MeV [2205.02529].

Taken together, these formulations show that “antineutron pseudopotential” is not a single unique function but a family of effective interactions: folded \(t\rho\) forms, local Woods–Saxon parameterizations, and microscopic Paris-based \(S\)- plus \(P\)-wave constructions. A plausible implication is that discrepancies among low-energy data can probe which ingredients—finite range, explicit surface absorption, or self-consistent subthreshold \(P\)-wave dynamics—are indispensable.

## 5. Fermi pseudopotentials for mirror reflection and ultracold-neutron storage

In neutron–antineutron oscillation experiments using slow or ultracold neutrons, the antineutron pseudopotential appears as the Fermi potential of the wall material. Starting from Enrico Fermi’s point-like pseudopotential for slow scattering off nuclei at positions \(R_j\),
\[
\hat V_F(r)=\sum_j \frac{2\pi\hbar^2}{m}\,b_{nA}\,\delta(r-R_j),
\]
one replaces the sum by an integral over number density \(N(r)\) and obtains
\[
V_{\rm eff}(r)=U(r)+iW(r),\qquad
U(r)=\frac{2\pi\hbar^2}{m}N(r)\Re b_{\bar nA},\qquad
W(r)=\frac{2\pi\hbar^2}{m}N(r)\Im b_{\bar nA}
\]
[1810.04988]. In the simplest mirror model, \(N(r)=N_0\) for \(z<0\) and zero outside, so \(V_{\rm eff}(z)\) is a constant complex step inside a semi-infinite reflector [1810.04988].

For a one-dimensional storage bottle model, the wall potential is written as
\[
U_{\bar n}(x)=
\begin{cases}
V_0'-iW_0', & x<-\ell/2,\\
0, & -\ell/2<x<+\ell/2,\\
V_0'-iW_0', & x>\ell/2,
\end{cases}
\]
while microscopically
\[
U_{\bar n}=V_{\bar n}-iW_{\bar n}
=\frac{2\pi\hbar^2}{m}\,N\,a_{\bar nA}
\]
[2508.17725]. Typical values envisaged are \(V_{\bar n}\sim O(10\text{–}100\,{\rm neV})\) and \(W_{\bar n}\sim O(1\text{–}10\,{\rm neV})\) [2508.17725].

For an antineutron with transverse energy \(e_\perp=\hbar^2k^2/(2m)\), the complex momentum inside the mirror is
\[
k_1=\sqrt{2m(e_\perp-U+iW)}/\hbar,
\]
and the specular reflection amplitude is
\[
r=\frac{k-k_1}{k+k_1}.
\]
The reflection probability and relative phase shift per bounce are then
\[
P=|r|^2,
\qquad
\Delta\phi=\arg r
\]
[1810.04988]. In the limit \(e_\perp\ll U\) and \(W\ll U\), one finds
\[
1-P\approx \frac{W}{2U}\sqrt{\frac{e_\perp}{U}},
\qquad
\Delta\phi\approx -\frac{W}{U}\sqrt{\frac{e_\perp}{U}}
\]
[1810.04988]. If the bounce frequency is \(f\), then
\[
\Gamma_{\rm ann}=f\alpha,\qquad \alpha\equiv1-P,
\qquad \frac{d\phi}{dt}=f\Delta\phi
\]
[1810.04988].

Nesvizhevsky et al. tabulate representative wall-material values, averaged over natural isotopes: carbon \(U=103\) neV, \(W=1.7\) neV; magnesium \(U=39\) neV, \(W=1.0\) neV; silicon \(U=48\) neV, \(W=13\) neV; nickel \(U=111\) neV, \(W=2.3\) neV; copper \(U=104\) neV, \(W=2.2\) neV; tungsten \(U=106\) neV, \(W=3.0\) neV; lead \(U=57\) neV, \(W=18.6\) neV [1810.04988]. The same work states that “good” mirror materials have \(U\sim(40\text{–}110)\) neV and \(W\lesssim(1\text{–}5)\) neV, so that \(W/U\ll1\), and identifies Cu and Ni as minimizing both phase slip and annihilation per bounce [1810.04988].

## 6. Role in neutron–antineutron oscillation searches and unresolved issues

In mirror-guided oscillation schemes, the \(n\)–\(\bar n\) two-state amplitude evolves under an effective Hamiltonian
\[
H=
\begin{pmatrix}
E_n & \epsilon\\
\epsilon & E_{\bar n}-i\,\Gamma_{\rm ann}/2
\end{pmatrix},
\]
with \(\epsilon=1/(2T_{n\bar n})\), \(\Gamma_{\rm ann}=f(1-P_{\bar n})\simeq f\alpha\), and \(E_n-E_{\bar n}=\Delta E\) [1810.04988]. For small \(\Gamma_{\rm ann}t\) and small total phase slip, the antineutron component grows proportionally to \(t\), and
\[
P_{n\to\bar n}(t)\simeq
e^{-\Gamma_{\rm ann}t}
\left|\int_0^t e^{i\Delta E t'}dt'\right|^2
\approx
\frac{\sin^2(\Delta E t/2)}{(\Delta E/2)^2}e^{-\Gamma_{\rm ann}t}
\xrightarrow[\Delta E\to0]{} t^2e^{-\Gamma_{\rm ann}t}
\]
[1810.04988]. The same paper states that coherence is lost once the accumulated phase slip reaches order unity and annihilation losses become large for \(t\gtrsim1/\Gamma_{\rm ann}\); choosing large \(U\) and small \(W\) therefore maximizes both the coherent observation time and the annihilation-limited time [1810.04988].

For UCN storage-bottle searches, the annihilation rate is written in terms of wall reflectivity and the neutron–antineutron relative phase shift. In the quasi-free limit,
\[
\Gamma_{\rm ann}
=
\frac{T}{\tau_n^2}
\frac{1-R_{\bar n}^2}{|e^{2i\Delta E T/\hbar}-\tilde R|^2}
\approx
\frac{T}{\tau_n^2}
\frac{1-R_{\bar n}^2}{|1-\tilde R|^2},
\qquad
\tilde R\equiv R_{\bar n}e^{i\Delta\varphi},
\]
and in the idealized limit \(\Delta E\to0\), \(\Delta\varphi\to0\),
\[
\Gamma_{\rm ann}\propto
\frac{T}{\tau_n^2}\frac{1+R_{\bar n}}{1-R_{\bar n}}
\]
[2508.17725]. The figure of merit is thus increased by large \(R_{\bar n}\), small \(\Delta\varphi\), and suppressed magnetic splitting \(\Delta E\) [2508.17725].

The unresolved issue on the nuclear side is that antineutron annihilation data at very low energies are not described by the simple optical potential that fits antiproton data well, and the observed \(\bar n\) cross sections rise more steeply than the naive \(1/v\) expectation [1402.3968]. That tension is the main controversy in the phenomenology summarized here. On the experimental side, the wall pseudopotential has only been studied indirectly thus far, and its optimization is stated to be crucial for maximizing oscillation sensitivity [2508.17725].

Two routes toward resolution are explicitly proposed in the literature. First, direct measurements of \(\bar p\) annihilation cross sections on the same nuclei and at the same momenta as existing \(\bar n\) data would isolate genuinely isospin-dependent dynamics and refine the optical-model picture [1402.3968; 1502.07127]. Second, for wall materials, the complex scattering length \(a_{\bar nA}\) could be inferred indirectly from antiprotonic-atom X-ray spectroscopy or directly from low-energy antineutron scattering if sufficiently slow antineutron beams become available [2508.17725]. This suggests that future progress depends equally on improved data and on more discriminating pseudopotential models.

Source: https://www.emergentmind.com/topics/antineutron-pseudopotential