∎

e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>e3e-mail<EMAIL_ADDRESS>11institutetext: Université Paris-
Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France 22institutetext: National
Centre for Nuclear Research, Warsaw, Poland

# Comparison of N̄N optical models

Jaume Carbonelle1,addr1  Guillaume Hupine2,addr1  Slawomir Wyceche3,addr2

(Received: date / Accepted: date)

###### Abstract

We, compare the strong part of the N̄N interaction obtained by the Nijmegen
partial wave analysis and the results of some of the most popular N̄N optical
potentials in configuration space. We have found severe discrepancies in most
of the partial waves, especially above $p_{Lab}$=400 MeV/c where the partial
wave analysis displays a resonant-like structure in the 31S0 and 33P0 waves.
Some theoretical difficulties to interpret this behaviour in terms of
dynamical resonances are pointed pout and an alternative explanation is
suggested. A much better stability is observed in the low energy parameters,
apart from some discrepancies due to the presence of near-threshold quasi-
bound states in particular waves. Large deviations have also been found
between the corresponding potentials, at short and medium-range ($r\gtrsim 1$
fm) distances.

###### Keywords:

Low energy antiproton physics Optical models Phase shifts PW analysis
Protonium

††journal: Eur. Phys. J. A

##  1 Introduction

In comparison with the Nucleon-Nucleon (NN) case, the Antinucleon-Nucleon-
(N̄N) interaction remains poorly known. The reason for that is, on one hand
the relatively limited number of N̄N low-energy data and on the other hand the
intrinsic difficulty of theoretically describing a system which has hundreds
of open annihilation many-body channels at rest. See for instance
REV_Annihilation_DGMF_PPNP29_1992 ; REV_LEAR_KBMR_PREP368_2002 ;
REV_LEAR_KBR_PREP413_2005  and references therein.

A rigorous theoretical approach of this physical problem in its full
complexity is far beyond our possibilities, both formal and computational, and
it will remain so probably for a long time. There are however phenomenological
ways to model the low energy N̄N physics and obtain a reasonable description
of the existing experimental data, provided one renounces to describe each
particular annihilation channel and by introducing a relatively large number
of parameters. A successful example is provided by the N̄N optical models,
which date from the early days of antiproton physics Levy_NCVIII_1958 , and
whose main properties have been recently reviewed in
REV_NNB_JMR_Frontiers_2020 ; REV_NNB_JMR_Handbook_2022 .

The first accurate description of the p̄p experimental results was provided by
the energy-dependent partial wave analysis of Nijmegen group
ZT_NNB_Nijm_PW_2012  (NPWA) which presents an almost perfect description
($\chi^{2}\approx 1$) of the existing data below $p_{Lab}<$925 MeV/c, although
after applying a severe rejection criteria. In this analysis, the long- and
medium-range N̄N interaction is given by a one- plus two-pion exchange
potential ($V_{\pi}\equiv V_{1\pi}+V_{2\pi}$) at N2LO chiral EFT detailed in
V2pi_NN_NPW_PRL82_1999 . This potential is matched at $b$=1.2 fm to a state
and energy-dependent complex boundary conditions which parametrise the short
range physics, in particular the very complex annihilation dynamics. This is
realised by fixing, for each energies E and partial wave
$\alpha=\\{T,L,S,J\\}$, the logarithmic derivative of the corresponding wave
function at $r$=$b$:
$P_{\alpha}(E)$=$b\left(\Psi^{\prime}_{\alpha}/\Psi_{\alpha}\right)_{r=b}$.
The parameters of the NPWA, i.e. the low energy constants (LEC’s) of $V_{\pi}$
($c_{1},c_{3},c_{4}$) and the complex boundary conditions $P_{\alpha}$, were
determined in ZT_NNB_Nijm_PW_2012  by a fit to the pp̄ scattering data. The
LEC’s found in this way, were compatible with previous determinations from
$pp$ V2pi_NN_NPW_PRL82_1999  and a combined fit of $pp$ and $pn$ scattering
data V2pi_NN_NPW_PRC67_2003 .

The possibility of performing a PW analysis of the N̄N data has been
questioned JMR_Comment_NIJMPW_1995 ; REV_NNB_JMR_Frontiers_2020 , as it
requires the determination of, at least, twice as many parameters as in the NN
case, the number of N̄N partial waves is higher than for NN, and the available
N̄N data are orders of magnitude less abundant. For instance, the N̄N S-matrix
for a tensor uncoupled states, is no longer determined by a real parameter
$\delta$ as in the unitary case ($S=e^{2i\delta}$), but by a complex quantity
$\delta_{C}=\delta_{R}+i\delta_{I}$ whose (positive) imaginary part
$\delta_{I}$ controls the inelastic processes through the parameter $\eta=\mid
S\mid=e^{-2\delta_{I}}$ ($0<\eta<1$), thus allowing the same formal expression
for the S-matrix ($S=e^{2i\delta_{C}}$). This criticism is sound and can
eventually rise some questions about the uniqueness of the solution,
especially when the inelasticity parameter $\eta$, and so the S-matrix itself,
are very small. There is, however, no doubt that the results presented in
ZT_NNB_Nijm_PW_2012  provide an excellent description of the selected data set
and constitute at least one reliable solution in the domain 100 MeV/c
$<p_{Lab}<$1000 MeV/c.

Once determined the parameters of the p̄p PW analysis, the authors of Ref.
ZT_NNB_Nijm_PW_2012  removed the Coulomb potential and the $n-p$ mass
difference ($\Delta_{0}\equiv m_{n}-m_{p}$), and obtained the strong N̄N
phase-shifts in the isospin symmetry, which are in fact the non trivial and
interesting part of the interaction. These results, which can be considered to
a large extent as being model independent, are extremely useful for a critical
comparison between the different models, without directly relying on the
experimental observables. The former, involve usually contributions of many
partial waves and can hide eventual significant disagreements among the
different interaction models. The strong N̄N Nijmegen phase shifts constitute
the basis of our further analysis.

The strong N̄N phase-shifts provided by the Nijmegen group were also the
starting point to determine the parameters of the most recent N̄N Jülich
potential Haidenbauer_JHEP_2017  111It is worth mentioning that, under this
denomination, one can include a series of previous works on N̄N interactions
developed since the 90’s, based on the G-parity transform of the meson-
exchange NN Bonn model VNNB_Julich_PRC44_1991  and even the N2LO version of
the chiral EFT N̄N potential Haidenbauer_JHEP_2014 . For shortness of the
notation, we will hereafter denote Jülich N̄N potential as the one described
in Ref. Haidenbauer_JHEP_2017 . This potential is based on the G-parity
transform of a previously established chiral EFT NN potential at N3LO
NN_N3LO_Epelbaum_2015 . It contains contributions from one- and two-pion
exchange and of contact terms with up to four derivatives. The annihilation
part is taken into account by introducing imaginary contact terms in each
partial wave, regularized by gaussian form factors. The potential is inserted
in a relativistic Lipmann-Schwinger equation to obtain the phase-shifts. The
low energy constants of the pion-exchange part were taken from the pion-
nucleon dynamics and the remaining ones, as well as the annihilation
constants, were adjusted to reproduce to the strong phase shifts and
inelasticity parameters of the Nijmegen PWA in the isospin basis. Supplemented
with the Coulomb and $\Delta m$ term, this model provides an equally good
description of the p̄p data as in the NPWA. Furthermore it has been extended
to describe the zero energy protonium results as well as the existing n̄p data
($T=1$).

The Jülich potential constitutes nowadays the most complete and accurate
description of the N̄N data, would it be at the price of a considerable number
of parameters ($\approx$ 90). However, it has been derived in momentum space
what makes difficult its implementation to study more complex systems, in
particular the very peripheral and loosely bound hydrogenic orbits of the p̄-A
systems. These Coulomb-like states constitute the cornerstone of the PUMA
research project PUMA_EPJA_2022  that requires reliable theoretical
predictions of the annihilation probabilities from some of them. A recent
application to the simple p̄-d case has been recently obtained LC_PLB820_2021
using a simplified (local) form of the Jülich potential. They led to
significantly different predictions with respect to other existing models and
it is not clear which part of these differences is a genuine prediction of the
potential itself or results from the simplifications. On the other hand the
strong non local character of the Jülich potential makes it difficult to be
used in configuration space calculations, where the few-nucleon scattering
problem can be more easily solved.

In view of further applications, but also for the sake of a theoretical
consistency, it is of the highest interest to examine the predictive power of
some of the most popular N̄N optical models formulated in configuration space,
by comparing them at the level of strong phase shifts as well as with the
recent – phase equivalent – Nijmegen PW and Jülich results. A previous
comparison devoted to protonium level shifts and scattering lengths was
published many years ago CIR_ZPA334_1989 ; CRW_ZPA343_1992  but to our
knowledge no systematic study has been performed at non zero energies.

Our goal in the present paper is thus to establish a detailed comparison of
the Nijmegen PW analysis and Jülich results with some selected optical models
widely used in the literature and formulated in configuration space. To this
aim we will consider the last updated version (2009) of the Paris potential
PARIS_PRC79_2009 , the Dover-Richard models DR1_PRC21_1980 ;
DR2_RS_PLB110_1982  published in 1980-82 and the Khono-Weise KW_NPA454_1985
potential formulated in 1986. They represent different degrees of complexity
in the theoretical description and, apart from being formulated in
configuration space, they have in common that: (i) they make full use of
conventional (not EFT) meson-exchange theory, (ii) they were constructed
before the NPWA ZT_NNB_Nijm_PW_2012 , and (iii) they were adjusted to a
restricted data set. At the end, they obtain a less accurate description of
the experimental results than NPWA and Jülich model, but they use a much
smaller number of parameters.

As we will see in what follows, there exist huge disagreements among the
partial wave predictions of the considered optical models, often hidden when
considering only integrated cross sections. They claim for an urgent
clarification of the N̄N interaction at the two-body level, both from the
theoretical as well as from the experimental point of view, before intending a
minimal model independent description of more complex systems, furthermore
involving off-shell properties of the interaction.

We will sketch in section 2 the main ingredients of the theoretical N̄N
formalism used in Refs. ZT_NNB_Nijm_PW_2012 ; Haidenbauer_JHEP_2017  as well
as in our own calculations with Paris, DR and KW optical models. Section 3 is
devoted to compare the strong phase shifts for the S and P partial waves, low
energy parameters and S- and P-wave protonium level shifts of these different
models. Section 4 contains some concluding remarks.

Figure 1: Integrated strong N̄N cross sections – elastic $\sigma_{e}$ (black),
annihilation $\sigma_{a}$ (red ), charge-exchange $\sigma_{ce}$ (green) and
their sum $\sigma_{t}$ (blue) – as functions of the N̄ laboratory momenta for
DR2 (dashed dotted line), KW (dashed line) and Paris 2009 (solid line) optical
models. The results of the Nijmegen Partial Wave analysis ZT_NNB_Nijm_PW_2012
are indicated by filled circles.

##  2 The formalism for N̄N optical models

The strong part of the N̄N force is derived in the isospin basis ($V_{T}$).
However, this basis is not adapted to computing low-energy p̄p scattering
processes due to the relevant role of Coulomb interaction and, in a less
extent, to the $n$-$p$ mass difference ($\Delta_{0}\equiv m_{n}-m_{p}$) which
couples, even asymptotically, the isospin states. One uses, instead, the so
called particle-basis where the $\mid p\bar{p}\rangle$ and $\mid
n\bar{n}\rangle$ states are coupled only by the short range ”charge-exchange”
potential. By adopting the isospin conventions Gasiorowicz_1966 ;
CIR_ZPA334_1989

$N={p\choose
n}\quad\bar{N}={-\bar{n}\choose+\bar{p}}\equiv\begin{array}[]{lcl}|1/2,+1/2>&=&-|\bar{n}>\cr|1/2,-1/2>&=&+|\bar{p}>\end{array}$
(1)

the particle basis is expressed in terms of N̄N isospin states $\mid T,T_{3}>$
as

$\begin{array}[]{lcl c
lcl}|p\bar{p}>&=&+{1\over\sqrt{2}}\left\\{|00>+|10>\right\\}\\\
|n\bar{n}>&=&+{1\over\sqrt{2}}\left\\{|00>-|10>\right\\}\\\
|p\bar{n}>&=&-|1,+1>\\\ |\bar{p}n>&=&+|1,-1>\end{array}$ (2)

The $\mid p\bar{p}>$ and $\mid n\bar{n}>$ states can be cast into a single
state vector

$\mid\Psi\rangle=\pmatrix{\Psi_{p\bar{p}}\cr\Psi_{n\bar{n}}}$

which, in the N̄N models that we will consider, obeys the non-relativistic
Schrodinger equation

$(E-H_{0})\mid\Psi\rangle=\hat{V}\;\mid\Psi\rangle$ (3)

where $E$ is the (non-relativistic) p̄p energy in the center of mass. The
potential matrix

$\hat{V}=\pmatrix{V_{p\bar{p}}&V_{ce}\cr V_{ce}&V_{n\bar{n}}}$ (4)

is expressed in terms of the isospin components $V_{T}$ and the p̄p Coulomb
potential

$ V_{c}(r)=-{\alpha\over r}$

as

$\displaystyle V_{p\bar{p}}$ $\displaystyle=$ $\displaystyle
V_{N\bar{N}}+V_{c}$ (5) $\displaystyle V_{n\bar{n}}$ $\displaystyle=$
$\displaystyle V_{N\bar{N}}+2\Delta_{0}$ (6) $\displaystyle 2V_{N\bar{N}}$
$\displaystyle=$ $\displaystyle V_{0}+V_{1}$ (7) $\displaystyle 2V_{ce}$
$\displaystyle=$ $\displaystyle V_{0}-V_{1}$ (8)

The kinetic energy is assumed to be channel-diagonal with the $p$-$n$ averaged
mass $m$

$H_{0}=-{\hbar^{2}\over m}\Delta\qquad m={m_{p}+m_{n}\over 2}$ (9)

After performing the PW expansion, the reduced radial wave functions $u_{i}$
obey a set of $n_{c}$ coupled differential equations

$u^{\prime\prime}_{i}+q_{i}^{2}u_{j}-\sum_{j=1}^{n_{c}}v_{ij}u_{j}=0$ (10)

where $i,j$ encodes the channel indexes $\\{\bar{p}p,\bar{n}n\\}$ as well as
the quantum number $\alpha=\\{L,S,J\\}$, $q_{i}$ the channel momenta in the
center of mass and

$v_{ij}={m}V_{ij}$

We use natural units ($\hbar=c=1$) along the paper. For the tensor uncoupled
states (1S0,1P1,3P0,…) $n_{c}$=2 and for the tensor coupled (3SD1,3PF2,…)
$n_{c}$=4.

Figure 2: N̄N 1S0 scattering phase shifts (degres) as functions of the N̄
laboratory momenta and for different optical models. Left panel for T=0 state
(11S0) and right one for T=1 (31S0). Solid lines correspond to the real part
$\delta_{R}$ and dashed lines to the (positive) imaginary part $\delta_{I}$.

The relations between the channel momenta are obtained by assuming the same
c.m. total energy ($\sqrt{s}$), which leads to:

${s\over 4}=m_{\alpha}^{2}+q_{\alpha}^{2}$

This gives

$\displaystyle q^{2}_{\bar{p}p}$ $\displaystyle=$ $\displaystyle mE$ (11)
$\displaystyle q^{2}_{\bar{n}n}$ $\displaystyle=$ $\displaystyle
q^{2}_{\bar{p}p}-2\Delta_{0}\;{m}$ (12)

We will denote hereafter by $q\equiv q_{\bar{p}p}$ the c.o.m. momenta of the
p̄p driving channel. Notice than when using the differential form (10), the
$n-p$ mass difference $\Delta_{0}$ is already included in the channel momenta
and must be removed from the potential $(\ref{Vnnb})$.

In the numerical calculations we used $m$= 938.28 MeV and
$\Delta_{0}=m_{n}-m_{p}$=1.2933 MeV. The n̄n channel is open at the p̄p center
of mass energy $E\geq$2.5866 MeV, i.e. $q$=0.2497 fm-1. The relation with the
laboratory momenta is given by

$p_{Lab}=2q\sqrt{1+\left({q\over m}\right)^{2}}\approx 2q$

that is $p_{Lab}$=98.54 MeV/c.

The strong N̄N potentials that we have considered in this work take the form

$V(r)=U(r)+W(r)$ (13)

where real $U$ is a G-parity transform of a NN potential regularised below
some cut-off radius $r_{c}$, and $W$ is the complex potential (eventually
containing also a real part) accounting for the annihilation.

The Paris N̄N model PARIS_PRC79_2009 ; CLLMV_PRL_82 ; PLLV_PRC50_1994 ;
ELLV_PRC59_1999  is based on the G-parity transform of the Paris NN potential
Paris_NN_Lacombe_PRD12_1975 ; Paris_NN_PRC21_1980 . It contains one- and two-
pion exchange (the latter via dispersion relations), plus $\omega$ and $A_{1}$
potentials as part of three-pion exchange. The real part $U$ has a central,
spin-spin, spin-orbit, tensor and quadratic spin-orbit terms. The first two
are energy-dependent, what results into seven scalar amplitudes for a given
isospin $T$. They are regularized below some distance $r_{c}$ ($r_{c}$=0.84 fm
or $r_{c}$=1.0 fm) by a cubic polynomial, whose coefficients introduce
adjustable parameters. All together this gives nine parameters for each
isospin. The annihilation potential $W$, derived in Bachir_PhD_Paris_1980 , is
purely imaginary and has a similar spin structure as the real part. It depends
on six parameters for each $T$. A particularity of the Paris potential is the
short range character of W ($r_{a}=1/2m_{N}\approx$ 0.1 fm). The total number
of parameters of the model is $\approx 30$ and that ensures a fairly good
description ($\chi^{2}$/datum $\approx$ 5) of most of the existing data,
without any selection criteria, including differential cross sections and
polarization observables.

Figure 3: 3SD1 N̄N bare phase shifts and inelasticities (upper panel) and
mixing parameters (lower panel), as functions of the N̄ laboratory momenta.

In Dover and Richard models – DR1 version DR1_PRC21_1980  and DR2 version
DR2_RS_PLB110_1982  – $U$ is taken from a simplified version of the NN Paris
potential Paris_NN_Lacombe_PRD12_1975  containing $\pi,2\pi$ and $\omega$
regularized below $r_{c}=0.8$ fm. DR models were adjusted to reproduce some
analytic parametrisations of the total, elastic, charge-exchange and
annihilation experimental integrated cross sections in the range
$0.4<p_{Lab}<0.9$ GeV/c with a $\chi^{2}$/data$\approx$0.5 for DR1. In
addition to $r_{c}$, there are only four parameters which control the
annihilation potential $W$.

In Kohno-Weise model KW_NPA454_1985 , U is taken from the NN Ueda potential
U_PTP_79  with $\pi,\rho,\omega,\sigma$ meson contributions, regularized below
$r_{c}$=1 fm by a $C^{1}$ matching to a Woods-Saxon potential. As for DR, the
parameters come only from W and are adjusted to reproduce the p̄p total
$(\sigma_{t})$, elastic $(\sigma_{e})$ and charge exchange ($\sigma_{ce}$)
cross sections in the region $200<p_{Lab}<700$ MeV/c. In this way this model
provides a good description of the forward p̄p elastic differential cross
sections at $p_{Lab}$=400, 500, 600 MeV/c, and p̄p elastic differential cross
sections at $p_{Lab}$=390,490,590 MeV/c as well as of differential ce at 490
and 590 MeV/c. No $\chi^{2}$ is given in this analysis.

In DR and KW models, the annihilation potential $W$ is local, energy- and
state-independent. It has the common form

$W(r)=-{W_{0}\over 1+e^{r-R\over a}}$ (14)

with the parameters given in Table 1.

| DR1 | DR2 | KW
---|---|---|---
W0 (GeV) | 21+20i | 0.5+ 0.5i | 1.2i
R (fm) | 0 | 0.8 | 0.55
a (fm) | 0.2 | 0.2 | 0.2
Table 1: Parameters of the Dover-Richard (DR1 and DR2 versions) and Khono-
Weise (KW) N̄N optical models.

These three optical models differ by their meson contents, the value of the
cut-off radius $r_{c}$, the regularization procedure as well as by their
annihilation potentials. They generate the very different potentials presented
in A. As an illustrative example, let us consider Figure 15 from this
Appendix, corresponding to the real part of the 11S0 potentials. They have in
common a strong attraction (200-400 MeV at $r=0.8$ fm) in T=0 channel, which
seems not required by the NPWA. On the other hand, the Paris potential
displays a strong repulsion below $r\approx 0.6$ fm as well as a repulsive
barrier at $r\approx 1$ fm that are absent in the other models. Despite of
that, they provide quite similar results for the integrated elastic
($\sigma_{e}$), annihilation ($\sigma_{a}$) and charge exchange
($\sigma_{ce}$) cross sections. This can be seen in Figure 1, where the
integrated strong cross sections of these three models (together with their
sum $\sigma_{t}=\sigma_{e}+\sigma_{a}+\sigma_{ce}$) are compared to each other
as well as to the NPW results ZT_NNB_Nijm_PW_2012 . The same agreement was
observed in the protonium S- P- and D- level shifts and widths as well as for
the strong and p̄p scattering lengths (see Refs. DR2_RS_PLB110_1982 ;
Bachir_ZPA325_1986 ; CIR_ZPA334_1989 ; CRW_ZPA343_1992 ). However, no
comparison has been done at the level of phase-shifts.

For the three considered models, Paris 2009, DR2 and KW, we have computed the
S-matrix in the energy range $0<p_{Lab}<1000$ MeV/c and for each PW state. We
have extracted the S-matrix real parameters and compared them with the results
of the Nijmegen PW analysis (Tabs VII-IX-X from Ref. ZT_NNB_Nijm_PW_2012 ).
The comparison with Jülich potential, adjusted to reproduce the former, would
be redundant except for the low energy parameters that were not given in the
NPWA ZT_NNB_Nijm_PW_2012  and that have been included in our discussion.

For the uncoupled states, the N̄N S-matrix is determined by a complex phase
shift $\delta_{C}=\delta_{R}+i\delta_{I}$ whose (positive) imaginary part
$\delta_{I}$ is unambiguously defined by the modulus of the S-matrix, the
inelasticity parameter $0<\eta<1$, according to

$S=e^{2i\delta_{C}}=e^{2i\delta_{R}}\;e^{-2\delta_{I}}\qquad\delta_{I}=-{1\over
2}\ln\eta$ (15)

Notice that the annihilation cross section in a given PW, is entirely
determined by $\eta$ as

$ \sigma_{a}=(2J+1){\pi\over 4q^{2}}\left[1-\eta^{2}\right]$ (16)

For the tensor-coupled states (e.g. 3SD1), the 2$\times$2 S-matrix can be
parametrised by 6 real parameters – two ”bare phase shifts” $\delta_{n}$, two
mixing parameters $\epsilon,\omega$ and two inelasticities $\eta_{n}$ –
according to the Bryan and Klarsfeld factorisation Bryan_PRC24_1981 ;
Klarsfeld_PLB126_1983 ; Bryan_PRC30_1984

$\pmatrix{S_{11}&S_{12}\cr S_{21}&S_{22}}=\pmatrix{e^{i\bar{\delta}_{1}}&0\cr
0&e^{i\bar{\delta}_{2}}}\;M\;\pmatrix{e^{i\bar{\delta}_{1}}&0\cr
0&e^{i\bar{\delta}_{2}}}$ (17)

where

$M=\pmatrix{\cos\epsilon&i\sin\epsilon\cr
i\sin\epsilon&\cos\epsilon}H\pmatrix{\cos\epsilon&i\sin\epsilon\cr
i\sin\epsilon&\cos\epsilon}$ (18)

and the matrix $H$, real and symmetric, contains the inelastic parameters
$\eta_{i}$ as eigenvalues

$
H=\pmatrix{\cos\omega&-\sin\omega\cr\sin\omega&\cos\omega}\pmatrix{\eta_{1}&0\cr
0&\eta_{2}}\pmatrix{\cos\omega&\sin\omega\cr-\sin\omega&\cos\omega}$ (19)

Unitary models are defined by the condition

$SS^{\dagger}=\bf 1$

to be fulfilled by (15) and (17). For uncoupled states, this implies
$\eta\equiv 1$ (or equivalently $\delta_{I}\equiv 0$). For tensor-coupled
states, this implies $\omega=0$, $\eta_{1}=\eta_{2}=1$ and so $H=\bf 1$. In
this case, (17) takes the usual Stapp-Ypsilantis-Metropolis (SYM) form
defining the standard bare phase shifts $\bar{\delta}_{n}$ and mixing
parameter $\bar{\epsilon}$ of the unitary case SYM_PR105_1957 .

It is worth mentioning that, in the non unitary case, the definition of
complex phase shifts for the coupled-channel states (3SD1,3PF2,…) is not free
from ambiguities. In fact, the inelasticity parameters can be negative: they
are only limited by the so-called ”unitarity condition” Klarsfeld_PLB126_1983

${\rm Tr}(1-SS^{\dagger})=2-{\rm Tr}(M^{2})=2-\eta_{1}^{2}-\eta_{2}^{2}>0$

which presumes nothing about their sign. We have found that in some of the
considered models one of the inelasticity parameters is indeed negative. This
happen at relatively high energy, when the mixing angles $\epsilon$ and
$\omega$ are large. The natural extension of the uncoupled case (15) to each
inelasticity parameter $\delta_{I,n}=-{1\over 2}\ln\eta_{n}$ poses a problem.
There are alternative ways to define the complex phase shifts, e.g. the
straightforward extension of the SYM parametrisation with complex parameters.
However, though being totally consistent, the relation with respect to the
previously defined parameters is not clear, even for the real part of the
phase shifts. Again, the differences appears at large values of the mixing
parameters, i.e. beyond the zero energy region. Because of that, the
definition of the low-energy parameters (scattering length and effective
range) remains unambiguous.

To get rid of this ambiguities, and to keep as closer as possible to the
results of NPWA, we have only displayed in the tensor-coupled case the real
bare shifts, together with the inelasticities and mixing parameters.

The practical determination of these parameters is more involved than in the
tensor decoupled case and we have followed the procedure described in Sect VII
of Ref. ZT_NNB_Nijm_PW_2012 .

Together with the phase shifts, the corresponding effective range functions
have been also computed and the corresponding Low Energy Parameters (LEP) have
been extracted.

##  3 Results

Figure 4: 1P1 N̄N scattering phase shifts as functions of the N̄ laboratory
momenta. We use the same conventions as in Fig. 2.

###  3.1 Phase shifts

We first present the strong N̄N complex phase shifts
$\delta_{C}=\delta_{R}+i\delta_{I}$ for the lowest partial waves as functions
of the N̄ laboratory momentum $p_{Lab}$. The different states
$\alpha\equiv\\{T,S,L,J\\}$ are alternatively denoted in the spectroscopic
notation by $\alpha\equiv^{2T+1,2S+1}L_{J}$.

The values corresponding to Nijmegen PW analysis are taken from Tab. VIII of
Ref. ZT_NNB_Nijm_PW_2012 . The other models KW, DR2 and Paris (2009) have been
computed by the authors directly from the potentials, with the original model
parameters. We emphasize that the Jülich model Haidenbauer_JHEP_2017  results
are, by construction, adjusted to the Nijmegen PW and there is no need to
included them.

There is an $\pm n\pi$ ambiguity in the definition of the phase shift $\delta$
which is formally solved by imposing $\delta(E\to+\infty)=0$ and by keeping
the same determination when the energy is decreased. Due to the sizable
strengths of the N̄N potentials, this recipe is however of little practical
interest since it imposes to start with the solution at very high energy and
go inwards in energy. In a unitary model (hermitian hamiltonian), another way
to fix the determination is by imposing the value at the origin to be
$\delta_{\alpha}$(E=0)=$n\pi$, where $n$ is the number of bound states in
channel $\alpha$. This impose the full knowledge of the spectrum for each
partial wave. On the other hand, the validity of this result, known as
Levinson theorem, is not well established in the non unitary systems like the
optical models we are considering in this work. Thus, and for the sake of
comparison, we have conventionally adjusted all the computed phase shifts to
the determination given in the Nijmegen PW analysis ZT_NNB_Nijm_PW_2012 .

Figure 5: 3P0 N̄N scattering phase shifts (degres) as functions of the N̄
laboratory momenta. We use the same conventions as in Fig. 2.

Figure 2 contains the results for the 1S0 state, left panel for isospin T=0
(11S0) and right panel for T=1 (31S0). The real part $\delta_{R}$ is in solid
line and the (positive) imaginary part $\delta_{I}$ in dashed line, both in
degres. Different colours have been used to disentangle the different models:
black for the NPW, blue for KW, red for Paris-2009. As one can see, there are
major differences between them, specially in $\delta_{R}$, which deserve some
comments.

For T=0 state (left panel), the real phase shifts of KW and DR models are
close to the NPWA ones up to $p_{Lab}\approx 700$ MeV/c and they both depart
dramatically from the Paris-2009 starting at very low energy. We recall the
reader that the slope of the phase shift at the origin is the scattering
length since $\delta_{\alpha}(q)\approx-a_{\alpha}q$ where $q$ is the center
of mass momentum, related to $p_{Lab}$ by $p_{Lab}=2q$. This difference could
be due to the fact that Paris potential has a near-threshold quasi-bound state
in this channel, absent in KW and Jülich interactions. Its binding energy was
$E=-4.8-26i$ MeV in Ref. PARIS_PRC79_2009 . By using two independent methods,
we confirm this state with a slightly different energy $E=-10.2-23.2i$ MeV. As
we will see in next subsection, the existence of this quasi-bound state is
supported by a different sign in the corresponding scattering lengths (see
Table 2). In this respect, a similar quasi-bound state is also present in DR2
model with $E=-138-320i$ MeV, much deeper in energy and leaving no trace in
the scattering region.

It is worth emphasizing, however, that there is no univocal relationship
between the sign of the scattering length (real part) and the existence of
bound states. For real potentials, the positive sign can be either a
consequence of a repulsive interaction or of an attractive interaction having
one (or several) bound states. The negative sign indicates always the
existence of an attraction but tells us nothing about the existence or non-
existence of a bound state, which will actually depend on the strength of the
attraction. The situation is even more delicate when using complex potentials
and additional informations are required to draw consistent conclusions.

In particular, we would like to notice that the very existence of a quasi-
nuclear state in the 1S0 N̄N state appears as a consequence of the sign of the
measured p̄p scattering length YC_PAD_EPJA57_2021 . It was shown (See Figure 7
of this reference) that for weak (and attractive) p̄p interactions – i.e.
using large values of the cutoff radius, $r_{c}>1.7$ fm – the sign of
Re[$a_{p\bar{p}}$] scattering length is negative. It becomes positive – and in
agreement with experiment – only when, by decreasing $r_{c}$, the interaction
is strong enough to create the first p̄p bound state. The $a_{p\bar{p}}$
involves however both isospin components and, from this single quantity, it is
not possible to conclude in which of the component it appears. Notice also
that the properties of such states, in particular their width, strongly depend
on the annihilation dynamics. When using short range annihilation potential,
as in Paris potential and in some coupled-channel unitary models (UCCM), the
widths are much smaller than when using annihilation potentials type eq. (14),
as in DR2, KW and Jülich potentials. One can find a discussion in Refs.
CDPS_NPA535_91  for KW and YC_PAD_EPJA57_2021  for UCCM.

The particular E-dependence of the Paris potential in the 11S0 state is also
observed in the imaginary phase shift $\delta_{I}$, which remains in a
reasonable agreement with other models only up to $p_{Lab}\approx 200$ MeV/c
and displays a maximum at $p_{Lab}\approx 600$ MeV/c.

When studying the $J/\Psi\to\gamma p\bar{p}$ decay of BES collaboration, the
authors of Ref. DLEW_PRC80_2009 ; DLW_PRC97_2018  interpreted the first peak
in the p̄p invariant mass in terms the above mentioned 11S0 quasi-bond state,
and the second one in terms of a resonant state 11S0 at $\approx$ 250 MeV
above the threshold, i.e. $p_{Lab}\approx$ 950 MeV/c. The peculiar form of the
Paris 11S0 potential depicted in Fig. 15, displaying a deep attractive well
and repulsive pocket at $r\approx$1 fm, can indeed accommodate an S-wave
resonance, but the height of the repulsive barrier is two times smaller than
the supposed resonance energy. On the other hand, no clear evidence of this
state is seen in the corresponding phase shifts. Only a vague structure is
noticeable in $\delta_{R}$ (red solid line) in the vicinity of
$p_{Lab}\approx$ 600 MeV/c, with a maxima of $\delta_{I}$ at almost the same
energy, that could be related.

For T=1 state (right panel), the same discrepancy in $\delta_{R}$ between the
Paris result and the other models is observed. However there is a major
difference between the NPWA and the other optical models: the sharp resonant-
like structure that the former manifests at $p_{Lab}\approx 600$ MeV/c, both
in the real and in the imaginary phase shifts. The existence of an S-wave
shape resonance would require a 31S0 potential with a repulsive bump at finite
distance, like for instance the one exhibited in Figure 15 for the Paris 2009
model, in the T=0 channel. But none of the considered models exhibit such a
behaviour (See Appendix).

Figure 6: One- ($V_{\pi}$) and two-pion ($V_{2\pi}$) exchange potentials in
singlet N̄N states. Results are taken from G-parity transform of the EFT
inspired NN Idaho potential Machleidt_Idaho_N3LO . The different orders up to
N3LO, are plotted separately for both isospin (T) channels. The sum
$V_{\pi}\equiv V_{1\pi}+V_{2\pi}$ is strongly attractive. Only the T=0 states
at NLO shows a short range repulsion.

The possibility for the NPWA to generate a resonance in this partial wave with
a purely attractive (single-channel) potential is difficult to understand.
Indeed, in their analysis the long- and medium-range part of the N̄N
interaction was parametrized by the one-pion ($V_{1\pi}$) plus the two-pion
($V_{2\pi}$) exchange potentials. According to the recent work from the Idaho
group on the EFT-NN interaction at N3LO Machleidt_Idaho_N3LO , $V_{2\pi}$ for
S=0 states is strongly attractive in both isospin channels, in agreement with
previous works V2pi_Kaiser_BW_NPA625_1997 . Let us remind that when going from
NN to N̄N system, $V_{1\pi}$ contribution change the sign while $V_{2\pi}$
remains unchanged.

We have displayed in Figure 6 the $V_{1\pi}$ and $V_{2\pi}$ contributions to
N̄N potential in spin singlet states for both T components. As one can see,
the only repulsion appears for T=0 at NLO, as in Paris potential, but becomes
increasingly attractive at higher orders. In NPWA, the attractive pion tail
($1\pi+2\pi$) is prolonged at $b=$1.2 fm with a boundary condition
corresponding to an also attractive square well (see Tab 1 from
ZT_NNB_Nijm_PW_2012 ). Thus, the overall 31S0 potential is attractive, as it
is the case of the other (KW, DR2, Paris) examined potentials. See Figure 15
in the Appendix. Furthermore the resonant-like bump takes place at center of
mass energy of $E\approx$ 100 MeV, which is quite a high energy for a single-
channel S-wave to produce a visible bump in the phase shifts.

It must be pointed out that other mechanism to mimic resonant like structures
in S-wave exist. For instance when a bound state, generated by the real part
of the interaction, moves into the positive energy region due to the
annihilation potential. The trajectory from the bound state to the continuum
region in the complex energy plane was examined in our previous work with the
KW potential CDPS_NPA535_91 . This can be illustrated by the complex energy
trajectory of the 11S0 state as function of the annihilation strength $W_{0}$.
When $W_{0}=0$ there is a bound state at E=-54.7 MeV. When the annihilation is
switched on, its imaginary part of the energy increases linearly with $W_{0}$
and the state is pushed out into the continuum. It reaches Re(E)$>$0 at
$W_{0}\approx 0.47$ GeV and has a width $\Gamma\approx$ 230 MeV. The energy
imaginary part continues to increase until the model value $W_{0}=$1.2 GeV.
Thus, the widths of the positive energy states ($E\sim$100 MeV) thus obtained
within this mechanism are very large (few hundreds of MeV) and cannot generate
structures like the one displayed in the right panel of Fig. 2.

Figure 7: Pole trajectory of a 11S0 bound state in KW model as function of the
strength of the imaginary part $W_{0}$. In absence of annihilation potential
$(W_{0}=0)$, there is a bound state with E=-54.7 MeV. The effect of the
annihilation potential is to generate a width and to pull out the state into
the continuum. When $W_{0}\approx$ 0.47 GeV, one has $Re(E)=0$ and
$Im(E)\approx$ 115 MeV. With the model parameter $W_{0}$=1.2 GeV, the width of
the state is $\sim 400$ MeV.

All these reasons above, make it difficult to understand the structure
displayed in Figure 2 (right panel) in terms of a resonance in the 31S0 PW,
especially at 100 MeV above threshold.

On the other hand, it is worth noticing that the Jülich potential nicely
reproduces the 13S0 phase shifts of the NPWA, that was attributed to an S-wave
resonance, although no further explanation in terms of the underlying
potential was given in their manuscript. This is a non trivial achievement,
that worked also reasonably well in their N2LO version Haidenbauer_JHEP_2014 ,
and shows the extreme flexibility of the EFT potential.

Interestingly, the same group, came to the conclusion JPsi_Julich_PRD91_2015 ;
JPsi_Julich_PRD98_2018  that in order to reproduce the BES results on the
$J/\Psi\to\gamma p\bar{p}$ decay they were forced to slightly modify their
original 31S1 potential. Once readjusted, the corresponding phase shifts do
not reproduce anymore the NPWA structure of Figure 2 (right panel) but are
very close to the – smoothly varying – KW results (blue curve). This happens
in the N2LO Haidenbauer_JHEP_2014  as well as in the N3LO versions of Jülich
potential Haidenbauer_JHEP_2017 . The authors conclude that the origin of the
near-threshold peaks observed in the BES experiment, may be explained by
assuming the existence of a N̄N 1S0 quasi-bound state, but in the T=1 rather
than in the T=0 channel, as claimed in DLEW_PRC80_2009 ; DLW_PRC97_2018 . Its
energy was estimated to be $E=-36.9-47.2i$ MeV. It could be relevant to notice
that all the examined models have indeed a bound state in this PW when
Im(V)=0. The imaginary part of the potential pushes KW one into continuum
while the DR2 and Paris ones remain still bound, although sizeably deeper than
in the Jülich potential : E=-430-346i MeV for DR2 and E=-184-171i MeV for
Paris.

The possible origin of the resonant-like structure manifested in the NPWA,
which is also manifested in other partial waves, will be discussed later.

Our last comment on this 1S0 state, is to remark that the imaginary phase
shifts $\delta_{I}$ agree reasonably well with each other up to
$p_{Lab}\approx 400$ MeV/c, where the resonant-like behaviour of the NPWA
starts showing up.

Figure 3 displays the S-matrix bare phase shifts and inelasticities (upper
panel) and the mixing parameters (lower panel) of the triplet tensor coupled
3SD1 state. The 3S1 bare phase shifts $\delta_{S}$ seem to be more stable than
the 1S0 ones, for both isospin states, although in this case the best
agreement is among KW, DR2 and Paris.

For T=0, the bare shifts $\delta_{S}$ are very close to each other up to
$p_{Lab}$=700 MeV/c, where DR2 displays some structure, crossing 90 degres in
increasing, what suggest a standard resonance driven by D-waves due to its
centrifugal barrier. The same structure is manifested in $\delta_{D}$ and the
mixing parameters. The inelasticities are less stable: $\eta_{S}$ (left panel)
departs sharply from NPW at $p_{Lab}\approx$500 MeV/c while $\eta_{D}$ the
dispersion among the models starts already at $p_{Lab}\approx$300 MeV/c. The
mixing parameter $\epsilon$ shows also strong deviations above 300 MeV/c while
$\omega$’s are relatively close to each other.

For T=1 (right panel) the S-wave inelasticity of Paris potential departs
sensibly from the other models from the zero energy region. The dispersion in
the mixing parameters is huge, with NPW and DR23 displaying a peculiar energy
dependence.

Figure 8: 3P1 N̄N scattering phase shifts (degres) as functions of the N̄
laboratory momenta, using same conventions than in Fig. 2

The 1P1 phase shifts are displayed in Fig. 4, using the same line and colour
conventions as in Fig. 2. At first glance, it seems that in this case the
global behaviour of the models is quite similar, at least at low energy.
However, we will see in the next section that this is not really the case:
major deviations exist from the origin but these differences are hidden in
this representation due to $q^{3}$ behaviour at small $q$.

The 3P0 results are displayed in Fig 5, with the same colour convention as in
Fig. 2 and 4. In this partial wave, the differences between the model
predictions (KW, DR2 and Paris-2009 ) – relatively close to each other – and
the results of the NPWA are dramatic.

For T=0, the deviations in $\delta_{R}$ start already at $q\approx 0$,
displaying a different concavity. This corresponds to a negative effective
range for the NPW (see Table 3). The imaginary phase shifts $\delta_{I}$,
start differing at $p_{Lab}=200$ MeV/c and the differences increase with the
energy.

For T=1, the low energy phase shifts of all models are quite in agreement,
including NPW results, what is manifested in the LEP’s displayed in Table 3.
Deviations start above $p_{Lab}=200$ MeV/c, where the NPW results display the
same resonant-like structure that the one observed in the 31S0 state, and
practically at the same value of $p_{Lab}$.

Contrary to the 31S0 case, the 33P0 potential has a centrifugal barrier which
can indeed acommodate a resonance, provided that the interaction is attractive
enough. However, it is not the case in any of the examined models – KW, DR2
and Paris – which are globally repulsive in this channel (See Fig. 19 from the
Appendix). This is in agreement with the corresponding scattering volumes in
Table 3, including the values of the Jülich potential : they are very stable
and have very small imaginary parts, as it corresponds to a repulsive
interaction. Nevertheless, the inner part of the NPW potential, used to define
the boundary conditions at b=1.2 fm, is an attractive square well with 160 MeV
depth. Even if beyond $r=b$ the long range part $V_{\pi}$ is slightly
repulsive, the effective potential (V+ centrifugal) in the vicinity of $r=1$
fm remains globally attractive ($-80$ MeV) and we cannot exclude that a
resonance could indeed be produced.

This will be in strong tension with all the potentials, but this is not the
only reason to be cautious with such a possibility. On one hand the maximum of
the centrifugal barrier (60 MeV at $r$=1.2 fm) is sensibly smaller than the
resonance energy ($p_{Lab}=600$ MeV/c, $E_{cm}\approx 95$ MeV). On the other
hand, the $\delta_{R}$ and $\delta_{I}$ curves of both T=1 states, 31S0 and
33P0, can be overimposed in the vicinity of the peak. Since it is difficult to
attribute such a coincidence to a dynamical effect, occurring in two
independent PW at the same energy, this behaviour suggest to look for an
alternative explanation. We will come back to this important point at the end
of this section.

The 3P1 results are displayed in Fig 8 with the colour convention of the
previous Figures. For the T=0 state (left panel), all the phase shifts display
quite a similar behaviour up to 300 MeV/c but the considered optical models
start to depart dramatically from NPWA at $p_{Lab}\approx$400 MeV/c. Below
that, it is one of the most stable channels.

For T=1 (right panel), the deviations are more sizeable already from the zero
energy region, specially in $\delta_{R}$ with Paris 2009. This difference is
due to the existence of a near-threshold quasi-bound state with
$E=-3.6-12.42i$ MeV, (slightly different form the value E=-4.5-9.0i MeV given
in PARIS_PRC79_2009 ) that is absent in the other models and is responsible
for a different sign of the scattering volume (see Table 3).

Finally, we show in Figure 9 the bare phase shifts and mixing parameters for
the 3PF2 partial wave.

For T=0, and in spite of some stability in the scattering lengths, the results
of $\delta_{P}$ falls in two different families: on one side the NPW and DR2
which are attractive and on the other KW and Paris that are repulsive. This
qualitative difference remains such in all the considered energy domain. The
same splitting is observed in the mixing parameters above $p_{Lab}\approx$200
MeV/c.

For T=1, a similar situation happens, with the $\delta_{P}$ values of NPW
evolving in the opposite direction than the rest of the models. Remarkably,
the mixing parameter $\epsilon$ remains stable up to $p_{Lab}\approx$600 MeV/c
while $\omega$ values start diverging at 300 MeV/c.

Figure 9: 3PF2 N̄N bare phase shifts and inelasticities (upper panel) and
mixing parameters (lower panel), as functions of the N̄ laboratory momenta.

As we already mentioned, the NPW results for the 31S0 and 33P0 states, display
the same kind of non trivial structure at $p_{Lab}\approx 600$ MeV/c both for
the real and the imaginary phases, while they are absent in the examined
optical models, with the exception of Jülich potential which reproduces it
well. It seems however unlikely, although not impossible, that a dynamical
effect could generate two resonances at the same energy, in different partial
waves having the same parameters, one in S-wave and other in P-waves.

Looking for a possible explanation of these structures, we noticed that this
energy region corresponds to a sharp maximum of $\delta_{I}$, that is to a
minimum of the inelasticity parameter $\eta$, which turns to be – in this
particular waves – very small. For the 31S0, for instance, one has
$\eta\approx 0.01$ at the minima, that is one order of magnitude smaller than
for the 11S0. The same is true for the 33P0 state, when compared to other
P-waves. This can be seen in Fig. 10, where the inelasticity parameter $\eta$
is plotted as function of $p_{Lab}$ for several states and where the
peculiarity of the 11S0 and 33P0 states is manifested.

Since

$\mid S_{\alpha}(E)\mid=e^{-2\delta^{\alpha}_{I}(E)}=\eta_{\alpha}(E)$

the scattering matrix of 11S0 is in modulus $\sim 10^{-2}$. And similar values
for the 33P0 one.

Given the (not estimated) errors of the PW results in the region of the
resonant-like structure, the S-matrix of these particular waves is in fact
compatible with zero, quite a different situation than for a real resonance
where the S-matrix would have rather a pole. It corresponds actually to the
”black sphere model” scattering, which is quite different from a resonant
scattering, although it produces indeed some structure in the phase shifts.

It remains to be seen wether this structures are an artefact of the analysis
or if they remain unavoidable conclusions of it.

In this respect, it could be pertinent to notice that both states where they
occur are $J=0$, they have very little statistical weight and could be
affected by large errors in their determination.

Furthermore the - very small - inelasticity parameter $\eta$ enters
quadratically as in the annihilation cross sections (16) from where it should,
in principle, be extracted. However, the possibility to extract from the data
analysis a significant signal of the order of 10-4 seems unrealistic. This is
specially true in an observable largely dominated by 3SD1 and 3PF2 partial
waves, at the considered values of $p_{Lab}\approx 600$ Mev/c.

On another hand to generate a model exhibiting zeros of the S-matrix,
practically at the same point of the real axis, in different partial waves
appears to be extremely artificial.

For all these reasons, we believe that the possibility for a PWA, to move from
one solution to another one in the vicinity of the minima of the inelasticity
parameter, should not be disregarded.

One can thus conjecture that the phase shifts in the vicinity of the
$\delta_{I}$ peak, could, in fact, be continued without exhibiting any
resonant-like behaviour, as it happens in all the discussed models, and as it
was considered in Refs. JPsi_Julich_PRD91_2015 ; JPsi_Julich_PRD98_2018  for
the 1S0 state. This would not eliminate all the before mentioned
inconsistencies among the N̄N optical models but will greatly simplify the
analysis in these two partial waves.

Figure 10: Inelasticity parameter $\eta$ in the NPW as function of the N̄
laboratory momenta.

While the above presentation of the phase shifts has some interest for a
global understanding of the interaction, it is not very useful for a detailed
comparison of the models at low energy. Apart from the poor determination of
the phase shifts themselves, the $L>0$ states have a low-energy behaviour like
$\delta(q)\approx-a_{L}q^{2L}$ which hides their contribution in this energy
region.

One can remove the ”centrifugal term” by redefining the reduced phase shifts
$\bar{\delta}=\delta/q^{2L}$, as it was done in CDPS_NPA535_91 , but we
believe it is more instructive to compare the effective range functions
$Z_{\alpha}$ and their dependence on the center of mass momenta $q$ for the
different partial waves. This will be done in the following section.

###  3.2 The Effective range functions and low energy parameters

Figure 11: Effective range function (20) for the N̄N 1S0 states as function of
the center of mass momentum squared (in fm-2). Upper figures correspond to T=0
and the lower ones to T=1. In both cases, the right figure is a zoom of the
left one, restricted to the low energy domain $q^{2}\in[0,0.2]$ fm-2
($p_{Lab}\leq$ 180 MeV/c), where the ERE of (20) is manifested by the linear
behaviour near the origin.

For the tensor decoupled states the effective range functions $Z_{\alpha}$
take the form

$ Z_{\alpha}(q^{2})=q^{2L+1}\cot\delta_{\alpha}=-{1\over a_{\alpha}}+{1\over
2}r_{\alpha}q^{2}+o(q^{4})$ (20)

It is interesting to plot this quantity as function of $q^{2}$, for it easily
allows to determine the validity region of the effective range expansion
(ERE), explicited in eq. (20), and given by the linearity domain near the
origin.

Notice that the value at the origin $Z(0)$

$Z(0)=Z(0)_{R}+i\;Z(0)_{I}={-a_{R}+ia_{I}\over\mid a\mid^{2}}$ (21)

is related to the scattering length as

$\displaystyle a_{R}$ $\displaystyle=$ $\displaystyle-\mid
a\mid^{2}Z(0)_{R}=-{Z(0)_{R}\over\mid Z(0)\mid^{2}}$ (22) $\displaystyle
a_{I}$ $\displaystyle=$ $\displaystyle\mid
a\mid^{2}Z(0)_{I}={Z(0)_{I}\over\mid Z(0)\mid^{2}}$ (23)

In particular one has imperatively $Z(0)_{I}<0$.

In what follows we will display $Z_{\alpha}(q^{2})$ for all the considered PW.
First (left panel), in the full energy domain $q^{2}\in[0,3]$ fm-2, which
correspond to $p_{Lab}\leq 700$ MeV/c. Next (right panel of the same figure),
presents a zoom in the low energy region $q^{2}\in[0,0.2]$ fm-2 ($p_{Lab}\leq$
180 MeV/c) to better exhibit the linearity domain and determine the low energy
parameters ($a_{\alpha},r_{\alpha}$), abusively denoted scattering ”length”
and effective ”range”. The scattering length is given by the $Z(0)$ value,
following eq. (21), and the effective range from (twice) the slope at the
origin.

The colour and model conventions are the same as those used for the phase
shifts: real part of $Z$ is plotted by solid lines and the imaginary part by
dashed lines.

The extracted LEP values are collected in Tables 2 and 3 for the considered
models and PWs. The NPW results given in ZT_NNB_Nijm_PW_2012  were limited to
$p_{Lab}\leq$ 100 MeV/c. We have quadratically extrapolated their values at
the origin by using the 3 lowest points. They are denoted by Nijm* in Tables 2
and 3 and may have only an indicative value, in particular by comparing them
to Julich results. Despite of this naive extrapolation, they all fulfill
$Z(0)_{I}<0$.

Figure 12: Effective range function (20) for the N̄N 1P1 state as function of
the center of mass momentum squared (in fm-2), with the same convention as in
figure 11.

The results for the 1S0 state are displayed in Fig. 11. The upper panel
correspond to T=0 and the lower one to T=1.

The particular behaviour of Paris results is manifested, both in the real as
well as the imaginary phase shifts in all the energy domain. One finds however
a qualitative agreement among the other models including NPW.

As one can see from the right upper figures, the $Z(q^{2})$ dependence of the
T=0 state for the Paris potential is totally flat in the full domain. For the
other models, the ERE is valid only at relatively low energy $q^{2}\leq 0.05$
fm-2, i.e is $p_{Lab}\approx 120$ MeV/c. Beyond this energy, the $q^{4}$ terms
neglected in (20) become relevant, and any linear extrapolation in $q^{2}$
would lead to wrong results.

The $Z(q^{2})$ dependence for the T=1 state (lower panels) is fairly smooth,
with no visible trace of the NPW resonant-like behaviour manifested in the
corresponding phase shift (right panel of Fig 2) at $p_{Lab}\approx$ 600 MeV/c
($q^{2}\approx$ 2.25 fm-2). The differences among the models are much smaller
than for T=0 (except for Paris potential) and lead to scattering length values
which are positive and consistent to each other within 15 % (see Table 2).

The effective range functions for the P-waves are displayed in Figures 12 to
14. In this representation the low-energy part is magnified with respect to
the phase shifts and one can see that, as it was the case for S-waves,
sizeable differences among the models themselves and with respect to the NPWA
emerge.

Figure 12 shows $Z(q)$ for the 1P1 state. For T=0, the results of the real
part (left upper panel) have a similar qualitative behaviour: monotonously
increasing from the origin until a maximum value and decreasing with a zero
crossing towards negative region. However, although the Paris, DR and KW
models are close to each other up to $q^{2}$=0.5 fm-2 ($p_{Lab}\approx$ 300
MeV/c), they differ significantly at high energy, specially with respect the
NPW results.

For T=1 (lower panels) the dispersion is even larger and starts at low energy.
Notice that the DR2 model displays a fast increasing near the origin which
suggest a near-threshold resonant state. It manifests also in the lower right
panel with an ERE breaking below $q^{2}=0.05$ fm-2.

Despite these differences in the model predictions, it is worth noticing the
remarkable stability of the real part of the scattering volumes for both
isospin states. They remain all between a 15% difference band, as one can see
in Table 3. A possible reason of this stability will be discussed at the end
of the section. The essential differences between models are in fact given by
their absorptive parts. These could, in principle, be settled by additional
measurements of the full fine structure in antiprotonic atoms. Unfortunately
this measurement still waits for its turn Gotta_Review_PPNP_2004 .

Figure 13: Effective range function (20) for the N̄N 3P0 state as function of
the center of mass momentum squared (in fm-2) and the same conventions as in
figure 11.

The results for 3P0 wave are shown in Figure 13. Again, for T=0 the real phase
shifts a general qualitative agreement is observed among DR, KW and Paris
models in a more extended energy region, with a departure from NPW already at
$q=0$. The validity of the ERE expansion (upper right panel) extends up to
$q^{2}=0.2$. Except for the Jülich model, the corresponding scattering volumes
have a large real part, attributed in CDPS_NPA535_91  to the existence of a
near-threshold bound or resonant state, and are in a very close agreement
($<$2%).

For T=1, the NPW resonance-like structure displayed in Fig 5 at
$p_{Lab}\approx$ 600 MeV/c leaves no trace in the effective range function
$Z$. However a similar structure – absent at the level of phase shifts – is
seen in $Z$ at $q^{2}=0.5$ fm-2, breaking any possible agreement with the
other optical models. The corresponding scattering lengths (real part) remain
close within 15% and the imaginary part is very small.

The 3P1 effective range functions $Z(q^{2})$ are displayed in Figure 14 for
both isospin channels. The T=0 state (upper panel) is the most stable partial
wave, for the real as well as for the imaginary part. This is probably due to
the 13P1 potential, repulsive in all the models, which also explains the small
imaginary part

The quantitative disagreements start only above $q^{2}\approx$1 fm-2
($p_{Lab}\approx$400 MeV/c) and increase with the energy. The ERE (right
panel) works perfectly in all the domain and the LEPs (both $a_{1}$ and
$r_{1}$) are in close agreement, with an almost vanishing imaginary part.

For T=1, the main difference comes Paris 2009, which displays a different
qualitative behaviour in all the energy domain, including the LEPs. As
mentioned, this particular feature is due to a quasi-bound state generated in
this model at E=-3.6-i12.4 MeV. The other optical models are in quite a good
agreement.

Figure 14: Effective range function (20) for the N̄N 3P1 state as function of the center of mass momentum squared (in fm-2), with the same convention as in figure 11. | $a_{0}$ | $r_{0}$ | $a_{0}$ | $r_{0}$
---|---|---|---|---
T=0 | 11S0 | 13SD1
Nijm* | -0.17 -1.01i | -6.9-2.9 i | – | –
Jülich | -0.21 -1.23i | – | 1.42-0.88i | –
Paris 09 |  1.27 -1.18i | -0.53+0.14i | 1.20-0.80i | –
KW | -0.03 -1.35i | -4.7-7.9i | 1.23-0.77i | –
DR2 |  0.10 -1.07i | -11-6.2i | 1.28-0.78i | –
T=1 | 31S0 | 33SD1
Nijm* | 1.02 -0.60i | 0.7-1.2i | – | –
Jülich | 1.05 -0.58i | – | 0.44-0.96i | –
Paris 09 | 0.76 -0.56i | 0.9-3.9i | 0.61-0.44i | –
KW | 1.07 -0.62i | 0.7-1.9i | 0.78-0.80i | –
DR2 | 1.20 -0.57i | 0.6-1.6i | 0.89-0.71i | –
Table 2: S-wave N̄N low energy parameters (in fm) for the considered optical models: Jülich results are taken from Tab 3 of Ref. Haidenbauer_JHEP_2017 , KW and DR2 from CRW_ZPA343_1992 , Paris 2009 have been recomputed and are in agreement with Benoit_Private . The values of Nijmegen are obtained by extrapolating the phase shifts from Figures 2 and 3. | $a_{1}$ | $r_{1}$ | $a_{1}$ | $r_{1}$ | $a_{1}$ | $r_{1}$ | $a_{1}$ | $r_{1}$
---|---|---|---|---|---|---|---|---
T=0 | 11P1 | 13P0 | 13P1 | 3PF2
Nijm* | -3.34-1.22i | 9.3-1.2i | -3.06-7.23i | -1.7-1.5i | 4.36-0.00i | -3.5-0.0i | – | –
Jülich | -2.87-0.36i | – | -2.83-7.82i | – | 4.61-0.05i | – | -0.74-1.13i | –
Paris 09 | -3.62-0.34i | 3.8-0.8i | -8.78-4.99i | 0.23-1.1i | 5.12-0.02i | -3.4-0.02 | -0.49-0.87i | –
KW | -3.36-0.62i | 3.7-1.6i | -8.83-4.45i | 0.25-0.97i | 4.73-0.08i | -3.5-0.1i | -0.46-1.09i | –
DR2 | -3.28-0.78i | 4.2-2.3i | -8.53-3.50i | 0.63-1.0i | 5.14-0.09i | -3.4-0.1i | -0.59-0.85i | –
T=1 | 31P1 | 33P0 | 33P1 | 3PF2
Nijm* | 0.66-0.18i | 3.3-20i | 2.33-0.92i | -10-0.7i | -2.02-0.70i | 4.7-2.8i | – | –
Jülich | 0.80-0.34i | – | 2.18-0.19i | – | -2.04-0.55i | – | -0.48-0.34i | –
Paris 09 | 1.00-0.77i | -3.7-9.8i | 2.74-0.00i | -5.2-0.01i | 0.28-4.11i | -3.0-2.0i | -0.13-0.21i | –
KW | 0.71-0.47i | -8.3-21i | 2.43-0.11i | -5.8-0.43i | -2.17-0.95i | 2.7-3.5i | -0.30-0.45i | –
DR2 | 1.02-0.43i | -11-10i | 2.67-0.15i | -5.4-0.53i | -2.02-0.70i | 4.6-3.9i | -0.04-0.53i | –
Table 3: P waves N̄N low energy parameters (in fm3) for the considered optical
models: Jülich results are taken from Tab 3 of Ref. Haidenbauer_JHEP_2017 , KW
and DR2 from CRW_ZPA343_1992 , Paris 2009 have been recomputed and are in
agreement with Benoit_Private . The values of Nijmegen are obtained by
extrapolating the phase shifts from Figures 2 and 3.

The low energy parameters ($a_{L}$ and $r_{L}$) of the examined partial waves
are collected in Table 2 for S-waves and Table 3 for P-waves. In view of these
results some general remarks can be drawn. They are in order:

  1. 1.

Despite the huge differences in the phase shifts described in this and the
previous sections, there is a remarkable stability in the ”qualitative” zero-
energy predictions, mainly the scattering lengths and volumes. We mean by that
their ”repulsive” or ”attractive character”, more precisely the sign of their
real part, the almost vanishing imaginary part of 13P1 and 33P0 states, or the
relatively small values of the 3PF2. This is specially true if one take into
account that none of these models has been adjusted in order to reproduce the
zero energy protonium results and that the potential themselves are very
different.

For the S-waves, only the 11S0 result of Paris and DR2 potentials have
different sign.

For the P-waves, the only exception in this general qualitative agreement is
the 33P1 state, again due to Paris potential which has a different sign. As we
have already noticed, the reason for this difference is in both cases related
to a near threshold quasi-bound state in the corresponding PW.

Furthermore, in most of the sates, this agreement is not only qualitative but
there is a reasonable quantitative agreement (between 10-20% with respect
their averaged) in the numerical values, except in some particular states and
models that we will detail below.

  2. 2.

A possible explanation for this astonishing stability in the real parts could
be the one-pion exchange dominance, as it was suggested by Ericson and Weise
for the NN case (see Sect 3.8 of Ref. Ericson_Weise_1988 ). Indeed, for a
tensor uncoupled state, the integral form for the scattering ”length” can be
written as

$a_{L}=\lim_{q\to 0}\;\;{1\over
q^{2L+2}}\;\int_{0}^{\infty}dr\;\hat{j}_{L}(qr)\;v(r)\;u_{L}(r)$ (24)

where $v={m\over\hbar^{2}}V$ is the corresponding potential, $\hat{j}_{L}$ the
reduce regular spherical Bessel and $u_{L}$ the reduced radial solution that
behaves asymptotically as

$u_{L}(r)=\hat{j}_{L}(qr)+\tan\delta_{L}\;\hat{n}_{L}(qr)$

According to these authors, a good approximation of $a_{L}$ (for L$>$0) is
provided by the Born approximation of the one-pion potential tail $v_{\pi}$,
that is:

$a^{B}_{L}(\pi)=\lim_{q\to 0}\;\;{1\over
q^{2L+2}}\;\int_{0}^{\infty}dr\;\mid\hat{j}_{L}(r)\mid^{2}\;v_{\pi}(r)$ (25)

By inserting the one-pion potential

$V_{\pi}(x)=c_{\pi}\left[\sigma\cdot\sigma+S_{12}\chi_{T}(x)\right]\;Y(x)\;\tau\cdot\tau$
(26)

with $x={m_{\pi}r\over\hbar}$,

$c_{\pi}={m_{\pi}\over 3}{g^{2}\over 4\pi}\left({m_{\pi}\over 2M}\right)^{2},$
$Y(x)={e^{-x}\over x},$

and

$\chi_{T}(x)=1+{3\over x}+{3\over x^{2}}\quad$

into eq. (25) one gets:

$\displaystyle
a^{B}_{L}(\pi)={c_{\pi}\over(2L+1)!!^{2}}\;\left({M\over\hbar^{2}}\right)\;\left({\hbar\over
m_{\pi}}\right)^{2L+3}(\tau\cdot\tau)$
$\displaystyle\bigl{\\{}(\sigma\cdot\sigma)(2L\\!+\\!1)!+S_{12}\bigl{[}(2L\\!+\\!1)!+3[(2L)!+(2L\\!\\!-\\!\\!1)!]\bigr{]}\bigr{\\}}$

The first remark about the later expression is the ”$\tau\cdot\tau$ rule”, i.e
the fact that the ratio of the scattering lengths of two isospin components of
the same PW is given by the value of $\tau\cdot\tau$ operator:
$\tau\cdot\tau$=-3 for T=0 and $\tau\cdot\tau$=1 for T=1. Indeed the real part
of the scattering volumes displayed in Table 3 (uncoupled states) roughly
fulfil this requirement. There are two exceptions: the results of Nijmegen-
Jülich 3P0 (in relative sizes) and the Paris 3P1 (in relative sizes and
signs).

By using the numerical values $m_{\pi}$=138.039 MeV, M=938.9183 MeV (averaged
pion and N masses) and $g^{2}/4\pi$=14.4, one obtains for the N̄N states the
results displayed in Table 4. The 3P2 results is decoupled from the 3F2 tensor
partner. With the recommended NPW $\pi NN$ coupling constant
$g^{2}/4\pi$=13.9, a reduction factor 0.965 must be used.

| $\sigma\cdot\sigma$ | $\tau\cdot\tau$ | $S_{12}$ | Re[$a_{L}$]
---|---|---|---|---
11P1 | -3 | -3 | 0 | -3.09
31P1 | -3 | 1 | 0 | 1.03
13P0 | 1 | -3 | -4 | -9.27
33P0 | 1 | 1 | -4 | 3.09
13P1 | 1 | -3 | 2 | 6.18
33P1 | 1 | 1 | 2 | -2.06
13P2 | 1 | -3 | -2/5 | 0
33P2 | 1 | 1 | -2/5 | 0
Table 4:  N̄N scattering volumess (fm3) as predicted by the pion dominance
from Ericson_Weise_1988

For 1P1 state, the Born pion values are close (15%) to the full results from
Table 3, except for KW where the difference is twice larger.

For 33P0, the differences are of the same order. Only the 13P0 Jülich result
shows a large discrepancy in the $\tau\cdot\tau$ rule.

For 33P1, the agreement is even better, except for the instructive Paris
result which differs substantially. Indeed the ”one-pion exchange dominance”
is based on the assumption that the scattering solution $u_{L}$ is close to
the free wave $\hat{j}_{L}$ in the dominant part of the integral (24). In case
of the existence of a bound state, as in the Paris model, $u_{L}$ has a node
and change its sign with respect to the free solution.

For 33PF2, the Born results from eq. (25) cannot directly be applied since
they were stablished for tensor uncoupled states. However for the single 3P2
state they predict a vanishing Re(a) and the small value of the full results
from Table 3 could be a trace of this compensation.

To close this remark, we would like to mention that while the ”one-pion
exchange dominance” is justified in the NN case, where it was stablished, it
has an uncertain applicability in the N̄N physics. Apart from disregarding the
annihilation physics, this approach will fail in presence of one or several
bound or resonant states, as it is the case in most of $V_{\bar{N}N}$ models.
We have seen an illustrative example in the 33P1 case with Paris results
having different sign. However the same breakdown of the ”one-pion exchange
dominance” can happen if there are two bound states, although keeping the same
sign. This can be the case of the 13P1 state with KW or the 13P0 with Jülich
where the $\tau\cdot\tau$-rule is badly violated.

  3. 3.

The imaginary part of S-waves is also remarkably stable within quite narrow
limits Im[a(11S0)]=1.18$\pm$ 0.17 fm, Im[a(31S0)]=0.60$\pm$0.03 fm and

Im[a(13SD1)]=0.82$\pm 0.05$ fm. Only the 33SD1 state presents some dispersion
essentially due to Paris result, with Im[a(33SD1)]=0.73$\pm 0.30$ fm.

The imaginary part of P-waves is much less stable, although some common
features are shared like the small values for the 13P1, due to its repulsive
character.

  4. 4.

Of particular interest is the 13P0 state, with a very large real part $\sim 9$
fm3 shared by DR, KW and Paris models (Jülich results are 3 times smaller),
and confirmed by protonium data. This large and negative value was attributed
in CDPS_NPA535_91  to the existence of a near-threshold state. However it
finds also a ”natural” explanation in terms of the ”pion dominance” in Table
4, which predicts Re(a)=-9.3 fm3. This, at first glance, puzzling situation
can be reconciled if one takes into account the result of Ref.
NN3P0_chiral_PRC1997 , according to which, in the chiral limit ($m_{\pi}$=0),
the NN 3P0 state (so T=1) has a zero energy virtual state, with a diverging
scattering volume. The existence of such a NN bound state, as well as some
related consequences in nuclear matter, is prevented by the short range NN
repulsion, which is however absent in the N̄N case and leave open such a
possibility.

  5. 5.

The most stable states are those with a repulsive potential. These are the
13P1 and 33P0. They have in common a small imaginary part both in $a_{1}$ and
$r_{1}$ since they are little sensitive to annihilation dynamics.

For 13P1, all models agree with a real part of $4.9\pm 0.3$ fm3 and an
imaginary part smaller than 0.1 fm3 For 33P0, they all agree with a real
scattering volume $2.5\pm 0.25$ fm3 and a small imaginary part $\sim 0.1$ fm3.
The Paris potential is a particular case: $V_{{}^{33}P_{1}}$ is attractive
with a quasi-bound state previously discussed. However $V_{{}^{13}P_{1}}$ is
even more attractive than the former but has a positive Re[$a_{{}^{13}P_{1}}$]
as for the repulsive models. This suggest the existence of a second bound
state for the 13P1 state which plays the role of an effective repulsion.

  6. 6.

Concerning the effective range values $r_{L}$ there is no any trace of
stability in the model predictions, which translate the fact that beyond the
zero energy region the examined N̄N optical models display larger differences.

state |  | Exp | Paris 2009 | Jülich | KW | DR2
---|---|---|---|---|---|---
1S0 | N̄N |  |  1.02 - i 0.87 | 0.42 - i 0.91 | 0.52 - i 0.99 | 0.65 - i 0.82
| p̄p |  0.493(92) - i 0.732(146) |  0.92 - i 0.67 | 0.50 - i 0.71 | 0.57 - i 0.77 | 0.68 - i 0.64
3SD1 | N̄N |  |  0.91 - i 0.62 | 0.93 - i 0.92 | 1.01 - i 0.79 | 1.09 - i 0.75
| p̄p | 0.933(45) - i 0.604(51) |  0.82 - i 0.50 | 0.90 - i 0.74 | 0.92 - i 0.63 | 0.98 - i 0.59
S-averaged | N̄N |  |  0.94 - i 0.68 | 0.80 - i 0.92 | 0.89 - i 0.84 | 0.98 - i 0.77
| p̄p | 0.823(57) - i 0.636(75) |  0.85 - i 0.54 | 0.80 - i 0.74 | 0.83 - i 0.67 | 0.90 - i 0.60
3P0 | N̄N |  | -3.02 - i 2.50 | -0.32 - i 4.01 | -3.20 - i 2.28 | -2.93 - i 1.83
| p̄p | -5.68(123) - i 2.45 (49) | -2.74 - i 2.46 | -0.32 - i 3.85 | -2.81 - i 1.99 | -2.53 - i 1.62
Table 5: Isospin averaged ($a_{\bar{N}N}$) and p̄p scattering lengths are
compared with those obtained from hydrogen atom level shifts and widths, in fm
for S and fm3 for P states. The p̄p values including Coulomb and $\Delta m$
corrections are taken from CRW_ZPA343_1992  for DR2 and KW, from
PARIS_PRC79_2009  for Paris and from Haidenbauer_JHEP_2017  for Jülich model.
The statistical averaged value for S-wave is defined as (1S0+3 3S1)/4 and is
given with averaged errors.

###  3.3 Hydrogen atoms

The measurement of level shifts and widths in Hydrogen atoms is an alternative
way to access the p̄p scattering lengths and volumes. A formula derived by
Trueman Trueman_NP26_1961  finds a connection between the protonium complex
level shifts and the Coulomb corrected p̄p scattering lengths. In the case of
antiprotonic hydrogen, due to large Bohr radius ($B\approx$ 57 fm), this
relation is essentially linear CRW_ZPA343_1992 .

The p̄p scattering lengths are obtained by coupling both T components by
Coulomb and $\Delta m$ corrections. One obtains however a reasonable
approximation, denoted $a_{\bar{N}N}$ to distinguish it from the exact value
$a_{\bar{p}p}$, by neglecting this coupling and isospin-averaging the results
of Tables 2 and 3, i.e:

$2\;a_{\bar{N}N}=a_{T=0}+a_{T=1}$ (27)

Table 5 shows the comparison between the computed values – $a_{\bar{N}N}$ and
$a_{\bar{p}p}$ – and those extracted from the atomic measurements Gotta_Balmer
via the Trueman relation. Notice that the inclusion of Coulomb and $\Delta m$
can represent up to a $\approx 30\%$ difference between $a_{\bar{N}N}$ and
$a_{\bar{p}p}$ values.

For S-waves the discrepancies existing in Table 2 within the different models,
mainly concerning the 11S0 state, are smeared out in the T and S-averaged
value, which is found to be in a nice agreement among them and with the
experimental value. A remarkably good agreement is also observed in the, non
trivial, 3SD1 state. The major problem to improve the situation for S-waves is
the 11S0 near-threshold quasi-bound state, present in Paris and probably DR2
models but absent in KW and Jülich ones. It results into a negative value of
the corresponding scattering length and that generates a factor 2 in the real
parts.

For P-waves, little is known experimentally. The measurement of the isolated
3P0 p̄p level shift Gotta_Review_PPNP_2004  seems to confirm the large value
of the 13P0 scattering volume displayed in Table 3, predicted by Paris, KW and
DR2 models. In fact the large value of Re[13P0]$\approx$-9 fm3 that they
predict, and that is averaged with a positive Re[33P0]$\approx$2.5 fm3, is
still underestimated for reproducing the experimental result. One would rather
need Re[13P0]$\approx$ -13 fm3. This is clearly in tension with the Jülich
prediction which is one order of magnitude smaller than the other models and
the experimental value.

Since the large values of Re[$a_{13P0}$] are predicted by the ”pion dominance”
described in the previous section, one could find an explanation of this
disagreement in the particular form of the one-pion potential (eq. 2.1 of
Haidenbauer_JHEP_2017 ), which includes a non-local relativistic correction
and results into a smaller effective $\pi NN$ coupling constant.

As polarisation experiments are missing, the atoms offer a unique possibility
to study the spin structure of interactions. Again, there are sizeable
differences between the models. Unfortunately these happen also in the
absorptive parts which are vital for the PUMA experiment. These should be
resolved on the side of theory and more important on the side of experiments.
The priority, we believe, should be given to the full resolution of the atomic
fine structure in Hydrogen and Deuterium. In particular the $2P$ state in
Hydrogen displays a clear 3P0 state, indicated in Table 5 and three other
states lumped together and difficult to resolve. See Ref
Gotta_Review_PPNP_2004  for a dedicated review. An improvement of this
resolution would be extremely helpful to eliminate the model differences in
the partial waves.

##  4 Conclusion

We have compared the strong N̄N phase shifts obtained in the Nijmegen Partial
Wave Analysis ZT_NNB_Nijm_PW_2012 , used to construct the chiral EFT Jülich
optical potential at N3LO Haidenbauer_JHEP_2017 , with some of the currently
used N̄N optical models in configuration space: Dover-Richard (DR2)
DR1_PRC21_1980 ; DR2_RS_PLB110_1982 , Kohno-Weise KW_NPA454_1985  and Paris
(updated version from 2009) PARIS_PRC79_2009 . For all these models we have
computed the strong phase shifts and extracted the low energy parameters
(scattering lengths and effective ranges). The corresponding potentials are
included in the Appendix. This comparison is limited to the S and P waves.

In spite of providing very close elastic, annihilation and charge-exchange
integrated cross sections (Figure 1), these models are not phase-equivalent:
large and systematic differences have been observed in almost all the partial
waves, among them and with respect to the NPWA.

In the low energy region one observes some stability in the scattering lengths
and volumes (Tables 2 and 3), in particular the ”repulsive” or ”attractive”
character, i.e. the sign of Re[$a_{L}$], which is respected by all models in
almost all partial waves. For P-waves this stability could be related to the
”one-pion exchange dominance”, that is scattering volumes roughly determined
by the pion Born term. It is however also manifested in S-waves, like the
surprising stability of the low energy parameters of the tensor coupled 13SD1
state. Exceptions are the 11S0 and 33P1 partial waves, due to the presence of
a neartheshold quasi-bound state in DR2 and Paris, and the 13P0 result of
Jülich model which underestimates the protonium experimental measurements.
Despite these isolated differences, the isospin- and spin-averaged values for
S-wave are in close agreement among the models themselves as well as with the
measured quantities (Table 5). The later concern mainly the p̄p measurements
and so are unable to disentangle a selected isospin component.

The differences worsen when increasing the energy, as it is already manifested
with the dispersion in the effective range values and more explicitly in the
phase-shifts (Figures 2, 3, 4, 5, 8 and 9) and zoomed in the corresponding
effective range functions. These increasing differences cannot be explained by
the relativistic kinematics implemented in the Nijmegen Partial Waves
Analysis, relating $p_{Lab}$ to the center of mass momentum, or in the Jülich
relativistic dynamical equation.

Our main conclusion in this work is that if the Nijmegen Partial waves
analysis must be considered as a reference, as it was the case for the Jülich
model Haidenbauer_JHEP_2017 , none of the examined optical potentials is
compatible with these results and require quite a severe adjustment. This
could be easily achieved if one limits to $p_{Lab}\leq$400 MeV/c, the main
obstacle lies in the position of the near-threshold quasi-bound states.

On the other hand, we have pointed out some anomalous behaviours of the
Nijmegen Partial Waves Analysis, also reported into the Jülich potential. They
manifest as a resonant-like structures of the phase shifts in the 31S0 and
33P0 states which takes place at the same – relatively high – energy and which
are difficult to interpret as having a dynamical origin, in particular in
terms of resonant states. Furthermore they coincide with an almost zero of the
S-matrix modulus (or inelasticity parameter) that can introduce a bias in the
analysis or a spurious change from one solution to another. The existence of
such structures in the phase shifts constitutes one of the major differences
with respect to the examined optical models. It would be of the greatest
interest to clarify this point or to better understand the underlying dynamics
of these states. It would also be interesting to decrease the lowest energy
value ($p_{Lab}$=100 MeV/c) and eventually incorporate the protonium zero-
energy data. This will not provide a magic solution of the observed
discrepancies but will clearly facilitate the agreement of the models, in
particular above $p_{Lab}=$500 MeV/c.

To have at our disposal a model-independent extraction of the strong N̄N phase
shifts, the non trivial part of the interaction, is of paramount importance to
the field. In this respect it would be also suitable to have at our disposal
an independent Partial Wave analysis, as it has been always the case in the
simpler NN case.

All the examined models roughly reproduce the experimental p̄p elastic,
inelastic and charge-exchange total cross sections, including some
differential cross sections. Unfortunately, these observables are computed at
relatively high energy, result from a coherent and incoherent sum of many
partial waves and hide the existing differences among them that have been
evidenced here.

It would be of the highest interest to the community to develop, in parallel
with more ambitious projects, an experimental program to measure the most
complete set of N̄N observables at energies $p_{Lab}<$200 MeV/c allowing to
determine the main partial waves (S,P,D) that control the low energy structure
calculations. In this energy domain we are not only faced to a bad ”quality of
data”, but to a total lack of experimental results.

As far as we will not have at our disposal a reliable determination of the N̄N
strong phase shifts for the lowest partial waves, would they be limited to a
restricted energy domain of few tens of MeV, any prediction concerning more
complex systems, like those of interest in PUMA project, could be strongly
model dependent.

## Acknowledgement

We are grateful for the support of the ”Espace de Structure et de réactions
Nucléaires Théorique” (ESNT, https://esnt.cea.fr) organizing a workshop and
welcoming the visit of S. Wycech at CEA Saclay This work was started during
our visit to National Center for Nuclear Research in Warsaw. We thank the
staff members of the theory group for their warm hospitality. We acknowledge
the support of the CNRS/IN2P3 French-Polish COPIN agreement. We are thankful
to Benoît Loiseau, Johann Haidenbauer and Ruprecht Machleidt for enlightening
discussions and for providing us with their respective potentials.

##  Appendix A Potentials in configuration space

Although not being observable we believe it could be instructive to compare
the potentials of the different models in a given partial wave. The Jülich
model being in momentum space and non local is not included. The Paris
potential is E-dependent and, except for the tensor-coupled states, we
selected some positive and negative arbitrary values of $E$. For the NPWA, the
inner part corresponds to the square well defining the boundary conditions at
$r$=1.2 fm. Beyond this value it is continued with the one- plus two-pion
(N2LO) exchange potentials.

As one can see in the following figures, the N̄N potentials exhibit quite
dramatic differences, making even difficult to asses wether the N̄N
interaction in a given PW is attractive or repulsive. This is in sharp
contrast with the NN case.

###  1S0

This partial wave is globally attractive in both isospins for all models, and
much stronger than for the NN case, specially in T=1. However in the NPWA,
there is no any need of short-range attraction in T=0. Paris potential
presents two peculiar differences with respect to the other potentials: the
strong short range repulsion, claimed to be imposed by phenomenology, and the
repulsive peak at 1 fm, which cannot be justified in terms of pion- or omega-
exchanges since they are shared by all models.

Figure 15: Real parts of 1S0 potentials for both isospins (T).

###  3SD1

The S-wave tensor-coupled state presents also some striking differences: the
13S1 potentials are strongly attractive wells, going from 500 MeV to several
GeV depth, while the NPWA is limited to 130 MeV. $V_{{}^{33}S_{1}}$ is also
deeply attractive in all models but turns to be slightly repulsive ($\approx$
50 MeV)in the NPWA.

Figure 16: Real parts of 3S1 and 3D1 potentials for both isospins (T).

The 3S${}_{1}\to^{3}$D1 transition potentials have in common that they are all
very strong but they display also sizeable differences. Notice that in the DR
and KW models the couplings dont vanish in the limit $r\to 0$, what spoils the
usual $r^{L+1}$ behaviour of the (reduced) radial wave functions.

Figure 17: 3S${}_{1}\to^{3}$D1 transition potentials for both isospins (T).
They are real in DR and KW models.

###  1P1

Apart form the centrifugal barrier, this potential is very close to the 1S0
one in all models. Their difference is due to the, attractive, Quadratic Spin-
Orbit term ($Q_{12}$), present in Paris and DR2 models but absent in KW. In
the short range part of NPWA, the vanishing 11S0 potential displayed in Figure
15, vanishes also in the 11P1 state, indicating that there is no any $Q_{12}$
contribution. However the strong (500 MeV) attraction present in the 31S0
state has now totally disappeared indicating rather an unexpected repulsion.

Figure 18: Real parts of 1P1 potentials for both isospins (T).

###  3P0

All models agree with a huge attraction in T=0 state, $\sim$ 1 GeV at $r$=1
fm. The NWPA does not require such a large attraction and the fit is done with
a potential depth of $\approx$ 100 MeV in the internal region, although
matched with a pion potential of 350 MeV.

For T=1, and in view of the repulsive scattering lengths, there is also a
general agreement in the repulsive character of the interaction, although the
direct inspection of the potentials requires some caution.

In KW model, the 33P0 potential is repulsive everywhere, while DR2 has an
attractive pocket below $r$=0.7 fm which is fully compensated by the
centrifugal term. Paris potential (at E=0) has also a deep attractive pocket
(-260 MeV) between $r$=0.5 fm and $r$=0.7 fm. It is almost totally compensated
by the centrifugal barrier, but there remains a shallow attractive pocket (-35
MeV) between 0.56 and 0.63 fm. Under these dynamical conditions there is no
room for developing a resonance, especially taking into account the repulsive
E-dependent amplitude at positive energies. The examined models are, thus,
globally repulsive.

However, the NPWA requires an overall attractive short range contribution of
$\approx$ 150 MeV, though matched at r=1.2 fm with a repulsive $V_{\pi}$.

Figure 19: Real parts of 3P0 potentials for both isospins (T).

###  3P1

This PW has repulsive scattering length in both isospins states. For T=0, KW,
DR are indeed repulsive (once the centrifugal barrier is included) but NPW has
an attractive pocket and Paris remains strongly attractive (2 GeV at $r$=0.5
fm).

Figure 20: Real parts of 3P1 potentials for both isospins (T).

###  3PF2

The P-waves tensor coupled state is the one exhibiting the largest differences
among models. The 3P2 component is attractive in bot isospin states for all
models but large differences in the strength are observed among them. The 3F2
is attractive and huge in all models but vanishes in the NPWA. For T=1, the
potential is repulsive in Paris 09 but attractive in all the other models: the
unique case where NPWA requires an attraction stronger than in all other
potentials

Figure 21: Real parts of 3P2 and 3F2 potentials for both isospins (T).

Concerning the 3P${}_{2}\to^{3}$F2 transition potentials, the same remarks as
for 3S$D_{1}$ partial wave are in place.

Figure 22: 3P${}_{2}\to^{3}$F2 transition potentials for both isospins (T). In
DR and KW models, they are real.

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