# Nuclear matter in $1+1$ dimensions

Robert M. Konik  Division of Condensed Matter Physics and Material Science,
Brookhaven National Laboratory, Upton, NY 11973-5000, USA  Marton Lajer
Division of Condensed Matter Physics and Material Science, Brookhaven National
Laboratory, Upton, NY 11973-5000, USA  Robert D. Pisarski  Department of
Physics, Brookhaven National Laboratory, Upton, NY 11973  Alexei M. Tsvelik
Division of Condensed Matter Physics and Material Science, Brookhaven National
Laboratory, Upton, NY 11973-5000, USA

###### Abstract

We review the solution of QCD in two spacetime dimensions. Following the
analysis of Baluni, for a single flavor the model can be analyzed using
Abelian bosonization. The theory can be analyzed in strong coupling, when the
quarks are much lighter than the gauge coupling. In this limit, the theory is
given by a Luttinger liquid.

##  I Introduction

This subject of this pedagogical article is not directly about keV warm dark
matter, to which this volume is devoted, but to other subjects on which Hector
worked before delving into cosmology, in particular his work on integrable
models in $1+1$ dimensions.

In particular, R.D.P. would like to express that I viewed Hector both as a
great physicist and as a true gentleman. Thus I am especially honored to
contribute to this volume, as I had both the joy of working with Hector
Boyanovsky  _et al._ (1998a, b), and to consider him as a good friend.

The subject of this paper is the behavior of Quantum ChromoDynamics (QCD) in
$1+1$ dimensions. This problem has been analyzed several times over the years,
and recently Lajer  _et al._ . The purpose of this article is to bring
together the results, which tend to span a rather wide range of methods. What
is interesting is that if one concentrates just upon the low energy
excitations for cold, dense QCD in $1+1$ dimensions, then the theory reduces
to that for a single, massless boson, which propagates with a speed less than
that of light. This is what is known as a Luttinger liquid.

We first sketch how to derive these results, and then conclude with some
speculations at to their possible relevance for cold, dense QCD in $3+1$
dimensions.

##  II QCD for a single flavor

We begin with the usual Lagrangian for QCD, where the quarks lie in the
fundamental representation of a $SU(N_{c})$ color group,

$L=\int\mbox{d}^{2}x\Big{[}-\frac{1}{4}{\rm
tr}G^{\mu\nu}G_{\mu\nu}+\bar{q}_{f,\sigma}\gamma^{\mu}D_{\mu,\sigma\sigma^{\prime}}q_{f,\sigma^{\prime}}+m\bar{q}_{f,\sigma}q_{f,\sigma}\Big{]}\;.$
(1)
$D_{\mu}=\partial_{\mu}-igA_{\mu}\;\;;\;\;G_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-ig[A_{\mu},A_{\nu}]\;.$
(2)

The gauge coupling $g$ has dimensions of mass in two spacetime dimensions; the
quark fields $\bar{q},q$ carry $a,b\ldots=1,...N_{f}$ flavor and
$\alpha,\beta\ldots=1,...N_{c}$ color indices.

In two dimensions, gauge fields are not propagating degrees of freedom, which
allows one to simplify the theory. It helps to choose the gauge $A_{0}=0$.
That still leaves $A_{x}$, which for simplicity we denote just as the color
matrix $A$. Baluni Baluni (1980) noted that one can further choose a “hybrid”
gauge, which vastly simplifies the analysis. First, to assume that the gauge
potential, $A$, is a off-diagonal matrix, and that the electric field, $E$, is
diagonal:

$A^{\alpha\beta}=0\;\;,\;\;\alpha=\beta\;\;\;;\;\;\;E^{\alpha\beta}=0\;\;,\;\;\alpha\neq\beta\;\;;\;\;E^{\alpha\alpha}=-\frac{1}{2}\left(e^{\alpha}-\frac{1}{N_{c}}\sum_{\beta=1}^{N_{c}}e^{\beta}\right)\;.$
(3)

This gauge is useful in imposing Gauss’ law, $D_{x}E=J_{0}$, where $J_{0}$ is
the quark current. Besides the contribution of the quark current, this also
involves the covariant derivative, $D_{x}$, and so the commutator of the gauge
potential with the electric field. In the hybrid gauge, however, the diagonal
elements of the electric field are directly proportional to the diagonal
elements of the quark current, while the off-diagonal elements of the gauge
potential are proportional to the off-diagonal elements of the quark current:

$\partial_{x}e^{\alpha}=j_{0}^{\alpha\alpha}\;\;;\;\;ig(e^{\alpha}-e^{\beta})A^{\alpha\beta}=j_{0}^{\alpha\beta}\;\alpha\neq\beta\;\;;\;\;j_{0}^{\alpha\beta}=\bar{q}^{\alpha}\gamma_{0}q^{\beta}\;.$
(4)

There is no sum over repeated indices: $j_{0}^{\alpha\alpha}$ is just a single
element of the diagonal quark current, with color $\alpha$. For the color
diagonal current, $A^{\alpha\beta}$ does not enter, because it is taken to be
purely off-diagonal. Similarly, for the elements of the color current which
are off-diagonal in color, the spatial derivative of the electric field does
not enter, because it is assumed to be purely diagonal.

In two dimensions there is no magnetic field, so the action for the gauge
field just involve the square of the electric field. Thus the above doesn’t
look very useful, since $j_{0}$ is proportional to the spatial derivative of
the electric field. This is where bosonization is useful, as the current
$j_{0}\sim\partial_{x}\phi$, where $\phi$ is a boson field. By Gauss’ law, in
the hybrid gauge the electric field $e^{\alpha}$ is naturally proportional to
the scalar field of bosonization.

The result for the Hamiltonian after bosonization is

$\displaystyle{\cal H}$ $\displaystyle=$ $\displaystyle{\cal H}_{0}+{\cal
H}_{\rm int}\;\;,$ $\displaystyle{\cal H}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\alpha=1}^{N_{c}}\pi_{\alpha}^{2}+2m\Lambda\left(1-\cos(2\sqrt{\pi}e^{\alpha})\right)\;,$
$\displaystyle{\cal H}_{\rm int}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{8\pi
N_{c}}\sum_{\alpha,\beta=1}^{N_{c}}\left(e^{\alpha}-e^{\beta}\right)^{2}+\Lambda^{2}\sum_{\alpha,\beta=1}^{N_{c}}\frac{\sin(2\sqrt{\pi}(e^{\alpha}-e^{\beta}))}{(e^{\alpha}-e^{\beta})}\;,$
(5)

where $\pi^{\alpha}$ is the momentum conjugate to the electric field
$e^{\alpha}$. We are sloppy about normalization, and in particular about
normal ordering. As typical with bosonization, most terms contain ultraviolet
divergences (from tadpole like diagrams), and are only well defined if normal
ordered. The terms in ${\cal H}_{0}$ are standard for bosonization. The term
in ${\cal H}_{\rm int}$ which is proportional to $(e^{\alpha}-e^{\beta})^{2}$
is related to the usual electric field term. The second term in ${\cal H}_{\rm
int}$ arises from the current-current interaction which the gauge field
induces. The mass scale $\Lambda$ arises from normal ordering, and previous
analysis took it as proportional to the gauge coupling, $\Lambda\sim g$. While
this at least ensures that the perturbative expansion is well defined, it is
not evident that this is consistent. In particular, it is not obvious how to
normal order expressions such as $\sin(\phi)/\phi$.

This form of the theory has been analyzed by several authors Steinhardt
(1980); Cohen  _et al._ (1983); Gepner (1985); Frishman and Sonnenschein
(1993); Abdalla and Abdalla (1996); Armoni  _et al._ (1998, 1999); Dempsey
_et al._ (2021). We assume a single flavor because with two or more flavors,
${\cal H}_{\rm int}$ involves the conjugate momenta, $\pi^{\alpha}$, as well
as the coordinates $e^{\alpha}$ Steinhardt (1980); Cohen  _et al._ (1983);
Gepner (1985); Frishman and Sonnenschein (1993). This significantly
complicates the analysis. For several flavors, it is more useful to adopt non-
Abelian bosonization Witten (1984); Gepner (1985); Frishman and Sonnenschein
(1993); Armoni  _et al._ (1998, 1999); Dempsey  _et al._ (2021); Lajer  _et
al._ (for a recent review of non-Abelian bosonization, see Ref. James  _et
al._ (2018)). Even so, the case of a single flavor is still most illustrative.

Returning to the present approach, even with a single flavor there are
$N_{c}-1$ coupled sine-Gordon models, with an peculiar coupling, from the last
term of ${\cal H}_{\rm int}$. A single sine-Gordon model has a rich spectrum
of excitations: both small fluctuations, analogous to mesons, and kinks and
anti-kinks, analogous to baryons and anti-baryons. With $N_{c}-1$ coupled
sine-Gordon models, the spectrum becomes even more convoluted. Notice,
however, that the fields for the lightest mesons are naturally proportional to
the $\sim e^{\alpha}-e^{\beta}$. From the above, their mass is $\sim g$, and
so this represents a set of mesons/glueballs. There are then baryons, given by
kinks Steinhardt (1980); Cohen  _et al._ (1983); Gepner (1985); Frishman and
Sonnenschein (1993) For further studies of the spectra of this model, see
Refs. Frishman and Sonnenschein (1993); Abdalla and Abdalla (1996); Armoni
_et al._ (1998, 1999); Dempsey  _et al._ (2021); Lajer  _et al._ . Certainly,
as a confining gauge theory, it is expected that all excitations are massive.

In Ref. Lajer  _et al._ , an alternate approach was taken. The gauge $A_{x}=0$
was taken, and the free gauge field integrated out:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{N_{c}}\int\mbox{d}x\Big{[}-i\bar{q}_{R,\alpha}\partial_{x}q_{R,\alpha}+i\bar{q}_{L,\alpha}\partial_{x}q_{L,\alpha}-m(\bar{q}_{R,\alpha}q_{L,\alpha}+\bar{q}_{L,\alpha}q_{R,\alpha})\Big{]}$
(6) $\displaystyle-$ $\displaystyle\pi
g^{2}\int\mbox{d}x\mbox{d}y\;\sum_{A=1}^{N_{c}^{2}-1}\;J^{A}_{0}(x)|x-y|J_{0}^{A}(y)\;;$

where $q_{L,R}=(1\pm\gamma_{5})q$ are the right- and left- moving components
of the quark field. The currents for right- and left- moving quarks,
$J_{R,L}^{A}$, are given by

$J_{0}^{A}=J_{R}^{A}+J_{L}^{A}\;\;,\;\;J^{A}_{R}=\bar{q}_{R,\alpha}\,(t)^{A}_{\alpha\beta}\,q_{R,\beta},~{}~{}J^{A}_{L}=\bar{q}_{L,\alpha}\,(t)^{A}_{\alpha\beta}\,q_{L,\beta}\;,$
(7)

with $(t^{A})_{\alpha\beta}$ matrices for the adjoint representation. The
chiral currents $J_{R,L}$ obey a $SU(N_{c})$ Kac-Moody algebra James  _et al._
(2018).

Introducing the chiral currents is especially useful when expanding about the
massless limit. With the fields
$q_{R,\alpha}\sim\exp(i\sqrt{4\pi}\varphi_{\alpha})$ and
$q_{L,\alpha}\sim\exp(i\sqrt{4\pi}\bar{\varphi}_{\alpha})$. After bosonizing
the current-current interaction, one obtains two terms in the potential. The
first is from the diagonal elements,

$V_{\rm Cartan}(x)=-\pi g^{2}(1-1/N_{c})\int
dy|x-y|\partial_{x}\Phi_{\alpha}(x)\partial_{y}\Phi_{\alpha}(y)=g^{2}(1-1/N_{c})\Phi_{\alpha}(x)\Phi_{\alpha}(x)\;,$
(8)

where $\Phi=\varphi+\bar{\varphi}$. The off-diagonal terms contribute

$\displaystyle V_{\rm off-diag}$ $\displaystyle=$
$\displaystyle\sum_{\alpha>\beta}\int
dy\frac{g^{2}}{4\pi}|x-y|^{-1}\Big{\\{}\cos[\sqrt{4\pi}[\varphi_{\alpha\beta}(x)-\varphi_{\alpha\beta}(y)]]$
(9) $\displaystyle+$
$\displaystyle\cos[\sqrt{4\pi}[\bar{\varphi}_{\alpha\beta}(x)-\bar{\varphi}_{\alpha\beta}(y)]]-2|x-y|^{2}\cos[\sqrt{4\pi}[\varphi_{\alpha\beta}(x)+\bar{\varphi}_{\alpha\beta}(y)]]\Big{\\}}$

with $\varphi_{\alpha\beta}=\varphi_{\alpha}-\varphi_{\beta}$.

It is useful to contrast these analyses with the solution of QCD at large
$N_{c}$ by ’t Hooft ’t Hooft (1974). Again, one goes to axial gauge, so that
the propagator for the gauge field reduces to that of a free field. Doing so,
it is possible to solve the Schwinger-Dyson equation for the quark propagator.
This demonstrates a confining spectrum. This has been extended to nonzero
quark density by Bringoltz Bringoltz (2009), who finds chiral density waves
for a massive quark.

There is a peculiarity in the diagonal potential, $V_{\rm Cartan}$. The
bosonized fields, $\varphi$ or $\bar{\varphi}$, are manifestly periodic.
However, $V_{\rm Cartan}$ is clearly not periodic. This is also present in the
previous form, the “mass” term in Eq. (5). It wasn’t apparent in this form,
nor the lack of periodicity appreciated. In terms of the $\varphi$ and
$\bar{\varphi}$ fields, though, it demonstrates that since periodicity is
lost, the only topologically non-trivial configurations are those which
involve the sum of all angles The only soft mode remaining is the sum of the
$\Phi_{\alpha}$’s,

$\Phi=\frac{1}{\sqrt{N_{c}}}\sum_{\alpha}\Phi_{\alpha}\;.$ (10)

This is completely unaffected by the potential terms in Eqs. (8) and (9),
which only involve the differences, $\varphi_{\alpha}-\varphi_{\beta}$, etc..

The ground state corresponds to the state where all fields are equal.
Projecting the mass term onto this vacuum gives a single sine-Gordon model:

${\cal
H}_{eff}=\frac{1}{2}\Big{\\{}\Pi^{2}+(\partial_{x}\Phi)^{2}\Big{\\}}+2\frac{\tilde{m}}{2\pi}\left[1-\cos\left(\sqrt{\frac{4\pi}{N_{c}}}\Phi\right)\right].$
(11)

In this expression $\tilde{m}$ is proportional to the original mass scale,
including the effects of renormalization by normal ordering. Naturally, the
projection assumes that the energy scale generated by the mass term is much
smaller than the energies of the mesonic fields. Besides the $U(1)$ field
$\Phi$, there are also color singlet excitations above $\Lambda_{QCD}$,
involving fluctuations of individual fields $\Phi_{\alpha}$ around the minimum
of the potential. By going to energies below the scale of the gauge coupling,
all of these massive degrees of freedom can be ignored.

This implies that in $1+1$ dimensions, dense QCD is much simpler than one
might expect. By bosonization, a nonzero chemical potential is incorporated
simply by shifting

$\Big{(}\frac{4\pi}{N_{c}}\Big{)}^{1/2}\Phi\rightarrow
2\,k_{0}x+\Big{(}\frac{4\pi}{N_{c}}\Big{)}^{1/2}\Phi\;\;,\;\;k_{0}=\frac{\mu}{N_{c}}\;.$
(12)

This follows directly because $j_{0}\sim\partial_{x}\phi$. It is only $\Phi$
is affected, as fermion number only couples to the global $U(1)$ symmetry for
fermion number. The resulting effective Lagrangian is then

${\cal
L}=\frac{1}{2}(\partial_{\mu}\Phi)^{2}-\frac{\tilde{m}}{2\pi}\cos\left(\sqrt{\frac{4\pi}{N_{c}}}\Phi+2k_{0}x\right),$
(13)

As mentioned above, in vacuum the spectrum of this model consists of a soliton
with mass $m_{s}$, and anti-soliton with the same mass, and $2N_{c}-2$
breathers, which are also massive. These are all gauge invariant states.

The chemical potential does not affect the system until $\mu>m_{s}$. At that
point, the solition becomes massless, while all other states, the anti-
solition and the breathers, remain massive. Analysis shows that at
$\mu>m_{s}$, this model renormalizes into that of a Luttinger liquid:

${\cal
L}_{eff}=\frac{\tilde{K}(\mu)}{2}\Big{[}v_{F}(\mu)^{-1}(\partial_{\tau}\Phi)^{2}+v_{F}(\mu)(\partial_{x}\Phi)^{2}\Big{]},$
(14)

This is a single, massless boson, which propagates with a speed less than that
of light.

The solution of the model is the following. The overall normalization of the
effective Lagrangian is the Luttinger parameter, $\tilde{K}$, while $v_{F}$ is
the Fermi velocity; both are functions of the chemical potential, $\mu$. The
extraction of these dependencies requires the exact solution of the sine-
Gordon model.

The limits of these parameters is easy to understand. At the edge of the Fermi
surface, where $k_{F}\rightarrow 0$, the Luttinger parameter
$\tilde{K}\rightarrow 1$. In contrast, the Fermi velocity $v_{F}\rightarrow
0$. This implies that the $\Phi$ field doesn’t propagate, as the spatial term
vanishes. For asymptotically high density, $k_{F}\rightarrow\infty$, the
Luttinger parameter $\tilde{K}\rightarrow 1/N_{c}$, and the Fermi velocity
$v_{F}\rightarrow 1$. This implies that in the limit of infinite $N_{c}$, that
there is a density at which $\tilde{K}$ goes from being of order one, as is
typical of dilute fermions, to of order $\sim 1/N_{c}$. This is only valid at
infinite $N_{c}$; at finite $N_{c}$, $\tilde{K}$ varies smoothly with density.

The solution for arbitrary chemical potential can be carried out through the
Thermodynamic Bethe Ansatz. This is valid for arbitrary fermion mass. It is
also possible to compute using perturbation theory in the mass parameter.
Details, and a solution for two flavors and colors, are given in Ref. Lajer
_et al._ .

For an arbitrary number of flavors, presumably the theory is always a
Luttinger liquid. The solution is rather more complicated, and involves non-
Abelian bosonization. For an arbitrary number of flavors, it is not direct to
solve the theory in weak coupling, when the mass is much larger than the gauge
coupling. Thus it is only a conjecture that the theory is a Luttinger liquid,
although in two dimensions it is most natural to expect.

It is an extraordinary feature of Fermi surfaces in two dimensions that the
excitations near the Fermi surface are not fermions, but bosons. This is only
possible because of bosonization in two dimensions.

It is impossible to resist speculating upon whether something analogus happens
in $3+1$ dimensions. In a quarkyonic phase McLerran and Pisarski (2007);
Andronic  _et al._ (2010); Kojo  _et al._ (2010a, b, 2012); Fukushima and Kojo
(2016); McLerran and Reddy (2019); Pisarski  _et al._ (2019); Jeong  _et al._
(2020); Duarte  _et al._ (2020a, b); Sen and Warrington (2020); Sen and
Sivertsen (2020); Zhao and Lattimer (2020); Pisarski  _et al._ (2020);
Pisarski (2021); Pisarski and Tsvelik (2021), while the free energy is that of
deconfined quarks and gluons, excitations near the Fermi surface are confined.
It is conceivable that this introduces a strong anisotropy into the system, so
that it is essentially one dimensional. If true, then there is a complicated
pattern of excitations which arise. Especially with two or more light flavors,
an involved pattern of chiral density waves can arise.

What is most intriguing, however, is whether the quarkyonic phase in $3+1$
dimensions is a Luttinger liquid. That is, a type of non-Fermi liquid. In
particular, are the excitations near the Fermi surface not controlled by
nucleons, but by (effective) bosons. The properties of an effective non-Fermi
liquid can be described by an (anisotropic) effective Lagrangian, and used to
compute the transport properties of a quarkyonic regime.

###### Acknowledgements.

The work of R.D.P. was supported by the U.S. Department of Energy under
contract number DE-SC0012704; by B.N.L. under the Lab Directed Research and
Development program 18-036; and by the U.S. Department of Energy, Office of
Science, National Quantum Information Science Research Centers, Co-design
Center for Quantum Advantage (C2QA) under contract number DE-SC0012704. The of
M. J., A.M.T., and R.M.K. was supported by the U.S. Department of Energy,
Office of Science, Materials Sciences and Engineering Division under contract
DE-SC0012704.

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# Energy-independent complex single-$P$-waves $NN$ potential
from Marchenko equation

N. A. Khokhlov<EMAIL_ADDRESS><EMAIL_ADDRESS>Peter the Great
St. Petersburg Polytechnic University, RU-195251 St. Petersburg, Russia.

###### Abstract

We extend our previous results of solving the inverse problem of quantum
scattering theory (Marchenko theory, fixed-$l$ inversion). In particular, we
apply an isosceles triangular-pulse function set for the Marchenko equation
input kernel expansion in a separable form. The separable form allows a
reduction of the Marchenko equation to a system of linear equations for the
output kernel expansion coefficients. We show that in the general case of a
single partial wave, a linear expression of the input kernel is obtained in
terms of the Fourier series coefficients of $q^{1-m}(1-S(q))$ functions in the
finite range of the momentum $0\leq q\leq\pi/h$ [$S(q)$ is the scattering
matrix, $l$ is the angular orbital momentum, $m=0,1,\dots,2l$]. Thus, we show
that the partial $S$–matrix on the finite interval determines a potential
function with $h$-step accuracy. The calculated partial potentials describe a
partial $S$–matrix with the required accuracy. The partial $S$–matrix is
unitary below the threshold of inelasticity and non–unitary (absorptive) above
the threshold. We developed a procedure and applied it to partial-wave
analysis (PWA) data of $NN$ elastic scattering up to 3 GeV. We show that
energy-independent complex partial potentials describe these data for single
$P$-waves.

quantum scattering, Marchenko theory, inverse problem, algebraic method,
numerical solution

###### pacs:

24.10.Ht, 13.75.Cs, 13.75.Gx

##  I INTRODUCTION

Extracting the interparticle interaction potential from scattering data is a
fundamental problem in nuclear physics. The various inverse problem of quantum
scattering (IP) methods are used for such extraction. Fitting parameters of
the phenomenological potential can solve this problem. Such an adjustment is
easily implemented in the case of a small number and low accuracy of the known
experimental data. An accurate description of a large number of experimental
data requires exact methods of IP solving. The development of such precise and
unambiguous methods remains a fundamental challenge Sparenberg1997 ;
Sparenberg2004 ; Kukulin2004 ; Pupasov2011 ; Mack2012 . The ill-posedness of
the problem significantly complicates its numerical solution.

The fixed-$l$ IP considered here is usually solved within Marchenko, Krein,
and Gelfand-Levitan theories Gelfand ; Agranovich1963 ; Marchenko1977 ;
Blazek1966 ; Krein1955 ; Levitan1984 ; Newton ; Chadan . In these approaches,
the IP is reduced to solving Fredhölm integral equations of the second kind.
H.V. von Gevamb and H. Kohlhoff successfully applied Marchenko and Gelfand-
Levitan theories to extract $NN$ partial potentials from PWA data Geramb1994 ;
Kohlhoff1994 . They used the PWA data up to the inelastic threshold
($E_{\text{lab}}\approx~{}280$ MeV) and approximated the corresponding partial
$S$–matrices (spectral densities) by rational fraction expansions (Padé
approximants). In this case, the input kernels of the integral equations are
represented as finite separable series of the Riccati-Hankel functions
products (separable kernels). A Fredhölm integral equation of the second kind
with a separable kernel is solved analytically. The partial potentials, in
this case, are also expressed through Riccati-Hankel functions (Bargman-type
potentials). A similar approach in frames of the Marchenko theory was used
later to extract optical model $NN$ partial potentials from elastic $NN$
scattering PWA data up to 3 GeV Khokhlov2006 ; Khokhlov2007 . This method for
solving the inverse problem includes four numerical procedures. The first
procedure is the PWA, the second procedure is the approximation of the
$S$-matrix, the third procedure is the solution of the integral equation, and
the fourth procedure is the differentiation of the output kernel to calculate
the potential. Errors of each step are accumulated, the estimation of the
error in the calculation of the potential is not an easy task. For $l\geq 1$,
Marchenko inversion is unstable for $r<1$ fm, and Gelfand-Levitan is unstable
for $r>8$ fm Geramb1994 ; Kohlhoff1994 . Description of PWA data within errors
for energies up to 3 GeV requires the use of high-order Padé approximants
Khokhlov2006 ; Khokhlov2007 . We found that an increase in the accuracy of the
approximation and, accordingly, an increase in the order of the Padé
approximant can lead to significant changes in the potential. The Padé
approximant for such a problem is not the best choice since approximants of
different orders giving close $S$-matrix values at the PWA points can differ
significantly between these points. Thus, the convergence of methods using
rational fraction expansions of $S$-matrix (spectral density) is not apparent.

This paper generalizes the algebraic method MyAlg1 ; MyAlg2  for solving the
IP. We expand the Marchenko equation input kernel into a separable series in
an isosceles triangular-pulse function set. We obtained a linear expression of
the expansion coefficients in terms of the Fourier series coefficients of
$q^{1-m}(1-S(q))$ ($m=0,1,\dots,2l$) functions on a finite range of the
momentum $0\leq q\leq\pi/h$. Thus, we solve the Marchenko equation with a
separable kernel, which can be performed analytically like in Refs. Geramb1994
; Kohlhoff1994 ; Khokhlov2006 ; Khokhlov2007 . Theory of the Fourier’s series
substantiates convergence of the procedure with decreasing step $h$.

$NN$ potentials used as an input for (semi)microscopic construction of nuclear
optical potentials (OPs) are usually real and only describe the $NN$ PWA data
below the inelastic threshold Burrows2020 ; Vorabbi2018 ; Vorabbi2017 ;
Guo2017 ; Vorabbi2016 . However, in the absence of a microscopic theory,
complex partial $NN$ potentials are required to describe nuclear reactions
with energies of the $NN$ relative motion above the threshold Khokhlov2007 .
In existing models, real partial potentials are modified by energy-dependent
imaginary terms which are zero below the threshold Khokhlov2006 ; Funk2001 .
The partial wave function is real below the threshold to within an
$r$-independent factor in such models. For realistic optical potentials, such
a restriction is unreasonable. Indeed, the optical model potential
(pseudopotential) definition does not guarantee that it will be Hermitian
Taylor ; Goldberger . One can only assert that Hamiltonian eigenfunctions
corresponding to eigenenergies below the threshold satisfy the Hermitianity
condition. However, the Hermitian condition

$\int\psi^{*}(V^{+}-V)\psi dx=0$ (1)

implies $V^{+}-V=0$ only if Eq. 1 holds for an arbitrary function $\psi$,
which is not true for optical potentials. Using the phase-equivalent Krein
transformations, one can obtain an energy-independent complex potential giving
a unitary matrix Agranovich1963 . The use of optical potentials limited by the
condition $V^{+}-V=0$ below the inelasticity threshold is a physically
unreasonable limitation. Previously, it was shown that the Marchenko theory
applies to the description of elastic $nD$ scattering from zero energy up to
energies well above the threshold. In this case, Marchenko theory produces
energy-independent complex partial $nD$ potentials Papastylianos1990 ; Alt1994
. They used a rational parametrization, the same as in Refs. Geramb1994 ;
Kohlhoff1994 ; Khokhlov2006 ; Khokhlov2007  for the unitary $S$–matrix. The
small value of the threshold (the deuteron’s binding energy
$E_{\text{c.m.}}\approx 2.226$ MeV) does not allow us to judge the
applicability of the Marchenko theory in the general case. We analyzed the
Marchenko theory Agranovich1963 ; Marchenko1977  ($l=0$), Blazek1966  ($l>0$)
and found that most of the theory applies to unitary and to non–unitary
$S$–matrices. Our algebraic form of the Marchenko equation allows calculating
the energy-independent complex local partial potential corresponding to a
partly unitary and partly non–unitary $S$–matrix. We previously applied this
approach to analyze ${}^{1}S_{0}$ data of elastic $NN$ scattering PWA MyAlg2 .
Extending this investigation, we analyze single $P$-waves $NN$ PWA data (up to
$E_{\text{lab}}\approx 3$ GeV) and show that these data are described by
energy-independent complex partial potentials. The reconstructed partial $S$
and $P$-waves potentials constitute energy-independent soft core OP that
describes elastic $NN$ scattering PWA data up to 3 GeV.

##  II Marchenko equation in an algebraic form

The radial Schrödinger equation is

$\left(\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}-V(r)+q^{2}\right)\psi(r,q)=0.$
(2)

The Marchenko equation Agranovich1963 ; Marchenko1977  is a Fredhölm integral
equation of the second kind:

$F(x,y)+L(x,y)+\int_{x}^{+\infty}L(x,t)F(t,y)dt=0$ (3)

The kernel function is defined by the following expression

$F(x,y)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}h_{l}^{+}(qx)[1-S(q)]h_{l}^{+}(qy)dq\\\
+\sum_{j=1}^{n_{b}}h_{l}^{+}\left(\tilde{q}_{j}x\right)M_{j}^{2}h_{l}^{+}\left(\tilde{q}_{j}y\right)\\\
=\frac{1}{2\pi}\int_{-\infty}^{+\infty}h_{l}^{+}(qx)Y(q)h_{l}^{+}(qy)dq$ (4)

where $h_{l}^{+}(z)$ is the Riccati-Hankel function, and

$Y(q)=\left[1-S(q)-i\sum_{j=1}^{n_{b}}M_{j}^{2}\left(q-\tilde{q}_{j}\right)^{-1}\right],$
(5)

Experimental data entering the kernel are

$\left\\{S(q),(0<q<\infty),\tilde{q}_{j},M_{j},j=1,\ldots,n\right\\},$ (6)

where $S(q)=e^{2\imath\delta(q)}$ is a scattering matrix dependent on the
momentum $q$. The $S$–matrix defines asymptotic behavior at
$r\rightarrow+\infty$ of regular at $r=0$ solutions of Eq. (2) for $q\geq 0;\
\tilde{q}_{j}^{2}=E_{j}\leq 0,E_{j}$ is $j$–th bound state energy
($-\imath\tilde{q}_{j}\geq 0$); $M_{j}$ is $j$-th bound state asymptotic
constant.

The potential function of Eq. (2) is obtained from the solution of Eq. 3

$V(r)=-2\frac{dL(r,r)}{dr}$ (7)

Many methods for solving Fredhölm integral equations use a series expansion of
the equation kernel. eprint7 ; eprint8 ; eprint9 ; eprint10 ; eprint12 ;
eprint13 ; eprint14 . We also use this approach.

We introduce auxiliary functions:

$F_{m}(z)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\frac{e^{\imath
qz}Y(q)dq}{q^{m}},$ (8)

then

$\frac{d^{k}F_{m}(z)}{dz^{k}}=\imath^{k}F_{m-k}(z),\ (k=1,2,\dots,m).$ (9)

We use transformations

$\hat{K}_{z,l}f(z)=z^{l+1}\left(-\frac{1}{z}\frac{d}{dz}\right)^{l}\left[z^{-1}f(z)\right]\equiv(-1)^{l}{\sum_{n=0}^{l}\frac{(2l-n)!}{n!(l-n)!}(-2z)^{n-l}\frac{d^{n}f(z)}{dz^{n}}}.$
(10)

From Abramowitz  (Eqs. (10.1.23)–(10.1.26)) we get

$\hat{K}_{z,l}e^{\pm\imath qz}=q^{l}h^{\pm}_{l}(qz).$ (11)

and

$\hat{K}_{y,l}\hat{K}_{x,l}F_{2l}(x+y)={\sum_{n_{1},n_{2}=0}^{l}\frac{(2l-n_{1})!}{n_{1}!(l-n_{1})!}\frac{(2l-n_{2})!}{n_{2}!(l-n_{2})!}(-2x)^{n_{1}-l}(-2y)^{n_{2}-l}\imath^{n_{1}+n_{2}}F_{2l-n_{1}-n_{2}}(x+y)}\\\
=\frac{1}{2\pi}\int_{-\infty}^{+\infty}h_{l}^{+}(qx)Y(q)h_{l}^{+}(qy)dq\equiv
F(x,y).$ (12)

Assuming the finite range $R$ of the potential function $V(r)$, we approximate
$F_{m}(x+y)$ as follows:

$\displaystyle F_{m}(x+y)\approx\sum_{k=-2N}^{2N}f_{m,k}H_{k}(x+y)$ (13)
$\displaystyle\approx\sum_{k,j=0}^{N}\Delta_{k}(x)f_{m,k+j}\Delta_{j}(y)$ (14)

where $f_{m,k}\equiv F_{m}(kh)$, $h$ is some step, and $R=Nh$. The used basis
sets are

$\left.\begin{array}[]{l}H_{0}(x)=\left\\{\begin{array}[]{lr}1&\text{if }0\leq
x\leq h,\\\ 0&\text{otherwise,}\end{array}\right.\\\
H_{n}(x)=H_{0}(x-hn).\end{array}\right\\}$ (15)
$\left.\begin{array}[]{l}\Delta_{0}(x)=\left\\{\begin{array}[]{lr}1-|x-0.25|/h&\text{if
}|x-0.25|\leq h,\\\ 0&\text{otherwise,}\end{array}\right.\\\
\Delta_{n}(x)=\Delta_{0}(x-hn).\end{array}\right\\}$ (16)

We use bases set $\Delta_{i}(x)\Delta_{j}(y)$ shifted by the vector
$(0.25h,0.25h)$ compared to the set used previously MyAlg1 ; MyAlg2 . The
basis sets are illustrated in Fig. 1.

Figure 1:  The basis set $H_{n}\equiv H_{n}(x+y)$ (Eq. (15) is shown as
trapezoid (triangle for $n=0$) regions where $H_{n}(x+y)=1$, and elsewhere
$H_{n}(x+y)=0$. The regions are bounded by lines $x=0$, $y=0$, and
$x+y=h(n-1)$. The basis set $\Delta_{i}(x)\Delta_{j}(y)$ (Eq. (16)) is shown
as projections (points) of the corresponding regular square pyramids apexes on
the $xy$–plane. $\Delta_{i}(x)\Delta_{j}(y)=1$ at $x=(0.25+i)h,\ y=(0.25+j)h$
(apex of the $ij$–pyramid). The pyramids bases are $(2h\times 2h)$ squares on
the $xy$–plain with sides parallel to the $x$ and $y$ axes.
$\Delta_{i}(x)\Delta_{j}(y)=0$ on sides of the corresponding squares (and
outside them).

Decreasing the step $h$, one can approach $F_{m}(x+y)$ arbitrarily close at
all points with both sets. Coefficients $f_{m,k}$ are same for both
approximations Eqs. (13), (14).

The Fourier transform of the basis set Eq. 13

$\tilde{\mathrm{H}}_{k}(q)=\int_{-\infty}^{\infty}\mathrm{H}_{k}(x)e^{-\imath
qx}dx=\frac{\imath\left(e^{-\imath qh}-1\right)}{qe^{\imath qhk}}.$ (17)

The Fourier transform of Eq. (8) yields

$\frac{Y(q)}{q^{m}}\approx\sum_{k=-2N}^{2N}f_{m,k}\tilde{\mathrm{H}}_{k}(q)=\sum_{k=-2N}^{2N}f_{m,k}\frac{\imath\left(e^{-\imath
qh}-1\right)}{qe^{\imath qhk}}.$ (18)

We rearrange the last relationship

$Y(q)/q^{m-1}=\imath\sum_{k=-2N}^{2N}f_{m,k}\left(e^{-\imath
qh}-1\right)e^{-\imath qhk}\\\
=\imath\sum_{k=-2N+1}^{2N}\left(f_{m,k-1}-f_{m,k}\right)e^{-\imath
qhk}+\imath\left(-f_{m,-2N}\right)e^{\imath qh2N}\\\
+\imath\left(f_{m,2N}\right)e^{-\imath qh(2N+1)}.$ (19)

Thus, the left side of the expression is represented as a Fourier series on
the interval $-\pi/h\leq q\leq\pi/h$.

$f_{m,k-1}-f_{m,k}=-\frac{\imath
h}{2\pi}\int_{-\pi/h}^{\pi/h}Y(q)\frac{e^{\imath qhk}dq}{q^{m-1}}$ (20)

for $k=-2N,\ldots,2N$. Recursive solving of the Eq. (20) from $k=2N+1$
($f_{m,2N+1}=0$) gives

$f_{m,k}=\frac{h}{\pi}\int_{-\pi/h}^{\pi/h}\left(\sum_{\nu=k+1}^{2N+1}e^{\imath
qh\nu}\right)\frac{Y(q)dq}{q^{m-1}}\\\ =-\frac{\imath
h}{2\pi}\int_{-\pi/h}^{\pi/h}\frac{e^{\imath qh(k+1)}\left(1-e^{\imath
qh(2N-k+1)}\right)}{\left(1-e^{\imath qh}\right){q^{m-1}}}Y(q)dq.$ (21)

The $F(x,y)$ is defined by $f_{m,k}$ $(m=0,1,\dots,2l)$, $k=0,1,\dots,2N$ from
Eqs. (12), and (14) as

$F(x,y)\approx\sum_{k,j=0}^{N}\Delta_{k}(x)F_{k,j}\Delta_{j}(y),$ (22)

where

$\displaystyle
F_{k,j}={{\sum_{n_{1},n_{2}=0}^{l}\frac{(2l-n_{1})!}{n_{1}!(l-n_{1})!}\frac{(2l-n_{2})!}{n_{2}!(l-n_{2})!}(-2(k+0.25)h)^{n_{1}-l}(-2(j+0.25)h)^{n_{2}-l}\imath^{n_{1}+n_{2}}f_{2l-n_{1}-n_{2},k+j}}}$
(23) $\displaystyle=-\frac{\imath
h}{2\pi}\int_{-\pi/h}^{\pi/h}h_{l}^{+}(q(k+0.25)h)\frac{e^{\imath
qh}\left(1-e^{\imath qh(2N-k-j+1)}\right)}{1-e^{\imath
qh}}Y(q)h_{l}^{+}\left(q(j+0.25)h\right)qdq.$ (24)

Thus, the range of known scattering data defines the value of $h$ and,
therefore, the inversion accuracy.

We solve the Eq. (3) substituting

$L(x,y)\approx\sum_{j=0}^{N}P_{j}(x)\Delta_{j}(y)$ (25)

Substitution of Eqs. (22) and (25) into Eq. (3) and linear independence of the
basis functions give

$\sum_{m=0}^{N}\left(\delta_{j\,m}+\sum_{n=0}^{N}\left[\int_{x}^{max((m+0.25)h,(n+0.25)h)}\Delta_{m}(t)\Delta_{n}(t)dt\right]F_{n,j}\right)P_{m}(x)=-\sum_{k=0}^{N}\Delta_{k}(x)F_{k,j}$
(26)

We define

$\zeta_{n\,m\,p}=\int_{(p+0.25)h}^{max((m+0.25)h,(n+0.25)h)}\Delta_{m}(t)\Delta_{n}(t)dt=\frac{h}{6}\left(2\delta_{n\,m}\left(\delta_{n\,p}+2\eta_{n\geq
p+1}\right)+\delta_{n\,(m-1)}\eta_{n\geq p}+\delta_{n\,(m+1)}\eta_{m\geq
p}\right),$ (27)

where $\delta_{k\,p}$ are the Kronecker symbols $\delta_{k\,p}$, and

$\eta_{a}=\left\\{\begin{array}[]{lr}1&\text{if }a\text{ is true},\\\
0&\text{otherwise,}\end{array}\right.$ (28)

Since $\Delta_{k}(hp)\equiv\delta_{k\,p}$, we finally get a system of
equations

$\sum_{m=0}^{N}\left(\delta_{j\,m}+\sum_{n=p}^{N}\zeta_{n\,m\,p}F_{n,j}\right)P_{p,m}=-F_{p,j},$
(29)

for $P_{k}(h(p+0.25))\equiv P_{p,k}$ $(p,k=0,\dots,N)$ ($j,p=0,\dots,N$).

Solution of Eq. (29) gives $P_{p,k}$. We calculate potential values at points
$r=hp$ $(p=0,\dots,N)$ from Eq. (7) by some finite difference formula.

We tested the developed approach by restoring the potential function
$V(r)=-3\exp(-3r/2)$ from the corresponding scattering data. Results are
presented in Figs. 2, 3, where $h=0.04$, $R=4$.

Figure 2: Data used to reconstruct $V(r)=V_{0}\exp(-ar)$, where $V_{0}=-3\
fm^{-2}=-124.5\ MeV$, $a=1.5\ fm^{-1}$. Angular orbital momentum $l=1$. Units
correspond to the $NN$ system.

Figure 3: Initial and reconstructed potentials: $V(r)=V_{0}\exp(-ar)$, where
$V_{0}=-3\ fm^{-2}=-124.5\ MeV$, $a=1.5\ fm^{-1}$. Units correspond to the
$NN$ system.

The input $S$–matrix was calculated at points shown in Fig. 2 up to $q=8$. The
$S$–matrix was interpolated by a quadratic spline in the range $0<q<8$. The
$S$–matrix was approximated as asymptotic $S(q)\approx\exp(-2i\alpha/q)$ for
$q>8$, where $\alpha$ was calculated at $q=8$.

##  III Energy-independent complex partial potentials

Two-particle relativistic potential models can be represented in a
non–relativistic form Keister1991 . Thus, the applicability of methods for
solving the inverse problem is not limited only to nonrelativistic quantum
mechanics. As we showed earlier MyAlg2 , modern data of the $NN$ partial-wave
analysis up to $E_{\text{lab}}=3$ GeV can be described by the energy-
independent complex partial potential for ${}^{1}S_{0}$ single wave at least.

After analyzing the Marchenko theory Agranovich1963 ; Marchenko1977 ,
including for the case of $l>0$ Blazek1966 , we found that Eqs. (2, 4) apply
to non–unitary $S$–matrices describing absorption. In this case, the absorbing
partial $S$–matrix should be defined as

$S(q)=\left\\{\begin{array}[]{lll}S_{u}(q)+S_{n}(q)&\text{ for }&q>0,\\\
S^{+}_{u}(-q)-S^{+}_{n}(-q)&\text{ for }&q<0,\end{array}\right.$ (30)

where superscript $+$ means hermitian conjugation. For $q>0$, we define

$S_{u}(q)=e^{2\imath\delta(q)},\ \
S_{n}(q)=-\sin^{2}(\rho(q))e^{2\imath\delta(q)},$ (31)

where $\delta(q)$ and $\rho(q)$ are phase shift and inelasticity parameter
correspondingly. With the $S$–matrix defined in this way, all Eqs. (24–29)
remain valid and allow calculating local and energy-independent OP from an
absorptive $S$–matrix and corresponding bound states’ characteristics.

##  IV Results and Conclusions

Extending our previous results MyAlg2 , we analyzed modern single $P$-wave
$NN$ phase shift data up to 3 GeV DataScat ; cite  (SW16, single-energy
solutions, Fig. 4).

Figure 4:  Data used to reconstruct $P$-wave partial potentials are phase
shifts $\delta$ (solid circles) and inelasticity parameters $\rho$ (solid
squares) (from Ref DataScat ; cite ). Circles and squares stand for $\delta$
and $\rho$ correspondingly calculated from the reconstructed partial
potentials.

We smoothed the phase shift and inelasticity parameter data for $q>3$ fm-1 by
the following functions, correspondingly:

$\delta(q)\sim\sum_{k=3}^{8}A_{k}/q^{k},\ \
\rho(q)\sim\sum_{k=1}^{9}A_{k}/q^{k},$ (32)

where the coefficients were fitted by the least-squares method. The SW16 data
and asymptotics (32) above $q>3$ fm-1 were used to calculate coefficients of
Eqs. (24) with $h=0.02$ fm corresponding to $q_{\text{max}}\approx 157.08$
fm-1.

The used data are described by energy-independent complex partial potentials
(Figs. 5, 6). Thus, we presented an IP solution algorithm for finite range
potentials (fixed-$l$, single partial wave inversion, in frames of the
Marchenko theory).

Our method gives a geometric glimpse into the $NN$ interaction. As noted
earlier by P. Fernández-Soler and E. Ruiz Arriola Fernandez-Soler2017 ,
”short-wavelength fluctuations/oscillations are inherent to the maximum energy
or CM momentum being fixed for the phase shift”. In this conclusion, they
refer to the results of our calculations Khokhlov2006 . Then, we did not pay
attention to this feature of the IP solution, and we considered the
oscillations an artifact of the used IP solution in which the potential is
explicitly, algebraically expressible in terms of Hankel functions.
Reconstructed ${}^{1}S_{0}NN$ partial potential MyAlg2  shows that short-
wavelength oscillations may indeed be a manifestation of a physical
phenomenon. There are also similar oscillations in the reconstructed $P$-wave
partial potentials (Fig. 6). Microscopic calculations also predict such
oscillations Wendt2012 .

We show that IP solutions give optical model energy independent $NN$ partial
potentials not only in the ${}^{1}S_{0}NN$ wave, but also in other single $NN$
partial waves. The optical model partial $NN$ potentials with a repulsive core
does not necessarily exhibit a strong energy dependence up to 3 GeV, as
previously stated in Fernandez-Soler2017 . Instead of the soft repulsive core
that was reconstructed in our model for the real part of the ${}^{1}S_{0}NN$
partial potential, the barriers at about 0.2-1.3 fm were reconstructed for the
real parts of the $P$-waves partial potentials.

The reconstructed $P$-wave $NN$ complex potentials may be requested from the
author in the Fortran code.

Figure 5:  Real and imaginary parts of the reconstructed partial potentials.

Figure 6:  Real and imaginary parts of the reconstructed partial potentials.

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