# Testing the $\phi$-nuclear potential in pion-induced $\phi$ meson production
on nuclei near threshold

E. Ya. Paryev
Institute for Nuclear Research, Russian Academy of Sciences,
Moscow 117312, Russia

###### Abstract

The near-threshold pion-induced $\phi$ production off nuclei has been studied
in the kinematical conditions of the HADES experiment using a collision model
based on the nuclear spectral function. Starting from the elementary reaction
${\pi^{-}}p\to{\phi}n$, absolute differential and integral cross sections for
the production of $\phi$ mesons off a light carbon and a heavy tungsten target
have been calculated and compared to the recently reported experimental cross
sections. The absolute cross section values are shown to be sensitive to the
effective nuclear scalar $\phi$ and neutron potentials while the transparency
ratio is governed by the ${\phi}N$ absorption cross section. The experimental
data are found to be consistent with an attractive ${\phi}$-nucleus potential
of $\approx$ -(50–100) MeV at normal nuclear matter density $\rho_{0}$ and an
effective ${\phi}N$ absorption cross section of $\approx$ 12–25 mb. The
extracted $\phi$-nucleus potential is deeper than previous theoretical and
experimental findings as well as those obtained in the present work within the
($t{\rho}$) approximation with accounting for Pauli correlations from the
spin-averaged $\phi$-$p$ scattering length reported in a recent ALICE
experiment.

## 1 Introduction

The study of the in-medium properties (effective masses and widths) of light
vector mesons $\rho$, $\omega$ and $\phi$ in nuclear matter via their
production in the collisions of hadron, heavy-ion and photon beams with
nuclear targets has received considerable interest in the last few decades
(see, for example, Refs. [1–5]) due to the hope to extract valuable
information on the partial restoration of chiral symmetry in hot or dense
nuclear matter [6, 7]. This restoration is characterized by a reduction of the
scalar quark condensate $<{\bar{q}}q>$ in the nuclear medium compared to its
vacuum value, which would lead to a modification of hadron properties. Among
these, the $\phi$ meson attracts particular interest because of its narrow
width in free space (4.3 MeV) and because it does not overlap with other light
resonances in the mass spectrum. This could allow for the measurement of any
modifications of its properties in nuclei. Theoretically, the properties of
the $\phi$ meson in nuclear matter have been extensively discussed based on
the QCD sum rules [6, 8, 9], on hadronic models [10–13] and on the Quark-Meson
Coupling model [14]. As is expected, the mass shift of the $\phi$ meson in
nuclear medium at threshold is small (about 1–3% of its vacuum mass at normal
nuclear matter density $\rho_{0}$), but its in-medium total width is
substantially increased (by a factor of ten at density $\rho_{0}$) compared to
the free space value of 4.3 MeV. The first evidence for the downward $\phi$
mass reduction of 3.4% (of about -35 MeV) and a width increase by a factor of
3.6 at this density was reported by the KEK-PS E325 experiment [15], where
$\phi$ mesons produced off C and Cu target nuclei with a 12 GeV proton beam
have been measured. At low energies, the production of $\phi$ mesons in nuclei
by photon and proton beams has also been studied experimentally by the LEPS
[16–19], CLAS [20–22] and ANKE [23] Collaborations at the SPring-8, JLab and
COSY facilities, respectively. Large in-medium ${\phi}N$ absorption cross
sections $\sigma_{{\phi}N}$ of about 35 mb [16], 16–70 mb [20] and 14–21 mb
[23] were extracted from the comparison of the data collected in these
experiments with the respective model calculations for average $\phi$ momenta
of about 1.8, 2 GeV/c and in the momentum range of 0.6–1.6 GeV/c. These cross
sections are to be compared to the total $\phi$–$N$ cross section of about 11
mb estimated by using the vector-meson dominance model in $\phi$
photoproduction off the proton at photon energies below 10 GeV [24]. The
$\phi$ production in high-energy heavy-ion collisions has been recently
investigated by the STAR [25, 26], PHENIX [27, 28] and ALICE [29]
Collaborations at RHIC and LHC, respectively. Recent reviews on the
measurements of $\phi$ meson production in high-energy heavy-ion experiments
are presented in Refs. [30, 31]. The HADES Collaboration observed deep
subthreshold $\phi$ production in 1.23 A GeV Au+Au collisions [32] with a
surprisingly high $\phi/K^{-}$ multiplicity ratio of 0.52$\pm$0.16, which
indicates a possibly attractive $\phi$ in-medium mass shift.

At low energies, the production of $\phi$ mesons in nuclei has also been
actively studied theoretically in proton–nucleus [23, 33–36], photon–nucleus
[24, 37–40] and heavy-ion [41] reactions, with the aim of obtaining
information on a possible modification of the $\phi$ properties in cold normal
and hot dense nuclear matters. Additional information on this modification,
complementary to that from proton–nucleus, photon–nucleus and heavy-ion
collisions, can be deduced from pion–nucleus reactions [5, 42]. In an attempt
to get valuable information on the behavior of $\phi$ mesons in the nuclear
medium, the near-threshold $\pi^{-}$ meson-induced $\phi$ production off 12C
and 184W target nuclei at an incident pion momentum of 1.7 GeV/c has been
recently investigated by the HADES Collaboration at SIS18/GSI [43]. For the
first time, the integrated $\phi$ production cross sections
(${\Delta}{\sigma}_{\rm C}^{\phi}$ and ${\Delta}{\sigma}_{\rm W}^{\phi}$) in
the HADES acceptance and the $\phi$ transparency ratio were measured for these
targets. The aim of this paper is to analyze these data within the collision
model, developed in Ref. [44] and based on the nuclear spectral function, for
incoherent direct $\phi$ production in the ${\pi^{-}}p\to{\phi}n$ reaction
assuming different scenarios for the ${\phi}N$ absorption cross section
$\sigma_{{\phi}N}$ and for the $\phi$ in-medium mass shift (or for the $\phi$
effective scalar nuclear potential). In doing so, we will follow strictly the
approach [44]. We briefly recall its main assumptions and describe, where
necessary, the corresponding extensions.

## 2 Direct $\phi$ meson production mechanism

The direct production of $\phi$ mesons in the kinematical conditions of the
HADES experiment in $\pi^{-}A$ ($A=^{12}$C and 184W) interactions at an
incident pion beam momentum of 1.7 GeV/c, corresponding to the excess energy
above the ${\phi}n$ production threshold $\sqrt{s_{\rm
th}}=m_{\phi}+m_{n}=1.959$ GeV ($m_{\phi}$ and $m_{n}$ are the $\phi$ meson
and final neutron bare masses, respectively) of 66 MeV, occurs in the
following $\pi^{-}p$ elementary process 111) Our calculations showed [44] that
this process dominates over secondary pion–nucleon $\phi$ production processes
in phi production off nuclei in the HADES acceptance window, corresponding to
the rapidity and transverse momentum intervals $0.4\leq y<1.0$ and $150\leq
p_{T}<650$ MeV/c, at an initial pion momentum of 1.7 GeV/c.):

$\pi^{-}+p\to\phi+n.$ (1)

Before going further, let us get a feeling, adopting relativistic kinematics,
about kinematic characteristics of $\phi$ mesons and neutrons allowed in this
process in the simpler case of a free target proton being at rest at an
incident pion momentum $p_{\pi^{-}}$ of interest. The kinematics of two-body
reaction with a threshold (as in our present case) tell us that the laboratory
polar $\phi$ and final neutron production angles $\theta_{\phi}$ and
$\theta_{n}$ vary from 0 to a maximal values $\theta^{\rm max}_{\phi}$ and
$\theta^{\rm max}_{n}$, correspondingly, i.e.:

$0\leq\theta_{\phi}\leq\theta^{\rm max}_{\phi},$ (2)
$0\leq\theta_{n}\leq\theta^{\rm max}_{n};$ (3)

where

$\theta^{\rm max}_{\phi}={\rm
arcsin}[(\sqrt{s}p^{*}_{\phi})/(m_{\phi}p_{\pi^{-}})],$ (4) $\theta^{\rm
max}_{n}={\rm arcsin}[(\sqrt{s}p^{*}_{n})/(m_{n}p_{\pi^{-}})].$ (5)

Here, the $\phi$ c.m. momentum $p^{*}_{\phi}$ is determined by the equation

$p_{\phi}^{*}=\frac{1}{2\sqrt{s}}\lambda(s,m_{\phi}^{2},m_{n}^{2}),$ (6)

in which the vacuum collision energy squared $s$ and function $\lambda(x,y,z)$
are defined, respectively, by the formulas

$s=(E_{\pi^{-}}+m_{p})^{2}-{p}_{\pi^{-}}^{2},\,\,\,\,E_{\pi^{-}}=\sqrt{m^{2}_{\pi^{-}}+p^{2}_{\pi^{-}}},$
(7)
$\lambda(x,y,z)=\sqrt{{\left[x-({\sqrt{y}}+{\sqrt{z}})^{2}\right]}{\left[x-({\sqrt{y}}-{\sqrt{z}})^{2}\right]}},$
(8)

and $p^{*}_{n}$ is the final neutron c.m. momentum. It is equal to the $\phi$
c.m. momentum $p^{*}_{\phi}$. The quantities $m_{p}$ and $m_{\pi^{-}}$,
entering into Eq. (7), denote the free space proton and $\pi^{-}$ meson
masses. From Eqs. (4), (5) one can get that

$\theta^{\rm max}_{\phi}=17.427^{\circ},\,\,\,\,\theta^{\rm
max}_{n}=18.963^{\circ}$ (9)

at an initial pion beam momentum of 1.7 GeV/c. Energy-momentum conservation in
the reaction (1), taking place in a free space, leads to two different
solutions for the laboratory $\phi$ meson and final neutron momenta $p_{\phi}$
and $p_{n}$ at given laboratory polar production angles $\theta_{\phi}$ and
$\theta_{n}$, belonging, correspondingly, to the angular intervals (2) and
(3):

$p^{(1,2)}_{\phi}(\theta_{\phi})=\frac{p_{\pi^{-}}\sqrt{s}E^{*}_{\phi}\cos{\theta_{\phi}}\pm(E_{\pi^{-}}+m_{p})\sqrt{s}\sqrt{p^{*2}_{\phi}-{\gamma^{2}_{\rm
cm}}{v^{2}_{\rm
cm}}m^{2}_{\phi}\sin^{2}{\theta_{\phi}}}}{(E_{\pi^{-}}+m_{p})^{2}-p^{2}_{\pi^{-}}\cos^{2}{\theta_{\phi}}},$
(10)
$p^{(1,2)}_{n}(\theta_{n})=\frac{p_{\pi^{-}}\sqrt{s}E^{*}_{n}\cos{\theta_{n}}\pm(E_{\pi^{-}}+m_{p})\sqrt{s}\sqrt{p^{*2}_{n}-{\gamma^{2}_{\rm
cm}}{v^{2}_{\rm
cm}}m^{2}_{n}\sin^{2}{\theta_{n}}}}{(E_{\pi^{-}}+m_{p})^{2}-p^{2}_{\pi^{-}}\cos^{2}{\theta_{n}}}.$
(11)

Here, ${\gamma_{\rm cm}}=(E_{\pi^{-}}+m_{p})/\sqrt{s}$, $v_{\rm
cm}=p_{\pi^{-}}/(E_{\pi^{-}}+m_{p})$,
$E_{\phi}^{*}=\sqrt{m^{2}_{\phi}+p^{*2}_{\phi}}$,
$E^{*}_{n}=\sqrt{m^{2}_{n}+p^{*2}_{n}}$ and sign ”+” in the numerators of Eqs.
(10), (11) corresponds to the first solutions $p^{(1)}_{\phi}$, $p^{(1)}_{n}$
and sign ”-” - to the second ones $p^{(2)}_{\phi}$, $p^{(2)}_{n}$. Inspection
of the expressions (10), (11) tells us that the first solutions
$p^{(1)}_{\phi}$ and $p^{(1)}_{n}$ as well as the second ones $p^{(2)}_{\phi}$
and $p^{(2)}_{n}$ have different dependencies, respectively, on the production
angles $\theta_{\phi}$ and $\theta_{n}$ within the angular intervals [0,
$\theta_{\phi}^{\rm max}]$ and [0, $\theta_{n}^{\rm max}]$. Namely, the former
drop and the latter ones increase as the production angles $\theta_{\phi}$ and
$\theta_{n}$ increase in these intervals (cf. Fig. 1) and

$p^{(1)}_{\phi}(\theta_{\phi}^{\rm max})=p^{(2)}_{\phi}(\theta_{\phi}^{\rm
max})=p_{\phi}(\theta_{\phi}^{\rm max}),$ (12) $p^{(1)}_{n}(\theta_{n}^{\rm
max})=p^{(2)}_{n}(\theta_{n}^{\rm max})=p_{n}(\theta_{n}^{\rm max}),$ (13)

where

$p_{\phi}(\theta_{\phi}^{\rm max})=\sqrt{E_{\phi}^{2}(\theta_{\phi}^{\rm
max})-m_{\phi}^{2}},\,\,E_{\phi}(\theta_{\phi}^{\rm max})={\gamma_{\rm
cm}}m_{\phi}^{2}/E_{\phi}^{*};$ (14) $p_{n}(\theta_{n}^{\rm
max})=\sqrt{E_{n}^{2}(\theta_{n}^{\rm
max})-m_{n}^{2}},\,\,E_{n}(\theta_{n}^{\rm max})={\gamma_{\rm
cm}}m_{n}^{2}/E_{n}^{*}.$ (15)

According to Eqs. (14), (15), for $p_{\pi^{-}}=1.7$ GeV/c we get then that
$p_{\phi}(\theta_{\phi}^{\rm max})=0.792$ GeV/c and $p_{n}(\theta_{n}^{\rm
max})=0.720$ GeV/c (cf. Fig. 1). This figure shows that the kinematically
allowed $\phi$ meson and final neutron laboratory momenta in the process
${\pi^{-}}p\to{\phi}n$ on a free target proton at rest vary within the
following momentum ranges at a given initial pion momentum:

$p^{(2)}_{\phi}(0^{\circ})\leq p_{\phi}\leq p^{(1)}_{\phi}(0^{\circ}),$ (16)
$p^{(2)}_{n}(0^{\circ})\leq p_{n}\leq p^{(1)}_{n}(0^{\circ}),$ (17)

where the quantities $p^{(1,2)}_{\phi}(0^{\circ})$ and
$p^{(1,2)}_{n}(0^{\circ})$ are defined, according to Eqs. (10), (11), as
follows:

$p^{(1,2)}_{\phi}(0^{\circ})={\gamma_{\rm cm}}E^{*}_{\phi}(v_{\rm cm}\pm
v^{*}_{\phi}),$ (18) $p^{(1,2)}_{n}(0^{\circ})={\gamma_{\rm
cm}}E^{*}_{n}(v_{\rm cm}\pm v^{*}_{n}).$ (19)

Here, $v^{*}_{\phi}=p^{*}_{\phi}/E^{*}_{\phi}$ and
$v^{*}_{n}=p^{*}_{n}/E^{*}_{n}$. Evidently, that

$p^{(1)}_{\phi}(0^{\circ})+p^{(2)}_{n}(0^{\circ})=p_{\pi^{-}},\,\,\,\,p^{(2)}_{\phi}(0^{\circ})+p^{(1)}_{n}(0^{\circ})=p_{\pi^{-}}.$
(20)

The relations (20) express the momentum conservation in the reaction (1),
taking place on a free target proton at rest, in the laboratory frame for zero
$\phi$ and neutron production angle. They correspond, respectively, to the
$\phi$ going forward (at 0∘) and neutron going backward (at 180∘) in the c.m.
system and vice versa. In line with Eqs. (18)–(19), for $p_{\pi^{-}}=1.7$
GeV/c we have:

$p^{(1)}_{\phi}(0^{\circ})=1.217~{}{\rm
GeV/c},\,\,\,p^{(2)}_{\phi}(0^{\circ})=0.548~{}{\rm GeV/c};$ (21)
$p^{(1)}_{n}(0^{\circ})=1.152~{}{\rm
GeV/c},\,\,\,p^{(2)}_{n}(0^{\circ})=0.483~{}{\rm GeV/c}$

and inequalities (16), (17) are then

$0.548~{}\leq p_{\phi}~{}\leq 1.217~{}{\rm GeV/c},$ (22) $0.483~{}\leq
p_{n}~{}\leq 1.152~{}{\rm GeV/c}.$ (23)

It is obvious that the binding of target protons and their Fermi motion will
distort the distributions of the final $\phi$ mesons and neutrons as well as
lead to a wider accessible momentum intervals compared to those given above by
Eqs. (22) and (23).

Figure 1: (Color online.) The kinematically allowed $\phi$ and $n$ momenta in
the free space ${\pi^{-}}p\to{\phi}n$ reaction in the laboratory system at an
incident pion momentum of 1.7 GeV/c against their exit angles with respect to
the beam direction in this system. Left and right vertical arrows mark the
maximum values of these angles allowed in the reaction at given pion momentum.
Upper and lower horizontal arrows indicate the values of the $\phi$ meson and
neutron momenta, corresponding to these maximum values.

As in Ref. [44], we will approximate the in-medium local effective masses
$m^{*}_{h}(r)$ of the final $\phi$ meson and neutron, participating in the
production process (1), with their average in-medium masses $<m^{*}_{h}>$
defined as:

$<m^{*}_{h}>=m_{h}+U_{h}\frac{<{\rho_{N}}>}{{\rho_{0}}}.$ (24)

Here, $m_{h}$ ($h=\phi,n$) is the rest mass of a hadron in free space, $U_{h}$
is the effective scalar nuclear potential, which the hadron sees inside the
nucleus, (or its in-medium mass shift) at normal nuclear matter density
${\rho_{0}}$, and $<{\rho_{N}}>$ is the average nucleon density. For the
target nuclei 12C and 184W, the ratio $<{\rho_{N}}>/{\rho_{0}}$ is equal to
0.55 and 0.76, respectively. The total energy $E^{\prime}_{h}$ of the hadron
in nuclear matter is expressed via its average effective mass $<m^{*}_{h}>$
and its in-medium momentum ${\bf p}^{\prime}_{h}$ by the equation:

$E^{\prime}_{h}=\sqrt{({\bf p}^{\prime}_{h})^{2}+(<m^{*}_{h}>)^{2}}.$ (25)

The momentum ${\bf p}^{\prime}_{h}$ is related to the vacuum hadron momentum
${\bf p}_{h}$ as follows:

$E^{\prime}_{h}=\sqrt{({\bf p}^{\prime}_{h})^{2}+(<m^{*}_{h}>)^{2}}=\sqrt{{\bf
p}^{2}_{h}+m^{2}_{h}}=E_{h},$ (26)

where $E_{h}$ is the hadron total energy in vacuum.

In view of the substantial uncertainties of the $\phi$ meson self-energy at
finite momenta in the momentum range (22) (cf. Refs.[13, 45] and [46]), we
will use in this work, as before in [44], for the effective potential
$U_{\phi}$ at these momenta the value corresponding to a mass drop of about
-2% at saturation density $\rho_{0}$, namely $U_{\phi}=-20$ MeV, predicted at
threshold by Hatsuda and Lee [6]. In addition, to extend the range of
applicability of our model and to see the sensitivity of the $\phi$ production
cross section of the direct process (1) to the potential $U_{\phi}$, we will
yet employ in our calculations three additional options for this potential,
namely:

$i)~{}U_{\phi}=-100~{}{\rm MeV},\,\,\,\,ii)~{}U_{\phi}=-70~{}{\rm
MeV},\,\,\,\,iii)~{}U_{\phi}=-50~{}{\rm MeV},$ (27)

motivated by the following low-energy information presently available in this
field. Recently, the spin-averaged scattering length of the $p$-$\phi$
interaction was extracted from two-particle correlations of $p$-$\phi$ pairs
measured in high-multiplicity $pp$ collisions at $\sqrt{s}=13$ TeV by the
ALICE Collaboration [47]. Its real part $f_{{\phi}p}$ is found to be
$f_{{\phi}p}=0.85\pm 0.37$ fm 222) It is interesting to note that this (real)
scattering length is an order of magnitude larger than that obtained from the
$\phi$ meson photoproduction data [22] from the CLAS experiment in Hall B of
JLab for the ${\gamma}p\to{\phi}p$ reaction near threshold in Ref. [48] using
the vector meson dominance, but it is comparable within errors to that
extracted in the spin 3/2 channel in Ref. [49] adopting the lattice HAL QCD
method.). An attractive optical low-energy $\phi$–nucleus potential depth at
saturation density $\rho_{0}$ can be estimated, using the impulse ($t{\rho}$)
approximation [4, 50] in the large $A$ limit and accounting for the Pauli
correlations term $C_{Pauli}(\rho_{0})$ at this density discussed for
$\Lambda$\- and $\Xi^{-}$-nuclear potential depths, respectively, in Refs.
[51] and [52, 53], as follows:

$U_{\phi}=-\frac{2\pi}{m_{\phi}}\left(1+\frac{m_{\phi}}{m_{N}}\right)C_{Pauli}(\rho_{0})f_{{\phi}p}\rho_{0},$
(28)

where

$C_{Pauli}(\rho_{0})=\left[1+{\alpha_{P}}\frac{3k_{F}}{2\pi}\left(1+\frac{m_{\phi}}{m_{N}}\right)f_{{\phi}p}\right]^{-1},$
(29)

with Fermi momentum $k_{F}=(3\pi^{2}\rho_{0}/2)^{1/3}$. The parameter
${\alpha_{P}}$ in Eq. (29) switches off (${\alpha_{P}}=0$) or on
(${\alpha_{P}}=1$) Pauli correlations in ${\phi}N$ in-medium multiple
scatterings. Taking the above real part $f_{{\phi}p}$, one gets that

$U_{\phi}\approx\left\\{\begin{array}[]{lll}-(70\pm 30)~{}{\rm MeV}&\mbox{for
${\alpha_{P}}=0$},\\\ &\\\ -32.0^{+5.5}_{-8.4}~{}{\rm MeV}&\mbox{for
${\alpha_{P}}=1$}.\end{array}\right.$ (30)

Here we have taken the density $\rho_{0}$ to be 0.16 fm-3. One sees that the
$\phi$ potential depth $U_{\phi}$, obtained in the case of disregarding Pauli
correlations (${\alpha_{P}}=0$), decreases in size by factor of about 2–3 upon
switching on these correlations (${\alpha_{P}}=1$). And our choice for this
depth covers fully the estimation (30) for two adopted values of the parameter
${\alpha_{P}}$ (see, also, the footnote 8) below). The effective scalar
neutron potential $U_{n}$, used in Eq. (24), can be determined from the
relation [4, 54]

$U_{n}=\frac{\sqrt{m_{n}^{2}+{p^{\prime}_{n}}^{2}}}{m_{n}}V_{NA}^{\rm SEP},$
(31)

where $V_{NA}^{\rm SEP}$ is the Schr${\ddot{\rm o}}$dinger equivalent
potential for nucleons at normal nuclear matter density. This potential is
momentum-dependent and can be parametrized as a function of the momentum
relative to the nuclear matter at rest by [54]:

$V_{NA}^{\rm SEP}(p_{n}^{{\prime}2})=\left(V_{1}-V_{2}{\rm
e}^{-2.3p_{n}^{{\prime}2}}\right);\,\,\,V_{1}=50~{}{\rm
MeV},\,\,\,V_{2}=120~{}{\rm MeV},$ (32)

where the momentum $p_{n}^{{\prime}}$ is measured in GeV/c. The potentials
(31) and (32) are shown in Fig. 2.

Figure 2: (Color online.) Momentum dependence of the effective scalar and
Schr${\ddot{\rm o}}$dinger equivalent neutron potentials at density $\rho_{0}$
(solid and dotted-dashed curves, respectively).

It is seen that these potentials are attractive at all momenta below of about
0.6 GeV/c with the value of $V_{NA}^{\rm SEP}(0)=U_{n}(0)=-70$ MeV, whereas
they are repulsive for higher momenta and increase monotonically with
increasing neutron momentum. It is worth noting that the inclusion of the
final neutron nuclear momentum-dependent potential (31) will result in a
reduction (enhancement) of the $\phi$ momentum distributions from nuclei at
low (high) meson momenta compared to the case with zero or positive neutron
potential at all phi momenta (see Figs. 3 and 4 given below). This is due to
the fact that, as shown above, the low-momentum (high-momentum) $\phi$ mesons
are produced in reaction (1) together with the high-momentum (low-momentum)
neutrons to balance the momentum of the incident pion beam. According to Fig.
2, these neutrons feel a repulsive (attractive) potential in the interior of
the nucleus, which leads to the above effects. In Ref. [44], for the effective
final neutron scalar potential $U_{n}$ we have used the momentum-independent
positive potential with value of $U_{n}\approx+25$ MeV, corresponding to its
in-medium momentum $p_{n}^{\prime}$ of 0.78 GeV/c, determined from Eqs. (26),
(31) and (32) for the average vacuum momentum $p_{n}$ of 0.8 GeV/c (cf. Eq.
(23)) in the kinematics of the HADES experiment. To simplify numerical
calculations, we will adopt again this potential as a basic case. In order to
see the sensitivity of the $\phi$ production differential and integral cross
sections from the one-step process (1) inside the HADES acceptance to the
neutron effective scalar potential $U_{n}$ as well as for the consistency
check of the HADES data analysis, we will also use in the calculations instead
the momentum-dependent one of Eqs. (31) and (32).

In what follows for the differential cross section for $\phi$ meson production
in ${\pi^{-}}A$ collisions from the direct process (1) we will use Eq. (21)
from Ref. [44] in these calculations, changing the limits of integration over
the $\phi$ laboratory polar production angle $\theta_{\phi}$ by 0∘ and 90∘ and
including in the integrand a factor $Q(p_{T},y)$, which eliminates the phase
space outside of the HADES acceptance. This procedure yields for the above
differential cross section the following expression:

$\frac{d\sigma_{{\pi^{-}}A\to{\phi}X}^{({\rm
prim})}(p_{\pi^{-}},p_{\phi})}{dp_{\phi}}=2{\pi}\left(\frac{Z}{A}\right)\left(\frac{p_{\phi}}{p^{\prime}_{\phi}}\right)\int\limits_{0}^{1}d\cos{{\theta_{\phi}}}I_{V}[A,\theta_{\phi}]\left<\frac{d\sigma_{{\pi^{-}}p\to{\phi}n}(p_{\pi^{-}},p^{\prime}_{\phi},\theta_{\phi})}{dp^{\prime}_{\phi}d{\bf\Omega}_{\phi}}\right>_{A}Q(p_{T},y),$
(33)

where

$I_{V}[A,\theta_{\phi}]=A\int\limits_{0}^{R}r_{\bot}dr_{\bot}\int\limits_{-\sqrt{R^{2}-r_{\bot}^{2}}}^{\sqrt{R^{2}-r_{\bot}^{2}}}dz\rho(\sqrt{r_{\bot}^{2}+z^{2}})\exp{\left[-\sigma_{{\pi^{-}}N}^{\rm
tot}A\int\limits_{-\sqrt{R^{2}-r_{\bot}^{2}}}^{z}\rho(\sqrt{r_{\bot}^{2}+x^{2}})dx\right]}$
(34)
$\times\int\limits_{0}^{2\pi}d{\varphi}\exp{\left[-\sigma_{{\phi}N}A\int\limits_{0}^{l(\theta_{\phi},\varphi)}\rho(\sqrt{x^{2}+2a(\theta_{\phi},\varphi)x+b+R^{2}})dx\right]}$

and the phase-space eliminating factor $Q(p_{T},y)$ is defined in the
following way:

$Q(p_{T},y)=Q(p_{T}){\cdot}Q(y),$ (35)

where

$Q(p_{T})=\left\\{\begin{array}[]{lll}1&\mbox{for $150\leq p_{T}<650~{}{\rm
MeV/c}$},\\\ &\\\ 0&\mbox{{\rm otherwise}};\end{array}\right.$ (36)
$Q(y)=\left\\{\begin{array}[]{lll}1&\mbox{for $0.4\leq y<1.0$},\\\ &\\\
0&\mbox{{\rm otherwise}}\end{array}\right.$ (37)

and

$p_{T}=p_{\phi}\sin{\theta_{\phi}},\,\,\,\,\,y=\frac{1}{2}\ln{\frac{E_{\phi}+p_{\phi}\cos{\theta_{\phi}}}{E_{\phi}-p_{\phi}\cos{\theta_{\phi}}}}.$
(38)

Here in Eq. (33),
$\left<\frac{d\sigma_{{\pi^{-}}p\to{\phi}n}(p_{\pi^{-}},p^{\prime}_{\phi},\theta_{\phi})}{dp^{\prime}_{\phi}d{\bf\Omega}_{\phi}}\right>_{A}$
is the off-shell inclusive differential cross section for the production of
$\phi$ mesons and neutrons with reduced masses $<m_{\phi}^{*}>$ and
$<m_{n}^{*}>$ and $\phi$ mesons with in-medium momentum ${\bf
p}_{{\phi}}^{{}^{\prime}}$ in reaction (1), averaged over the Fermi motion and
binding energy of the protons in the nucleus. The quantities $\rho(r)$ and
$\sigma_{{\pi^{-}}N}^{\rm tot}$, entering into Eq. (34), denote, respectively,
the target nucleon density, normalized to unity, and the total cross section
of the free ${\pi^{-}}N$ interaction. For the rest of notation see Ref. [44].

Figure 3: (Color online.) Momentum differential cross sections for the
production of $\phi$ mesons from the primary ${\pi^{-}}p\to{\phi}n$ channel in
the interaction of $\pi^{-}$ mesons with momentum of 1.7 GeV/c with 184W
nuclei for different values of the ${\phi}N$ absorption cross section
indicated in the figure, imposing (see Eqs. (36)–(38)) the HADES spectrometer
kinematical cuts on the laboratory $\phi$ momenta and production angles and
for $\phi$ meson effective scalar potential $U_{\phi}=-20$ MeV at density
$\rho_{0}$. The solid and dashed-dotted curves are calculations, assuming that
the secondary neutron feels the momentum-independent effective scalar
potential $U_{n}=+25$ MeV and the momentum-dependent potential at this density
in the nucleus (see text). Figure 4: (Color online.) The same as in Fig. 3,
but for the $\phi$ meson effective scalar potential $U_{\phi}=-100$ MeV at
density $\rho_{0}$. Figure 5: (Color online.) 2d-plot showing the calculated
$\phi$ meson total production cross sections on carbon and tungsten target
nuclei in the HADES spectrometer acceptance window for an incident $\pi^{-}$
meson momentum of 1.7 GeV/c for values of the ${\phi}N$ absorption cross
section of 0 (only for $U_{\phi}=-20$ MeV case), 5, 7.5, 10, 15, 20, 25, 30,
35 mb as well as for different values of the $\phi$ meson effective scalar
potential at density $\rho_{0}$ indicated in the plot and for secondary
neutron nuclear potential $U_{n}=+25$ MeV at this density (full points) in
comparison with the data from the HADES experiment [43]:
${\Delta}{\sigma}_{\rm C}^{\phi}=(41\pm 6.2)$ $\mu$b, ${\Delta}{\sigma}_{\rm
W}^{\phi}=(112\pm 19.5)$ $\mu$b. The rectangular box in the plot corresponds
to these data.

In our present calculations, for the cross section for $\phi$ meson absorption
by target nucleons $\sigma_{{\phi}N}$, entering into the expression (34),
which describes (multiplied by factor $Z/A$) the effective number of protons
involved in the ${\pi^{-}}p\to{\phi}n$ elementary reaction, we will adopt the
following representative options: $\sigma_{{\phi}N}=$ 0, 5, 7.5, 10, 15, 20,
25, 30 and 35 mb, covering in view of the aforementioned the bulk of the low-
energy theoretical and experimental information presently available in this
field. The absorption cross section $\sigma_{{\phi}N}$ can be extracted in our
case, in particular, from a comparison of the total (integral) cross sections
for $\phi$ meson production, calculated on the basis of Eq. (33) inside the
HADES acceptance, and measured ones [43] on 12C and 184W target nuclei at pion
beam momentum of 1.7 GeV/c, namely: ${\Delta}{\sigma}_{\rm C}^{\phi}=(41\pm
6.2)$ $\mu$b and ${\Delta}{\sigma}_{\rm W}^{\phi}=(112\pm 19.5)$ $\mu$b. An
alternative way to estimate this cross section would be through a direct fit
of the relative transparency ratio, measured by HADES Collaboration, by the
transparency ratio $T_{A}$ defined as the ratio between the total $\phi$
production cross section in the HADES acceptance on a heavy nucleus (184W) and
on a light one (12C), viz.:

$T_{A}=\frac{12}{184}\frac{{\Delta}{\sigma}_{\rm
W}^{\phi}}{{\Delta}{\sigma}_{\rm C}^{\phi}}.$ (39)

Below, we will employ these absolute and relative observables in our HADES
data analysis.

## 3 Results and discussion

First, we consider the momentum dependencies of the absolute differential
cross sections for $\phi$ meson production from the direct process (1) in
${\pi^{-}}^{184}$W reactions in the HADES acceptance window for the incident
pion momentum of 1.7 GeV/c. These cross sections have been calculated in the
scenarios of collisional broadening of the $\phi$ meson characterized by the
values of $\sigma_{{\phi}N}=5$, 20 and 35 mb, for the momentum-independent
positive final neutron optical potential with central value of
$U_{n}\approx+25$ MeV and a momentum-dependent scalar nuclear potential (31),
(32) and assuming an in-medium $\phi$ mass shifts of -20 and -100 MeV at
saturation density $\rho_{0}$. In line with Ref. [44], the calculations have
been performed, assuming the angular distribution of $\phi$ mesons in reaction
(1) to be isotropic in the ${\pi^{-}}p$ c.m.s. and using for its vacuum total
cross section the following expression [44]:

$\sigma_{{\pi}^{-}p\to{\phi}n}(\sqrt{s},\sqrt{s_{\rm
th}})=\left\\{\begin{array}[]{ll}0.47\left(\sqrt{s}-\sqrt{s_{\rm
th}}\right)~{}[{\rm mb}]&\mbox{for $\sqrt{s_{\rm th}}<\sqrt{s}<2.05~{}{\rm
GeV}$},\\\ &\\\ 23.7/s^{4.4}~{}[{\rm mb}]&\mbox{for $\sqrt{s}\geq 2.05~{}{\rm
GeV}$}.\end{array}\right.$ (40)

Here, $\sqrt{s}$ and $\sqrt{s_{\rm th}}$ are the free center-of-mass energy
and the free threshold energy, respectively. These dependencies are shown,
respectively, in Figs. 3 and 4. They show clearly that the inclusion of the
momentum-dependent scalar nuclear potential (31), (32) leads to a modification
of the $\phi$ meson momentum distributions obtained for the momentum-
independent one of $U_{n}\approx+25$ MeV. As noted above, the inclusion
results indeed in a reduction of these distributions at $\phi$ momenta below
of about 1.1 GeV/c and in their enhancement for higher momenta. However, what
is very important – the areas under the solid and dotted-dashed curves in
Figs. 3 and 4 at given ${\phi}N$ absorption cross section, which are the
integrated differential $\phi$ production cross sections
(${\Delta}{\sigma}_{\rm W}^{\phi}$) within the HADES acceptance turned out to
be almost equal to each other. Our calculations show that the differences
between them are less than 1%. This means that the use of the final neutron
momentum-independent scalar potential of $U_{n}\approx+25$ MeV at density
$\rho_{0}$ in calculating the $\phi$ total (integral) production cross
sections inside the HADES acceptance of our main interest is well justified
333) In addition to this point, our calculations have shown that, for example,
for $\phi$-nuclear potential with central depth of -70 MeV increase (decrease)
of the effective scalar final neutron momentum-independent potential $U_{n}$
at density $\rho_{0}$ from $U_{n}=+25$ MeV to $U_{n}=+50$ MeV (from
$U_{n}=+25$ MeV to $U_{n}=0$ MeV) leads to reduction (enhancement) of the
total cross sections for $\phi$ meson production inside the HADES acceptance
on target nuclei of interest by about 20–25%. Such notable deviations from the
$\phi$ total production cross sections obtained for the momentum-independent
neutron potential with central depth of $U_{n}=+25$ MeV and, correspondingly,
from the cross sections determined in the reference case of use the momentum-
dependent one (31), (32) indicate that our basic choice for final neutron
optical potential is optimal.). It should be also pointed out that the
adopting in the calculations an anisotropic form of the $\phi$ angular
distribution in the ${\pi^{-}}p$ c.m.s. with a slope of $b_{\phi}=2.1$ GeV-2
as in [55] leads to only insignificant corrections of the phi total production
cross section in the HADES acceptance. The differences are about 5%, as we
found. This implies that employing a $\phi$ isotropic c.m.s. distribution in
calculations of the $\phi$ integral production cross sections in ${\pi^{-}}A$
reactions in kinematics of the HADES experiment is also very well justified
and the predictions, obtained within our approach, are robust enough.
Inspection of figures 3 and 4 tells us that the obtained results for the
$\phi$ momentum distributions for 184W and, hence, those for the integrated
differential $\phi$ production cross sections within the HADES acceptance also
depend strongly on the $\phi$–nucleon absorption cross section and on the
$\phi$-nuclear potential. Thus, for example, assuming an attractive $\phi$
potential of -100 MeV at normal nuclear matter density leads to an enhancement
of the $\phi$ production cross sections for outgoing $\phi$ meson momenta of
around 1.0 GeV/c (at which they are the largest) by factor of about 1.6 as
compared to those obtained for the potential $U_{\phi}=-20$ MeV. Since in our
near-threshold calculations, contrary to the cases of proton- and photon-
induced $\phi$ production in nuclei, there is no need to know the meson
creation cross section off the neutron as well as in view of the above-
mentioned, we can hope to impose reasonable constraints on these quantities
using the HADES data.

Figure 6: (Color online.) Transparency ratio $T_{A}$ as a function of the
${\phi}N$ absorption cross section for the combination 184W/12C in the HADES
spectrometer acceptance window for an incident $\pi^{-}$ meson momentum of 1.7
GeV/c as well as for different values of the $\phi$ meson effective scalar
potential at density $\rho_{0}$ indicated in the inset and for secondary
neutron nuclear potential $U_{n}=+25$ MeV at this density in comparison with
the value of 0.18$\pm$0.04 measured in the HADES experiment [43]. The measured
central value of $T_{A}$ of 0.18 and error bar are indicated in the figure by
horizontal solid and dotted lines, respectively.

In this context, the calculated $\phi$ meson total production cross sections
on carbon and tungsten target nuclei inside the HADES acceptance,
corresponding to the $\phi$ momentum range of about 0.5–1.3 GeV/c (cf. Figs.
3, 4), for an incident pion momentum beam of 1.7 GeV/c as well as for
different values of the ${\phi}N$ absorption cross section $\sigma_{{\phi}N}$
and $\phi$-nuclear potential $U_{\phi}$ at density $\rho_{0}$ are shown in
Fig. 5 in comparison with the HADES data [43]. One can nicely see that the
relatively shallow attractive $\phi$–nucleus potential of $U_{\phi}$ $\simeq$
-20 MeV (or $\phi$ in-medium mass shift of about 2%), predicted in Ref. [6]
close to threshold within the QCD sum rule approach, is excluded by the data
for all adopted values of the ${\phi}N$ absorption cross section
$\sigma_{{\phi}N}$ 444) Full points should be inside the rectangular box.).
Looking closely at Fig. 5, one finds that the following constraints can be set
for $\sigma_{{\phi}N}$ and $U_{\phi}$: $\sigma_{{\phi}N}\geq 10$ mb and
$U_{\phi}\leq-50$ MeV. For realistic values of the cross section
$\sigma_{{\phi}N}$ $\approx$ 10–25 mb [21, 23, 24] a comparison of the
calculated and measured $\phi$ production cross sections indicates a
sufficiently attractive $\phi$–nucleus potential with central depth of
$U_{\phi}$ $\approx$ -(50–100) MeV, which is deeper than that of $U_{\phi}$
$\approx$ -(35$\pm$7) MeV extracted in the KEK-PS E325 experiment [15] from
the data collected on a copper target in the low ${\beta}{\gamma}$ region of
$\phi$ mesons (${\beta}{\gamma}<1.25$). Since
${\beta}{\gamma}=p_{\phi}/m_{\phi}$ and $m_{\phi}$ $\approx$ 1.02 GeV, the
condition ${\beta}{\gamma}<1.25$ corresponds to the laboratory $\phi$ momenta
below of about 1.3 GeV/c studied also in the HADES experiment (see above). On
the other hand, an attractive $\phi$–nucleus potential of $U_{\phi}$ $\approx$
-(50–100) MeV inferred from the present analysis of the HADES
$\pi^{-}$–nucleus data is consistent with the potential depth of $\approx$
-(70$\pm$30) MeV estimated above (see Eq. (30)), using the spin-averaged
$\phi$-$p$ scattering length obtained from the study of the $p$-$\phi$
momentum correlations in the recent ALICE experiment [47] in the case of
disregarding Pauli correlations. But it is also deeper than that of
$U_{\phi}\approx-32.0^{+5.5}_{-8.4}$ MeV obtained upon switching on these
correlations. For the proton and neutron densities, $\rho_{p}(r)$ and
$\rho_{n}(r)$, in the cases of the ${}_{6}^{12}$C and ${}_{74}^{184}$W target
nuclei considered we have used in our present calculations, respectively, the
harmonic oscillator and the Woods–Saxon distributions with the same radial
parameters for protons and neutrons [44]. They were normalized to $Z$ for
protons ($\rho_{p}(r)=Z\rho(r)$) and $N=A-Z$ for neutrons
($\rho_{n}(r)=N\rho(r)$). This is reasonable for light carbon nucleus, but
might be questionable for heavy tungsten nucleus [51]. To see the sensitivity
of the results of calculations of the $\phi$ meson total production cross
sections on the W nucleus in the HADES acceptance of our main interest to the
choice of proton and neutron densities, we adopted also for protons the two-
parameter Fermi density distribution, inferred from nuclear charge
distribution for this target nucleus [56]. For the neutron density
$\rho_{n}(r)$, which is not known to sufficient accuracy, the ’skin’ and
’halo’ forms of Ref. [57] were used. Their radial parameters were determined
from the r.m.s. radius $r_{n}$ of $\rho_{n}(r)$, which assumes in line with
Eq. (8) 555) In this equation the parameters $\gamma$ and $\delta$ were chosen
as $\gamma=1.05$ fm, $\delta=-0.035$ fm and the r.m.s. radius $r_{p}$ was set
equal to that of the known nuclear charge density [56].) of [57] larger value
than that $r_{p}$ for proton density distribution. This replacement of the
proton and neutron densities with the same radial forms by the latter ones
made little difference: the calculated $\phi$ total production cross sections
in the HADES acceptance are reduced only by a small fractions about 5 and 8%
for the ’halo’ and ’skin’ shapes of $\rho_{n}(r)$, which cannot change our
conclusion, made above, about the $\phi$-nuclear potential central depth.

For the cross section $\sigma_{{\pi^{-}}N}^{\rm tot}$ entering Eq. (34) we
adopt in our present calculations, as before in Ref. [44], the value of 35 mb,
representing this cross section in the case of interaction of 1.7 GeV/c pion
with a free nucleon at rest [58]. But since the nucleons in the nucleus are in
Fermi motion, one needs to carry out its averaging over this motion. For this
purpose, it is sufficient in principle to perform an average of the
${\pi^{-}}N$ total cross section over the nucleon Fermi motion occurring along
the $\pi^{-}$ beam axis [59]. To evaluate the effective pion momentum range,
accessible in this averaging, and the momentum dependence of the above cross
section in this range, we assume the local Fermi gas model for nucleons in the
nucleus. Using for the momentum of the target nucleon the Fermi momentum
$k_{F}=210$ MeV/c 666) Corresponding to the average nucleon density
$\rho_{0}/2$.) and for its binding energy the total binding energy per nucleon
$\epsilon_{A}=7$ MeV as well as for its total in-medium energy the expression
(11) from [44], we get that the effective initial pion momentum is 2.04 and
1.27 GeV/c for employed nucleon momentum directed, respectively, opposite and
along to the incoming pion beam 777) The effective incident pion momentum is
determined from the condition that the ${\pi^{-}}N$ center-of-mass energy
squared available in the free space ${\pi^{-}}N$ interaction, taking place on
the nucleon at rest, is the same as that in-medium one, obtained by using Eq.
(10) from Ref. [44] for the pion momentum of 1.7 GeV/c and for the considered
kinematical characteristics of the intranuclear nucleon.) . In the pion
momentum range of 1.27–2.04 GeV/c the ${\pi^{-}}p$ total cross section is
practically momentum-independent: here, the deviations from adopted in the
calculations the value of 35 mb for it are only about 1–5% [58]. The
${\pi^{-}}n$ total cross section (which is equal to the ${\pi^{+}}p$ one due
to the isospin considerations) has in this momentum range a weak momentum
dependence with somewhat larger but still small deviations $\sim$ 1–15% from
the value of 35 mb also used for this cross section. It is obvious that the
averaging of such weak dependencies over the Fermi motion of the intranuclear
nucleons will lead only to the small corrections of the above value of 35 mb,
employed in the calculations for the ${\pi^{-}}p$ and ${\pi^{-}}n$ total cross
sections. These corrections are expected to be no more than the deviations
indicated above. They cannot also change our conclusion about the central
depth of the $\phi$ optical potential since, for example, the change of this
value by maximal deviations $\sim$ 10–15% leads to the alteration of the
calculated $\phi$ total production cross sections in the HADES acceptance, as
the calculations have shown, by only the small fractions of about 7–10%.

It is interesting to consider the possibility of extracting the $\phi$–nucleon
absorption cross section $\sigma_{{\phi}N}$ from the transparency ratio
defined by Eq. (39) and measured in the HADES experiment [43], since this
observable is insensitive to an in-medium massdrop of the meson [60]. Figure 6
shows the calculated transparency ratio $T_{A}$ for tungsten and carbon for
$\phi$ mesons produced in the primary ${\pi^{-}}p\to{\phi}n$ reaction channel
in the HADES acceptance by 1.7 GeV/c $\pi^{-}$ mesons as a function of this
absorption cross section in comparison with the measured data point of
0.18$\pm$0.04 [43]. The transparency ratio is calculated on the basis of Eq.
(39), using the results given in Fig. 5 which were obtained for both the
adopted options for the ${\phi}N$ absorption cross section and for the $\phi$
meson and final neutron effective scalar nuclear potentials. It is seen from
this figure that the differences between all calculations corresponding to
different adopted options for these potentials are indeed insignificant,
especially between options $U_{\phi}=-20$ MeV and $U_{\phi}=-70$ MeV, for all
employed values of the $\phi$–nucleon absorption cross section
$\sigma_{{\phi}N}$. With this and accounting for the findings of Fig. 5, one
can conclude that the measured transparency ratio favors the ${\phi}N$
absorption cross section $\sigma_{{\phi}N}\geq 12$ mb, the fit of its central
value of 0.18 is achieved for $\sigma_{{\phi}N}$ $\approx$ 20–25 mb for the
considered $\phi$–nucleus potential central depths. The larger values of the
cross section $\sigma_{{\phi}N}$ that are required to accommodate the large
experimental uncertainty are rejected by the results given in Fig. 5 for these
potential depths.

Thus, we come to the conclusion that a consistent description of the HADES
integral data presented in Figs. 5 and 6 can be reached for realistic values
of this cross section of the order of $\sigma_{{\phi}N}$ $\approx$ 12–25 mb
only if $U_{\phi}$ $\approx$ -(50–100) MeV. The imaginary part of the
$\phi$–nucleus potential $W_{\phi}$ at density $\rho_{0}$ can be estimated
from the expression $W_{\phi}=(1/2){\hbar}c{\beta}\sigma_{{\phi}N}{\rho_{0}}$,
where $\beta$ is the velocity of the $\phi$ meson in the laboratory [4]. For
an average $\phi$ momentum of 800 MeV/c and $\sigma_{{\phi}N}$ $\approx$ 20–25
mb we obtain $W_{\phi}=$ 20–25 MeV which is a factor 2-3 smaller than the real
part of the $\phi$–nucleus potential and thus favourable for the experimental
observation of $\phi$-nucleus bound states. At least, the real part of the
potential seems to be sufficiently deep to allow for the formation and
population of $\phi$ mesic states [14, 61] even if the $\phi$ meson is not
produced in recoilless kinematics 888) It is worth noting that the structure
and formation of the $\phi$ mesic nuclei have been studied in Ref. [61] also
for deep real $\phi$–nucleus potential with central depths of -40, -70 and
-100 MeV at threshold keeping the weak absorptive potential strength. It was
found that one can observe a clear peak structure only if this depth $\leq$
-70 MeV.). The experimental search for such states is planned in the future
J-PARC experiment E29 using antiproton annihilation on nucleus in elementary
reaction ${\bar{p}}p\to{\phi}{\phi}$ for incident antiproton beam momentum of
1.1 GeV/c [62, 63]. It should be also noted that the J-PARC E16 Collaboration
also intends [63] to study the possible mass shift of the $\phi$ meson in cold
nuclear matter in the $pA\to{\phi}X$ reaction, using a 30 GeV proton beam,
more systematically via the $\phi$ dilepton decay channel with statistics two
orders of magnitude higher than the KEK-PS E325 experiment [15]. Moreover,
when the high intensity high resolution secondary beam line will be realized
at J-PARC, an experimental study of $\phi$ mesons in nuclear matter can be
performed [63] employing the reaction ${\pi^{-}}A\to{\phi}nX$, like that for
$\omega$ mesons in the reaction ${\pi^{-}}A\to{\omega}nX$ within the J-PARC
E26 experiment. A $\pi^{-}$ beam with momentum $\sim$ 2 GeV/c will be used to
induce the ${\pi^{-}}p\to{\phi}n$ elementary reaction. If the neutron is
detected at the forward angle, slow $\phi$ mesons can be selected. About 10
times more $\phi$ mesons compared to the E16 experiment are expected to be
collected for ${\beta}{\gamma}<0.5$ [63]. Therefore, forward neutron
measurement may lead to the observation of the $\phi$–nucleus bound states and
$\phi$ in-medium modification. One may hope that the results obtained in the
present work can be useful in planning these experiments.

## 4 Conclusions

In this paper we studied the near-threshold pion-induced production of $\phi$
mesons off nuclei in the kinematical conditions of the HADES experiment,
recently performed at GSI. The calculations have been performed within a
collision model based on the nuclear spectral function. The model accounts for
the direct ${\pi^{-}}p\to{\phi}n$ $\phi$ production process as well as for the
impact of the nuclear effective scalar $\phi$ and secondary neutron mean-field
potentials (or their in-medium mass shifts) on this process. We have
calculated the absolute differential and integral cross sections in the HADES
acceptance for the production of $\phi$ mesons off a light carbon and a heavy
tungsten target by $\pi^{-}$ mesons with momentum of 1.7 GeV/c. The
calculations have been performed allowing for different options for the
${\phi}N$ absorption cross section $\sigma_{{\phi}N}$ and different scenarios
for the $\phi$ meson and secondary neutron effective scalar potentials at
normal nuclear matter density $\rho_{0}$. We also calculated the dependence of
the ”integral” transparency ratio for the $\phi$ mesons on the ${\phi}N$
absorption cross section $\sigma_{{\phi}N}$ for the combination 184W/12C. We
demonstrated that the transparency ratio for the $\phi$ mesons has, contrary
to the absolute integral $\phi$ production cross sections, a weak sensitivity
to the real $\phi$ meson nuclear potential. The comparison of the calculated
integral $\phi$ production cross sections and transparency ratio with the
HADES data suggests this potential to be attractive with a central value
$\approx$ -(50–100) MeV for realistic values of an effective ${\phi}N$
absorption cross section of the order of $\sigma_{{\phi}N}\approx$ 12–25 mb at
$\phi$ momenta of about of 0.5–1.3 GeV/c studied in the HADES experiment.
There is some tension with some theoretical and experimental findings of
attractive potentials with a depth of $\approx$ -(20–30) MeV. Further
experimental and theoretical investigations on the topic of interest will be
helpful to better understand the $\phi$ meson in-medium properties.

Note added.-When preparing the present paper for submission to the arXiv, we
became aware of a re-analysis [64] of the experimental $p$-$\phi$ correlation
function measured by ALICE [47], using as input recent lattice calculations of
the $N$-$\phi$ interaction in the spin 3/2 channel by the HAL QCD
Collaboration [49]. A constrained fit of the experimental data allows to
determine the spin 1/2 channel of the $p$-$\phi$ interaction with evidence for
the formation of a $p$-$\phi$ bound state in this channel with sufficiently
large binding energy in the range of 14.7–56.6 MeV. This supports also the
findings of our present study.

Acknowledgments

The author is very grateful to Volker Metag for initiation of this study, for
detailed discussions with him during it as well as for his essential
contribution to the results presented in this publication.

## References

  * [1] R. S. Hayano and T. Hatsuda, Rev. Mod. Phys. 82, 2949 (2010);
arXiv:0812.1702 [nucl-ex].

  * [2] S. Leupold, V. Metag, and U. Mosel, Int. J. Mod. Phys. E19, 147 (2010);
arXiv:0907.2388 [nucl-th].

  * [3] G. Krein, A. W. Thomas, and K. Tsushima, Prog. Part. Nucl. Phys. 100, 161 (2018);
arXiv:1706.02688 [hep-ph].

  * [4] V. Metag, M. Nanova, and E. Ya. Paryev, Prog. Part. Nucl. Phys. 97, 199 (2017);
arXiv:1706.09654 [nucl-ex].

  * [5] L. Tolos and L. Fabbietti, Prog. Part. Nucl. Phys. 112, 103770 (2020);
arXiv:2002.09223 [nucl-ex].

  * [6] T. Hatsuda and S. H. Lee, Phys. Rev. C 46, R34 (1992).
  * [7] G. E. Brown and M. Rho, Phys. Rev. Lett. 66, 2720 (1991).
  * [8] F. Klingl, N. Kaiser, and W. Weise, Nucl. Phys. A 624, 527 (1997);
arXiv:hep-ph/9704398.

  * [9] P. Gubler and W. Weise, Phys. Lett. B 751, 396 (2015);
arXiv:1507.03769 [hep-ph].

  * [10] E. Oset and A. Ramos, Nucl. Phys. A 679, 616 (2001);
arXiv:nucl-th/0005046.

  * [11] D. Cabrera and M. J. Vicente Vacas, Phys. Rev. C 67, 045203 (2003);
arXiv:nucl-th/0205075.

  * [12] F. Klingl, T. Waas, and W. Weise, Phys. Lett. B 431, 254 (1998);
arXiv:hep-ph/9709210.

  * [13] D. Cabrera, A. N. Hiller Blin, and M. J. Vicente Vacas, Phys. Rev. C 95, 015201 (2017);
arXiv:1609.03880 [nucl-th].

  * [14] J. J. Cobos-Martinez et al., Phys. Lett. B 771, 113 (2017) [arXiv:1703.05367 [nucl-th]];
J. J. Cobos-Martinez et al., Phys. Rev. C 96, 035201 (2017) [arXiv:1705.06653
[nucl-th]].

  * [15] R. Muto et al., Phys. Rev. Lett. 98, 042501 (2007);
arXiv:nucl-ex/0511019.

  * [16] T. Ishikawa et al., Phys. Lett. B 608, 215 (2005);
arXiv:nucl-ex/0411016.

  * [17] W. C. Chang et al., LEPS Collaboration, Phys. Lett. B 658, 209 (2008);
arXiv:nucl-ex/0703034.

  * [18] W. C. Chang et al., LEPS Collaboration, Phys. Lett. B 684, 6 (2010);
arXiv:0907.1705 [nucl-ex].

  * [19] W. C. Chang et al., LEPS Collaboration, Phys. Rev. C 82, 015205 (2010);
arXiv:1006.4197 [nucl-ex].

  * [20] M. H. Wood et al., Phys. Rev. Lett. 105, 112301 (2010);
arXiv:1006.3361 [nucl-ex].

  * [21] T. Mibe et al., Phys. Rev. C 76, 052202(R) (2007) [arXiv:nucl-ex/0703013];
X. Qian et al., Phys. Lett. B 680, 417 (2009) [arXiv:0907.2668 [nucl-ex]];
X. Qian et al., Phys. Lett. B 696, 338 (2011) [arXiv:1011.1305 [nucl-ex]].

  * [22] B. Dey et al., CLAS Collaboration, Phys. Rev. C 89, 055208 (2014);
arXiv:1403.2110 [nucl-ex].

  * [23] A. Polyanskiy et al., Phys. Lett. B 695, 74 (2011) [arXiv:1008.0232 [nucl-ex]];
M. Hartmann et al., Phys. Rev. C 85, 035206 (2012) [arXiv:1201.3517 [nucl-
ex]].

  * [24] A. Sibirtsev et al., Eur. Phys. J. A 29, 209 (2006);
arXiv:nucl-th/0606044.

  * [25] M. Wada, for the STAR Collaboration, A. Phys. Pol. B Supp 5, 249 (2012);
arXiv:1210.2744 [nucl-ex].

  * [26] Y. Zhou, for the STAR Collaboration, Sci Post Phys. Proc. 10, 038 (2022);
arXiv:2110.15530 [nucl-ex].

  * [27] U. A. Acharya et al., PHENIX Collaboration, Phys. Rev. C 106, 014908 (2022);
arXiv:2203.06087 [nucl-ex].

  * [28] N. J. Abdulameer et al., PHENIX Collaboration, Phys. Rev. C 107, 014907 (2023);
arXiv:2207.10745 [nucl-ex].

  * [29] B. Abelev et al., ALICE Collaboration, Phys. Rev. C 91, 024609 (2015).
  * [30] Md. Nasim et al., Adv. High Energy Phys. 2015, 197930 (2015).
  * [31] Md. Nasim, EPJ Web Conf. 259, 03001 (2022);
arXiv:2201.03816 [nucl-ex].

  * [32] J. Adamczewski-Musch et al., HADES Collaboration, Phys. Lett. B 778, 403 (2018);
arXiv:1703.08418 [nucl-ex].

  * [33] V. K. Magas, L. Roca, and E. Oset, Phys. Rev. C 71, 065202 (2005);
arXiv:nucl-th/0403067.

  * [34] E. Ya. Paryev, Eur. Phys. J. A 23, 453 (2005);
E. Ya. Paryev, J. Phys. G: Nucl. Part. Phys. 36, 015103 (2009);
arXiv:0810.1124 [nucl-th].

  * [35] A. V. Akindinov et al., J. Phys. G: Nucl. Part. Phys. 37, 015107 (2010);
arXiv:0911.4616 [nucl-ex].

  * [36] A. Sibirtsev, H.-W. Hammer, and U.-G. Meissner, Eur. Phys. J. A 37, 287 (2008);
arXiv:0802.3373 [nucl-th].

  * [37] P. Muehlich, T. Falter, C. Greiner, et al., Phys. Rev. C 67, 024605 (2003);
arXiv:nucl-th/0210079.

  * [38] D. Cabrera et al., Nucl. Phys. A 733, 130 (2004);
arXiv:nucl-th/0310054.

  * [39] P. M$\ddot{\rm u}$hlich and U. Mosel, Nucl. Phys. A 765, 188 (2006);
arXiv:nucl-th/0510078.

  * [40] D. Cabrera et al., Phys. Rev. C 96, 034618 (2017);
arXiv:1706.08064 [nucl-th].

  * [41] T. Song, J. Aichelin and E. Bratkovskaya, Phys. Rev. C 106, 024903 (2022);
arXiv:2205.10251 [nucl-th].

  * [42] Ye. S. Golubeva, L. A. Kondratyuk, and W. Cassing, Nucl. Phys. A 625, 832 (1997);
arXiv:nucl-th/9710071.

  * [43] J. Adamczewski-Musch et al., HADES Collaboration, Phys. Rev. Lett. 123, 022002 (2019);
arXiv:1812.03728 [nucl-ex].

  * [44] E. Ya. Paryev, Chin. Phys. C 42 (8), 084101 (2018);
arXiv:1806.00303 [nucl-th].

  * [45] D. Cabrera, A. N. Hiller Blin, and M. J. Vicente Vacas, JPS Conf. Proc. 13, 020006 (2017);
arXiv:1609.09371 [nucl-th].

  * [46] H. Kim and P. Gubler, Phys. Lett. B 805, 135412 (2020);
arXiv:1911.08737 [hep-ph].

  * [47] S. Acharya et al., ALICE Collaboration, Phys. Rev. Lett. 127, 172301 (2021).
  * [48] I. Strakovsky, L. Pentchev, and A. Titov, Phys. Rev. C 101, 045201 (2020);
arXiv:2001.08851 [hep-ph].

  * [49] Y. Lyu, T. Doi, T. Hatsuda et al., Phys. Rev. D 106, 074507 (2022);
arXiv:2205.10544 [hep-lat].

  * [50] J. Schaffner-Bielich, I. N. Mishustin, and J. Bondorf, Nucl. Phys. A 625, 325 (1997);
arXiv:nucl-th/9607058.

  * [51] E. Friedman and A. Gal, Phys. Lett. B 837, 137669 (2023) [arXiv:2204.02264 [nucl-th]];
E. Friedman and A. Gal, EPJ Web Conf. 271, 06002 (2022) [arXiv:2301.03923
[nucl-th]].

  * [52] E. Friedman and A. Gal, Phys. Lett. B 837, 137640 (2023);
arXiv:2209.01606 [nucl-th].

  * [53] E. Friedman and A. Gal, EPJ Web Conf. 271, 03002 (2022);
arXiv:2212.12283 [nucl-th].

  * [54] E. Ya. Paryev, J. Phys. G: Nucl. Part. Phys. 43, 015106 (2016);
arXiv:1503.09007 [nucl-th].

  * [55] E. Ya. Paryev, Nucl. Phys. A 1023, 122452 (2022);
arXiv:2205.00728 [hep-ph].

  * [56] C. W. de Jager, H. de Vries, C. de Vries, At. Data Nucl. Data Tables 14, 479 (1974).
  * [57] E. Friedman and A. Gal, Phys. Rep. 452, 89 (2007);
arXiv:0705.3965 [nucl-th].

  * [58] V. Flaminio et al., Compilation of cross sections, I: $\pi^{+}$ and $\pi^{-}$ induced reactions. CERN-HERA, 83-01 (1983).
  * [59] C. Gobbi, C. B. Dover, and A. Gal, Phys. Rev. C 50, 1594 (1994).
  * [60] E. Ya. Paryev, J. Phys. G: Nucl. Part. Phys. 40, 025201 (2013);
arXiv:1209.4050 [nucl-th].

  * [61] J. Yamagata-Sekihara et al., Prog. Theor. Phys. 124, 147 (2010);
arXiv:1001.2235 [nucl-th].

  * [62] H. Ohnishi et al., Acta Phys. Polon B 45, 819 (2014).
  * [63] K. Aoki, for the J-PARC E16 Collaboration, Contribution to the Xth Workshop on Particle Correlations and Femtoscopy (WPCF2014);
arXiv:1502.00703 [nucl-th].

  * [64] E. Chizzali, Y. Kamiya, R. D. Grande, T. Doi, L. Fabbietti, T. Hatsuda, and Y. Lyu, arXiv:2212.12690 [nucl-ex].