#  Inclusive  e+e- production in collisions of pions with protons and nuclei
in the second resonance region of baryons

R. Abou Yassine6,13, J. Adamczewski-Musch5, O. Arnold10,9, E.T. Atomssa13, M.
Becker11, C. Behnke8, J.C. Berger-Chen10,9, A. Blanco1, C. Blume8, M.
Böhmer10, L. Chlad14,e, P. Chudoba14, I. Ciepał3, S. Deb13, C. Deveaux11, D.
Dittert6, J. Dreyer7, E. Epple10,9, L. Fabbietti10, P. Fonte1,a, C. Franco1,
J. Friese10, I. Fröhlich8, J. Förtsch18, T. Galatyuk6,5, J. A. Garzón15, R.
Gernhäuser10, R. Greifenhagen7,c, M. Grunwald17, M. Gumberidze5, S.
Harabasz6,b, T. Heinz5, T. Hennino13, C. Höhne11,5, F. Hojeij13, R. Holzmann5,
M. Idzik2, B. Kämpfer7,c, K-H. Kampert18, B. Kardan8, V. Kedych6, I. Koenig5,
W. Koenig5, M. Kohls8,d, J. Kolas17, B. W. Kolb5, G. Korcyl4, G. Kornakov17,
R. Kotte7, W. Krueger6, A. Kugler14, T. Kunz10, R. Lalik4, K. Lapidus10,9, S.
Linev5, F. Linz6,5, L. Lopes1, M. Lorenz8, T. Mahmoud11, L. Maier10, A.
Malige4, J. Markert5, S. Maurus10, V. Metag11, J. Michel8, D.M. Mihaylov10,9,
V. Mikhaylov14,f, A. Molenda2, C. Müntz8, R. Münzer10,9, M. Nabroth8, L.
Naumann7, K. Nowakowski4, J. Orliński16, J.-H. Otto11, Y. Parpottas12, M.
Parschau8, C. Pauly18, V. Pechenov5, O. Pechenova5, K. Piasecki16, J.
Pietraszko5, T. Povar18, P. Prościnki4, A. Prozorov14,e, W. Przygoda4, K.
Pysz3, B. Ramstein13, N. Rathod17, P. Rodriguez-Ramos14,f, A. Rost6,5, A.
Rustamov5, P. Salabura4, T. Scheib8, N. Schild6, K. Schmidt-Sommerfeld10, H.
Schuldes8, E. Schwab5, F. Scozzi6,13, F. Seck6, P. Sellheim8, J. Siebenson10,
L. Silva1, U. Singh4, J. Smyrski4, S. Spatarog, S. Spies8, M. Stefaniak5, H.
Ströbele8, J. Stroth8,5,d, C. Sturm5, K. Sumara4, O. Svoboda14, M. Szala8, P.
Tlusty14, M. Traxler5, H. Tsertos12, O. Vazquez-Doce10,9, V. Wagner14, A.A.
Weber11, C. Wendisch5, M.G. Wiebusch5, J. Wirth10,9, A Wladyszewska4, H.P.
Zbroszczyk17, E. Zherebtsova5,h, M. Zielinski4, P. Zumbruch5 (HADES
collaboration)
1LIP-Laboratório de Instrumentação e Física Experimental de Partículas ,
3004-516 Coimbra, Portugal
2AGH University of Science and Technology, Faculty of Physics and Applied
Computer Science, 30-059 Kraków, Poland
3Institute of Nuclear Physics, Polish Academy of Sciences, 31342 Kraków,
Poland
4Smoluchowski Institute of Physics, Jagiellonian University of Cracow, 30-059
Kraków, Poland
5GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany
6Technische Universität Darmstadt, 64289 Darmstadt, Germany
7Institut für Strahlenphysik, Helmholtz-Zentrum Dresden-Rossendorf, 01314
Dresden, Germany
8Institut für Kernphysik, Goethe-Universität, 60438 Frankfurt, Germany
9Excellence Cluster ’Origin and Structure of the Universe’ , 85748 Garching,
Germany
10Physik Department E62, Technische Universität München, 85748 Garching,
Germany
11II.Physikalisches Institut, Justus Liebig Universität Giessen, 35392
Giessen, Germany
12Frederick University, 1036 Nicosia, Cyprus
13Laboratoire de Physique des 2 infinis Irène Joliot-Curie, Université Paris-
Saclay, CNRS-IN2P3. , F-91405 Orsay , France
14Nuclear Physics Institute, The Czech Academy of Sciences, 25068 Rez, Czech
Republic
15LabCAF. F. Física, Univ. de Santiago de Compostela, 15706 Santiago de
Compostela, Spain
16Uniwersytet Warszawski - Instytut Fizyki Doświadczalnej, 02-093 Warszawa,
Poland
17Warsaw University of Technology, 00-662 Warsaw, Poland
18Bergische Universität Wuppertal, 42119 Wuppertal, Germany

a also at Instituto Politécnico de Coimbra, Instituto Superior de Engenharia
de Coimbra, 3030-199 Coimbra, Portugal
b also at Helmholtz Research Academy Hesse for FAIR (HFHF), Campus Darmstadt,
64390 Darmstadt, Germany
c also at Technische Universität Dresden, 01062 Dresden, Germany
d also at Helmholtz Research Academy Hesse for FAIR (HFHF), Campus Frankfurt ,
60438 Frankfurt am Main, Germany
e also at Charles University, Faculty of Mathematics and Physics, 12116
Prague, Czech Republic
f also at Czech Technical University in Prague, 16000 Prague, Czech Republic
g also at Dipartimento di Fisica and INFN, Università di Torino, 10125 Torino,
Italy
h also at University of Wrocław, 50-204 Wrocław, Poland

###### Abstract

Inclusive e+e- production has been studied with HADES in $\pi^{-}$ \+ p,
$\pi^{-}$ \+ C and $\pi^{-}+\mathrm{CH}_{2}$ reactions, using the GSI pion
beam at $\sqrt{s_{\pi p}}$ = 1.49 GeV. Invariant mass and transverse momentum
distributions have been measured and reveal contributions from Dalitz decays
of $\pi^{0}$, $\eta$ mesons and baryon resonances. The transverse momentum
distributions are very sensitive to the underlying kinematics of the various
processes. The baryon contribution exhibits a deviation up to a factor seven
from the QED reference expected for the dielectron decay of a hypothetical
point-like baryon with the production cross section constrained from the
inverse $\gamma$ n$\rightarrow\pi^{-}$ p reaction. The enhancement is
attributed to a strong four-momentum squared dependence of the time-like
electromagnetic transition form factors as suggested by Vector Meson Dominance
(VMD). Two versions of the VMD, that differ in the photon-baryon coupling,
have been applied in simulations and compared to data. VMD1 (or two-component
VMD) assumes a coupling via the $\rho$ meson and a direct coupling of the
photon, while in VMD2 (or strict VMD) the coupling is only mediated via the
$\rho$ meson. The VMD2 model, frequently used in transport calculations for
dilepton decays, is found to overestimate the measured dielectron yields,
while a good description of the data can be obtained with the VMD1 model
assuming no phase difference between the two amplitudes. Similar descriptions
have also been obtained using a time-like baryon transition form factor model
where the pion cloud plays the major role.

##  I Introduction

Radiative transitions of an excited baryon (R) to a nucleon (N) and a virtual
massive photon ($\gamma^{*}$), converting to a dilepton pair (Dalitz decays),
carry important information about the electromagnetic structure of the baryon
in the time-like region. It is encoded in electromagnetic Transition Form
Factors (eTFF) that describe the baryon-photon interaction vertex. Magnetic,
electric and Coulomb form factors are the most common representation of eTFFs
used in the description of R $\to$ N$\gamma^{*}$ transitions. Generally, form
factors are analytic functions of the squared four momentum transfer $q^{2}$
which, in the Dalitz decay process, is the dilepton mass
$q^{2}=M_{\ell^{+}\ell^{-}}^{2}$ [1]. A strong enhancement of an eTFF as a
function of increasing $M_{\ell^{+}\ell^{-}}$ is expected by the Vector Meson
Dominance (VMD) as an effect of the intermediate light vector mesons
$\rho/\omega/\phi$. Such an enhancement of the effective eTFF, defined as the
sum of squares of the moduli of the form factors, has been observed by HADES
for the first time for baryonic transitions in the second resonance region
($i.e.$ baryon masses around 1.5 GeV/$c^{2}$) by means of pion-nucleon
scattering [2]. There, the discussion of the results focused on the exclusive
channel $\pi^{-}$p $\to$ ne+e-, the extraction of the effective eTFF and
comparison of microscopic models for baryonic transitions to data.

In this paper, we extend the presentation of dielectron production to
inclusive distributions obtained from $\pi^{-}+p$ and $\pi^{-}$ \+ C
reactions. They provide an important reference for the understanding of
dielectron production off nuclei and in heavy-ion collisions in the few-$A$
GeV energy region. Indeed, in such reactions decays of baryon resonances are
of particular interest since they provide contributions from primary $N+N$ and
secondary $\pi$+N collisions. These contributions are dominant for invariant
masses above the $\eta$ meson mass and need to be known before drawing
conclusions on in-medium effects. For example, studies of dielectron
production in p + Nb collisions at $E_{\mathrm{kin}}=3.5$ GeV revealed a
significant cold matter effect, observed as an enhancement in the invariant
mass distribution below the $\rho$ meson pole w.r.t. the p + p reference
measured at the same energy [3, 4]. One possible explanation of this effect
was attributed to production and Dalitz decay of baryonic resonances. Such a
mechanism was corroborated by microscopic transport calculations (GiBUU)
assuming a two-step decay of baryonic resonances $R\rightarrow
N\rho\rightarrow Ne^{+}e^{-}$ [5]. Also descriptions of dielectron production
in heavy-ion collisions in other transport models include such processes [6,
5, 7]. The two-step decay scheme strongly relies on the applicability of VMD
to baryon Dalitz decays. Moreover, it assumes that the intermediate $\rho$
meson saturates the baryon-virtual photon transition amplitude (“strict VMD”).
However, as pointed out by several authors [8, 1, 9], such a strict VMD
approach overestimates radiative decays of baryon resonances. Consequently,
various extensions of VMD including either excited vector mesons or a two-
component photon coupling scheme, i.e. direct and via intermediate vector
mesons, were suggested to cure this problem. It is therefore important to test
those approaches using an elementary process with sufficient sensitivity.

In principle the VMD hypothesis can be directly tested for baryons by
measuring Dalitz decays. They, however, are not easy to measure due to the low
branching ratios ($BR\sim 10^{-5}$) and because of ambiguities in the
assignment of the contributing resonances. The data obtained with the High
Acceptance Di-Electron Spectrometer (HADES) in proton-proton reactions at
$E_{\mathrm{kin}}=1.25$ GeV, i.e. below the $\eta$ meson production threshold,
allowed to identify the $\Delta(1232)$ Dalitz decay [10]. The analysis
indicates a mass dependence of the effective eTFF in line with VMD. Measured
e+e- invariant mass distributions show an enhancement of about a factor two at
invariant masses $M_{\rm{e}^{+}\rm{e}^{-}}\approx 400$ MeV/$c^{2}$ w.r.t.
predictions for point-like resonances, signaling an important contribution
from off-shell $\rho$’s [10]. Measurements of p + p reactions at higher
energy, $E_{\mathrm{kin}}=3.5$ GeV, also indicated an enhancement at higher
invariant masses, presumably due to contributions from higher mass resonances.
However, determination of the individual resonance contributions depends both
on the resonance production amplitudes and on the $\rho$ meson couplings to
the various baryon transitions. Constraints on the resonant production
amplitudes were obtained from one-pion production [3] but the isolation of
individual transitions in p + p collisions was not possible because of the
overlap of the various baryon resonances excited in the collision and since
branching ratios to the $\rho$n channel are poorly known.

Pion-nucleon reactions offer a much better sensitivity. Baryon resonances with
a mass equal to $\sqrt{s_{\pi N}}$, can be excited in the s-channel and hence
can be studied more directly. Indeed, recent HADES experiments with the GSI
secondary pion beam allowed for a Partial Wave Analysis (PWA) with the Bonn-
Gatchina framework (Bn-Ga) of the two-pion production in $\pi^{-}$p in the
second resonance region, around $\sqrt{s_{\pi p}}=1.49$ GeV [11]. This
analysis allowed for the extraction of N(1440), N(1520) and N(1535) resonance
contributions and the determination of the branching ratios for decays into
various 2$\pi$ final states ($\Delta\pi,\sigma$ N, $\rho$N). The results for
the $\rho$N decay were particularly important for calculating the dielectron
invariant mass distributions and the extraction of effective eTFF of the
resonances published in [2]. As will be discussed below, this information is
also crucial for the interpretation of the inclusive and exclusive e+e-
production in context of the various VMD approaches mentioned above. We will
investigate in particular two versions of VMD, the first one (VMD1 or two-
component VMD) using a two-coupling ($\gamma$+$\rho$) scheme based on the
$\gamma\rho$ Lagrangian of [12] and the second assuming baryon-virtual photon
coupling only via $\rho$ based on the Lagrangian of [13] (VMD2 or ”strict”
VMD). These two versions were compared in theoretical studies of dielectron
production in pion-induced reactions and brought about significantly different
results [14]. In a recent publication of the GiBUU group, calculations using
VMD2 are presented and compared to the data measured in the exclusive channel
[15].

To scrutinize these different approaches we present a comprehensive comparison
of the two VMD versions to the inclusive and previously published exclusive
data for the two collision systems $\pi^{-}$ \+ p and $\pi^{-}$ \+ C. In
addition to VMD models, we also use an eTFF model which has been extended to
the time-like region [16, 17]. We also present details of the data-driven
model calculations applied to the description of the exclusive $\pi^{-}$p
$\to$ ne+e- channel [2]. As the data available for e+e- emission in $\pi$N
reactions is very scarce and limited to the region of the $\Delta(1232)$
resonance or below [18, 19, 20], our data fill an important gap.

The structure of our paper is the following: Sec. II and III describe the
experimental set-up and data analysis procedure, Sec. IV presents the
ingredients of the simulations. Inclusive and exclusive dielectron
distributions are compared to simulations in Sec. V, and conclusions are drawn
in Sec. VI.

##  II Experimental set-up

A secondary pion beam with a central momentum of $p_{\mathrm{beam}}=0.685$
GeV/$c$ was used. On average $10^{5}$ $\pi^{-}$/s with a momentum spread of
$\pm 1.7~{}\%$ were transported by the beam line up to the target [21]. The
contamination, dominated by electrons from external conversion, reaches about
$9.6\,\%$. A 4.6 cm long polyethylene (CH2) cylinder and a 3-fold segmented
stack of carbon (C) foils, each with a diameter of $12$ mm and equally spaced
over a distance of $4.6$ cm, were used in separate runs. A diamond detector
with an active area of $14\times 14\,$mm2, placed in front of the target,
provided a start signal for time-of-flight measurements in HADES [22]. A
coincidence of the start signal with a multiplicty of hits ($\geq 2$) in HADES
triggered the data acquisition. HADES has excellent reconstruction
capabilities for e+e- pairs. It consists of six identical sectors covering the
full azimuthal and the polar angles from $18^{\circ}$ to $85^{\circ}$ with
respect to the beam axis. Each sector contains a hadron-blind Ring Imaging
CHerenkov (RICH) detector for electron identification, two sets of Mini-Drift
Chambers (MDCs) and a time-of-flight wall (TOF) accompanied by a pre-shower
detector at lower polar angles (18∘ to $45^{\circ}$). The time-of-flight (TOF)
is measured by Resistive Plate Counters and scintillators covering polar
angles smaller and larger than $45^{\circ}$, respectively. Momentum vectors of
produced particles are calculated from the deflection of tracks measured in
the MDCs, with one set in front of and the other one behind a magnetic-field
region provided by the six coils of the super-conducting toroid. The four-
momentum vectors of the positron (e+) and electron (e-) are reconstructed with
a precision of $1-2~{}\%$. Electron identification is provided by combining
the signals measured by the RICH, the tracking and TOF systems. More details
on particle identification (PID) and tracking in HADES can be found in [22,
23].

##  III Data analysis

Electron pair candidates (like-sign and opposite-sign) were selected by an
algorithm which requested at least two fully reconstructed tracks in the MDCs,
assigned velocities measured in TOF of $v/c>0.9$ and momenta larger than $100$
MeV/$c$. Those tracks were extrapolated back to determine the intersection
points with the spherical mirror of the RICH [24]. The ring width ($\sigma$)
was estimated and calculated as a function of the primary vertex position, the
polar angle and the azimuthal angle. Finally, a region of interest (ROI) was
defined to search for fired pads on the RICH pad plane by smearing the ring
radius as $r=r_{0}\pm 3\sigma$ around the position derived from the
intersection points. At least three pads inside ROI were requested to qualify
the track as electron candidate. Furthermore, it was demanded that the “point”
of closest approach of the electron track to the beam axis was located inside
the target and only one hit is found in the start detector. More details on
the e+ and e- reconstruction in this experiment can be found in [25, 26].

The combinatorial background (CB) is estimated bin-by-bin using the
geometrical average of the numbers of like-sign pairs reconstructed in the
same event:
$N_{\mathrm{CB}}=2\sqrt{N_{\rm{e}^{+}\rm{e}^{+}}N_{\rm{e}^{-}\rm{e}^{-}}}$. If
for a given bin (of the invariant mass or transverse momentum) either
$N_{\rm{e}^{+}\rm{e}^{+}}$ or $N_{\rm{e}^{-}\rm{e}^{-}}$ is zero, the
arithmetic sum is used:
$N_{\mathrm{CB}}=N_{\rm{e}^{+}\rm{e}^{+}}+N_{\rm{e}^{-}\rm{e}^{-}}$.

Figure 1: Invariant mass distribution of signal pairs (black discs) and
combinatorial background (CB, red triangles) measured in the HADES acceptance
and corrected for detection efficiency. Statistical errors are shown as
vertical bars.

In Fig. 1, the CB distribution as a function of the invariant mass is compared
to the correlated signal obtained after CB subtraction. It can be clearly
observed that the contribution of the CB to the total e+e- yield is
significant only around 120 MeV/$c^{2}$, where a strong contribution of
correlated background arises from double-conversion of both photons from the
$\pi^{0}$ decay. In total, 16 340 signal e+e- pairs have been reconstructed in
reactions on the CH2 target. To derive distributions of dielectrons from
$\pi^{-}$ \+ p reactions in the polyethylene target, the contribution from
pion interactions with carbon needs to be subtracted. This was achieved by
separately measuring the interactions of pions with the carbon target. The
respective normalization factor for carbon data was estimated using high
statistics events with pions from measurements on the carbon and polyethylene
targets. The dedicated subtraction procedure is described in detail in [11].

Figure 2: Ratio of e+e- production cross sections measured on the carbon to
polyethylene targets (black discs) as a function of
$M_{\rm{e}^{+}\rm{e}^{-}}$. The data is for the inclusive analysis and show
statistical (vertical lines) and systematic uncertainties (boxes). The
horizontal width reflects the bin width. The various curves display the
results of e+e- simulations for the eTFF (green), VMD2 (black) and VMD1
(magenta) models without (dotted lines) and with the missing mass cut (full
lines). For details of the simulations see Sec. IV.

The ratio
$\left(\mathrm{d}\bar{\sigma}/\mathrm{d}M_{\rm{e}^{+}\rm{e}^{-}}\right)_{\rm{C}}/\left(\mathrm{d}\bar{\sigma}/\mathrm{d}M_{\rm{e}^{+}\rm{e}^{-}}\right)_{\rm{CH_{2}}}$
between the inclusive dielectron production cross sections for reactions on
CH2 and on C is displayed in Fig. 2 as a function of the invariant mass. Here
and throughout the text, $\bar{\sigma}$ denotes cross sections integrated over
the HADES acceptance. The ratio of inclusive yields each integrated over the
whole invariant mass range is $R_{\mathrm{C/CH}_{2}}=0.59\pm 0.02$ (stat)
$\pm$ 0.01 (syst), where the dominant systematic error results from the
relative normalization of the carbon and polyethylene measurements. This value
corresponds to a ratio of $\pi^{-}$ \+ C to $\pi^{-}$ \+ p yields
$R_{\mathrm{C/H}}$= $2R_{\mathrm{C/CH2}}/(1-R_{\mathrm{C/CH2}})=2.88\pm 0.18$
(stat) $\pm 0.12$ (syst). It should be stressed that the main contribution to
this averaged ratio comes from the region below $M_{\rm{e}^{+}\rm{e}^{-}}$=
140 MeV/$c^{2}$, which corresponds to the $\pi^{0}$ Dalitz decay. However, no
significant dependence of the ratio is observed as a function of invariant
mass (see Fig. 2). The ratios based on the simulations are discussed in Sec.
IV.

The reconstructed $\pi^{-}$ \+ p elastic scattering yields $N_{\text{el}}$ of
known cross sections $\sigma_{\text{el}}$ (from SAID) were used to calculate
the integrated accepted luminosity (see for details [11]). The cross sections
for the dielectron production on CH2, proton and $\mathrm{C}$ (obtained from
CH2 after subtraction of the contribution from proton interactions), are given
inside the HADES acceptance after correction for the efficiency accounting for
losses in electron-positron detection and reconstruction. The efficiency
correction factors $\epsilon(p,\theta,\phi$) were calculated as 3-dimensional
functions of the electron momentum, the polar and the azimuthal angles. These
efficiencies are understood as detection efficiency, as they include
corrections for tracks which were in the acceptance, defined by
$acc(p,\theta,\phi$) (see below), but did not traverse the active area of all
tracking and PID detectors used for reconstruction. Electron tracks were
processed with GEANT simulations of the HADES detection system and
reconstructed using the same algorithm as employed for the experimental data
analysis. These simulations are also used to calculate resolution effects on
the reconstructed electron momenta. They are particularly important for
modeling the e+e- missing mass distribution, presented in [2], used for the
exclusive channel separation. The electron momentum resolution is
parameterized as a function of the true momentum (GEANT input) and the
emission angles. These momentum smearing functions together with the
acceptance matrices $acc(p,\theta,\phi$), calculated separately for positrons
and electrons, constitute the HADES filter. The filter, available on request,
allows to compare any model to the data measured in the HADES acceptance.

##  IV Simulation inputs

In the collision-energy range investigated with our experiment, several
sources of e+e- pairs are expected to contribute and were accordingly
implemented in simulations using the PLUTO event generator [27]. The most
important are (i) Dalitz decays of neutral pions ($\pi^{0}$$\to\gamma$ e+e-
with $BR=1.2~{}\%$), originating from single or double production (ii) $\eta$
Dalitz decays ($\eta\to\gamma$e+e-) with $BR=6.9\times 10^{-3}$ and (iii)
exclusive $\pi^{-}$p $\to$ ne+e- reactions with contributions from the
R$\rightarrow$ e+e-n (Dalitz decay of point-like baryon) and
$\rho\,(\rightarrow$ e+e-) n final states. These two contributions are
described in sub-sections A and B, respectively.

For the pion production cross sections we use data from earlier measurements
and the SAID data base ([28, 29, 30, 31]). The excitation function of $\eta$
close to the production threshold was measured in [32] and included in our
simulations. Less important are channels with $\Delta$ Dalitz decays in
$\Delta^{+,0}\pi^{-,0}$ final states (with $BR=4.2\cdot 10^{-5}$). The
respective production cross section were derived from the PWA analysis of the
two-pion production [11]. The individual contributions of these channels is
about two orders of magnitude smaller than the $\eta$ channel and was skipped
in simulations. Table I summarizes the production cross sections of various
final states applied in the simulations.

Table 1: List of cross sections for various final states in $\pi^{-}$p collisions. Final state |  $\sigma$ [$mb$]
---|---
$\pi^{0}$n | $9.0\pm 0.8$
$\pi^{0}$$\pi^{0}$n | $1.9\pm 0.1$
$\pi^{-}$$\pi^{0}$p | $4.0\pm 0.5$
$\eta$ n | $0.63\pm 0.2$
$\Delta^{+}\pi^{-}$ | $0.78\pm 0.08$
$\Delta^{0}\pi^{0}$ | $1.0\pm 0.1$
$\rho$ n | $1.35\pm 0.20$
n$\gamma$ | $0.22$

To estimate dielectron production in $\pi^{-}$ \+ CH2 collisions, the reaction
channels listed above must also be calculated for $\pi^{-}$ \+ C collisions.
In our simulations they were modeled using a spectator model described in Sec.
IV.3.

###  IV.1 $\pi^{-}$ p $\rightarrow$ R $\rightarrow$ n e+e-: QED reference

The baryon Dalitz decays were modeled using results of the PWA of Bonn-
Gatchina (Bn-Ga) group on $\gamma$ n $\to\pi^{-}$ p and $\pi^{-}\,$p
$\rightarrow\pi\pi\,$N reactions. As the PWA of both $\gamma$ n and $\pi\pi$
final states in the $\pi^{-}$ \+ p reaction indicate that the dominant
resonant contributions are s-channel N(1520) ($J^{P}$= 3/2-) and N(1535)
($J^{P}$= 1/2-) [11] formation, our calculation of dielectron production was
conducted for Dalitz decays of these states We used detailed balance to
calculate the cross section of the inverse
$\pi^{-}\mathrm{p}\rightarrow\gamma$ n reaction
($\sigma({\pi^{-}\mathrm{p}\rightarrow\mathrm{n}\gamma})=220$ $\mu$b).
Assuming the formation of a baryon resonance (N∗) with a given spin and parity
in the $\pi^{-}$ \+ p entrance channel, the differential cross section for its
Dalitz decay ($\mathrm{N}^{*}\rightarrow$ ne+e-) was calculated as a function
of the invariant mass using equations derived in [1]:

$\displaystyle\frac{\mathrm{d}\Gamma^{\rm{N}^{\star}\rightarrow\rm{N}e^{+}e^{-}}}{\mathrm{d}M_{\rm{e}^{+}\rm{e}^{-}}}=\frac{2\alpha}{3\pi
M_{\rm{e}^{+}\rm{e}^{-}}}\Gamma^{\rm{N}^{\star}\rightarrow\rm{N}\gamma^{*}}(M_{\rm{e}^{+}\rm{e}^{-}}).$
(1)

$\Gamma^{\rm{N}^{\star}\rightarrow\rm{N}\gamma^{*}}(M_{\rm{e}^{+}\rm{e}^{-}})$
is related to the radiative decay width
$\Gamma^{\rm{N}^{\star}\rightarrow\rm{N}\gamma}$ as follows:

$\displaystyle\Gamma^{\rm{N}^{\star}\rightarrow\rm{N}\gamma^{*}}(q^{2})=\Gamma^{\rm{N}^{\star}\rightarrow\rm{N}\gamma}\frac{\sigma_{+}^{3}\sigma_{-}}{m_{+}^{3}m_{-}}\frac{\left|G_{T}(q^{2})\right|^{2}}{\left|G_{T}(0)^{2}\right|^{2}},$
(2)

where $m_{\pm}=M_{\mathrm{R}}\pm M_{\mathrm{N}}$ and
$\sigma_{\pm}^{2}=m_{\pm}^{2}-q^{2}$. $M_{\mathrm{R}}$ is the resonance mass,
$M_{\mathrm{N}}$ the nucleon mass and $\alpha$ the fine-structure constant.
The form factor $\left|G_{T}\right|$ is a combination of the squared moduli of
the electric ($G_{E}$), magnetic ($G_{M}$) and Coulomb ($G_{C}$) form factors
and defined by

$\displaystyle\left|G_{T}(q^{2})\right|^{2}$
$\displaystyle=2\left|G_{E}(q^{2})\right|^{2}+\frac{q^{2}}{M_{\rm{R}}^{2}}\left|G_{C}(q^{2})\right|^{2}$
(3)

for $J^{P}=1/2^{-}$, and

$\displaystyle\begin{split}\left|G_{T}(q^{2})\right|^{2}=&\left|G_{E}(q^{2})\right|^{2}+3\left|G_{M}(q^{2})\right|^{2}\\\
&+\frac{q^{2}}{2M_{\rm{R}}^{2}}\left|G_{C}(q^{2})\right|^{2}\end{split}$ (4)

for $J^{P}=3/2^{-}$. We note that Eq. (2) is strictly valid for both
$J^{P}=1/2^{-}$ and 3/2-. Other $J^{P}$ configurations would lead to steeper
mass distributions (for details, see [1]).

The electric, magnetic and Coulomb form factors are commonly used by models of
eTFFs which provide their mass dependence. In case of simulations based on the
eTFF model of [33, 16, 17] we have used the respective form factors taken from
calculations for the N(1520) and N(1535) transitions, which have a very
similar invariant mass dependence. However, in the limit of point-like
baryons, representing in our simulations the QED reference, we have used
covariant form factors $g_{1},~{}g_{2}$ and $~{}g_{3}$. These form factors are
directly connected to the baryon-photon interaction vertex and can be assumed
constant while $G_{E},~{}G_{M}$ and $G_{C}$ are related by transformations
which depend on the dielectron invariant mass. The respective relations are
given by equations (III.18/19) and (III.24/25) in [1]. A careful analysis of
the dependence of the Dalitz decay width on the covariant form factors was
presented in [34] (see Fig. 1 there). It shows that for $J^{P}=1/2^{-}$ and
$J^{P}=3/2^{-}$ transitions the dominant contributions originate from terms
related to $g_{1}$ and $g_{2}$, respectively. Moreover, in the latter case
these two contributions are of very similar shape. Interference effects
between the contributions of various isospins might also affect the invariant
mass distribution of pairs but are negligible in our case. Therefore, in
calculations we have used just one form factor.

The integrated cross section for the $\pi^{-}$p $\to$ ne+e- reaction is
sensitive to the differential cross section at small
$M_{\rm{e}^{+}\rm{e}^{-}}$, which does not depend strongly on the $J^{P}$
configurations. Hence, the Dalitz decay width at small
$M_{\rm{e}^{+}\rm{e}^{-}}$ can be approximated as

$\displaystyle\Gamma_{\mathrm{QED}}^{\rm{N}^{\star}\rightarrow\rm{N}e^{+}e^{-}}\approx\frac{2\alpha}{3\pi}\Gamma^{\rm{N}^{\star}\rightarrow\rm{N}\gamma}\log\bigg{(}\frac{m_{-}}{2m_{\rm{e}}}\bigg{)},$
(5) where $m_{\rm{e}}$ is the electron mass. For $M_{\rm{R}}\approx$ 1.49
GeV/$c^{2}$
$\displaystyle\Gamma_{\mathrm{QED}}^{\rm{N}^{\star}\rightarrow\rm{N}e^{+}e^{-}}\sim
1.34\ \alpha\ \Gamma^{\rm{N}^{\star}\rightarrow\rm{N}\gamma}.$ (6)

A similar relation can be derived for the cross sections

$\displaystyle\sigma({\pi^{-}\rm{p}\rightarrow\rm{N}e^{+}e^{-}})\approx 1.34\
\alpha\ \sigma({\pi^{-}\rm{p}\rightarrow\rm{n}\gamma}).$ (7)

Finally, with $\sigma({\pi^{-}\mathrm{p}\to\mathrm{n}\gamma})=220~{}\mu$b, we
obtain $\sigma({\pi^{-}\mathrm{p}\to\mathrm{n}e^{+}e^{-}})=2.1\,\mu$b as
normalization for the Dalitz decays. Taking into account this cross section
value and Eq. (1) for the differential distribution as a function of
$M_{\rm{e}^{+}\rm{e}^{-}}$, a well defined reference (“QED”) is obtained,
which is in agreement with measurements of the $\pi^{-}\mathrm{p}\,\to$
n$\gamma$ cross section. In the case of the eTFF model, the simulations were
normalized, at small invariant masses, where the effect of mass dependence of
the form factors is negligible, to the same values as the QED simulations,
i.e. according to Eq. (5).

###  IV.2 $\pi^{-}\rm{p}\to R\to n\rho(\to e^{+}e^{-}$): VMD models

In this section we discuss the two-step decay process using the two VMD
versions introduced above. Both approaches take into account the $\rho$ decay
but with different expressions for the differential decay widths, as described
below. The mass distribution of the produced $\rho$ is constrained by the
results of the PWA of two-pion production [11].

####  IV.2.1 $\rho$ spectral function and partial decay widths

The $\rho$ meson spectral function has the form

$\displaystyle\mathcal{A}(M)\sim\frac{M^{2}\Gamma_{\text{tot}}(M)}{(M_{\rho}^{2}-M^{2})^{2}+M^{2}\Gamma^{2}_{\text{tot}}(M)},$
(8)

where $M$ is the $\rho$ mass and $M_{\rho}$ is its pole value. The total width
$\Gamma_{\text{tot}}(M)$ is the sum of the $\rho\to$ $\pi^{+}\pi^{-}$ and
$\rho\to$ e+e- partial decay widths:

$\displaystyle\Gamma_{\text{tot}}(M)=\Gamma_{\pi^{+}\pi^{-}}(M)+\Gamma_{e^{+}e^{-}}(M).$
(9)

Here, it is assumed that Eq. (8) describes the $\rho$-meson spectral function
also below the two-pion threshold. The spectral function, as defined in Eq.
(8), derives from the relation between the total width and the imaginary part
of the self-energy $\Sigma_{\rho}(M)$:

$\displaystyle\operatorname{Im}(\Sigma_{\rho}(M))=M\Gamma_{\text{tot}}(M).$
(10)

To be consistent with the Bn-Ga PWA approach, we used the following
expressions of the decay widths :

$\displaystyle\Gamma_{\pi^{+}\pi^{-}}(M)=\Gamma_{0}\left(\frac{M_{\rho}}{M}\right)^{2}\left(\frac{q_{\pi\pi}}{q_{0,\pi\pi}}\right)^{3}\frac{1+R^{2}q_{0,\pi\pi}^{2}}{1+R^{2}q_{\pi\pi}^{2}},$
(11)

where $q_{\pi\pi}=\frac{1}{2}(M^{2}-4m_{\pi}^{2})^{1/2}$ is the center-of-mass
momentum for the two-pion decay of the $\rho$ meson with the running mass $M$
and $q_{0,\pi\pi}$ is the value for $M=M_{\rho}$. The cut-off parameter is
$R=0.8\,$fm and the width at the pole of the resonance is $\Gamma_{0}$.
Equation (11) is similar to the Manley parameterization, commonly used in
transport approaches, which, however, uses a softer cut-off value of $R=1\,$fm
[15].

For the $\rho\to$ e+e- decay, we use the VMD1 (two-coupling scheme) and VMD2
(strict) models, as defined in [35], which are based on two different
Lagrangians for the $\gamma$-hadron coupling. In VMD1 it is described by two
terms representing a direct $\gamma$ and $\gamma-\rho$ transition with
independent coupling constants. For baryon resonances the coupling constants
can be fixed by fitting the radiative and the N$\rho$ decay widths separately.
In the case of VMD2 the transition is only mediated via $\rho$ mesons. The
parts of the two Lagrangians describing the $\rho-\gamma$ coupling are given
by

$\displaystyle\mathcal{L}_{\rho\gamma}^{\text{VMD1}}=\frac{-e}{2g_{\rho}}F^{\mu\nu}\rho^{0}_{\mu\nu},\
\mathcal{L}_{\rho\gamma}^{\text{VMD2}}=\frac{-eM_{\rho}^{2}}{g_{\rho}}\rho^{0}_{\mu}A^{\mu},$
(12)

where Fμν and $\rho^{0}_{\mu\nu}$ are the electromagnetic and $\rho$ meson
field tensors, $A^{\mu}$ is the photon field, $g_{\rho}$ the dimensionless
$\rho-\gamma$ coupling constant ($g_{\rho}$=5.96) and $e$ the electron charge.
The respective partial decay widths are given by

$\displaystyle\mathrm{VMD1:\ }$
$\displaystyle\Gamma(M_{\rm{e}^{+}\rm{e}^{-}})_{\text{VMD1}}=$
$\displaystyle\Gamma_{0}\ M_{\rm{e}^{+}\rm{e}^{-}}/M_{\rho}$ (13)
$\displaystyle\mathrm{VMD2:\ }$
$\displaystyle\Gamma(M_{\rm{e}^{+}\rm{e}^{-}})_{\text{VMD2}}=$
$\displaystyle\Gamma_{0}\left(M_{\rho}/M_{\rm{e}^{+}\rm{e}^{-}}\right)^{3}.$
(14)

We wish to stress that these expressions for the partial widths derive from
the choice of total width. If the relation
$\operatorname{Im}(\Sigma_{\rho})=M_{\rho}\Gamma_{\text{tot}}(M)$ is used
instead of Eq.( 10), then the spectral function reads:

$\displaystyle\mathcal{A}(M)\sim\frac{M_{\rho}^{2}\Gamma_{\text{tot}}(M)}{(M_{\rho}^{2}-M^{2})^{2}+M_{\rho}^{2}\Gamma^{2}_{tot}(M)}$
(15)

and the expressions of the partial decay widths differ by a factor
$M$/$M_{\rho}$ w.r.t. Eqs. (11), (13) and (14). Simulations of the $\pi^{-}$p
$\to$ ne+e- and $\pi^{-}$p $\to$ n$\pi^{+}\pi^{-}$ were performed using a
modified version of the Pluto event generator [27] with expressions for the
decay widths given in Eqs. (11), (13) and (14).

####  IV.2.2 $\mathbf{\rho}$ production in $\pi^{-}p$ collisions

Figure 3: Invariant mass distribution of pion pairs from $\rho$ meson decay in
$\pi^{-}$p $\to$ n$\pi^{+}\pi^{-}$ reaction obtained from the PWA (black
circles)[11]. Vertical bars account for the systematic uncertainties of the
analysis (statistical errors are negligible). The curves display the result of
a fit using simulated distributions for s- (dashed green) and d-wave (dash-
dotted green) $\rho$-n production. The sum is shown as solid green curve. The
vertical lines show the range in invariant mass used for the fit. The solid
black curve displays the total $\rho$ mass distribution for the $\pi^{-}$ \+ C
case normalized to the same $\rho$ production cross section as for the
$\pi^{-}$ \+ p reaction. For the dashed black curve, only $\rho$ mesons
selected by the e+e- analysis, with the missing mass cut 900 $<$
$M_{\rm{miss}}$(MeV/$c^{2}$) $<$ 1030 , are considered. Figure 4: The
distribution of the participant proton momentum for the quasi-free $\pi^{-}$
\+ p$\to$ n$\rho$ reaction in carbon is shown for all events (dashed curve)
and after the mising mass selection 900 $<$ $M_{\rm{miss}}$(MeV/$c^{2}$) $<$
1030 (solid curve). The spectrum is normalized to the free $\pi^{-}$p $\to$
n$\rho$ cross section.

A realistic description of the $\rho$ meson mass distribution in the $\pi^{-}$
\+ p reaction is essential for a consistent prediction of the $\pi^{-}$p $\to$
ne+e- differential cross sections. In the simulations, the mass distribution
of produced $\rho$ mesons was fixed to the one obtained in the PWA for the
$\pi^{-}$p $\to$ n$\pi^{+}\pi^{-}$ reaction at the same energy [11]. According
to the PWA, the total cross section for this channel amounts to
$\sigma_{\rho}=1.35\pm 0.12$ mb and is dominated by s-channels with about
$50~{}\%$ contribution from N(1520). The corresponding differential cross
section is displayed as black circles in Fig. 3, where the errors represent
the systematic uncertainties of the analysis. We stress that this $\rho$ meson
mass distribution comprises only the low mass tail of the spectral function
given by Eq. (8) as a consequence of the restricted phase space for $\rho$
production in $\pi^{-}$p $\to$ n$\rho$ at $\sqrt{s_{\pi p}}=1.49$ GeV. For the
calculation of the total width, the $\rho\,\to\,$ e+e- partial decay width was
chosen following the VMD2 model (Eq. (14)). However, the actual choice of the
distribution function affects the $\pi^{+}\pi^{-}$ production yield only very
close to the threshold. Furthermore, the $\rho$ meson mass distribution is
affected by the available phase space in the reaction and depends on the
orbital momentum in the $\rho-$N system: for s-wave production no dependence
on the $\rho$ momentum in the center-of-mass of the $\pi^{-}$p $\to$ n$\rho$
reaction ($p_{\text{CM}}$) is expected while, for d-wave production, events
were further weighted by a factor $p_{\text{CM}}^{2}$. The invariant mass
distributions from PWA were used to determine the relative contribution of
d-wave (dashed dotted green curve) to s-wave (dashed green curve) by means of
a fit, as shown in Fig. 3. The fit was performed in the range of invariant
masses [320, 490] MeV/$c^{2}$in order to limit the sensitivity to resolution
effects or to the incident pion beam momentum distribution. The best agreement
with the PWA solution is obtained with a 10$\pm$ 2 % d-wave contribution.

####  IV.2.3 $e^{+}e^{-}$ distributions

The $\rho$ production model described above was used to generate e+e- events
from the $\rho$ decay with two options for the dielectron partial decay
widths: VMD1 or VMD2 (given in Eqs. (13) and (14), respectively). The VMD2
model provides by itself a description of the full $\pi^{-}$p $\to$ ne+e-
process. In the case of VMD1, the vanishing $\rho\to~{}$e+e- partial decay
width for $M_{\rm{e}^{+}\rm{e}^{-}}$$\rightarrow 0$ is compensated by a
contribution corresponding to the coupling of the photon to point-like
baryons. The latter is given by the QED reference introduced above. As the
phase between the $\rho$ and $\gamma$ amplitudes is unknown, we fixed it to
have the maximum yield, to better describe the data. The yields generated with
the VMD1 $\rho$ decay model and with the QED reference models, respectively,
were therefore simply added.

###  IV.3 Modelling of quasi-free $\pi^{-}+p$ reactions in the carbon nucleus

Figure 5: Dielectron differential production cross sections as a function of
the e+e- invariant mass obtained from simulations of baryon Dalitz decay are
shown in the full solid angle for the $\pi^{-}$ \+ p reaction (long dashed
line), for the $\pi^{-}$ \+ C reaction with 900 $<$
$M_{\rm{miss}}$(MeV/$c^{2}$) $<$ 1030 (dash-dotted line) and for the quasi-
free reaction (normalized by $Z_{\text{eff}}$, solid line). The results
obtained for the $\pi^{-}$ \+ C reaction without the missing mass cut are
shown for comparison (dotted lines). The orange curves correspond to the eTFF
model and the black curves display the QED reference.
Figure 6:  Same as Fig. 5 but for the $\rho$ contribution calculated with VMD2
(black curves) and VMD1 (orange curves) models.

To describe the e+e- production off the carbon and the polyethylene targets,
interactions with carbon nuclei were modeled using a quasi-free participant
spectator model, $i.e$ the ${}^{12}\mathrm{C}$ nucleus is represented as a
spectator on-shell ${}^{11}\mathrm{B}$ and a participant off-shell proton. The
proton momentum distribution is taken into account in PLUTO as obtained from
quasi-free ${}^{12}\mathrm{C}(e,e^{\prime}p)$ measurements [36] and is
displayed in Fig. 4. A condition on the e+e- missing mass 900 $<$
$M_{\rm{miss}}$(MeV/$c^{2}$) $<$ 1030 w.r.t. pion-nucleon system was applied
in the analysis to select the quasi-free $\pi^{-}$p $\to$ ne+e- exclusive
production channel (see Fig. 1 of [2]). This cut reduces the contribution of
large proton momenta in the carbon nucleus. In addition, it suppresses the
contribution of mechanisms beyond the quasi-free approach (inelastic
processes, rescatterings, short range correlations, …).

For the $\eta$ production off ${}^{12}\mathrm{C}$, the steep increase of the
cross section as a function of the energy above the production threshold has
been taken into account within the participant-spectator model. It results in
a similar average cross section for $\pi^{-}$ \+ p interactions in
${}^{12}\mathrm{C}$ as for the p target (see [37] for details). In addition to
the scaled yields, the e+e- distributions obtained for ${}^{12}\mathrm{C}$ are
affected by the difference in phase space, due to the $p$ momentum in
${}^{12}\mathrm{C}$. For the $\eta$ and $\pi^{0}$ Dalitz decays and for the
QED reference and eTFF model, this effect just distorts the high invariant
mass region, as shown for the two latter models in Fig. 5.

The case of the VMD models is different. Indeed, the $\rho$ mass distribution
is significantly affected by the difference in available energy in the
$\pi^{-}$ \+ p center-of-mass for the carbon target, which results from the
participant spectator model and favors $\rho$ mesons with lower masses, as
shown in Fig. 3. Due to this effect, the e+e- yields in the region below 400
MeV/$c^{2}$ are increased by a factor of 6.1 (instead of 2.9) - see Fig. 6
(dotted lines). The effect of the missing mass cut, which is used for the
analysis to select the exclusive quasi free $\pi^{-}$p $\to$ ne+e- channel, is
also illustrated in Fig. 6 (dash-dotted lines). It can be observed that the
cut reduces the difference between the shapes of the distributions in proton
and carbon. This is due to the fact that the distortion of the invariant mass
distribution is mostly due to the large proton momenta, which are suppressed
by the cut, as shown in Fig. 4.

Differential e+e- cross sections on the polyethylene targets are deduced from
the corresponding ones on p and C:

$\displaystyle\left(\frac{d\sigma}{dM_{\rm{e}^{+}\rm{e}^{-}}}\right)_{\mathrm{CH}_{2}}=\left(\frac{d\sigma}{dM_{\rm{e}^{+}\rm{e}^{-}}}\right)_{\mathrm{C}}+2\left(\frac{d\sigma}{dM_{\rm{e}^{+}\rm{e}^{-}}}\right)_{\mathrm{H}}.$
(16)

To deduce from the CH2 measurements a cross section comparable to a free cross
section, we define a ”quasi-free” cross section as:

$\displaystyle\left(\frac{\mathrm{d}\sigma}{\mathrm{d}M_{\rm{e}^{+}\rm{e}^{-}}}\right)_{qf}=\left(\frac{\mathrm{d}\sigma}{\mathrm{d}M_{\rm{e}^{+}\rm{e}^{-}}}\right)_{\mathrm{CH}_{2}}/R_{\mathrm{CH}_{2}/\mathrm{H}}\quad,$
(17)

with $R_{\mathrm{CH}_{2}/\mathrm{H}}=2+R_{\mathrm{C/H}}$, following from
Eq.(16). Taking $R_{\mathrm{C/CH}_{2}}$ = 0.59, as found experimentally, one
obtains $R_{\mathrm{C/H}}$= 2.9 and $R_{\mathrm{CH}_{2}/\mathrm{H}}$ = 4.9.

Figure 7: Inclusive differential cross sections, in the HADES acceptance and
corrected for efficiency, as a function of the e+e- invariant mass for
$\pi^{-}$ \+ p , $\pi^{-}$ \+ C and $\pi^{-}$ \+ CH2 reactions are shown by
symbols. Total and systematic uncertainties are shown by bars and boxes,
respectively. The black, green and magenta solid curves display the results of
the simulations for the ”VMD2”, ”VMD1 constr.” and ”eTFF model”, which use the
same $\pi^{0}$ and $\eta$ contributions (dashed cyan and dashed orange curves,
respectively). Contributions from $\rho$ decay are shown separately for VMD2
(black dashed) and VMD1 (green dot-dashed). VMD1 constr. is a coherent sum of
the QED reference (dot-dashed blue) and $\rho$ decay. For the eTFF model, the
contribution of the baryon Dalitz decay with eTFF (dashed magenta) is also
displayed. Figure 8:  Inclusive differential cross sections as a function of
the transverse momentum $p_{\rm{T}}$ of the e+e- pair for $\pi^{-}$ \+ p
(first column), $\pi^{-}$ \+ C (second column) and $\pi^{-}$ \+ CH2 (third
column) reactions are shown by symbols in the HADES acceptance after
efficiency corrections. Total and systematic errors are shown by bars and
boxes, respectively. Pairs with $M_{\rm{e}^{+}\rm{e}^{-}}$ $<$140 MeV/$c^{2}$,
140 $<$ $M_{\rm{e}^{+}\rm{e}^{-}}$ (MeV/$c^{2}$)$<$ 300, and 300 $<$
$M_{\rm{e}^{+}\rm{e}^{-}}$ (MeV/$c^{2}$) $<$ 600 are selected. The color code
for the simulation curves is as in Fig. 7, except for the $\pi^{0}$
contribution, which is shown separately for the one and two-pion processes as
cyan dash-dotted and dashed lines, respectively.

The distributions for the quasi-free process were calculated for the CH2
target within the missing mass cut (900 $<$ $M_{\rm{miss}}$(MeV/$c^{2}$) $<$
1030 ) and normalized by $Z_{\rm{eff}}+2=4.9$. As one can see in Figs. 5 and
6, the quasi-free distributions (solid lines) are very similar to the ones for
the $\pi^{-}$ \+ p reaction (long dashed), except for invariant masses above
400 MeV/$c^{2}$ where the aforementioned effects of high momentum tails in the
proton momentum distribution in carbon is still visible. Below this value, the
shapes of the distribution are very similar and the difference is about 20 %.
The ratios of the simulated dielectron differential yields from C and CH2
targets are shown in Fig. 2 for the eTFF, VMD2 and VMD1 models in comparison
to the data. Solid lines represent the ratio for the inclusive distributions,
while the dotted lines show the ratio after the missing mass cut. One can note
that the missing mass cut reduces the ratio for the VMD2. The higher value of
the ratio for the VMD2 model is due to the phase space effect, as discussed
above.

##  V Results

###  V.1 Inclusive distributions

Dielectron differential cross sections in the HADES acceptance as a function
of the e+e- invariant mass for $\pi^{-}$ \+ p (top panel), $\pi^{-}$ \+ C
(middle panel) and $\pi^{-}$ \+ CH2 (bottom panel) reactions are shown by
symbols in Fig. 7. Total and systematic errors are shown by vertical bars and
boxes, respectively. For the $\rho$ decay, the two versions of VMD were
considered in our simulations and are compared separately to the data: VMD1
(dot-dashed green) and VMD2 (dashed black). In case of VMD1, the dielectron
mass distribution is vanishing at the two-pion mass threshold while in the
other one it is strongly increasing with the decreasing invariant mass
according to the $1/M^{3}$ dependence of the partial decay width. The data are
compared to simulations described above and to the eTFF model of [33, 16]
(dashed magenta) with added contributions from the meson Dalitz decays (solid
magenta). The QED reference representing the contribution expected for point-
like baryons is shown by a dot-dashed blue curve. As expected, it is smoothly
converging to the eTFF model at small masses. The $\pi^{0}$ mass region is
well described in all cases, mainly by contributions originating from single
and double pion production (cyan dashed), except for the calculation with
VMD2, where some overshoot is visible. This overshoot is clearly apparent in
the mass region above the pion mass. In the case of VMD1 (solid green), the
coherent sum of the QED reference (point-like baryon, dot-dashed blue) and
VMD1 $\rho$ (dot-dashed green) contributions with constructive interference
describes the data very well.

Differential cross sections as a function of the transverse momentum of the
e+e- pair are shown in Figs. 8, for $\pi^{-}$\+ p (first column), $\pi^{-}$\+
C (second column) and $\pi^{-}$ \+ CH2 (third column) reactions for three bins
of the invariant mass: $M_{\rm{e}^{+}\rm{e}^{-}}$$<$ 140 MeV/$c^{2}$,
$140<$$M_{\rm{e}^{+}\rm{e}^{-}}$(MeV/$c^{2}$) $<300$ and 300 $<$
$M_{\rm{e}^{+}\rm{e}^{-}}$( MeV/$c^{2}$) $<600$, respectively. Below the pion
mass, the distributions are dominated by the $\pi^{0}$ Dalitz decay
contribution, which is well described in our simulations. However, it is
interesting to note that the data show already in this region a good
sensitivity to the exclusive process $\pi^{-}$p $\to$ ne+e- which contributes
at the highest $p_{\rm{T}}$. A clear overshoot is observed for the VMD2 model,
while the data are fairly well reproduced by the VMD1 (coherent sum) and eTFF
models.

In the intermediate mass region, the distributions are broader. The
simulations clearly depict two structures, due to the $\eta$ Dalitz decay
(orange dashed line) and the exclusive $\pi^{-}$p $\to$ ne+e- process at the
lowest and highest $p_{\rm{T}}$, respectively. The $\eta$ contribution, which
is particularly sensitive to the pion beam momentum spread in the case of the
$\pi^{-}$ \+ p interactions and to the proton momentum distribution in the
case of $\pi^{-}$ \+ C interactions, is satisfactorily described in the
simulation for the three reactions. For the highest $p_{\rm{T}}$, where the
exclusive process dominates, the difference between the models is striking.
The VMD2 model overshoots the yield in the high $p_{\rm{T}}$ region by about a
factor 3, while the other models provide a much better description of the
data. The calculation for the point-like baryon Dalitz decay (blue dot-dashed
line) is also shown in the picture as a reference. It clearly shows that, for
140 $<$ $M_{\rm{e}^{+}\rm{e}^{-}}$ (MeV/$c^{2}$)$<$ 300 and high $p_{\rm{T}}$,
the most important ingredient of these models is not the eTFF, which has a
small effect, but the constraints provided by the $\pi^{-}$p $\to$ n$\gamma$
cross sections. One should also note some excess of the yield above models at
low $p_{\rm{T}}$ which might be explained by the $\pi^{-}$p $\to$ ne+e-
bremsstrahlung, not included in the simulations.

In the highest invariant mass bin, which is mostly due to the exclusive
$\pi^{-}$p $\to$ ne+e-process, the three models are much closer. The overshoot
of the data by the VMD2 model is reduced, while the VMD1 and eTFF models are a
bit below the data, so that the quality of the description of the data is
similar in all models for this invariant mass region. In this region, which is
closest to the $\rho$ meson pole, albeit still very much off-shell, the data
are strongly in excess with respect to the point-like calculation. This points
to the strong effect of the eTFF and VM contribution which is described in a
realistic way in the three models, as discussed in more details below.

###  V.2 Exclusive $\pi^{-}p\rightarrow e^{+}e^{-}n$ distributions.

Figure 9:  Dielectron invariant mass distributions for the $\pi^{-}$p $\to$
ne+e-X reaction integrated over the pair missing mass range [900-1030]
MeV/$c^{2}$ for the free (first column) and quasi-free reactions on C (second
column) and CH2 (third column). First row: ( a), c) and e)): Full triangles
show experimental yields (d$\bar{\sigma}/$d$M_{\rm{e}^{+}\rm{e}^{-}}$)
integrated in the HADES acceptance with total and systematic errors displayed
as bars and boxes. The curves show the simulations for point-like baryon
Dalitz decay (“QED reference”, blue dash-dotted curve), $\pi^{0}$ Dalitz decay
(cyan dashed curve) and the sum (red dashed curve). Second row : (b), d) and
f)): Data (full squares) are shown after subtraction of the $\pi^{0}$ Dalitz
decay contribution and acceptance corrections. Blue dot-dashed curve: QED
reference, black dashed area: VMD2 model with d-wave contribution varied from
0 (full curve) to 10 % (dashed curve), green full curve: same for VMD1 models
with constructive sum of $\rho$ and $\gamma$ contributions, dot-dashed green
curve: $\rho$ contribution from the VMD1 model. The effect of varying the
d-wave contribution between 0 and 10% is negligible for VMD1. Calculations
using the time-like eTFF model (magenta dashed curve) are also shown.

The results on the exclusive dielectron production channel $ne^{+}e^{-}$ have
been reported in [2] for the CH2 case. Here we recall the main findings and
present comparisons to the two VMD models and the eTFF model for the proton
and carbon case separately. The selection of the exclusive channel is achieved
by the cut on the dielectron missing mass 900 $<$ $M_{\rm{miss}}$
(MeV/$c^{2}$) $<$ 1030\. For invariant masses $M_{\rm{e}^{+}\rm{e}^{-}}$
$>\,M_{\pi^{0}}$, a clear peak centered around the neutron mass is visible in
the reactions on the proton and carbon (see Fig. 2 in [2]). The mass
distributions are very well described by simulations with the eTFF model and
the two-component model. In particular, the description of the neutron peak on
carbon provides evidence for the quasi-free production. However, at smaller
invariant masses, the $\pi^{0}$ Dalitz decay contribution is not fully
suppressed by the missing mass cut. This is very clearly seen for the three
reactions in the first row of Fig. 9. This contribution, being well described
by the simulation, can be subtracted to further select the free or the quasi-
free $\pi^{-}$p $\to$ ne+e-reactions. The corresponding data points are
displayed for the three reactions $\pi^{-}$ \+ p, $\pi^{-}$ \+ C and $\pi^{-}$
\+ CH2 in Fig. 9 (lower row) and compared to the predictions of the two VMD
versions. The VMD2 version is found to give a reasonable description of yields
measured for the highest invariant masses, but strongly overestimates the
yields below 350 MeV/$c^{2}$. VMD2 overshoots the data in the low mass region
and also does not provide the right slope. On the other hand, the two-
component model with a constructive interference between amplitudes with
$\rho$ and direct $\gamma$ couplings (full green curve) provides a very good
description. The hatched areas visualises the effect of the $\rho$-n d-wave
contribution, which is mostly significant for the VMD2 model.

Conclusions from the exclusive channel are consistent with the one obtained
from the analysis of the inclusive channel. It can indeed be observed that the
application of the VMD2 approach for the dielectron production at low
invariant masses strongly overestimates the production yields. This
observation clearly points to the limitation of the ”strict VMD” model in the
baryon sector, previously suggested by some theoretical considerations, which
can here be confirmed experimentally, thanks to the sensitivity offered by
pion induced reactions. Recent calculations based on VMD2 [15] show a similar
slope as a function of the invariant mass, but a lower yield, due to a lower
cross section for the $\rho$ production.

##  VI Conclusions

We presented results on inclusive e+e- production in $\pi^{-}$ \+ p, $\pi^{-}$
\+ C and $\pi^{-}$ \+ CH2 reactions in the second resonance region which
complement the study of the exclusive e+e- production discussed in [2] for the
CH2 target. The differential distributions as a function of the invariant mass
and transverse momentum measured for the inclusive reaction within the HADES
acceptance clearly show the contributions from Dalitz decays of neutral meson
($\pi^{0}$, $\eta$) and baryonic resonances. The former can be well described
by using the known elementary cross sections, Transition Form-Factors for the
meson decays and assuming a quasi-free process for the production off carbon
nuclei, with a number of participant protons close to 3. In particular, the
transverse momentum distributions are very sensitive to the different
kinematics of the various processes. The invariant mass distributions indicate
that the contribution from baryon resonances strongly deviates from a point-
like behavior and highlights the contribution from off-shell $\rho$ mesons.
This effect was taken into account in the simulations within two
implementations of the Vector Meson Dominance model: VMD1, which assumes a
twofold photon-hadron coupling scheme, both direct and via $\rho$, and VMD2
which uses only the $\rho$ meson coupling. It was shown by detailed simulation
studies that these two scenarios lead to very different predictions for the
dielectron production. In the calculations, the PWA of $\rho$ meson production
in $\pi^{-}$p $\to$ n$\pi^{+}\pi^{-}$ provided a very important constraint.
The comparison of the simulations to the data reveals, for the first time, the
importance of VMD in the description of Dalitz decays in the baryonic sector.
The calculations assuming only direct photon coupling to point-like baryons,
representing the QED reference, strongly underestimate the dielectron
production in the mass region above the pion mass. However, a coherent sum of
the QED reference and the $\rho$ meson contribution calculated within the VMD1
approach describes the invariant mass and transverse momentum distributions
very well. Similarly do the calculations based on microscopic models of baryon
electromagnetic Transition Form Factors (etFF) [16, 17] with a photon coupling
to a quark core and a pion cloud, where the latter plays a dominant role. On
the contrary, the VMD2 model, assuming the saturation of the transition via
intermediate $\rho$ mesons strongly overestimates the measured dielectron
yields.

The present study fully confirms the conclusions of [2] on the eTFFs derived
from the studies of quasi-free $\pi^{-}$ \+ p reactions on CH2 target and
extends them to the $\pi^{-}$ \+ p and $\pi^{-}$ \+ C systems. As discussed in
[2], a missing mass selection allows to select the exclusive e+e- production
on a quasi free proton, which provides a better sensitivity to study eTFFs and
VDM models. We have presented acceptance corrected exclusive e+e- invariant
mass distributions and compared to the two VMD models. The data are well
described by the VMD1 approach, in contrast to VMD2 which strongly overshoots
the measured dielectron yields. These results call for microscopic approaches
taking into account constraints from both measurements of $\rho$ production in
pion-nucleon reactions and pion photoproduction, as discussed in [2].

These results constitute a unique basis of information for e+e- production
processes in the second resonance region. They provide important constraints
for the model calculations of dielectron production in elementary and heavy-
ion collisions and shed light on the electromagnetic structure of baryon
transitions in the time-like region.

###### Acknowledgements.

We appreciated the theoretical tools provided by A. Sarantsev, G. Ramalho and
T. Peña. We are also very grateful for the interesting discussions on
theoretical aspects with K. Gallmeister and U. Mosel. We acknowledge support
from SIP JUC Cracow, Cracow (Poland), National Science Center,
2016/23/P/ST2/040 POLONEZ, 2017/25/N/ST2/00580, 2017/26/M/ST2/00600; TU
Darmstadt, Darmstadt (Germany), DFG GRK 2128, DFG CRC-TR 211, BMBF:05P18RDFC1,
HFHF, ELEMENTS:500/10.006, VH-NG-823,GSI F&E, ExtreMe Matter Institute EMMI at
GSI Darmstadt; Goethe-University, Frankfurt (Germany), BMBF:05P12RFGHJ,
GSIF&E, HIC for FAIR (LOEWE), ExtreMe Matter Institute EMMI at GSI Darmstadt;
TU München, Garching (Germany), MLL München, DFG EClust 153, GSI TMLRG1316F,
BmBF 05P15WOFCA, SFB 1258, DFG FAB898/2-2; JLU Giessen, Giessen (Germany),
BMBF:05P12RGGHM; IJCLab Orsay, Orsay (France), CNRS/IN2P3, P2IO Labex, France;
NPI CAS, Rez, Rez (Czech Republic), MSMT LM2018112, LTT17003, MSMT OP VVV
CZ.02.1.01/0.0/0.0/18 046/0016066.
The following colleagues from Russian institutes did contribute to the results
presented in this publication but are not listed as authors following the
decision of the HADES Collaboration Board on March 23, 2022: A. Belyaev, O.
Fateev, M. Golubeva, F. Guber, A. Ierusalimov, A. Ivashkin, A. Kurepin, A.
Kurilkin, P. Kurilkin, V. Ladygin, A. Lebedev, M. Mamaev, S. Morozov, O.
Petukhov, A. Reshetin, A. Sadovsky, A. Shabanov and A. Taranenko.

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Johnson Sub-graphs]On sub-graphs of the Johnson $\mathcal{J}(n,k)$ graph induced by particle systems on simple graphs.

J. W. Fischer]  Jens Walter Fischer$^{\spadesuit\clubsuit}$
Jens Walter FISCHER,
Institut de Mathématiques de Toulouse. CNRS UMR 5219.

Université Paul Sabatier

118 route
de Narbonne, F-31062 Toulouse cedex 09<EMAIL_ADDRESS>
Understanding the topology of the state space has proven to be extremely efficient for dynamical systems with a continuous state space. On the other hand, for particle systems on finite simple graphs, it has not yet been subject to deep investigation due to combinatorial hurdles and existing efficient spectral theoretic approaches to the analysis of classical particle systems. In the context of complex systems with heterogeneous interactions of particles, these techniques can break down due to intractability. This work provides a tool box of results on the topology of state spaces of particles systems under the sole conditions that exactly one particle moves at a time and no two particles may occupy the same vertex at the same time. The Johnson $\mathcal{J}(n,k)$ graph yields the overarching structure and we prove that any particle system may be embedded as a dynamical system on a sub-graph of $\mathcal{J}(n,k)$.

$^{\spadesuit}$  Université de Toulouse

$^{\clubsuit}$ University of Potsdam

Key words : Johnson graph, Intersection Graph Theory, Vertex-induced Sub-graphs, Combinatorics, Complex Systems

MSC 2010 :

This is work in progress. It is not necessarily in a perfect form.
Additional results may be added over time.

EDIT 02.11.2022: It came to my attention that identical concepts were introduced in <cit.> as symmetric $k$th power of a graph and in <cit.> as Token Graphs. Our results, while of fundamental nature, are still complementary to what has been discussed in <cit.> and <cit.> as well as to work based on those two publications.

§ INTRODUCTION

§ DEFINITON AND GENERAL PROPERTIES

§ GENERAL PROPERTIES OF KPG

§ KPGS INDUCED BY REGULAR GRAPHS

§ OUTLOOK

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Graph isomorphism, Walks, Spectrum
We consider symmetric powers of a graph. In particular, we show that the
spectra of the symmetric square of strongly regular graphs with the same
parameters are equal. We also provide some bounds on the spectra of the
symmetric squares of more general graphs. The connection with generic
exchange Hamiltonians in quantum mechanics is discussed in Appendix A.

titleSymmetric squares of graphs
journaltitleJournal of Combinatorial Theory, Series B

Institute of Mathematical Statistics

Bernoulli-Laplace diffusion, Eigenvalues, Exclusion process, Markov
chains, Poincare inequalities
pages696 730
titleComparison Theorems for Reversible Markov Chains
journaltitleThe Annals of Applied Probability

For a graph G and integer k ≥ 1, we define the token graph Fk(G) to be
the graph with vertex set all k-subsets of V(G), where two vertices are
adjacent in Fk(G) whenever their symmetric difference is a pair of adjacent
vertices in G. Thus vertices of Fk(G) correspond to configurations of k
indistinguishable tokens placed at distinct vertices of G, where two
configurations are adjacent whenever one configuration can be reached from
the other by moving one token along an edge from its current position to an
unoccupied vertex. This paper introduces token graphs and studies some of
their properties including: connectivity, diameter, cliques, chromatic
number, Hamiltonian paths, and Cartesian products of token graphs.

titleToken Graphs
journaltitleGraphs and Combinatorics

* Invalid format of field 'month'

titleThe Dense k -Subgraph Problem

* Invalid format of field 'month'

titleApplications of product colouring
journaltitleActa Mathematica Academiae Scientiarum Hungarica

* Invalid format of field 'month'

titleComputing Vertex Connectivity: New Bounds from Old
journaltitleJ Algorithms

Cambridge University Press

seriesAustralian Mathematical Society Lecture Series
titleThe Petersen Graph

titleOn Finding Dense Subgraphs

International World Wide Web Conferences Steering Committee

densest subgraph problem, graph algorithms, graph mining, near-clique
booktitleProceedings of the 24th International Conference on World
Wide Web
seriesWWW '15
titleThe K-Clique Densest Subgraph Problem
Florence, Italy

* Can't use 'location' + 'address'

titleGraph puzzles, homotopy, and the alternating group
journaltitleJournal of Combinatorial Theory, Series B

§ SPECIAL CASES FOR $\BAR{D}$-REGULAR GRAPH

To begin with, we consider the cases $k=2$ and $k=\bar{n}-2$. In these cases due to the isomorphism $\mathfrak{L}_k\stackrel{\sim}{=}\mathfrak{L}_{\bar{n}-k}$ we only have to make claims about one or the other. It turns out that the degree set $D_k$ is given by $D_k=\{2\bar{d},2(\bar{d}-1)\}$ since for $k=2$ the two particles can be neighbors and block, therefore, an adjacent vertex mutually, or not. We define the level sets with respect to the degree as $\mathfrak{V}_{D_k;2l}:=\{\mathfrak{v}\in\mathfrak{V}_k|\,\mathrm{deg}_k(\mathfrak{v})=2l\}$ for $l\in\{\bar{d},\bar{d}-1\}$. Remark that $|\mathfrak{V}_{D_k;2(\bar{d}-1)}|=|E|=\frac{\bar{n}\bar{d}}{2}$. An important question to ask is how large subsets of $\mathfrak{V}_k$ can be chosen when taking into account weights given by the degree sequence on said subset.
We consider to this end for $\mathfrak{U}\subset\mathfrak{V}_k$ the map

\begin{equation*}
\mathfrak{f}(\mathfrak{U}):=\dfrac{\bar{d}|\mathfrak{V}_{D_k;2(\bar{d}-1)}\cap\mathfrak{U}|+(\bar{d}+1)|\mathfrak{V}_{D_k;2\bar{d}}\cap\mathfrak{U}|}{\frac{\bar{n}\bar{d}^2}{2}+ \frac{\bar{n}(\bar{n}-1-\bar{d})(\bar{d}+1)}{2}}
\end{equation*}

which represents the weighted sum of the vertices in $\mathfrak{U}$ where each weight corresponds to the quotient of the number of smaller and larger degrees then $\mathrm{deg}_k(\mathfrak{v})$. As there are only two sets $\mathfrak{V}_{D_k;2(\bar{d}-1)}$ and $\mathfrak{V}_{D_k;2\bar{d}}$ which constitutes $\mathfrak{V}_k$ and vertices in $\mathfrak{V}_{D_k;2(\bar{d}-1)}$ have a smaller weight than vertices in $\mathfrak{V}_{D_k;2\bar{d}}$, the size of $|\mathfrak{U}|$ is maximal with subject to the constraint $\mathfrak{f}(\mathfrak{U})\leq 2^{-1}$ if $\mathfrak{U}$ contains as many states with small degree as possible. Indeed, the size of can be characterized by

\begin{equation}
\max_{\mathfrak{U}\subseteq \mathfrak{V}_k, \mathfrak{f}(\mathfrak{U})\leq 2^{-1}}|\mathfrak{U}| = \begin{cases} \dfrac{\bar{d}\bar{n}}{2} + y_{\ast},& 0\leq \bar{d}\leq \frac{1}{2}\left(\bar{n}-1+\sqrt{\bar{n}-1}\sqrt{\bar{n}-1+4}\right)\\
y^{\ast} ,& otherwise
\end{cases}
\end{equation}

where we define

\begin{align*}
y_{\ast} &= \left\lfloor\dfrac{\bar{n}(\bar{n}-1-\bar{d})}{4}-\dfrac{\bar{d}^2\bar{n}}{4(\bar{d}+1)}\right\rfloor;\quad y^{\ast} = \left\lfloor\dfrac{\bar{d}+1}{\bar{d}}\dfrac{\bar{n}(\bar{n}-1-\bar{d})}{4}+\dfrac{\bar{d}\bar{n}}{4}\right\rfloor.
\end{align*}

We outline a proof in what follows. First assume that

\begin{equation*}
\bar{d}\leq \frac{1}{2}\left(\bar{n}-1+\sqrt{\bar{n}-1}\sqrt{\bar{n}-1+4}\right).
\end{equation*}

Recall that $D_k=\{d\in\mathbb{N}| \exists\mathfrak{v}\in\mathfrak{V}_k:\mathrm{deg}_k(\mathfrak{v})=d\}$ and for $l\in D_k$ that $\mathfrak{V}_{D_k;l}:=\{\mathfrak{v}\in \mathfrak{V}_k|\mathrm{deg}_k(\mathfrak{v})=l\}$. In the case $k=2$, there are exactly two distinct degrees in $\mathfrak{V}_k$ and, hence, only two sets $\mathfrak{V}_{D_k;l}$, namely, $\mathfrak{V}_{D_k;2(\bar{d}-1)}$ and $\mathfrak{V}_{D_k;2\bar{d}}$. Indeed, in the discussed case, we can derive $|\mathfrak{U}^{\ast}|:=\max_{\mathfrak{U}\subseteq \mathfrak{V}_k, \mathfrak{f}(\mathfrak{U})\leq 2^{-1}}|\mathfrak{U}|$ explicitly, which becomes important since $|\mathfrak{S}|\leq |\mathfrak{U}^{\ast}|$. Considering the representation of sub-sets of size $2$ of a vertex set as an edge, we obtain that $|\mathfrak{V}_{D_k;2(\bar{d}-1)}|=\frac{\bar{n}\bar{d}}{2}$ and, consequently, $|\mathfrak{V}_{D_k;2(\bar{d}-1)}|=\frac{\bar{n}(\bar{n}-1-\bar{d})}{2}$. We can now consider the two cases for $\max_{\mathfrak{U}\subseteq \mathfrak{V}_k, \mathfrak{f}(\mathfrak{U})\leq 2^{-1}}|\mathfrak{U}|$ which correspond to the cases $\frac{\bar{n}\bar{d}^2}{2}\leq \frac{\bar{n}(\bar{n}-1-\bar{d})(\bar{d}+1)}{2}$ and $\frac{\bar{n}\bar{d}^2}{2}\geq \frac{\bar{n}(\bar{n}-1-\bar{d})(\bar{d}+1)}{2}$, respectively.Consequently, in the first case, the maximal set $\mathfrak{U}$ satisfies $|\mathfrak{V}_{D_k;2(\bar{d}-1)}|\leq|\mathfrak{U}|$ and writing $|\mathfrak{U}|=|\mathfrak{V}_{D_k;2(\bar{d}-1)}|+y=\frac{ \bar{d}\bar{n}}{2}+y$ for some non-negative integer $y$ we obtain that for all $y\leq y_{\ast}$, for $y_{\ast}$ as in the claim, the inequality

\begin{align*}
\mathfrak{f}(\mathfrak{U})&=\dfrac{\frac{\bar{n}\bar{d}^2}{2}+(\bar{d}+1)y}{\frac{\bar{n}\bar{d}^2}{2}+ \frac{\bar{n}(\bar{n}-1-\bar{d})(\bar{d}+1)}{2}}\vspace{5pt} \leq \dfrac{\frac{\bar{n}\bar{d}^2}{2}+(\bar{d}+1)\left(\frac{\bar{n}(\bar{n}-1-\bar{d})}{4}-\frac{\bar{d}^2\bar{n}}{4(\bar{d}+1)}\right)}{\dfrac{\bar{n}\bar{d}^2}{2}+ \frac{\bar{n}(\bar{n}-1-\bar{d})(\bar{d}+1)}{2}}
\end{align*}

is satisfied where the last term is smaller or equal one half. Let $\mathfrak{A}_{y_{\ast}}\subset \mathfrak{V}_{D_k;2\bar{d}}$ be any subset of size $y_{\ast}$. Then, choosing $\mathfrak{U}=\mathfrak{V}_{D_k;2(\bar{d}-1)}\cup \mathfrak{A}_{y_{\ast}}$ we obtain a subset $\mathfrak{U}\subset \mathfrak{V}_k$ of size $\frac{\bar{d}\bar{n}}{2}$ with $\mathfrak{f}(\mathfrak{U})\leq 2^{-1}$ and any larger subset $\mathfrak{W}$ gives $\mathfrak{f}(\mathfrak{W})> 2^{-1}$. We have therefore found the size maximum of any subset $\mathfrak{U}$ satisfying $\mathfrak{f}(\mathfrak{U})\leq 2^{-1}$. In the second case, we can choose $\mathfrak{U}\subset\mathfrak{V}_{D_k;2(\bar{d}-1)}$ and

\begin{equation*}
\mathfrak{f}(\mathfrak{U})=\dfrac{\frac{\bar{n}\bar{d}y}{2}}{\frac{\bar{n}\bar{d}^2}{2}+ \frac{\bar{n}(\bar{n}-1-\bar{d})(\bar{d}+1)}{2}}.
\end{equation*}

The remaining arguments can be made analogously to the first case. Due to the symmetry We discuss now special cases $\bar{d}\in\{2,\bar{n}-2\}$. In these cases explicit quantitative statements can be made about the structure of $\mathfrak{L}_k$ for any $k\in\{1,\hdots,\bar{n}-1\}$. We lead into this section with the case $\bar{d}=2$.
We consider first the case $\bar{d}=2$ and analyze the structure of the underlying graph $\mathfrak{L}_k$. This means we are working with a generalized exclusion process on a cycle graph as illustrated in Figure <ref>. We assume that there are $k\leq \frac{\bar{n}}{2}$ particles since by symmetry $\mathfrak{L}_k\stackrel{\sim}{=}\mathfrak{L}_{\bar{n}-k}$ we cover the remaining cases as well.

[scale = 0.8]
[x=0,y=0,color = blue,size = 0.5]z
[x=-2,y=1,color = blue,size = 0.5]z
[x=2,y=3,color = blue,size = 0.5]z

A particle configuration of $3$ particles on a cycle graph of length $7$.

To find the degrees in $\mathfrak{L}_k$ we can write any configuration $\mathfrak{v}$ of particles as a vector which contains as entries the size of a connected component of $\mathfrak{L}_{\mathfrak{v}}$. Assume that there are $c'_{\mathfrak{v}}$ connected components of $\mathfrak{L}_{\mathfrak{v}}$ and denote their respective sizes by $k_i$. Each connected component contains exactly $k_i-1$ edges due to the structure of the underlying cycle graph $L$. Therefore, we obtain that

\begin{equation*}
\mathrm{deg}_k(\mathfrak{v}) = 2k-2\sum_{i=1}^{c'_{\mathfrak{v}}} (k_i -1) = 2 c'_{\mathfrak{v}}.
\end{equation*}

Furthermore, since $k\leq \frac{\bar{n}}{2}$ we can construct for any $l \in\{1, \hdots,k\}$ a configuration $\mathfrak{v}$ satisfying $c'_{\mathfrak{v}} = l$. This implies that $D_k = \{2l|l\in\{1,\hdots,k\}\}$. The remaining task consist in finding $|\mathfrak{V}_{D_k;d}|$ for any $d\in D_k$ which is equivalent to finding the number of vertex induced $k$ sub-graphs of $L$ containing exactly $\frac{d}{2}$ edges. Figure <ref> shows that this problem is related to finding all unique configurations of two colored balls on a necklace with a fixed number of each color.
To explain this further, we recall that any configuration $\mathfrak{v}$ can be translated to a vector of length $c'_{\mathfrak{v}}$ where each entry corresponds to the length of one connected component of $L_{\mathfrak{v}}$. We can, hence, identify the problem with finding all distinct colored necklaces of length $c'_{\mathfrak{v}}$ with colorings $\{(x_i,y_i)_{i=1}^{c'_{\mathfrak{v}}}|\sum_{i=1}^{c'_{\mathfrak{v}}}(x_i,y_i)=(k,\bar{n}-k)\}$, a problem solved by Pòlya's enumeration theorem. The number of configurations induced by each distinct necklace is then given by the size of its orbit under the symmetric group $S_{c'_{\mathfrak{v}}}$.
From this we can derive $|\mathfrak{V}_{D_k;2 c'_{\mathfrak{v}}}|$ which gives the first part of the puzzle.

Let $L$ be a cycle on $\bar{n}$ vertices and $k\in\left\{1,\hdots,\left\lfloor\frac{\bar{n}}{2}\right\rfloor\right\}$. Then, for $l\in\{0,\hdots,k\}$ the identity

\begin{equation}
|\mathfrak{V}_{D_k;2 l}|=\binom{k-1}{l-1}\binom{\bar{n}-k-1}{l-1}\dfrac{\bar{n}}{l}
\end{equation}

is satisfied.

First, note that

\begin{align*}
\binom{k-1}{l-1}\binom{\bar{n}-k-1}{l-1}\dfrac{\bar{n}}{l} &= \binom{k-1}{l-1}\binom{\bar{n}-k-1}{l-1}\dfrac{\bar{n}-k+k}{l}\\
&=  \binom{k-1}{l-1}\binom{\bar{n}-k-1}{l-1}\dfrac{k}{l} + \binom{k-1}{l-1}\binom{\bar{n}-k-1}{l-1}\dfrac{\bar{n}-k}{l}\\
&=  \binom{k}{l}\binom{\bar{n}-k-1}{l-1} + \binom{k-1}{l-1}\binom{\bar{n}-k}{l}.
\end{align*}

This underlines the symmetry of the problem in $k$ and $\bar{n}-k$. Based on Figure <ref> we identify the problem with counting necklaces of two colors, which are fully defined by the sets $\mathfrak{v}\in\mathfrak{V}_k$. The colors are assumed to be blue and red, where blue represents an occupied site and red an unoccupied site. We want to calculate the number of distinct blue-red colored necklaces respecting rotational symmetry, i.e., two necklaces $\mathfrak{v},\mathfrak{w}\in \mathfrak{V}_k$ are considered identical if and only if $\mathfrak{v}=\mathfrak{w}$. By fixing one vertex of the cycle, without loss of generality $v_1\in V$, we can consider necklaces starting with a blue beat, continuing then counterclockwise.

Assume that $v_1$ is occupied, i.e, carries a blue beat. Place $k$ blue beats on a line of length $k$ and select $l$ of them without replacement. There are $\binom{k}{l}$. Now, separate $\bar{n}-k$ in $l$ positive parts, i.e, create a vector $(n_1,\hdots,n_l)$ with $n_i\in\mathbb{N}^{\ast}$ and $\sum_{i=1}^l n_i=\bar{n}-k$, which is possible in $\binom{\bar{n}-k-1}{l-1}$ ways. Place $n_i$ red beats after the $i$-th drawn blue beat. This construction gives, consequently, all necklaces starting with a blue beat and, thus, there are $\binom{k}{l}\binom{\bar{n}-k-1}{l-1}$ such necklaces.

By symmetry there are $\binom{k-1}{l-1}\binom{\bar{n}-k}{l}$ necklaces starting with a red beat. Summing both terms yields the claim.

It turns out that the case $\bar{d}=\bar{n}-2$ is much more forgiving in terms of the difficulty of deriving $|\mathfrak{V}_{D_k;d}|$ for some fixed $d\in D_k$ which in the previous case demanded the utilization of results from combinatorics. In what follows we can reduce the problem to considerations on the graph complement of a $\bar{d}$-regular graph with $\bar{d}=\bar{n}-2$. In Figure <ref> one can see the translation from the graph $L$ to its complement $L^c$ in the sens that $L^c$ contains exactly the edges which are not contained in $L$ while preserving the vertex set. This renders $L^c$ a disjoint union of paths of length $1$, exhibiting $\frac{\bar{n}}{2}$ connected components.

[scale = 0.8]
[Latex-Latex, ultra thick] (3,2) – (6,2);

Translation from the $4$-regular graph $L$ on six vertices to the disjoint union of three paths of length $1$, which corresponds to the complement graph of $L$, denoted by $L^c$.

Let $L$ be a $\bar{n}-2$ regular graph and $k\in\left\{1,\hdots,\frac{\bar{n}}{2}\right\}$. Then,

\begin{equation}
D_k = \left\{k(\bar{n}-2)-k(k-1) + 2l\;\bigg|\;l\in\left\{0,\hdots,\left\lfloor\frac{k}{2}\right\rfloor\right\}\right\}
\end{equation}

and, consequently, any degree is even.

We employ the idea based on Figure <ref> that it is sufficient to consider the graph complement $L^c$.	Let $d\in D_k$. Then there exists a $\mathfrak{v}\in\mathfrak{V}_k$ such that $\mathrm{deg}_k(\mathfrak{v}) = d$. Since $\mathfrak{v}\subset V$ it also spans a vertex induced sub-graph in $L^c$. Denote by $l$ the number of paths in $L^c$ which are fully contained in $\mathfrak{v}$, i.e., both vertices of the path are elements of $\mathfrak{v}$. Then, $l\in \left\{0,\hdots,\left\lfloor\frac{k}{2}\right\rfloor\right\}$ and

\begin{equation}
d = \mathrm{deg}_k(\mathfrak{v}) = k(\bar{n}-2) -  k(k-1) + 2l.
\end{equation}

Since $\bar{n}-2$ is even, $k(k-1)$ is even and so is $2l$ for any positive integer $l$ we obtain that all degrees are even.

Based on this correspondence between $L$ and $L^c$ we can also derive the size of each $\mathfrak{V}_{D_k;d}$ for $d\in D_k$. Additionally, we can derive $l_{\mathfrak{v}}:=|\{\mathrm{deg}_k(\mathfrak{w})|\mathrm{deg}_k(\mathfrak{w})>\mathrm{deg}_k(\mathfrak{v})\}|$. Indeed, if $\mathrm{deg}_k(\mathfrak{v})=k(\bar{n}-2) -  k(k-1) + 2l$, then $l_{\mathfrak{v}}=\left\lfloor\frac{k}{2}\right\rfloor-l$.

Let $L$ be a $\bar{n}-2$ regular graph and $k\in\left\{1,\hdots,\frac{\bar{n}}{2}\right\}$. Then, for $j_d\in\left\{0,\hdots,\left\lfloor\frac{k}{2}\right\rfloor\right\}$ the set, corresponding to the degree $	d = k(\bar{n}-2) -  k(k-1) + 2j_d\in D_k$, satisfies the identity

\begin{equation}
|\mathfrak{V}_{D_k;d}|=\sum_{l=0}^k \binom{\frac{\bar{n}}{2}}{l}\binom{l}{j_d}\binom{\frac{\bar{n}}{2}-l}{k-l-j_d}.
\end{equation}

Fix $j_d\in\left\{0,\hdots,\left\lfloor\frac{k}{2}\right\rfloor\right\}$. We write the vertices $v\in V$ in $\frac{\bar{n}}{2}$ columns and two rows where two vertices connected by an edge in $L^c$ belong to the same column. First, place $l$ particles in the first row. We have $\binom{\frac{\bar{n}}{2}}{l}$ possibilities to do so. To obtain a configuration $\mathfrak{v}$ with $\mathrm{deg}_k(\mathfrak{v})=k(\bar{n}-2) -  k(k-1) + 2j_d$ we have to place precisely $j_d$ of the remaining $k-l$ particles in the second row among the neighbors of the already occupied vertices. We have $\binom{l}{j_d}$ to obtain this. Finally, we distribute the remaining $k-l-j_d$ particles on the remaining $\frac{\bar{n}}{2}-l$ vertices in the second row which is possible in $\binom{\frac{\bar{n}}{2}-l}{k-l-j_d}$ ways. To obtain the total, we sum over all possibilities for $l\in\{0,\hdots,k\}$ which leads to the formula

\begin{equation*}
|\mathfrak{V}_{D_k;d}|=\sum_{l=0}^k \binom{\frac{\bar{n}}{2}}{l}\binom{l}{j_d}\binom{\frac{\bar{n}}{2}-l}{k-l-j_d}.
\end{equation*}

We could demonstrate that for certain special cases of $k$ and $\bar{d}$ it is possible to obtain explicit results on the structure of $\mathfrak{L}_k$ associated to the underlying graph $L$. In particular, we exploited in the cases where we fixed $\bar{d}$, that there is exactly one regular graph with this degree. Deviating from these special cases immediately yields non-uniqueness of the underlying graph $L$. For example, consider the case $\bar{n}=6$ and $\bar{d}=3$. Then, there are already two graphs with vastly different properties, which satisfy this condition. One of them even is bipartite, while the other is not, which has by Proposition <ref> important influences on the structure of the associated $\mathfrak{L}_k$.