# Effective spectral function of vector mesons via lifetime analysis

Renan Hirayama  Helmholtz Forschungsakademie Hessen für FAIR (HFHF), GSI
Helmholtzzentrum für Schwerionenforschung, Campus Frankfurt, Max-von-Laue-Str.
12, 60438 Frankfurt am Main, Germany  Frankfurt Institute for Advanced
Studies, Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany  Jan
Staudenmaier  Frankfurt Institute for Advanced Studies, Ruth-Moufang-Strasse
1, 60438 Frankfurt am Main, Germany  Institut für Theoretische Physik, Goethe
Universität, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany
Department of Physics and Astronomy, Wayne State University, Detroit MI 48201,
United States  Hannah Elfner  Helmholtz Forschungsakademie Hessen für FAIR
(HFHF), GSI Helmholtzzentrum für Schwerionenforschung, Campus Frankfurt, Max-
von-Laue-Str. 12, 60438 Frankfurt am Main, Germany  Frankfurt Institute for
Advanced Studies, Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany
Institut für Theoretische Physik, Goethe Universität, Max-von-Laue-Strasse 1,
60438 Frankfurt am Main, Germany  GSI Helmholtzzentrum für
Schwerionenforschung, Planckstr. 1, 64291 Darmstadt, Germany

###### Abstract

Effective spectral functions of the $\rho$ meson are reconstructed by
considering the lifetimes inside different media using the hadronic transport
SMASH (Simulating Many Accelerated Strongly-interacting Hadrons). Due to
inelastic scatterings, resonance lifetimes are dynamically shortened
(_collisional broadening_), even though the employed approach assumes vacuum
resonance properties. Analyzing the $\rho$ meson lifetimes allows to quantify
an effective broadening of the decay width and spectral function, which is
important in order to distinguish dynamical effects from additional genuine
medium modifications to the spectral functions, indicating e.g. an onset of
chiral symmetry restoration. The broadening of the spectral function in a
thermalized system is shown to be consistent with other theoretical
calculations. The effective $\rho$ meson spectral function is also presented
for the dynamical evolution of heavy-ion collisions, finding a clear
correlation of the broadening to system size, which is explained by an
observed dependence of the width on the local hadron density. Furthermore, the
difference in the results between the thermal system and full collision
dynamics is explored, which may point to non-equilibrium effects.

##  I Introduction

Under extreme thermodynamic conditions the restoration of chiral symmetry is
expected in Quantum Chromodynamics (QCD). The spectral functions of chiral
partners become degenerate as a consequence. The most accessible of these
chiral pairs is composed of $(\rho,a_{1})$, the vector and axial-vector mesons
with the lowest masses. In particular, the $\rho$ meson’s direct dilepton
decay allows direct experimental access to its in-medium modifications of
hadronic properties. Therefore, a large experimental and theoretical effort
has been undertaken to study the behavior of the spectral function of the
$\rho$ meson, in and out of thermal equilibrium.

Experimentally, QCD and particularly the properties of resonances in a hot and
dense medium are studied through nucleus-nucleus collisions. Dileptons are
particularly attractive to study in-medium properties, since they are not
subject to the strong force and only interact electromagnetically with the
surrounding fireball, escaping the medium essentially undisturbed.

The dilepton emission was therefore studied by various experiments, starting
with the CERES/NA45 experiment at CERN-SPS at high beam energies Wessels  _et
al._ (2003), exhibiting an excess attributed to the direct radiation of the
fireball. Afterwards, the NA60 experiment Arnaldi  _et al._ (2006) settled the
debate about the nature of this excess: it was found consistent with a strong
broadening of the $\rho$ spectral function with no apparent mass shift. The
latter being predicated by a handful of theoretical models as the ‘dropping
mass’ scenario along with an increasing width Pisarski (1982); Hatsuda and Lee
(1992); Li  _et al._ (1995); Leupold  _et al._ (1998). More recently, the
HADES experiment at GSI investigated the kinematic regime of low beam energies
for p+p, p+A, and A+A collisions. An excess in the measured electron pair
yield $M_{ee}\sim 0.15$-$0.5$ GeV/c2 Agakichiev  _et al._ (2007); Agakishiev
_et al._ (2012a, b, c, 2011), in agreement with previous results from the DLS
experiment at BEVALAC Porter  _et al._ (1997) was found, again showing an
excess yield connected to strong coupling of the $\rho$ meson to the baryonic
sector Agakichiev  _et al._ (2010); Rapp  _et al._ (2009). Future experiments,
such as CBM at FAIR, will add more high-quality experimental data on dilepton
emission in the high net-baryon density region, by probing the intermediate
energy range Höhne  _et al._ (2014).

Different theoretical techniques are employed to study the vector meson
spectral functions. A well-established approach ensues from hadronic many-body
theory Rapp  _et al._ (1997); Rapp and Wambach (1999); Rapp (2005); Peters
_et al._ (1998), evaluating the $\rho$ propagator including several
modifications to its self-energy. Convolving it with a simple fireball model
resulted in a dilepton yield consistent to SPS data van Hees and Rapp (2008,
2006). One more recent advance comes from the Functional Renormalization Group
approach, a non-perturbative framework capable of taking critical fluctuations
into account Tripolt  _et al._ (2014); Rennecke (2015), and therefore is
suited to study medium modifications in cold and dense nuclear matter close to
the liquid-gas phase transition, as well as the effects of the corresponding
critical endpoint Jung  _et al._ (2017); Tripolt  _et al._ (2021).

Transport approaches allow to connect theoretical approaches for the spectral
function to experimental measurements. They describe the full evolution in a
collision and, therefore, have the advantage of allowing direct access to the
spectral information of particles at all times. In some approaches, the
resonances can propagate _off-shell_ , and their spectral function can
dynamically change, with the constraint of a vacuum behavior in the absence of
surrounding matter. One example is (P)HSD Cassing and Juchem (2000);
Bratkovskaya and Cassing (2008), which computes the spectral function at each
time step via transition amplitudes; another is GiBUU Buss  _et al._ (2012);
Larionov  _et al._ (2020), in which the invariant mass of the particle becomes
an independent variable, determined self-consistently through the evolution.
Other transport approaches, such as UrQMD Bass  _et al._ (1998), deal with
_on-shell_ resonances always assuming vacuum properties. SMASH (Simulating
Many Accelerated Strongly-interacting Hadrons) Weil  _et al._ (2016), the
hadronic transport used for this work, falls in this category as well.

A possibility to supplement direct medium-modifications to the spectral
function to vacuum hadronic transport approaches is the coarse-graining
method, where the local energy and net-baryon densities of an “average event”
are converted into temperature and baryo-chemical potential. They are used as
input parameters for a medium modifications e.g. when calculating
electromagnetic radiation Endres  _et al._ (2016a, b); Staudenmaier  _et al._
(2018). This improves agreement with experimental measurements as shown for
dilepton radiation with the SMASH approach in Staudenmaier  _et al._ (2018). A
description of the resonance dynamics solely based on on-shell propagation is
not sufficient; instead, a mixed approach including the coarse-graining method
leads to a better agreement for large collision systems.

Nevertheless, even with a resonances description based on vacuum properties,
the dynamical evolution of resonances is dramatically different in vacuum and
in medium. Of particular interest for this work is the shortening of resonance
lifetimes by inelastic scatterings i.e. absorption inside the medium, often
referred to as _collisional broadening_. A shortening of the lifetime ($\tau$)
translates to an effective, dynamically-generated increase of the width
($\Gamma$) and subsequently broadening of the spectral function ($\Gamma_{\rm
eff}={1}/{\tau_{\rm eff}}$).

The goal of this work is to investigate this dynamical broadening on the
example of the $\rho$ meson quantitatively with the transport approach SMASH.
The effective width and spectral function is reconstructed by analyzing the
resonance lifetimes. Both equilibrium and non-equilibrium systems are studied
and compared to assess the role of the different dynamics.

This paper is organized as follows: Sec. II describes SMASH, the hadronic
transport approach used in this work, and details its relevant features. Sec.
III defines how the collisional broadening is calculated, with the results for
different scenarios presented in Sec. IV and summarized in Sec. V.

##  II Model Description

For the investigation of the $\rho$ meson in this work, the hadronic transport
approach SMASH-2.1 Weil  _et al._ (2016); Oliinychenko  _et al._ (2021) is
employed. It allows to access the full phase space information at all times.
Particles can be followed individually and their lifetimes and interactions
are directly accessible. Hadrons evolve in spacetime according to an effective
solution of the relativistic Boltzmann equation. Particle species, their pole
masses $M_{0}$, and corresponding decay widths $\Gamma_{0}$ are taken from the
Particle Data Group Zyla  _et al._ (2020) up to $M_{0}\sim 2.3\ \mathrm{GeV}$.
Hadrons with $\Gamma_{0}\leq 10\ \mathrm{keV}$ are considered stable,
otherwise they are regarded as resonances with a non-singular vacuum spectral
function, and can decay with probability given by the mass-dependent decay
width. Only two-body decays and scatterings are included in the calculations
with a geometrical collision criterion in order to maintain detailed balance.
Resonances go through a $1\to 2$ decay, or are absorbed, either in a $2\to 1$
resonance formation or a $2\to 2$ inelastic collision Weil  _et al._ (2016).

Equilibrium properties are studied in infinite matter calculations. This is
achieved by using a finite box with periodic boundary conditions. The initial
multiplicities are sampled from a Poisson distribution, simulating a grand-
canonical ensemble. Particle momenta are sampled from a Maxwell-Boltzmann
distribution with the given $(T,\mu_{B})$, which approximates thermal and
chemical equilibrium. For nucleus-nucleus collisions, the nuclei travel
towards each other along the longitudinal axis with a given kinetic energy,
and offset in the transverse axis by a given impact parameter. For the initial
condition, the positions of nucleons in each nucleus are sampled according to
the Woods-Saxon distribution, without Fermi momentum. Densities in this work
are computed in the Eckart rest frame with a Gaussian smearing. For the hadron
density, each particle in SMASH has the same unitary weight.

###  II.1 Decay widths

The hadronic decay widths in SMASH follow the treatment of Manley and Saleski
(1992), where the two-body decay $R\to ab$ has a mass-dependent width of

$\Gamma^{\mathrm{dec}}_{R\to ab}(m)=\Gamma_{R\to
ab}^{0}\frac{\rho_{ab}(m)}{\rho_{ab}(M_{0})},$ (1)

where $m$ is the off-shell mass, $M_{0}$ and $\Gamma_{R\to ab}^{0}$ are the
pole mass and corresponding width, and $\rho(m)$ is a parametrization,
described in full detail in (Weil  _et al._ , 2016).

The proper lifetime of a resonance with mass $m$ is defined as

$\tau=\frac{1}{\Gamma^{\mathrm{dec}}(m)},$ (2)

where $\Gamma^{\mathrm{dec}}(m)$ is the _total_ decay width, computed as the
sum of the partial widths (1) over all decay channels $\\{ab\\}$. The
probability for the resonance to decay in a sufficiently small time interval
in its rest frame is

$P(\mathrm{decay\ in\ }\Delta t)=\frac{\Delta
t}{\tau}=\Gamma^{\mathrm{dec}}(m)\Delta t.$ (3)

SMASH uses this probability at each timestep to decay resonances. When it
happens, a decay channel $R\to X$ is randomly chosen from the list of possible
processes, with a mass-dependent branching ratio $\Gamma^{\mathrm{dec}}_{R\to
X}(m)/\Gamma^{\mathrm{dec}}(m)$.

###  II.2 Spectral function

The spectral function of a resonance relates to the imaginary part of its
propagator, and hence carries information about the mass probability
distribution. In general, it can depend on temperature and density; however,
such in-medium modifications are currently neglected in SMASH, and vacuum
properties are assumed for all particles. The spectral function of a resonance
is given by the relativistic Breit-Wigner distribution

$\mathcal{A}(m)=\frac{2\mathcal{N}}{\pi}\frac{m^{2}\Gamma^{\mathrm{dec}}(m)}{(m^{2}-M_{0}^{2})^{2}+m^{2}\Gamma^{\mathrm{dec}}(m)^{2}},$
(4)

where $\mathcal{N}$ is a normalization factor. When a resonance is formed,
SMASH samples its mass using (4) within the available phase-space. The mass
distribution in a simple gas in equilibrium, for instance, amounts to folding
(4) with a thermal distribution:

$\frac{1}{N}\derivative{N}{m}\
(m;T,\mu)\propto(mT)^{3/2}e^{(\mu-m)/T}\mathcal{A}(m).$ (5)

Depending on kinematic limitations, different channels for the production of a
$\rho$ will be available, which can lead to interesting non-thermal structures
in the mass distribution Schumacher  _et al._ (2006); Vogel and Bleicher
(2006).

##  III Collisional broadening

In a hadronic medium, absorption of particles decreases the average lifetime
of resonances compared to that in vacuum. Such a decrease can be considered as
an effective increase of the total decay width, consequently widening the
spectral function. This effect is known as _collisional broadening_. In a
medium, absorptions are the main mechanism that determines resonance
lifetimes.

The effective total width is computed by extracting the average lifetime of a
collection of particles, and inverting (2) to define

$\Gamma^{\mathrm{eff}}=\frac{1}{\left\langle\tau\right\rangle}=\left\langle\frac{t_{f}-t_{i}}{\gamma}\right\rangle^{-1},$
(6)

where $\gamma$ is the Lorentz factor of the resonance with respect to the
computational system, computed with the momentum of the resonance, which is
taken from the interaction history provided by SMASH along with the initial
and final times $t_{i,f}$. The average in (6) can be computed differentially,
for instance depending on the invariant mass or the local hadron density. The
additional contribution to the width, the collisional width, is defined as
$\Gamma^{\mathrm{col}}=\Gamma^{\mathrm{eff}}-\Gamma^{\mathrm{dec}}$.

In order to study the dynamical effects on the spectral function, the mass-
dependent $\Gamma^{\mathrm{eff}}(m)$ replaces the regular vacuum decay width
$\Gamma^{\mathrm{dec}}$ in (4). As particles have a shorter average lifetime
in the medium due to absorption processes, $\Gamma^{\mathrm{eff}}$ is larger
than $\Gamma^{\mathrm{dec}}$. The obtained “dynamic” spectral function
$\mathcal{A}^{\mathrm{dyn}}(m)$ is therefore broader than ${A}(m)$. Since
resonances that are annihilated in absorptions do not decay, this quantity can
be thought of as the spectral function of $\rho$ mesons which effectively
contribute to the dilepton yields in Staudenmaier  _et al._ (2018).

In order to allow for a comparison between different systems, the spectral
function must be properly normalized. This is not trivial since higher masses
are increasingly rare, so the support of $\mathcal{A}^{\mathrm{dyn}}(m)$ is
not infinite as in the vacuum. Each dynamic spectral function is normalized to
the integral of the vacuum Breit-Wigner in the available support. This
introduces a small scaling error, which has no impact on the analyses below.

##  IV Results

###  IV.1 Thermal systems

First, the collisional broadening of $\rho$ mesons is computed inside a box
with an equilibrated hadron gas. This allows for an assessment of the
thermodynamic behavior of the effective width, as well as for the comparison
of $\mathcal{A}^{\mathrm{dyn}}$ to well-established model calculations of full
in-medium modifications Rapp and Wambach (1999); van Hees and Rapp (2008). The
box is set to initialize as explained in Sec. II at different temperatures
$T\in\\{120,150,180\\}$ MeV, each with three baryochemical potential values
$\mu_{B}\in\\{0,330,450\\}$ MeV. Results are only considered after $t=10^{4}$
fm, which was checked to guarantee equilibration.

Figure 1: Probability distribution of $\rho$ meson masses in thermal
equilibrium, at $\mu_{B}=330\ \mathrm{MeV}$ and $T\in\\{120,150,180\\}\
\mathrm{MeV}.$ SMASH production shown in the solid histogram, and the thermal
folding (5) in dashed lines. The vacuum Breit-Wigner (4) is shown in black
dash-dotted lines.

The mass distribution of $\rho$ mesons is shown to closely match the folding
(5) in Fig. 1, as the simulated matter indeed corresponds to a thermalized
hadron gas. The thermal weight exponentially favors the creation of smaller
masses in comparison to the vacuum (4), and there is not much difference
between the production at the selected temperatures. The value of
baryochemical potential does not alter the distribution – at least in this
region of the phase diagram –, therefore only $\mu_{B}=330$ MeV is displayed.
This is consistent with the folding, in which the dependence on $\mu_{B}$ is
cancelled by the normalization. In simpler systems exact matches between the
analytic expectation and SMASH were found, and the slight deviations can be
attributed to additional production channels in the full hadron gas via the
$N^{*}(1520)$ resonance.

Figure 2: Effective width of $\rho$ mesons in thermal equilibrium. Error bands
are statistical.

Applying (6) to the $\rho$ mesons separated in bins of mass, the mass-
dependent effective width, shown in Fig. 2, is extracted.
$\Gamma^{\mathrm{eff}}(m)$ depends strongly on the thermodynamic conditions of
the system, in contrast to the mass distribution of Fig. 1. Lower masses are
more affected by changes in the thermodynamic parameters, since the cross-
section for $2\to 1$ and $2\to 2$ processes decreases with the masses of
incoming particles in this energy range Weil  _et al._ (2016). Hence, these
heavier $\rho$ mesons are less likely to be absorbed. The baryochemical
potential is only relevant below the pole mass of $M_{0}=776$ MeV; an increase
in $\mu_{B}$ favors the creation of baryons, suggesting that their coupling to
the $\rho$ dominates the low-mass region.

Figure 3: (Upper) Dynamic spectral function of $\rho$ mesons in thermal
equilibrium. (Lower) Momentum-integrated in-medium spectral function for
$\mu_{B}=330$ MeV Rapp and Wambach (1999).

Figure 3 shows a comparison between the dynamic spectral function from SMASH
and the modifications to the $\rho$ meson propagator from a full in-medium
model Rapp and Wambach (1999). Qualitatively, the behavior of
$\mathcal{A}^{\mathrm{dyn}}(m)$ is similar to the full in-medium spectral
function: it is broadened with increasing temperature, and the peak mass
shifts slightly. Notably, the baryochemical potential makes almost no
difference in the dynamic spectral function.

However, there are quantitative differences present. The high-mass tail is
less broadened in SMASH, and the opposite happens in the low-mass tail. This
is likely due to the “tree-level” character of hadronic transport. Quantum
corrections – loops – are taken into account in the matching of elementary
cross-sections to experimental data; that is, only vacuum corrections are
correctly described. The medium effects on these diagrams are not present,
unlike in the in-medium model description, which modifies the propagator self-
consistently including interference terms.

###  IV.2 Collision systems

Next, the emergence of collisional broadening is studied in the off-
equilibrium matter created by low-energy nuclear collisions following the
selection of the HADES collaboration. Experiments assess the medium effects in
these systems, by analyzing the excess dilepton yields in comparison to a
hadronic cocktail Arnaldi  _et al._ (2006). The present study is restricted to
low beam energies, where the evolution is appropriately described by the
kinetic transport approach. Several nuclear systems are considered to obtain
insight about how the broadening depends on system size, centrality, and beam
energy.

Figure 4: Nominal spectral function of $\rho$ mesons in different collision
systems.

The mass distribution of $\rho$ mesons, shown in Fig. 4, for nuclear
collisions at HADES beam energies is not following a thermal folding, because
the system is very far from equilibrium. There is an enhanced production peak
at $\sim 0.5\ \mathrm{GeV}$, stemming from the decay $N^{*}(1520)\to N+\rho$,
since the mass of this resonance does not allow to produce a pole-mass $\rho$
Schumacher  _et al._ (2006); Vogel and Bleicher (2006). This is most apparent
in the C+C collisions at $E_{\mathrm{kin}}=1$ AGeV, where the low beam energy
changes the preferred production channel. This effect is purely kinematic,
with this peak populated by low-momentum $\rho$ mesons Reichert and Bleicher
(2022), since the production of resonances does not take lifetimes nor in-
medium effects into account.

Figure 5: Effective width of $\rho$ mesons in different collision systems.
Error bands are statistical.

The effective width shown in Figure 5 reveals system size as a dominant
factor. In more central collision the dynamic width is larger, as evidenced by
the four centrality classes of Au+Au collisions. Similarly, heavier nuclei
reveal a stronger collisional broadening. The p+p collision follows the vacuum
decay, since there is essentially no medium with which the $\rho$ mesons
interact formed.

However, system size is not the only factor, given that a Au+Au collision at
30-40% broadens as much as a central Ag+Ag, even though it has far fewer
participants on average111Using the Woods-Saxon model with an inelastic NN-
cross section of $\sigma_{NN}=30$ mb, a 30-40% Au+Au collision has $\sim 98$
participants on average, whereas a central Ag+Ag collision has $\sim 168$..
The beam energy is playing a role as well: the faster velocity of the ions in
the larger system lets the medium dissipate faster, leading to fewer binary
collisions overall. This is also seen in C+C collisions, the width at
$E_{\mathrm{kin}}=1$ AGeV is larger than at $2$ AGeV.

Figure 6: Dynamic spectral function of $\rho$ mesons in different collision
systems.

Unlike in the thermal system, the broadening below $m\sim 0.5$ GeV always
decreases, because most low-mass resonances are created in the later stages of
the collision, when the medium is already dilute. Furthermore, the peak mass
of the dynamic spectral function, shown in Fig. 6, does not shift to a great
extent, as these nuclear systems do not reach temperatures comparable to the
selected box values of Fig. 3.

#### Time evolution

It is also interesting to observe how the collisional broadening evolves with
the expansion of the fireball. This is seen by computing (6) as a function of
the time $t_{i}$ at which the resonance is created (in the computational
frame). The corresponding width is shown in Fig. 7, normalized by the width at
pole.

Figure 7: Time evolution of the effective width of $\rho$ mesons in different
collision systems.

One way to understand this evolution is by probing the lifetime of the medium
itself. This is done in Fig. 8, where $n_{0}=0.16~{}\mathrm{fm}^{-3}$ is the
nuclear ground state density, by using the average hadron density
$n_{\mathrm{h}}$ as a proxy. The densities in SMASH are computed at
interactions, so the hadron density is chosen to be calculated at $t_{i}$.

Figure 8: Time evolution of the average hadron density at the interaction in
which a $\rho$ meson is removed from the system.

During the first few fm of the collision, the effective width grows as the
nuclei traverse each other, until the number of binary collisions is maximal.
This occurs when $n_{\mathrm{h}}$ is largest and so the position of this
maximum changes with nuclear mass and beam energy.

As the medium expands, it becomes more dilute and consequently the effective
width decreases. In light of this, the difference in the effective widths of
30-40% Au+Au and central Ag+Ag systems – that have a similar mass-dependent
width in Figure 5 – becomes evident. Having more participating nucleons, the
initial broadening in the latter is larger, but it decreases faster than in
the former.

At later times, particles are essentially travelling in vacuum, hence
$\Gamma^{\mathrm{eff}}\approx\Gamma^{\mathrm{dec}}$. Since higher masses have
higher vacuum decay widths (as per Figs. 2 and 5), and $\left\langle
m\right\rangle<M_{0}$ as seen in Fig. 9, the widths in all systems fall to
small values less than $\Gamma_{0}=0.149$ GeV. This also explains why, for
instance, the width in a C+C collision falls below the one in p+p at this
stage.

Figure 9: Time evolution of the average $\rho$ meson mass in different nuclear
collisions.

#### Density dependence

Figures 7 and 8 in the previous subsection suggest a monotonic dependence of
the effective width on the local hadron density, as seen in Figure 10. Here,
the density is computed at the end-interaction (at time $t_{f}$) of the
resonance, so as to probe the medium conditions where the $\rho$ is absorbed
or decays. This is an approximate way of probing
$\Gamma^{\mathrm{eff}}(n_{\mathrm{h}})$, as it does not consider how the
density changes during propagation.

Figure 10: Density dependence of the effective width of $\rho$ mesons in
different collision systems.

A near universal relation is recognizable, with most systems following similar
curves, while in Figs. 5 and 7 the widths of the $\rho$ mesons are distinct in
different systems. This universality is reminiscent of the semiclassical
calculation $\Gamma^{\mathrm{col}}\sim\gamma n_{N}\left\langle
v\sigma_{VN}^{\mathrm{tot}}\right\rangle$, used in models in which the mass is
not constant and the collisional width is proportional to the local nucleon
density Bratkovskaya and Cassing (2008); Larionov  _et al._ (2020).

A deviation to this appears in the p+p and C+C systems, when the density is
high. This is because the density calculation starts to break down for very
dilute system and an influence of the reaction partners in the specific binary
interaction cannot be ruled out as there are never many particles close to the
interaction point, potentially being the reason for the spurious high
densities seen for such small systems.

###  IV.3 Non-equilibrium effects

Having quantified the collisional broadening in these two different scenarios,
it is tempting to ask: how does the effective width in a non-equilibrated
collision system compare to the thermalized value of a box?

Figure 11: Collisional width of a central Au+Au collision at
$E_{\mathrm{kin}}=1.23$ GeV, in the full phase and restricted to a region of
constant $(T,\mu_{B})$; in a box initialized with these thermodynamic
parameters.

There is a spacetime region of a heavy-ion collision in which the temperature
and chemical potential222These are computed by assuming the HRG equation of
state with local densities of energy and baryon number. are nearly constant
Staudenmaier  _et al._ (2018). This allows to compare the effective width in a
collision system with a box at the same thermodynamic conditions (shown in
Figure 11). In the central cell of a Au+Au collision, the system remains
stable in the same thermodynamic state of around
$(T,\mu_{B})=(80,900)\mathrm{MeV}$ early in the collision, between $7\leq
t\leq 15\ \mathrm{fm}$ Staudenmaier  _et al._ (2018). The system is much
denser on average in this spacetime region, so the width is correspondingly
enhanced. Since this restriction removes the late-stage resonances, the
collisional width at the hadronic threshold is no longer approximately zero. A
thermalized hadron gas at the same $(T,\mu_{B})$ manifests a similar
broadening for masses above $0.5$ GeV. Below this value, the collision system
displays a larger broadening, meaning that the non-equilibrium character in a
heavy-ion system results in a suppresion of lifetimes of low-mass resonances.

The suppression is explained by considering the hadronic content of each
scenario in Fig. 12. The invariant yield is computed by considering the
average number of the particles over the relevant region of the phase space,
and normalizing by the size of that region. The species displayed are first
ordered by multiplicity in the central cell of the Au+Au collision until
$N^{*}(1535)$, then by the multiplicity of remaining species in the box.

During the interval of constant thermodynamic conditions, the central cell of
the nuclear system is composed mostly of baryons, being dominated by nuclear
and $\Delta$-baryon resonances. As discussed in the introduction Agakichiev
_et al._ (2010); Rapp  _et al._ (2009); Salabura and Stroth (2021), these
couple strongly with low-mass $\rho$ mesons, enhancing the effective width.
These particles are not as abundant in the thermal gas. Instead, the energy is
distributed in the form of lighter hyperons, since the multiplicity is
$\propto e^{-m/T}$ in an equilibrated system. Such particles do not couple as
much with the $\rho$, and hence do not increase the width in the thermal box.

##  V Conclusion

In this work, the collisional broadening of $\rho$ mesons was investigated and
quantified by computing their effective width via lifetime analysis. The
employed transport approach, SMASH Weil  _et al._ (2016), relies on vacuum
properties of hadrons, so that the mass distribution is given by a vacuum
Breit-Wigner function adjusted to the kinematically available energy. The
“dynamic” spectral function is computed using the effective width, as the
collisional broadening emerges from absorption processes as part of the
evolution of the hadronic medium.

First, an infinite hadron gas in equilibrium has been calculated for different
temperatures and baryochemical potentials. The resulting spectral functions is
compared to full in-medium model calculations Rapp and Wambach (1999), which
take into account modifications to self-energies and higher order interactions
and a qualitatively similar broadening is observed, suggesting that a
significant contribution to medium modifications of the $\rho$ meson comes
from hadronic interactions irrespective of chiral symmetry restoration, which
is not directly present in SMASH. Numerically, however, the shapes in Fig. 3
are different. This evidences that the collisional broadening alone is not
sufficient to reproduce the experimental dilepton yield in heavy-ion
collisions, so other methods using full in-medium modifications need to be
applied, such as coarse-graining Staudenmaier  _et al._ (2018).

Furthermore, the emergence of collisional broadening is studied in non-
equilibrium systems created by nuclear collisions (pp, CC, ArKCl, AgAg, AuAu)
at different beam energies and centrality classes. The effective width
exhibits a clear dependence on system size, as a larger medium enhances the
broadening. It also reveals that larger beam energies leads to smaller widths,
which is understood through the time evolution of the system. These
observations are caused by an universal dependence on the local hadron
density.

Lastly, the two scenarios are compared in order to assess non-equilibrium
effects. This has been achieved by simulating a box with similar thermodynamic
conditions to those present in a spacetime region of Au+Au collisions. Above
$m\sim 0.5$ GeV the observed broadening is similar, while below it the
collision system exhibited an enhanced width, which is explained by the
different hadronic composition of the two scenarios.

The lifetime analysis used throughout this work can be used to understand how
inelastic scatterings of vector mesons affect the decay width dynamically and
quantify this effect in a transport approach in contrast to genuine in-medium
modifications. In the future, it will be interesting to investigate if
assumptions about the resonance properties, such as the exact way to calculate
the decay probability also affect the effective width and therefore
potentially the emission of dileptons.

## Acknowledgments

This work was supported by the Helmholtz Forschungsakademie Hessen für FAIR
(HFHF) and in part by the National Science Foundation (NSF) within the
framework of the JETSCAPE collaboration, under grant numbers ACI-1550228 and
OAC-2004571. The authors also acknowledge the support by the State of Hesse
within the Research Cluster ELEMENTS (Project ID 500/10.006), and by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project
number 315477589 – TRR 211. Computational resources have been provided by the
GreenCube at GSI.

Figure 12: Multiplicity of each species in the thermodynamically stable region
of a Au+Au collision, and in a box under the same $(T,\mu_{B})$.

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# Endomorphisms and derivations of the measure algebra of commutative
hypergroups

Żywilla Fechner, Eszter Gselmann and László Székelyhidi

###### Abstract

Endomorphisms of the measure algebra of commutative hypergroups are
investigated. We focus on derivations and higher order derivations which are
closely related to moment function sequences of higher rank. We describe the
exact connection between those higher order derivations which are
endomorphisms of the measure algebra if it is considered as a module over the
ring of continuous functions.

##  1 Introduction

Moment function sequences on hypergroups have been defined in [2] (see also
[7, 8, 9, 10]). In [3] and [4] moment function sequences of higher rank have
been defined and investigated. In this paper we present results which explain
a connection between moment function sequences of higher rank and higher order
derivations of the measure algebra of commutative hypergroups. For the
discussion on derivations on abstract structures we refer to [5] or [6] It
turns out that moment function sequences generate higher order derivations,
which are not just linear operators of the measure algebra, but they are
endomorphisms of the measure algebra when considered as a module over the ring
of continuous functions. And conversely: every higher order derivation of this
property generates a moment function sequence of higher rank.

##  2 Measure algebra

Let $X$ be a commutative hypergroup. The space of all complex valued
continuous functions on $X$ will be denoted by $\mathcal{C}(X)$. Equipped with
the pointwise linear operations and with the topology of uniform convergence
on compact sets it is a locally convex topological vector space. We recall
that its topological dual $\mathcal{C}^{*}(X)$ can be identified with the
measure algebra of $X$: this is the space $\mathcal{M}_{c}(X)$ of all
compactly supported complex Borel measures on $X$ equipped with the linear
operations (addition and multiplication by complex numbers) and with the
convolution of measures defined by

$\int_{X}f\,d(\mu*\nu)=\int_{X}\int_{X}f(x*y)\,d\mu(x)\,d\nu(y)$

for each function $f$ in $\mathcal{C}(X)$. If we equip $\mathcal{M}_{c}(X)$
with weak*-topology and then it is a locally convex topological space, in
fact, a topological algebra. The topological dual of $\mathcal{M}_{c}(X)$ can
be identified with $\mathcal{C}(X)$: for each weak*-continuous linear
functional $\Lambda:\mathcal{M}_{c}(X)\to\mathbb{C}$ there exists a unique
continuous function $\varphi$ in $\mathcal{C}(X)$ such that

$\Lambda(\mu)=\int_{X}\varphi\,d\mu$

holds whenever $\mu$ is in $\mathcal{M}_{c}(X)$ (see [1], Section 3.14). We
can write $\varphi(\mu)=\Lambda(\mu)$. In particular, we have
$\varphi(\delta_{x})=\varphi(x)$ for each $x$ in $X$. For instance, the
multiplicative functionals of the measure algebra arise in this manner exactly
from the exponentials of $X$.

Besides the linear structure of the measure algebra we can consider it as a
module over the ring $\mathcal{C}(X)$ of continuous functions on $X$: the
action of $\varphi$ in $\mathcal{C}(X)$ on $\mathcal{M}_{c}(X)$ is defined by
the multiplication of the measure $\mu$ by $\varphi$ defined as

$\langle\varphi\cdot\mu,f\rangle=\int_{X}f\varphi\,d\mu.$

Clearly, $\varphi\cdot\mu$ is in $\mathcal{C}(X)$. We simply write
$\varphi\mu$ for $\varphi\cdot\mu$.

If we say that a map $F:\mathcal{M}_{c}(X)\to\mathcal{M}_{c}(X)$ is a module
homomorphism, then we mean that it is an endomorphism of $\mathcal{M}_{c}(X)$
as a $\mathcal{C}(X)$-module. In particular, we have $F(\varphi\mu)=\varphi
F(\mu)$ for each $\mu$ in $\mathcal{M}_{c}(X)$ and $\varphi$ in
$\mathcal{C}(X)$. Obviously, this implies homogeneity with respect to
multiplication with complex numbers, which can be identified with constant
functions. A module homomorphism is called continuous, if it is continuous
with respect to the weak*-topology. A multiplicative module homomorphism is a
module homomorphism, which is also a homomorphism of the algebra structure of
$\mathcal{M}_{c}(X)$, that is, it preserves convolution.

A derivation of the measure algebra usually defined as a continuous linear
operator $D:\mathcal{M}_{c}(X)\to\mathcal{M}_{c}(X)$ satisfying the additional
property

$D(\mu*\nu)=D\mu*\nu+\mu*D\nu$ (1)

for each $\mu,\nu$ in $\mathcal{M}_{c}(X)$. In this paper, however, we shall
consider a modified version of this concept. Namely, instead of linearity we
require that $D$ is a continuous module homomorphism of $\mathcal{M}_{c}(X)$
as a module over the ring of continuous functions $\mathcal{C}(X)$. In other
words, besides (1) we assume that

$D(\mu+\nu)=D\mu+D\nu$ (2)

and

$D(\varphi\mu)=\varphi D\mu$ (3)

holds for each $\mu,\nu$ in $\mathcal{M}_{c}(X)$ and $\varphi$ in
$\mathcal{C}(X)$. The latter two properties express the fact that $D$ is a
module homomorphism.

Suppose that $r$ is a positive integer. We say that tha family
$(D_{\alpha})_{\alpha\in\mathbb{N}^{r}}$ of continuous module homomorphisms of
$\mathcal{M}_{c}(X)$ is a higher order derivation of rank $r$ if for each
$\alpha$ in $\mathbb{N}^{r}$ we have

$D_{\alpha}(\mu*\nu)=\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}D_{\beta}\mu*D_{\alpha-\beta}\nu$
(4)

whenever $\mu,\nu$ is in $\mathcal{M}_{c}(X)$. The order of $D_{\alpha}$ is
defined as $|\alpha|$. In particular, $D_{0}$ satisfies

$D_{0}(\mu*\nu)=D_{0}\mu*D_{0}\nu,$ (5)

that is, $D_{0}$ is an algebra homomorphism and module homomorphism
simultaneously. We call such a mapping a multiplicative module homomorphism.
Moreover, for each $\alpha$ with $|\alpha|=1$ we have

$D_{\alpha}(\mu*\nu)=D_{0}\mu*D_{\alpha}\nu+D_{\alpha}\mu*D_{0}\nu$ (6)

holds. In the case $D_{0}=id$, the identity homomorphism, then $D_{\alpha}$ is
a derivation. In general, a continuous module homomorphism $D$ satisfying (6)
in place of $D_{\alpha}$ with a given $D_{0}$ which is a multiplicative
algebra homomorphism, is called a $D_{0}$-derivation. Thus ordinary
derivations are exactly the $id$-derivations.

##  3 Module homomorphisms

We begin with the characterization of continuous module homomorphism on
$\mathcal{M}_{c}(X)$.

######  Theorem 1.

Let $X$ be a commutative hypergroup. Every continuous module homomorphism $F$
on $\mathcal{M}_{c}(X)$ has the form

$\langle F(\mu),f\rangle=\int_{X}f\cdot\varphi\,d\mu$ (7)

with a unique function $\varphi$ in $\mathcal{C}(X)$. Conversely, each mapping
of the form given by (7) is a module homomorphism of $\mathcal{M}_{c}(X)$.

###### Proof.

Given a function $\varphi$ in $\mathcal{C}(X)$ we define the mapping
$F:\mathcal{M}_{c}(X)\to\mathcal{M}_{c}(X)$ by

$\langle
F(\mu),f\rangle=\int_{X}f\cdot\varphi\,d\mu=\langle\varphi\mu,f\rangle$

for each $\mu$ in $\mathcal{M}_{c}(X)$ and $f$ in $\mathcal{C}(X)$. Clearly,
$F$ is additive and weak*-continuous. Moreover, for each $\psi$ in
$\mathcal{C}(X)$ we have

$\langle
F(\psi\mu),f\rangle=\int_{X}f\cdot\varphi\,d(\psi\mu)=\int_{X}f\varphi\psi\,d\mu=\int_{X}f\psi\cdot\varphi\,d\mu=$
$\int_{X}f\psi\,d(F(\mu))=\int_{X}f\,d(\psi F(\mu))=\langle\psi
F(\mu),f\rangle,$

hence

$F(\psi\mu)=\psi F(\mu),$

which proves that $F$ is a continuous module homomorphism.

For the converse, let $F:\mathcal{M}_{c}(X)\to\mathcal{M}_{c}(X)$ be a
continuous module homomorphism. We define
$\Lambda:\mathcal{M}_{c}(X)\to\mathbb{C}$ by

$\Lambda(\mu)=\langle F(\mu),1\rangle$

for each $\mu$ in $\mathcal{M}_{c}(X)$. Clearly, $\Lambda$ is a linear mapping
from $\mathcal{M}_{c}(X)$ into $\mathbb{C}$. We show that $\Lambda$ is also
weak*-continuous. Indeed, if $(\mu_{i})_{i\in I}$ is a net converging to the
zero measure in the weak*-topology, then the net $(F(\mu_{i}))_{i\in I}$
converges to the zero measure n the weak*-topology as well, by the
weak*-continuity of $F$. It follows that the net $(\langle
F(\mu_{i}),1\rangle)_{i\in I}$ converges to zero – in other words, the net
$(\Lambda(\mu_{i}))_{i\in I}$ converges to zero, hence $\Lambda$ is
weak*-continuous. We infer that $\Lambda$ is a linear functional of the space
$\mathcal{M}_{c}(X)$. It is known, that $\Lambda$ arises from a function in
$\mathcal{C}(X)$ – more exactly, there exists a unique continuous function
$\varphi:X\to\mathbb{C}$ such that

$\Lambda(\mu)=\int_{X}\varphi\,d\mu$

holds for each $\mu$ in $\mathcal{M}_{c}(X)$. Then we conclude for each $f$ in
$\mathcal{C}(X)$:

$\int_{X}f\,d(\varphi\mu)=\int_{X}f\varphi\,d\mu=\int_{X}\varphi\,d(f\mu)=\Lambda(f\mu)=\langle
F(f\mu),1\rangle=$ $\langle
fF(\mu),1\rangle=\int_{X}1\,d(fF(\mu))=\int_{X}f\,dF(\mu),$

that is $F(\mu)=\varphi\mu$, which is equation (7).

∎

We infer that the general form of a continuous endomorphism of the measure
algebra, as a $\mathcal{C}(X)$-module, is exactly multiplication by a
continuous function. The following theorem describes the continuous
multiplicative module homomorphisms of the measure algebra.

######  Theorem 2.

Let $X$ be a commutative hypergroup. Every continuous nonzero multiplicative
module homomorphism $F$ on $\mathcal{M}_{c}(X)$ has the form

$\langle F(\mu),f\rangle=\int_{X}f\cdot m\,d\mu$ (8)

for some exponential $m$ in $\mathcal{C}(X)$. The exponential $m$ is uniquely
determined by $F$. Conversely, each mapping of the form given by (8) is a
continuous multiplicative module homomorphism of $\mathcal{M}_{c}(X)$.

###### Proof.

The mapping $F$ defined by (8) is a continuous module homomorphism of
$\mathcal{M}_{c}(X)$, by the previous theorem. We show that if $m$ is an
exponential, the it is also multiplicative. Indeed, for each $\mu,\nu$ in
$\mathcal{M}_{c}(X)$ and $\varphi$ in $\mathcal{C}(X)$, we have

$\langle
F(\mu*\nu),\varphi\rangle=\int_{X}m\varphi\,d(\mu*\nu)=\int_{X}\int_{X}m(x*y)\varphi(x*y)\,d\mu(x)\,d\nu(y)=$
$\int_{X}\int_{X}\varphi(x*y)\,d(m\cdot\mu)(x)\,d(m\cdot\nu)(y)=\int_{X}\int_{X}\varphi(x*y)\,dF(\mu)(x)\,dF(\nu)(y)=$
$\int_{X}\varphi\,d(F(\mu)*F(\nu))=\langle F(\mu)*F(\nu),\varphi\rangle.$

For the converse, let $F$ be a nonzero continuous multiplicative module
homomorphism of $\mathcal{M}_{c}(X)$. Then, by the previous theorem, it has
the form

$F(\mu)=\varphi\cdot\mu$

with some continuous function $\varphi:X\to\mathbb{C}$. For $x,y$ in $X$ we
have, by the multiplicativity of $F$:

$\varphi(x*y)=\int_{X}\varphi(z)\,d(\delta_{x}*\delta_{y})(z)=\langle
F(\delta_{x}*\delta_{y}),1\rangle=\langle
F(\delta_{x})*F(\delta_{y}),1\rangle=$
$\int_{X}\,d(F(\delta_{x})*F(\delta_{y}))=\int_{X}\,dF(\delta_{x})\cdot\int_{X}\,dF(\delta_{y})=$
$\int_{X}\varphi\,d\delta_{x}\cdot\int_{X}f\,d\delta_{y}=\varphi(x)\varphi(y).$

It follows $\varphi(o)=\varphi(o)\varphi(o)$. If $\varphi(o)=0$, then
$\varphi=0$, and $F=0$, which is excluded. Hence $\varphi(o)=1$, and $\varphi$
is an exponential. Our theorem is proved. ∎

##  4 Derivations

In this section we describe the connection between higher order derivation and
generalized moment function sequences.

######  Theorem 3.

Let $X$ be a commutative hypergroup and $r$ a positive integer. The family
$(D_{\alpha})_{\alpha\in\mathbb{N}^{r}}$ of self-mappings on
$\mathcal{M}_{c}(X)$ is a continuous higher order derivation of order $r$ if
and only if there exists a generalized moment function sequence
$(\varphi_{\alpha})_{\alpha\in\mathbb{N}^{r}}$ of rank $r$ such that

$\langle D_{\alpha}\mu,f\rangle=\int_{X}f\cdot\varphi_{\alpha}\,d\mu$ (9)

holds for each $\mu$ in $\mathcal{M}_{c}(X)$, $f$ in $\mathcal{C}(X)$ and
$\alpha$ in $\mathbb{N}^{r}$.

###### Proof.

Let $(\varphi_{\alpha})_{\alpha\in\mathbb{N}^{r}}$ be a generalized moment
function sequence of rank $r$ on $X$, and we define $D_{\alpha}\mu$ for each
$\mu$ in $\mathcal{M}_{c}(X)$, $f$ in $\mathcal{C}(X)$ and $\alpha$ in
$\mathbb{N}^{r}$ by equation (9). Obviously,
$D_{\alpha}:\mathcal{M}_{c}(X)\to\mathcal{M}_{c}(X)$ is a linear mapping for
each $\alpha$ in $\mathbb{N}^{r}$. The weak*-continuity of $D_{\alpha}$
follows easily from the continuity of the functions $\varphi_{\alpha}$.

From Theorem 1, it follows immediately, that $D_{\alpha}$ is a module
homomorphism for each $\alpha$ in $\mathbb{N}^{r}$. Now we show that
$(D_{\alpha})_{\alpha\in\mathbb{N}^{r}}$ is a continuous higher order
derivation of order $r$. Given $\alpha$ in $\mathbb{N}^{r}$, $\mu,\nu$ in
$\mathcal{M}_{c}(X)$ and $f$ in $\mathcal{C}(X)$ we have

$\langle
D_{\alpha}(\mu*\nu),f\rangle=\int_{X}f\cdot\varphi_{\alpha}\,d(\mu*\nu)=\int_{X}f(x*y)\cdot\varphi_{\alpha}(x*y)\,d\mu(x)\,d\nu(y)=$
$\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\int_{X}\int_{X}f(x*y)\varphi_{\beta}(x)\varphi_{\alpha-\beta}(y)\,d\mu(x)\,d\nu(y)=$
$\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\int_{X}\int_{X}f(x*y)\,d(\varphi_{\beta}\mu)(x)\,d(\varphi_{\alpha-\beta}\nu)(y)=$
$\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\int_{X}\int_{X}f(x*y)\,dD_{\beta}\mu(x)\,dD_{\alpha-\beta}\nu(y)=$
$\int_{X}f\,d\Big{(}\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}D_{\beta}\mu*D_{\alpha-\beta}\nu\Big{)}=\langle\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}D_{\beta}\mu*D_{\alpha-\beta}\nu,f\rangle,$

which proves our statement.

Now we prove the converse statement. Let
$(D_{\alpha})_{\alpha\in\mathbb{N}^{r}}$ be a continuous higher order
derivation of order $r$ on $\mathcal{M}_{c}(X)$. We define

$\varphi_{\alpha}(x)=\langle D_{\alpha}\delta_{x},1\rangle$

for each $\alpha$ in $\mathbb{N}^{r}$ and $x$ in $X$. If $x$ tends to $x_{0}$
in $X$, then $\delta_{x}$ converges to $\delta_{x}{x_{0}}$ in the
weak*-topology, hence, by the weak*-continuity of $D_{\alpha}$,
$\varphi_{\alpha}(x)$ tends to $\varphi_{\alpha}(x_{0})$. This proves the
continuity of $\varphi_{\alpha}$. On the other hand, for each $\alpha$ in
$\mathbb{N}^{r}$ and $x,y$ in $X$ we have

$\varphi_{\alpha}(x*y)=\langle D_{\alpha}\delta_{x*y},1\rangle=\langle
D_{\alpha}(\delta_{x}*\delta_{y}),1\rangle=$
$\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\langle
D_{\beta}\delta_{x}*D_{\alpha-\beta}\delta_{y},1\rangle=\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\int_{X}1\,d(D_{\beta}\delta_{x}*D_{\alpha-\beta}\delta_{y})=$
$\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\int_{X}\int_{X}1\,dD_{\beta}\delta_{x}\,dD_{\alpha-\beta}\delta_{y}=\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\int_{X}1\,dD_{\beta}\delta_{x}\cdot\int_{X}1\,dD_{\alpha-\beta}\delta_{y}=$
$\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\langle
D_{\beta}\delta_{x},1\rangle\cdot\langle
D_{\alpha-\beta}\delta_{y},1\rangle=\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\langle\varphi_{\beta}(x)\cdot\varphi_{\alpha-\beta}(y),$

and our theorem is proved. ∎

##  5 Derivations of the Fourier algebra

Let $X$ be a commutative hypergroup. We denote by $\tilde{X}$ the set of all
exponentials on $X$. For each measure $\mu$ in $\mathcal{M}_{c}(X)$, we define
the Fourie–Laplace transform of $\mu$ as the function
$\hat{\mu}:\tilde{X}\to\mathbb{C}$ by

$\hat{\mu}(m)=\int_{X}\check{m}\,d\mu.$

The set of all Fourier–Laplace transforms is called the Fourier algebra of $X$
and it is denoted by $\mathcal{A}(X)$. It is well-known that the
Fourier–Laplace transform is bijective between $\mathcal{M}_{c}(X)$ and
$\mathcal{A}(X)$. This makes it possible to equip a topology on
$\mathcal{A}(X)$ in the obvious way, that makes the bijection between
$\mathcal{M}_{c}(X)$ and $\mathcal{A}(X)$ a homeomorphism. Using this
bijection we can study derivations on the Fourier algebra as well. First we
consider the Fourier algebra a module over the ring $\mathcal{C}(X)$ in the
following way: for each $\varphi$ in $\mathcal{C}(X)$ and $\mu$ in
$\mathcal{M}_{c}(X)$ we define

$\varphi\cdot\hat{\mu}=(\varphi\mu)\,\hat{}.$

With this operation $\mathcal{A}(X)$ is a module over $\mathcal{C}(X)$. A
module homomorphism of $\mathcal{A}(X)$ into itself is an additive mapping
$F:\mathcal{A}(X)\to\mathcal{A}(X)$ such that

$F(\varphi\cdot\hat{\mu})=\varphi\cdot F(\hat{\mu})$

holds for each $\mu$ in $\mathcal{A}$ and $\varphi$ in $\mathcal{C}(X)$.
Accordingly, the family of module homomorphisms
$(\partial_{\alpha})_{\alpha\in\mathbb{N}^{r}}$ is a higher order derivation
of rank $r$, if it satisfies

$\partial_{\alpha}(\hat{\mu}\cdot\hat{\nu})=\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\partial_{\beta}\hat{\mu}\cdot\partial_{\alpha-\beta}\hat{\nu}$
(10)

for each $\mu,\nu$ in $\mathcal{M}_{c}(X)$. The following theorem is obvious.

######  Theorem 4.

Let $X$ be a commutative hypergroup, and $r$ a positive integer. Then the
family $(D_{\alpha})_{\alpha\in\mathbb{N}^{r}}$ is a higher order derivation
of rank $r$ on $\mathcal{M}_{c}(X)$ if and only if the family of mappings
$\hat{D}_{\alpha}:\mathcal{A}\to\mathcal{A}$ defined by

$\hat{D}_{\alpha}\hat{\mu}=(D_{\alpha}\mu)\,\hat{}$ (11)

for each $\alpha$ in $\mathbb{N}^{r}$ and $\mu$ in $\mathcal{M}_{c}(X)$ is a
higher order derivation of rank $r$ on $\mathcal{A}(X)$.

###### Proof.

The only thing to show is that $D_{\alpha}$ and $\hat{D}_{\alpha}$ are module
homomorphisms simultaneously. If $D_{\alpha}$ is, then have for each $\varphi$
in $\mathcal{C}(X)$:

$\hat{D}_{\alpha}(\varphi\cdot\hat{\mu})=\hat{D}_{\alpha}(\varphi\mu)\,\hat{}=(D_{\alpha}(\varphi\cdot\mu))\,\hat{}=(\varphi\cdot
D_{\alpha}\mu)\,\hat{}=\varphi\cdot(D_{\alpha}\mu)\,\hat{}=\varphi\cdot\hat{D}_{\alpha}\hat{\mu},$

hence $\hat{D}_{\alpha}$ is a module homomorphism, too. Conversely, if
$\hat{D}_{\alpha}$ is a module homomorphism, then for each $\varphi$ in
$\mathcal{C}(X)$ we have:

$(D_{\alpha}(\varphi\cdot\mu))\,\hat{}=\hat{D}_{\alpha}(\varphi\mu)\,\hat{}=\hat{D}_{\alpha}(\varphi\cdot\hat{\mu})=\varphi\cdot\hat{D}_{\alpha}\hat{\mu}=$
$\varphi\cdot(D_{\alpha}\mu)\,\hat{}=(\varphi\cdot D_{\alpha}\mu)\,\hat{}\,,$

which implies $D_{\alpha}(\varphi\cdot\mu)=\varphi\cdot D_{\alpha}\mu$, by the
injectivity of the Fourier–Laplace transform. The equivalence of the
derivation properties can be verified by a simple calculation, hence the
theorem is proved. ∎

##  6 Examples

In this section we present several examples of applications of our results.

######  Example 1.

Let $G=\mathbb{R}$, and we consider the functions
$\varphi_{k,\lambda}(x)=x^{k}e^{\lambda x}$ for $k=0,1,\dots$, where $x$ is in
$\mathbb{R}$ and $\lambda$ is in $\mathbb{C}$. Clearly, they form a
generalized moment sequence of rank $1$ as

$\varphi_{n,\lambda}(x+y)=(x+y)^{n}e^{\lambda(x+y)}=\sum_{k=0}^{n}\binom{n}{k}x^{k}e^{\lambda
x}y^{n-k}e^{\lambda
y}=\sum_{k=0}^{n}\binom{n}{k}\varphi_{k,\lambda}(x)\varphi_{n-k,\lambda}(y).$

This moment function sequence generates the following higher order derivation
on the space $\mathcal{M}_{c}(\mathbb{R})$:

$\langle D_{k}\mu,f\rangle=\int_{\mathbb{R}}x^{k}e^{\lambda x}f(x)\,d\mu(x).$

We have

$\langle D_{k}(\psi\mu),f\rangle=\int_{\mathbb{R}}x^{k}e^{\lambda
x}f(x)\cdot\psi(x)\,d\mu(x)=\int_{\mathbb{R}}x^{k}e^{\lambda
x}f(x)\,d(\psi\mu)(x)=\langle\psi D_{k}(\mu),f\rangle$

hence $D_{k}$ is a module homomorphism for each $k=0,1,\dots$. In particular,
for $\lambda=0$

$\langle D_{k}\mu,1\rangle=\int_{\mathbb{R}}x^{k}\,d\mu(x),$

which corresponds to the classical moments of the measures.

######  Example 2.

Now let $X=D(\theta)$ be the two-point hypergroup on the set $\\{0,1\\}$ with
$0<\theta\leq 1$, where $0$ is the identity. We have two exponentials on this
hypergroup: the identically $1$ function: $m_{0}\equiv 1$, and the function

$m_{1}(x)=\begin{cases}1&\enskip\text{for}\enskip x=0\\\
-\theta&\enskip\text{for}\enskip x=1.\end{cases}$

The only generalized moment sequence of any rank with $\varphi_{0}$ is
trivial: $\varphi_{\alpha}\equiv 0$ for $|\alpha|>0$. Accordingly, the only
higher order derivation on $\mathcal{M}_{c}(X)$ is the trivial, $D_{\alpha}=0$
for $|\alpha|>0$.

######  Example 3.

For a less trivial example we consider the polynomial hypergroup $X$ generated
by the sequence of polynomials $(P_{n})_{n\in\mathbb{N}}$. Let $z$ be
arbitrary in $\mathbb{C}$ and we define $\varphi_{k}(n)=P_{n}^{(k)}(z)$ for
$k,n=1,2,\dots$, then $(\varphi_{k})_{k\in\mathbb{N}}$ is a moment function
sequence. Indeed,

$\varphi_{k}(n*m)=\sum_{l=|n-m|}^{n+m}c(m,n,l)\varphi_{k}(l)=\sum_{l=|n-m|}^{n+m}c(m,n,l)P_{l}^{(k)}(z)=$
$\big{(}P_{n}(z)\cdot
P_{m}(z)\bigr{)}^{(k)}=\sum_{j=0}^{k}\binom{k}{j}P_{n}^{(j)}(z)\cdot
P_{m}^{(k-j)}(z)=\sum_{j=0}^{k}\binom{k}{j}\varphi_{j}(n)\varphi_{k-j}(m).$

By Theorem 3, this moment function sequence generates the higher order
derivation $(D_{k})_{k\in\mathbb{N}}$ of rank one defined by

$\langle D_{k}\mu,f\rangle=\int f(n)\cdot P_{n}^{(k)}(z)\,d\mu(n)$

for each $k=0,1,\dots$ and $f:\mathbb{N}\to\mathbb{C}$. Observe, that

$\langle
D_{k}\mu,1\rangle=\int_{X}P_{n}^{(k)}(z)\,d\mu(n)=\hat{\mu}^{(k)}(z),$

that is, the higher order derivation $(D_{k})_{k\in\mathbb{N}}$ corresponds to
the higher order differentiation of the Fourier–Laplace transform with respect
to the parameter $\lambda$, which represents the exponential $n\mapsto
P_{n}(z)$. Let $z=0$, and let $(D_{k})_{k\in\mathbb{N}}$ denote the higher
order derivation which corresponds to the moment function sequence
$\varphi_{k}(n)=P_{n}^{(k)}(0)$. Observe that, for each $\mu$ in
$\mathcal{M}_{c}(X)$, $\hat{\mu}$ is a polynomial. Hence we have for each
$\lambda$ in $\mathbb{C}$:

$\hat{\mu}(\lambda)=\sum_{k}\frac{\hat{\mu}^{(k)}(0)}{k!}\lambda^{k}=\sum_{k}\frac{\lambda^{k}}{k!}\int_{X}P_{n}^{(k)}(0)\,d\mu(n)=\sum_{k}\frac{\lambda^{k}}{k!}\langle
D_{k}\mu,1\rangle,$

that is, the higher order derivation determines $\hat{\mu}$ for each $\mu$.
This can be considered as a kind of ”Taylor formula”.

######  Acknowledgment.

The research of E. Gselmann has partially been carried out with the help of
the project 2019-2.1.11-TÉT-2019-00049, which has been implemented with the
support provided from NRDI (National Research, Development and Innovation Fund
of Hungary), financed under the TÉT funding scheme.
The research of E. Gselmann and L. Székelyhidi has been supported by the NRDI
(National Research, Development and Innovation Fund of Hungary) Grant no. K
134191.

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