11institutetext: Lev D. Beklemishev 22institutetext: Steklov Mathematical
Institute, Gubkina str. 8, 119991 Moscow, Russia, 22email<EMAIL_ADDRESS>33institutetext: Fedor N. Pakhomov 44institutetext: Steklov Mathematical
Institute, Gubkina str. 8, 119991 Moscow, Russia 55institutetext: Ghent
University, Krijgslaan 281 S8, 9000 Ghent, Belgium, 55email:
<EMAIL_ADDRESS>
# Automatic structures and the problem of natural well-orderings

Lev D. Beklemishev0000-0002-2949-0600 and
Fedor N. Pakhomov The work of Fedor Pakhomov was funded by the FWO grant
G0F8421N.

###### Abstract

We explore the idea of using automatic and similar kind of presentations of
structures to deal with the conceptual problem of natural proof-theoretic
ordinal notations. We conclude that this approach still does not meet the
goals.

###### Keywords:

proof theory, ordinal notation, automatic structure, Caucal hierarchy

##  1 Introduction

This paper is written for the Festschrift volume dedicated to the 75th
anniversary of Johann Makowsky. While thinking on the topic that would be
appropriate for this volume, we decided to go for one that would link the
interests of the authors in proof theory and some topics that play a role in
Johann’s own work: classes of structures with decidable (MSO) theories and
interpretations. We decided to record our attempts to explore one particular
approach to the (in)famous problem in proof theory — the problem of canonicity
of proof-theoretic ordinal notation systems. Even though this problem is well-
known, few written accounts and discussions of it exist in the literature.

Arguably, historically the first encounter with this problem occurs in the
famous work of Alan Turing “System of logics based on ordinals” Tur39
presenting the content of his PhD dissertation. Substantial discussions are
found in the papers by Georg Kreisel, in particular KraOZ, and especially
Solomon Feferman Fefbug who describes it as one of the three problems that
‘bug him’.

The problem requires delineating canonical, well-behaved ordinal notation
systems from pathological ordinal notation systems, such as a computable well-
ordering of order type $\omega$ such that the associated induction scheme
implies the consistency of Peano arithmetic (Kreisel’s example). All examples
of such pathological well-orderings have some external notions (such as
consistency statements) encoded into them; however, it is unclear exactly what
is required from an ordinal notation system to forbid such counterexamples and
at the same time to be sufficiently general.

The absence of a good mathematical solution of this problem is not only
annoying, but it makes the basic question what constitutes an ordinal analysis
of a formal theory intuitive rather than fully rigorous. The commonly used
expression ‘to calculate the proof-theoretic ordinal of a theory’ does not
really have a definite meaning: it is not clear in which terms the result of
this ‘calculation’ needs to be specified. Thus, ordinal analysis, the
characterization of proof-theoretic ordinals of theories, is sometimes
described as an art Rat06; Rat99. We usually recognize individual ordinal
notation systems arising in concrete situations as natural, the simplest one
is the system of ordinal notation for $\varepsilon_{0}$ based on Cantor normal
forms. Many other systems are surveyed, e.g., in Fefbug; Poh; Rat99. However,
a general mathematical definition of a ‘natural ordinal notation system’ is
lacking.

At this point we would like to remark that the term ‘natural ordinal notation
system’ may be somewhat misleading. Our understanding of this problem is
purely mathematical: the goal is not to explicate the vague notion of
‘naturality’ in a philosophical sense of the word, but rather to find a
suitably general definition or a framework that would delineate a wide class
of ordinal notation systems (well-ordering representations) suitable for
proof-theoretic analysis.

Our main goal in this note is to consider this problem from the point of view
of the theory of automatic structures. We will motivate this approach and see
where it will lead us to. Our attempts do not really solve the problem,
however we believe that there is value in studying failures. In the process we
learn new insights, and ultimately such a study may help to point us in the
right direction.

##  2 Turing

Arguably the first discussion of the _problem of natural ordinal notations_ ,
as we would call it today, is found in the work of Alan Turing Tur39. There,
Turing studied transfinite progressions of theories based on iteration of the
process of extending a theory by consistency assertions and by some more
general reflection principles. The main goal of this study was to obtain a
classification of arithmetical sentences111Turing considered sentences of
logical complexity at most $\Pi_{2}^{0}$ that he called ‘number-theoretic
theorems’. according to the stages of this process. To quote:

> _We might also expect to obtain an interesting classification of number-
> theoretic theorems according to “depth”. A theorem which required an ordinal
> $\alpha$ to prove it would be deeper than one which could be proved by the
> use of an ordinal $\beta$ less than $\alpha$. However, this presupposes more
> than is justified. _

The last sentence here apparently indicates that Turing realized that his
results fall short of the stated goal. To explain this, we first remark that
to construct a Turing progression

$T_{0}:=T,\quad T_{\alpha+1}=T_{\alpha}+\mathrm{Con}(T_{\alpha}),\quad
T_{\lambda}=\bigcup_{\alpha<\lambda}T_{\alpha},\text{if
$\lambda\in\mathrm{Lim}$},$

one really needs to associate theories $T_{\alpha}$ with ordinal notations (or
some kind of constructive representations) rather than with ordinals $\alpha$
in the set-theoretic sense. In order to formulate consistency assertions
$\mathrm{Con}(S)$ according to the recipe of Gödel, the axiom set of a theory
$S$ must be r.e. and represented in the language of arithmetic by a
$\Sigma_{1}$-formula. Thus, the arithmetical formula
$\mathrm{Con}(T_{\alpha})$ must somehow refer to the ordinal $\alpha$, and one
has to deal with computable and arithmetized ordinal representations rather
than with the ordinals themselves. With this understanding, Turing showed how
to accurately define such a progression for a given ordinal notation system.
This was further elaborated by Feferman in 1962 Fef62. Both Turing and
Feferman used Kleene’s universal ordinal notation system ${\mathcal{O}}$ for
this purpose.

One of the main results of Turing’s paper can be stated (in modern terms) as
follows, where $|a|$ denotes the order type of $a\in{\mathcal{O}}$.

######  Theorem 2.1 (Turing)

For each true $\Pi^{0}_{1}$-sentence $\pi$ there is an ordinal notation
$a\in{\mathcal{O}}$ such that $|a|=\omega+1$ and $T_{a}$ proves $\pi$.

Turing’s completeness result is a mixed blessing. The negative side of it is
that any true $\Pi_{1}^{0}$-sentence can be proved already at stage $\omega+1$
of a suitable Turing progression. These include all sentences of the form
$\mathrm{Con}(T_{b})$ with $|b|$ much larger than $\omega+1$, which entails
that the theories $T_{a}$ heavily depend on particular ordinal representations
rather than on their order types. Thus, the idea of a meaningful
classification of sentences according to the progression stages breaks down.
Turing put it in a remarkably pessimistic form:

> _This completeness theorem as usual is of no value. Although it shows, for
> instance, that it is possible to prove Fermat’s last theorem with
> $\Lambda_{P}$ (if it is true) yet the truth of the theorem would really be
> assumed by taking a certain formula as an ordinal formula_222$\Lambda_{P}$
> is progression based on iteration of consistency, and ‘ordinal formulas’ are
> his ordinal notations..

Further in the paper Turing suggests a partial way out, a careful selection of
specific ordinal notations.

> _We can still give a certain meaning to the classification into depths with
> highly restricted kinds of ordinals. Suppose that we take a particular
> ordinal logic $\Lambda$ and a particular ordinal formula $\Psi$ representing
> the ordinal $\alpha$ say (preferably a large one), and that we restrict
> ourselves to ordinal formulae of the form $\mathrm{Inf}(\Psi,a)$.333These
> formulas define initial segments of $\alpha$. We then have a classification
> into depths, but the extents of all the logics which we so obtain are
> contained in the extent of a single logic._

Thus, Turing essentially suggests to consider linearly ordered subsets of
${\mathcal{O}}$ defined by specific ‘highly restricted’ ordinal notations. It
seems likely that he means here particular notations such as, for example, the
one for $\varepsilon_{0}$ based on Cantor normal forms, in other words,
ordinal notation systems we would describe as ‘natural’. However, he admits
that by doing this we have to give up the idea of classification of _all_ true
$\Pi_{1}^{0}$ sentences. We would refer to this proposal below as _“limited
Turing’s program.”_

Turing did not pursue this suggestion any further than that. This kind of
approach has much later reappeared in the works of U. Schmerl Schm; Sch82, it
was taken up in Bek99b and other papers with many positive results. Turing’s
idea has been used to classify, for example, $\Pi_{n}^{0}$-consequences of
specific theories, such as $\mathsf{PA}$ and its predicative extensions. These
results were indeed based on highly specific ‘natural’ ordinal notations.

We know many examples of natural ordinal notation systems for fairly large
constructive ordinals. Several types of such notation systems are reviewed in
Feferman’s paper Fefbug. However, we still lack a general understanding of
what constitutes a ‘natural’ ordinal notation system. Essentially the same
problem appears in the study of proof-theoretic ordinals based on Gentzen’s
approach.

##  3 Proof-theoretic ordinals

Since Gentzen gave his consistency proof of Peano arithmetic by transfinite
induction for the ordinal $\varepsilon_{0}$, much of the work in proof theory
has been the exploration of the relationships between formal theories and
well-orderings Kra68; Tak; Poh; Poh09; Rat99. A proof-theoretic study of a
formal theory usually culminates in the calculation of its _proof-theoretic
ordinal_ , that is, a well-ordering representing a bound on the strength of
the system. With this ordinal, the other important characteristics of a formal
theory, such as its class of provably total computable functions, are often
connected.

Although it is not always duly emphasised in proof-theoretic literature, there
are different ways of associating ordinals to theories. They are sensitive to
different levels of logical complexity and lead, in general, to inequivalent
notions of proof-theoretic ordinals. The most common notion, known since
Gentzen and prevalent in the work on proof theory, is the so-called
_$\Pi_{1}^{1}$ -ordinal_ which is defined as the supremum of order types of
recursive well-orderings that are provably well-founded in a given formal
theory. Of course, for this definition to be applicable, the language of the
theory must be able to define recursive relations and to express the well-
foundedness property of such relations (which is $\Pi_{1}^{1}$-complete).
Thus, we usually assume the language to contain at least that of first-order
arithmetic with free second-order variables (denoting arbitrary sets of
natural numbers). A fundamental result of Gentzen stated in these terms is
that the $\Pi_{1}^{1}$-ordinal of $\mathsf{PA}(X)$, a version of Peano
arithmetic in the language expanded by free set variables, is
$\varepsilon_{0}$.

The $\Pi_{1}^{1}$-ordinal is a well-defined and robust measure of proof-
theoretic strength, however it is not very sensitive: Extension of a theory by
true $\Sigma_{1}^{1}$-axioms does not change its $\Pi_{1}^{1}$-ordinal KraOZ;
Rat99. In particular, theories with the same $\Pi_{1}^{1}$-ordinal may have
vastly different consistency strength and the classes of provably total
computable functions. For example, $\mathsf{PA}$ and
$\mathsf{PA}+\mathrm{Con}(\mathsf{ZF})$ have the same $\Pi_{1}^{1}$-ordinal.
Moreover, Gentzen’s consistency proof for Peano arithmetic is not exactly
captured by the characterization of its $\Pi_{1}^{1}$-ordinal: Whereas a proof
of $\mathrm{Con}(\mathsf{PA})$ by transfinite induction on $\varepsilon_{0}$
entails that the well-foundedness of $\varepsilon_{0}$ is unprovable in
$\mathsf{PA}(X)$ (by Gödel’s second incompleteness theorem), the converse
cannot in general be concluded.

Attempts to define the notions of proof-theoretic ordinals relevant for
arithmetical complexity classes such as $\Pi_{2}^{0}$ and $\Pi_{1}^{0}$ only
succeed provided some natural system of ordinal notation is given (as in
Turing’s limited program), see Bek99b for a discussion of various proposals of
this kind. Without such an assumption, these attempts invariably fail. This
issue was treated by Georg Kreisel who constructed a number of pathological
counterexamples in order to demonstrate such failures.

##  4 Pathological well-orderings

Given an arithmetically definable binary relation $\prec$ and a class of
formulas ${\mathcal{F}}$ we denote by $\mathrm{TI}(\prec;{\mathcal{F}})$ the
schema of transfinite induction for $\varphi\in{\mathcal{F}}$:

$\forall x\>(\forall y\prec x\>\varphi(y)\to\varphi(x))\to\forall
x\>\varphi(x).$

$\mathrm{TI}(\prec;{\mathcal{F}})$ is true in $\mathbb{N}$ iff $\prec$ is
well-founded w.r.t. ${\mathcal{F}}$-definable sets. As remarked by Kreisel,
Gentzen’s consistency proof for $\mathsf{PA}$ was naturally formalizable in
Primitive Recursive Arithmetic $\mathsf{PRA}$ extended by
$\mathrm{TI}(\prec;\Delta_{0})$, where $\prec$ is the canonical primitive
recursive well-ordering of order type $\varepsilon_{0}$, and $\Delta_{0}$ is
the class of primitive recursive arithmetical formulas.

Now we turn to pathological well-orderings. As our starting point we consider
the simplest example from Kra68.

###### Proposition 1

For any true $\Pi_{1}^{0}$ sentence $\pi$, there is a primitive recursive
well-ordering $\prec_{\pi}$ of order type $\omega$ such that
$\mathsf{PRA}+\mathrm{TI}(\prec_{\pi};\Delta_{0})\vdash\pi$.

###### Proof

. If $\pi$ has the form $\forall x\>\pi_{0}(x)$ with $\pi_{0}\in\Delta_{0}$,
then one can define:

$x\prec_{\pi}y\stackrel{{\scriptstyle\text{def}}}{{\iff}}(x<y\land\forall
z<x\>\pi_{0}(z))\lor(y<x\land\exists z<y\>\neg\pi_{0}(z)).$

It is easy to see that, once $\pi$ is false and $n$ is the minimal natural
number such that $\neg\pi_{0}(n)$, the ordering $\prec_{\pi}$ has a
$\Delta_{0}$-definable subset $\\{y:y>n\\}$ without the least element. Hence
$\mathrm{TI}(\prec_{\pi};\Delta_{0})$ is false. Formalizing this in
$\mathsf{PRA}$ shows
$\mathsf{PRA}+\mathrm{TI}(\prec_{\pi};\Delta_{0})\vdash\pi$. On the other
hand, since $\pi$ is true in the standard model of arithmetic, the witness of
$\neg\pi$ does not exist and $\prec_{\pi}$ is, in fact, isomorphic to $\omega$
(but not provably so unless one can prove $\pi$). $\Box$

This effect is essentially the same as the one observed in the above quotation
from Turing: If $\pi$ is Fermat’s last theorem (or, better now, some
$\Pi_{1}^{0}$-equivalent of Riemann’s Hypothesis), the truth of $\pi$ is
encoded in the fact that $\prec_{\pi}$ is a well-ordering. This rather cheap
trick is to be compared with Gentzen’s theorem stating that
$\mathrm{Con}(\mathsf{PA})$ is provable in
$\mathsf{PRA}+\mathrm{TI}(\prec;\Delta_{0})$ where $\prec$ is Cantor’s
canonical notation system for the ordinal $\varepsilon_{0}$, but not any
proper initial segment of it.

Kreisel’s example immediately shows the inadequacy of the naive definition of
a proof-theoretic ordinal of a theory $T$ as the order type of the shortest
primitive recursive well-ordering $\prec$ such that
$\mathrm{TI}(\prec,\Delta_{0})$ over $\mathsf{PRA}$ proves $\mathrm{Con}(T)$:
Take $\mathrm{Con}(T)$ for $\pi$ and observe that the ordinal of $T$ then
equals $\omega$ irrespectively of $T$.

###### Remark 1

Kreisel’s example shows that $\mathrm{TI}(\prec,\Delta_{0})$ can be
pathologically strong. However, there are other exmaples showing that this
schema can be pathologically weak Kra68; Bek00. For example, Kreisel
demonstrated that there are non-wellfounded primitive recursive linear
orderings $\prec$ such that $\mathsf{PA}$ proves the schema of transfinite
induction on $\prec$ for arbitrary arithmetical formulas
$\mathrm{TI}(\prec,\Pi^{0}_{\infty})$. Accordingly, for all
$\alpha<\omega_{1}^{CK}$, one can construct a primitive recursive well-
ordering $\prec$ of order type $\alpha$, such that
$\mathrm{TI}(\prec,\Pi^{0}_{\infty})$ is provabe in $\mathsf{PA}$. This
phenomenon is closely related to the construction of linear orderings well-
founded w.r.t. hyperarithmetical sets, but not actually well-founded, see
Har68.

Other important uses of ordinals in proof theory occur in the constructions of
subrecursive hierarchies Ros84. In turn, the latter are used to characterize
provably total computable functions of theories BW and as technical tools to
prove the independence of certain combinatorial theorems KS81.

An important hierarchy of functions is the so-called _fast-growing hierarchy_
$(F_{\alpha})_{\alpha<\Lambda}$ (also known as the extended Grzegorczyk
hierarchy) LW:

  1. 1.

$F_{0}(x)=x+1$,

  2. 2.

$F_{\alpha+1}(x)=\underbrace{F_{\alpha}(\ldots(F_{\alpha}}\limits_{\mbox{\footnotesize$x$-times}},(x))\ldots)$

  3. 3.

$F_{\lambda}(x)=F_{\lambda[x]}(x)$, for limit ordinals $\lambda$.

Here $\Lambda$ is supposed to be a countable ordinal, and $\lambda[\cdot]$,
for each limit ordinal $\lambda<\Lambda$, denotes a _fundamental sequence_ ,
that is, a monotonically increasing sequence of ordinals $\lambda[n]$ such
that $\mathsf{sup}_{n<\omega}\lambda[n]=\lambda$. Once a system of fundamental
sequences is fixed, the functions $F_{\alpha}$ are uniquely defined for each
$\alpha<\Lambda$.

If we want the functions $F_{\alpha}$ to be computable, then $\Lambda$ should
be represented as a computable well-ordering; the set
$\Lambda\cap\mathsf{Lim}$ and the functions $x+1$ and $x[y]$ should be
computable as well.

The typical expectation for non-pathological computable systems of ordinal
notations and fundamental sequences is that the growth rate of $F_{\alpha}$
reflects the size of $\alpha$, e.g., $F_{\omega}$ is expected to be of the
growth rate of the Ackermann function. The fast-growing hierarchy is often
used to characterize provably total computable functions of theories. For
natural ordinal notation systems, the functions $F_{\alpha}$ are provably
total in $\mathsf{PA}$, if $\alpha<\varepsilon_{0}$, while
$F_{\varepsilon_{0}}$ grows faster than any $\mathsf{PA}$-provably total
computable function.

Examples of computable ordinal notation systems leading to pathological
subrecursive hierarchies are well-known and going back at least to Fef62a (see
also the references therein). We give an easy example of this kind here.

###### Proposition 2

For any total computable function $g$ there is a (polytime) computable well-
ordering of order type $\omega+1$ and a fundamental sequence for $\omega$ such
that the growth rate of $F_{\omega}$ is faster than $g$.


# Decoding the flow evolution in Au+Au reactions at $1.23A$ GeV
using hadron flow correlations and dileptons

Tom Reichert  Institut für Theoretische Physik, Goethe Universität Frankfurt,
Max-von-Laue-Strasse 1, D-60438 Frankfurt am Main, Germany  Helmholtz Research
Academy Hesse for FAIR (HFHF), GSI Helmholtz Center for Heavy Ion Physics,
Campus Frankfurt, Max-von-Laue-Str. 12, 60438 Frankfurt, Germany  Oleh Savchuk
GSI Helmholtzzentrum für Schwerionenforschung GmbH, Planckstr. 1, 64291
Darmstadt, Germany  Frankfurt Institute for Advanced Studies (FIAS), Ruth-
Moufang-Str.1, D-60438 Frankfurt am Main, Germany  Bogolyubov Institute for
Theoretical Physics, 03680 Kyiv, Ukraine  Facility for Rare Isotope Beams,
Michigan State University, East Lansing, MI 48824 USA  Apiwit Kittiratpattana
Institut für Theoretische Physik, Goethe Universität Frankfurt, Max-von-Laue-
Strasse 1, D-60438 Frankfurt am Main, Germany  Suranaree University of
Technology, University Avenue 111, Nakhon Ratchasima 30000, Thailand
Pengcheng Li  School of Nuclear Science and Technology, Lanzhou University,
Lanzhou 730000, China  School of Science, Huzhou University, Huzhou 313000,
China  Jan Steinheimer  Frankfurt Institute for Advanced Studies (FIAS), Ruth-
Moufang-Str.1, D-60438 Frankfurt am Main, Germany  Mark Gorenstein  Frankfurt
Institute for Advanced Studies (FIAS), Ruth-Moufang-Str.1, D-60438 Frankfurt
am Main, Germany  Bogolyubov Institute for Theoretical Physics, 03680 Kyiv,
Ukraine  Marcus Bleicher  Institut für Theoretische Physik, Goethe Universität
Frankfurt, Max-von-Laue-Strasse 1, D-60438 Frankfurt am Main, Germany
Helmholtz Research Academy Hesse for FAIR (HFHF), GSI Helmholtz Center for
Heavy Ion Physics, Campus Frankfurt, Max-von-Laue-Str. 12, 60438 Frankfurt,
Germany

###### Abstract

We investigate the development of the directed, $v_{1}$, and elliptic flow,
$v_{2}$, in heavy ion collisions in mid-central Au+Au reactions at
$E_{\mathrm{lab}}=1.23A$ GeV. We demonstrate that the elliptic flow of hot and
dense matter is initially positive ($v_{2}>0$) due to the early pressure
gradient. This positive $v_{2}$ transfers its momentum to the spectators,
which leads to the creation of the directed flow $v_{1}$. In turn, the
spectator shadowing of the in-plane expansion leads to a preferred decoupling
of hadrons in the out-of-plane direction and results in a negative $v_{2}$ for
the observable final state hadrons. We propose a measurement of $v_{1}-v_{2}$
flow correlations and of the elliptic flow of dileptons as methods to pin down
this evolution pattern. The elliptic flow of the dileptons allows then to
determine the early-state EoS more precisely, because it avoids the strong
modifications of the momentum distribution due to shadowing seen in the
protons. This opens the unique opportunity for the HADES and CBM
collaborations to measure the Equation-of-State directly at 2-3 times nuclear
saturation density.

##  I Introduction

Heavy-ion collisions provide an excellent opportunity to study nuclear matter
in a controlled laboratory setting allowing to recreate conditions which were
present in the early universe or are present in neutron stars. One of the main
goals of nuclear research is the precise extraction of the nuclear Equation-
of-State (EoS) at large baryon densities which are found in the interior of
neutron stars. The link between astrophysical observations on a scale of
$10^{15}$ m and collisions of heavy atomic nuclei at nearly the speed of light
on a scale of $10^{-15}$ m comes as a surprise. Remarkable resemblance between
binary neutron star mergers (BNS merger) and heavy ion collisions (HICs) has
been found in Ref. Most:2022wgo . Most intriguing is the fact that BNS mergers
and HICs are subject to the same microscopic interactions, namely Quantum-
Chromo-Dynamics (QCD). In order to better understand astrophysical objects, it
is of utmost importance to pin down the nuclear Equation-of-State with high
precision, desirably in a neutron rich system.

HICs range from center of mass energies of 2 GeV at GSI’s SIS18 accelerator up
to several hundred GeV at BNL-RHIC or CERN-SPS or even to 13 TeV at CERN-LHC.
However, the density range suited for BNS comparison will be probed in the
upcoming FAIR facility which is currently constructed near Darmstadt, Germany.
In this energy regime, the densities in central heavy ion collisions reaches
from 2-6 times nuclear saturation density OmanaKuttan:2022the . Currently, the
lower range of this regime can already be probed in the currently running
HADES experiment at GSI. Here, information about the Equation-of-State can be
extracted via measurements of harmonic flow of hadrons Danielewicz:2002pu ;
LeFevre:2015paj ; Reichert:2022gqe , investigations of the speed of sound
Bedaque:2014sqa ; Tews:2018kmu ; Steinheimer:2012bp  or strangeness
enhancement or flow KAOS:2000ekm ; Fuchs:2000kp ; Hartnack:2005tr ;
HADES:2018noy ; Chlad:2020kpm  and many more. Especially the second flow
coefficient $v_{2}$ (called elliptic flow) is perfectly suited to investigate
the EoS at large densities. It is well known from higher beam energies that
the measured elliptic flow in the final state is directly connected to the
pressure gradients exerted by the EoS. A detailed comparison of flow
measurements at BNL’s RHIC STAR:2000ekf ; PHENIX:2003iij  with hydrodynamic
simulations Huovinen:2001cy ; Song:2007fn ; Luzum:2008cw ; Romatschke:2007mq
lead to the renowned finding that the deconfined phase of QCD (a Quark-Gluon-
Plasma, QGP) resembles the most perfect liquid.

At low energies the situation is more challenging: here the incoming baryon
currents cannot decouple quickly enough to allow free expansion driven solely
by the pressure gradients. Instead, the residues of the incoming nuclei (the
spectators) are blocking the emission of particles in-plane during the
compression phase. This leads to the well known squeeze-out effect of
nucleons. Therefore, the final elliptic flow of protons is negative in the
range $\sqrt{s_{\mathrm{NN}}}=2-4$ GeV, which is confirmed by measurements at
FOPI FOPI:2004bfz ; FOPI:2011aa , EOS/E895 E895:1999ldn ; E895:2000maf , E877
E877:1997zjw , STAR-BES STAR:2014clz ; STAR:2020dav ; STAR:2021ozh .

Even though the magnitude of the negative elliptic flow is still sensitive to
the EoS Steinheimer:2022gqb , a detailed understanding of the dynamics which
lead to this dependency and its connection to the directed flow is still
missing.

In this article we aim to understand the precise expansion dynamics during the
compression stage and how the initial positive elliptic flow is observed as
negative elliptic flow in the final state. This will allow to extract
information on the Equation-of-State in the most dense stage of the reaction.
We analyze the flow of Au-Au collisions at 1.23$A$ GeV kinetic beam energy, in
line with recent HADES measurements HADES:2020lob ; HADES:2022osk , and
investigate the contributions to the final observable flow. Finally we propose
two distinct measurements (flow correlations and dilepton flow) to test our
findings.

##  II Model setup and flow extraction

For the present study we use the Ultra-relativistic Quantum Molecular Dynamics
(UrQMD) model Bass:1998ca ; Bleicher:1999xi ; Bleicher:2022kcu  in its most
recent version (v3.5). UrQMD is a microscopic transport simulation based on
the explicit propagation of hadrons in phase-space. The imaginary part of the
interactions is modeled via binary elastic and inelastic collisions, leading
to resonance excitations and decays or color flux-tube formation and their
fragmentation. The real part of the interaction potential is implemented via
different equations of state, where in the present work a realistic chiral
mean field EoS is used, see OmanaKuttan:2022the . In its current version,
UrQMD includes a large body of baryonic and mesonic resonances up to masses of
4 GeV. The model is well established in the GSI energy regime. For recent
studies of the bulk dynamics, we refer the reader to Reichert:2020uxs ;
Reichert:2021ljd . For the analysis of the integrated harmonic flows at SIS
energies see Hillmann:2018nmd ; Hillmann:2019wlt ; Reichert:2022gqe ;
Reichert:2022yxq .

The flow coefficients are identified with the Fourier coefficients in the
series expansion of the azimuthal angle distribution which can be written as

$\frac{\mathrm{d}N}{\mathrm{d}\phi}=1+2\sum\limits_{n=1}^{\infty}v_{n}\cos(n(\phi-\Psi_{RP})),$
(1)

in which $v_{n}$ is n-th order flow coefficient, $\phi$ is the azimuthal angle
and $\Psi_{RP}$ is the angle of the reaction plane. The HADES experiment uses
a forward wall to reconstruct the event plane from the spectator nucleons
Kardan:2017knj . In the simulation, the spectator event plane is fixed and
$\Psi_{RP}=0$ is used for the present analysis of the simulation. The flow
coefficients are then calculated as

$v_{n}=\langle\cos(n(\phi-\Psi_{RP}))\rangle,$ (2)

where the average $\langle\cdot\rangle$ is taken over all nucleons in a fixed
rapidity or transverse momentum range in a given event.

The reader should be aware that this definition is different than the one
usually used at the highest beam energies where the flow components are
defined with respect to specific event planes. The methodical difference has
its origin in the way that flow develops at low beam energies. Here, the
initial squeeze out of nucleons leads to a large negative elliptic flow
component with respect to the reaction plane defined by the spectators. The
initial compression phase is decisive for the observed flow and therefore the
scaling of the initial eccentricity with final state flow known from large
collision energies PHOBOS:2006fqf  is not as straightforward. Recently,
scaling could be established in Refs. HADES:2022osk ; Reichert:2022imz ,
however with a negative sign. Thus, it can be expected that also the flow
correlations will differ significantly from what was observed at high beam
energies, which opens up new possibilities to study the equation of state
responsible for the formation of flow.

##  III Results

All results were obtained by simulating 20-30% peripheral Au+Au collisions at
$E_{\rm beam}=1.23A$ GeV kinetic beam energy with the UrQMD (v3.5) model. We
employ the model with the Chiral Mean Field (CMF) Equation-of-State as it was
shown to yield the best description of available data (cf. Ref.
Steinheimer:2022gqb ). We note that in this energy regime, the results of the
CMS-EoS are very similar to the ones obtained by a hard EoS Hillmann:2018nmd ;
Hillmann:2019wlt . We focus our analysis on participating nucleons and also
exclude nucleons that are bound in light clusters. It has been shown
Hillmann:2019wlt  that both effects need to be taken into account to reliably
describe the measured data on nucleon flow HADES:2020lob ; HADES:2022osk . The
centrality is selected via impact parameter cuts following previous Monte-
Carlo Glauber simulations HADES:2017def .

###  III.1 Flow development of the system

In contrast to ultra-relativistic energies probed at RHIC and LHC where the
initial baryon-currents decouple quickly allowing the overlap region to
propagate its spatial anisotropy to the final state momentum space anisotropy
PHOBOS:2006fqf , low energy heavy-ion collisions are very different. Here, the
baryon-currents of the initial impinging nuclei are present over the whole
course of the compression and expansion phase, which results in a space and
time dependent interplay between pressure gradients, spectator blocking,
compression and expansion phase and corrections due to transport coefficients
such as shear viscosity Reichert:2020oes  which influence the development of
the finally observable flow.

Figure 1: [Color online] The directed flow coefficient
$v_{1}^{*}\equiv\left\langle{\rm sign}(y)\cdot v_{1}\right\rangle$ calculated
from all baryons (black circles) which are present in the whole phase space at
time $t$ and of the nucleons emitted at time $t+\Delta t$ (red triangles) from
20-30% peripheral Au+Au collisions at a kinetic beam energy of 1.23$A$ GeV
from UrQMD. Figure 2: [Color online] The elliptic flow coefficient $v_{2}$
calculated from all baryons (black circles) which are present in the whole
phase space at time $t$ and of the nucleons emitted at time $t$ (red
triangles) from 20-30% peripheral Au+Au collisions at a kinetic beam energy of
1.23$A$ GeV from UrQMD.

It is thus convenient to start the investigation by discussing how flow
develops over the course of the collision in the whole system before we later
quantify the different contributions to the final observable flow at freeze-
out. For this we simulate semi-peripheral (corresponding to $6.6\leq b\leq
8.1$ fm, cf. Ref. HADES:2017def ) Au+Au collisions with UrQMD. In this section
we integrate over all baryons present at a specified time $t$ (the time is
defined in the computational frame, i.e. the center-of-mass frame of the whole
system) to extract the time development of the harmonic flow coefficients
$v_{1}^{*}$ and $v_{2}$. We show the flow coefficients $v_{1}^{*}$ in Fig. 1
and $v_{2}$ in Fig. 2 calculated from all baryons which are present in the
whole phase space at time $t$ as a black solid line with full circles. In
addition we also show the directed and elliptic flow only of nucleons being
emitted at given time $t+\Delta t$ as a red solid line with full triangles. It
should be noted that, since the directed flow, $v_{1}$, averaged over all
nucleons is exactly zero due to momentum conservation, we show the rapidity
weighted averaged directed flow defined as $v_{1}^{*}\equiv\left\langle{\rm
sign}(y)\cdot v_{1}\right\rangle$ where the average runs over all baryons and
$y$ and $v_{1}$ correspond to the rapidity and directed flow of each baryon.

First, we observe that the time dependence of the directed flow calculated
from all participating baryons present in the system (black line with circles)
exerts a strong increase during the collision. During the initial compression
stage the curve is consistent with zero until $\approx 5-7$ fm/c when the
density reaches 2-3 times saturation density and the CMF EoS becomes repulsive
OmanaKuttan:2022the . Here, maximum compression is reached OmanaKuttan:2022the
. At this time the development of the directed flow component begins due to a
large pressure gradient in impact parameter direction. The pressure gradient
pushes the nucleons in in-plane direction, which is reflected in a rapid
development of large $v_{1}^{*}$ over the course of the next 7 fm.

After 15 fm, the directed flow $v_{1}^{*}$ of all baryons present in the
system reaches a plateau around $v_{1}^{*}\approx 0.18$ in line with measured
data HADES:2020lob ; HADES:2022osk . This tells that the bounce-off is
generated in a rather short time frame and thus probes the system at maximal
compression, i.e. $\rho_{\mathrm{B}}/\rho_{0}\approx 2-3$ Reichert:2020oes ;
OmanaKuttan:2022the .

The directed flow of the first nucleons that decouple kinetically from the
system is strongly negative. This finding can be understood, because the
nucleons which propagate in-plane will scatter with the residue of the
projectile nucleus still propagating forward and therefore the only direction
free for decoupling has a negative $v_{1}^{*}$ at initial contact.

Turning to the elliptic flow $v_{2}$ (shown in Fig. 2) supports this line of
arguments. The elliptic flow of the whole system is zero until maximal
compression is achieved at the time of full geometric overlap at 7 fm. This
means that the system’s momentum space remains isotropic in the transverse
plane in momentum space from initial contact until full overlap. At overlap
time the compression is maximal with values around
$\rho_{\mathrm{B}}/\rho_{0}=2-3$ Reichert:2020oes ; OmanaKuttan:2022the .
Although measurements by the HADES collaboration HADES:2020lob  show in
agreement with theoretical models Hillmann:2019wlt ; Mohs:2020awg  that the
elliptic flow is negative in the final state (i.e. at kinetic freeze-out) at
this energy, the elliptic flow of the whole system extracted at different
times starts to become positive after full overlap is reached and only turns
negative at even later stages of the evolution of the system. This can be
interpreted as follows: At the time of maximal compression the overlap zone
resembles an almond shape which is known also from high energies. The spatial
pressure gradient of this shape exerts a force over a large surface in in-
plane direction (i.e. aligned with the impact parameter). The participating
baryons in the center of the system thus try to expand in-plane with positive
$v_{2}$ but they cannot decouple because the spectators are still
blocking111Here the phrase ’blocking’ should not be understood as hard wall
blocking, but as a deceleration or momentum transfer of the initially
expanding system to the (semi-)spectator matter. expansion in this direction.
The resulting momentum transfer to the (semi-)spectators next to the central
collision zone then generates the observed $v_{1}^{*}$ (as discussed above).
However, the observed hadrons are emitted out-of-plane, because only in this
direction the emission is not blocked and therefore the elliptic flow of the
participating nucleons turns negative at a slightly later time around 15 fm
which is exactly where the directed flow reaches its plateau.

Contrary to the $v_{2}$ of all baryons in the system, the elliptic flow of the
emitted nucleons is always strongly negative even though the full systems
$v_{2}$ is zero or positive over the first 15 fm. In other words even though
more nucleons are flowing in in-plane direction than out-of-plane, those in-
plane are blocked from emission by the spectators and can not be observed as
free hadrons in the detector.

This underlines the importance of shadowing to understand flow at SIS
energies: the pressure gradient generated by the Equation-of-State behaves
similar to the highest beam energies and tries to generate expansion with
positive $v_{2}$. However, the momentum flow in $x$-direction gets absorbed by
nucleons close to the central reaction zone and the bypassing spectators
generating a strongly positive $v_{1}^{*}$ (the bounce-off, or rather a “push-
away”) and leads to an isotropization of the momentum flow in $x$-direction.
In contrast, the nucleons which can freeze-out dominantly only with momenta in
$y$-direction create a negative $v_{2}$. This also explains how a softening of
the early EoS, causing less early expansion and smaller early $v_{2}$, leads
to a smaller final $v_{1}^{*}$.

The tight connection between $v_{1}^{*}$ and $v_{2}$ should therefore be
observable in their correlation. Following our previous study in Ref.
Reichert:2022gqe  we define event classes by triggering on the integrated
final state elliptic flow of nucleons $\langle v_{2}\rangle_{|y|\leq 0.5}$. As
discussed, a smaller $v_{2}$ leads to less momentum transfer to the
(semi-)spectators and thus the integrated final elliptic flow of protons
should have a positive correlation with the mid-rapidity slope of the directed
flow. Fig. 3 shows the mid-rapidity slope of the directed flow of nucleons at
kinetic freeze-out as a function of event-class selected integrated $v_{2}$
values at mid-rapidity from 20-30% peripheral Au+Au collisions at a kinetic
beam energy of 1.23$A$ GeV.

Figure 3: [Color online] The mid-rapidity slope of the directed flow of
nucleons at kinetic freeze-out as a function of event-class selected
integrated $v_{2}$ values at mid-rapidity from 20-30% peripheral Au+Au
collisions at a kinetic beam energy of 1.23$A$ GeV from UrQMD.

A clear positive correlation between $v_{2}$ and
$\mathrm{d}v_{1}/\mathrm{d}y|_{y=0}$ is found. This result supports the idea
that the initial expansion in $x$-direction exerts a momentum transfer to the
(semi-spectator) nucleons close to the central reaction zone and thus
enforcing a strong correlation between the initial elliptic flow and the final
directed flow.

How can such a scenario be tested further? In the following section we suggest
dileptons as a second observable to explore the early stage expansion pattern.

Figure 4: [Color online] The time distribution of nucleons at kinetic freeze-
out (dotted black line) and of emitted dileptons (solid red line) from 20-30%
peripheral Au+Au collisions at a kinetic beam energy of 1.23 $A$GeV from
UrQMD.

###  III.2 Time of decoupling and measurement via dileptons

In an experimental setup the time dependence of heavy-ion collisions cannot be
accessed by hadronic observables. On the other hand, dilepton emission from
the hot and dense phase offers time integrated information of the whole
evolution history Shuryak:1978ij ; McLerran:1984ay ; Bratkovskaya:1996qe ;
Rapp:1999ej . Dileptons are perfect candidates because they only interact
electromagnetically and thus very weakly in comparison to the strong
interaction Rapp:2014hha . Recent dilepton measurements by the HADES
collaboration HADES:2019auv  have shown great potential for future high
precision measurements of the properties of the matter produced
Savchuk:2022aev .

To calculate the dilepton emission from our simulated collisions we will
employ a well known coarse graining method Bravina:1998pi ; Bravina:1999dh ;
Huovinen:2002im ; Endres:2013daa ; Endres:2014zua ; Endres:2015fna ;
Endres:2015egk ; Galatyuk:2015pkq ; Endres:2016tkg ; Reichert:2020yhx ;
Reichert:2020oes  where event averaged spatial distributions of the baryon and
energy-momentum density are used to calculate the time dependent in-medium
dilepton emission using state-of-the-art vector meson spectral functions
Rapp:1999us ; vanHees:2007th ; Rapp:2013nxa . In particular we run UrQMD-CMF
events at a fixed impact parameter of $b=6.6$ fm and apply the coarse graining
method described in Savchuk:2022aev .

From the coarse grained simulation we can directly extract the time dependence
of the dilepton emission rate, at mid-rapidity, and compare it to the time
evolution and decoupling rate of the nucleons. In Fig. 4 the normalized
freeze-out time distributions of the nucleons are shown as a dotted black line
while the emission time distribution of the dileptons is shown as a solid red
line. One can clearly observe that the dilepton emission time distribution is
much more narrow and centered at 10-12 fm. In contrast, the nucleon kinetic
freeze-out time distribution is very broad, peaks at 18-20 fm and is strongly
skewed towards later times. By comparison with the time development of the
directed and elliptic flow shown previously in Figs. 1 and 2 one can see that
the dileptons are emitted exactly during the most hot and dense phase which is
the time when $v_{1}*$ is just generated and $v_{2}$ is positive. The
nucleons, on the other hand, mainly decouple when the elliptic flow of the
whole system has already turned negative due to immense shadowing and the
directed flow has reached its plateau.

A similar idea has been put forward in Refs. Chatterjee:2005de ;
Chatterjee:2007xk ; Chatterjee:2008tp ; Deng:2010pq ; vanHees:2011vb ;
Mohanty:2011fp ; Vujanovic:2013jpa ; Vovchenko:2016ijt  considering RHIC and
LHC energies where the early $v_{2}$ of the partonic phase can be accessed via
measurement of the flow of direct photons e.g. at PHENIX PHENIX:2011oxq  or at
ALICE Lohner:2012ct ; ALICE:2018dti .

We can now compare the elliptic flow of the dileptons with the hadrons at
kinetic freeze-out. In particular the proton, $\pi$ and $\rho$(770) flows are
of special interest. The pions probe the kinetic decoupling stage while the
$\rho$ either decays into a dilepton pair within the medium or into a $\pi\pi$
pair at kinetic freeze-out allowing us to study the final state and early flow
in a mass range not dominated by pions.

Figure 5: [Color online] The transverse momentum dependence of the elliptic
flow of $\pi$ (brown triangles), protons (dark blue circles), final state
$\rho$(770) (orange stars) and dileptons at invariant masses of $M=0.15$ GeV
(red line) and $M=0.6$ GeV (green line) in peripheral Au+Au collisions at a
kinetic beam energy of 1.23$A$ GeV from UrQMD.

Figure 5 displays the transverse momentum dependence of the elliptic flow
$v_{2}$ at mid-rapidity of the emitted dilepton radiation at invariant masses
of $M=0.150$ GeV (dash-dotted red line) and $M=0.600$ GeV (green line) in
comparison to the elliptic flow of protons (dark blue dashed line with
circles), pions (brown dashed line with triangles) and final state $\rho$
mesons (dashed orange line with stars).

As expected the elliptic flow of protons, pions and the $\rho$ mesons is
negative and decreasing with increasing transverse momentum. (The calculations
are also in agreement with the recent HADES measurements for protons,
deuterons, tritons and preliminary pions at mid-rapidity HADES:2022osk .) The
dilepton $v_{2}$ on the other hand shows the opposite behavior: It probes the
system during the most hot and dense phase when the pressure gradients
generate expansion in in-plane direction and thence their elliptic flow is
positive and increases with increasing transverse momentum. The dilepton flow
at both invariant masses increases up to 3% at 1 GeV transverse momentum which
is in good alignment with the elliptic flow of the whole system (cf. Fig. 2).
One can further observe hydrodynamic flow scaling in the dileptons where
larger masses are shifted to higher $p_{\mathrm{T}}$ Fries:2003kq .

We conclude that the above discussed effect of a positive elliptic flow
experienced by all particles in the central collision zone exerted by the
pressure gradients during the most dense phase can be seen in the flow of
vector mesons decaying to dileptons throughout the time evolution. In
contrast, stable hadrons and resonances that can be reconstructed in their
hadronic decay channels are only sensitive to the last generation of particles
at kinetic freeze-out and are therefore subject to a negative observable
elliptic flow value. This allows to measure the nuclear Equation-of-State
directly at 2-3 times saturation density using dileptons in the $\rho$-mass
excess region.

##  IV Discussion

Before we conclude, we would like to address that similar time evolutions of
the elliptic flow where presented in Refs. LeFevre:2016vpp ; Fevre:2019xao ;
Gao:2022shr  within a QMD model at much lower energies, confirming the general
expansion pattern of matter at such collision energies. However, the previous
studies did not make any attempts to link the specific anisotropic expansion
dynamics of the matter at the highest baryon densities to experimentally
accessible observables. Here, we present for the first time two distinct
observables that allow to pin down the expansion of the matter during the most
dense stages, namely dilepton emission and flow correlations. Therefore, we
suggest to scrutinize $v_{1}-v_{2}$ correlations and the positive elliptic
flow of dileptons in the region of the $\rho$ mass as distinct signals for an
initial expansion in positive $x$-direction, which is later turned by
shadowing into a negative elliptic flow with an expansion predominantly along
the $y$-direction.

##  V Conclusion

We presented a detailed study of the time evolution of flow in Au-Au
collisions at a beam energy of $1.23A$ GeV employing the Ultra-relativistic
Quantum-Molecular-Dynamics model (UrQMD v3.5). The investigations revealed
that, similar to known flow patterns at ultra-relativistic beam energies, the
initial overlap geometry does lead to a positive elliptic flow during the
early evolution of the system. This positive initial $v_{2}$ acts as the
source of the directed flow, $v_{1}$. The negative $v_{2}$, observed in the
final state hadrons, is mainly due to shadowing effect. We suggest to use
$v_{1}-v_{2}$ correlations to probe the described connection between the
different flow components and to confirm the predicted initial state expansion
geometry. We further suggest to use dileptons as another independent probe to
observe the early expansion of the matter in $x$-direction, which will be
signaled by a positive $v_{2}$ for dileptons in contrast to a negative $v_{2}$
(already observed) for protons.

As the early flow is a direct reaction of the system to the initial pressure
gradient, and thus the Equation-of-State at the highest densities, the
measurement of the dilepton elliptic flow may provide direct access to the
high density EoS of QCD matter in a clearer way than possible before.

###### Acknowledgements.

The authors thank Behruz Kardan and Christoph Blume as well as Arnaud Le Fèvre
and Jörg Aichelin for fruitful discussion about ther previous work. This
article is part of a project that has received funding from the European
Union’s Horizon 2020 research and innovation programme under grant agreement
STRONG – 2020 - No 824093. OS thanks the GSI for funding. Computational
resources were provided by the Center for Scientific Computing (CSC) of the
Goethe University and the “Green Cube” at GSI, Darmstadt. This project was
supported by the DAAD (PPP Thailand). This research has received funding
support from the NSRF via the Program Management Unit for Human Resources &
Institutional Development, Research and Innovation [grant number B16F640076].

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