Constraining the equation of state with identified particle spectra
Akihiko Monnai Jean-Yves Ollitrault
nucl-th hep-lat hep-ph nucl-ex






Institut de physique théorique, Université Paris Saclay, CNRS, CEA, F-91191 Gif-sur-Yvette, France 


Institut de physique théorique, Université Paris Saclay, CNRS, CEA, F-91191 Gif-sur-Yvette, France 



We show that in a central nucleus-nucleus collision, the variation of
the mean transverse mass with the multiplicity is 
determined, up to a 
rescaling, by the variation of the energy over entropy ratio as a
function of the entropy density, thus providing a direct link between
experimental data and the equation of state. 
Each colliding energy thus probes the equation of state at an
effective entropy 
density, whose approximate value is 19 fm^-3 for Au+Au collisions
at 200 GeV and 41 fm^-3 for Pb+Pb collisions at 
2.76 TeV, corresponding to temperatures of 227 MeV and 279 MeV if
the equation of state is taken from lattice calculations.  
The relative change of the mean transverse mass as a function of the 
colliding energy gives a direct measure of the pressure over energy
density ratio P/ϵ, at the corresponding effective density. 
Using RHIC and LHC data, we obtain P/ϵ=0.21± 0.10, in
agreement with the lattice value P/ϵ=0.23 in the corresponding
temperature range. 
Measurements over a wide range of colliding energies using a single detector
with good particle identification would help reducing the error. 


Constraining the equation of state with identified 
  particle spectra
    Jean-Yves Ollitrault
    December 30, 2023
======================================================================






§ INTRODUCTION


One of the motivations for studying 
nucleus-nucleus collisions at high energies is to probe experimentally
the equation of state of QCD matter <cit.>. 
Ultrarelativistic collisions probe the phase diagram at vanishing
chemical potential: at high temperatures, hadrons merge into a
quark-gluon plasma.   
It was originally hoped that this change occurred through a
first-order phase transition <cit.>.  
However, it was progressively understood that it is a smooth, analytic 
crossover <cit.>, and that a phase
transition, if any <cit.>, can only take place at high
baryon density <cit.>. 
The equation of state of baryonless QCD matter is now known 
precisely from lattice simulations with 
physical quark masses <cit.>. 
The goal of this paper is to understand the imprints of the equation
of state on heavy-ion data, in particular transverse momentum
spectra. 

Relativistic hydrodynamics <cit.> plays a central role in
our understanding of heavy-ion observables in the soft sector.  
Its simplest version is ideal hydrodynamics <cit.>, which
describes most of the qualitative features seen in transverse
momentum spectra, elliptic flow, and interferometry
radii <cit.>. This simple description can be refined
by taking into account finite-size corrections due to
viscosity <cit.> which are important for azimuthal
anisotropies <cit.>. 
The equation of state lies at the core of the hydrodynamic
description, and the vast majority of modern hydrodynamic
calculations <cit.>,
which give a satisfactory description of soft observables, 
use as an input an equation of state from lattice QCD calculations. 

While the success of hydrodynamics suggests that 
equilibration takes place to some 
degree <cit.>, 
most dynamical calculations predict that the system produced in the
early stages of a heavy ion collision is far from chemical
equilibrium, typically with overpopulation in gluon
numbers <cit.> and underpopulation in quark
numbers <cit.>.  
The resulting effective equation of state might differ significantly
from that calculated in lattice QCD, and it is important to understand
what experimental data tell us about the equation of state,
beyond a comparison between different lattice
results <cit.>.
It has been recently shown that a simultaneous fit of several observables to
hydrodynamic calculations constrains the equation of state
to some extent <cit.>.
However, this recent study uses a
systematic, Bayesian framework, and the nature of the 
relationships between model parameters and observables remains obscure. 
Further Bayesian studies have
shown <cit.> that  
interferometry radii and transverse momentum spectra are the 
observables which are most sensitive to the equation of state, but
they are still unable to provide a simple picture of how this
dependence takes place. Another related approach is 
to use a deep learning method to distinguish the crossover and first-order
phase transitions in equations of state from heavy-ion particle 
spectra <cit.>.

We show that for central collisions, the variation of the mean
transverse mass per particle as a function of the multiplicity density
dN/dy (which itself depends on the collision energy √(s)) 
reproduces, up to proportionality factors, the variation of energy
over entropy ratio ϵ/s as a function of the entropy density
s <cit.>.
We illustrate our point by discussing an ideal experiment in
Sec. <ref>. 
We then carry out detailed hydrodynamic simulations using a variety of
equations of state. 
The equations of state are presented in Sec. <ref>. 
Results from hydrodynamic calculations are discussed in
Sec. <ref>. 
Calculations are compared with experimental data from RHIC and LHC 
in Sec. <ref>. 




§ AN IDEAL EXPERIMENT
 


In order to illustrate our picture, we first describe a simple 
ideal experiment: the fluid is initially at rest in thermal
equilibrium at temperature T_0 in a container of arbitrary shape, and
large volume 
V. At t=0, the walls of the container disappear and the fluid
expands freely into the vacuum. If V is large enough, 
this expansion follows the laws of ideal hydrodynamics. 
At some point, the fluid transforms into N particles. We assume for
simplicity that this transformation occurs at a single freeze-out
temperature T_f <cit.>.

The thermodynamic properties at the initial
temperature T_0 can be easily be reconstructed by measuring the
energy E and the number of particles N at the end of the
evolution, provided that the initial volume V is known. 
The total energy E is conserved throughout the evolution,
hence the initial energy density is: 

    ϵ(T_0)=E/V.

For simplicity, we assume throughout this paper that the net baryon
number is negligible (which corresponds to high-energy
collisions) so that the energy density depends solely on the
temperature. 

The initial entropy density can be inferred from the final number of
particles N. 
Ideal hydrodynamics conserves the total entropy S. 
The fluid is transformed into particles at the freeze-out temperature
T_f, and the multiplicity N is directly proportional to the
entropy.[Both the multiplicity N and the entropy S are scalar quantities,
hence, the entropy per particle only depends on the freeze-out
temperature T_f, not on the fluid velocity.] 
Therefore, the initial
entropy density is related to the final multiplicity through the
relation:

    s(T_0)= (S/N)_T_fN/V,

The volume dependence cancels in the energy per particle: 

    ϵ(T_0)/s(T_0)=(N/S)_T_fE/N,

One can repeat the experiment for several values of the initial 
density, and plot the energy per particle E/N as a function of N/V. 
One thus obtains a plot of ϵ/s versus s, 
which gives access to the equation of state. 
Note that Eqs. (<ref>) and (<ref>) do not involve
the fluid velocity pattern, 
which depends on the shape of the initial volume. 
Hydrodynamic modeling only enters through the entropy per particle at
freeze-out (S/N)_T_f. This ideal experiment thus allows one to measure
the equation of state for temperatures larger than T_f. 
Based on a similar picture, Van Hove <cit.> argued that
the transition from a hadronic gas to a quark-gluon plasma should
result in a flattening of the mean transverse momentum ⟨
p_T⟩ as a function of the multiplicity.
It has been recently attempted to extract an approximate equation of
state from recent pp and pp̅ collision data  on this
basis <cit.>.

The little liquid produced in an ultrarelativistic nucleus-nucleus
collision has similarities with this ideal experiment if one cuts a
thin slice perpendicular to the collision axis and looks at its
evolution in the transverse plane. 
The initial transverse velocity is initially zero, and the fluid 
expands freely into the vacuum right after the collision takes place. 
The two main differences are:


  * The initial temperature profile is not
uniform in a box but has a non-trivial transverse structure.

  * The slice expands in the longitudinal direction and its energy
  decreases as a result of the work of the longitudinal
  pressure <cit.>  exerted by neighboring slices: dE=-PdV.

As we shall see, both effects can be taken care of by appropriately
redefining the volume V and the temperature T_0, and replacing the
energy per particle E/N with the mean transverse mass,
where the transverse mass is defined by m_T=√(p_T^2+m^2). 
Eqs. (<ref>) and (<ref>) are replaced with: 

    s(T_ eff)   =    a 1/R_0^3dN/dy, 
    ϵ(T_ eff)/s(T_ eff)   =   b⟨ m_T⟩,

where R_0 is a measure of the transverse radius, which will be
defined in Sec. <ref>, T_ eff is an effective 
temperature taking into account the longitudinal cooling 
(T_ eff<T_0), and dN/dy is the multiplicity per unit rapidity,
and a and b are dimensionless parameters whose values are
independent of the equation of state and of the colliding energy. 
Their values will be determined in Secs. <ref> using
hydrodynamic calculations, which take into account the 
longitudinal cooling and the inhomogeneity of the initial profile. 

By measuring the mean transverse mass and the multiplicity density in
a given system at different colliding energies, one obtains the
variation of ⟨ m_T⟩ as a function of dN/dy. 
Neglecting the energy dependence of the transverse size R_0 
(this will be justified in Sec. <ref>), 
the slope of this curve in a log-log
plot is the ratio of pressure over energy density, P(T_
  eff)/ϵ(T_ 
  eff) <cit.>. 
Using Eqs. (<ref>), one obtains

    dln⟨ m_T⟩/dln dN/dy=
    .dlnϵ-dln s/dln s|_T_ eff=
    .P/ϵ|_T_ eff,
 
where we have used the thermodynamic identities dϵ=Tds and
ϵ+P=Ts. 
Note that the dependence on the unknown coefficients a and b
cancels in this expression. 
One thus obtains a measure of the ratio P/ϵ of 
the quark-gluon matter produced in the collision from data alone. 
The entropy density s(T_ eff) at which this ratio is measured,
however, depends on the  coefficient a, which can only be obtained
through detailed hydrodynamic simulations. These will be carried out in
Sec. <ref>. 





§ EQUATIONS OF STATE







The equation of state of QCD is characterized
by a transition  from a hadronic, confined system at low temperatures
to a phase dominated by colored degrees of freedom at high
temperatures. It has been determined precisely through lattice
calculations <cit.>. 
Lattice calculations are carried out at zero baryon chemical
potential, and the matter produced at central rapidity in high-energy
collisions also has small net baryon number. We therefore choose to 
neglect net baryon density in the present study. 

In lattice calculations, one first computes the trace
anomaly I≡ϵ-3P  as a function of the
temperature T, where ϵ is the energy density and P the
pressure. Other quantities are then determined through the
thermodynamic relations:  

    P/T^4   =   ∫_0^T I/T^5 dT , 
    ϵ   =   I+3P , 
    
    s   =   ϵ+P/T.

The equation of state used in hydrodynamic calculations is
constrained, on the low-temperature side, by the condition that it
matches that of the hadron resonance gas created at
the end of the evolution <cit.>. 
All the equations of state used in this paper match the hadron
resonance gas for temperatures smaller than 140 MeV, which is the
freeze-out temperature of our hydrodynamic calculation. 
We choose to vary the high-temperature part along two different
directions: either by varying the high-temperature limit of P/T^4,
which is proportional to the number of degrees of freedom of the
quark-gluon plasma (denoted as equation of state (EOS) A, B, L and C in Fig. <ref> (a) where EOS L corresponds to the lattice QCD-based equation of state), or by varying the
temperature range over which the transition occurs (denoted as equation of state (EOS) D, E, L and F in Fig. <ref>
(b)). The parameterization is explicated in
Appendix <ref>. 
We thus span a range of equations of state around the lattice value. 
Note that the error on P/T^4 from lattice calculations is smaller
than 0.1 for all T <cit.>. We explore a much
wider range of equations of state. 






According to the picture outlined in Sec. <ref>, 
heavy-ion collisions measure the variation of the 
energy over entropy ratio as a function of the
entropy density. 
This variation is displayed in Fig. <ref> for the various
equations of state displayed in Fig. <ref>. 
Note that the ratio ϵ/s is closely related to the
temperature <cit.>:

    3T/4<ϵ/s<T,

where the lower bound corresponds to the ideal gas limit
P=ϵ/3 and the upper bound to P=0. 
Thus, the variation of  ϵ/s as a function of s is
essentially the variation of the temperature with the entropy density. 
In the high-temperature phase, s∝ν T^3, where ν is the
effective number of degrees of freedom  of the quark gluon plasma. 
More degrees of freedom implies a smaller temperature, for the same entropy
density, which explains why the order of the curves is inverted in
Fig. <ref> compared to Fig. <ref>. 




§ HYDRODYNAMIC CALCULATIONS



In this section, we carry out hydrodynamical simulations in order to
determine the mapping between observables and the equation of state
according to Eq. (<ref>). 
We model the evolution of the fluid near midrapidity and assume
boost invariance in the longitudinal 
direction <cit.>. 
We solve the transverse expansion numerically using a (2+1)-dimensional 
code <cit.>. 
The initial transverse velocity is assumed to be zero  at the proper
time τ_0=0.4 fm/c at which the hydrodynamic expansion starts. 
This small value of τ_0 accounts for the early transverse
expansion <cit.>, 
irrespective of whether or not hydrodynamics is applicable at early 
times <cit.>. 

Initial conditions are defined by the initial transverse density profile. 
The most important quantity involving initial conditions in this
study is the effective radius R_0 defined by: 

    R_0^2≡ 2(⟨ | x|^2⟩-|⟨
      x⟩|^2),

where x is the position in the transverse plane, and 
angular brackets denote an average value weighted with the 
initial entropy density: 

    ⟨ F( x)⟩≡∫ F( x) s( x,τ_0)d^2 x/∫ s( x,τ_0)d^2 x.

The normalization factor 2 in Eq. (<ref>) ensures that one
recovers the correct result for a uniform entropy density profile within a
circle of radius R_0. 

In the ideal experiment described in Sec. <ref>, the
mapping between observables and the equation of state is independent
of the shape of the initial volume. 
For this reason, one expects that most of the dependence on the shape
of the initial density profile is through the radius R_0. 
This has been checked in detail in studies of transverse momentum
fluctuations <cit.>, where it was shown
that the mean transverse momentum in hydrodynamics is sensitive to
initial state fluctuations only through fluctuations of R_0. 
We have checked it independently by comparing two standard models of 
initial conditions, the Monte Carlo Glauber model <cit.> and the
MCKLN <cit.> model, as will be explained below. 
The default setup of our hydrodynamic calculation uses a Monte Carlo
Glauber simulation of 0-5% most central Au+Au collisions 
where the energy density is a sum of contributions of
binary collisions, and the contribution of each collision is a
Gaussian of width 0.4 fm centered half way between the colliding
nucleons. The resulting density profile is centered, and then averaged over a
large number of events in order to obtain a smooth profile <cit.>. 
The normalization of the density profile determines the multiplicity
dN/dy. We run each calculation with 5 different normalizations
spanning a range which covers the LHC and RHIC data which will be used in Sec. <ref>. 









 §.§ Ideal hydrodynamics




We first carry out ideal hydrodynamic simulations 
for all the equations of
state displayed in Fig. <ref>. 
The fluid is converted into hadrons through the standard Cooper-Frye
freeze-out procedure <cit.> at a temperature T_f=140 MeV. 
We include all hadron resonances with M<2.25 GeV, 
and compute ⟨ m_T⟩ and dN/dy directly at freeze-out,
before resonances decay. Our goal here is to mimic as closely as 
possible the ideal experiment outlined in Sec. <ref>. 

The symbols in Fig. <ref> correspond to the right-hand side 
of Eq. (<ref>), where the
dimensionless parameters a and b have been fitted 
to achieve the best possible agreement with the 
left-hand side. 
There are 5 points for each equation of state, which
correspond to different initial temperatures. 
The overall agreement is excellent, and shows that the variation of 
⟨ m_T⟩ as a function of (1/R_0^3)(dN/dy) is determined
by the equation of state. 

In order to test that this mapping is independent of initial
conditions, we have carried out a calculation with MCKLN initial
conditions. 
While both models give values of R_0 that differ by 5%, they yield
the same value of ⟨ m_T⟩ when compared at the same value
of (1/R_0^3)dN/dy. 


Let us now comment on the order of magnitude of the fit parameters a
and b. 
First, compare Eq. (<ref>) and the second line of
Eq. (<ref>). 
The entropy per particle at freeze-out before decays is
(S/N)_T_f=6.5 in this calculation.
The transverse mass of a particle is smaller than its energy, since it
does not include the longitudinal momentum p_z. 
The relevant longitudinal momentum here is that relative to the fluid,
which cannot be measured, since data are integrated over all 
fluid rapidities. 
The value of b=0.202  is slightly larger than (N/S)_T_f=0.154, 
and thus compensates for the loss of longitudinal momentum. 

We now discuss the order of magnitude of a. The main difference
between the ideal experiment described in Sec. <ref> and
the real experiment is that the energy of the fluid slice decreases 
as a result of the work done by the longitudinal pressure. 
In ideal hydrodynamics, this cooling is only significant at 
early times: After the transverse expansion sets in, the pressure
decreases very rapidly, the work becomes negligible
and the energy stays constant. 
A rough, but qualitatively correct, picture is that the expansion is
purely longitudinal during a time τ_ eff and that the energy is
conserved for τ>τ_ eff <cit.>. 
For dimensional reasons, τ_ eff=fR_0, where f is of order
unity. 
The volume at τ_ eff is V=π R_0^2 τ_ eff=π fR_0^3.
Inserting this value into Eq. (<ref>) and identifying the
right-hand side with the first line of Eq. (<ref>), one obtains
f≃ 0.5, in agreement with the value obtained in previous
calculations <cit.>. 
Ideal hydrodynamics thus probes the equation of state at a time
τ_ eff∼ 0.5R_0, which is the typical time at which transverse 
flow and elliptic flow develop <cit.>.





 §.§ Resonance decays





The largest correction to the naive ideal fluid picture comes from
decays occurring through strong or electromagnetic interactions, 
which occur after freeze-out, but before the 
daughter particles reach the 
detectors.
We compute particle spectra after strong and 
electromagnetic decays, but before weak decays.
Decays are treated in Ref. <cit.>, by assuming
that the decay rate is proportional to the invariant phase space. 
After decays, the only remaining particles are pions, kaons, nucleons
and strange baryons. 
In this preliminary study, we neglect strange baryons, which are a
small fraction of the total number of particles, and are 
identified in separate analyses <cit.>. 
We therefore evaluate the multiplicity dN/dy and the mean transverse
mass including only pions, kaons, and (anti)nucleons, both charged and
neutral. 
As shown in Fig. <ref>, 
decays increase the multiplicity by 40%. 
They also conserve the total energy, so that ⟨ m_T⟩
decreases, while the product ⟨ m_T⟩ dN/dy only changes 
by a few percent.  




Since the increase of dN/dy due to decays depends solely on the
freeze-out temperature, but is independent of the colliding energy and
the equation of state, 
decays amount to further rescalings of ⟨ m_T⟩ and 
dN/dy. They can be taken into account by modifying the values of
the coefficients a and b in Eq. (<ref>).
We again determine the values of a and b through a simultaneous
least-square fit to all equations of state. The result is shown in 
Fig. <ref>, where only the equations of state of
Fig. <ref> (a) are shown. 
After rescaling, the effective entropy density of the fluid is
unchanged: locations of symbols in Fig. <ref> (a) and
Fig. <ref> are identical to within less than 0.5%. 
The fact that they are identical confirms that Eqs. (<ref>)
reconstruct thermodynamic properties of the fluid. 

A more realistic description of the hadronic stage should include not
only decays, but also rescatterings, for instance by coupling
hydrodynamics to a transport
code <cit.>. 
It has been recently shown <cit.> that transverse momentum
spectra are remarkably independent of the temperature at which one
switches from the hydrodynamic to the transport description, which
implies that our results would be unchanged if we switched from a
hydrodynamic description to a transport calculation at a temperature
larger than 140 MeV. Below 140 MeV, effects of hadronic scatterings
are suppressed due to the lower density. Our choice of T_f allows us
to roughly reproduce observed particle ratios, in agreement with 
Ref. <cit.>. This is important as the mean m_T, averaged
over all particle species, strongly depends on particle ratios. 





 §.§ Viscosity



We finally study viscous corrections to the ideal fluid picture.
We use “minimal” shear viscosity η/s=1/4π <cit.>
and bulk viscosity ζ/s = 2 (1/3 - c_s^2) η/s <cit.>
based on the gauge-string correspondence, where
c_s is the sound velocity. The relaxation times are also
conjectured in the holographic approach <cit.>.
Viscosity modifies the equations of motion of the
fluid <cit.>, 
and the momentum distribution of particles at
freeze-out <cit.>. 
We show both effects separately in Fig. <ref>.
The main effect of viscosity is to 
increase the multiplicity for a given initial condition, which is
a consequence of the entropy increase due to dissipative processes. 
On the other hand, the value of ⟨ m_T⟩ changes little, 
which is due to a partial cancellation between effects of shear
viscosity (which increases ⟨ m_T⟩) and bulk viscosity 
(which decreases ⟨ m_T⟩) <cit.>.


The values of ⟨ m_T⟩ and dN/dy can again be matched to
the equation of state through Eqs. (<ref>). 
We again determine the values of a and b which give the best
simultaneous fit to all equations of state. 
The result is displayed in Fig. <ref>, where only the
equations of state of Fig. <ref> (a) are shown. 
This figure shows that viscous corrections do not alter qualitatively
the ideal fluid picture, and that the variation of the mean
transverse mass with the multiplicity density is still driven by the
equation of state in the presence of viscosity. 
Comparison with Fig. <ref> shows that symbols are shifted to
the right, which means that for the same initial temperature, viscous
hydrodynamics results in a higher effective entropy density. 
The reason is that entropy is produced in the off-equilibrium
processes. 

The actual value of the shear and bulk viscosity are not known
precisely. Since a and b depend slightly on the viscosity, 
the uncertainty on the viscosity translate into an uncertainty on the
mapping of experimental data onto the equation of state through
Eq. (<ref>). 
The upper bound on constant η/s from heavy-ion data is typically
0.2 <cit.>. It has been recently noted that the
inclusion of bulk viscosity tends to lower the preferred value 
of the shear viscosity <cit.>, so that η/s<0.2 seems
conservative. 
We assume that viscous corrections are proportional to the viscosity, 
therefore the uncertainty can be inferred from the difference between
our viscous and ideal calculations. 
The uncertainty on a is 7% and amounts on an uncertainty on
the effective entropy density s_ eff. 
The uncertainty on b is 11% and is essentially an uncertainty on the
corresponding temperature. 
Note, however, that the dependence on a and b cancels in the
logarithmic slope, Eq. (<ref>), and the ratio
P/ϵ can be determined precisely even if transport
coefficients are not precisely determined. 



§ COMPARISON WITH DATA

 





We now discuss to what extent existing data constrain the equation of
state. 
Both dN/dy and ⟨ m_T⟩ require spectra of pions, kaons
and protons. 
Such data have been published by STAR <cit.> and
PHENIX <cit.> at the Relativistic Heavy Ion Collider
(RHIC) and by ALICE <cit.> at the Large Hadron Collider (LHC).
PHENIX and ALICE data for protons are corrected for the contamination
from weak Λ decays, while STAR data are not. 
We correct STAR data by assuming that a fraction 35%± 10% of
protons come from Λ decays, as determined by the PHENIX
analysis <cit.>. 
Particles are only identified within a limited p_T range, which
depends on the experiment, and spectra must be extrapolated in order
to obtain dN/dy and ⟨ m_T⟩. 
These extrapolations are discussed in Appendix <ref>. 
The data we use are for charged particles, and we need ⟨
m_T⟩  and dN/dy for all hadrons, including neutral ones. 
Yields of neutral particles are obtained assuming isospin symmetry. 
The resulting values of ⟨ m_T⟩ and dN/dy are given in
Table <ref>. 
For 200 GeV, we include both STAR and PHENIX measurements, which are
slightly different, but compatible within errors. 

In order to convert the multiplicity dN/dy into a density, one needs
an estimate of the initial transverse size R_0. 
This quantity, which represents the mean square radius of the initial
density profile, is not measured and can only be estimated in a
model. As we shall see, it turns out to be the largest source of uncertainty
when constraining the equation of state from data. 
In particular, the uncertainty from R_0 is larger than the
uncertainty from transport coefficients. 

We discuss how we estimate R_0. 
Note that the transverse size fluctuates event to event, even in a narrow
centrality window <cit.>.
Ideally, we would like to estimate the average value over events of
(1/R_0^3)dN/dy. Since the input available from experiment is an average
of dN/dy, for the sake of simplicity, we estimate the average value of
R_0 over many events to divide dN/dy for our analyses.
We use the same Monte Carlo Glauber model as in our hydrodynamic
calculation. 
The resulting values, averaged over many events, are given in
Table <ref>. The 
MCKLN model <cit.> gives values 5% smaller, which
implies that the density is 15% larger. 
This shows that the uncertainty on the transverse size is significant. 


However, the variation of R_0 with colliding energy for a given
system is small, so that the evolution of the density is mostly
driven by the increase in the multiplicity dN/dy. 
Therefore, uncertainties on R_0 cancel 
when comparing two different collision energies. 
The variation of the mean transverse mass with dN/dy directly gives
the ratio P/ϵ, as shown by Eq. (<ref>). 
As pointed out in Sec. <ref>, 
uncertainties from the viscosity also cancel in this 
energy dependence. Using PHENIX and ALICE data, which span a wide
range of dN/dy, and taking into account the different sizes of Au
and Pb nuclei, Eq. (<ref>) gives

    .P/ϵ|_T_ eff=0.21± 0.10,

where the error is solely from experiment. 

The only significant theoretical uncertainty is on the effective
temperature T_ eff at which this ratio is measured.  
We provide in Table <ref> the values of the effective
entropy density s_ eff given by Eq. (<ref>), where a is
given by our viscous hydrodynamic calculation. 
The value at 5.02 TeV, where identified particle spectra are not yet
published, is obtained by assuming that the relative
increase in dN/dy from 2.76 TeV equals that of
dN_ch/dη, that is, 20% <cit.>. 
As discussed in Sec. <ref>, the uncertainty on s_
  eff from transport coefficients is 7%, and that 
from the transverse size R_0 is at least 15%. 

The value of the temperature T_ eff corresponding to s_
  eff  can only be obtained if the equation of state is known. 
The values in the last column of Table <ref> correspond to
the lattice equation of state. 
Lattice calculations give P/ϵ=0.23 for a temperature half-way
between the values of T_ eff corresponding to 200 GeV and
2.76 TeV. The experimental value, Eq. (<ref>), is
compatible with the lattice result. 
Experiments at √(s)=5.02 TeV, for which identified particle
spectra are yet unpublished, will probe the equation of state at a
temperature close to 300 MeV. 
Note that the theoretical uncertainty of ≃ 20% on s_ eff
translates into an uncertainty  ∼ 15 MeV on the effective
temperature at the LHC, which is dominated by the uncertainty on the
initial transverse radius R_0. 





Figure <ref> shows the comparison between experimental data
and the values obtained from the equation of state through
Eqs. (<ref>), where a and b are taken from our viscous
hydrodynamic calculation (see Fig. <ref>). 
With the minimal viscosity chosen in this calculation, LHC data
slightly favor the equation of state C, which has a larger pressure
than the lattice equation of state. With a higher viscosity, however,
the lattice equation of state would be preferred. 
Equations of state A and B are ruled out: as already well known,
heavy-ion data favor a soft equation of state. 
Note that current experiments only probe the equation of
state up to T∼ 300 MeV (see Table <ref>). 




§ CONCLUSIONS



We have shown that in central nucleus-nucleus collisions, the
variation of the mean 
transverse mass as a function of the multiplicity density is, up to
rescaling factors, driven by the variation of the energy over entropy
ratio ϵ/s as a function of the entropy density s. 
Each collision energy probes the equation of state at a different
entropy density s_ eff, which corresponds 
roughly to the average density at a time 
τ_ eff∼ 3 fm/c. RHIC and LHC experiments 
probe the equation of state for temperatures up to ∼ 300 MeV.

The largest source of uncertainty at the theoretical level is the initial
transverse size R_0. 
The uncertainty from unknown transport coefficients (shear and bulk
viscosity) is twice smaller. 
These theoretical uncertainties cancel if one measures the evolution of the mean 
transverse mass as a function of collision energy, which gives direct
access to the pressure over energy density ratio P/ϵ of the
quark-gluon plasma. 

This analysis requires precise experimental data on identified
particle spectra. 
One could think of replacing the transverse mass with the transverse
momentum, and the rapidity by the pseudorapidity, which was the
original idea of Van Hove <cit.>, and would allow to
work with unidentified particles. However, we have checked that the
mapping onto the equation of state is not as good in this case. 

The  value of P/ϵ obtained from the evolution of spectra from
RHIC to LHC energies is compatible with the lattice equation of 
state, but with large errors. Carrying out an energy scan at the LHC
with a single detector would greatly improve the quality of the
measurement. 













§ ACKNOWLEDGEMENTS

We thank Matt Luzum and Michele Floris for discussions and Jean-Paul
Blaizot for useful comments on the manuscript. 
AM is supported by JSPS Overseas Research Fellowships.







§ VARYING THE EQUATION OF STATE
 



The equation of state is constructed by connecting the trace 
anomaly of the hadron resonance gas model smoothly to 
that of lattice QCD <cit.>. To systematically 
generate variations of the equation of state, modification 
is made through two factors c_w and c_h in the QGP 
phase for our analyses. The expression reads:

    I(T)    =   1/2[ 1 - tanh(T-T_s/Δ T_s) ] I_HRG(T) 
       +   c_h/2[ 1 + tanh(T-T_s/Δ T_s) ] I_lat(T_w),

where T_w = T_s + c_w(T-T_s). c_w and c_h are 
associated with the width and the magnitude of I(T) in the 
QGP phase, respectively. c_w = 1 and c_h = 1 
recover the lattice QCD result. The hadronic equation of state 
is left untouched because, as mentioned earlier, the 
Cooper-Frye formula requires that kinetic theory reproduces 
the equation of state used in the hydrodynamic model at 
freeze-out for energy-momentum conservation. When one 
chooses T_s = 160 MeV and Δ T_s = 0.1 T_s, 
this is satisfied at and below T = 140 GeV.





The pressure is obtained through the thermodynamic relations
(<ref>). Since the trace anomaly is integrated, c_w 
and c_h have to be modified simultaneously to shift the pseudo-critical 
temperature and change the effective number of degrees 
of freedom in the pressure or the entropy density (Fig.<ref>). 


We first consider a set of equation of state with different numbers 
of QGP degrees of freedom by choosing (c_w,c_h) = 
(2,0.5), (1.5, 0.75), (1,1), and (0.5,1.25). They are labeled 
as EOS A, B, L and C, respectively. The normalized pressure 
as a function of the temperature for each equation of state is plotted 
in Fig. <ref> (a). It is note-worthy that we consider an 
equation of state which exceeds the Stefan-Boltzmann limit 
with the last parameter set (0.5,1.25). We also vary the 
pseudo-critical temperature by setting the parameters to 
(c_w,c_h) = (2,1.5), (1.5, 1.25), (1,1), and (0.5,0.75) as 
shown in Fig. <ref> (b), which are labeled as EOS D, E, L and F. 
The equation of state becomes harder for larger T_c 
because it is fixed on the hadronic side.




§ IDENTIFIED PARTICLE SPECTRA AT RHIC AND LHC
 


In order to estimate the mean transverse mass per particle from
experimental data, we use as input p_T spectra of 
identified charged hadrons in the central rapidity region. 
More specifically, we use data for charged pions, charged kaons,
protons and antiprotons, which are shown as symbols in
Fig. <ref>.  
These plots show the probability distribution of p_T near
midrapidity, dN/dp_Tdy. 
Experimental data are shown as symbols. 
Pion and kaon yields increase smoothly with
collision energy as expected. 
This does not appear to hold for proton and antiprotons, but the
reason is simply that STAR data for protons and antiprotons include,
in addition to primary particles, secondary products of weak Λ
and Λ̅ decays. 
Apart from this difference, PHENIX and STAR data at 200 GeV are
compatible within error bars.  

The effect of the net baryon number becomes visible at the lower
energies:  it results in more protons than antiprotons at midrapidity,
and also slightly more K^+ than K^- because 
the strangeness chemical potential is non-vanishing in 
the presence of the net baryon chemical potential μ_B owing to 
the strangeness neutrality condition. 
While the differences between particles and antiparticles are linear
in μ_B, the total multiplicities
are even functions of μ_B, hence effects of net baryon number only
appear to order μ_B^2. We assume that they are negligible down to
62.4 GeV.  


Particles are identified only in a limited p_T range which depends on
the experiment. In order to evaluate the mean m_T, we need to 
extrapolate the measured spectrum to the whole p_T range. 
These extrapolations are done with blast-wave fits <cit.>. 
For ALICE data, we fit each particle species 
independently, as in the experimental paper <cit.>. 
The resulting values of dN/dy and ⟨ p_T⟩ are given in
Tables <ref> and <ref>. They are very close to
the values in the experimental paper. The small differences, which are
much smaller than 
error bars, can be ascribed to different fitting algorithms. 
For sake of consistency, we also use blast-wave fits to extrapolate PHENIX
data <cit.>. 
The resulting values of dN/dy and ⟨
p_T⟩ differ somewhat from the experimental values which use a
different extrapolation scheme, but are compatible within error bars. 
For STAR data, the p_T range is too limited to fit each particle
species independently: therefore, we follow the recommendation of the
experimental paper <cit.> and carry out a simultaneous
fit for kaons and (anti)protons. 
For pions, however, we carry out an independent blast-wave fit as for 
PHENIX data. 
Agreement between STAR and PHENIX pion yields at 200 GeV is much
better than in the corresponding experimental papers, which suggests
that the differences were mostly due to the different extrapolation 
methods. 

Finally, the values of ⟨ m_T⟩, which are needed in this
paper, are listed in Table <ref>. 























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