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PoS(CORFU2018)200

Evolution of multiplicity ﬂuctuations in heavy ion

collisions

Radka Sochorová∗

ˇ

Ceské vysoké uˇcení technické v Praze, FJFI, Bˇrehová 7, 115 19 Praha 1, Czech Republic

E-mail: sochorad@fjfi.cvut.cz

Boris Tomášik

Univerzita Mateja Bela, Tajovského 40, 97401 Banská Bystrica, Slovakia

ˇ

Ceské vysoké uˇcení technické v Praze, FJFI, Bˇrehová 7, 115 19 Praha 1, Czech Republic

E-mail: boris.tomasik@fjfi.cvut.cz

The evolution of multiplicity distribution of a species which undergoes chemical reactions can

be described with the help of a master equation. We study the master equation for a ﬁxed tem-

perature, because we want to know how fast different moments of the multiplicity distribution

approach their equilibrium value. We particularly look at the 3rd and 4th factorial moments and

their equilibrium values from which central moments, cumulants and their ratios can be calcu-

lated. Then we study the situation in which the temperature of the system decreases. We ﬁnd

out that in the non-equilibrium state, higher factorial moments differ more from their equilibrium

values than the lower moments and that the behaviour of the combination of the central moments

depends on the combination we choose. If one chooses to determine the chemical freeze-out

temperature from the measured values of higher moments, these effects might jeopardise the cor-

rectness of the extracted value.

Corfu Summer Institute 2018 "School and Workshops on Elementary Particle Physics and Gravity"

(CORFU2018)

31 August - 28 September, 2018

Corfu, Greece

∗Speaker.

c

Copyright owned by the author(s) under the terms of the Creative Commons

Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/

PoS(CORFU2018)200

Evolution of multiplicity ﬂuctuations in heavy ion collisions Radka Sochorová

1. Motivation

The main motivation of this work is that measured moments of the multiplicity distribution

for various sorts of particles are used for the determination of the hadronisation parameters of hot

QCD matter in ultrarelativistic heavy-ion collisions. We assume that there are still some inelas-

tic scatterings after hadronisation that may drive the multiplicity distribution out of equilibrium.

We demonstrate how the different moments depart away from their equilibrium values. If such

moments were measured and interpreted as if they were equilibrated, we would obtain different ap-

parent temperatures from different moments. For the description of the evolution of the multiplicity

distribution we use a master equation. Because our aim is to study the ﬂuctuations of multiplicities,

we actually study an ensemble of ﬁreballs and the time evolution of the multiplicity distribution

across the ensemble.

2. Relaxation of factorial moments

We consider a binary reversible process a1a2←→b1b2with a6=b. Such a reaction is relevant

for the investigation of rare species production. We note that the involved species are not identical

to each other and it is important to say that b-particles carry conserved charge while a-particles do

not. We will also assume that we have a sufﬁciently large pool of a-particles. The pool basically

does not change during this chemical process.

Now we can write the master equation [3] for P

n(t), the probability of ﬁnding npairs b1b2. It

has the following form

dP

n(t)

dt =G

VhNa1ihNa2i[P

n−1(t)−P

n(t)] −L

Vhn2P

n(t)−(n+1)2P

n+1(t)i,(2.1)

where n=0,1,2,3,... and Vis proper volume of the reaction. For a thermal distribution of particle

momentum, G≡ hσGviis gain term (describes creation) and L≡ hσLviis loss term (describes

annihilation). These two terms are averaged cross-sections.

The probability P

nwhich is described by eq. (2.1) increases when a pair of b1b2is produced

from the state with (n−1)pairs or a pair is destroyed from the state with (n+1)pairs. On the

other hand, it decreases with creation or annihilation of a pair from the state with npairs.

When we want to study thermalisation, then it is useful to cast the equation into dimensionless

form with the help of the time variable τ=tL/V, where V/L=τc

0is so-called relaxation time. Now,

G,Vand Lare constant so the master equation formulated in this dimensionless time must describe

the approach towards equilibrium. In terms of the relaxation time, the evolution is universal and

same for all reactions. Instead of the constants G,L,hNa1iand hNa2iwe use ε=GhNa1ihNa2i/L.

The master equation can be converted into a partial differential equation for a generating func-

tion [3]

g(x,τ) =

∞

∑

n=0

xnP

n(τ),(2.2)

where xis an auxiliary variable.

From the derivative of the generating function we can determine the equilibrium values of the

factorial moments.

1

PoS(CORFU2018)200

Evolution of multiplicity ﬂuctuations in heavy ion collisions Radka Sochorová

If we multiply eq. (2.1) by xnand sum over n, we ﬁnd that [3]

∂g(x,τ)

∂ τ = (1−x)(xg00 +g0−εg),(2.3)

where g0=∂g/∂x. The generating function obeys the normalisation condition

g(1,τ) =

∞

∑

n=0

P

n(τ) = 1.(2.4)

The equilibrium solution, geq(x), must not depend on time, thus it obeys the following equation

xg00

eq +g0

eq −εgeq =0.(2.5)

The solution that is regular at x=0 is then given by

geq(x) = I0(2√εx)

I0(2√ε).(2.6)

The average number of b1b2pairs per event in equilibrium is given by

hNieq =g0

eq(1) = √εI1(2√ε)

I0(2√ε).(2.7)

Here, I0(x)and I1(x)are the Bessel functions. Higher derivatives give us then the equilibrium

values of the factorial moments which are deﬁned as (their scaled values)

F2=hN(N−1)i

hNi2(2.8)

F3=hN(N−1)(N−2)i

hNi3(2.9)

F4=hN(N−1)(N−2)(N−3)i

hNi4.(2.10)

Equilibrium values of the 2nd [3,4], 3rd and 4th [6] factorial moments (not scaled) are then

F2,eq =−1

2√εI1(2√ε)

I0(2√ε)+1

2εI2(2√ε) + I0(2√ε)

I1(2√ε)(2.11)

F3,eq =3

4√εI1(2√ε)

I0(2√ε)−3

4ε1+I2(2√ε)

I0(2√ε)

+1

4ε3/2I3(2√ε) + 3I1(2√ε)

I0(2√ε)(2.12)

F4,eq =−15

8√εI1(2√ε)

I0(2√ε)+15

8εI2(2√ε)

I0(2√ε)+1

−3

4ε3/23I1(2√ε) + I3(2√ε)

I0(2√ε)+1

8ε23+4I2(2√ε) + I4(2√ε)

I0(2√ε).(2.13)

Now we study the relaxation of the multiplicity distribution with the help of the master equa-

tion. For numerical calculations binomial initial conditions are used

P

0(τ=0) = 1−N0(2.14)

P

1(τ=0) = N0(2.15)

P

n(τ=0) = 0 for n>1,(2.16)

2

PoS(CORFU2018)200

Evolution of multiplicity ﬂuctuations in heavy ion collisions Radka Sochorová

0

2

4

6

8

10

0 1 2 3 4 5

0 1 2 3 4 5

Fn / Fn,eq (τ)

τ

2nd factorial moment

3rd factorial moment

4th factorial moment

Figure 1: Time evolution of scaled factorial moments divided by their equilibrium values for constant

temperature and ε=0.1.

where N0=0.005 (N0=hNi(τ=0)).

The evolution of the 2nd, 3rd and 4th scaled factorial moments divided by their equilibrium

values is shown in Figure 1. The value of the parameter εhas been set to 0.1. Note that we obtained

qualitatively similar results also for other values of ε. We can see that all moments relax within the

same dimensionless time τand that higher moments differ more from their equilibrium values than

lower moments.

3. Higher moments in a cooling ﬁreball

Master equation deﬁned in dimensionless time τcan be used only for constant temperature.

However, we want to study a more realistic case in which the ﬁreball will cool down. A change of

temperature implies a variation of the relaxation time. Hence, we have to use the original master

equation which is deﬁned by eq. (2.1) and evaluate the creation and annihilation terms for each

temperature. In order to place ourselves into an interesting regime, we have choosen the reaction

system π+n←→ K+Λ. At present we shall use a parametrisation of the cross-section [5]

σΛK

πN=

0fm2√s<√s0

0.054(√s−√s0)

0.091 fm2√s0≤√s<√s0+0.09GeV

0.0045

√s−√s0fm2√s≥√s0+0.09GeV

(3.1)

where √s0is the threshold energy of the reaction and the energies are given in GeV.

The evolution starts at T=165 MeV. At this temperature, where the hadronisation happens, the

system is generated in chemical equilibrium. We further calculate how the multiplicity distribution

changes.

We use a simple toy model in which the temperature and volume behave like in 1D longitudi-

nally boost-invariant expansion (Bjorken scenario).

3

PoS(CORFU2018)200

Evolution of multiplicity ﬂuctuations in heavy ion collisions Radka Sochorová

0

0.2

0.4

0.6

0.8

1

3 4 5 6 7 8 9

0.165 0.138 0.122 0.112 0.105

factorial moments

t [fm/c]

T [GeV]

<N(N−1)(N−2)(N−3)>/<N>4

<N(N−1)(N−2)(N−3)>eq./<N>eq. 4

<N(N−1)(N−2)>/<N>3

<N(N−1)(N−2)>eq./<N>eq. 3

<N(N−1)>/<N>2

<N(N−1)>eq./<N>eq. 2

Figure 2: Evolution of scaled factorial moments for gradual decrease of the temperature from 165 MeV

to 100 MeV. This graph is plotted for 15 pions and 10 neutrons and for 50 times enlarged cross-section.

Thick lines represent the evolution of moments according to the master equation and thin lines represent the

equilibrium values calculated for each temperature.

The temperature drops according to

T3=T3

0

t0

t(3.2)

all the way down to the ﬁnal temperature T=100 MeV. Motivated by femtoscopic measurements

we set the ﬁnal time to 10 fm/c. This leads to t0=2.2 fm/c.

The effective system volume grows linearly

V(t) = V0

t

t0

,(3.3)

where the initial volume we set V0=125 fm3.

If the chemical processes under investigation are much faster than the characteristic time scale

of the expansion, then the multiplicity distribution will be always adapted to the ambient temper-

ature. If, on the other hand, chemistry is much slower than the expansion, then the distribution

will barely change. Hence, the interesting regime, where non-equilibrium evolution is expected, is

when the reaction rate and the expansion rate are roughly of the same order. In order to investigate

such a regime, we scale up the cross-section and in the next part of this work we also investigate

the inﬂuence of density dependence of the masses.

The evolution of scaled factorial moments for a gradual decrease of the temperature is shown

in Figure 2. We can see that since the temperature is decreasing, the moments change, but the

reaction rate is too low to keep them in equilibrium.

4. The apparent freeze-out temperature

We can now demonstrate the potential danger in case of extraction of the freeze-out temper-

4

PoS(CORFU2018)200

Evolution of multiplicity ﬂuctuations in heavy ion collisions Radka Sochorová

0

0.2

0.4

0.6

0.8

1

3 4 5 6 7 8 9

0.165 0.138 0.122 0.112 0.105

factorial moments

t [fm/c]

T [GeV]

<N(N−1)(N−2)(N−3)>/<N>4

<N(N−1)(N−2)(N−3)>eq./<N>eq. 4

<N(N−1)(N−2)>/<N>3

<N(N−1)(N−2)>eq./<N>eq. 3

<N(N−1)>/<N>2

<N(N−1)>eq./<N>eq. 2

Figure 3: Evolution of scaled factorial moments for gradual change of temperature from 165 MeV to

100 MeV. This graph is plotted for 15 pions and 10 neutrons and for 50 times enlarged cross-section.

Thick lines represent the evolution of moments according to the master equation and thin lines represent

equilibrium values calculated for each temperature.

ature from the different moments. Suppose that we observe the ﬁnal values of factorial moments

that the system eventually achieves in its non-equilibrium evolution, as shown in Figure 3. Suppose

further, that we wrongly assume that the system is still thermalised. This would mean that it has

evolved along the thin lines in Fig. 3. Now we ask at what temperature would a thermalised system

lead to the observed value of the factorial moments. The actual observed ﬁnal value of a thick line

is thus projected horizontally on the corresponding thin line (Figure 3) and the apparent temper-

ature is read off from the abscissa. We can see that such a procedure can lead to different values

of the apparent temperature if different moments are used. We can also see that higher factorial

moments seem to indicate lower temperatures than lower moments.

In data analysis, central moments are often used, which are deﬁned as

µ1=hNi=M(4.1)

µ2=hN2i−hNi2=σ2(4.2)

µ3=h(N−hNi)3i(4.3)

µ4=h(N−hNi)4i.(4.4)

Often, one uses their combinations, like the skewness

S=µ3

µ3/2

2

(4.5)

or the kurtosis

κ=µ4

µ2

2−3.(4.6)

5

PoS(CORFU2018)200

Evolution of multiplicity ﬂuctuations in heavy ion collisions Radka Sochorová

0

1

2

3

4

5

0.165 0.138 0.122 0.112 0.105

µ1

T [GeV]

equil.

200 x σ

100 x σ

50 x σ

15 x σ

0

0.5

1

1.5

2

µ2

0

0.5

1

µ3

0

5

10

15

3456789

µ4

t [fm/c]

Figure 4: Evolution of the ﬁrst four central moments (from top to bottom). Different curves (different

colours) on the same panel show results for cross sections scaled by different factors. Solid lines represent

the equilibrium values.

6

PoS(CORFU2018)200

Evolution of multiplicity ﬂuctuations in heavy ion collisions Radka Sochorová

0

0.1

0.2

0.3

0.4

0.5

0.165 0.138 0.122 0.112 0.105

S

T [GeV]

equil.

200 x σ

100 x σ

50 x σ

15 x σ

0

0.1

0.2

0.3

0.4

3456789

κ

t [fm/c]

Figure 5: Evolution of the skewness (upper panel) and the kurtosis (lower panel). Different curves (different

colours) on the same panel show results for different cross sections. Solid lines represent the equilibrium

values.

We thus investigate the evolution of the central moments and their combinations when the

ﬁreball cools down. It turns out that it is very difﬁcult to extract the exact freeze-out temperature

from the non-equilibrium values.

We can see in Figure 4and Figure 5that while the central moments are decreasing, the co-

efﬁcients of skewness and kurtosis are increasing in the scenario of temperature decrease due to

boost-invariant expansion.

5. Decreasing mass of Λhyperon

In the previous part of this work we assumed that the involved masses and cross-section do not

depend on density. The gain term of our reaction π++n←→ K++Λis small because of the rather

higher threshold, which is about 530 MeV above the masses of the incoming particles, while the

temperature is lower than 165 MeV. It means that the reaction rate might increase if the threshold

is lowered, for example through a decrease of the hyperon mass in baryonic matter.

7

PoS(CORFU2018)200

Evolution of multiplicity ﬂuctuations in heavy ion collisions Radka Sochorová

Therefore, we now explore the possibility of decreasing the Λhyperon mass. It means that also

the threshold for the reaction is lowered and its rate may grow due to the increase of the available

phase space.

We assume a simple dependence of the hyperon mass on baryon density ρB

mΛ(ρB) = ρ0−ρB

ρ0

mΛ0+ρB

ρ0

mp0.(5.1)

Note that the hyperon mass becomes identical to that of proton mp0at the highest baryon density

ρ0at which our calculation starts. Hyperon mass returns to the vacuum value mΛ0if baryon density

vanishes.

Density in our toy model also evolves according to one-dimensional longitudinally boost-

invariant expansion

ρ=ρ0

t0

t,(5.2)

where ρ0=0.08 fm−3.

For this scenario we present the volume-independent ratios which are often measured. These

are, e.g.

R32 =µ3

µ2

=Sσ(5.3)

R42 =µ4

µ2−3µ2=κσ2.(5.4)

Results for the case with density-dependent mass of hyperon are plotted in Figure 6.

We can see that also for the time evolution of moments and their combinations for density-

dependent mass the central moments are decreasing while the coefﬁcients of skewness and kurtosis

are increasing. Only weak time dependence is seen for the volume independent ratios Sσand κσ 2.

Thus in real collisions, where non-equilibrium evolution is likely, it is very difﬁcult to determine

the unique freeze-out temperature from the measured moments.

Nevertheless, in realistic ﬁreballs there are also other channels that can change the numbers of

kaons and/or lambdas so we have to expect that moments may change stronger than in our work.

6. Conclusion

If chemical equilibrium is broken, higher factorial moments of multiplicity distribution differ

more from their equilibrium values than the lower moments. Evolution of chemical reaction off

equilibrium may show different temperatures for different orders of the factorial or central moments

(or their combinations). We demonstrated this on the reaction π++n←→ K++Λ. The behaviour

of the combination of the central moments depends on which combination of moments we choose.

Hence, one should be very careful when extracting the freeze-out temperature from higher

moments.

Acknowledgments

This work was supported by the grant 17-04505S of the Czech Science Foundation (GAˇ

CR).

BT also acknowledges the support by VEGA 1/0348/18.

8

PoS(CORFU2018)200

Evolution of multiplicity ﬂuctuations in heavy ion collisions Radka Sochorová

0

1

2

3

4

5

6

7

0.165 0.138 0.122 0.112 0.105

µ1

T [GeV]

equil.

µ1

0

0.5

1

1.5

µ3

equil.

µ3

0

0.1

0.2

0.3

0.4

S

equil.

S

0.3

0.35

0.4

0.45

0.5

3456789

Sσ

t [fm/c]

equil.

Sσ

0

1

2

3

4

5

6

7

0.165 0.138 0.122 0.112 0.105

µ2

T [GeV]

equil.

µ2

0

10

20

30

µ4

equil.

µ4

0

0.1

0.2

0.3

0.4

κ

equil.

κ

3456789

0

0.1

0.2

κσ2

t [fm/c]

equil.

κσ2

Figure 6: Central moments, skewness, kurtosis and volume-independent ratios Sσand κσ2for the scenario

with density-dependent mass of Λand the decreasing temperature. Thick solid lines represent numerically

calculated evolution and thin dotted lines represent equilibrium values at the given temperature.

9

PoS(CORFU2018)200

Evolution of multiplicity ﬂuctuations in heavy ion collisions Radka Sochorová

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