Pis’ma v ZhETF, vol. 111, iss. 3, pp. 147 - 148

February 10

The role of the chiral phase transition in modelling the kaon to pion

ratio

A. V. Friesen⁺¹⁾, Yu. L. Kalinovsky^+∗, V. D. Toneev⁺

⁺Joint Institute for Nuclear Research, 141980 Dubna, Russia

^∗State University “Dubna”, 141982 Dubna, Russia

Submitted 27 November 2019

Resubmitted 12 December 2019

Accepted 12

December 2019

DOI: 10.31857/S0370274X20030017

The search of a quark-gluon plasma (QGP), where

∑

g_V

( qγ_µλ^a q )² + L_det - U(Φ,

Φ; T),

(1)

hadrons dissolve and quarks are supposed to be free

a=0

and deconfined, is difficult due to the short QGP life-

time. Various signals were proposed for detection of the

with the Kobayashi-Maskawa-t’Hooft (KMT) interac-

QGP phase, and the “horn”, which appears in the ra-

tion L_det = g_D{det [q(1 + γ₅)q] + det [q(1 - γ₅)q]} and

tio of positive charged kaon to pion, was supposed be

the effective potential U(Φ,

Φ; T), which describes the

one of them. Nowdays the picture of this peak becomes

confinement/deconfinement properties [5-7].

more clear from the experimental side: the peak appears

In the model, the current quark propagating in the

in the ratio of positive charged kaons and pions at the

chiral condensate develops a quasi-particle mass that

collision energy

√s_NN ∼ 7-10 GeV for the high-size

leads to spontaneous chiral symmetry breaking. At that

systems in Au+ Au and Pb+ Pb collisions [1]. With

time the quark is coupled to a homogeneous background

decreasing system size, the sharp peak becomes lower

field representing the Polyakov loop dynamics. At low

and for Be + Be, p + p collisions the ratio demonstrates

chemical potential the chiral symmetry is restored when

smooth behaviour [2]. It is in agreement now, that the

the dynamically generated quark mass drops as a func-

quick rise in K⁺/π⁺ ratio at low energies is associated

tion of temperature and chemical potentials in half of its

with the phase transition in medium. The microscopic

value. At low chemical potential and high temperature

transport model with involving partial restoration of

the chiral symmetry restoration is soft and it is sup-

chiral symmetry at the early stages of the collision re-

posed to be a crossover. At high chemical potential and

produces experimental data and predicts smoothing of

low temperature the gap equation has three solutions

the peak with decreasing system size [3]. The authors

and the first order chiral phase transition is discussed.

showed that the partial chiral symmetry restoration was

This picture can be changed in the Polyakov loop ex-

responsible for a quick increase in the K⁺/π⁺ ratio

tended Nambu-Jona-Lasinio (PNJL) model when the

at low energies. The jump in the ratio after reaching

vector interaction is added. With increasing vector cou-

maximum could be explained as a result of a QGP for-

pling g_V, the domain of the first order transition de-

mation during collision, as the deconfinement transition

creases until it completely disappears. Varying the vec-

(the strangeness quark mass tends to its current value)

tor coupling, we can check if the change of the type of

makes the strangeness yield independent on energy [4].

the phase transition affects the behaviour of the kaon

In present work, we discuss the chiral phase tran-

to pion ratio in the low temperature and high chemical

sition and in-medium behaviour of the pseudo-scalar

potential (low energy) region.

mesons in the framework of the SU(3) Polyakov loop

The study of pseudo-scalar mesons is interesting

exended Nambu-Jona-Lasinio model with vector inter-

since they due to their Goldstone boson nature are as-

action:

sociated with the breaking of the chiral symmetry and

are sensible to the medium. Meson masses are defined

L = q(iγµ D_µ - m - γ₀µ)q +

by the Bethe-Salpeter equation at P = 0

∑

g_S

[(qλ^aq)²

+ (qiγ₅λ^aq)²]-

1 - P_ijΠ^Pij(P₀ = M,P = 0) = 0.

(2)

a=0

When the meson mass exceeds the sum of masses of

¹⁾e-mail: avfriesen@theor.jinr.ru

its constituents (P₀ > m_i + m_j ), the meson turns into

Письма в ЖЭТФ том 111 вып. 3 - 4

2020

147

148

A. V. Friesen, Yu. L. Kalinovsky, V. D. Toneev

the resonance state and the Mott transition occurs. In

this case, the complex properties of the integrals have

to be taken into account and the solution has to be

defined in the form P₀ = M_M -¹² iΓ_M . Though at the

zero chemical potential pions and kaons are degener-

ated, their masses split with increasing density. At the

nonzero chemical potential and low T , the mass splitting

in charged multiplets appears due to the excitation of

the Dirac sea is modified by the presence of the medium

[8-10].

For the effective models the ratio of the particle num-

ber can be calculated in terms of the ratio of the number

densities of mesons (n_K± /n_π± ) with

∫ ∞

n_M± = p²dp

√

(3)

e^β(

p²+mM±∓µM±) - 1

Fig. 1. The K⁺/π⁺ ratios for different values of gV

the chemical potential for pions is a phenomenological

hight. The structure is more sensitive to the slope of the

parameter µ_π = 0.135 GeV, and the chemical potential

phase transition curve (see [11]) and the matter proper-

for kaons can be defined as µ_K = µ_u - µ_s.

ties. For example, when the strange chemical potential

The experimental data are shown as a function of

is zero µ_S (µ_K ) = 0, the K⁺/π⁺ ratio shows smooth be-

the collision energy

√s_NN which never appears as a

haviour. When the strangeness neutrality is introduced,

parameter in effective models. We introduced the pa-

the K⁺/π⁺ ratio does not show a peak structure [8, 11].

rameter T/µ_B, where (T, µ_B) are taken on the phase

The work was supported by the Russian Foundation

diagram along the phase transition curve assuming that

for Basic Research, grant # 18-02-40137.

chiral phase transition can be considered as chemical

freeze-out, and rescaled both experimental and theoret-

Full text of the paper is published in JETP Letters

ical results (see [8, 11]). The main difference between

journal. DOI: 10.1134/S0021364020030017

the choice of T and µ_B along the phase transition line

is whether we are in the crossover region or in the first-

order transition region. In the region of the first-order

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Nucl. Phys. A 967, 35 (2017).

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836

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(2017).

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shifts the crossover pattern to higher chemical poten-

Phys. Pol. B 42, 307 (2011).

tials. It can be seen in the Fig. 1 that the absence of

5. A. V. Friesen, Yu. L. Kalinovsky, and V. D. Toneev, Int.

the first order phase transition domain leads only to a

J. Mod. Phys. A 27, 1250013 (2012).

changing in the peak hight in the K⁺π⁺ ratio.

6. A. V. Friesen, Yu. L. Kalinovsky, and V. D. Toneev, Int.

In the PNJL model we can show how K/π ratio

J. Mod. Phys. A 30, 1550089 (2015).

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curve and discuss in this way how the chiral phase tran-

8. A. V. Friesen, Yu. L. Kalinovsky, and V. D. Toneev,

sition can affect to the K/π behaviour. Using our pre-

Phys. Rev. C 99, 045201 (2019).

vious study we can conclude that the peak appears in

9. M. Lutz, A. Steiner, and W. Weise, Nucl. Phys. A 574

the range of low temperatures and high baryon chemical

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10. P. Costa, M. C. Ruivo, Yu. L. Kalinovsky, and

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the type of phase transition in the high density region,

11. A. V. Friesen, Yu. L. Kalinovsky, and V. D. Toneev,

as the replacement of the the first order transition to

PEPAN Lett. 16, 681 (2019).

the soft crossover only leads to a changing in the peak

Письма в ЖЭТФ том 111 вып. 3 - 4

2020