Supported by the National Natural Science Foundation of China (11475085, 11535005, 11690030, 11574145)

^{3}and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

Within the framework of the Dyson-Schwinger equations and by means of Multiple Reflection Expansion, we study the effect of finite volume on the chiral phase transition in a sphere, and discuss in particular its influence on the possible location of the critical end point (CEP). According to our calculations, when we take a sphere instead of a cube, the influence of finite volume on phase transition is not as significant as previously calculated. For instance, as the radius of the spherical volume decreases from infinite to 2 fm, the critical temperature

Article funded by SCOAP^{3}

It is widely believed that with the increase of temperature and/or chemical potential, the strongly-interacting matter undergoes a phase transition from hadronic matter to quark-gluon plasma (QGP), which can be studied in relativistic heavy-ion collisions (RHIC) at CERN (France/Switzerland), BNL (USA), and GSI (Germany) [

It should be noted that many previous calculations of the location of CEP are based on an infinite thermodynamical system. However, the QGP system produced in RHIC has undoubtedly a finite volume. The homogeneous volume before freeze-out in Au-Au and Pb-Pb collisions ranges between approximately
^{3} [^{3} , as estimated in Ref. [

It is well known that when the volume of a strongly interacting system is small enough, not only its size but also its shape have an important impact on the QCD phase transition. However, it should be noted that in most calculations, for the sake of convenience, a cube was used to simulate the fireball produced in RHIC, thus ignoring the influence of different shapes on the phase transition. Therefore, in order to get closer to the real shape of the fireball produced in a RHIC experiment, we use a sphere to study, by means of MRE [

This paper is organized as follows: In Sec. 2, we give a brief introduction to the quark gap equation in a finite spherical volume at finite temperature and finite chemical potential. In Sec. 3, we study the effect of finite volume on the chiral phase transition, especially its influence on the behavior of CEP. Finally, we give a brief summary in Sec. 4.

DSE is a suitable QCD-connected non-perturbative method and it is widely used in the studies of hadron physics [^{①}

Here, we work in the Euclidean space, and take

where

where

Next, we extend the quark gap equation to finite

where

where

which can be used to verify the accuracy of numerical calculations. In our calculations, we ignore the function

We are now ready to introduce the quark gap equation in a finite spherical volume, and for taking the finite volume effects into account we consider the MRE formalism [

where

The curvature contribution is given by Madsen

which takes into account the finite quark mass. It should be noted that there are different interpretations of

For

where

where now

To solve the quark gap equation, truncations are inevitable. Here, we employ the Rainbow truncation [

which is widely used in the studies of hadron physics and QCD phase diagram. We also employ the widely used gluon propagator, as in Refs. [

where

and

The related parameters,
^{2} [

In this section, we study the chiral phase transition in a finite spherical volume, and especially discuss its influence on the location of CEP. We first solve the quark gap equation. The procedure is to insert Eqs. (7,13,14,15) into Eq. (12), multiply each side by

These coupled equations can be numerically solved by iteration. In

(color online)

The crossover behavior can be further studied by the chiral susceptibility which is defind as [

Its volume dependence is plotted in

(color online) Chiral susceptibility

(color online)

We plot the volume dependence of CEP in

(color online) Volume dependence of CEP.

Within the framework of DSE, we consider, for the first time, the influence of the finite volume on the chiral phase transition in a sphere. For taking the finite volume effects into account, we consider the MRE formalism, in which the surface and curvature effects are also properly incorporated. This formalism has been used to study the thermodynamic quantities in the PNJL model [