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Nontopological Soliton in the Polyakov Quark Meson Model

Hong Mao (毛鸿 ) Department of Physics, Hangzhou Normal University

With: Jinshuang Jin （ HZNU）

XMU, Xiamen, 21/11/2015

arXiv: 1508.03920

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Outline Motivation Effective potential and the nontopological soliton Nucleon static properties at finite temperature and

density Phase diagram and QCD thermodynamics at zero

chemical potential Summary and discussion

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I. Motivation1.1Thermodynamics on Hadron-Quark Phase Transition

Mixed Mixed phasephase

We assume the firs-order phase transition

the degrees of freedom are

nucleons, and mesons

the degrees of freedom are

quarks and gluons

EOS

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How about the crossover transition?

T.K.Herbst,et al., PRD88,014007(2013)

chiral effective models with including quark and meson fluctuationsvia the functional renormalization group

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What nontopological soliton models can do?

In hadron phase, the nucleon naturally arises as a nontopological soliton (bound state), and the model has been proven to be a successful approach of the description of nucleon static properties

In quark phase, the quarks are set to be free with dissolution of the soliton

1.2 Why the nontopological Soliton model with the Polyakov loop

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Still suffering from two serious problems:

Models predict a first-order phase transition in whole phase diagram

the critical temperature is extremely low (Tc ~110MeV) as compared with lattice data.

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II. Effective potential and the nontopological soliton

2.1 The Polyakov Quark Meson Model with two flavors

with

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Parameters fixing

we take m= 472 MeV and g = 4.5 as the typical values. And the actual value of T0 for two quark favors is T0 = 208MeV.

H.Mao,et al., PRC88,035201(2013)

g = 4.5 gives Mq=419 MeV and R=0.877fm

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2.2 The thermodynamic grand potential in the presence of appropriately renormalized fermionic vacuum

with (1) pure mesonic section

(2) Temperature-dependent section

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Minimizing the thermodynamical potential with respective to order parameters

We can determine the behavior of the chiral order parameter and the Polyakov loop expectation values as a function of T and μ.

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The normalized chiral order parameter and the Polyakov loop expectation values as a function of temperature for μ= 0 MeV and μ = 320 MeV. The solid curves are for μ =0 MeV and the dashed curves are for μ= 320 MeV.

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2.3 The relationship of the form of potential function and the existence of stable soliton

For a general quartic form potential

With the bag constant

R.Goldam and L.Wilets, PRD25,1951(1982)

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Three typical forms for the potential U(σ) as a function of σ field. f = 3 and f = are the two limiting cases in which the stable soliton bag still exists. Units on the vertical and horizontal axes are arbitrary

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A key criterion:

in order to ensure the stability of the two vacuum states and guarantee the existence of the stable soliton solution, it is indispensable that the potential of the σ field must has three distinct extrema.

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2.4 Effective potential in PQM model(1)the mesonic field direction

we treat the σ field as a variable in the grand canonical potential while fixing the Polyakov loop fields on their expectation values

(2) the Polyakov loop filed direction

the Polyakov loop fields are consider as variables when the σ field will maintain their expectation value all the time

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Our results:

(1)as long as the temperature is not lager than the chiral critical temperature , there truly exist the stable soliton solutions in the model for both crossover and first-order phase transitions.

(2)We do not have a bag-like soliton solutions for the Polyakov loop variables. On the other hand, the Polyakov loop variables will always develop their expectation values in whole space.

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III. Nucleon static properties at finite temperature and density

3.1 the Euler-Lagrange equations of motion

the boundary conditions

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3.2 the stability of the soliton

the stability of the soliton should be checked carefully by comparing the total energy of the system in thermal medium with the energy of three free constituent quarks in system.

if stable

if unstable

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IV. Phase diagram and QCD thermodynamics at μ= 0

4.1 Phase diagram

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4.2 QCD thermodynamics at μ= 0 as a prototype

An idealistic consideration:

(a) A nontopological soliton model (NS)By taking the hadronic phase as a noninteracting hadron gas composed of nucleons and mesons with the effective masses in hadron phase.

By taking the hadronic phase as a noninteracting quantum gas composed of constituent quarks (unbound) and mesons with the effective masses both in quark phase.

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M.I.Gorenstein and S.N.Yang, PRDD52,5206 (1995).

The last term B is introduced in order to recover the thermodynamical consistency of the system, since the nucleons are treated as the chiral solitons with a temperature-dependent masses.

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(b) The Polyakov Quark Meson Model (PQM)

30Thermodynamic pressures are scaled by the QCD Stefan-Boltzmann (SB) limit

lattice data are taken from A. Ali Khan et al. [CP-PACS Collaboration], PRD64,074510 (2001)

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V. Summary and Discussion Hadron-quark phase transition in a unified modelIn the framework of the nontopological soliton model rooted in PQM model,

we are working on description the hadron quark phase transition self-consistently by instead of RMF-PNJL or RMF-MIT methods.

Some interesting results.(1) as long as the temperature is not lager than the chiral critical

temperature, we always have the stable soliton solutions.

(2) two critical temperatures always satisfy

(3) For zero chemical potential and .

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Future studies To improve the present study by considering the interaction

between the nucleons and mesons in hadron phase and the interaction between the quarks and mesons in quark phase.

To include the center-of mass (c.m.) correction to nucleon properties.

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Thanks!