summary of the doctor thesis studying of the phase transition in linear sigma model

The aims of thesis - Studying of the phase structure of LSM and LSMq with two different forms of symmetry breaking term: the standard case and non – standard case... - Studying of phase

MINISTRY OF EDUCATION AND TRAINING MINISTRY OF SCIENCE AND TECHNOLOGY

VIETNAM ATOMIC ENERGY INSTITUTE

NGUYEN VAN THU

STUDYING OF THE PHASE TRANSITION

IN LINEAR SIGMA MODEL

A SUMMARY OF THE DOCTOR THESIS

Speciality: Theoritical and mathematical physics

Code : 62.44.01.01

Scientific supervisors

PROF DR TRAN HUU PHAT

DR NGUYEN TUAN ANH

HANOI, 2011

THIS THESIS WAS COMPLETED AT INSTITUTE FOR NUCLEAR SCIENCE AND TECHNIQUE – VIETNAM ATOMIC ENERGY

INSTITUTE

Scientific supervisor: PROF DR TRAN HUU PHAT

DR NGUYEN TUAN ANH

Prof Dr Dang Van Soa

This thesis will be defended in the Scientific Counsil of Vietnam Atomic

Energy Institute held on May 28, 2012

THIS THESIS MAY BE FOUND AT THE VIETNAM NATIONAL

LIBRARY AND ATOMIC ENERGY LIBRARY

1

INTRODUCTION

1 The research topic

The phase structure of QCD plays an impotant role in morden physics, attracting intense experimental and theoretical investigations

Some theories and models are used in order to study the phase structure

of QCD, for example, chiral pertubative theory, Nambu-Jona-Lasinio (NJL) model, Poliakov-NJL (PNJL) model, linear sigma model (LSM)

Up to now the study of linear sigma model is still not complete It is the

reasons why we choose subject “Studying of the phase transition in linear sigma model”

2 History of problem

Studying of D K Campell, R F Dashen, J T Manassah is the first paper, in which they studied LSM with two different forms of the symmetry breaking term (standard case and non-standard case) but they are restricted only within tree-level approximation

In higher order approximation, present papers are researched in Fock (HF) approximation, expanded N – large or isospin chemical potential (ICP) is neglected The study of the non-standard case is so far still absent

Hatree-When constituent quarks are presented, in the framework of NJL and PNJL models the researchs are quite complete Meanwhile the linear sigma model with constituent quarks (LSMq) the present researchs only consider the case in which ICP is vanished

The studies of chiral phase transition in compactified space – time are in first stage so far

3 The aims of thesis

- Studying of the phase structure of LSM and LSMq with two different forms of symmetry breaking term: the standard case and non – standard case

2

- Studying of the effect from neutrality condition on the phase structure of LSM and LSMq

- Studying of the chiral phase transition in compactified space – time

4 The subject, research problems and scope of thesis

- Studying of the phase structure of LSM at finite value of temperature T and

different forms of symmetry breaking term

- Studying of phase structure of LSMq at finite value of temperature, ICP and quark chemical potential (QCP) with and without neutrality condition and two different forms of symmetry breaking term

- Studying of the chiral phase transition in compactified space – time when ICP is zero

5 The method

In this thesis we combine the mean – field theory and effective action Cornwall – Jackiw – Tomboulis (CJT) in order to research the phase structure

of LSM and LSMq

6 The contribution of thesis

This thesis has many contributions in morden physics

7 The structure of thesis

The thesis includes 133 pages, 106 figures and 61 references Besides introduction, conclusion, appendices and references, this consists of 3 chapters: Chapter 1 Phase structure of linear sigma model without constituent quarks Chương 2 Phase structure of linear sigma model with constituent quarks

3

CHAPTER 1 PHASE STRUCTURE OF LINEAR SIGMA MODEL

WITHOUT CONSTITUENT QUARKS 1.1 The linear sigma model

- Lagrangian

- The standard form

- The non – standard form

1.2 Phase structure in standard case

1.2.1 Chiral phase transition in case isospin chemical potential is vanishing

1.2.1.1 Chiral limit

In two – loop expanded and HF approximation, there Goldstone bosons are not preserved

In order to preserve Goldstone bosons we introduced improved Hatree – Fock (IHF) approximation In this approximation we obtain

- The gap equatiion

4

Fig 1.1 The chiral condansate

as a function of temperature

Fig 1.2 The evolution of effective potential

versus u From the top to bootom the graphs correspond to T = 200 MeV, Tc = 136.6 MeV

200 400 600

expanded 2-loop approximation there is no Goldstone boson Using IHF approximation becomes Goldstone boson and we get

- The gap equation

- SD equations

- The numerical computation

gives the phase diagram The

phase diagram in Fig 1.8

Fig 1.8 Phase diagram in

-plane compares with those

form HF approximation and

expanded N-large In IHF

approximation, the solid and

dashed lines correspond to first

and second-order phase transition

0 50 100 150 200 250 300 0

50 100 150 200 250 300

- The phase diagram

1.3 Phase structure in non – standard case

Calculations in tree – level approximation give Goldstone boson for component However, in HF approximation with 2-loop expanded gives

no Golstone boson Employing IHF approximation in order to preserve Goldstone boson we lead

- The gap equations

1.4 The effect from neutrality condition

pion-decay processes

- The neutrality condition

- Basing on above equations, we calculate numerically in order to study the effect from neutrality condition on the phase structure with two different forms of symmetry breaking term

- In these numerical computation we set electron mass to be zero

Fig 1.20 The phase diagram of

pion condensate

Fig 1.24 The phase diagram of chiral condensate

8

1.4.1 The standard case

Fig 1.25 The pion condensate in

chiral limit within neutrality condition

(solid line) and without neutrality

condition (dashed line) at = 300

MeV

Fig 1.26 The pion condensate in chiral limit with neutrality condition Starting from the top the lines correspond to = 0, 1/4, 1/2

Fig 1.27 The pion condensate in

physical world The solid, dashed and

dotted lines correspond to = 0, 1/4,

1/2

Fig 1.28 The chiral condensate in physical world The solid and dashed lines correspond to = 0, 1/4

1 In the standard case:

- We affirm that in chiral limit the chiral phase transition is second – order

It is clearly answer about a question which has been disputing for a long time

transition of pion condensate is second – order The chiral symmetry gets

2 In the non – standard case, this is the first time the phase structrure of LSM has completely considered in high order approximation of effective potential

3 The effects from neutrality on phase structure are studied in detial

Fig 1.30 The pion condensate versus

T The solid (dashed) line corresponds

to with (without) neutrality conditiion

Dashed line is ploted at = 200MeV

Fig 1.32 The chiral condensate versus T The solid (dashed) line corresponds to with (without) neutrality conditiion Dashed line is ploted at = 100MeV

10

CHAPTER 2 PHASE STRUCTURE OF LINEAR SIGMA MODEL

WITH CONSTITUENT QUARKS 2.1 The effective potential in mean – field theory

- Lagrangian

- The effective potential in mean – field theory (MFT)

2.2 The standard case - The gap equations

2.2.1 Chiral limit

v  0

v  0

0 20 40 60 80 100 120 140

12

2.3 Non – standard case

- The gap equations

Fig 2.20 Chiral condensate in region

From the right to left = 0,

100, 200, 220MeV

Fig 2.21 Phase diagram of chiral condensate in region

Fig 2.24 Chiral condensate at = 150

MeV From the right to left T = 0, 50,

100 MeV

Fig 2.27 Chiral condensate at = 300

MeV From the right to left T = 0, 50,

50 100 150 200

0.05 0.10 0.15 0.20 0.25

f

13

2.3.1 Region

2.3.2 Region

2.4 The effects from neutrality condition

- The matter must be stable under the weak processes like

Fig 2.36 The pion condensate as a

function of T at = 0 and = 192

MeV

Fig 2.34 Phase diagram v = 0 From

the bottom to top = 138, 200, 300 MeV

Fig 2.41 The chiral condensate as a

14

- The neutrality condition reads as

- The electron mass is neglected in our numerical computation

2.4.1 The standard case

2.4.2 The non – standard case

Fig 2.53 Phase diagram v = 0 with > and neutrality condition (solid line) and without neutrality condition at = 200 MeV (dashed line)

Fig 2.47 Phase diagram v = 0 in

chiral limit The solid and dashed

lines correspond to with and without

neutrality condition and = 232.6

MeV)

Fig 2.49 Phase diagram u = 0 in

physical world From the bottom to top = 0, 0.25, 0.3 The solid (dashed) line corresponds to first (second) – order phase transition

500 1000 1500 2000

 MeV

plane has a CEP, which separates first and second – order of phase transition This result is suitable with those prediction of LQCD

3 The effects form neutrality on phase structure are completely considered

16

CHAPTER 3 CHIRAL PHASE TRANSITON IN COMPACTIFIED

SPACE - TIME 3.1 Chiral phase transition without Casimir effect

3.1.1 The effective potential and gap equations

- The potential

- The effective potential in MFT

- Neglecting the Casimir energy

- The dispersion relation

twisted quark (TQ)

- The gap equation

3.1.2 Numerical computation

3.1.2.1 Chiral limit

- In chiral limit we set

- At = 50 MeV the phase diagram obtained from numerical computation for UQ and TQ are ploted in Fig 3.3

3.1.2.2 Physical world

Fig 3.3 Phase diagram of chiral condensate in chiral limit at

= 50 MeV for UQ (left) and TQ (right)

Fig 3.6b Phase diagram of

chiral condensate for UQ in

physical world at = 50 MeV

Fig 3.9b Phase diagram of chiral condensate for TQ in physical world at = 50 MeV

18

- The results are similar for different value of

- In physical world, chiral phase transition for UQ has both first – order and crossover Two kinds of phase transition are sapareted by a CEP For

TQ chiral phase transition is always the crossover

3.2 Chiral phase transition driven by Casimir effect

3.2.1 Casimir energy

- The Casimir energy

- Using Abel-Plana relation we calculate Casimir energy for UQ

And for TQ

- Taking to account Casimir energy the effective potential has the form

for UQ and

for TQ

19

Fig 3.12a Phase diagram of chiral condensate for UQ in chiral limit From the top to bottom the graphs correspond to = 0, 100 MeV

MeV The solid, dashed and

dotted lines correspond to a = 0,

0.152, 0.253 fm-1

Fig 3.11b Chiral condensate of

TQ in chiral limit at = 100

MeV The solid, dashed and

dotted lines correspond to a = 0,

0.253, 0.507 fm-1

Fig 3.12b Phase diagram of chiral condensate for TQ in chiral limit From the top to bottom the graphs correspond to = 0, 100 MeV

Fig 3.14a Chiral condensate of UQ in

physical world at = 50 MeV The

solid, dashed, dotted lines correspond

to a = 0, 0.253, 1.014 fm-1

Fig 3.15a Phase diagram of chiral condensate for UQ in physical world From the top the lines correspond to

= 0, 50 MeV

Fig 3.14b Chiral condensate of TQ in

physical world at = 50 MeV The solid,

dashed, dotted lines correspond to a = 0,

0.253, 1.014 fm-1

Fig 3.15b Phase diagram of chiral condensate for TQ in physical world From the top the graphs correspond to = 0, 50 MeV

21

a) For UQ

- This result shows that u approaches to 0 when a increases, it means that

Hohenberg theorem is satisfied

b) For TQ

- This result of TQ shows

- In this case the anti-periodic boundary condition is equivalent to the present of external field and Hohenberg theorem is satisfied, too

Hình 3.17 The a dependence of chiral condensate in chiral limit for UQ at = 50

MeV and T = 100 MeV (solid line), 150 MeV (dashed line), 200 MeV (dotted line)

Fig 3.18 The a dependence of chiral condensate in chiral limit for TQ at = 50 MeV The solid, dashed, dotted lines correspond to T = 50, 80, 100 MeV (left panel) and T = 150, 200, 250 MeV (right panel)

22

CONCLUSION

In this thesis we have investigated systematically the phase structure

of the linear sigma model by means of the improved Hatree – Fock approximation, where Goldstone theorem is preserved and self-consistancy

of theory is satisfied Among many results obtained the most remarkable

results are in order:

1 We found the chiral phase diagram of the linear sigma model in which the pion condensation was incorporated into consideration This is the major success of the thesis Moreover we proved that the chhiral phase transition in chiral limit is second – order if the Goldstone theorem was respected

2 Taking into account the present of quarks, the phase diagram in

- plane has a CEP, this result coincides with prediction of LQCD

3 The critical temperature of chiral phase transition depends on the length of compactified space – time

23

LIST OF PAPERS RELATE TO THIS THESIS

1 Tran Huu Phat and Nguyen Van Thu, Phase structure of the linear

sigma model with the non-standard symmetry breaking term, J

Phys G: Nucl and Part 38, 045002, 2011

2 Tran Huu Phat and Nguyen Van Thu, Phase structure of the linear

sigma model with the standard symmetry breaking term, Eur

Phys J C 71, 1810 (2011)

3 Tran Huu Phat, Nguyen Van Thu and Nguyen Van Long, Phase

structure of the linear sigma model with electric neutrality

constraint, Proc Natl Conf Nucl Scie and Tech 9 (2011), pp

246-256

4 Tran Huu Phat, Nguyen Van Long and Nguyen Van Thu,

Neutrality effect on the phase structure of the linear sigma model with the non-standard symmetry breaking term, Proc Natl Conf

Theor Phys 36, (2011), pp 71-79

5 Tran Huu Phat and Nguyen Van Thu, Casimir effect and chiral

phase transition in compactified space-time, submitted to Eur

Phys J C

6 Tran Huu Phat and Nguyen Van Thu, Phase structure of linear

sigma model without neutrality (I), Comm Phys Vol 22, No 1

(2012), pp 15-31

7 Tran Huu Phat and Nguyen Van Thu, Phase structure of linear

sigma model with neutrality (II), Comm Phys., to be published

8 Tran Huu Phat and Nguyen Van Thu, Phase structure of linear

sigma model with constituent quarks: Non-standard case,