Summary of the doctor thesis: Studying of the phase transition in linear sigma model

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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 the effect from neutrality condition on the phase structure of LSM and LSMq; studying of the chiral phase transition in compactified space

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Nội dung Text: Summary of the doctor thesis: Studying of the phase transition in linear sigma model

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 First referee: Prof. Dr. Nguyen Xuan Han Second referee: Prof. Dr. Nguyen Vien Tho Third referee: 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
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 Hatree- 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. 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. 1
- 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 isospin chemical potential with and without neutrality condition and two 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. Chapter 3. Chiral phase transition in compactified space – time. 2
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 tree – level approximation pions are Goldstone bosons. 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 - Numerical computation with parameters MeV, MeV, MeV. 3
20 1.0 15 0.8 10 VMeV 3 .fm 0.6 u f 5 0.4 0 0.2 5 0.0  10 20 40 60 80 100 120 140 0 20 40 60 80 100 T MeV uMeV Fig. 1.1. The chiral condansate Fig. 1.2. The evolution of effective potential as a function of temperature. versus u. From the top to bootom the graphs correspond to T = 200 MeV, Tc = 136.6 MeV và T = 100 MeV. 1.2.1.2. Physical world - The gap equation - Schwinger–Dyson (SD) equations - Numerical results 1.2 1.0 0.8 u f 0.6 0.4 0.2 0.0 100 200 300 400 500 T MeV Fig. 1.3. The chiral condensate as a function of T in physical world. 4
800 600 M M  ,  MeV 400 200 M 0 0 50 100 150 200 250 300 T MeV Fig. 1.4. The evolution of effective masses of pion and sigma versus temperature. 1.2.2. Phase structure at finite T and 1.2.2.1. Chiral limit In tree – level approximation is Goldstone boson. In HF and 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 300 gives the phase diagram The phase diagram in Fig. 1.8. v0 250 IHF C Large N Fig. 1.8. Phase diagram in 200 T MeV -plane compares with those 150 form HF approximation and expanded N-large. In IHF 100 HF approximation, the solid and 50 dashed lines correspond to first v0 and second-order phase transition. 0 0 50 100 150 200 250 300 5  I MeV
1.2.2.2. Physical world - The gap equations - SD equations - The phase diagram 300 250 v 0 IHF 200 Large N T MeV 150 100 HF 50 v0 0 0 100 200 300 400 I MeV Fig. 1.13. Phase giagram of pion condensate in physical world. This result is compared with those in HF and expanded N-large. 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 - SD equations 6
- In Figs. 1.20 and 1.24 we plot the phase diagrams are obtained from numerical computation for pion and chiral condensates 300 200 250 v0 180 200 160 u0 T MeV T MeV 150 140 100 v0 120 u0 50 m 100 0 0 50 100 150 0 100 200 300 400  I MeV  I MeV Fig. 1.20. The phase diagram of Fig. 1.24. The phase diagram of pion condensate. chiral condensate. 1.4. The effect from neutrality condition - The whole system is neutral in broken phase if it is in equilibrium with the 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. 7
1.4.1. The standard case 1.0 1.4 1.2 0.8 1.0 0.6 vT v0 0.8 v f  0.6 0.4 0.4 0.2 0.2 0.0 0.0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 T MeV MeV Fig. 1.25. The pion condensate in Fig. 1.26. The pion condensate in chiral limit within neutrality condition chiral limit with neutrality condition. (solid line) and without neutrality Starting from the top the lines condition (dashed line) at = 300 correspond to = 0, 1/4, 1/2. MeV. 0.6 1.0 0.5 0.8 0.4 0.6 u f  v f  0.3 0.4 0.2 0.2 0.1 0.0 0.0 100 120 140 160 180 200 0 100 200 300 400 500 I MeV  I MeV Fig. 1.27. The pion condensate in Fig. 1.28. The chiral condensate in physical world. The solid, dashed and physical world. The solid and dashed dotted lines correspond to = 0, 1/4, lines correspond to = 0, 1/4. 1/2. 8
1.4.2. The non – standard case 1.0 1.0 0.8 0.8 0.6 uT u0 v T v0 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0 50 100 150 200 0 50 100 150 200 T  MeV T MeV Fig. 1.30. The pion condensate versus Fig. 1.32. The chiral condensate T. The solid (dashed) line corresponds versus T. The solid (dashed) line to with (without) neutrality conditiion. corresponds to with (without) Dashed line is ploted at = 200MeV. neutrality conditiion. Dashed line is ploted at = 100MeV. 1.5. The comments 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. - In physical world, the pion condensate appears at and phase transition of pion condensate is second – order. The chiral symmetry gets restored at high values of T for fixed and of for fixed T. 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. 9
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 - Parameters of model: = 138 MeV, = 500 MeV, = 93 MeV, = 12, = 5.5 MeV, . 2.2.1. Chiral limit 10
140 120 100 v 0 T MeV 80 60 v0 40 20 0 0 50 100 1 50 2 00 25 0 300 MeV Fig. 2.5. The evolution of pion Fig. 2.5. Phase diagram of pion condensate at = 100 MeV. condensate. From the bottom to top the graphs correspond to = 100, 200, 300 MeV 2.2.2. Physical world Fig. 2.9. The evolutioin of pion Fig. 2.12. Phase diagram v = 0 condensate at = 0, = 192 at = 50 MeV. MeV. 11
200 1.0 0.8 150 T  MeV 0.6 CEP u f  100  0.4 50 0.2 0.0 0 0 50 100 150 200 250 300 350 0 100 200 300 400 500 600 T  MeV   MeV Fig. 2.20. Chiral condensate in region Fig. 2.21. Phase diagram of chiral . From the right to left = 0, condensate in region . 100, 200, 220MeV. 1.0 0.25 0.8 0.20 0.6 0.15 u f  u f  0.4 0.10 0.2 0.05 0.0 0.00 0 100 200 300 400 500 600 0 100 200 300 400 500 600  MeV  MeV Fig. 2.24. Chiral condensate at = 150 Fig. 2.27. Chiral condensate at = 300 MeV. From the right to left T = 0, 50, MeV. From the right to left T = 0, 50, 100 MeV. 100 MeV. 2.3. Non – standard case - The gap equations - Parameters = 0 và . 12
2.3.1. Region 1.0 140 120 v0 0.8 100 v T v 0 T MeV 0.6 LQCD 80 LSMq 0.4 PNJL 60 v0 40 0.2 20 0.0 0.6 0.7 0.8 0.9 1.0 1.1 1.2 0 T TC 0 50 100 150 200 250   MeV Fig. 2.36. The pion condensate as a Fig. 2.34. Phase diagram v = 0. From function of T at = 0 and = 192 the bottom to top = 138, 200, 300 MeV. MeV. 2.3.2. Region 100 u0 80 T MeV 60 u0 40 20 0 0 50 100 150 200  MeV Fig. 2.41. The chiral condensate as a Fig. 2.45. Phase diagram of chiral function of T and . condensate in -plane. 2.4. The effects from neutrality condition - The matter must be stable under the weak processes like 13
. - The neutrality condition reads as - The electron mass is neglected in our numerical computation. 2.4.1. The standard case 2000 140 u0 120 1500 100 v0 T MeV T  MeV 80 1000 60 v 0 40 500 M N   20 u0 0 0 0 50 100 150 200 250 300 0 500 1000 1500 2000 MeV  MeV Fig. 2.47. Phase diagram v = 0 in Fig. 2.49. Phase diagram u = 0 in chiral limit. The solid and dashed physical world. From the bottom to lines correspond to with and without top = 0, 0.25, 0.3. The solid neutrality condition and = 232.6 (dashed) line corresponds to first MeV). (second) – order phase transition. 2.4.2. The non – standard case 140 120 v0 100 T MeV 80 60 v0 40 20 0 0 50 100 150 200 250  MeV Fig. 2.53. Phase diagram v = 0 with > and neutrality condition (solid line) and without neutrality condition at = 200 MeV (dashed line). 14
120 u0 100 80 T  MeV 60 u0 40 20 m 0 0 20 40 60 80 100 120  I MeV Fig. 2.55. Phase diagram u = 0 in region < with neutrality condition. 2.5. The comments 1. This is the first time the phase structure of LSMq is considered versus ICP, QCP and temperature. Meanwhile the current quark mass is included in our study. 2. One of the important resluts we obtained is phase diagram in - 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. 15
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 in which for untwisted quark (UQ) and for 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. 16
Fig. 3.3. Phase diagram of chiral condensate in chiral limit at = 50 MeV for UQ (left) and TQ (right). - Characteristics of the phase diagram at different value of is the same as at MeV. - In chiral limit, chiral phase transition of UQ is always first – order, meanwhile for TQ chiral phase transition has both the first and second – order and of course it exists a critical point C. 3.1.2.2. Physical world Fig. 3.6b. Phase diagram of Fig. 3.9b. Phase diagram of chiral condensate for UQ in chiral condensate for TQ in physical world at = 50 MeV. physical world at = 50 MeV. 17
- 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. 18