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 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 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. 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 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. 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 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. 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 và T = 100 MeV. 20 40 60 80 100 120 140 0.0 0.2 0.4 0.6 0.8 1.0 T MeV u f  0 20 40 60 80 100 10 5 0 5 10 15 20 uMeV V  M eV .fm 3  Fig. 1.3. The chiral condensate as a function of T in physical world. 100 200 300 400 500 0.0 0.2 0.4 0.6 0.8 1.0 1.2 T MeV u f  1.2.1.2. Physical world - The gap equation - Schwinger–Dyson (SD) equations - Numerical results 5 Fig. 1.4. The evolution of effective masses of pion and sigma versus temperature. M M 0 50 100 150 200 250 300 0 200 400 600 800 T MeV M ,  M eV  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 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. v  0 v  0 IHF Large N HF C 0 50 100 150 200 250 300 0 50 100 150 200 250 300 IMeV T M eV  6 1.2.2.2. Physical world - The gap equations - SD equations - 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 - SD equations Fig. 1.13. Phase giagram of pion condensate in physical world. This result is compared with those in HF and expanded N-large. v  0 v  0 IHF Large N HF 0 100 200 300 400 0 50 100 150 200 250 300 I MeV T M eV  7 m v  0 v  0 0 100 200 300 400 0 50 100 150 200 250 300  I MeV T M eV  u  0 u  0 0 50 100 150 100 120 140 160 180 200  I MeV T M eV  - In Figs. 1.20 and 1.24 we plot the phase diagrams are obtained from numerical computation for pion and chiral condensates 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. 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. 0 50 100 150 200 250 300 0.0 0.2 0.4 0.6 0.8 1.0 T MeV v T v 0  0 50 100 150 200 250 300 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 MeV v f  0 100 200 300 400 500 0.0 0.2 0.4 0.6 0.8 1.0 I MeV u f  100 120 140 160 180 200 0.0 0.1 0.2 0.3 0.4 0.5 0.6 I MeV v f  9 0 50 100 150 200 0.0 0.2 0.4 0.6 0.8 1.0 T MeV v T v 0  0 50 100 150 200 0.0 0.2 0.4 0.6 0.8 1.0 T MeV u T u 0  1.4.2. The non – standard case 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. 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 - Parameters of model: = 138 MeV, = 500 MeV, = 93 MeV, = 12, = 5.5 MeV, . 2.2.1. Chiral limit 11 2.2.2. Physical world Fig. 2.5. The evolution of pion condensate at = 100 MeV. Fig. 2.5. Phase diagram of pion condensate. From the bottom to top the graphs correspond to = 100, 200, 300 MeV v  0 v  0 0 50 100 1 50 2 00 25 0 300 0 20 40 60 80 100 120 140 MeV T M eV  Fig. 2.9. The evolutioin of pion condensate at = 0, = 192 MeV. Fig. 2.12. Phase diagram v = 0 at = 50 MeV. 12 2.3. Non – standard case - The gap equations - Parameters = 0 và . 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, 100 MeV. 0 50 100 150 200 250 300 350 0.0 0.2 0.4 0.6 0.8 1.0 T MeV u f   CEP 0 100 200 300 400 500 600 0 50 100 150 200 MeV T M eV  0 100 200 300 400 500 600 0.0 0.2 0.4 0.6 0.8 1.0  MeV u f  0 100 200 300 400 500 600 0.00 0.05 0.10 0.15 0.20 0.25  MeV u 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 function of T and . Fig. 2.45. Phase diagram of chiral condensate in -plane. LQCD LSMq PNJL 0.6 0.7 0.8 0.9 1.0 1.1 1.2 0.0 0.2 0.4 0.6 0.8 1.0 T TC v T v 0  v  0 v  0 0 50 100 150 200 250 0 20 40 60 80 100 120 140 MeV T M eV  u  0 u  0 0 50 100 150 200 0 20 40 60 80 100 MeV T M eV  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). v  0 v  0 0 50 100 150 200 250 300 0 20 40 60 80 100 120 140 MeV T M eV  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.   u  0 u  0 M N 0 500 1000 1500 2000 0 500 1000 1500 2000  MeV T M eV  v  0 v  0 0 50 100 150 200 250 0 20 40 60 80 100 120 140  MeV T M eV  15 Fig. 2.55. Phase diagram u = 0 in region < with neutrality condition. u  0 u  0 m 0 20 40 60 80 100 120 0 20 40 60 80 100 120  I MeV T M eV  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. 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 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. 17 - 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. 3.3. 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. 3.2.2. Numerical computation 3.2.2.1. Chiral limit - In chiral limit, chiral condensate of UQ is first – order everywhere, meanwhile for TQ it exists both first and second - order. Fig. 3.11a. Chiral condensate of UQ in chiral limit at = 100 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. 20 3.2.2.2. Physical world - In physical world chiral phase transition of TQ has only crossover, meanwhile for UQ it shows both first – order and crossover. 3.3. The comments After discussing about the results, we find relation between the chiral phase transition and Hohenberg theorem. For example, we consider in chiral limit at = 50 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. Some outlooks: 1. Study phase structure of QCD in Polyakov – LSM in order to study phase structure of QCD in high QCP region. 2. Due to the fact that the critical temperature of phase transition depends on the compactifcation length then the present study might be helpful to explore many physical properties of high temperature superconductors, and, moreover, it can be also applied to studying the Bose - Einstein condensation in (2D + ) - dimensional space. 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, 24 Scientific Journal, Hanoi University of Education 2, to be published. 9. Tran Huu Phat and Nguyen Van Thu, Chiral phase transition in compactified space-time, submitted to The 37th National Conference on Theoritical Physics.