Lecture Overview... Modern Problems in Nuclear Physics I

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1 Lecture Overview... Modern Problems in Nuclear Physics I D. Blaschke (U Wroclaw, JINR, MEPhI) G. Röpke (U Rostock) A. Sedrakian (FIAS Frankfurt, Yerevan SU) 1. Path Integral Approach to Partition Function (3) - Scalar fields, neutral and charged, Bose condensate - Femionic fields, Bosonization Strongly interacting nuclear matter: Walecka model for gas-liquid transition Nambu--Jona-Lasinio model for quark matter: Chiral symmetry breaking 2. Basic Green's functions approximations (3) Ideal gas; Hartree-Fock approximation; Polarization function (RPA) 3. Strongly interacting quark matter (3) NJL model beyond meanfield: Bound states (pions, kaons, nucleons ) Beth-Uhlenbeck EoS and Mott effect 4. Strong correlations in the Green's function approach (3) Bound states; Cluster expansion; Pairing & Superconductivity Four series of Exercises/Seminars with: Georgy Kolomiytsev (NRNU (MEPhI))

2 Lecture Overview... Modern Problems in Nuclear Physics II D. Blaschke (U Wroclaw, JINR, MEPhI) G. Röpke (U Rostock) A. Sedrakian (FIAS Frankfurt, Yerevan SU) 1. Physics of Neutron Stars I Phenomenology, EoS and Structure (3) 2. Statistical Model and HIC I NSE, Clusters in Nuclei, HIC & Femto-Nova (3) 3. Superfluidity in Nuclear Matter I Nuclear vs. Quark Matter and Atoms in Traps (1) 4. Physics of Neutron Stars II Cooling and Superfluidity, Nuclear vs. Quark Matter (2) 5. Superfluidity in Nuclear Matter II BEC-BCS Crossover. NSR Theory (2) 6. Statistical Model and HIC II RHIC and CERN Experiments, Chemical Freeze-out (1) Four series of Exercises/Seminars with: Georgy Kolomiytsev (NRNU (MEPhI))

3 Modern problems in Nuclear Physics 2018 / 19 (I) David.Blaschke@gmail.com (Wroclaw University & JINR Dubna & MEPhI Moscow) 1. Path integral approach to the partition function 2. Ideal quantum gases - neutral and charged scalar fields: Bose condensation - fermion fields; Hubbard-Stratonovich trick: bosonization (- gauge fields and blackbody radiation*) 3. Strongly interacting matter: Walecka model for gas-liquid transition 4. Nambu Jona-Lasinio model for quark matter: Chiral symmetry breaking Literature: J.I. Kapusta: Finite Temperature Field Theory (Cambridge University Press, 1989) K. Yagi, T. Hatsuda, Y. Miyake: Quark-Gluon Plasma (Cambridge University Press, 2005)

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6 NuPECC Long Range Plan 2017; CEP in the QCD phase diagram: HIC vs. Astrophysics A. Andronic, D. Blaschke, et al., Hadron production..., Nucl. Phys. A 837 (2010) 65-86

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19 Exercise: Calculation of Dirac determinant, =

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28 2 2 2 Isospin degeneracy (proton=neutron) 4 4 4

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30 Chiral phase transition, criticality and all that Based on K. Yagi, T. Hatsuda, Y. Miake: Quark-Gluon Plasma, CUP 2005 A. SQM-2015 in Dubna

31 Chiral phase transition Order parameter? Operator not invariant under chiral rotation Thermal expectation value is a measure of dynamical breaking of chiral symmetry. As temperature increases, quark pairing is dissociated by thermal fluctuations and the Transition from the NG phase to the Wigner phase takes place. Analogy to electron pairing in metallic superconductors: Nambu and Jona-Lasinio (1961) Questions: 1) What is the critical temperature of the chiral phase transition? 2) What will be the order of the phase transition? 3) What will be the observable phenomena associated with the chiral transition?

32 Chiral phase transition Order parameter in hot/dense matter: Consider QCD partition function! High temperature expansion Low temperature expansion (Gell-Mann Oakes Renner)

33 The Nambu Jona-Lasinio (NJL) Model Simplest version, two-flavors: Grassmannian integration: det A = exp(tr ln A)

34 NJL Model, Meanfield approximation Stationary solution: Space-time independent and real scalar meanfield: Gap equation: Dynamically generated fermion mass: Thermodynamic potential (free energy density): Dispersion relation: Interaction energy Energy of quarks in Dirac sea Interpretation: Relation of minimum with dynamical quark mass and chiral condensate in the chiral limit (m=0): Quark degeneracy factor: Entropy term

35 Dynamical symmetry breaking in NJL at T=0 Low energy model, restriction of momentum integral to k < Λ, and scale all dimensionful quantities by Λ, rewrite We expanded the exact expression of free energy around σ ~ 0 and defined Gap equation from d feff(σ,0)/d σ = 0 is given by: Solution is given by the Lambert function W(z), satisfying W ew = z. Asymptotic expansion near critical point gives (Exercise):

36 Symmetry restoration at T 0 Using the high-temperature expansion one obtains: With the critical temperature: The behaviour of the order parameter (chiral condensate) near T~Tc can be found as: For Λ >> σ, T we have (Exercise): For typical values of dynamical mass M0~ MeV, we obtain:

37 Symmetry restoration at μ 0 (T=0)

38 Symmetry restoration at μ 0 (T=0) First-order phase transition: Order parameter M jumps at

39 Meanfield theory and Landau functional Order of the phase transition Consider the partition function in the thermodynamic limit, where K={Kl} is A set of generalized parameters, such as temperature, chem. Potential, coupling constants, external fields and so on. Depending on the nature of the discontinuity of The phase transition is either first order or continuous: across the phase boundary, Let us consider an order parameter field σ(x), so that the partition function is: For a uniform system, we introduce the Landau function:

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41 Meanfield theory and Landau function Second order phase transition Stationarity condition: Define critical exponents: Result: Solutions:

42 Meanfield theory, Landau functions First oder transition, driven by cubic interaction:

43 Meanfield theory, Landau functions Tricritical behaviour with sextet interaction: Both, a and b can change sign and may be parametrized as The point (a,b)=(0,0) is called the tricritical point (TCP), behaviour around it governed by t, reduced temperature, and an independent parameter s. Three stationary solutions for h=0 (Fig. a) are given by: For h 0 (Fig. b) the second order critical line disappears and the point where the first order line terminates is called critical end point (CEP).

44 Meanfield theory, Landau functions Tricritical behaviour with sextet interaction: Three-dimensional parameter space (Figure), with a TCP at (a, b, h)=(0, 0, 0) One second order line for (a, b, h)=(0, b>0, 0) Two second order lines for: (a, b, h)=(0, b<0, h>0) (a, b, h)=(0, b<0, h<0) Starting from the TCP One may define the n-critical point By introducing terms up to σ n In the Landau function

45 29 member countries!! (MP1304) New! Kick-off: Brussels, November 25, 2013

46 21 member countries! (CA15213) Theory of HOt Matter in Relativistic Heavy-Ion Collisions New: THOR! Kick-off: Brussels, October 17, 2016

47 Name, First name Khusnutdinov, Radmir Lukianov, Pavel Postnikov, Valentin Gallyamov, Renat Akhmetov, Fedor Shatalov, Nazar Matveevski, Konstantin Chulkov, Ilya Devyatyarov, Kyrill address hatach(at)mail.ru p.d.lukianov(at)yandex.ru postnikow.valentin(at)yandex.ru pehat96(at)yandex.ru f.o.akhmetov(at)gmail.com satalovnazar(at)gmail.com matveevskiykonstantin(at)gmail.com chulkov.ilya96(at)gmail.com ivanforlife9(at)gmail.com Schedule for Winter semester 2018/ , 10:15 11:00, 11:05 11:50, 12:10 13:05, , 08:30 09:10, 09:15 10:00, ,?? , 14:15 15:45 (?) , 10:15 13:30, ,?? , 14:15 15: , 10:15 13: ,?? , 14:15 15: , 10:15 13: ,?? Lectures David Blaschke Lectures Exercises Georgy Kolomiytsev Lectures Gerd Roepke Lectures Exercises Georgy Kolomiytsev Lectures David Blaschke Lectures Exercises Georgy Kolomiytsev Lectures Armen Sedrakian Lectures Exercises Georgy Kolomiytsev