PHASE TRANSITIONS IN QED THROUGH THE SCHWINGER DYSON FORMALISM
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1 PHASE TRANSITIONS IN THROUGH THE SCHWINGER DYSON FORMALISM Spyridon Argyropoulos University of Athens Physics Department Division of Nuclear Physics and Elementary Particles Supervisor: C.N. Ktorides Athens 2009
2 Introduction : relativistic quantum field theory Describes interactions between Weak coupling: Perturbation theory Strong coupling: non-perturbative phenomena (phase transitions)
3 Scope Non-perturbative formulation of (The Schwinger Equations) Study of phase transitions in through the SDE (bifurcation theory, Hammerstein integral equations)
4 Plan of the Talk 1. Introductory Remarks Quantum Electrodynamics Symmetry Breaking Renormalization and Critical Phenomena 2. The Schwinger Equations SDE Renormalization The M,F,G system 3. Chiral symmetry breaking in Hammerstein integral equations Quenched Unquenched
5 Introductory Remarks Schwinger equations Chiral symmetry breaking in Symmetry Breaking Renormalization Critical Phenomena Quantum Electrodynamics Vertex
6 Θραύση Symmetry συµµετρίας Breaking Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Επανακανονικοποίηση Renormalization Critical Κρίσιµα Phenomena φαινόµενα Επανακανονικοποίηση Κρίσιµα φαινόµενα Κβαντική Quantum Ηλεκτροδυναµική Electrodynamics Perturbation Theory Interacting System: Tomonaga Schwinger Equation: Scattering matrix: Dyson expansion of the Scattering matrix
7 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Κβαντική Symmetries Ηλεκτροδυναµική Chiral symmetry Symmetry Breaking Renormalization Critical Phenomena Chirality: In the chiral limit Right-handed p Left-handed p s s
8 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Dynamical Ηλεκτροδυναµική Symmetry θραύση συµµετρίας Breaking Θραύση Symmetry συµµετρίας Breaking Επανακανονικοποίηση Renormalization Critical Κρίσιµα Phenomena φαινόµενα Επανακανονικοποίηση Κρίσιµα φαινόµενα Nambu-Jona-Lasinio model Eq. of motion: Mass determined by dynamics! Order parameter: Asymmetrical phase: Appearance of Ν 2 massless pseudoscalar bosons
9 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Renormalization Ηλεκτροδυναµική θραύση Group συµµετρίας Symmetry Breaking Renormalization Critical Phenomena
10 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Οµάδα Renormalization επανακανονικοποίησης Ηλεκτροδυναµική θραύση Group συµµετρίας Θραύση Symmetry συµµετρίας Breaking Επανακανονικοποίηση Renormalization Critical Κρίσιµα Phenomena φαινόµενα Επανακανονικοποίηση Κρίσιµα φαινόµενα Fixed point of RG flow (Bifurcation point) Critical RG flow
11 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Non perturbative Ηλεκτροδυναµική θραύση methods συµµετρίας SDE Renormalization The M,F,G system Why do we need non perturbative methods; Dyson expansion of S matrix For we cannot resort to perturbation theory
12 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Οι The εξισώσεις Schwinger Ηλεκτροδυναµική θραύση Schwinger συµµετρίας equations Dyson Οι SDE εξισώσεις SD Επανακανονικοποίηση Renormalization Το The σύστηµα M,F,G system M,F,G SDE for the fermion propagator SDE for the photon propagator
13 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Κανονικός The Schwinger Ηλεκτροδυναµική φορµαλισµός θραύση συµµετρίας equations Οι SDE εξισώσεις SD Επανακανονικοποίηση Renormalization Το The σύστηµα M,F,G system M,F,G Vertex function: Vacuum Polarization Tensor Fermion self-energy operator
14 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Το The ανάπτυγµα Dyson Ηλεκτροδυναµική expansion θραύση Dysonσυµµετρίας SDE Renormalization The M,F,G system
15 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Το The ανάπτυγµα Dyson Ηλεκτροδυναµική expansion θραύση Dysonσυµµετρίας Οι SDE εξισώσεις SD Επανακανονικοποίηση Renormalization Το The σύστηµα M,F,G system M,F,G
16 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Επανακανονικοποίηση Renormalization Ηλεκτροδυναµική θραύση συµµετρίας Οι SDE εξισώσεις SD Επανακανονικοποίηση Renormalization Το The σύστηµα M,F,G system M,F,G Taylor expansion Define
17 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Επανακανονικοποίηση Renormalization Ηλεκτροδυναµική θραύση συµµετρίας Fermionic field renormalization function Οι SDE εξισώσεις SD Επανακανονικοποίηση Renormalization Το The σύστηµα M,F,G system M,F,G Dressing Near the pole
18 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Επανακανονικοποίηση Renormalization Ηλεκτροδυναµική θραύση συµµετρίας Κανονικός Οι SDE εξισώσεις φορµαλισµός SD Συναρτησιακός Επανακανονικοποίηση Renormalizationφορµαλισµός Το The σύστηµα M,F,G system M,F,G Taylor expansion
19 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική The M,F,G Ηλεκτροδυναµική system θραύση συµµετρίας SDE Renormalization The M,F,G system
20 Hammerstein equations Quenched Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Unquenched Αυθόρµητη Κβαντική Integral equations Ηλεκτροδυναµική θραύση of συµµετρίας Hammerstein type Hammerstein equation Phase transition Bifurcation solutions Hammerstein equations [J.J. Kozak et al., J. Chem. Phys. 52 (1970) 2416] [J.J. Kozak, Adv. Chem. Phys. 40 (1979) 229] [E.E. Tareeva, Theor. Math. Phys. 21 (1974) 1189]
21 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Θεωρία Bifurcation Διακλαδώσεων Ηλεκτροδυναµική Theory θραύση συµµετρίας Εξισώσεις Hammerstein Hammerstein equations Quenched µε σβέση Unquenched χωρίς σβέση Phase transition Bifurcation solutions
22 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Θεωρία Bifurcation Διακλαδώσεων Ηλεκτροδυναµική Theory θραύση συµµετρίας Εξισώσεις Hammerstein Hammerstein equations Quenched µε σβέση Unquenched χωρίς σβέση Maskawa Nakajima theorem There exists a non-trivial solution iff Where is the smallest eigenvalue of the linearized gap equation: [Prog. Theor. Phys. 52 (1974) 1326]
23 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Quenched Ηλεκτροδυναµική approximation θραύση συµµετρίας Hammerstein equations Quenched Unquenched Fermion SDE Bare Vertex Approximation: Quenched approximation: Rainbow approximation
24 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Εξίσωση Gap equation χάσµατος Ηλεκτροδυναµική θραύση συµµετρίας Εξισώσεις Hammerstein Hammerstein equations Quenched µε σβέση Unquenched χωρίς σβέση Landau gauge Rainbow approximation Linearization
25 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Εξίσωση Gap equation χάσµατος Ηλεκτροδυναµική θραύση συµµετρίας Εξισώσεις Hammerstein Hammerstein equations Quenched µε σβέση Unquenched χωρίς σβέση
26 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Εξίσωση Gap equation χάσµατος Ηλεκτροδυναµική θραύση συµµετρίας Εξισώσεις Hammerstein Hammerstein equations Quenched µε σβέση Unquenched χωρίς σβέση General solution: BC: Miransky scaling
27 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Ευσταθειοποίηση Vacuum stabilization Ηλεκτροδυναµική θραύση κενού συµµετρίας Εξισώσεις Hammerstein Hammerstein equations Quenched µε σβέση Unquenched χωρίς σβέση : (2N 2 Tachyons) Chirally symmetric phase is unstable : Dynamical Chiral Symmetry Breaking Unstable Vacuum Ν 2 Pseudoscalar Tachyons Ν 2 Pseudoscalar NG bosons : Ν 2 Scalar Tachyons Ν 2 massive scalar bosons Dynamical mass generation stabilizes the vacuum
28 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Διαστατική Dimensional Ηλεκτροδυναµική µετάλλαξη θραύση Transmutation συµµετρίας In the absence of interactions Εξισώσεις Hammerstein Hammerstein equations Quenched µε σβέση Unquenched χωρίς σβέση In the presence of interactions Anomalous dimension of : Supercritical phase: 4-fermion interactions become relevant (renormalizable) Gauged Nambu-Jona-Lasinio
29 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Πόλος Landau Landau Pole Ηλεκτροδυναµική θραύση συµµετρίας Εξισώσεις Hammerstein Hammerstein equations Quenched µε σβέση Unquenched χωρίς σβέση Landau Pomeranchuk - Fradkin Zero Charge Situation Landau Pole Trivial Local Limit
30 Εξισώσεις Hammerstein Hammerstein equations Quenched µε σβέση Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Unquenched χωρίς σβέση Αυθόρµητη Κβαντική UV σταθερό stable Ηλεκτροδυναµική fixed θραύση σηµείο pointσυµµετρίας Gell-Mann-Low: Existence of local limit UV stable fixed point [Phys. Rev. 95 (1954) 1300] Miransky scaling: α c : 2 nd order chiral transition α c : UV stable fixed point Different universality class [V.A. Miransky, Nuovo Cim. 90A (1985) 149]
31 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Διάγραµµα Phase diagram Ηλεκτροδυναµική φάσης θραύση συµµετρίας Εξισώσεις Hammerstein Hammerstein equations Quenched µε σβέση Unquenched χωρίς σβέση Quenched Subcritical phase Supercritical phase Every value of the coupling α defines a trivial IR stable fixed point The theory is scale invariant Chiral symmetry breaking new UV divergence: α c UV stable fixed point non-trivial local limit
32 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Critical coupling Ηλεκτροδυναµική θραύση constant συµµετρίας Hammerstein equations Quenched Unquenched [D. Atkinson et al, IJMPA 7 (1992) 7629] [J. Oliensis, P.W. Johnson, PRD 42 (1990) 656] [K. Kondo et al, MPLA 7 (1992) 1509] (Charge screening reduces the interaction strength) Existence of critical number of flavours Lattice 4: Relevant parameters space may include [J.B. Kogut et al, PLB 232 (1989) 235] [E. Dagotto, Nucl. Phys. (Proc. Suppl.) B4 (1988) 607] [J.B. Kogut et al, PLB 232 (1989) 235]
33 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Κλάση Universality οικουµενικότητας Ηλεκτροδυναµική θραύση class συµµετρίας Εξισώσεις Hammerstein Hammerstein equations Quenched µε σβέση Unquenched χωρίς σβέση Lattice 4 : Mean-field critical exponents The operator acquires a large anomalous dimension [M. Göckeler et al, NPB 371 (1992) 713] [M. Göckeler et al, NPB 334 (1990) 527] [G. Schierholz, NPB (Proc. Sup.) 20 (1991) 623] gnjl, unquenched mean-field universality class [K. Kondo, IJMPA 6 (1991) 5447]
34 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Διάγραµµα Phase diagram Ηλεκτροδυναµική φάσης θραύση συµµετρίας Εξισώσεις Hammerstein Hammerstein equations Quenched µε σβέση Unquenched χωρίς σβέση Unquenched (gnjl) Flows terminate at with different no UV stable fixed point only for Complete screening (trivial local limit) Strong coupling phase Chiral symmtery breaking (Yukawa-type interaction) Zero charge situation Theory describes free condensates at the local limit [K. Kondo, IJMPA 6 (1991) 5447]
35 Θραύση Chiral χειραλικής symmetry συµµετρίας breaking στην in Αυθόρµητη Κβαντική Διάγραµµα Phase diagram Ηλεκτροδυναµική φάσης θραύση συµµετρίας Εξισώσεις Hammerstein Hammerstein equations Quenched µε σβέση Unquenched χωρίς σβέση Possibility of existence of non-trivial local limit Magnetic forces suppress vacuum polarization (possibly) absence of complete screening [K. Kondo et al, PLB 220 (1989) 270] [J. Kogut et al, PLB 213 (1988) 56] If complete screening occurs low-energy effective theory (weak phase: ) (strong phase: )
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