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Real time correlation function from analytic continuation Based on Finite temperature Nicolas Wink Pawlowski, Strodthoff, NW, arxiv:1710.xxxx Self-consistent spectral functions In collaboration with J. M. Pawlowski Jung, Pawlowski, von

  1. Real time correlation function from analytic continuation Based on Finite temperature Nicolas Wink Pawlowski, Strodthoff, NW, arxiv:1710.xxxx Self-consistent spectral functions In collaboration with J. M. Pawlowski Jung, Pawlowski, von Smekal NW, work in prog. Formal development Pawlowski, NW, work in prog. 1 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  2. Motivation Why real time correlation functions? PACS-CS collaboration PRL, 115 (2015) no.11, 112002 Christiansen, Haas, Pawlowski, Strodthoff Bound state spectrum Transport coefficients 2 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  3. Outline • Correlation functions and their analytic continuation (simple truncations) • Results for the O(N) model • Extension to self consistency and reformulation as integral equation 3 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  4. Continuation procedure From imaginary to real times Schwinger-Keldysh contour Matsubara contour Continuation from Matsubara frequencies Use analyticity constrains and KMS condition to obtain real time correlation functions form Matsubara formalism 4 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  5. Continuation procedure Illustrative example – one-loop perturbation theory Two bosonic fields with Calculate for Calculate Matsubara sum 5 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  6. Continuation procedure Illustrative example Bosonic occupation number Replace sum by contour integral: 6 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  7. Continuation procedure Illustrative example Identify ambiguity of the analytic continuation Mathematically rigorous Baym and Mermin, Journal of Mathematical Physics 2, 232 (1961) Analytic off the imaginary axis Correct decay behaviour at infinity Unique physical analytic continuation identified by setting everywhere 7 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  8. Continuation procedure Generalisation to the FRG No new conceptual problems Kamikado, Strodthoff, von Smekal, Wambach, Eur.Phys.J. C74, 2806 (2014) Tripolt, Strodthoff , von Smekal, Wambach, Phys.Rev. D89, 034010 (2014) Regulator poles No changes Additional poles Pawlowski, Strodthoff, Phys.Rev. D92 (2015) 8 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  9. Results O(N)-model Application to the O(N)-Model Vacuum : Effective description of the lightest mesons Finite Temperature : Calculate spectral functions of the O(N) model 9 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  10. Results O(N)-model Application to the O(N)-Model Temperature evolution of the spectral function Pawlowski, Strodthoff, NW, arxiv:1710.xxxx 10 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  11. Results O(N)-model Application to the O(N)-Model Temperature evolution of the spectral function Pawlowski, Strodthoff, NW, arxiv:1710.xxxx 11 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  12. Results O(N)-model Pion Application to the O(N)-Model Imaginary part of the retarded two-point function Pawlowski, Strodthoff, NW, arxiv:1710.xxxx 12 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  13. Results O(N)-model Sigma meson Application to the O(N)-Model Imaginary part of the retarded two-point function Pawlowski, Strodthoff, NW, arxiv:1710.xxxx 13 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  14. Results O(N)-model Pion meson Application to the O(N)-Model Finite temperature spectral function for various external momenta Pawlowski, Strodthoff, NW, arxiv:1710.xxxx 14 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  15. Results O(N)-model Sigma meson Application to the O(N)-Model Finite temperature spectral function for various external momenta Pawlowski, Strodthoff, NW, arxiv:1710.xxxx 15 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  16. Self consistency Open problems so far • How to handle branch cuts and vertices • Small values of ε makes numerical calculations hard • Self-consistency at finite temperature next to impossible Retarded Greens function Generic structure real part Generic structure imaginary part 16 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  17. Self consistency Extension to branch cuts Gluon vacuum polarization Integration contour Analytic structure propagator Analytic structure gluon polarization diagram Pawlowski, Strodthoff, NW, arxiv:1710.xxxx 17 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  18. Self consistency Extension to branch cuts Gluon vacuum polarization Integration contour Integration contour Analytic structure gluon polarization diagram Pawlowski, Strodthoff, NW, arxiv:1710.xxxx 18 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  19. Self consistency Extension to branch cuts Gluon vacuum polarization Multiply with Multiply with Finite temperature Integration contour Integration contour Recovers unique analytic continuation Pawlowski, Strodthoff, NW, arxiv:1710.xxxx 19 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  20. Self consistency Spectral representation Gluon vacuum polarization Contour integral along branch cut Cuts have “Spectral” representation Directly use spectral representation of propagators Generalized spectral representation of propagator Mass poles, Regulator poles, …. Integration contour Solve Matsubara sum analytically Choose correct analytic continuation trivially by hand Pawlowski, Strodthoff, NW, arxiv:1710.xxxx 20 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  21. Integral equation FRG as Integral equation Structure of the FRG Rewrite as integral equation Conveniently solved by iteration Analytic structure manifest with spectral representations Mass poles, Regulator poles, …. • Taking the limit ε → 0 makes numerical calculations easier • All poles of the regulated propagator equations close in Minkowski space-time • Vertices work the same (more later) Pawlowski, NW, work in progress Jung, Pawlowski, von Smekal, NW, work in progress 21 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  22. Integral equation Application to scalar field preliminary Jung, Pawlowski, von Smekal, NW, work in progress 22 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  23. Integral equation Application to scalar field preliminary Jung, Pawlowski, von Smekal, NW, work in progress 23 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  24. Integral equation Application to scalar field preliminary Jung, Pawlowski, von Smekal, NW, work in progress 24 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  25. Integral equation Application to scalar field preliminary Plotted with ε >0 for convenience Jung, Pawlowski, von Smekal, NW, work in progress 25 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  26. Spectral representation Spectral representation propagator Spectral representation How to construct spectral representations: 1. Fourier transform correlation function to time domain 2. Make θ -functions from time ordering explicit 3. Rewrite θ -functions as integral representation (Fourier space) 4. Fourier transform correlation function back to frequency domain 5. Use KMS condition/cyclicity to regroup terms 6. Obtain spectral functions by inversion Evans, Phys.Lett. B249 (1990) Evans, Nucl.Phys. B374 (1992) Bodeker, Sangel, JCAP 1706 (2017) Pawlowski, NW, work in progress 26 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  27. Spectral representation Spectral representation Propagator Spectral representation: Spectral function: Three-point function Spectral representation: preliminary Spectral functions: Degenerate for a identical fields Evans, Phys.Lett. B249 (1990) Evans, Nucl.Phys. B374 (1992) Bodeker, Sangel, JCAP 1706 (2017) Pawlowski, NW, work in progress 27 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

  28. Spectral representation Three-point function Analytic continuations Consider Constrained by Analytically continue with Two-point function Identities: Retarded and Advanced Identities: There are n-point functions and of which are independent Evans, Nucl.Phys. B374 (1992) Number of different analytic continuations unknown for general n Hou, Wang, Heinz, J.Phys. G24 (1998) Pawlowski, NW, work in progress 28 Nicolas Wink (ITP Heidelberg) Cold Quantum Coffee (Heidelberg 2017)

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