A holographic approach to QCD The worldline formalism Adrian - PowerPoint PPT Presentation

A holographic approach to QCD – The worldline formalism Adrian Koenigstein Institut für Theoretische Physik, Johann Wolgang Goethe-Universität, Max-von-Laue-Str. 1, 60438 Frankfurt am Main 12 Dezember 2016 in close cooperation with Dennis D. Dietrich Adrian Koenigstein (ITP) Worldline holography 12.12.2016 1 / 31

Overview Structure: 1. Holography in theoretical physics – Motivation 2. Maldacena’s conjecture (AdS/CFT), and AdS/QCD 3. The worldline formalism Adrian Koenigstein (ITP) Worldline holography 12.12.2016 2 / 31

Motivation Adrian Koenigstein (ITP) Worldline holography 12.12.2016 3 / 31

Motivation Classical optical holography: Adrian Koenigstein (ITP) Worldline holography 12.12.2016 4 / 31

Motivation Classical optical holography: interference and diffraction of light 3 D information stored in 2 D diffraction patterns holograms only pretend three dimensionality → only surface structure is depicted Holography in theoretical physics: abstract: two theories in different dimensions same amount of information in both theories mapping between theories = holographic mapping theories are “dual” to each other Adrian Koenigstein (ITP) Worldline holography 12.12.2016 5 / 31

Motivation An analogy (by Maldacena) : Take two copies of a movie, on a 70 mm film and on a DVD. linear ribbon of celluloid plastic disc, thin metal layer information = frames of movie information = “pits” and scenes “lands” in the metal Adrian Koenigstein (ITP) Worldline holography 12.12.2016 6 / 31

Motivation Black hole thermodynamics : Einstein field equations: R µν − 1 2 g µν R = 8 π T µν . Schwarzschild solution → classical field configuration → S = 0? Bekenstein-Hawking entropy ∼ surface of black-hole horizon A H S = 4 π M 2 = 1 4 A H Adrian Koenigstein (ITP) Worldline holography 12.12.2016 7 / 31

Maldacena’s conjecture (AdS/CFT), and AdS/QCD Adrian Koenigstein (ITP) Worldline holography 12.12.2016 8 / 31

Maldacena’s conjecture (AdS/CFT), and AdS/QCD Holographic principle (Maldacena, ’t Hooft, Gubser, Klebanov, Polyakov, Witten, ...) For theories of quantum field theory and gravitation, every description of the dynamics within a spacetime volume has an equivalent description on its surface. Both theories can appear to be completely unrelated and their connection (the holographic mapping) be of arbitrary complexity. Why is this useful? How can we find this equivalent description (the dual theory)? How are both theories connected? How can we imagine such a duality? Adrian Koenigstein (ITP) Worldline holography 12.12.2016 9 / 31

Maldacena’s conjecture (AdS/CFT), and AdS/QCD Holography and QCD : Theory of (quantum) gravity Quantum chromodynamics: ? perturbation theory ? works for high energies fails for low energies ? effective models ? high computational effort ? boundary theory theory in the spacetime volume How can we find an appropriate holographic dual theory to QCD? Adrian Koenigstein (ITP) Worldline holography 12.12.2016 10 / 31

Maldacena’s conjecture (AdS/CFT), and AdS/QCD Maldacenca’s approach: Search for symmetries! → high energies: QCD ≈ invariant under conformal transformations : Translations: x µ → x ′ µ = x µ + a µ → → → → → Adrian Koenigstein (ITP) Worldline holography 12.12.2016 11 / 31

Maldacena’s conjecture (AdS/CFT), and AdS/QCD Maldacenca’s approach: Search for symmetries! → high energies: QCD ≈ invariant under conformal transformations : Lorentz transformations: x µ → x ′ µ = Λ µ ν x ν → → → → → Adrian Koenigstein (ITP) Worldline holography 12.12.2016 12 / 31

Maldacena’s conjecture (AdS/CFT), and AdS/QCD Maldacenca’s approach: Search for symmetries! → high energies: QCD ≈ invariant under conformal transformations : Dilations (scale transformations): x µ → x ′ µ = ρ x µ → → → → → Adrian Koenigstein (ITP) Worldline holography 12.12.2016 13 / 31

Maldacena’s conjecture (AdS/CFT), and AdS/QCD Maldacenca’s approach: Search for symmetries! → high energies: QCD ≈ invariant under conformal transformations : Special conformal transformations: x µ → x ′ µ x ′ 2 = x µ x 2 + b µ → → → → → Adrian Koenigstein (ITP) Worldline holography 12.12.2016 14 / 31

Maldacena’s conjecture (AdS/CFT), and AdS/QCD Higher dimensional space: Five dimensional anti-de Sitter space (AdS 5 ) − d T 2 � d x µ d x µ d s 2 = � 4 T 2 T reflects the conformal transformations in Minkowski spacetime d s 2 = d x µ d x µ as its isometries . Adrian Koenigstein (ITP) Worldline holography 12.12.2016 15 / 31

Maldacena’s conjecture (AdS/CFT), and AdS/QCD Holography and QCD : Theory of (quantum) gravity Quantum chromodynamics: Theory on AdS 5 spacetime. perturbation theory ? works for high energies fails for low energies ? effective models ? high computational effort ? boundary theory theory in the spacetime volume But, what about the particle content, the dynamics, etc.? Adrian Koenigstein (ITP) Worldline holography 12.12.2016 16 / 31

Maldacena’s conjecture (AdS/CFT), and AdS/QCD Maldacena’s conjecture – AdS/CFT correspondence A Type II B string theory on an AdS 5 × S 5 space is holographically dual to an N = 4 SU ( N ) Super Yang-Mills Theory living on the four-dimensional boundary (Minkowski space) of the AdS 5 space. well-established and well-tested nice to have − → but not QCD and not nature shares lots of properties with QCD strong weak duality How can we profit from this correspondence for QCD? Adrian Koenigstein (ITP) Worldline holography 12.12.2016 17 / 31

Maldacena’s conjecture (AdS/CFT), and AdS/QCD Adrian Koenigstein (ITP) Worldline holography 12.12.2016 18 / 31

The worldline formalism Adrian Koenigstein (ITP) Worldline holography 12.12.2016 19 / 31

The worldline formalism Original usage: mathematical tool alternative description in QFT method to calculate Feynman diagrams study of anomalies The discovery (by Dennis Dietrich): AdS 5 structure appears naturally leads to a 5 D holographic description of QFT Our goal: find a dual theory to QCD a strict derivation of AdS/QCD?! currently: reproduce existing holographic models of QCD Adrian Koenigstein (ITP) Worldline holography 12.12.2016 20 / 31

The worldline formalism Simplifications (in this talk): omit: spin, quark mass, color, flavor, higher loops a scalar flavor coupled to a vector source V Starting point: 1-loop effective action w (all connected diagrams): w = − 1 2 Tr ln( − D 2 ) , where � 2 . D 2 = ( ∂ µ − i V µ ) 2 = − � ˆ p µ − V µ (ˆ x ) Adrian Koenigstein (ITP) Worldline holography 12.12.2016 21 / 31

The worldline formalism The fifth extra dimension (Schwinger proper time): Use the integral representation of the logarithm. � ∞ d T T e − Ta + normalization , ln( a ) = − ε> 0 results in � ∞ � − D 2 (ˆ � = w = − 1 d T � e TD 2 (ˆ p ) � x , ˆ 2 Tr ln x , ˆ p ) . 2 T Tr ε> 0 Adrian Koenigstein (ITP) Worldline holography 12.12.2016 22 / 31

The worldline formalism We rewrite the trace as an quantum mechanical path-integral, � T � 2 � � � � � � e TD 2 (ˆ p ) � 0 d τ p µ − V µ ( x ) − ip ( τ ) · ˙ x ( τ ) x , ˆ − Tr = [ . . . ] = [ d x ] [ d p ] e . P and integrate out the Gaussian momentum integrals, � T N x 2 � � � ˙ 0 d τ 4 − i ˙ x · V ( x ) = [ d x ] e . (4 π ) 2 T 2 P Adrian Koenigstein (ITP) Worldline holography 12.12.2016 23 / 31

The worldline formalism → → → → → Adrian Koenigstein (ITP) Worldline holography 12.12.2016 24 / 31

The worldline formalism Finally we have, � ∞ d T � d 4 x 0 w = 2 T 3 L , ε> 0 � T � � y 2 ˙ N � 4 − i ˙ y · V ( x 0 + y ) 0 d τ L ≡ [ d y ] e . (4 π ) 2 P The volume element is now the volume element of AdS 5 . − dT 2 � dx µ dx µ 1 ds 2 = � � − → | g | = 2 T 3 . 4 T 2 T (D. Dietrich, Phys.Rev. D89 (2014) no.10, 106009) Adrian Koenigstein (ITP) Worldline holography 12.12.2016 25 / 31

The worldline formalism A field theory for V on AdS 5 : expand and solve path integral contractions are w.r.t. AdS 5 -metrics �� ∞ # n , n ′ ( g ◦◦ ) n ( ∂ ◦ ) n ′ [ V ◦ ( x 0 )] n ′ . � d 5 x � | g | w = ε> 0 n , n ′ But: 1. V T components are missing. 2. V µ ( x 0 ) seems not to depend on T yet. 3. Derivatives ∂ T in T -direction are missing. (D. Dietrich, Phys.Rev. D94 (2016) no.8, 086013) Adrian Koenigstein (ITP) Worldline holography 12.12.2016 26 / 31

The worldline formalism The final result is a fully fledged action for V on AdS 5 �� ∞ # n , n ′ ( g •• ) n ( ∇ • ) n ′ [ V • ( x 0 , T )] n ′ . � d 5 x � | g | w = ε> 0 n , n ′ The lowest order contribution (second order) is the free theory of the vector field in AdS 5 �� ∞ � | g | V MN V MN + selfinteraction terms . d 5 x w ⊃ # ε> 0 Adrian Koenigstein (ITP) Worldline holography 12.12.2016 27 / 31

The worldline formalism → → → → → Adrian Koenigstein (ITP) Worldline holography 12.12.2016 28 / 31