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

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A holographic approach to QCD The worldline formalism Adrian Koenigstein Institut fr Theoretische Physik, Johann Wolgang Goethe-Universitt, Max-von-Laue-Str. 1, 60438 Frankfurt am Main 12 Dezember 2016 in close cooperation with Dennis

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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

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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

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Motivation

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 3 / 31

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Motivation

Classical optical holography:

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 4 / 31

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Motivation

Classical optical holography: interference and diffraction of light 3D information stored in 2D 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

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Motivation

An analogy (by Maldacena): Take two copies of a movie, on a 70 mm film and on a DVD. linear ribbon of celluloid information = frames of movie scenes plastic disc, thin metal layer information = “pits” and “lands” in the metal

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 6 / 31

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Motivation

Black hole thermodynamics: Einstein field equations: Rµν − 1 2 gµνR = 8πTµν . Schwarzschild solution → classical field configuration → S = 0? Bekenstein-Hawking entropy ∼ surface of black-hole horizon AH S = 4πM2 = 1 4 AH

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 7 / 31

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Maldacena’s conjecture (AdS/CFT), and AdS/QCD

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 8 / 31

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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

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Maldacena’s conjecture (AdS/CFT), and AdS/QCD

Holography and QCD: Quantum chromodynamics: perturbation theory

works for high energies fails for low energies

effective models high computational effort boundary theory Theory of (quantum) gravity ? ? ? ? ? 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

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Maldacena’s conjecture (AdS/CFT), and AdS/QCD

Maldacenca’s approach: Search for symmetries! → high energies: QCD ≈ invariant under conformal transformations: Translations: xµ → x′µ = xµ + aµ → → → → →

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 11 / 31

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Maldacena’s conjecture (AdS/CFT), and AdS/QCD

Maldacenca’s approach: Search for symmetries! → high energies: QCD ≈ invariant under conformal transformations: Lorentz transformations: xµ → x′µ = Λµ

νxν

→ → → → →

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 12 / 31

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Maldacena’s conjecture (AdS/CFT), and AdS/QCD

Maldacenca’s approach: Search for symmetries! → high energies: QCD ≈ invariant under conformal transformations: Dilations (scale transformations): xµ → x′µ = ρxµ → → → → →

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 13 / 31

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Maldacena’s conjecture (AdS/CFT), and AdS/QCD

Maldacenca’s approach: Search for symmetries! → high energies: QCD ≈ invariant under conformal transformations: Special conformal transformations: xµ → x′µ

x′2 = xµ x2 + bµ

→ → → → →

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 14 / 31

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Maldacena’s conjecture (AdS/CFT), and AdS/QCD

Higher dimensional space: Five dimensional anti-de Sitter space (AdS5) ds2 =

dxµdxµ

T − dT 2 4T 2

  • reflects the conformal transformations in

Minkowski spacetime ds2 = dxµdxµ as its isometries.

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 15 / 31

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Maldacena’s conjecture (AdS/CFT), and AdS/QCD

Holography and QCD: Quantum chromodynamics: perturbation theory

works for high energies fails for low energies

effective models high computational effort boundary theory Theory of (quantum) gravity Theory on AdS5 spacetime. ? ? ? ? theory in the spacetime volume But, what about the particle content, the dynamics, etc.?

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 16 / 31

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Maldacena’s conjecture (AdS/CFT), and AdS/QCD

Maldacena’s conjecture – AdS/CFT correspondence

A Type IIB string theory on an AdS5 × S5 space is holographically dual to an N = 4 SU(N) Super Yang-Mills Theory living on the four-dimensional boundary (Minkowski space) of the AdS5 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

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Maldacena’s conjecture (AdS/CFT), and AdS/QCD

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 18 / 31

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The worldline formalism

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The worldline formalism

Original usage: mathematical tool alternative description in QFT method to calculate Feynman diagrams study of anomalies The discovery (by Dennis Dietrich): AdS5 structure appears naturally leads to a 5D 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

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The worldline formalism

Simplifications (in this talk):

  • mit: 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 2Tr ln(−D2) , where D2 = (∂µ − iVµ)2 = −

ˆ

pµ − Vµ(ˆ x)

2 .

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 21 / 31

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The worldline formalism

The fifth extra dimension (Schwinger proper time): Use the integral representation of the logarithm. ln(a) = −

∞

ε>0

dT T e−Ta + normalization , results in w = −1 2Tr ln

− D2(ˆ

x, ˆ p)

= ∞

ε>0

dT 2T Tr

  • eTD2(ˆ

x, ˆ p)

.

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 22 / 31

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The worldline formalism

We rewrite the trace as an quantum mechanical path-integral, Tr

  • eTD2(ˆ

x, ˆ p)

= [. . .] =

  • P

[dx]

  • [dp] e

T

0 dτ

  • −
  • pµ−Vµ(x)

2

−ip(τ)·˙ x(τ)

  • .

and integrate out the Gaussian momentum integrals, = N (4π)2T 2

  • P

[dx] e

T

0 dτ

  • ˙

x2 4 −i ˙

x·V (x)

  • .

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 23 / 31

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The worldline formalism

→ → → → →

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 24 / 31

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The worldline formalism

Finally we have, w =

  • d4x0

∞

ε>0

dT 2T 3 L , L ≡ N (4π)2

  • P

[dy] e

T

0 dτ

  • ˙

y2 4 −i ˙

y·V (x0+y)

  • .

The volume element is now the volume element of AdS5. ds2 =

dxµdxµ

T − dT 2 4T 2

  • −

→

  • |g| =

1 2T 3 .

(D. Dietrich, Phys.Rev. D89 (2014) no.10, 106009)

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 25 / 31

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The worldline formalism

A field theory for V on AdS5: expand and solve path integral contractions are w.r.t. AdS5-metrics w =

∞

ε>0

d5x

  • |g|
  • n,n′

#n,n′ (g◦◦)n(∂◦)n′[V◦(x0)]n′ . But:

  • 1. VT components are missing.
  • 2. Vµ(x0) 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

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The worldline formalism

The final result is a fully fledged action for V on AdS5 w =

∞

ε>0

d5x

  • |g|
  • n,n′

#n,n′ (g••)n(∇•)n′[V•(x0, T)]n′ . The lowest order contribution (second order) is the free theory of the vector field in AdS5 w ⊃ #

∞

ε>0

d5x

  • |g| VMNVMN + selfinteraction terms .

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 27 / 31

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The worldline formalism

→ → → → →

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 28 / 31

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The worldline formalism

Figure: The squared masses of the first few ρ resonances versus their consecutive number n. The straight line is the fit m2

n ∼ n.

(A. Karch et al., Phys.Rev. D74 (2006) 015005)

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 29 / 31

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The worldline formalism

Summary There are dualities between CFT’s and string theory. Deploying the symmetries of CFT’s and string theories, dualities can be found. The worldline formalism connects sources in QFT with fields in AdS5. (QFT is not a CFT.)

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 30 / 31

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The worldline formalism

Outlook/Work in progress: consider spin 1/2 quarks inclusion of other sources inclusion of flavor and color (gauge fields) reproduction of bottom up holographic models is possible study the connection to FRG and renormalization

Adrian Koenigstein (ITP) Worldline holography 12.12.2016 31 / 31