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  • .

    Gauge/Gravity Duality and Particle Physics:

    New approaches to strongly coupled sectors

    Johanna Erdmenger

    Julius-Maximilians-Universität Würzburg

    1

  • Motivation and Outline

    Gauge/Gravity Duality

    New development in quantum field theory and string theory

    Generalizes the AdS/CFT correspondence

    Maps strongly coupled gauge theories to weakly coupled gravity theories

    Intrinsic fundamental interest for gravity

    Applications: Low-energy QCD

    Applications: Composite Higgs

    2

  • Motivation

    Applications also in nuclear and condensed matter physics

    Example: Strongly coupled hot and dense medium / quark-gluon plasma

    AdS/CFT prediction for ratio shear viscosity/entropy density agrees with measurements at RHIC, LHC

    η

    s =

    1

    ~ kB

    Kovtun, Son, Starinets 2004

    3

  • Motivation

    Gauge/Gravity Duality for particle physics:

    Non-supersymmetric, confining gauge theories are obtained by deforming the original AdS/CFT correspondence with relevant operators

    Embedding in string theory guarantees control of parameters

    UV completeness

    Duality allows to calculate masses and decay constants in strongly coupled theories with spontaneous chiral symmetry breaking

    4

  • Applications within particle physics

    QCD at low energies: Chiral symmetry breaking and confinement

  • Applications within particle physics

    QCD at low energies: Chiral symmetry breaking and confinement

    Composite Higgs: The Higgs particle as a pseudo-Goldstone boson

  • Applications within particle physics

    QCD at low energies: Chiral symmetry breaking and confinement

    Composite Higgs: The Higgs particle as a pseudo-Goldstone boson

    Deep inelastic scattering at very small x: Uncover universal behaviour

    5

  • Composite Higgs

    Dynamical electroweak symmetry breaking (Weinberg 1976)

    Higgs as pseudo-Goldstone boson of global symmetry (Kaplan, Georgi 1984)

    SSB of global SU(4) symmetry (phenomenologically viable) (Agashe, Contino, Pomarol; Cacciapaglia, Ferretti; Gherghetta ....)

    mixing between top quark and composite fermions

    slow running of couplings

  • Composite Higgs

    Dynamical electroweak symmetry breaking (Weinberg 1976)

    Higgs as pseudo-Goldstone boson of global symmetry (Kaplan, Georgi 1984)

    SSB of global SU(4) symmetry (phenomenologically viable) (Agashe, Contino, Pomarol; Cacciapaglia, Ferretti; Gherghetta ....)

    mixing between top quark and composite fermions

    slow running of couplings

    Gauge/gravity duality: Calculation of masses and decay constants for strongly coupled theories with SSB

    Gauge/gravity duality models constrained by string theory (Different from Randall-Sundrum models)

    6

  • Gauge/Gravity Duality

    Duality:

  • Gauge/Gravity Duality

    Duality:

    Gauge/Gravity Duality:

    A theory without gravity is dual to a gravity theory.

    7

  • Gauge/gravity duality

  • Gauge/gravity duality

    Conjecture which follows from a low-energy limit of string theory

    Duality:

    Quantum field theory at strong coupling ⇔ Theory of gravitation at weak coupling

    Holography:

    Quantum field theory in four dimensions ⇔ Gravitational theory in five dimensions

    8

  • Foundations: Gauge/gravity duality

    Best understood example: AdS/CFT correspondence

    AdS: Anti-de Sitter space, CFT: Conformal field theory

    9

  • Anti-de Sitter Space

    Space of constant negative curvature, has a boundary ds2 = e2r/Ldxµdx

    µ + dr2 Figure source: Institute of Physics, Copyright: C. Escher

    10

  • Conformal field theory

    Quantum field theory

    in which the fields transform covariantly under conformal transformations

    Conformal coordinate transformations:

    Preserve angles locally: dx′µdx′µ = Ω2(x)dxµdxµ

    Correlation functions are determined up to a small number of parameters J.E., Osborn ’97

    In AdS/CFT correspondence: Conformal field theory in 3+1 dimensions: N = 4 SU(N) Super Yang-Mills theory (global symmetry SO(4, 2)× SU(4))

    11

  • AdS/CFT correspondence

    ‘Dictionary’ Gauge invariant field theory operators ⇔ Classical fields in gravity theory

    Symmetry properties coincide

    Test: (e.g.) Calculation of correlation functions

    12

  • AdS/CFT correspondence

    Field-operator correspondence:

    〈e ∫ ddxφ0(~x)O(~x)〉CFT = Zsugra

    ∣∣∣ φ(0,~x)=φ0(~x)

    Generating functional for correlation functions of particular composite operators in the quantum field theory

    coincides with

    Classical tree diagram generating functional in supergravity

    13

  • AdS/CFT correspondence

    String theory origin⇒ Ten dimensions

    AdS5 × S5

    Symmetries of field theory and geometry coincide: SO(4, 2)× SO(6)

    Internal manifold determines field content

    14

  • String theory origin of the AdS/CFT correspondence

    near-horizon geometry AdS x S

    5 5

    D3 branes in 10d

    duality

    ⇓ Low energy limit

    Supersymmetric SU(N) gau- ge theory in four dimensions (N →∞)

    Supergravity on the space AdS5 × S5

    15

  • AdS/CFT correspondence

    16

  • Book on gauge/gravity duality

    17

  • Generalized AdS/CFT Correspondence: Gauge/Gravity Duality

    Generalization of AdS/CFT to quantum field theories of experimental relevance?

  • Generalized AdS/CFT Correspondence: Gauge/Gravity Duality

    Generalization of AdS/CFT to quantum field theories of experimental relevance?

    Prototype candidate: Low-energy QCD

  • Generalized AdS/CFT Correspondence: Gauge/Gravity Duality

    Generalization of AdS/CFT to quantum field theories of experimental relevance?

    Prototype candidate: Low-energy QCD

    SU(3) gauge theory with matter (gluons and quarks)

    Strongly coupled at low energies⇒ mesons, baryons

    Beta function negative

  • Generalized AdS/CFT Correspondence: Gauge/Gravity Duality

    Generalization of AdS/CFT to quantum field theories of experimental relevance?

    Prototype candidate: Low-energy QCD

    SU(3) gauge theory with matter (gluons and quarks)

    Strongly coupled at low energies⇒ mesons, baryons

    Beta function negative

    Relax symmetry requirements of original AdS/CFT in controlled way

    Add quark degrees of freedom Additional D-brane probes

    18

  • Chiral symmetry breaking within generalized AdS/CFT

    Combine the deformation of the supergravity metric with the addition of brane probes:

    Dual gravity description of chiral symmetry breaking and Goldstone bosons

    J. Babington, J. E., N. Evans, Z. Guralnik and I. Kirsch,

    “Chiral symmetry breaking and pions in non-SUSY gauge/gravity duals”

    Phys. Rev. D 69 (2004) 066007 [arXiv:hep-th/0306018].

    19

  • Applications to QCD-like theories: Light mesons

    Babington, J.E., Evans, Guralnik, Kirsch PRD 2004

    Gravitational realization of

    Spontaneous chiral symmetry breaking

    New ground state given by quark condensate 〈ψ̄ψ〉

    Spontaneous symmetry breaking→ Goldstone bosons (Mesons)

    20

  • Light mesons

    Babington, J.E., Evans, Guralnik, Kirsch PRD 2004

    Add D7-Branes (eight-dimensional surfaces) to ten-dimensional space

    π pseudoscalar meson mass: From fluctuations of D-brane

    ρ vector meson mass: From fluctuations of gauge field on D-brane

    21

  • Comparison to lattice gauge theory

    Mass of ρ meson as function of π meson mass2 (for N →∞)

    0 0.25 0.5 0.75 1

    (mπ / mρ0) 2

    1

    1.2

    1.4

    m ρ /

    m ρ0

    Lattice extrapolation AdS/CFT computation N= 3 N= 4 N= 5 N= 6 N= 7 N=17

    22

  • Comparison to lattice gauge theory

    Gauge/Gravity Duality: J.E., Evans, Kirsch, Threlfall ’07, review EPJA

    Lattice gauge theory: Lucini, Del Debbio, Bali, Panero et al ’13

    Result Gauge/Gravity Duality:

    mρ(mπ)

    mρ(0) = 1 + 0.307

    ( mπ mρ(0)

    )2

    Result Lattice Gauge Theory (Bali, Bursa ’08): Slope 0.341± 0.023

    23

  • Gauge/gravity dual model for composite Higgs

    J.E., Evans, Porod

    Toy model: Composite Higgs similar to η′ of QCD

    Introduce fermions into gravity action

    Given by Dirac-Born-Infeld action of string theory

    S f D7

    = TD7

    2

    ∫ d 8 ξ √ − det gAB Ψ̄P−Γ

    A ( DA +

    1

    2× 8× 5! FNPQRSΓ

    NPQRS (iσ2) ΓA

    ) Ψ

    Abt, J.E., Evans, Rigatos to appear

  • Gauge/gravity dual model for composite Higgs

    J.E., Evans, Porod

    Toy model: Composite Higgs similar to η′ of QCD

    Introduce fermions into gravity action

    Given by Dirac-Born-Infeld action of string theory

    S f