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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

4π

~ 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