Heidelberg, June 2008 Volker Schomerus - DESY Hamburg -

Heidelberg, June 2008

Volker Schomerus

- DESY Hamburg -

Of Mesons and Metals – Of Mesons and Metals – Bethe & the 5th Dimension Bethe & the 5th Dimension

Introduction: Mesons Introduction: Mesons

Even perturbative physics difficult

flux tubes

~ strings

Can meson physics be modeled by strings ?

Non-perturbative QCD physics remains to be

understood → confinement, sQG plasma ...

But: String amplitudes are too soft

(↔ extended nature) ↔experiment!

QCD → description of perturbative

regime through gauge theory (GT).

Similar issues for SM of many body systems

Loops & legs

The 5The 5thth Dimension Dimension

● Large Nc exp. of GT↔String theory [‘t Hooft]

● Flux tube extension ↔ 5th dim. ? [Polyakov]

Maldacena ’97: Many strongly coupled 4D GTs

possess a weakly coupled description through

strings in a curved 5D geometry.

compare high-T

low-T dualities

Geometrization of Quantum Physics

String theoretic descriptions for non-relativistic

3D systems are being investigatedcrit cold atomsstrange metals

Particle Theory String Theory

Δ = L • L

Laplacian

H = ∑ Li • Li+1

1D Spin Chain

Bethe, Metals and StringsBethe, Metals and Strings

non-linear σ model

continuum limit

Solution method: (thermodynamic) Bethe Ansatz

e.g. spherically sym-metric problem in ED

Spectrum from solutions of non-linear integral equations

Plan:Plan:

Part I Part II

Part III

Strings & Gauge TheoryStrings & Gauge Theory

Closed strings and SUGRAClosed strings and SUGRA

Closed string theory in background X

Infinite tower of vibrational modes M2 ~ n/ℓs

String interaction through 3-vertex gs

String length

At low energies (E « ℓs ): massless closed str.

modes behave like gravitons 10D SUGRA

String

coupling

2

-1

Solitonic & Dirichlet p-branesSolitonic & Dirichlet p-branes

D-branes are objects on

which open strings can

end [Polchinski]

10-dim SUGRA has

solutions describing

massive & charged

objects localized on

p+1 - dim. surfaces.

[black branes]

Branes in string theory ?

Open strings & gauge theoryOpen strings & gauge theory

At low energies (E«ℓs ):

massless modes of the

open string behave like

gauge bosons + matter

on the world-volume of

D-branes. p+1- dimensional

gauge theory

a .. b N

Aab

-1

Gauge-String theory dualities Gauge-String theory dualities

• Closed string in curved 10-dimensional space

• Gauge theory loop classical string theory !

Holography← Δy →

Depending on Δy

one side simpler

Δy << ℓs: gauge th

↔ closed string th

High redshift ↔ soft strings ↔ meson resonances

Example: AdSExample: AdS5 5 / CFT/ CFT4 4 duality duality

N=4 4D S Yang-Mills Strings in AdS5 x S5

U(Nc) gauge field A; 6 scalars Φ

Gauge theory on stack Strings in their near

of Nc D3 - branes horizon geometry

par.: par.: λλ=g=g22YM YM NNc c ; N; Ncc R4 / ℓs

4=λ ; gs=λ/Nc

Gauge inv. operatorsGauge inv. operators Closed string states

Anomalous dimensionsAnomalous dimensions Mass of string mode

…... …...

pert. Gauge Theory

pert. String TheoryNc

λ

ls/R

gs

LatticeGauge Theory

[‘t Hooft]

[Polyakov, Maldacena]

Map of PhysicsMap of Physics

Quantum Gravity

Strings & Spin Strings & Spin ChainsChains

Particle vs. String GeometryParticle vs. String GeometryParticle in S1 with radius R:

String on S1: momentum winding

oscillations

q = e-β

Does Z(q) encode spectrum of operator ?

String Geometry & MagnetsString Geometry & Magnets

HXXZ ↔ String spectrum on S1 with radius R

Pauli matrices

L ∞

Spectrum of many 1D magnets is known !Factorized scattering → Bethe Ansatz [Bethe 31] ...

1D anisotropic spin ½ Heisenberg magnet:

Luttinger liquid

String geometry & String geometry & σσ-models-models

ΣXμ

e.g. S1

↔ Spectrum of Hamiltonian of 1D Quantum

Field Theory defined by the action: σ-model

Strings ~ MagnetsL→∞ ~ Sigma Models

1D Systems for Gauge Theory1D Systems for Gauge Theory

Supermagnets

Supercoset σ-Models

Magnets must have same symmetries as GT

Anisotropic Heisenberg chain has SO(2) sym

OR

σ-Model on PSU(2,2|4)/SO(1,4) x SO(5) …

V rep of PSU(2,2|4)L → ∞

Sym of N=4 SYM

Conclusion: Toolkit of String Geometry

.. is toolkit of 1d quantum

systems

~ systems of

2d stat mech

Sigma Models

Bethe Ansatz

Integrability

Yangians

Spin Chainsaffine algebras

non-linear integral equationsYang-Baxter equation

numerical studies

Successful recent applications to N=4 SYM theory:

anomalous dimensions and gluon amplitudes

all loops 0,1,∞ loops

Scattering in N=4 SYM theoryScattering in N=4 SYM theory

n-gluon Scattering Amplitude

(MHV, color ordered,planar)

p1

p2 p3

p4

n-gluon SA depends

on 3n-10 variables:

s = (p1+p2)2

t = (p2+p3)2

s tcutoff coupling

known Finite Remainder

BDS conjecture: [Bern et al.]

Holds for n = 4 ! known from ST!

Gluon Scattering in AdS gravityGluon Scattering in AdS gravity

Gluon SA at strong coupling:

Given by area of a 2D surface

ending on the polygon P{pj}

& pulled by gravity into AdS

[Alday, Maldacena]

Confirms n = 4 gluon BDS amplitude &new prediction for SA with n > 5 gluons

Kinematic data

..but surface very hard to find

Reformulation through TBAReformulation through TBAThermodynamik Bubble Ansatz [Alday,Gaiotto,Maldacena]

~ calculation of vacuum energy in 1D quantum systems

m, .. - parameters Y - density of part./holes A - energy

Kinematic data kernel fct K known

Area from nonlinear integral equations (NLIE):

Amplitudes 2012

Organizers:

R. Boels, G. Heinrich,

J. Henn, P. Mastrolia

J. Plefka, V. Schomerus

Hamburg, Mar 5-9 2012

If you want to see more come to …

SummarySummary

Strings in 5D

for mesons

Strings from

magnets

Gluon scattering from TBA