II Russian-Spanish Congress “Particle and Nuclear Physics at all scales and Cosmology”, Saint Petersburg, Oct. 4, 2013

RECENT ADVANCES IN THE BOTTOM-UP RECENT ADVANCES IN THE BOTTOM-UP HOLOGRAPHIC APPROACH HOLOGRAPHIC APPROACH TO QCDTO QCD

Sergey Afonin

Saint Petersburg State University

A brief introduction

AdS/CFT correspondence – the conjectured equivalence between a string theorydefined on one space and a CFT without gravity defined on conformal boundary ofthis space.

Maldacena example (1997):Type IIB string theory onin low-energy (i.e. supergravity)approximation

55AdS S

YM theory on AdS boundary4

in the limit 1YMg N

AdS/QCD correspondence – a program to implement such a duality for QCD following the principles of AdS/CFT correspondence

Up

dow

n

Bottom

up

String theory

QCD

We will discuss

Basic property: Algebra of SO(4,2) group and that of isometries of AdS5 coincide

(4, 2) :SO Equivalence of energy scales The 5-th coordinate – (inverse) energy scale

[Witten; Gubser, Polyakov, Klebanov (1998)]

Essence of the holographic method

generating functional effective action

Operators in a 4D gauge theory Classical fields in 5D dual theory

In the sence that the corresponding sources Boundary values

One postulates:

The correlation functions are given by

Mass spectrum: Poles of the two-point correlator

Alternative way for finding the mass spectrum is to solve e.o.m.

The output of the holographic models: Correlators

An important example of dual fields for the QCD operators (R=1):

Main assumption of AdS/QCD: There is an approximate 5D holographic dual for QCD

Here

The holographic correspondence dictates the relation

A typical model (Erlich et al., PRL (2005); Da Rold and Pomarol, NPB (2005))

For

Hard wall model:

At one imposes certain gauge invariant boundary conditions on the fields.

Equation of motion for the scalar field

Solution independent of usual 4 space-time coordinates

bare quark massquark condensate

hereAs the holographicdictionary prescribes

Denoting

the equation of motion for the vector fields are (in the axial gauge)

where

The spectrum of normalizable modes is given by zeros of Bessel function, thus the asymptotic behavior is

nm n

that is not Regge like 2nm n

due to chiral symmetry breaking

Soft wall model (Karch et al., PRD (2005))

The IR boundary condition is that the action is finite at

To have the Regge like spectrum:

To have AdS space in UV asymptotics:

The mesons of arbitrary spin J can be considered, the spectrum has the form

But! No natural chiral symmetry breaking!

Self-consistent extension to the arbitrary intercept: Afonin, PLB (2013)

Some applications

Meson, baryon and glueball spectraLow-energy strong interactions (chiral dynamics)Hadronic formfactorsThermodynamic effects (QCD phase diagram)Condensed matter (high temperature superconductivity etc.)...

Deep relations with other approaches

Light-front QCDSoft wall models: QCD sum rules in the large-Nc limitHard wall models: Chiral perturbation theory supplemented by infinite number of vector and axial-vector mesons

Holographic description of thermal and finite density effects

Basic ansatz - corresponds to

One uses the Reissner-Nordstrom AdS black hole solution

where is the charge of the gauge field.

The hadron temperature is identified with the Hawking one:

The chemical potential is defined by the condition

The critical temperature and density (deconfinement) can be found from the condition ofcomplete dissociation of meson peaks in the correlators. The typical critical temperature at zerochemical potential for the light flavors lies about 200 MeV, for heavy ones does near 550 MeV.

Some examples of phase diagrams

He et al., JHEP (2013)

Colangelo et al., EPJC (2013)

Hadronic formfactorsDefinition for mesons:

Electromagnetic formfactor:

In the holographic models for QCD:

Brodsky, de Teramond, PRD (2008)

Linear spectrum and quark masses

The dependence of A and B on the quark masses? Afonin, Pusenkov, PLB (2013)

Basic construction: The no-wall holographic model (Afonin, PLB (2009))

The result:

From the ω-meson trajectory:

From the holography:

Charmonim: Bottomonium:

In the heavy-quark limit:

Interpretation: When a non-relativistic quark is created and moves with the velocity v in the c.m.frame, should compensate its kinetic energy

The binding energy grows linearlywith the quark mass!

In the limit

This coincides with a prediction of the Lovelace-Shapiro dual amplitude!

Thank you!