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QFTHEP'2010, Golitsyno, September 8-15, 2010

A five-dimensional effective model for excited light mesons
S.S. Afonin St.-Petersburg State University

Based on S.S. Afonin, Int. J. Mod. Phys. A25 (2010) 3933 (arXiv:0908.0457 [hep-ph])


A brief reminder
AdS/CFT correspondence ­ the conjectured equivalence between a string theory defined on one space and a CFT without gravity defined on conformal boundary of this space. Maldacena example (1997): Type IIB string theory on AdS5 in low-energy (i.e. supergravity) approximation

вS

5



=4

YM theory on AdS boundary

in the limit

gYM N C >> 1
String theory
Up down Bottom up

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

We will discuss

QCD

SO(4, 2) : Equivalence of energy scales



The 5-th coordinate ­ (inverse) energy scale


A typical model (Erlich et al., PRL 95, 261602 (2005))

For

Hard wall model:

The fifth coordinate corresponds to the energy scale: Because of the conformal isometry of the AdS space, the running of the QCD gauge coupling is neglected until an infrared scale . 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 where M is identified with the quark mass matrix and with the condensate.


Denoting

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

where

Due to chiral symmetry breaking They have normalizable solutions only for discrete values of 4d momentum


Some conceptual problems:
1. Matching to QCD is implemented in the UV region (small z) where QCD is weakly coupled. 2. The spectrum is drastically shaped by the UV region contrary to what we expect from QCD. In addition, it does not have Regge form. A popular phenomenological solution ­ soft wall model,

S = d x g e
5

(z)

L

However, such a 5D background cannot follow from a solution of 5D Einstein equations.

Alternative mechanism? ­ Higgs one


Vacuum sector

By assumption, is dual to the gluonic field strength tensor square whose vev causes the anomaly in the trace of energy-momentum tensor,

Making use of the scaling

the action can be rewritten as

By assumption the selfinteraction is weak


The classical equation of motion is

We assume that the vacuum solution does not depend on the usual space-time coordinates,

The equation above has then a kink solution

Translational invariance along the z-direction is broken! Thus, different energy scales are now not equivalent. The effect is essential at large z (small energies) but disappears at small z (high energies)


Consider the particle-like excitations by varying Assuming with and retaining only linear part,

There are two normalizable discrete states,

Continuum begins at

"glueball"?


Coupling to bosons
For simplicity, we consider the scalar case only

Making the rescaling above and

the corresponding Lagrangian reads

Consider the particle-like excitations


The discrete spectrum is

where F is hypergeometric function and

The continuum sets in at


An example of numerical fit
Fitting the masses of -mesons by the curve

one obtains

which gives

while in the model


Coupling to fermions

Here

After our rescaling and

Let us find particle-like excitations components

for the left and right

The equation is known to possess a normalizable zero-mode solution

This mode is located near

There is also an asymptotic solution


Conclusions
We have proposed a new way for description of meson spectrum within holographic models for QCD based on the Higgs mechanism Next steps ­ extension of the presented scheme to the AdS background and simultaneous description of the chiral symmetry breaking with all related physics.