Progress report:

Holographic chiral symmetry

O.Bergman, S. SekiB. BurringtonV. Mazoalso O. Aharony and S.Yankielowicz and K.

Peeters and M. Zamaklar

Introduction

• QCD admits at low energies: Confinement and chiral symmetry breaking.

• Apriori there is no relation between the two phenomena.

• Except of lattice simulations the arsenal of non perturbative field theory tools is quite limited.

• Gauge/gravity duality is a powerful method to deal with strongly coupled gauge theories.

• There are several stringy ( gravitational) models with a dual field theory in the same universality class as QCD

• Confinement is easily realized, flavor chiral symmetry is not.

• Fundamental quarks can be incorporated via probe branes. First were introduced in duals of coulomb phase. (Karch Katz).

• Quarks in confining scenarios were introduced into the KS confining background using D7 braens ( Sakai Sonnenscehin)

• Models based on Witten’s near extremal D4 branes with D6 ( Kruczenski et al). Also Erdmenger et al

• A model that admits full flavor chiral symmetry breaking by incorporating D8 and anti D8 branes to Witten’s model. Sakai and Sugimoto

• In this work we examine the issues of • (i) chiral symmetry breaking, current algebra quark mass and the GOR relation via a “tachyonic DBI” action

• (ii) Back-reaction on the background and the stability of the SS model.

• (iii) Thermal phase structure using non critical string background. In particular we discuss the transitions • confinement/deconfinement• chiral symmetry breaking/ restoring • We determine the thermal spectra of meson and their dissociation• (no) drag on mesons and breakup due to velocity

Witten’s model of near extremal D4 branes

•The periodicity of x4

•The high temperature background

•The periodicity of t

•The parameters of the gauge theory are given in the sugra

5d coupling 4d coupling glueball mass

String tension

•The gravity picture is valid only provided that >> R

•At energies E<< 1/R the theory is effectively 4d.

•However it is not really QCD since Mgb ~ MKK

•In the opposite limit of R we approach QCD

The Sakai Sugimoto model

• The basic underlying brane configuration is -|------|-

• In the limit of Nf < <Nc the Sugra background is that of the near horizon limit of the near extremal D4 branes with Nf probe D8 branes and Nf probe anti D8 branes.

• The strings between the D4 branes and the D8 and anti D8 branes

D4- D8 strings L – left chiral fermions in ( Nf, 1 ,Nc) of U(Nf )x U(Nf )x U(Nc)

D4- anti-D8 strings R – right chiral fermions in (1, Nf ,Nc) of U(Nf )x U(Nf)xU(Nc)

• Note that it is a chiral symmetry and not an U(Nf)xU(Nf) of Dirac fermions. This is due to the fact that the there is no transverse direction to the D8 branes.

• The same applies to D4 branes in 6d non critical model ( Casero Paredes J.S)

D8 anti D8

D4

Outline

• Quark mass, condensate from tachyonic DBI

• Back reaction of the flavor branes

• Phases of thermal QCD from Non critical strings

1. Quark mass, chrial symmetry breaking and tachyonic DBI

• In the Sakai Sugimoto model the quarks are massless and there is no apparent way to add a current algebra mass.

• Even in the generalized model the pion mass is zero and henceso is the C.A quark mass

• u0 -u corresponds to constituentquark mass

• It is not clear what is the sourceof the chiral symmetry breakingand in particular it is not associated with an expectation value of a bifundamental

• One would like to have a holographic dual of the GOR relation• That states that

To understand the dynamics of the chiral symmetryBreaking we incorporate a complex bi-fundamental“Tachyon”. Discussed also by Casero Kiritsis Paredes• We start with an action proposed by Garousifor a separated parrallel Dp and anti- D p branes• This action is obtained by generalizing Sen’s actionfor non-BPS branes.•The action reads

Setting the gauge fields to zero in the non compactcase leaves us with the following tachyonic DBI action

where

For small u the bi-fundamental T will become tachyonic

The brane locations and

The filed T has a localized tachyonic mode at small u

The corresponding EOM for L and T are

For T=0 the EOM are those of the Sakai Sugimoto modelFrom the point of view of the potential of T this is a non stablesolution. We are after a solution with T(u) and L(u) such that when the tachyon condenses the brane anti brane separationvanishes

Profile of T(u)

T diverges

IR asymptotic solution

We expand the equations around u=u0 and find

As expected

The tachyon diverges at u=u0 where the brane anti brane merge

UV asymptotic solution

We put a cutoff and expand the solution around it.We than compute “physical quantities ”

and check that they are independent of this cutoff. Strictlywe cannot take it to infinity since at this region the stringcoupling becomes large.

Requiring that the perturbation are small implies

We identify the non-normalizable solution with the mass

The Hamiltonian density has the form

then the condensate is determined from

by differentiating with respect to mq

And since

the condensate is given by

To compute the spectrum of the vector mesonsand of the pions we analyze the spectrum offluctuations of the flavor gauge fields that live onthe probe branes

We go over to vector and axial gauge fields

and use the unitary gauge where the tachyonIs real

The A+ sectorWe dimensionally reduce the 9d action

to a 5 one, and plug the background

We parameterize the guage fields

After solving the eigenvalues problemand reducing to 4d we get

Thus we find a spectrum of massive vectormesons

The A- sector

We make the following decompositions

In a similar manner the 5d action now is

In our gauge the w0 are the pions

The 4d action of the A- now reads

Thus we see that the pions are massive

The mass of the pion is determined byeigenvalue equation

The pion decay constant and the GOR relation

We evaluate the pion decay constantfby computing the correlator of axialVector currents Erlich, Katz,Son,Stefanov

The effective action is

By taking we find

• Finally we get

which in leading order in mq/<qq> translates intothe Gell-Mann Oakes Rener relation

2. From probe to fully back reaction

• In the Sakai Sugimoto model flavor is introduced by incorporating N f D8 anti-D8 branes. This is done in the probe approximation based on having Nf<<Nc .

• The profile of the probe branes is determined by solving the DBI EOM. In the probe approximation the configuration is stable ( no tachyonic modes).

• The motivation to go beyond the probe approximation is tow follded: (a) To check that the back-reaction on the background does not destroy the

stability. (b) To determine the flavor dependence on properties that are extracted

from the background like string tension, beta function, the viscosity /entropy density etc.

The action of the back-reacted system is

Massive IIA action

DBI action

Action of form

x4(x4-)]

Induced metric

The EOM of the metric, dilaton, and RR forms are

The solution for the parameter M and F10

The jumps are at the locations of the branes

Our basic assumption is that the back-reaction is a small perturbation controled by the small parameter

The bulk term in the massive IIA can be neglected sinceit is of order

•The delta functions are codimension one which leads to an harmonic function ( absolute value x4) which is finite at the location of the delta function

We take the following ansatz for the background

We plug it to the EOM and assume an expansion

Original background perturbation

• To simplify the computation we replace the cigarOf the (u,x4) directions with a cylinder that asymptote to it

Which means that we omitted the factor

f(u)= u/u)^3 from the background

Note that for large u this factor is irrelevant

We separate the equations using the following variables

F1 and F2 are invariant under x4 and u coordinate transformations

We Fourier decompose in x4

Finally defining the equations read

The general solution after enforcing the convergence of the Fourier sum takes the form

The behavior at large u

The general behavior is of spikes around the locations of the branes.

The spike structure ocours here only for much larger u

The perturbed dilaton at small u

te that the solution developes a duble notch behavior betweenthe cusp solution of very small u (red) and the spikes of large u

The perturbed A at small u

•We were able to solve the equations also without the Fourier decomposition. It can be shown that the generalstructure of the of the solution is of the following form

•The implications of the solution on the stabilityIs still not clear to us.

where

In particular for F2 we get

3 .Thermal phases of QCD from non critcal string model

In 2006 with O. Aharony and S. Yankielowicz we have analyzed the hologrphic thermal phase structure of QCD based on thermalizing the Sakai Sugimoto model.

With K. Peeters and M. Zamaklar we analyzed the spectrum of thermal mesons and the ( no ) drag of mesons.

With V. Mazo we have done a similar analysis based on

A model of non critical D4 color branes with Nf D4 and anti D4 flavor branes.

The non critical near extremal D4 brane

• The color and flavor barnes are

The flavor probe action is

Review of the bulk thermodynamics•We introduce temperature by compactifying the Euclidean time direction with periodicity =1/T and imposing anti-periodic boundary conditions on the fermions.

•We use amodel of near extremal D4 branes (either ciritical or non critical).

There is already a compact direction x4 so in our thermal model ( t, x4) are compact.

•There are only two such smooth SUGRA backgrounds

• At any given T the background that dominates is the one that has a lower free energy, namely, lower classical SUGRA action ( times T).

•The classical actions diverge. We regulate them by computing the difference between the two actions.

•It is obvious that the two actions are equal for =2 R , thus at Td =1/2 R there is a first order phase transition .

•The transition is first order since the two solutions continue to exist both below and above the transition.

•It is easy to see that for T<1/2 R the background with a thermal factor on X4 dominates, and above it the one with the thermal factor on t.

•The interpretation of the phase transition is clear. The order parameters are:

(i)low temperature the string tension Tst ~gtt gxx(um=u ) >0 confinement

high temperature the string tension Tst ~gtt gxx(um=u ) =0 deconfinement

(ii) Discrete spectrum versus continuum and dissociation of glueballs.

(iii) Free energy ~ Nc2 at high temperature Nc

0 at low temperature

(iv) Vanishing/non vanishing Polyakov loop ( string wrapping the time direction)

•The dominant phase for small /R, due to the symmetry under

T > 1/2 R, is a symmetric phase Aharony, Minwalla Weismann

R

The phase diagram of the pure glue theory

Symmetric

phase

• At the UV the D8 and anti D8 are separated U(Nf)L x U(Nf)R

• In the IR they merge together spontaneous breaking U(Nf)D

• To verify this we analyze the DBI probe brane action

he solution of the corresponding equation of motion is

Low temperature phase

The low temperature phase with flavor

The high temperature deconfining phase•Recall that the action has now the thermal factor on the t direction

•The equation of motion admits a solution similar to the oneof the low temperature domain, namely with chiral symmetry breaking

•However there is an additional stable solution of two disconnected stacks of branes.

This obviously corresponds to chiral restoration.

This is possible since at u=uT the t cycle shrinks to zero and the D8 branes can smoothly end there.

Chiral symmetry breaking/restoring

•The configuration with the lower free energy ~DBI action dominates

•The action diverges but can be regulated by computing the differencebetween the chiral symmetry breaking and restoring solutions

where y=u/u0

•We solve it numerically and find

•For yT > yTc ~ 0.735 S > 0 | |

•For yT < yTc ~ 0.735 S < 0 U

The action difference S as a function of yT (~LT)

S

yT

symmetry broken

symmetry

restored

•The symmetry breaking/restoring phase transition, just like the conf/decon Is a first order phase transition

Phase diagram- •We express the critical point in terms of the physical quantities T,L

Summary

• We introduce a current algebra mass to the quark, associate the chiral symmetry breaking to an expectation value of a bi-fundamental tachyon and derived the GOR relation.

• We analyzed the leading order backreaction on the Sakai Sugimoto model

• We realized that the thermal phase structure derived from a non critical holomorphic model is very similar to the result from the ciritical thermalized Sakai Sugimoto model.

Outline

Bulk thermodynamics of Witten’s model- phases of “YM theory” dual

Adding quarks in the fundamental representation

The low temperature phase of the SS model – confinement,

The high temperature phase –deconfinement.

The phase diagram, intermediate phase of deconfinement and chiral symmetry breaking

• The spectrum of the thermal mesons of the various phases

• The dissociation of mesons at high temperature• Boosting mesons, ( no) drag, critical velocity

•The parameters of the gauge theory are given in the sugra

•The gravity picture is valid only provided that >> R

•In fact near the D8 branes the condition is >> L

•At energies E<< 1/R the theory is effectively 4d.

•However it is not really QCD since Mgb ~ MKK

•In the opposite limit of R we approach QCD

•Thus there is a family of solutions parametrized by u0 >u

•A special case is the u0 = u , or L= R (Sakai Sugimoto)

•We can parameterize the solution instead in terms of L

•For small values of L the action depends on L as follows

Thermal Mesons

n general mesons are strings that start and end on a D8 brane

•For low spin these mesons correspond to the fluctuations ofthe fields that reside on the probe branes.

•Embedding scalars pseudo scalar mesons•U(Nf) gauge fiedls vector mesons

•Higher spin mesons are described by semi-classical stringyConfigurations Kruczenski, Pando Zayas, J.S, Vaman

High spin Stringy meson

Low spin Mesons in the confining phase

•The structure of the mesonic spectrum is like in zero temperature.

• We expand the 5d probe gauge fields

•The four dimensional action of the vector fluctuations reads

•The spectrum includes massless “Goldstone pions” associated with the symmetry breaking

•The mass eigenvalues are determined from

•There are no deconfined quarks

•The spectrum of massive mesons is discrete and indpendent of T

Scaling and the M~1/L relation.

•For “short mesons” one can determine the mass scale of the meson with no computation from a scaling argument

•In this limit and

•We can rewrite the eigenvalue equation in terms of a dimensionless quantity y in the form

Which implies that the right hand side is also dimensionless and thus

Meson mass as a function of the constituent quark mass

• If we identify the vertical parts of the spinning string as

massive quark end points, the energy of these segments corresponds to the quark constituent mass given by

•The numerical results show that the meson mass is linaer with the constituent mass

M2 as a function of the excitation number n

n

• Thus, the meson mass behaves like M~n ( and not M^2~n)

Low spin mesons in the intemediate phase

•To determine the thermal masses we consider spatially homogeneous modes

•The probe brane action reduces to

•The spectrum is determined by numerical ``shooting” for

Low phaseLow phase

Intermediate Intermediate

•The masses in the intermediate phase are smaller than in the low one

•They admit the non stringy behavior of m ~ n

Masses of the low lying mesons as a function of the temperature

•The masses decrease as a function of the temeperature

•The behavior is in qualitative agreement with lattice calculations.

High spin stringy meson•A meson is a spinning string that start and ends on a probe brane

•The relevant part of the metric of the deconfining background

•The classical spinning string

•The DBI action for such a configuration reads

•The corresponding equation of motion

•Thus the spinning string has to be above these curves

•For the action to be real or

•The numerical solutions of the profiles of the strings

•In fact this is nothing but that the speed of light is f(u)

•Dissociation temperature of high spin mesons

•There is a maximal value of the spin as a function of the temperature

•Zero temperature

Zero temperature

•High temperature

High temperature

•The mass dependence as a function of the temperature

for the high spin mesons is similar to that of the low spine ones

•Mesons in the chiraly restored phase

Recall in this phase there are two separated stacks of branes

and hence chiral symmetry is restored.

•The meson masses are determined by

and the normalizability condition

•The pion is no longer normalizable and hence disappear from the spectrum

•There are light deconfined stringy quarks in this phase

• If we expand the equation around uT we can solve analytially

the equation and can compare to the shooting results

•The solution :

no) Drag effect on mesons moving in the plasma

•It was shown recently that a single quark ( a string from the probebrane to the horizon) suffers from a drag when moving :The string is bended and there is momentum flow into the horizon and one has to apply force on the string at the flavor brane

•This does not happen to the spinning string and even not to a moving spinning string. The string bends but there is no drag

•A suitable ansatz of the string is of the form

•The action becomes

•The condition for a real action is

•The solutions of the EOM are above this threshold and hencethe mesons do not suffer any drag

•There is a critical velocity beyond which a state with a given spin has to dissociate.•Similarly the 4d size of the meson decreases with the velocity

Found also (analytically) by Liu, Rajagopal and U. Wiedermann

• We constructed a holographic model of the thermal phases of QCD

• It is based on thermalizing the Sakai Sugimoto mode

• Both the conf/deconf and breaking/restoring

are first order phase transition

• For small L/R there is an intermediate phase of

deconfinement and chiral symmetry breaking

• We computed the thermal spectrum of mesons

Summary

• In the low temperature phase the masses are temperature independent

•The low spin mesons admit M~ n behavior (unlike the stringy form M2~ n )

•In the intermediate phase the masses are temperature dependent similar

to lattice results.

•The same qualitative behavior occurs also for spinning string mesons

•There is a dissociation phenomenon of mesons, at any given temperature there is a maximal possible string.

•There is no drag on meson.

•There is a velocity dependence of the maximal spin and 4d size

Thermal

Mass square of first state as a function of u0

Adding quarks in the fundamental representation- The Sakai Sugimoto model model