Progress report :

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Progress report Holographic chiral symmetry
O.Bergman, S. Seki B. Burrington V. Mazo also O.
Aharony and S.Yankielowicz and K. Peeters and M.
Zamaklar
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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.

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• 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 Wittens 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 Wittens 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

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Wittens model of near extremal D4 branes

• The periodicity of x4

• The high temperature background

• The periodicity of t

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• 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
  l5 gtgt R
• At energies Eltlt 1/R the theory is effectively
  4d.
• However it is not really QCD since Mgb MKK

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The Sakai Sugimoto model

• The basic underlying brane configuration is
  --------
• In the limit of Nf lt ltNc 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 y L left chiral
  fermions in ( Nf, 1 ,Nc) of U(Nf )x U(Nf )x
  U(Nc)
• D4- anti-D8 strings yR 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)

D4
D8
anti D8
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Outline

• Quark mass, condensate from tachyonic DBI
• Back reaction of the flavor branes
• Phases of thermal QCD from Non critical strings

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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 hence
• so is the C.A quark mass
• u0 -u L corresponds to constituent
• quark mass
• It is not clear what is the source
• of the chiral symmetry breaking
• and 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

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• To understand the dynamics of the chiral symmetry
• Breaking we incorporate a complex bi-fundamental
• Tachyon. Discussed also by Casero Kiritsis
  Paredes
• We start with an action proposed by Garousi
• for a separated parrallel Dp and anti- D p
  branes
• This action is obtained by generalizing Sens
  action
• for non-BPS branes.
• The action reads

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Setting the gauge fields to zero in the non
compact case leaves us with the following
tachyonic DBI action
and
where
The brane locations
For small u the bi-fundamental T will become
tachyonic
The filed T has a localized tachyonic mode at
small u
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The corresponding EOM for L and T are
For T0 the EOM are those of the Sakai Sugimoto
model From the point of view of the potential of
T this is a non stable solution. We are after a
solution with T(u) and L(u) such that when the
tachyon condenses the brane anti brane
separation vanishes
Profile of T(u)
T diverges
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IR asymptotic solution
We expand the equations around uu0 and find
As expected The tachyon diverges at uu0 where
the brane anti brane merge
UV asymptotic solution
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We put a cutoff and expand the solution around
it. We than compute physical quantities and
check that they are independent of this cutoff.
Strictly we cannot take it to infinity since at
this region the string coupling becomes large.
Requiring that the perturbation are small implies
We identify the non-normalizable solution with
the mass
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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
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To compute the spectrum of the vector mesons and
of the pions we analyze the spectrum
of fluctuations of the flavor gauge fields that
live on the probe branes
We go over to vector and axial gauge fields
and use the unitary gauge where the tachyon Is
real
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The A sector
We dimensionally reduce the 9d action to a 5
one, and plug the background
We parameterize the guage fields
After solving the eigenvalues problem and
reducing to 4d we get
Thus we find a spectrum of massive vector mesons
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The A- sector
In a similar manner the 5d action now is
We make the following decompositions
In our gauge the w0 are the pions
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The 4d action of the A- now reads
Thus we see that the pions are massive
The mass of the pion is determined by eigenvalue
equation
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The pion decay constant and the GOR relation
We evaluate the pion decay constant fp by
computing the correlator of axial Vector currents
Erlich, Katz,Son,Stefanov
The effective action is
By taking
we find
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• Finally we get

which in leading order in mq/ltqqgt translates
into the Gell-Mann Oakes Rener relation
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2. From probe to fully back reaction

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

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The action of the back-reacted system is
Action of A9 form
Massive IIA action
DBI action
d (x4)d(x4-p)
Induced metric
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The EOM of the metric, dilaton, and RR forms are
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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 since it 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

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We take the following ansatz for the background
We plug it to the EOM and assume an expansion
perturbation
Original background
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• To simplify the computation we replace the
  cigar
• Of the (u,x4) directions with a cylinder that
  asymptote to it

Which means that we omitted the factor f(u)
1-(uL/u)3 from the background Note that for
large u this factor is irrelevant
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We separate the equations using the following
variables
F1 and F2 are invariant under x4 and u coordinate
transformations
We Fourier decompose in x4
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Finally defining the
equations read
The general solution after enforcing the
convergence of the Fourier sum takes the form
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The behavior at large u
The general behavior is of spikes around the
locations of the branes.
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The spike structure ocours here only for much
larger u
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The perturbed dilaton at small u
Note that the solution developes a duble notch
behavior between the cusp solution of very small
u (red) and the spikes of large u
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The perturbed A at small u
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• We were able to solve the equations also without
  the
• Fourier decomposition. It can be shown that the
  general
• structure of the of the solution is of the
  following form

where
In particular for F2 we get

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

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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.

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The non critical near extremal D4 brane

• The color and flavor barnes are

The flavor probe action is
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Review of the bulk thermodynamics

• We introduce temperature by compactifying the
  Euclidean time direction with periodicity b1/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

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• 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
  b 2p R , thus at Td 1/2p 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 Tlt1/2 pR the
  background with a thermal factor on X4
  dominates, and above it the one with the thermal
  factor on t.

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• The interpretation of the phase transition is
  clear. The order parameters are
• (i)low temperature the string tension Tst gtt
  gxx(umuL ) gt0 ? confinement
• high temperature the string tension Tst gtt
  gxx(umuT ) 0 ? deconfinement
• (ii) Discrete spectrum versus continuum and
  dissociation of glueballs.
• (iii) Free energy Nc2 at high temperature Nc0
  at low temperature
• (iv) Vanishing/non vanishing Polyakov loop (
  string wrapping the time direction)
• The dominant phase for small l /R, due to the
  symmetry under
• T ?gt 1/2p R, is a symmetric phase Aharony,
  Minwalla Weismann

Tlt? 1/2 pR
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The phase diagram of the pure glue theory
Symmetric phase
----------------
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Low temperature 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

• The solution of the corresponding equation of
  motion is

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The low temperature phase with flavor
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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 one
• of 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 uuT the t cycle
  shrinks to zero and the
• D8 branes can smoothly end there.

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Chiral symmetry breaking/restoring
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• The configuration with the lower free energy
  DBI action dominates
• The action diverges but can be regulated by
  computing the difference
• between the chiral symmetry breaking and
  restoring solutions

where yu/u0

• We solve it numerically and find

• For yT gt yTc 0.735 DS gt 0
  

• For yT lt yTc 0.735 DS lt 0
  U

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The action difference DS as a function of yT
(LT)
DS
c symmetry restored
yT
c symmetry broken

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

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Phase diagram-

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

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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.

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Outline

• Bulk thermodynamics of Wittens 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

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• The parameters of the gauge theory are given in
  the sugra
• The gravity picture is valid only provided that
  l5 gtgt R
• In fact near the D8 branes the condition is l5
  gtgt L
• At energies Eltlt 1/R the theory is effectively
  4d.
• However it is not really QCD since Mgb MKK
• In the opposite limit of l5 ltlt R we approach QCD

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• Thus there is a family of solutions parametrized
  by u0 gtuL.
• A special case is the u0 uL , or Lp 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

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Thermal Mesons

• In general mesons are strings that start and end
  on a D8 brane
• For low spin these mesons correspond to the
  fluctuations of
• the 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 stringy
• Configurations Kruczenski, Pando Zayas, J.S,
  Vaman

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High spin Stringy meson
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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 c symmetry breaking
• The mass eigenvalues are determined from
• There are no deconfined quarks
• The spectrum of massive mesons is discrete and
  indpendent of T

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Scaling and the M1/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
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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

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M2 as a function of the excitation number n
M2
n

• Thus, the meson mass behaves like Mn ( and not
  M2n)

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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

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

• They admit the non stringy behavior of m n

Low phase
Low phase
Intermediate
Intermediate
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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.

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High spin stringy meson

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

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• 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

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• For the action to be real

or

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

• Thus the spinning string has to be above these
  curves

• The numerical solutions of the profiles of the
  strings

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• Dissociation temperature of high spin mesons

Zero temperature
High temperature

• Zero temperature

• High temperature

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

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• The mass dependence as a function of the
  temperature
• for the high spin mesons is similar to that of
  the low spine ones

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• 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

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• If we expand the equation around uT we can
  solve analytially
• the equation and can compare to the shooting
  results

• The solution

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(no) Drag effect on mesons moving in the plasma

• It was shown recently that a single quark ( a
  string from the probe
• brane 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

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• The condition for a real action is

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

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• 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
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Summary

• 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 c 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

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• 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

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Thermal
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Mass square of first state as a function of u0
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Adding quarks in the fundamental representation-
The Sakai Sugimoto model model
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