On Holographic stringy Baryons

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On Holographic (stringy ) Baryons

• 
  Crete June 09
• 
  work done with V. Kaplunovsky
• G.
  Harpaz ,N. Katz and S.Seki

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Introduction

• Holography is a useful tool in discussing the
  physics of glueballs and meson.
• Baryons can be described as a semi-classical
  stringy configurations.
• In large N baryons require a special treatment.
  This leads to the description in terms of
  skyrmions.
• The holographic duals of baryons are instantons
  of a five dimensional flavor gauge theory.
• Relating SUGRA predictions to a stringy picture.
• Modern stringy baryons versus the old picture
• We will put emphasis on comparison to data.

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Outline

• The Regge trajectories revisited
• Stringy holographic baryons
• Does the baryonic vertex have a trace in data
• The stability of stringy baryons, simulation
• Confining background- the Sakai Sugimoto model
• Baryons as flavor gauge instantons
• Baryonic properties in a genrealized model
• Attraction between nucleons
• Summary - Are we back in square one?

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Regge trajectories revisited

• Since excited baryons as we will see have a
  shape of a single string, lets discuss first
  stringy mesons.
• On the probe branes there are only scalars and
  vectors so there are no candidates for higher
  spin mesons.
• Apart from special tayllored models SUGRA does
  not admit the linearity of M2 n
• Mesons and baryons admit Regge behavior
• M2 J
• and hence are described by semi-classical strings.

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Regge trajectories of baryons
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The Regge mesons are described by
semi-classical strings that end on the probe
branes in the confining background
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• We solve the the equations of motion associated
  with the NG action in the confining background.
• An approximate solution takes the form of ___
• The same relations between the Mass and the
  angular momentum follow from a system of an
  open string with massive endpoints in flat
  space-time.
• This is similar to old models of mesons that
  include a string with massive endpoints

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• In the small mass limit wR -gt 1
• In the large mass limit wR -gt 0

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Quantization of the string

• There is no exact expression of the quatization
  of the string with the massive endpoints.
• For the the low mass case one can use the
• intercept of the massless case so that

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Fit of the first r trajectory
Low mass trajectory
High mass trajectory
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Fit of the first b b trajectory
Low mass trajectory
High mass trajectory
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• Obviously the approximation of low mass
  trajectory yields a better fit for the r meson
  trajectories and the high mass has a lower c2 for
  the b bar b mesons.
• The best fit for all the light trajectories was
  found for the following parameters ( preleminry)
• msep 0.1 GEV
• Tst 0.17 GEV2
• a 0.94 GEV-2
• For the b quark
• msep 5 GEV

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Fit to the Regge trajectories of baryons
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• What are the deviations of the full holographic
  model from the toy model?
• For mesons of not so large J and hence also L
  there are deviations from the __ configuration
  .
• The ends of the string are charged under U(Nf)
  gauge interaction. For a single probe brane U(1)
  the constraint equation is modified and as a
  result we find that the constant term (
  intercept) gets a shift. The charge is
  proportional to the string coupling which is a
  function of u0 and hence of the string endpoint
  mass!

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• Combining the low spin spectra ( scalar and
  vector) from brane fluctuations with the high
  spin spectra from stringy configurations imposes
• a puzzle since the mass of the formers
• Mm 1/R
• while the tension of the string
• Tst (1/R)2 l
• Since for small curvature we need l gtgt 1, there
  is a large unacceptable gap between low and high
  spin mesons.
• This implies that will eventually have to work
  with curvature of order one.

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Quark masses

• We have encounter the end of the string mass
• Mmes Tst L m1sep m2sep
• msep is neither mQCD nor constituent mass
• GOR relation tells us that
• mp2 mQCDltqqgt/fp2
• In the SS model mp 0 ltqqgt 0 mQCD 0
• In the generalized SS with u0 gt uL
• mp 0 ltqqgt
  0 mQCD 0
• msep
  mQCD

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• To turn on a QCD mass or more generally an (
  (non-local) operator that breaks explicitly
  chiral symmetry
• One can either introduce a tachyonic DBI
  (Casero, Kiritsis and Paredes Bergman, Seki J.S,
  Dhar Nag
• or introduce an open Wilson line (Aharony
  Kutasov)
• Both admit the GOR relation

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Holographic Baryons

• How to identify a baryon in holography ?
• Since a quark corresponds to a string, the
  baryon has to be a structure with Nc strings
  connected to it.
• Witten proposed a baryonic vertex in AdS5xS5 in
  the form of a wrapped D5 brane over the S5.
• On the world volume of the wrapped D5 brane there
  is a CS term of the form

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• Strings end on the boundary external
  baryon
• 
  /flfflav
  
• Strings end on a flavor brane dynamical
  baryons

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• The flux of the five form
• It implies that there is a charge Nc for the
  abelian gauge field. Since in a compact space one
  cannot have non-balanced charges there must be N
  c strings attached to it.

 
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Possible experimental trace of the baryonic
vertex?

• We have seen that the Nucleon states furnish a
  Regge trajectory.
• For Nc3 a stringy baryon may be similar to the
  Y shape old stringy picture. The difference is
  the massive baryonic vertex.

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• The effect of the baryonic vertex in a Y shape
  baryon on the Regge trajectory is very simple.
  It affects the Mass but since if it is in the
  center of the baryon it does not affect the
  angular momentum
• We thus get instead of
• J ames M2 a0 ? J
  abar(M-mbv)2 a0
• and similarly for the improved trajectories with
  massive endpoints
• Comparison with data shows that the best fit is
  for mbv 0 and abar ames

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• Thus we are led to a picture where the baryon is
  a single string with a quark on one end and a
  di-quark ( a baryonic vertex) at the other end.
• This is in accordance with stability analysis
  which shows that a small instability in one arm
  will cause it to shrink so that the final state
  is a single string

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Stability analysis of classical stringy baryons

• t Hooft (Sharov) showed that the classical Y
  shape three string configuration is unstable
• We have examined Y shape strings with massive
  endpoints and with a massive baryonic vertex in
  the middle.
• The analysis included numerical simulations of
  the motions of mesons and Y shape baryons under
  the influence of symmetric and asymmetric
  disturbance.
• We also performed a parturbative analysis

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The conclusion from both the simulations and the
perturbative analysis is that indeed the Y shape
string configuration is unstable to asymmetric
deformations. Thus an excited baryon is an
unbalanced single string with a quark on one side
and a diquark and the baryonic vertex on the
other side.
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Baryons in confining SUGRA backgrounds

• Holographic baryons duals of QCD-like baryons
  have to include baryonic vertex embedded in a
  gravity background dual to the YM theory with
  flavor branes that admit chiral symmetry breaking
• A suitable candidate is the Sakai Sugimoto model
  which is based on the incorporation of D8 anti D8
  branes in Wittens model

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Wittens model-a prototype of confining model

• A way to get a confining background is to cut
  the radial direction and introduce a scale.
• One approach is indeed to cut by hand an Ads
  space. This is not a solution of the SUGRA
  equations of motion. People use it to examine
  phenomenological properties (AdsQCD)
• The approach of Witten was to compactify one
  coordinate of D3 (D4) brane background with a
  cigar-like solution.

R
UL
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• One imposes anti-periodic boundary conditions on
  fermions. This kills supersymmetry.
• In the dual gauge theory the gauginos and the
  scalars acquire a mass 1/R and hence in the
  small R limit they decouple and we are left only
  with the gauge fields.
• For a Dp brane, in the small R limit we
  loose one space dimension and we end up with a
  pure gauge theory in p-1 space dimensions.
• The gravity theory associated with D3 branes
  namely the AdS5xS5 case compactified on a
  circle is dual to a pure YM theory in 3d (
  with KK contamiation)
• The same mechanism for near extremal D4 branes
  yields a dual theory of pure YM in 4d.

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D4
D4
R
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• The gauge theory and sugra parameters are
  related via
• 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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• To add fundamental quarks one adds flavor branes.
• Lets go for a moment from the SUGRA background
  back to the brane configuration.
• If we add to the original stack of Nc D3 ( or D4
  ) branes another set of Nf Dp branes there will
  be strings connecting the D3 (D4) and Dp
  branes.
• These strings map in the dual field theory to
  bifundamental quarks that transform as the(Nc,
  Nf) representation of the gauge symmetry
  U(Nc)xU(Nf)
• For Nc gtgt Nf the U(Nf) can be treated as a
  global symmetry and hence we get fundamental
  quarks.

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• Coming back to the SUGRA background, in the case
  of Nc gtgt Nf we can safely neglect the
  backreaction of the additional branes on the
  background. Thus we have introduced in fact
  flavor probe branes into a background gravity
  model dual of a YM (SYM) theory. This is the
  gravity analog of using a quenched approximation
  in lattice gauge theories.

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• We would like to introduce probe flavor branes to
  Wittens model.
• What type of Dp branes should we add D4, D6 or D8
  branes?
• How do we incorporate a full chiral flavor global
• symmetry of the form U(Nf)xU(Nf), with left and
  right handed chiral quarks?

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Adding flavor probe branes
The mass is the string endpoint masss discussed
before
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• U(Nf)xU(Nf) global flavor symmetry in the UV
  calls for two separate stacks of branes.
• To have a breakdown of this chiral symmetry to
  the diagonal U(Nf)D we need the two stacks of
  branes to merge one into the other.
• This requires a U shape profile of the probe
  branes.
• The opposite orientations of the probe branes at
  their two ends implies that in fact these are
  stacks of Nf D8 branes and a stack of Nf anti
  D8 branes. ( Thus there is no net D8 brane
  charge)
• This is the Sakai Sugimoto model.

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We see that the model admits chiral symmetry
U(Nf)xU(Nf) in the UV which is broken to a
diagonal one U(Nf)D in the IR.
qL
qR
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• We place the two endpoints of the probe branes on
  the compactified circle. If there are additional
  transverse directions to the probe branes then
  one can move them along those directions and by
  that the strings will aquire length and the
  corresponding fields mass. Thus this will
  contradict the chiral symmetry which prevents a
  mass term.
• Thus we are forced to use D8 branes that do not
  have additional transverse directions.
• The fact that the strings are indeed chiral
  follows also from analyzing the representation
  of the strings under the Lorentz group

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Stringy baryons in the SS model

• The baryonic vertex will now be wrapped D4
  branes over the S4 .
• The Lorentz structure of the strings is
  determined by the DD, NN, DN b.c
• In the approximation of flat space one finds a
  degeneracy between the R and NS ground state
  energies thus the bosonic and fermionic are
  degenerate.

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• The location of the baryonic vertex in the radial
  direction is determined by static
  equillibrium.
• The energy is a decreasing function of xuB/uL
  and hence it will be located at the tip of the
  flavor brane

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• It is interesting to check what happens in the
  deconfining phase.
• For this case the result for the energy is
• For xgtxcr low temperature stable baryon
• For xltxcr high temperature disolved
  baryon
• The baryonic vertex falls into the black hole

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Baryons as instantons

• In the SS model the baryon takes the form of an
  instanton in the 5d U(Nf) gauge theory.
• The instanton is the BPST instanton in the
  ( xi,z) 4d curved space. In the leading order
  in l it is exact.
• For Nf 2 the SU(2) yields a run away potential
  and the U(1) has an opposite nature so that one
  finds a stable size but unfortunately on the
  order of l-1/2 so stringy effects cannot be
  neglected in the large l limit.

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Baryons in the Sakai Sugimoto model

• The probe brane world volume 9d 5d upon
  
• Integration over the S4. The 5d DBI CS is
  expanded
• where

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• One decomposes the gauge fields to SU(2) and U(1)
• In a 1/l expansion the leading term is the YM
• Ignoring the curvature the solution of the SU(2)
  gauge field with baryon instanton 1 is the
  BPST instanton

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• Upon introducing the CS term ( next to leading in
  1/l, the instanton is a source of the U(1) gauge
  field that can be solved exactly.
• Rescaling the coordinates and the gauge fields,
  one determines the size of the baryon by
  minimizing its energy

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• Performing collective coordinaes semi-classical
  analysis the spectra of the nucleons and deltas
  was extracted.
• In addition the mean square radii, magnetic
  moments and axial couplings were computed.
• The latter have a similar ( maybe better)
  agreement with data then the skyrme calculations.
• The results depend on one parameter the scale.
• Comparing to real data for Nc3, it turns out
  that the scale is different by a factor of 2 from
  the scale needed for the meson spectra.

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• With the generalized scenario with non trivial
  msep namely for u0 different from uL we found
• that the size scales in the same way with l
• We can match the meson and baryon spectra and
  properties with one scale
• ML 1GEV and y0 0.94
• Obviously this is unphysical since by definition
  
• y0gt1.
• This may signal that the Sakai Sugimoto picture
  of baryons has to be modified ( Baryon
  backgreaction, DBI expansion,)

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One flavor baryons

• Both from the point of view of QCD and of the
  stringy configuration there is no reason why
  there should not be also baryons for Nf 1.
• However, there is no non-trivial instanton in the
  abelian gauge theory of Nf 1.
• This is presumably the analog of no Skyrme model
  for one flavor.

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Holographic Nuclear force

• Hashimoto Sakai and Sugimoto showed that there is
  a hard core repulsive potential between two
  baryons ( instantons) due to the abelian
  interaction of the form
• VU(1) 1/r2
• In nuclear physics one believes that there is
  repulsion between nucleons due to exchange of a
  isovector particle ( omega) and an attraction due
  to exchange of an isoscalar ( sigma)

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• We expect to find a holographic attraction due to
  the interaction of the instanton with the
  fluctuation of the embedding which is the dual of
  the scalar fields.
• The attraction term should have the form
• Lattr fTrF2
• In the antipodal case ( the SS model) there is a
  symmetry under dx4 -gt -dx4 and since
  asymptotically x4 is the transverse direction
• fdx4
• such an interaction term does not exist.

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• Indeed the 5d effective action for AM and f is
• Since the instanton is small we can set uu0
• Thus there is also an atraction potential
• Vscalar 1/r2
• The ratio of the attraction to repulsive
  potential
• Since the instanton is small u u0

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Summary and conclusions

• We have discussed properties of baryons that
  follow from the holographic SUGRA picture as well
  as
• their stringy description.
• Unfortunately to bridge the SUGRA and stringy
  pictures requires t Hooft parameter ( and hence
  curvature ) of order 1. ( This may hint for
  non-critical strings)
• The modern stringy picture is not so different
  than the old one.

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• The stringy picture for a baryon with high spin
  seems to be that of a single string with a quark
  and a di-quark
• Baryons as instantons lead to a picture that is
  similar to the Skyrme model.
• From the results for baryons made out of quarks
  with string end point masses we deduce that the
  naïve instanton picture should be improved.
• We showed that on top of the repulsive hard core
  due to the abelian field there is an attraction
  potential due the scalar interaction.

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1. Holographic mesons

• Steps needed to create holographic mesons
• Allocate a gravity dual of confining gauge
  dynamics in particular pure YM theory.
• Add flavor probe branes to incorporate
  fundamental quarks.
• Identify the modes on the flavor branes that
  correspond to the various types of mesons
• Compute the spectrum and examine its dependence
  on the excitation number the string endpoint
  mass, Parity and Charge conjugation .

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Regge trajectories of mesons

• Rotating bosinic string admits Regge behavior

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D8
D8
L
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Nf
Nc