Transcript Slide 1

On Holographic (stringy )
Baryons
Imperial College August 09
work done with V. Kaplunovsky
G. Harpaz ,N. Katz and S.Seki
Introduction
Holography is a useful tool in discussing the physics of
glueballs and mesons.
Baryons can be described as a semi-classical stringy
configurations.
In large N baryons require a special treatment. This leads
for instance to a 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.
Outline
The Regge trajectories of mesons 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?
Regge trajectories revisited
Since excited baryons, as we will see later,
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.
Regge trajectories of baryons
The holographic Regge mesons are
described by semi-classical strings that end
on the flavor probe branes in the ``confining
background”
We solve the the equations of motion
associated with the Nambu-Goto action in a
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 spacetime.
This is similar to old models of mesons that
include a string with massive endpoints
In the small mass limit wR -> 1
In the large mass limit wR -> 0
Quark masses
We refer to the mass parameter as “string
endpoint mass”
Mmes~ Tst L + m1sep + m2sep
msep is neither mQCD nor constituent mass
GOR relation tells us that
mp2~ mQCD<qq>/fp2 _
In the SS model mp =0  q q  0 mQCD =0
In the generalized SS with_ u0 > uL
mp =0  q q  0 mQCD =0
msep
mQCD
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
Semi-classical quantization
So far we have described the classical string
To quantize it we introduce quantum fluctuations
Tseytlin
Canonical quantization via the Virasoro constraints
The frequencies wn=Kn are given by
sym
anti-sym
A is given in terms of the endpoint velocities
There is no exact expression of the quatization of the
string with the massive endpoints.
For the low mass case one can use the
intercept of the massless case so that
For high mass we find
Fitting to experimental data
Holography is valid in ceretain limits like large N
and large l
The confining backgrounds like SS is dual to a
QCD-like theory.
Nevertheless with some “Huzpa” and since we
related the holographic model to a simple toy
model, we compare the holographic model with
experimental data of mesons and deduce the
parameters
Tst , msep, a0(D)
Fit of the first r trajectory
Low mass trajectory
High mass trajectory
Fit of the first b b trajectory
Low mass trajectory
High mass trajectory
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
Fit to the Regge trajectories of baryons
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!
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 >> 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.
Holographic stringy 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
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.
Strings end on the boundary
external baryon
/
fflavor brane
Strings end on a flavor brane
dynamical baryons
Possible experimental trace of the baryonic vertex?
We have seen that the Nucleon
states furnish a Regge trajectory.
For Nc=3 a stringy baryon may be
similar to the Y shape “old” stringy
picture. The difference is the massive
baryonic vertex.
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= a’mes M2 + a0 
J= a’bar(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 a’bar ~ a’mes
Thus we are led to a picture where the baryon is
a single string with a quark on one end and a diquark (+ 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
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
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.
Baryons in confining
SUGRA backgrounds
Holographic baryons have to include a 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 Witten’s model
Witten’s modela 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
R
(D4) brane background with a “cigar-like” solution.
UL
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.
D4
D4
R
•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 >> 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 l5  R we approach QCD
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 >> Nf the U(Nf) can be treated as a global
symmetry and hence we get fundamental quarks.
Coming back to the SUGRA background, in the case of
Nc >> 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.
We would like to introduce probe flavor branes
to Witten’s 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?
Adding flavor probe branes
The mass is the string endpoint
masss discussed before
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.
We “see” that the model admits
chiral symmetry U(Nf)xU(Nf) in the
UV which is broken to a diagonal
qL
one U(Nf)D in the IR.
qR
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
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.
The location of the baryonic vertex in the radial direction is
determined by ``static equillibrium”.
The energy is a decreasing function of x=uB/uL and hence it will
be located at the tip of the flavor brane
It is interesting to check what happens in the
deconfining phase.
For this case the result for the energy is
For x>xcr low temperature stable baryon
For x<xcr
high temperature disolved baryon
The baryonic vertex falls into the black hole
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.
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
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
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
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 Nc=3, it turns out that
the scale is different by a factor of 2 from the
scale needed for the meson spectra.
With the generalized non-antipodal with non
trivial msep namely for u0 different from uL with
general z =u0 / uKK
we found that the size scales in the same way
with l. We computed also the baryonic
properties
Mean square radii of baryons
The flavor guage fields are parameterized as
On the boundary the gauge action is
The L and R currents are given by
The relevant field strength is
The baryonic density is given by
where the eigenfunctions obey
The Yukawa potential is
Finally the mean square of the baryonic radius
as a function of MKK and z
We can match the meson and baryon spectra
and properties with one scale
ML = 1GEV and z =u0 / u L= 0.94
Obviously this is unphysical since by definition
z>1.
This may signal that the Sakai Sugimoto picture
of baryons has to be modified ( Baryon
backreaction, DBI expansion, coupling to
scalars)
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.
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
isoscalar mesons: a vector particle ( omega) and
an attraction due to exchange of an scalar (
sigma)
The various regions of the nuclear interaction.
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 ~fTr[F2]
In the antipodal case ( the SS model) there is a
symmetry under dx4 -> -dx4 and since
asymptotically x4 is the transverse direction
f~dx4
such an interaction term does not exist.
Indeed the 5d effective action for AM and f is
For instantons F=*F so there is a competition
between
repulsion
A TrF2
attraction
fTr F2
Thus there is also an attraction potential
Vscalar ~ 1/r2
The ratio between the attraction and repulsion in
the intermediate zone is
Nuclear potential in the far zone
We have seen the repulsive hard core and
attraction in the intermediate zone.
To have stable nuclei the attractive potential has
to dominate in the far zone.
In holography this should follow from the fact
that the isoscalar scalar is lighter that the
corresponding vector meson.
In SS model this is not the case.
Maybe the dominance of the attraction
associates with two meson exchange( sigma?).
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.
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.
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 .
Regge trajectories of
mesons
•Rotating bosinic string admits Regge behavior
D8
L
D8
Nf
Nc