On Holographic (stringy ) Baryons

Imperial College August 09 work done with V. Kaplunovsky G. Harpaz ,N. Katz and S.Seki

IntroductionHolography 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.

OutlineThe 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 space-time.

This is similar to old models of mesons that include a string with massive endpoints

In the small mass limit R -> 1

In the large mass limit R -> 0

Quark massesWe refer to the mass parameter as “string endpoint mass”

Mmes~ Tst L + m1sep + m2

sep

msep is neither mQCD nor constituent mass

GOR relation tells us that

m2~ mQCD<qq>/f2

In the SS model m=0 mQCD =0

In the generalized SS with u0 > u

m=0 mQCD =0

msep mQCD

_

0 qq

0_

qq

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 quantizationSo far we have described the classical string

To quantize it we introduce quantum fluctuations Tseytlin

Canonical quantization via the Virasoro constraints

The frequencies n=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 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, D)

Fit of the first trajectory

Low masstrajectory High mass trajectory

Low mass trajectory High mass trajectory

Fit of the first b b trajectory

Obviously the approximation of low mass trajectory yields a better fit for the meson trajectories and the high mass has a lower 2 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

’ ~ 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

Since for small curvature we need >> 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= ’mes M2 + J= ’bar(M-mbv)2 +

and similarly for the improved trajectories with massive endpoints

Comparison with data shows that the best fit is for mbv =0 and ’bar ~ ’mes

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

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 singlestring 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 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.

U

R

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 >> 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

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.

qL qR

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.

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/u 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 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 -1/2 so stringy effects cannot be neglected in the large 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/ 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/, 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 u with general u0uKK

we found that the size scales in the same way with 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

We can match the meson and baryon spectra and properties with one scale

M= 1GEV and u0u= 0.94

Obviously this is unphysical since by definition

>1.

This may signal that the Sakai Sugimoto picture of baryons has to be modified ( Baryon backreaction, DBI expansion, coupling to scalars)

One flavor baryonsBoth 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 ~Tr[F2]

In the antipodal case ( the SS model) there is a symmetry under x4

-> -x4 and since asymptotically x4 is the transverse direction

x4

such an interaction term does not exist.

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 D8L

Nc

f