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A low-dimensional analogue of holographic baryons

TL;DR: In this article, a low-dimensional analogue of the Sakai-Sugimoto model is introduced, in which the bulk soliton can be approximated by a flat space sigma model instanton.
Abstract: Baryons in holographic QCD correspond to topological solitons in the bulk. The most prominent example is the Sakai-Sugimoto model, where the bulk soliton in the five-dimensional spacetime of AdS-type can be approximated by the flat space self-dual Yang-Mills instanton with a small size. Recently, the validity of this approximation has been verified by comparison with the numerical field theory solution. However, multi-solitons and solitons with finite density are currently beyond numerical field theory computations. Various approximations have been applied to investigate these important issues and have led to proposals for finite density configurations that include dyonic salt and baryonic popcorn. Here we introduce and investigate a low-dimensional analogue of the Sakai-Sugimoto model, in which the bulk soliton can be approximated by a flat space sigma model instanton. The bulk theory is a baby Skyrme model in a three-dimensional spacetime with negative curvature. The advantage of the lower-dimensional theory is that numerical simulations of multi-solitons and finite density solutions can be performed and compared with flat space instanton approximations. In particular, analogues of dyonic salt and baryonic popcorn configurations are found and analysed.

Summary (1 min read)

1 Introduction

  • As the soliton is small compared to the curvature scale then it can be approximated by a flat space self-dual Yang-Mills instanton with a small size [3, 4].
  • Multi-solitons in the Sakai-Sugimoto model (including solitons at finite density) are unlikely to have any continuous symmetries, so numerical field theory computations would require a fully four-dimensional computation that is beyond current capabilities.
  • It is argued that, with increasing density, this dyonic salt arrangement turns into a cubic crystal of half-instantons that is dual to the well-known Skyrme crystal.
  • The results provide evidence to support the validity of these ideas within the Sakai-Sugimoto model.

3 Solitons at finite density

  • As mentioned in the introduction, the study of solitons at finite density in the SakaiSugimoto model has attracted some recent attention in attempts to understand dense QCD within a holographic setting.
  • These results show that the double chain solution has a lower energy than the single chain solution once the density is greater than about twice the optimal density.
  • Both approaches therefore confirm the analogue of baryonic popcorn in their low-dimensional model.
  • A minimization over the parameters in such an instanton approximation yields the blue curve in Figure 5, which agrees with the field theory results.
  • All these results suggest that as the density is increased further then the number of soliton chains increases and eventually the configuration begins to resemble a portion of a two-dimensional lattice rather than the one-dimensional chain that arises at the optimal density.

4 Conclusion

  • The advantage of the low-dimensional theory is that several aspects that one would like to study in the Sakai-Sugimoto model, but are currently not tractable, can be investigated exactly.
  • This provides further support for the use of self-dual Yang-Mills instantons in approximating bulk solitons in the Sakai-Sugimoto model.
  • Analogues of dyonic salt and baryonic popcorn configurations have been found, providing strong evidence for their relevance in the study of holographic baryons.
  • Note that the results in that paper show that this can produce similar results to the inclusion of a baby Skyrme term.

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Deposited in DRO:
20 March 2014
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Citation for published item:
Bolognesi, S and Sutclie, P.M. (2014) 'A low-dimensional analogue of holographic baryons.', Journal of
physics A : mathematical and theoretical., 47 (13). p. 135401.
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http://dx.doi.org/10.1088/1751-8113/47/13/135401
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DCPT-13/39
A low-dimensional analogue of
holographic baryons
Stefano Bolognesi and Paul Sutcliffe
Department of Mathematical Sciences, Durham University, Durham DH1 3LE, U.K.
Email: s.bolognesi@durham.ac.uk & p.m.sutcliffe@durham.ac.uk
February 2014
Abstract
Baryons in holographic QCD correspond to topological solitons in the bulk. The
most prominent example is the Sakai-Sugimoto model, where the bulk soliton in the
five-dimensional spacetime of AdS-type can be approximated by the flat space self-dual
Yang-Mills instanton with a small size. Recently, the validity of this approximation
has been verified by comparison with the numerical field theory solution. However,
multi-solitons and solitons with finite density are currently beyond numerical field
theory computations. Various approximations have been applied to investigate these
important issues and have led to proposals for finite density configurations that include
dyonic salt and baryonic popcorn. Here we introduce and investigate a low-dimensional
analogue of the Sakai-Sugimoto model, in which the bulk soliton can be approximated
by a flat space sigma model instanton. The bulk theory is a baby Skyrme model in
a three-dimensional spacetime with negative curvature. The advantage of the lower-
dimensional theory is that numerical simulations of multi-solitons and finite density
solutions can be performed and compared with flat space instanton approximations.
In particular, analogues of dyonic salt and baryonic popcorn configurations are found
and analysed.
1

1 Introduction
A common feature of all models of holographic QCD is that baryons correspond to
topological solitons in the bulk. The foremost example of a top-down theory with a string
theory embedding is the Sakai-Sugimoto model [1, 2], which results in a Yang-Mills theory
with a Chern-Simons term in a five-dimensional bulk spacetime of AdS-type. As there
is an identification between baryon number and instanton charge, the study of baryons is
equivalent to the construction of Yang-Mills-Chern-Simons solitons in curved space, with a
prescribed instanton number.
At large ’t Hooft coupling the Yang-Mills term dominates over the Chern-Simons term
and the soliton has a small size, determined by balancing curvature and Chern-Simons
contributions to the action. As the soliton is small compared to the curvature scale then it
can be approximated by a flat space self-dual Yang-Mills instanton with a small size [3, 4].
The validity of this approximation has recently been confirmed by numerical field theory
computations [5], by exploiting the SO(3) symmetry of the static single soliton to reduce
the computation in four-dimensional space to a reduced theory in a two-dimensional space.
A subtlety was revealed regarding the large distance nonlinear properties of the soliton tail,
but this aspect is not relevant for the current paper that concerns multi-solitons in close
proximity. Multi-solitons in the Sakai-Sugimoto model (including solitons at finite density)
are unlikely to have any continuous symmetries, so numerical field theory computations
would require a fully four-dimensional computation that is beyond current capabilities.
The construction of solitons at finite density is a crucial aspect for understanding the
important issue of dense QCD. In the limit of a large number of colours, which is the regime
of holographic QCD, cold nuclear matter becomes a crystalline solid, although the details
of this are still to be understood. It should be possible to capture this behaviour via a
bulk soliton desription within holographic QCD and in particular within the Sakai-Sugimoto
model. However, the lack of numerical computations has led to various approximate methods
being employed to describe this phase, as follows.
Calorons, which are flat space self-dual Yang-Mills instantons with a periodic direction,
can split into monopole constituents if the period is smaller than the instanton size. This
fact, together with a point particle approximation, has led to the suggestion [6] that the
appropriate soliton crystal consists of pairs of dyons with opposite charges arranged in a salt-
like configuration. It is argued that, with increasing density, this dyonic salt arrangement
turns into a cubic crystal of half-instantons that is dual to the well-known Skyrme crystal.
In a different study, making use of approximations involving flat space calorons and dilute
instantons, it has been proposed [7, 8] that with increasing density a series of transitions
takes place, dubbed baryonic popcorn, where the three-dimensional soliton crystal develops
additional layers in the holographic direction. Unfortunately, even classical field theory
computations are not yet available to test these ideas in the Sakai-Sugimoto model.
In this paper we introduce and investigate a low-dimensional analogue of the Sakai-
Sugimoto model. The bulk theory is defined in a three-dimensional spacetime with negative
curvature, and is an O(3) sigma model with a baby Skyrme term that plays the role of the
Chern-Simons term in the higher dimensional theory. It is well-known that instantons in
2

planar sigma models are natural low-dimensional analogues of Yang-Mills instantons. If the
coefficient of the baby Skyrme term is small then the soliton has a small size and may be
approximated by an instanton of the flat space sigma model. The advantage of the lower-
dimensional theory is that numerical simulations of multi-solitons and finite density solutions
can be performed and compared with predictions using flat space instanton approximations.
In particular, analogues of dyonic salt and baryonic popcorn configurations are found and
analysed. The results provide evidence to support the validity of these ideas within the
Sakai-Sugimoto model.
2 Solitons of a holographic baby Skyrme model
Consider a (D + 2)-dimensional spacetime with a metric of the form
ds
2
= H(dt
2
+ dx
2
1
+ . . . + dx
2
D
) +
1
H
dz
2
, (2.1)
where
H(z) =
1 +
z
2
L
2
p
. (2.2)
The warp factor H(z), multiplying the (D +1)-dimensional Minkowski spacetime of the dual
boundary theory, depends only on the additional holographic coordinate z. The constant p
is to be specified later and L determines the curvature length scale and can be set to unity
by an appropriate choice of units.
The metric of the Sakai-Sugimoto model [1, 2] corresponds to the choice D = 3 and p =
2
3
.
In this case the spacetime has a conformal boundary as z and the scalar curvature is
R =
16(4z
2
+ 3)
9(1 + z
2
)
4/3
, (2.3)
with the properties that R 0 and R is finite (in fact zero) as z .
In this paper we are interested in a low-dimensional analogue with D = 1, so that the
bulk spacetime is three-dimensional with coordinates t, x, z. In this case, for general p, the
scalar curvature is
R = 2p(1 + z
2
)
p2
(5p 2)z
2
+ 2
. (2.4)
For this spacetime to have finite curvature with R 0 requires the restriction
2
5
p 1.
Later we shall see that a convenient choice is p =
1
2
, but for now we consider a general value
of p within the above interval.
The action of the massless O(3) baby Skyrme model in the above spacetime is
S =
Z
1
2
g
µν
µ
φ ·
ν
φ +
κ
2
4
g
µν
g
αβ
(
µ
φ ×
α
φ) · (
ν
φ ×
β
φ)
g dx dz dt, (2.5)
3

where φ = (φ
1
, φ
2
, φ
3
) is a three-component unit vector and greek indices run over the bulk
spacetime components t, x, z. The first term in (2.5) is that of the O(3) sigma model and
the second term, with constant coefficient κ
2
, is the baby Skyrme term [9].
The associated static energy is
E =
1
2
Z
1
H
|
x
φ|
2
+ H |
z
φ|
2
+ κ
2
|
x
φ ×
z
φ|
2
H dx dz, (2.6)
and the boundary condition is that φ (0, 0, 1) as x
2
+ z
2
. As we shall see, this
theory has bulk topological solitons that share many analogous features to those in the
Sakai-Sugimoto model. The analogue of the baryon number is the integer-valued topological
charge
B =
1
4π
Z
φ · (
x
φ ×
z
φ) dx dz, (2.7)
which defines the instanton number of the planar sigma model.
Using the fact that the baby Skyrme term contribution to the energy is non-negative and
H 1, together with the obvious inequality
1
H
x
φ ±
Hφ ×
z
φ
2
0, (2.8)
yields the Bogomolny bound E 4π|B|.
In flat space (H = 1) without a baby Skyrme term (κ = 0) this inequality is attained
by the instanton solutions of the O(3) sigma model (for a review see [10]). To write these
instanton solutions explicitly it is convenient to use the equivalent formulation of the O(3)
sigma model in terms of the CP
1
sigma model. This is obtained by defining the Riemann
sphere coordinate W = (φ
1
+
2
)/(1φ
3
), obtained by stereographic projection of φ. In terms
of this variable, instanton solutions are given by W a holomorphic function of ζ = x + iz.
The instanton solutions with finite B > 0 are given by W (ζ) a rational function of degree B,
where the degree of the numerator is larger than that of the denominator in order to satisfy
the above boundary condition. Taking into account the global U(1) symmetry associated
with the phase of W , this leaves an instanton moduli space M
B
of dimension 4B 1.
The radially symmetric sigma model instanton with topological charge B and centre at
the origin is given by W = (ζ)
B
, where the positive real constant µ is the arbitrary size
of the instanton. Converting back to the O(3) sigma model formulation this solution is
φ = (sin f cos(Bθ), sin f sin(Bθ), cos f), (2.9)
where r, θ are polar coordinates in the (x, z)-plane and f(r) is the radial profile function
f = cos
1
r
2B
µ
2B
r
2B
+ µ
2B
. (2.10)
If we require that this field configuration has finite energy in the curved spacetime theory
(2.6) then this places a further restriction on the power p, as follows. The large r behaviour
4

Citations
More filters
Journal ArticleDOI
TL;DR: In this article, the Sakai-Sugimoto model is extended to the decompactified limit of the instanton model, and a first-order baryon onset and chiral restoration at high density are possible.
Abstract: We discuss homogeneous baryonic matter in the decompactified limit of the Sakai-Sugimoto model, improving existing approximations based on flat-space instantons. We allow for an anisotropic deformation of the instantons in the holographic and spatial directions and for a density-dependent distribution of arbitrarily many instanton layers in the bulk. Within our approximation, the baryon onset turns out to be a second-order phase transition, at odds with nature, and there is no transition to quark matter at high densities, at odds with expectations from QCD. This changes when we impose certain constraints on the shape of single instantons, motivated by known features of holographic baryons in the vacuum. Then, a first-order baryon onset and chiral restoration at high density are possible, and at sufficiently large densities two instanton layers are formed dynamically. Our results are a further step towards describing realistic, strongly interacting matter over a large density regime within a single model, desirable for studies of compact stars.

21 citations

References
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TL;DR: In this paper, a holographic dual of four-dimensional, large N_c QCD with massless flavors was constructed by placing N_f probe D8-branes into a D4 background, where supersymmetry is completely broken.
Abstract: We present a holographic dual of four-dimensional, large N_c QCD with massless flavors. This model is constructed by placing N_f probe D8-branes into a D4 background, where supersymmetry is completely broken. The chiral symmetry breaking in QCD is manifested as a smooth interpolation of D8 - anti-D8 pairs in the supergravity background. The meson spectrum is examined by analyzing a five-dimensional Yang-Mills theory that originates from the non-Abelian DBI action of the probe D8-brane. It is found that our model yields massless pions, which are identified with Nambu-Goldstone bosons associated with the chiral symmetry breaking. We obtain the low-energy effective action of the pion field and show that it contains the usual kinetic term of the chiral Lagrangian and the Skyrme term. A brane configuration that defines a dynamical baryon is identified with the Skyrmion. We also derive the effective action including the lightest vector meson. Our model is closely related to that in the hidden local symmetry approach, and we obtain a Kawarabayashi-Suzuki-Riazuddin-Fayyazuddin-type relation among the couplings. Furthermore, we investigate the Chern-Simons term on the probe brane and show that it leads to the Wess-Zumino-Witten term. The mass of the \eta' meson is also considered, and we formulate a simple derivation of the \eta' mass term satisfying the Witten-Veneziano formula from supergravity.

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TL;DR: Fujiwara et al. as mentioned in this paper investigated the interactions among the pion, vector mesons and external gauge fields in the holographic dual of massless QCD and obtained the coupling constants by performing both analytic and numerical calculations, and compare them with experimental data.
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802 citations

Journal ArticleDOI
TL;DR: In this article, the authors considered aspects of dynamical baryons in a holographic dual of QCD that is formulated on the basis of a D4/D8-brane configuration.
Abstract: We consider aspects of dynamical baryons in a holographic dual of QCD that is formulated on the basis of a D4/D8-brane configuration. We construct a soliton solution carrying a unit baryon number and show that it is obtained as an instanton solution of four-dimensional Yang-Mills theory with fixed size. The Chern-Simons term on the flavor D8-branes plays a crucial role of protecting the instanton from collapsing to zero size. By quantizing the collective coordinates of the soliton, we derive the baryon spectra. Negative-parity baryons as well as baryons with higher spins and isospins can be obtained in a simple manner.

335 citations

Book
01 Jun 1989
TL;DR: In this article, the saddly point method has been applied to functional integrals in quantum mechanics, including path integrals with fermions, in the case of instantons.
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Journal ArticleDOI
TL;DR: In this paper, the axial coupling between baryons and pions and the magnetic dipole moments were estimated in the five-dimensional setting, and it was shown that the instanton interpretation implies a particular magnetic coupling.
Abstract: We study baryons in a holographic model of QCD by Sakai and Sugimoto, realized as small instantons with fundamental string hairs. We introduce an effective field theory of the baryons in the five-dimensional setting, and show that the instanton interpretation implies a particular magnetic coupling. Dimensional reduction to four dimensions reproduces the usual chiral effective action, and, in particular, we estimate the axial coupling ${g}_{A}$ between baryons and pions and the magnetic dipole moments, both of which are proportional to ${N}_{c}$. We extrapolate to finite ${N}_{c}$ and discuss subleading corrections.

226 citations

Frequently Asked Questions (2)
Q1. What are the contributions mentioned in the paper "A low-dimensional analogue of holographic baryons" ?

Here the authors introduce and investigate a low-dimensional analogue of the Sakai-Sugimoto model, in which the bulk soliton can be approximated by a flat space sigma model instanton. 

In this paper the authors have introduced and investigated a holographic baby Skyrme model that can be used to study several low-dimensional analogues of bulk solitons in the Sakai-Sugimoto model, describing holographic baryons. The advantage of the low-dimensional theory is that several aspects that one would like to study in the Sakai-Sugimoto model, but are currently not tractable, can be investigated exactly. This provides further support for the use of self-dual Yang-Mills instantons in approximating bulk solitons in the Sakai-Sugimoto model. As the baby Skyrme model is easier to investigate the authors have chosen this route in the current paper, but it might be of interest to investigate the vector meson model to see if additional phenomena can be obtained.