Baryons and Baryonic Matter in Holographic QCD from Superstring

1
Baryons and Baryonic Matter in Holographic QCD
from Superstring
H. Suganuma (Kyoto U.), K. Nawa (Osaka U.) and T.
Kojo (BNL)
Abstract We study baryons and baryonic matter in
holographic QCD using a multi-D-brane system
starting from Type IIA superstring theory. We
study baryon in holographic QCD for the first
time as chiral soliton solution in the
four-dimensional meson effective action derived
from multi-D-brane system. We also study QCD
phase transition at finite density in holographic
QCD by investigating the chiral soliton on S3.
References K. Nawa, H. Suganuma and T.
Kojo Baryons in Holographic QCD Phys. Rev.
D75, 086003 (2007). Brane-induced Skyrmion
Prog. Theor. Phys. Suppl. 168, 231 (2007).
Baryons with Holography Mod. Phys.
Lett. A (2008). Brane-induced Skyrmion on S3
Prog. Theor. Phys. Suppl. (2008).
2
Superstring, D-brane, Gauge Theory and
Supergravity
Superstring theory is well-defined in 10
dimensional space-time, and has Dp-brane as a
(p1)dimensional soliton-like object of
fundamental strings. On N-folded D-brane,
U(N) Gauge Theory can be constructed, where open
string behaves as U(N) gauge field. Around
N-folded D-brane, Supergravity field is formed,
because D-brane is massive and is the source of
gravity field.
On D-brane
U(N) Gauge Theory
Around D-brane
Gravity
Open string on N-folded D-brane behaves as U(N)
gauge field
closed string around D-brane behaves as graviton
3
Holography
On D-brane, gauge theory Is constructed.
Maldacena (1997)
Dp-braneNc
On the other hand, D-brane behaves as a
Gravitational source around it.
(p1) dim. Gauge Theory
10dim.
4
Construction of Non-SUSY Gauge Theory
SUSY Breaking is realized by Boundary Condition
(Witten, 1998)
N-folded D-brane system leads to SUSY gauge
theory, reflecting superstring nature.
To obtain Non-SUSY gauge theory, we break SUSY by
spatial S1 compactification of N-folded D brane.
(cf. thermal SUSY breaking)
open stringU(N) gauge field Aµ
open stringU(N) gauge field Aµ
The inverse radius of spatial S1
compactification is called as Kalza-Klein mass
MKK. - impose periodic boundary condition for
bosons - impose anti-periodic boundary
condition for fermions SUSY is explicitly broken
by the boundary condition. Gaugino mass
becomes O(MKK ) and scalar fields become massive
by quantum correction.
Non-SUSY U(N) Gauge Theory can be constructed on
spatial S1 compactified N-folded Dbrane at
larger scale than Kalza-Klein mass scale 1/MKK.
For the argument of QCD, Kalza-Klein mass MKK is
cutoff scale and is taken as 1GeV.
5
Gravitational Description of YM theory with
Gauge/Gravity correspondence
S1 compactified N-folded D4-brane
Gravity Solution around D4 brane
xµ(t, x, y, z)
t
Spatial one-dimension is S1 compactified Radius
of S1 MKK-1
u distance from D4 brane
On the S1 compactified N-folded D4-brane, only
the gauge field Aµ remains to be massless, and
non-SUSY 4 dim.YM theory is realized in
low-energy region. On the other hand, the
effect of D4-brane can be also described by
Gravity Field around it, under the hypothesis of
Gauge/Gravity correspondence. Thus, 4 dim. YM
theory or the gauge sector of QCD is introduced
using D4 brane, and it is transferred into
Gravitational background field in holographic
framework.
6
Construction of massless QCD in Holographic QCD
T.Sakai and S.Sugimoto, Prog. Theor. Phys. 113,
843 (2005)
Following Sakai-Sugimoto, we construct massless
QCD (quarks and gluons) using the following
D4/D8/D8-brane system, where D4-brane gives color
and Gluons and D8-brane gives flavor. Quarks,
which have color and flavor, appear at the cross
point between D4 and D8.
Index on D4 brane ( color )
Index on D8 brane ( flavor )
10dim.
quark (L)
D4-braneNC
D8-braneNf (L)
4-8
4-4
gluon
4-8
D8-braneNf (R)
quark (R)
In Holographic QCD, Color and Flavor are
described as different physical objects,
different D-branes.
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Note that Holographic QCD is based on Large Nc ,
i.e., Nc gtgt Nf. Since mass of N-folded D-brane is
proportional to sheet number N, Nc-folded
D4-brane is extremely massive in Large Nc. Only
D4-brane is replaced by Gravitational background,
remaining D8 brane.
Gravitational background
D4Nc
D8, D8Nf
D8, D8Nf
8-8
4-8
4-4
4-8
Nc-folded D4-brane becomes Gravitational
Background using Gauge/Gravity duality. The
system becomes D8 brane in the presence of
Gravitational Background of D4 brane. Note
that, in this framework, Color degrees of freedom
around D4-brane is hidden. The system is
described by non-colored and flavored degrees of
freedom like hadrons.
8
In the presence of D4 gravity background,
D8-brane leads to Dirac-Born-Infeld (DBI) action
at the leading order.
D8-brane 9dim. manifold
surface tension
dilaton field
D4 SUGRA backgroud
Integration over 4-dim. angle dO4 around D4 brane
9 4 5

5 dim. Flavored Yang-Mills theory in Curved Space
The remaining 5th dimension is distance
parameter u (or z) from D4 brane. The color
degrees of freedom of D4-brane is replaced by
Gravitational background, and only flavor
degrees of freedom remains.
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Infrared effective theory of QCD derived from
D8-brane in the presence of D4 gravity
background

5 dim. Flavored Yang-Mills theory in Curved Space
mode expansion in 5th direction, z
5 dim. gauge field from 8-8 string
pion
(axial) vector mesons
Through the mode expansion in 5th direction,
4dim. effective theory of QCD is derived as
Meson theory, that is, theory of pion and (axial)
vector meson.
Hidden local symmetry.
This effective theory describes many
phenomenological features of mesons
Bando, et al.,1985
Vector meson dominance.
Sakurai, 1969
KSRF relation.
T.Sakai and S.Sugimoto, Prog. Theor. Phys. 113,
843 (2005)
Kawarabayashi et al., 1966
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Baryons in Holographic QCD
How do baryons describe in Holographic QCD with
Large-Nc?
In Large-Nc, QCD is reduced to weakly
interacting theory of mesons and glueballs, so
that baryons do not appear as fundamental degrees
of freedom.
tHooft, Nucl. Phys. B72, 461(1974) B75,
461(1974)
11
K. Nawa et al., Phys. Rev. D75, 086003 (2007)
We derive 4dim. effective action of pions and
?mesons in Holographic QCD.
Note that this meson action is derived from QCD.
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Note also that the Skyrme term appears from
Holographic QCD without appearance of other 4- or
more derivative terms.
2-derivative term

4-derivative term

,
,
Instability of Skyrme soliton.
I. Zahed and G.E. Brown, Phys. Rep. 142, 1
(1986)
Effective action of leading order of Holographic
QCD
Topological Chiral Soliton picture for baryon can
be directly derived from QCD in the Holographic
framework.
Baryon as a Chiral Soliton
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Baryon as Chiral Soliton
In the Chiral Soliton picture, baryon appears as
hedgehog soliton.
Hedgehog soliton with B 1
T.H.R. Skyrme, Proc. R. Soc. A260, 127 (1961)
For baryon, pion profile function F(r) has
topological boundary condition
Boundary condition
Baryon number current (Goldstone-Wilczek current)
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We derive the Euler-Lagrange equations, which
are coupled nonlinear differential equations of
Pion and Rho-meson profiles, F(r) and G(r).
Field Equation for Baryons in Holographic QCD
Under the Topological Boundary conditionsF(r
0)p, F(r 8)0, we solve this complicated field
equation and obtain Chiral Soliton Solution with
Baryon number 1.
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Baryon as Chiral Soliton
K. Nawa, H. Suganuma, and T. Kojo, Phys. Rev.
D75, 086003 (2007)
Hedgehog soliton with B 1
p
?
pion profile F(r)
?-meson profile G(r)
Stable Hedgehog Soliton solution for baryon
exists. We call it Brane-induced Skyrmion, since
it is basically Skyrmion. ?-meson field appears
in the core region of baryon.
c.f. Baryon from an instanton in 5dim. YM theory
Baryon shrinks but they include infinite modes
beyond UV cutoff MKK
D.K.Hong et al., PRD76, 061901(2007) H.Hata
et al., PTP117,1157(2007) K.Y.Kim, S.J.SIn,
I.Zahed (2008)
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Active ?-meson component in the core of a Baryon
(Skyrmion)
Contribution of ?-meson interaction terms to the
energy density for the brane-induced Skyrmion
?-meson components are rather active in the core
region of baryon
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Baryon as Chiral Soliton
K. Nawa, H. Suganuma, and T. Kojo, Phys. Rev.
D75, 086003 (2007)
Hedgehog soliton with B 1
p
?
pion profile F(r)
?-meson profile G(r)
Using Experimental inputs for pion decay constant
and ?-meson mass, we estimate the mass and
radius of hedgehog baryon M834MeV, R0.37fm ?
fp92.7MeV, m?776MeV (i.e., MKK948MeV)
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Baryonic Matter
QCD phase diagram
Non-abelian nature gives various phases of QCD
at T ?B.
Deconfinement phase (QGP)
RHIC LHC-Alice
Chiral condensed phase
170MeV
Lattice QCD
Precursor of CSC
Diquark BEC
2SC
Confined phase
CFL
Cross-over
0
Neutron star
We consider application of holographic QCD to
finite density QCD, which is difficult to study
with lattice QCD.
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Baryonic Matter in holographic QCD
Holographic QCD is based on Large Nc argument. In
Large Nc, kinetic energy and quantum effect of
baryons become higher order, so that the baryonic
matter becomes Static Solid-like Soliton Matter.
Baryon Mass
G.tHooft, Nucl. Phys. B72, 461(1974) B75,
461(1974)
E.Witten, Nucl. Phys. B160, 57 (1979)
Baryon Kinetic energy
zero-point quantum fluctuation
Quantum effect
splitting
G.S.Adkins et al., Nucl. Phys. B228, 552
(1983)
I.Klebanov, Nucl. Phys. B262, 133 (1985)
large Nc
Static Solid-like Brane-induced Skyrmion Matter
Brane-induced Skyrmion
20
Single Brane-induced Skyrmion on S3
Static Brane-induced Skyrmion Matter
Compactification of unit cell
Following Manton, we use a mathematical trick to
simulate the soliton matter by single soliton
on S3.
Single Brane-induced Skyrmion on S3
We investigate single Brane-induced Skyrmion on
S3 in Holographic QCD.
N.S.Manton and P.J.Ruback, Phys. Lett. B181, 137
(1986)
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Total Energy Density of a baryon on S3 and
Delocalization
Total energy density of Brane-induced Skyrmion on
S3 with radius R in ANW unit
Radius R v.s. baryon number density ?B
When R goes to small, Baryonic Soliton is
gradually Delocalized, and becomes Homogeneous
at a critical radius Rc. This correspond to the
Delocalization of baryonic soliton at High
Density.
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Delocalization of Baryonic Soliton
Total energy density of Brane-induced Skyrmion on
S3 with radius R in ANW unit
Single Brane-induced Skyrmion on S3
Delocalization of Brane-induced Skyrmion
chiral restoration
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Localization parameter
We define Localization parameter as spatial
fluctuation of Energy density.
Localization parameter
Spatially Localized
Spatially Homogeneous
Low-density
high-density
For small R, Localization parameter goes to zero,
and the system becomes homogeneous. This result
indicates deconfinement phase transition at high
density.
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Spatially averaged Chiral Condensate
cf. Spontaneous magnetization
For small R, Spatially averaged Chiral Condensate
goes to zero. This result indicates chiral
symmetry restoration at high density.
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?-meson and pion contributions at high density
?-meson profile decreases and disappears at
critical density.
Only pion profile remains near ?c !
Note that Pion Field cannot vanish, since Pion
Field has Topological Boundary Condition for
Chiral Soliton.
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Summary
We have studied baryons and baryonic matter in
holographic QCD using a multi-D-brane system
in superstring theory. We have performed the
first study of baryon in holographic QCD, and
have obtained chiral soliton solution for baryon.
We have found active?meson in the core region
of baryon. We have also studied finite-density
QCD phase by investigating the chiral soliton
on S3 in holographic QCD. We have found
Delocalization of Baryonic Soliton, which would
correspond to Deconfinement and Chiral Symmetry
Restoration, at the critical densiy of about
7?0. For high density baryons, ? meson profile
decreases and disappears, and only pion
profile remains at the critical density. We have
found Swelling of Baryons in Dense Matter, which
leads the reduction of N-? mass splitting in
the framework of chiral soliton.