1. in collaboration with Y. Nakagawa and K. Matsumoto
1+1 Large Nc QCD and its Holographic Dual ~Baryons in Single Flavor (Nf=1) World
H. Suganuma (Kyoto U.)
in collaboration with Y. Nakagawa and K. Matsumoto
Abstract: We study Baryons in low-dimensional
Large Nc Holographic QCD with Single Flavor (Nf=1)
using a D-brane system formulated in superstring theory.
In particular, Single-Flavor 1+1 QCD can be formulated with an S1 compactified D2/D8/D8bar branes, and its baryon
can be expressed as a topological configuration.
However, unlike 1+3 QCD, we find that the baryonic soliton is unstable in Nf=1 1+1 QCD against a scale transformation.
MENU2016, 27 July 2016, Kyoto University

2. Introduction ~ Non-Perturbative QCD physics
What is QCD ?
-QCD is an SU(Nc) gauge theory of quarks and gluons.
-QCD exhibits confinement and chiral symmetry breaking.
-QCD leads to mesons (including glueballs) and baryons
as observable elementary excitations.

3. Introduction ~ Non-Perturbative QCD physics
How to Solve QCD ?
-QCD is very difficult to solve analytically.
-Lattice QCD is a first-principle calculation,
but has several weak points:
　1. chiral limit: zero-mass pions require infinite volume lattice.
2. state information (e.g. wave function) is limited:
because of path integral formalism.
3. finite density: because of sign problem
-Most Model approaches are not QCD-based,
and cannot be derived from QCD.

4. Introduction ~ Non-Perturbative QCD physics
Holographic QCD (HQCD)
T.Sakai and S.Sugimoto, Prog. Theor. Phys. 113, 843 (2005).
- Holographic QCD has a “direct” connection to QCD,
and is QCD-based.
- Holographic QCD is equivalent to infrared QCD
in large Nc and strong ’t Hooft coupling l ,
via gauge/gravity correspondence.
- Holographic QCD is successful to reproduce many
hadron phenomenology such as vector meson dominance,
KSRF relation, Hidden Local Symmetry,
GSW model, Skyrme soliton picture.
- Holographic QCD is usually formulated
in chiral limit.
- No sign problem in the finite density.

5. Introduction ~ Non-Perturbative QCD physics
Baryons in Holographic QCD (HQCD)
- Holographic QCD is described by mesons,
such as pions, vector mesons and axial-vector mesons.
- In Holographic QCD, Large Nc is used, so that
HQCD reduces a weak interacting theory of mesons,
and baryons do not appear as explicit degrees of freedom.
- In a standard argument of Large Nc ,
Baryon is described as Skyrmion
　i.e. Topological Chiral Soliton of mesons
(mainly Nambu-Goldstone bosons).
[E. Witten, Nucl. Phys. B160, 57 (1979).]
- In HQCD, Baryons are spatially-extended topological objects.
NG bosons
Baryon

6. Baryons in Large Nc QCD with various Nf
NG bosons
For Nf ≧ 2
- Surely, in Our Real world with Nf ≧ 2,
it is possible to describe Baryons
as Topological Chiral Solitons,
according to nontrivial homotopy group of P3 (SU(Nf)A)=Z.
Baryon
So, everything
looks consistent
in our world! !
- BUT, in Single Flavor (Nf=1) world, Baryons cannot be
described as topological objects, because of P3 (U(1)A)=1.

7. Single Flavor (Nf=1) world
- QCD with Single Flavor (Nf=1) is a possible theory, and
can be actually investigated by Lattice QCD.
- Actually, if the Higgs coupling to d-quark and s-quark
were large enough, Single Flavor world would be realized.
- In Single Flavor world, there appear massive PS meson
h’(ug5u), vector meson w (ugmu), and also a baryon D++ (uuu). - -
- Note however that, in Single Flavor world, Baryons cannot be
described as topological objects, because of P3 (U(1)A)=1.
- So, in Single Flavor world, it is difficult to
describe Baryons with mesons in Large Nc ,
where baryons do not appear explicitly ?
For Nf = 1
This is an
open problem.
Baryon

8. Baryons in 1+1 QCD with Single Flavor (Nf=1)
- We note that 1+1 Single Flavor QCD and its Holographic Dual
have Topological Objects, corresponding to the nontrivial
homotopy group P1 (U(1))=Z, as will be shown.
- Then, in 1+1 Single-Flavor QCD,
as a natural possibility,
Baryons can be definitely described
as the Topological Objects in Large Nc,
like 1+3 QCD with Nf ≧ 2.
1+1 QCD with Nf = 1
Baryon ?
- This is the motivation to investigate
1+1 Single-Flavor QCD and its Baryons,
especially in large Nc.

9. Holographic QCD

10. Emergence of Gauge Theory on D-brane,
Superstring theory is formulated in 10 dimensional space-time, and has Dp-brane as a (p+1)dimensional soliton-like object of strings.
On Nc D-brane, U(Nc) Gauge Theory is constructed,
where open string behaves as U(Nc) gauge field.
Around Nc D-brane, Supergravity field is formed,
because D-brane is massive and is the source of gravity field.
On D-brane
U(Nc) Gauge Theory
Around D-brane
Gravity Theory
Open string on Nc D-brane behaves as U(Nc) gauge field
Closed string around D-brane behaves as graviton

11. λ=NcgYM2 :’tHooft coupling
Holography
On D-brane, gauge theory
Is constructed.
[Maldacena (1997)]
Dp-brane×Nc
On the other hand, D-brane behaves as a
Gravitational source around it.
(p+1) dim. Gauge Theory
D-brane = Gravitational Source
[Polichinsky (1995)]
Gravity field depends on distance from D brane.
Then, one more coordinate appears in gravity side.
((p+1)+1) dim. Supergravity Theory
: low energy
: weak coupling of string
: strong coupling
Strong-Weak Duality
(S-duality)
: large Nc
λ=NcgYM2 :’tHooft coupling
・Remarkably, there is Strong-Weak Duality:
strong coupling of one side corresponds to
weak coupling of the other side.
・Nonperturbative quantities of Large Nc QCD can
be calculated with classical gravitational theory.
10dim.

12. Construction of Non-SUSY SU(N) gauge theory
Similar to Thermal SUSY breaking,
Supersymmetry can be removed by S1-compactification
with normal boundary condition
(periodic for bosons, anti-periodic for fermions).
[E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998).]
SUSY
Non-SUSY τ

13. Color degrees of freedom Flavor degrees of freedom
Holographic QCD corresponding to 1+3 QCD
T. Sakai and S. Sugimoto, Prog. Theor. Phys. 113, 843 (2005).
Using D4/D8/D8-branes, massless 1+3 QCD can be constructed.
Here, Nc D4-brane gives Color and Gluons and D8-brane gives Flavor.
Left (Right) Quarks appear at the cross point between D4 and D8 (D8bar).
Index on D4 brane ( color )
Color degrees of freedom
Flavor degrees of freedom
Index on D8 brane ( flavor )
10dim.
: quark (L)
D4-brane×NC
D8-brane×Nf (L)
4-8
4-4
: gluon
4-8
D8-brane×Nf (R)
: quark (R)

14. Gravitational Background
In Large Nc limit, Nc D4-brane is extremely massive
and is replaced by Gravitational background,
via gauge/gravity correspondence.
D4-brane is replaced by
Gravitational Background
D4-brane×N C
D8×N f
8-8 D8
4-8
4-4
4-8
10dim. space time

15. Gauge/gravity correspondence
1+3 QCD
1+3 QCD can be
constructed on a
D4/D8/D8-brane.
D4/D8/D8bar
D4-brane×N C
D8×N f
4-8
4-4
4-8
10dim. space time
Gauge/gravity correspondence
8-8 D8
This D4/D8/D8-brane
becomes 1+4 dim theory in Flavor Space.
1+4 dim theory in Flavor Space
T. Sakai and S. Sugimoto, PTP 113, 843 (2005).
: extra spatial dimension

16. Baryon as Topological Chiral Soliton
In Holographic QCD, Baryon appears as Topological Hedgehog Soliton.
Hedgehog soliton with B = 1
T.H.R. Skyrme, Proc. R. Soc. A260, 127 (1961).
P3 (SU(Nf)A)=Z
K. Nawa, H. Suganuma, and T. Kojo, Phys. Rev. D75, (2007).
For baryon, pion profile function F(r)
has topological boundary condition:
Boundary
condition
Baryon number current
(Goldstone-Wilczek current)

17. Baryon as Chiral Soliton
K. Nawa, H. Suganuma, and T. Kojo, Phys. Rev. D75, (2007).
Hedgehog soliton with B = 1 π ρ
pion profile F(r)
ρ-meson profile G(r)
In previous work, we performed first holographic study of Hedgehog Baryon.
For the holographic study of B=2 H Dibaryon, K.Matsumoto will present.

18. Holographic QCD corresponding to
1+1 Single Flavor QCD

19. Color degrees of freedom Flavor degrees of freedom
Holographic QCD corresponding to 1+1 QCD
H.-U. Yee and I. Zahed, JHEP 033 (2011).
Using D2/D8/D8-branes, massless 1+1 QCD can be constructed.
Here, Nc D2-brane gives Color and Gluons and D8-brane gives Flavor.
Left (Right) Quarks appear at the cross point between D2 and D8 (D8bar).
Index on D2 brane ( color )
Color degrees of freedom
Flavor degrees of freedom
Index on D8 brane ( flavor )
10dim.
D2-brane×NC
: quark (L)
D8-brane×Nf (L)
2-8
2-2
: gluon
2-8
D8-brane×Nf (R)
: quark (R)

20. Gravitational Background
In Large Nc limit, Nc D2-brane is extremely massive
and is replaced by Gravitational background,
via gauge/gravity correspondence.
D2-brane is replaced by
Gravitational Background
D2-brane×N C
D8×N f
8-8 D8
2-8
2-2
2-8
10dim. space time

21. Gauge/gravity correspondence
1+1 QCD
1+1 QCD can be
constructed on a
D2/D8/D8-brane.
D2/D8/D8bar
D2-brane×N C
D8×N f
2-8
2-2
2-8
10dim. space time
Gauge/gravity correspondence
8-8 D8
This D2/D8/D8-brane
becomes 1+2 dim theory in Flavor Space.
1+2 dim theory in Flavor Space
H.-U. Yee and I. Zahed, JHEP 033 (2011).
: extra spatial dimension
unit

22. Analysis of 1+2 Holographic QCD corresponding to1+1 Nf=1 QCD
In this 2D spatial system, there is a topological charge called
the Pontryagin index, which should be an integer:
From holographic viewpoint, this corresponds to Baryon Number.
H.Hata et al., PTP117,1157(2007).
Direct analogue is Quantization of
Magnetic Flux in 2D space.
Typical example: Abrikosov vortex
in Type-II Superconductor.

23. We consider 1+2 Holographic QCD (1+1 Nf=1 QCD)
under the topological constraint:
Here, we take temporal gauge A0=0, which leads to
ordinary canonical quantization.
The mass of Topological Soliton (Baryon):
Non-negative

24. From this mass formula
we consider ground-state soliton (the lowest baryon state)
under the topological constraint:
Since the topological condition does not act on the electric field,
we can take , and the formula becomes simple:

25. As a result in low-D HQCD, we find that
the baryonic soliton is generally unstable
against some scale transformation,
and the soliton swells infinitely.
Let’s check this!

26. Suppose we obtain the solution which
minimizes the mass
and satisfies the topological condition:
For the solution, its mass should be a minimum on
any small variation consistent with the topological condition.

27. We consider a “scaled configuration” as a simple variation:
This scaled configuration includes the solution at l=1, and satisfies the topological condition:

28. We consider a “scaled configuration” as a simple variation:
The mass of this scaled configuration is
l times the original mass :
Then, the mass becomes smaller continuously to zero as l goes to zero from unity.
This corresponds to a swelling of the solution.

29. Thus, in Holographic QCD of 1+1 Single-Flavor QCD,
baryonic soliton is unstable
against this type of scale transformation,
and the soliton swells infinitely.
No stable baryons in 1+1 Single-Flavor QCD in large Nc?
So, Baryons in Single-Flavor World seems puzzling
also in 1+1 QCD. (cf. puzzling in 1+3 QCD)

30. Comparison with Abrikosov Vortex in Type-II Superconductor
Pontryagin index
Swelling
Shrinkage
Superconductor has photon field A and Cooper scalar field j.
On scale transformation,
- Photon-field contribution is to promote Swelling of the soliton.
- Scalar-field contribution is to promote Shrinkage of the soliton.
Because of these two opposite effects,
Abrikosov vortex is stable
against scale transformation.

31. Scale Instability of Baryons in HQCD of 1+1 QCD
Pontryagin index
Swelling
HQCD of 1+1 QCD has only “flavor-space vector field A” in 1+2 dim
at the leading order of 1/Nc and 1/l.
On scale transformation,
Vector-field contribution is to promote Swelling of the soliton.
Because of this one-side effect,
Soliton is unstable against scale transformation.

32. Summary We have studied Baryons in low-dimensional
Large Nc Holographic QCD with Single Flavor (Nf=1)
using a D-brane system formulated in superstring theory.
In particular, Single-Flavor 1+1 QCD can be formulated with an S1-compactified D2/D8/D8bar branes, and its baryon can be expressed as a topological configuration.
Unlike 1+3 QCD, however, we have found that the baryonic soliton is unstable in Nf=1 1+1 QCD
against a scale transformation.
Thus, Baryons in Single-Flavor World seems puzzling also in 1+1 QCD.

33. Thank you !