1. Quark Mass in Holographic QCD
30th April, Galileo Galilei Institute, Florence
Quark Mass in Holographic QCD
　Koji Hashimoto (RIKEN)
arXiv/ (hep-th)
w/ T.Hirayama, Feng-Li Lin （National Taiwan Normal U）
and Ho-Ung Yee （ICTP Trieste）
Note : Significant overlap with [Aharony, Kutasov ]

3. 1. Problem of Sakai-Sugimoto
Cause
Solution
Benefit

4. Sakai-Sugimoto model　 　　　　　　
Best model of holographic QCD : Sakai-Sugimoto model
Massless QCD on D4
Gauge/Gravity
(AdS/CFT)
Correspondence
Gravity Dual
Composite operator states
in the bulk

5. Problem of the Sakai-Sugimoto model
Quarks are massless!
String
QCD
L-Flavor
D4-D4 Gluon
D4-D8 L-quark
D4-D8 R-quark
R-Flavor
Color
Quark mass is important in QCD……
Explicit vs Spontaneous chiral sym. breaking
Chiral condensate
GOR relation between pion and quark mass
Strange dynamics

6. Causes of the problem 　　　　　　
(a) There is no room to make the D8s separate from D4
(b) L and R quarks live on different locations in 10dim. L
Quark mass term : R

7. Our Solution to the problem
(1) Worldsheet instanton
Yukawa interactions
in string phenomenology Dp Dp Dp =
(2) Extended Technicolor
Techni-sector
QCD sector
techni-quarks
“W-boson”

8. Our results 　　　　
We derive
GOR relation
Chiral condensate
Chiral lagrangian
Low energy sector of the SS model is drastically refined!
Importance (future directions)
Strange physics
Finite temperature / finite density QCD

9. 1. Problem and Our Solution 2. Brief Review of Sakai-Sugimoto
Plan of today’s talk
1. Problem and Our Solution
2. Brief Review of Sakai-Sugimoto
3. Quark mass, Pion mass
4. Chiral condensate, Flavor dep.
6 slides
5 slides
5 slides

10. 2. Sakai-Sugimoto : Review

11. QCD Holographic QCD Gravity -dual
AdS/CFT：　Equivalence of two ways to describe D-branes
[Maldacena(97)]
QCD
Gravity
-dual
Open string （gauge theory）
Closed string （gravity）
Holographic QCD
Low energy effective action of
open strings on N D3-branes
Closed string in blackbrane background of N D3-branes
10 dim. supergravity
in curved backgrounds
4dim. gauge theory
Corresp.

12. Sakai-Sugimoto model (hep-th/0412141)
Open string side (D-branes)
・　Nc D4-branes wrap with radius
Gauginos satisfy anti-periodic boundary condition
　　  Massless gluons at low energy, SU(Nc) gauge
[Witten]
・　Nf D8 intersecting D4s  Nf massless L-quarks
・　Nf D8 intersecting D4s  Nf massless R-quarks
Massless QCD is brane-engineered at low energy
D8 x D8 gauge group = Chiral symmetry

13. Closed string side (gravity)
D8s : probe (approximation valid at )　　　　　　
Near horizon geometry of black 4-brane solution on which
fermions satisfy the anti-periodic b.c. is
[Witten]
Geometry truncated at
 D8 and D8 are connected
 Spontaneous Chiral sym. br.

14. ＝ Once the correspondence is applied… D8 D4
・ D8 → bound states of quarks（Mesons, Baryons）
　　　KK modes of gauge fields on the D8 → Meson
　　　D8-brane action → Chiral lagrangian
・ gravity → bound states of gluons（Glueballs）

15. Meson sector in the SS model
D8-brane action on the curved background
Induced metric: z
Redefinition of coordinates：
KK modes of 　　　　  Vector mesons
A KK mode of  massless Pion

16. Going to gauge and integrating
D8-brane action = Chiral lagrangian
Sugra parameters are

17. 3. Quark Mass, Pion Mass

18. Extended technicolor  quark mass
Technicolor sector :
QCD
Technicolor
Mediator
Quark mass
Techni-quark
W-induced
4-Fermi
Condensation

19. Brane configuration of the Technicolor
ref. [Hirayama, Yoshioka (07)]
Technicolor sector ||
Additional D4s
Strongly coupled
Technicolor sector ||
Another throat

20. Worldsheet instanton
Double-line =
Worldsheet instanton

21. Quark mass
Worldsheet instanton
W-boson mass :
Quark mass :

22. Pion mass
Gravity dual has 2 throats
 Non-trivial 1-cycle
on the D8-brane
Boundary coupling
to D8 gauge field b.g.
Worldsheet instanton
Chiral lagrangian
GOR relation

23. Chiral Condensate, Flavor Dependence

24. D6 instead of the D4’
Difficulties in the introduction of the D4’ :
Joining two gravity solutions?
Decoupling of the overall NG boson?
Coefficient c to compute chiral condensate?
We introduce
a D6 ending on D8 and D8
intead of the D4’ D6
A lot simpler!

25. Virtue of the D6
D6 is a probe.
No need of new b.g.
D6 is given as a spike
solution of the D8 DBI.
D6 spike
D6 charge is magnetic (1,-1)
under the chiral symmetry :
 Explicit breaking of the chiral symmetry
Worldsheet instanton for the quark mass
is given in flat spacetime 

26. Chiral lagrangian
Worldsheet instanton
looks same.
D4 throat
D6 spike
: effective coupling

27. Chiral condensate, flavor dependence
On each D8, D6 ends
2D8
Standard chiral lagrangian

28. Numerics ( just for illustration )
Pion / quark mass
From pion decay constant and rho meson mass,
We substitute ,
Cf) Experiments :
Chiral condensate
Cf) Lattice :

29. 5. Conclusions

30. Our solution to the massless quark problem
Extended technicolor + worldsheet instanton
 Quark mass, pion mass
GOR relation
We have derived
Chiral condensate
Chiral lagrangian
Low energy sector of the SS model is refined!

31. Relation to previous attempts
Tachyon condensation
(Higgs mechanism)
Quark mass
[Casero, Kiritsis, Peredes (07)]
[Bergman, Seki, Sonnenschein (07)]
[Dhar, Nag (07)]
D8-D8 string : “tachyon”
Gauge invariant term
Difficulty : Tachyon action?
… No consistent truncation?

32. “Tachyon” picture and our solution
Tachyon condensation = partial annihilation of D8D8
(non-normalizable mode of “bulk tachyon” quark mass)
The same worldsheet instanton gives the quark mass =
Higgs mechanism and technicolor
are s-t channel dual to each other! =

33. Future directions 　　　　
Strange physics, K/pi mass difference
…. 2 instantons
Finite temperature / finite density QCD
Running charm in QGP?
Deconfinement transition temperature?
More precise evaluation of the instanton amplitudes
Vector meson mass?