Analysis of QCD via Supergravity

1. Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai 2/14-16, 2005

2. Sorry, my talk is not going beyond SM. But, many features are common to models with branes, extra-dimensions, etc. So, please don’t sleep now.

3. ① Introduction ★ D-brane open string Dp-brane (p+1) dimensional plane etc. gauge field (p+1) dim gauge theory is realized on the Dp-brane

4. ★ gauge/gravity duality cf) particle Solution of Einstein eq. D-brane Supergravity solution Conjecture The information of the gauge theory on the D-brane is somehow encoded in the supergravity solution.

5. Type IIB string theory on Super Yang-Mills The (strongly coupled) gauge theory can be analyzed via (classical) SUGRA ! ※ SUGRA is good when Typical example（AdS/CFT correspondence） [Maldacena 1997] D3-brane dual ( in R = 1 unit )

6. Today, I’d like to discuss an attempt to analyze QCD using this technique.

7. Plan of Talk String theory ② QCD ③ Sugra Hadron physics ④

8. ② Construction of QCD Consider D4-brane on massless modes are gauge field scalar fermion unwanted Impose become massive ( SUSY is broken ) also acquire mass via 1-loop effect low energy 4 dim pure Yang-Mills theory ★ Construction of 4 dim pure Yang-Mills theory [Witten 1998]

9. ★ Adding quarks 4 dim QCD with massless quarks [cf) S.S.-Takahashi 2004]

10. ③ SUGRA description of QCD ★“Probe approximation” [Karch-Katz 2002] We assume Only D4-brane is replaced with the SUGRA solution. pairs are treated as probes. cf) light heavy probe

11. where radial direction restricted to flat D4 solution D4 D4 Fortunately, corresponding SUGRA solution is known.

12. ④ Exploration Now we are ready to explore ! highlights • Chiral symmetry breaking • Unification of mesons • Comparison to experimental data • Construction of baryons

13. ★ Chiral symmetry breaking and must be connected in the D4 background. interpreted as the chiral symmetry breaking ! connected As before, we replace D4 with the SUGRA sol.

14. ★ Where are mesons? The low energy effective theory on the D8-brane is a 9 dim gauge theory. Gauge field: There is an symmetry ※ Today, we only consider the singlet states, for simplicity. (mesons in realistic QCD live in this sector) reduced to 5 dim gauge theory closed string glueballs ∃ flavor indices open string mesons

15. D8-brane action This 5 dim Yang-Mills theory is considered as the effective theory of mesons. Insert the SUGRA solution [cf) Son-Stephanov 2003]

16. mode expansion some complete sets • To diagonalize kinetic & mass terms of , ★ choose as eigen functions satisfying ★ choose as • In addition, we have one more normalizable mode (first we consdier case)

17. eaten eaten are unified in the 5 dim gauge field ! ( is consistent with this interpretation.) Using these, we obtain eaten massive vector meson massless scalar meson In summary,

18. ★ Comments on is even func. is pseudo-scalar are even func. are vector are odd func. are axial-vector ★ Consistent with our interpretation. ★ Prediction: vector and axial-vector mesons appear alternately. ( … not established) Parity : Charge conjugation : and

19. ★ mass spectrum ★ Let us solve this numerically, to see if our model is too bad or not. We should not be too serious, since the approximation is very crude. The mass of the n-th meson is given by where is the eigen value of the equation

20. Results : (our model) +4 ％ (experiment) (our model) -17 ％ (experiment) Not ｂad !

21. We partially fix the gauge s.t. Residual gauge symmetry s.t. This is interpreted as constant ★ generalization to

22. transforms as interpreted as the pion field Define

23. We can move to gauge by the gauge tr. with The pion field appears in the boundary condition. Then, mode exp. is given by where is defined by and ★ gauge

24. ★ effective action If we turn off vector mesons for simplicity, we obtain This is the Skyrme model ! Inserting the mode exp. into the D8-brane action we obtain the effective action for the mesons written in terms of and .

25. In particular, the 3 pt coupling is calculated as (our model) (experiment, KSFR relation) • Turning on ρmeson ( ), • we can calculate the couplings of π and ρ.

26. ★ Baryon as Skyrmion The pion field • Skyrme proposed [Skyrme 1961] Baryon Soliton in Skyrme model ( Skyrmion ) defines the winding number This is interpreted as the baryon number charge.

27. ★ Baryon as wrapped D-brane • There is a source of electric charge on D4 F-string should be attached. Bound state of quarks Baryon Baryons in the AdS/CFT context is constructed by wrapped D-branes D4 [Witten 1998] In our case, the baryon is a D4-brane wrapped on the . F1

28. D4 within D8 instanton on D8 D8 D8 D4 Using and the boundary condition [Son-Stephanov 2003] Wrapped D4 instanton on D8 Skyrmion ■ ∴ • In our model, the wrapped D4 can be embedded in D8. [Douglas 1995]

29. ⑤ Summary • The baryons are constructed as wrapped D4 instanton on D8 Skyrmion ★ QCD is constructed in string theory. ★ Though the approximation is still very crude, our model catches various qualitative features of QCD and provides some new insights in the low energy hadron physics. • Chiral symmetry breaking is understood geometrically. • Mesons are unified in the 5 dim gauge field. ★ The numerical results are also encouraging.

30. ★ Other topics • Chiral anomaly and axial anomaly are reproduced. • WZW term is obtained from CS-term on the D8 • The Witten-Veneziano formula for the η’meson mass is derived.

31. Fin