Holographic Mesons

1. Holographic Mesons RCM & Thomson (hep-th/0605017) Kruczenski, Mateos, RCM & Winters (hep-th/0304032)

2. Outline: • AdS/CFT with D7-branes • fundamental matter Karch + Katz (hep-th/0205236) holographic mesons Kruczenski, Mateos, RCM + Winters (hep-th/0304032) 2. Gauge/Gravity with probe branes “universal” meson spectrum RCM + Thomson (hep-th/0605017) Arean + Ramallo (hep-th/0602174) 3. Observations/Future Directions

3. Maldacena; Witten; Gubser, Klebanov & Polyakov; ….. AdS/CFT from D3-branes: d=10 Type IIb superstrings in AdS5 X S5 with Nc units of RR flux equivalent to all adjoint fields! d=4 N =4 U(Nc) super-Yang-Mills supergravity limit ~ large-N with strong `t Hooft coupling

4. (Karch and Katz) Fundamental fields: decoupling limitof Nc D3-branes with Nf D7-branes D7 D7 x r=L D3 L black brane with d=4 U(Nc) o.s., d=8 U(Nf) o.s. & bifundamental 3-7 o.s. Low-energy limit with Reduces brane theory to field theory Minimum mass for 3-7 strings: Decouples asymptotic (closed) strings Gauge coupling on D7’s: • throat geometry: AdS5 X S5 with Nf D7’s • brane theory →N =4 U(Nc) SYM with fundamental matter

5. Field theory: N=4 U(Nc) super-Yang-Mills vector: 4 Weyl fermions: 3 complex scalars: adjoint in U(Nc) coupled to: Nf massive N=2 hypermultiplets 2 complex scalars: 2 Weyl fermions: fund. in U(Nc) & global U(Nf) • SUSY: N =4 → N =2 • SO(6) → SO(4) = SU(2)L X SU(2)R

6. Geometry: equator AdS boundary S5 D7 S3 horizon SO(4) = rotational symmetry of S3 pole

7. Geometry: equator AdS boundary S5 D7 S3 horizon Mesons, bound states of fundamental fields, dual to open string states supported by D7-brane pole

8. Gauge/gravity dictionary: supergravity modes: flavor indices Dq-brane modes: Probe approximation: Nf /Nc → 0 The above construction does not take into account the “gravitational” back-reaction of the D7-branes! → considering large-Nc limit with Nf fixed (see, however: Burrington et al; Kirsch & Vaman; Casero, Nunez & Paredes)

9. Kruczenski, Mateos, RCM and Winters (hep-th/0304032) Mesons: lowest lying open string states are excitations of the massless modes on D7-brane: vector, scalars (& spinors) their dynamics is governed by usual worldvolume action: • free spectrum: • expand action to second order in fluctuations • solve linearized eq’s of motion by separation of variables Veff discrete spectrum r

10. Kruczenski, Mateos, RCM and Winters (hep-th/0304032) Meson spectrum: = radial AdS quantum # = angular quantum # on S3 = R-charge massive supermultiplets with bosons and fermions

11. Itzhaki, Maldacena, Sonnenschein & Yankielowcz (p<5) Gauge/Gravity Duality from Dp-branes: d=10 Type II superstrings in Dp-brane throat with Nc units of RR flux equivalent to d=p+1 N =4 U(Nc) super-Yang-Mills dimensionless effective coupling: YM coupling dimensionful !

12. Itzhaki, Maldacena, Sonnenschein & Yankielowcz (p<5) Gauge/Gravity Duality from Dp-branes: d=10 Type II superstrings in Dp-brane throat with Nc units of RR flux dilaton and curvature run! equivalent to d=p+1 N =4 U(Nc) super-Yang-Mills dimensionless effective coupling: Supergravity only in intermediate region:

13. Itzhaki, Maldacena, Sonnenschein & Yankielowcz (p<5) Gauge/Gravity Duality from Dp-branes: d=10 Type II superstrings in Dp-brane throat with Nc units of RR flux equivalent to all adjoint fields! d=p+1 N =4 U(Nc) super-Yang-Mills dimensionless effective coupling: Supergravity only in intermediate region:

14. Fundamental matter (with small number of flavours) introduce probe branes e.g., D3/D7 system generalizes to Dp/D(p+4) (p+1)-dim. gauge theory with fundamental matter r, S3 ( embedded in S8–p ) Other systems: Dp/D(p+2) (p+1)-dim. gauge theory with fundamental matter on codim. 1 defect r, S2 ( embedded in S8–p )

15. Other systems: Dp/D(p+2) (p+1)-dim. gauge theory with fundamental matter on codim. 1 defect r, S2 ( embedded in S8–p ) Dp/Dp (p+1)-dim. gauge theory with fundamental matter on codim. 2 defect r, S1 ( embedded in S8–p ) (have focused on SUSY configurations for stability)

16. Fundamental fields: Decoupling limitof Nc Dp-branes with Nf Dq-branes Low-energy limit with U(Nc) adjoint Field theory: U(Nc) super-Yang-Mills coupled to Nfmassive hypermultiplets fund. in U(Nc) & global U(Nf) ( SUSY: N =4 → N =2 ) Gravity theory: Dp-throat containing Dq probe brane

17. Geometry: equator S8-p Dq Sn Free quarks appear with mass: pole

18. Geometry: equator S8-p Dq Sn Mesons, bound states of fundamental fields, dual to open string states supported by Dq-brane pole

19. (RCM & Thomson; Arean & Ramallo) Mesons: Calculation of masses for lowest lying mesons as for D3/D7 Veff effective radial potential discrete spectrum r Detailed results numerical as radial ODE has no analytic solution (except in D3 background) Spectra have universal features: discrete, deeply bound, mass gap ) ( with mass scale dictated by Holography!

20. Holography: (p+2)-dim. gravity (p+1)-dim. gauge theory radius energy (Horowitz & Polchinski) • gravity physics is local in radius gauge physics must be local in energy! Probe brane meson physics focused at r =L supergravity wave-packet

21. Holography: (Peet & Polchinski) String energy: Supergravity energy: Precisely matches meson energy scale with U=mq ! Locality in energy preserved between open and closed strings! Probe brane meson physics focused at r =L supergravity wave-packet

22. Mesons: Spectra have universal features: discrete, deeply bound, mass gap ) ( with mass scale dictated by Holography!

23. Holographic meson scale: • SUSY versus nonSUSY: SUSY embedding: no brane bending nonSUSY embedding: brane bends bare mass = constituent mass bare mass ≠ constituent mass (eg, Sakai & Sugimoto model) Holographic meson scale set by constituent mass! • Goldstone modes: symmetry breaking may require • massless modes

24. Conclusions/future directions: • holographic meson spectra have universal features: • discrete, deeply bound, mass gap • mass scale dictated by holography • meson interactions: scale as in large N

25. Meson interactions: Continue expansion of D7-brane action beyond 2nd order • substitute and integrate out r and S3 4th order: agrees with standard large N Applies for general holographic theory of course, for fixed : glueballs mesons

26. Conclusions/future directions: • holographic meson spectra have universal features: • discrete, deeply bound, mass gap • mass scale dictated by holography • meson interactions: scale as in large N • form factors: proportional to Mgap Hong, Yoon & Strassler (hep-th/0312071)

27. Hong, Yoon & Strassler (hep-th/0312071) Form factors in D3/D7: • exam mesons with various external probes: where • “charge radius” given by Mgap in particular, does NOT scale with quantum numbers ) (surprise since • similar results should apply in general holographic theory • (since Mgap sets scale of radial profile in all cases)

28. Observations/future directions: • holographic meson spectra have universal features: • discrete, deeply bound, mass gap • mass scale dictated by holography • meson interactions: scale as in large N • form factors: proportional to Mgap Hong, Yoon & Strassler (hep-th/0312071) • degeneracies broken for nonconformal backgrounds

29. Degeneracies in meson spectra: • for D3 background/conformal gauge theory with • D7/D5/D3 probe branes, spectra only depend on e.g., = radial AdS quantum # = angular quantum # on S3 = R-charge • NOT true for nonconformal backgrounds: <U> grows <U> shrinks geff grows with <U> geff decreases with <U> in accord with M ~ mq / geff

30. Observations/future directions: • holographic meson spectra have universal features: • discrete, deeply bound, mass gap • mass scale dictated by holography • meson interactions: scale as in large N • form factors: proportional to Mgap Hong, Yoon & Strassler (hep-th/0312071) • degeneracies broken for nonconformal backgrounds • beyond probe approximation; Nf /Nc finite (see: Burrington et al; Kirsch & Vaman; Casero, Nunez & Paredes)

31. Beyond probe approximation; Nf /Nc finite • intriguing hint from Erdmenger & Kirsch (hep-th/0408113): study D2/D6 with Nf /Nc finite in far infrared M2/M6 geometric singularity scaling of meson masses match probe results: M ~ mq / geff ~ mq / Nc • perhaps no surprise given demands of holography

32. Observations/future directions: • holographic meson spectra have universal features: • discrete, deeply bound, mass gap • mass scale dictated by holography • meson interactions: scale as in large N • form factors: proportional to Mgap Hong, Yoon & Strassler (hep-th/0312071) • degeneracies broken for nonconformal backgrounds • beyond probe approximation; Nf /Nc finite (see: Burrington et al; Kirsch & Vaman; Casero, Nunez & Paredes) • mesons with large spin

33. Mesons with large J (in D3/D7 system): Classical rotating open strings attached to D7-brane Nambu-Goto action: Ansatz: eom: bc: T (string ends to D7)

34. Spinning meson spectrum: solve E(J) numerically: nonrelativistic Coulomb bound states Regge behaviour applies in general with:

35. Observations/future directions: • holographic meson spectra have universal features: • discrete, deeply bound, mass gap • mass scale dictated by holography • meson interactions: scale as in large N • form factors: proportional to Mgap Hong, Yoon & Strassler (hep-th/0312071) • degeneracies broken for nonconformal backgrounds • beyond probe approximation; Nf /Nc finite (see: Burrington et al; Kirsch & Vaman; Casero, Nunez & Paredes) • mesons with large spin

36. Danielsson, Keski-Vakkuri & Kruczenski Holography: “shadows on the wall” AdS/CFT dictionary allows us to see “glue clouds” associated with quark/meson states eg, SUGRA dilaton is dual to ; so evaluate asymptotic dilaton sourced by particular (macroscopic) string configuration for “static quark” (infinite straight string): in agreement with the Wilson loop calculation for the potential between such external quarks Maldacena; Rey & Yee

37. Dynamical “Quarks”: easy to repeat calculations for finite mass quarks AdS boundary D7 horizon Static/infinite-mass quarks

38. Holography: for finite mass “quark” (straight string ending on D7): for large |x| for small |x| where ?

39. Holographic Mesons: A exotic hybrid states Profile A:

41. Holographic Mesons: A B Profile B:

43. Holographic Mesons: A C B Profile C:

46. Conclusions: Flavor physics is fun!