Holographic stringy hadrons

1. O. Aharony, O. Bergman, V. Kaplunovsky , S. Yankielowicz, B. Burrington, A. Dymarsky,G. Harpaz, N. Katz, U. Kol, Y. Kinar, S. Kuperstein, M. Kruczenski, D. Melnikov, O. Mintakevich, L. Pando-Zayas, K. Petters,U. Schreiber, S. Seki, D. Vaman, M. Zamaklar Cobi Sonnenschein Tel Aviv university Natal August 2013 Holographic stringy hadrons

2. Lecture 1- Confining gravitational backgrounds • Confining Wilson loop • Witten’s model • Screening ‘t Hooft loop • Holographic Entanglement entropy and confinement • Glueballsepctra

3. Lecture 2- From WL to the string theory of QCD • From classical holographic Wilson lines to the W.L of QCD • The quantum energy of the NG string • On the effective action of long strings • Quantum correction of the hologrphic WL

4. Lecture 3- Fundamental quarks and mesons • Adding probe branes • Chiral symmetry • The Sakai Sugimoto model • Mesons and the fluctuations of probe branes • The spectrum of vector and scalar mesons • Quark masses and the GOR relation

5. Lecture 4-Stringy holographic mesons • Stringy holographic mesons versus fluctuations of flavor branes • From classical holographic rotating string to rotating string with massive endpoints • Regge trajectories revisited • CNN, the lund model and the decay of stringy mesons

6. Lecture 5- From string baryons to flavor instantons • The basic concept of holographic baryon • Stringy baryons • Does the baryonic vertex have a trace in data • The stability of stringy baryons, simulation • The Regge trajectories revisited • Confining background the Sakai Sugimoto model • Baryons as flavor gauge instantons • Baryons in genrealized Sakai Sugimoto model

7. Lecture 6- Holographic Nuclear matter, instanton chains and popcorn • A brief review of static properties of Baryons • Nuclear interaction: repulsion and attraction • The DKS model and the binding energy puzzle • Chains of baryons-generalities • The 1d and 3d toy models of point charges. • Exact ADHM 1d chain of instantons • The two instanton approximation • Phase transitions between lattice structures • The phase diagram of QCD at large Nc

8. Lecture 1- Confining gravitational backgrounds ,

9. The modern view of the string theory of QCD of string hadrons is based on two novel concepts • D branes • Maldacena correspondence between string theory on AdS5xS5 and N=4 SYM in 4d.

10. In reality color gauge dynamics is confining at the strong coupling limit and not conformal • There is zero supersymmetries • Our goal is thus to pave the road from the duality of the conformal and maximally supersymmetric theory to string( gravity)/gauge duality for a theory as close as possible to the pure YM theory and QCD and in particular to hadron physics

11. Confinement in gauge dynamics • It is well known that quarks and gluons are confined and hence not asymptotic states. • There are several manifestations of confinement already in the pure glue theory: • The Wilson loop has an area law behavior • The ‘t Hooft loop admits a screening nature • The physical states are colorless glueballs with a discrete spectrum and a mass gap • Hence the free energy does not scale with Nc

12. When temperature is introduced one finds that at low temperature the theory is in a confining phase whereas at high temperature the theory is in a deconfining phase • The deconfining phase is characterized by: • Wilson line which scales with the perimeter • The spectrum is continuous • The free energy behaves like Nc^2 • There is a non trivial expectation to the Polyakov loop

13. Confining Wilson Loop • In SU(N) gauge theories one defines the following gauge invariant operator where C is some contour • The quark – antiquark potential can be extracted from a strip Wilson line • The signal for confinement is E ~ Tst L

14. Stringy Wilson loop • The natural stringy gravity dual of the Wilson line ( which obeys the loop equation) is where is the renormalized Nambu Goto action, namely the renormalized world sheet area

15. The basic set up is a d dimensional space-time with the metric where x|| are p space coordinates on a Dp brane and s and xT are radial coordinate and transverse directions. • The corresponding Nambu Goto action is • Upon using the gauge the NG action is where

16. The Hamiltonian equation of motion is • The separation distance between the string endpoints ( the quark antiquark)

17. The NG action diverges. It is renormalized by • So that the remormalized quark antiquark potential is

21. Sufficient conditions for confinement • We thus conclude that the a sufficient condition for confinement is if either

24. Witten’s model-a prototype of confining model • We have seen that a way to get a confining background is to cut the radial direction and introduce a scale. • One approach is indeed to cut by hand an Ads space. This is not a solution of the SUGRA equations of motion. People use it to examine phenomenological properties (AdsQCD) • The apparoch of Witten was to compactify one coordinate and find a “cigar-like” solution.

25. One imposes anti-periodic boundary conditions on fermions. This kills supersymmetry. • In the dual gauge theory the gauginos and the scalars acquire a mass ~T and hence in the large T limit they decouple and we are left only with the gauge fields. • Since in large T b goes to zero so that we loose one space dimension and we end up with a pure gauge theory in p-1 space dimensions. • The gravity theory associated with D3 branes namely the Ads5xs5 case compactified on a circle is dual to pure YM theory in 3d • The same mechanism for D4 branes yields a dual theory of Pure YM in 4d.

27. The gauge theory and sugra parametrs are related via • 5d coupling 4d coupling glueball mass • String tension • The gravity picture is valid only provided that l5 >> R • At energies E<< 1/R the theory is effectively 4d. • However it is not really QCD since Mgb ~ MKK • In the opposite limit of l5 << R we approach QCD

28. Quantum fluctuations • Introduce quantum fluctuations around the classical configuration • Consequently the quantum corrected Wilson loop reads where are the fluctuations left after gauge fixing. • The corresponding correction to the free energy is

29. Wilson loop in flat space-time • Consider the bosonic action of a string in flat space-time with boundary conditions fixed at • The static NG action reads • The classical quark antiquark potential is • Thus this Wilson loop associates with linear confinement

30. Bosonic quantum fluctuations • The action of the bosonic fluctuations is • The Eigenvalues of the are • The free energy is given by • Zeta function regularization yields the following potential • This is the famous Luscher term

31. Fermionic fluctuations • For the Green Schwarz superstring there are also fermionic fluctuations. • The fermionic part of the k gauged fixed GS action is Weyl Majorana spinor SO(9,1) gamma matices i,j=1,2 • Thus the fermionic operator is and squaring it gives • The total B and F contribution to the free energy is • No QM correction and vanishing Luscher term

32. Can the WL be evaluated exactly? • The space-time energy of a bosonic string is • Hence for the lowest tachyonic state L0=0 we get • The energy is therefore • When expanded we recover the linear and Luscher terms

33. Comparing the exact bosonic string picture to lattice calculations • The quark antiquark potential for SU(3) and SU(6) gauge theories in 2+1 dimensions were extracted in lattice calculations.

34. The energies of the lowest 7 states as a function of L In comparison to the NG predictions. Lessons from the comparison (i) The NG string fits nicely (ii) The string “resides” in “flat space time”

35. One imposes anti-periodic boundary conditions on fermions. This kills supersymmetry. • In the dual gauge theory the gauginos and the scalars acquire a mass ~T and hence in the large T limit they decouple and we are left only with the gauge fields. • Since in large T b goes to zero so that we loose one space dimension and we end up with a pure gauge theory in p-1 space dimensions. • The gravity theory associated with D3 branes namely the Ads5xs5 case compactified on a circle is dual to pure YM theory in 3d • The same mechanism for D4 branes yields a dual theory of Pure YM in 4d.

36. Back to the fluctuations for a confining background • As a prototype we take the near extremal Ads5xs5 which in the limit of small radius is the dual to 3d pure YM theory. • The bosonic operators are .

37. The operators of the transverse fluctuations correspond to massless modes but the longitudinal normal mode is a massive mode. So altogether there are 7 bosonic massless modes

38. Had the fermionic modes been those of flat space time then the total coefficient in front of the Luscher term whould have been +8-7=+1. • This means a repulsive “Culomb” like potential.This contradicts gauge dynamics. • However, in the near extremal case due to the coupling to the RR flux the square of the fermionic operator is • Thus we get an attarctive Luscher term

39. For Wilson lines of distances which are much larger than the radius of the S^5 than the fluctuations of the latter coordinates are massive and will not contribute to the lowest level of the Luscher term. • In this case we will have only 2 transverse direction that contribute to the Luscher term

40. Does the confining WL behave like a QCD WL? Yes! Effectively it behaves like a string in flat space time. This is due to the |_| shape and the cancelation of the vertical segments. A more precise scalling argument verifies this confining

41. The ‘t Hooft loop –monopole anti-monopole potential • It is well known that in a theory were the quarks are confined, the monopoles are screened ( and wise versa). • Just as a Wilson loop with a fundamental string describes the quark anti-quark pair, the hodge dual of a string namely a D1 brane is the stringy description of a ‘t Hooft loop • In Witten’s model ( or its non-critical analog) we are in type IIA so the role of the D1 brane is played by a D2 brane that wraps the compactified cycle.

42. The action of a D2 brane is the DBI action. The worldvolume of it is along (t,x,t). • Similar to the Wilson line we solve the e.o.m • The separation distance is

43. Again we have to substruct the masses of the monopole pair, namely of D2 brane that strech from the boundary to the horizon • The renormalized energy is • Since this energy is positive it means that the configuration of two parallel D2 branes is favorable and hence the system is screened

44. Entanglment entropy and confining backgrounds • Consider a d + 1 dimensional quantum field theory on R^(d+1) in its vacuum state |0>. • Divide the d dimensional space into two complementary regions Where Il is a segment of length l • The entanglement entropy between the regions A and B is defined as the entropy seen by an observer in A who does not have access to the degrees of freedom in B, or vice versa

45. It can be calculated by tracing the density matrix of the vacuum, ρ0 = |0><0|, over the degrees of freedom in B and forming the reduced density matrix The quantum entanglement entropy is It is has an l independent UV divergence

46. In d+1 QFT the finite dependent part of the EE is negative and behaves like • A natural question is how can one calculate the EE in holography • A simple geometrical method was proposed by Ryu and Takanayagi: for a d+1 dim CFT with AdSd+2 gravity dual find the minimal area d-dimensional surface γ in AdSd+1 such that the boundary of γ coincides with the boundary of A ,

47. For the case of 2d CFT one has on Ads3

48. In non-conformal theories, the volume of the 8 −d compact dimensions and the dilaton are in general not constant. • A natural generalization of the holographic EE to the corresponding 10d dimensional geometries • The EE is found by minimizing the action over all surfaces with a boundary which is that of A. • The holographic computation serves two goals: (i) Computing the EE for strongly coupled theories (ii) Provide a criterion for confining background

49. There are multiple of local minima for the action for any given l • One of them is a pair of disconnected cigars extended in R^(d-1) and separated by l • A second one is a connected surface, in which the two cigars are connected by a tube whose width depends on l. • The EE is identified with the absolute minimum. • For the disconnected case SA does not depend on l

50. For the connected case for l<lcrit it is smaller than the disconnected so it dominates. It is of order N^2 • For l>lcrit the disconnected wins and it behaves line N^0 • Thus we can view this a phase transition which is characteristic to large N confining theories. • We demonstrate this in the case of Witten’s model