Lecture 2- From WL to the string theory of QCD

1. Lecture 2- From WL to the string theory of QCD Natal July 2013 ,

2. Introduction • We have discussed the classical stringy description in holography of the confining Wilson line. • Next we need a stringy description of the WL in the full range of separation distances. In particular the string should admit a Coulomb quark anti-quark potential at short distances. • We therefore examine the full constraints on the holographic WL and check to what extent holographic models obey them • We then address the issue of the quantum WL string in flat space time and curved one. • We state what is known about the effective long string of QCD

3. Outline • 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 holgrphic WL

4. I. From classical holographic Wilson lines to the W.L of QCD

5. Toward the holographic Wilslon line of QCD • We have identified the conditions for a confining background. • We described the WL and other confining properties in several holographic models. • The WL of QCD admits confining behavior at long distances but at short distances is Coulombic. • Can we find in holography a model that admits both properties ?

6. The conditions for asymptotic Coulomb behavior • We have seen the general conditions on holographic models to admit confinement for large strings. • What are the conditions to have Coulomb behavior at short separation distance • We expand f(u) and g(u) around the boundary

7. The conditions for asymptotic Coulomb behavior • We assume here that f(u) is power-law divergent at the boundary and that g(u) asymptotes a constant. • For small separation l we find • We thus conclude that we need k=2, namely • Obviously this is obeyed by an asymptotic

8. Renormalization • Generically, the bare energy of the string in the holographic construction isdivergent • In the string picture it follows from the fact that the boundary is at radial infinity. • From the field theory it is due to the infinitely heavy quark anti-quark pair. • There are three different ways to renormalize the energy

9. Holographic renormalization • The idea behind this scheme is to expand the bare energy around the boundary, identify the infinite terms and subtract them from the original expression of the bare energy Now let us divide the integration into two parts such that f (u) ≫ f (u0) above um so we can simplify this expression

10. Holographic renormalization • The second term does not include boundary infinities . • We subtract the energy with

11. The Legendre transform scheme • We take that the renormalized energy is the Legendre transform • The subtracted part of the energy is • Changing x to u and substituting the momentum As above

12. The mass subtraction scheme • Simply subtract the bare mass of two free quarks, each of them is represented by a straight string stretched from the boundary to the horizon • Let separate the integral into two regions • where above um, g is approximately stabilized on its boundary value

13. The mass subtraction scheme • The subtracted energy • where A is a constant finite number that depends on the exact form of the horizon • In the non critical AdSn+1

14. The mass subtraction scheme

15. Wilson lines of holographic models • So far we have discussed the general conditions for confinement at long distances and Coulomb at short ones and the renormalization. • We now apply this analysis to few prototype models • We address Witten’s model for D3 and D4 branes, the non-critical Ads6 model and the Klebanovstrassler model. One can similarly discuss the hard and soft wall models and the improved HQCD models of Kiritsis et al

16. 1. CompactifiedDpbrane ( Witten’s model) • For Dpbrane background with one spatial coordinate compactified on S^1 we have • Thus for small distances • So only for p=3 we have 1/l behavior. However, for long distances it describes a 3d FT.

17. 2. The compactified non-critical Ads background • The non critical analog of the Dpbrane background with one spatial compact coordinate has the metric • This translates to • Thus the two conditions are obeyed • To have a 4d dual theory we can take n=5

18. 3.The Klebabovstrassler background • The sugra solution of the deformed conifold reads • The 6d deformed conifold metric is • where

19. The KlebanovStrassler background • The condition for confinement is obeyed for the minimum at t=0 • But asymptotically it does not follow our ansatz. However it is asymptotically Ads and indeed for small l it has a 1/l behavior with log correction

20. Comparison to lattice results: (i) measurement units • If a system is characterized by one scale ls then any physical quantity will be • If there are two scales then • If the two scales are infinitely separated than there should be certain choices of p1 and p2 such that Q is finite. • In our case we have

21. Comparison to lattice results: (i) measurement units • On comparing results between systems with different scales, one has to express them in terms of universal physical quantities • The energy of the WL and the separation distance l are our key players. In certain models they are expressed as

22. Comparison to lattice results: (i) measurement units • Instead for the comparison we will express them in terms of the physical quantities : glueball mass and string tension • In our favorite holographic models it reads

23. Comparison to lattice results: (i) measurement units • We compare the holographic WL to those derived in lattice simulations for SU(2) and SU(3) YM theories. • In lattice the length is related to the string tension • The lowest glue ball mass is related to the string tension as • So we can write the length as

24. Comparison to lattice results • We compare the full Wilson line behavior of lattice SU(2), SU(3) and the holographic models of compactified D3, D4 models, non-critical models and the KS model • We did it using two methods: (i) Fixing the zero energy point (ii) Comparing the confining asymptotic behavior.

25. Quark anti-quark potential- fixed zero energy point

26. Quark anti-quark potential – confining asymptotics

27. Comparison to lattice results • We compute the c2 values of the correspondence • Thus the Ads6 non-critical model and the Klebanovstrassler model are better.

28. The constant term of the Wilson line • So far we discussed the long s l and short 1/l behavior. What about the intermediate behavior? • In holography we may have in fact • The energy can always be shifted by a constant but is meaningful.

29. The constant term of the Wilson line • The values of the different holographic models in the zero energy fixing. • Thus all the models have • An interpretation can be of physical masses which are left after the subtraction

30. The constant term of the Wilson line • Let us check what is the optimal value from lattice. • We adjust the lattice WL to • And extract the value of CL • We use theoretical calculation at 0,1,2 loops of perturbation theory to see how much CL is changing when we go to higher order expansion.

31. The constant term of the Wilson line • It is subtle to determine Cs but we see that CL of the lattice has the same sign and is similar to the one of the non-critical model and KS

32. II. The quantum energy of the NG string

33. The quantum energy of the NG string • So far we have discussed classical strings in various holographic setups. • It is time to quantize the string • We start with a derivation of the energy of the quantum string in flat space time • We then discuss quantum fluctuations of classical holographic string

34. 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.

35. 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”

36. III. On the effective theory of long strings

37. The Nambu-Goldstone bosons • The effective theory of long string is characterized by the massless modes associated with the fact that the string breaks translational invariance and Lorentz invariance; • For a straight string moving in D space-time dimensions the symmetry breaking pattern2 is ISO(D − 1, 1) → ISO(1, 1)×SO(D−2). • This leads to (D−2) Nambu-Goldstone bosons (NGBs).

38. Schemes of the effective theory • Aharony and Komargodski analyzed the effective string theory using 3 approaches: • (i) Coordinate reparameterizationinvariant scheme • (ii) Manifestly unitary formalism in the static gauge • (iii) Orthogonal gauge, ws conformal invariant.

39. Reparameterization invariant formalism • The reparameterizationLagrangian is expressed in terms of the induced metric • And the second fundamental form • At weight=0 it is just the NG action • The deviations at weight=2 are

40. The effective theory of long strings • The main results can be stated in terms of deviations from the square root formula • The main result is that for closed strings there are no deviations from the formula before the order 1/L5 • The deviation at the order 1/L5 is model-independent, and proportional to (D − 26) • In is only the excited states that deviate at this order

41. The effective theory of long strings • While the deviations at order 1/L5 are universal, beyond this order the coefficients a(k)n are expected to be model dependent • For the first closed string excited states (with one left-moving and one right-moving excitation on the world-sheet) is given by

42. Summary of lecture 2- • We introduce the conditions for WL of QCD for any separation distance • We checked in confining backgrounds • We performed a comparison with Lattice • We discussed various renormalization schemes • We derive the exact quantum energy • On the effective theory of long string • On quantum fluctuations of holographic strings.

43. 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

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

45. Fermionic fluctuations • For the Green Schwarz superstring there are also fermionic fluctuations. • The fermionic part of the k gauged fixed GS action is WeylMajoranaspinor SO(9,1) gamma maticesi,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

46. Back to the fluctuations for a confining background • As a prototype we take the near extreemalAds5xs5 which in the limit of small radius is the dual to 3d pure YM theory. • The bosonic operators are .

47. 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

48. 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 extreemalcase due to the coupling to the RR flux the square of the fermionic operator is • Thus we get an attractive Luscher term

49. Summary of lecture 2- • We derived the conditions for a holographic WL of QCD • We described various renormalization schemes • We compared lattice simulation to holographic confining models. • We computed the exact quantum energy of a string in flat space time • We computed the contribution of Quantum Fluctuations to holographic WL