1. The effective action of the confining string Zohar Komargodski Weizmann Institute of Science, Israel and Institute for Advanced Study, NJ, USA Based on work (to be published) with: Ofer Aharony, Adam Schwimmer

2. 2 Outline 1)Motivations 2)The effective action of long confining strings and constraints from Lorentz invariance 3)Comparison with lattice results 4)Another prediction

3. 3 Motivations Strongly coupled gauge theories often have flux tubes. These are “fat” strings within which the color gauge potential lines are confined. It is very easy to imagine that the theory also has a mass gap so that the very low energy fluctuations are described by small deformations of this flux tube, without emitting particles into the bulk. We can define a local 1+1 dimensional effective action via an expansion in the energy of waves on the flux tube over the mass scale of the gap.

4. 4 As an example, we can consider pure Yang- Millls theory in 3 or 4 dimensions. This theory is confining and has flux tubes. The spectrum of the flux tubes has been simulated extensively on the lattice. Consequently, there is a lot of data to compare to and we can see how far we can get with the tools given by the effective action. Motivations

5. 5 Instead of a model-independent analysis of the effective action, one can look for string backgrounds, where the spectrum of the flux tube can be computed through the weakly coupled gravitational dual. To do this in YM theory we would need to understand highly curved stringy backgrounds. But important lessons can be learned from theories which do have weakly curved duals. Bottom-Up or Top-Down ?

6. 6 We will concentrate on the bottom-up approach, analyzing light excitations of the QCD string (for a long string). This is systematic, model-independent, but has the usual limitation of an effective-action approach. Bottom-Up or Top-Down ?

7. 7 What is the effective action on a long QCD string ? Like any other solitonic object, we find massless NGBs on the worldvolume. The broken symmetries are translation symmetries in the X i (i=2,…,D-1) (for string stretching along X 0,X 1 ). Also all the Lorentz generators in SO(D)/(SO(D-2)xSO(2)) are broken. XiXi X 0,X 1 L

8. 88 What is the effective action on a long QCD string ? The Nambu-Goldstone theorem is a little subtle, we only get “one set” of NGBs associated to translation symmetries X i (i=2,…,D-1). We do not need anymore NGBs because with a space dependent profile for the X i we can also realize small boosts. The reason is that Lorentz transf. and translation symmetries are both related to the E-M tensor. Assuming the absence of other massless worldsheet fields, the effective action is given by some L (X i ).

9. 9 This action must obey all the symmetries. Translations imply that it is only a function of d a X i, but it is further constrained by Lorentz invariance. Lorentz invariance is realized non-linearly: This puts strong constraints on the form of the effective action. Not any function of derivatives is allowed. What is the effective action on a long QCD string ?

10. In analogy to pion physics, we can think of X i as if they have dimension -1, and therefore dX i are dimensionless. Terms related by the nonlinear symmetry are of the same “scaling.” The analogue of the Wess-Zumino term in pion physics is some function of dX i One finds this function, like the WZ term, is unique 10 What is the effective action on a long QCD string ?

11. 11 What is the effective action on a long QCD string ? The scale T is the tension. The action is identified with the familiar NG action. It has “scaling” zero. Corrections have higher derivatives ddX i One can use this effective action to compute the spectrum of excitations of some long string. A closed form expression can be found

12. The first allowed correction to the NG action This must be accompanied by terms with the same scaling but more fields to maintain nonlinear Lorentz invariance. One can check how this contributes to the effective action and predict the following: 12 What is the effective action on a long QCD string ?

13. 13 The deviations from Nambu-Goto generically start at order 1/L 5, while for the ground state there is a cancelation in the partition function and it first deviates only at order 1/L 7. For D=3 the operator we wrote is zero, and one can check that the first nontrivial operator leads to modifications from Nambu-Goto at order 1/L 7. What is the effective action on a long QCD string ?

14. 14 Some lattice results The best lattice results for pure Yang-Mills theory in 2+1 dimensions are (for gauge group SU(6)) : (Athenodorou, Bringoltz, Teper)

15. The NG prediction fits the lattice results fantastically well. At first sight this is puzzling because we do not expect NG to be the full theory of the flux tube. Our analysis explains this “experimental” fact neatly: the first deviations are expected to be too small for the current precision available. Soon to be observed ? Can we predict the next correction to NG or is it model-dependent? 15 Some lattice results

16. 16 Examples from String Theory Many examples of holographic confining backgrounds are weakly curved and weakly coupled in some limit (Witten,MN,KS). The confining string sits in the IR; expanding its action (in the Green-Schwarz formalism) in static gauge, we find that some of the bosonic fields and all unprotected fermionic fields are massive. At small curvature the theory is weakly coupled, so we can integrate out these massive fields at one- loop, and obtain corrections to Nambu-Goto.

17. 17 We compute these corrections by looking at scattering amplitudes of the massless X i fields on the worldsheet. Many diagrams appear (Aharony&Karzbrun) At one-loop, all these backgrounds have the same action, just with different values for the boson and fermion masses. Thus, a single computation captures the corrections in all known weakly coupled holographic confining backgrounds. Examples from String Theory

18. 18 The first deviation from Nambu-Goto is found in d 6 X 4 terms, as expected based on nonlinearly realized Lorentz symmetry. One finds that the coefficient is independent of any of the details of high energy physics. It is given by c 4 =(26-D)/(192  ). The structure of this coefficient is reminiscent of anomalies. In pion physics one correction is the WZW term, and it is very robust – depends on anomalies only. Perhaps the FIRST correction to NG is such a universal object. Examples from String Theory

19. 19 Comments on a different gauge We have looked at the problem from the point of view of static gauge. But there should be some diff-invariant description. We imagine fixing all the diffs but the conformal transformations. Then the action is 1+1 conformal field theory for all and the leading action is

20. The relation with static gauge arises via the BRST conditions – one has to impose that all the with annihilate physical states and identify states modulo null states. This kills two degrees of freedom and gives a positive definite Hilbert space, isomorphic to static gauge. But we know that unless this BRST problem does not eliminate all the ghosts! On the other hand, flux tubes at definitely exist. 20 Comments on a different gauge

21. As Polchinski+Strominger (1991) found, even with just 4 bosons the central charge can be fixed to c=26 by adding a term singular for But since we are interested in expanding around long strings, such a singularity is not devastating. (Of course, as a theory of “fundamental” strings, this is meaningless.) 21 Comments on a different gauge

22. 22 Comparison of Static Gauge and Conformal Gauge This operator that we have to add in conformal gauge is very similar to the one in static gauge In fact they coincide if one naively goes from static to conformal gauge. So this strongly suggest that indeed the leading correction to NG is universal and should be observed soon.

23. However to really show the equivalence one has to quantize the theories to high order in. This is still not done (in progress) but the first steps have already been taken by Drummond. Also based on some intuition from string theory one may worry that perhaps the conformal-gauge theory may not be even local so the comparison is not always legitimate. Thus, for now, the claim that there is a universal correction to NG is just a plausible conjecture 23 Comparison of Static Gauge and Conformal Gauge

24. 24 Nonlinear Lorentz invariance places strong constraints on the effective action of flux tubes, with the leading deviation at six- or eight-derivative order (depending on whether the string is in 3 or more dimensions). Measuring such deviations on the lattice is challenging but interesting. Lattice precision studies so far revealed no statistically significant deviations from NG, and this is neatly explained by non-linear Lorentz invariance. We conjecture that the leading deviation from NG is universal (depending only on the number of dimensions via ). Conclusions

25. 25 Establish firmly the existence of this universal correction to NG and to understand whether it is related to some anomaly in static gauge. Additional massless fields on the worldsheet (e.g. confining strings in supersymmetric gauge theories; interesting field theoretic realizations exist, e.g. Shifman&Yung and others.) Open strings – what boundary terms are allowed in the effective action (in general and in holographic backgrounds) ? See Aharony&Field. More Open Questions