1. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 1 An approximate vacuum state of temporal-gauge Yang–Mills theory in 2+1 dimensions (in collaboration with Jeff Greensite) J. Greensite, ŠO, Phys. Rev. D 77 (2008) 065003, arXiv:0707.2860 [hep-lat]

2. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 2 Introduction Confinement is the property of the vacuum of quantized non-abelian gauge theories. In the hamiltonian formulation in D=d+1 dimensions and temporal gauge:

3. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 3 At large distance scales one expects: Halpern (1979), Greensite (1979) Greensite, Iwasaki (1989) Kawamura, Maeda, Sakamoto (1997) Karabali, Kim, Nair (1998) Property of dimensional reduction: Computation of a spacelike loop in d+1 dimensions reduces to the calculation of a Wilson loop in Yang-Mills theory in d Euclidean dimensions.

4. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 4 Strong-coupling lattice-gauge theory – systematic expansion: Greensite (1980) At weak couplings, one would like to similarly expand: For g ! 0 one has simply: Wheeler (1962)

5. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 5 A possibility to enforce gauge invariance: No handle on how to choose f’s.

6. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 6 Suggestion for an approximate vacuum wavefunctional

7. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 7 Support #1: Free-field limit (g ! 0)

8. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 8 Support #2: Zero-mode, strong-field limit D. Diakonov (private communication to JG) Let’s assume we keep only the zero-mode of the A-field, i.e. fields constant in space, varying in time. The lagrangian is and the hamiltonian operator The ground-state solution of the YM Schrödinger equation, up to 1/V corrections:

9. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 9 Now the proposed vacuum state coincides with this solution in the strong-field limit, assuming The covariant laplacian is then and the eigenvalues of M are

10. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 10 Then The argument can be extended also to 3+1 dimensions.

11. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 11 Support #3: Dimensional reduction and confinement What about confinement with such a vacuum state? Define “slow” and “fast” components using a mode-number cutoff: Then:

12. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 12 Effectively for “slow” components we then get the probability distribution of a 2D YM theory and can compute the string tension analytically (in lattice units): Non-zero value of m implies non-zero string tension  and confinement! Let’s revert the logic: to get  with the right scaling behavior ~ 1/  2, we need to choose

13. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 13 Why m 0 2 = - 0 + m 2 ? Samuel (1997)

14. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 14 Support #4: Non-zero m is energetically preferred Take m as a variational parameter and minimize with respect to m: Assuming the variation of K with A in the neighborhood of thermalized configurations is small, and neglecting therefore functional derivatives of K w.r.t. A one gets:

15. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 15 Abelian free-field limit: minimum at m 2 = 0 → 0.

16. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 16 Non-abelian case: Minimum at non-zero m 2 (~ 0.3), though a higher value (~ 0.5) would be required to get the right string tension. Could (and should) be improved!

17. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 17 Support #5: Calculation of the mass gap To extract the mass gap, one would like to compute in the probability distribution: Looks hopeless, K[A] is highly non-local, not even known for arbitrary fields. But if - after choosing a gauge - K[A] does not vary a lot among thermalized configurations … then something can be done.  Numerical simulation Numerical simulation

18. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 18 Numerical simulation of |  0 | 2 Define: Hypothesis: Iterative procedure:

19. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 19 Practical implementation: choose e.g. axial A 1 =0 gauge, change variables from A 2 to B. Then Spiral gauge 1. given A 2, set A 2 ’=A 2, 2. the probability P [A;K[A’]] is gaussian in B, diagonalize K[A’] and generate new B- field (set of Bs) stochastically; 3. from B, calculate A 2 in axial gauge, and compute everything of interest; 4. go back to the first step, repeat as many times as necessary. All this is done on a lattice. Of interest: Eigenspectrum of the adjoint covariant laplacian. Connected field-strength correlator, to get the mass gap: For comparison the same computed on 2D slices of 3D lattices generated by Monte Carlo.

20. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 20 Eigenspectrum of the adjoint covariant laplacian

21. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 21 Mass gap

22. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 22

23. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 23 Summary (of apparent pros) Our simple approximate form of the confining YM vacuum wavefunctional in 2+1 dimensions has the following properties: It is a solution of the YM Schrödinger equation in the weak-coupling limit … … and also in the zero-mode, strong-field limit. Dimensional reduction works: There is confinement (non-zero string tension) if the free mass parameter m is larger than 0. m > 0 seems energetically preferred. If the free parameter m is adjusted to give the correct string tension at the given coupling, then the correct value of the mass gap is also obtained.

24. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 24 Open questions (or contras?) Can one improve (systematically) our vacuum wavefunctional Ansatz? Can one make a more reliable variational estimate of m? Comparison to other proposals? Karabali, Kim, Nair (1998) Leigh, Minic, Yelnikov (2007) What about N-ality? Knowing the (approximate) ground state, can one construct an (approximate) flux-tube state, estimate its energy as a function of separation, and get the right value of the string tension? How to go to 3+1 dimensions? Much more challenging (Bianchi identity, numerical treatment very CPU time consuming). The zero-mode, strong-field limit argument valid (in certain approximation) also in D=3+1. Comparison to KKN N-ality Flux-tube state

25. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 25 Comparison to KKN The KKN string tension following from the above differs from string tensions obtained by standard MC methods, and the disagreement worsens with increasing .

26. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 26 N-ality Dimensional reduction form at large distances implies area law for large Wilson loops, but also Casimir scaling of higher- representation Wilson loops. How does Casimir scaling turn into N-ality dependence, how does color screening enter the game? A possibility: Necessity to introduce additional term(s), e.g. a gauge- invariant mass term Cornwall (2007) … but color screening may be contained! Strong-coupling: Greensite (1980)

27. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 27 Guo, Chen, Li (1994)

28. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 28

29. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 29 Flux-tube state The simplest guess: where (- D 2 ) is the covariant laplacian (on a time-slice) in the fundamental representation. The energy of such a state for a given quark-antiquark separation can be computed from:

30. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 30 On a lattice: Work is in progress. A more elaborate Ansatz, motivated by the gluon-chain model: Thorn (1979), Greensite (1985), Greensite, Thorn (2002)

31. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 31 Thank you for attention, and Axel and Christian for invitation to this beautiful place and workshop which has been such a refreshment in the hard life of a theorist!

32. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 32 Spiral gauge

33. 19.3.2008 415th Wilhelm und Else Heraeus Seminar: "Quarks and Hadrons in Strong QCD" St. Goar, Germany, March 17-20, 2008 33 Dimensions