1. Constructing worm-like algorithms from Schwinger- Dyson equations Pavel Buividovich (ITEP, Moscow and JINR, Dubna) Humboldt University + DESY, 07.11.2011

2. Lattice QCD is one of the main tools Lattice QCD is one of the main tools to study quark-gluon plasma to study quark-gluon plasma Interpretation of heavy-ion collision experiments: Interpretation of heavy-ion collision experiments: RHIC, LHC, FAIR,… RHIC, LHC, FAIR,… But: baryon density is finite in experiment !!! Dirac operator is not Hermitean anymore exp(-S) is complex!!! Sign problem Monte-Carlo methods are not applicable !!! Try to look for alternative numerical simulation strategies Motivation: Lattice QCD at finite baryon density

3. Lattice QCD at finite baryon density: some approaches Taylor expansion in powers of μ Taylor expansion in powers of μ Imaginary chemical potential Imaginary chemical potential SU(2) or G 2 gauge theories SU(2) or G 2 gauge theories Solution of truncated Schwinger-Dyson equations in a fixed gauge Solution of truncated Schwinger-Dyson equations in a fixed gauge Complex Langevin dynamics Complex Langevin dynamics Infinitely-strong coupling limit Infinitely-strong coupling limit Chiral Matrix models... Chiral Matrix models... “Reasonable” approximations with unknown errors, BUT No systematically improvable methods!

4. Quantum field theory: Path integrals: sum over paths vs. sum over fields Sum over fields Euclidean action: Sum over interacting paths Perturbative expansions

5. Worm Algorithm [Prokof’ev, Svistunov] Monte-Carlo sampling of closed vacuum diagrams: Monte-Carlo sampling of closed vacuum diagrams: nonlocal updates, closure constraint Worm Algorithm: sample closed diagrams + open diagram Worm Algorithm: sample closed diagrams + open diagram Local updates: open graphs closed graphs Local updates: open graphs closed graphs Direct sampling of field correlators (dedicated simulations) Direct sampling of field correlators (dedicated simulations) x, y – head and tail of the worm Correlator = probability distribution of head and tail Applications: systems with “simple” and convergent perturbative expansions (Ising, Hubbard, 2d fermions …) Applications: systems with “simple” and convergent perturbative expansions (Ising, Hubbard, 2d fermions …) Very fast and efficient algorithm!!! Very fast and efficient algorithm!!!

6. Worm algorithms for QCD? Attracted a lot of interest recently as a tool for QCD at finite density: Y. D. Mercado, H. G. Evertz, C. Gattringer, ArXiv:1102.3096 – Effective theory capturing center symmetry Y. D. Mercado, H. G. Evertz, C. Gattringer, ArXiv:1102.3096 – Effective theory capturing center symmetry P. de Forcrand, M. Fromm, ArXiv:0907.1915 – Infinitely strong coupling P. de Forcrand, M. Fromm, ArXiv:0907.1915 – Infinitely strong coupling P. de ForcrandM. Fromm P. de ForcrandM. Fromm W. Unger, P. de Forcrand, ArXiv:1107.1553 – Infinitely strong coupling, continuos time W. Unger, P. de Forcrand, ArXiv:1107.1553 – Infinitely strong coupling, continuos timeP. de ForcrandP. de Forcrand K. Miura et al., ArXiv:0907.4245 – Explicit strong-coupling series … K. Miura et al., ArXiv:0907.4245 – Explicit strong-coupling series …

7. Worm algorithms for QCD? Strong-coupling expansion for lattice gauge theory: confining strings [Wilson 1974] Strong-coupling expansion for lattice gauge theory: confining strings [Wilson 1974] Intuitively: basic d.o.f.’s in gauge theories = confining strings (also AdS/CFT etc.) Intuitively: basic d.o.f.’s in gauge theories = confining strings (also AdS/CFT etc.) Worm something like “tube” Worm something like “tube” BUT: complicated group-theoretical factors!!! Not known explicitlyStill no worm algorithm for non-Abelian LGT (Abelian version: [Korzec, Wolff’ 2010]) BUT: complicated group-theoretical factors!!! Not known explicitlyStill no worm algorithm for non-Abelian LGT (Abelian version: [Korzec, Wolff’ 2010])

8. Worm-like algorithms from Schwinger- Dyson equations Basic idea: Schwinger-Dyson (SD) equations: infinite hierarchy of linear equations for field correlators G(x 1, …, x n ) Schwinger-Dyson (SD) equations: infinite hierarchy of linear equations for field correlators G(x 1, …, x n ) Solve SD equations: interpret them as steady-state equations for some random process Solve SD equations: interpret them as steady-state equations for some random process G(x 1,..., x n ): ~ probability to obtain {x 1,..., x n } G(x 1,..., x n ): ~ probability to obtain {x 1,..., x n } (Like in Worm algorithm, but for all correlators)

9. Example: Schwinger-Dyson equations in φ 4 theory

10. Schwinger-Dyson equations for φ 4 theory: stochastic interpretation Steady-state equations for Markov processes: Steady-state equations for Markov processes: Space of states: Space of states: sequences of coordinates {x 1, …, x n } sequences of coordinates {x 1, …, x n } Possible transitions: Possible transitions:  Add pair of points {x, x} at random position at random position 1 … n + 1 1 … n + 1  Random walk for topmost coordinate coordinate  If three points meet – merge  Restart with two points {x, x} No truncation of SD equations No truncation of SD equations No explicit form of perturbative series No explicit form of perturbative series

11. Stochastic interpretation in momentum space Steady-state equations for Markov processes: Steady-state equations for Markov processes: Space of states: Space of states: sequences of momenta {p 1, …, p n } sequences of momenta {p 1, …, p n } Possible transitions: Possible transitions:  Add pair of momenta {p, -p} at positions 1, A = 2 … n + 1 at positions 1, A = 2 … n + 1  Add up three first momenta (merge) (merge) Restart with {p, -p} Restart with {p, -p} Probability for new momenta: Probability for new momenta:

12. Diagrammatic interpretation History of such a random process: unique Feynman diagram BUT: no need to remember intermediate states Measurements of connected, 1PI, 2PI correlators are possible!!! In practice: label connected legs Kinematical factor for each diagram: q i are independent momenta, Q j – depend on q i Monte-Carlo integration over independent momenta Monte-Carlo integration over independent momenta

13. Normalizing the transition probabilities Problem: probability of “Add momenta” grows as (n+1), rescaling G(p 1, …, p n ) – does not help. Problem: probability of “Add momenta” grows as (n+1), rescaling G(p 1, …, p n ) – does not help. Manifestation of series divergence!!! Manifestation of series divergence!!! Solution: explicitly count diagram order m. Transition probabilities depend on m Solution: explicitly count diagram order m. Transition probabilities depend on m Extended state space: {p 1, …, p n } and m – diagram order Extended state space: {p 1, …, p n } and m – diagram order Field correlators: Field correlators: w m (p 1, …, p n ) – probability to encounter m-th order diagram with momenta {p 1, …, p n } on external legs w m (p 1, …, p n ) – probability to encounter m-th order diagram with momenta {p 1, …, p n } on external legs

14. Normalizing the transition probabilities Finite transition probabilities: Finite transition probabilities: Factorial divergence of series is absorbed into the growth of C n,m !!! Factorial divergence of series is absorbed into the growth of C n,m !!! Probabilities (for optimal x, y): Probabilities (for optimal x, y):  Add momenta:  Sum up momenta + increase the order: increase the order: Otherwise restart Otherwise restart

15. Critical slowing down? Transition probabilities do not depend on bare mass or coupling!!! (Unlike in the standard MC) No free lunch: kinematical suppression of small-p region (~ Λ IR D )

16. Resummation Integral representation of C n,m = Γ(n/2 + m + 1/2) x -(n-2) y -m : Integral representation of C n,m = Γ(n/2 + m + 1/2) x -(n-2) y -m : Pade-Borel resummation. Borel image of correlators!!! Pade-Borel resummation. Borel image of correlators!!! Poles of Borel image: exponentials in w n,m Poles of Borel image: exponentials in w n,m Pade approximants are unstable Pade approximants are unstable Poles can be found by fitting Poles can be found by fitting Special fitting procedure using SVD of Hankel matrices Special fitting procedure using SVD of Hankel matrices No need for resummation at large N!!!

17. Resummation: fits by multiple exponents

18. Resummation: positions of poles Two-point function Connected truncated four-point function 2-3 poles can be extracted with reasonable accuracy

19. Test: triviality of φ 4 theory in D ≥ 4 Renormalized mass: Renormalized coupling: CPU time: several hrs/point (2GHz core) [Buividovich, ArXiv:1104.3459]

20. Large-N gauge theory in the Veneziano limit Gauge theory with the action Gauge theory with the action t-Hooft-Veneziano limit: t-Hooft-Veneziano limit: N -> ∞, N f -> ∞, λ fixed, N f /N fixed N -> ∞, N f -> ∞, λ fixed, N f /N fixed Only planar diagrams contribute! connection with strings Only planar diagrams contribute! connection with strings Factorization of Wilson loops W(C) = 1/N tr P exp(i ∫dx μ A μ ): Factorization of Wilson loops W(C) = 1/N tr P exp(i ∫dx μ A μ ): Better approximation for real QCD than pure large-N gauge theory: meson decays, deconfinement phase etc. Better approximation for real QCD than pure large-N gauge theory: meson decays, deconfinement phase etc.

21. Large-N gauge theory in the Veneziano limit Lattice action: Lattice action: No EK reduction in the large-N limit! Center symmetry broken by fermions. Naive Dirac fermions: N f is infinite, no need to care about doublers!!!. Basic observables: Basic observables:  Wilson loops = closed string amplitudes  Wilson lines with quarks at the ends = open string amplitudes Zigzag symmetry for QCD strings!!! Zigzag symmetry for QCD strings!!!

22. Migdal-Makeenko loop equations Loop equations in the closed string sector: Loop equations in the open string sector: Infinite hierarchy of quadratic equations! Markov-chain interpretation?

23. Loop equations illustrated Quadratic term

24. Nonlinear Random Processes [Buividovich, ArXiv:1009.4033] Also: Recursive Markov Chain [Etessami,Yannakakis, 2005] Let X be some discrete set Let X be some discrete set Consider stack of the elements of X Consider stack of the elements of X At each process step: At each process step:  Create: with probability P c (x) create new x and push it to stack  Evolve: with probability P e (x|y) replace y on the top of the stack with x  Merge: with probability P m (x|y 1,y 2 ) pop two elements y 1, y 2 from the stack and push x into the stack  Otherwise restart

25. Nonlinear Random Processes: Steady State and Propagation of Chaos Probability to find n elements x 1... x n in the stack: Probability to find n elements x 1... x n in the stack: W(x 1,..., x n ) W(x 1,..., x n ) Propagation of chaos [McKean, 1966] Propagation of chaos [McKean, 1966] ( = factorization at large-N [tHooft, Witten, 197x]): ( = factorization at large-N [tHooft, Witten, 197x]): W(x 1,..., x n ) = w 0 (x 1 ) w(x 2 )... w(x n ) W(x 1,..., x n ) = w 0 (x 1 ) w(x 2 )... w(x n ) Steady-state equation (sum over y, z): Steady-state equation (sum over y, z): w(x) = P c (x) + P e (x|y) w(y) + P m (x|y,z) w(y) w(z) w(x) = P c (x) + P e (x|y) w(y) + P m (x|y,z) w(y) w(z)

26. Loop equations: stochastic interpretation Stack of strings (= open or closed loops)! Wilson loop W[C] ~ Probabilty of generating loop C Possible transitions (closed string sector): Create new string Append links to string Join strings with links Join strings …if have collinear links Remove staples Probability ~ β Create open string Identical spin states

27. Loop equations: stochastic interpretation Stack of strings (= open or closed loops)! Possible transitions (open string sector): Truncate open string Probability ~ κ Close by adding link Probability ~ N f /N κ Close by removing link Probability ~ N f /Nκ Probability ~ N f /N κ Hopping expansion for fermions (~20 ord.) Strong-coupling expansion (series in β) for gauge fields (~ 5 orders) Strong-coupling expansion (series in β) for gauge fields (~ 5 orders) Disclaimer: this work is in progress, so the algorithm is far from optimal...

28. Sign problem revisited Different terms in loop equations have different signs Different terms in loop equations have different signs Configurations should be additionally reweighted Configurations should be additionally reweighted Each loop comes with a complex-valued phase Each loop comes with a complex-valued phase (+/-1 in pure gauge, exp(i π k/4) with Dirac fermions ) (+/-1 in pure gauge, exp(i π k/4) with Dirac fermions ) Sign problem is very mild (strong-coupling only)? Sign problem is very mild (strong-coupling only)? for 1x1 Wilson loops For large β (close to the continuum): For large β (close to the continuum): sign problem should be important sign problem should be important Large terms ~β sum up to ~1 Large terms ~β sum up to ~1

29. Temperature and chemical potential Finite temperature: strings on cylinder R ~1/T Finite temperature: strings on cylinder R ~1/T Winding strings = Polyakov loops ~ quark free energy Winding strings = Polyakov loops ~ quark free energy No way to create winding string in pure gauge theory No way to create winding string in pure gauge theory at large-NEK reduction at large-NEK reduction Veneziano limit: Veneziano limit: open strings wrap and close open strings wrap and close Chemical potential: Chemical potential: Strings oriented Strings oriented in the time direction in the time direction are favoured are favoured κ -> κ exp(+/- μ) No signs or phases!

30. Phase diagram of the theory: a sketch High temperature (small cylinder radius) OR Large chemical potential Numerous winding strings Nonzero Polyakov loop Deconfinement phase

31. Summary and outlook Diagrammatic Monte-Carlo and Worm algorithm: useful strategies complimentary to standard Monte-Carlo Stochastic interpretation of Schwinger-Dyson equations: a novel way to stochastically sum up perturbative series Advantages: Implicit construction of perturbation theory Implicit construction of perturbation theory No truncation of SD eq-s No truncation of SD eq-s Large-N limit is very easy Large-N limit is very easy Naturally treats divergent series No sign problem at μ≠0 No sign problem at μ≠0Disadvantages: Limited to the “very strong- coupling” expansion (so far?) Limited to the “very strong- coupling” expansion (so far?) Requires large statistics in IR region Requires large statistics in IR region QCD in terms of strings without explicit “stringy” action!!!

32. Summary and outlook Possible extensions: Weak-coupling theory: Wilson loops in momentum space? Combination with Renormalization-Group techniques? A study of large-N models of quantum gravity is feasible (IKKT model etc.) Extension to SU(3): 1/N expansion? Explicit resummation?

33. Thank you for your attention!!! References: ArXiv:1104.3459 (φ 4 theory) ArXiv:1104.3459 (φ 4 theory)1104.3459 ArXiv:1009.4033, 1011.2664 (large-N theories) ArXiv:1009.4033, 1011.2664 (large-N theories)1009.40331011.26641009.40331011.2664 Some sample codes are available at: Some sample codes are available at:http://www.lattice.itep.ru/~pbaivid/codes.html

34. Back-up slides

35. Some historical remarks “Genetic” algorithm vs. branching random process Probability to find some configuration of branches obeys nonlinear equation Steady state due to creation and merging Recursive Markov Chains [Etessami, Yannakakis, 2005] Also some modification of McKean-Vlasov-Kac models [McKean, Vlasov, Kac, 196x] “Extinction probability” obeys nonlinear equation [Galton, Watson, 1974] “Extinction of peerage” Attempts to solve QCD loop equations [Migdal, Marchesini, 1981] “Loop extinction”: No importance sampling