(ITEP, Moscow and JINR, Dubna) Stochastic solution of Schwinger-Dyson equations: an alternative to Diagrammatic Monte-Carlo [ArXiv:1009.4033, 1104.3459, 1011.2664] Pavel Buividovich (ITEP, Moscow and JINR, Dubna) Lattice 2011, Squaw Valley, USA, 13.07.2011

Motivation Look for alternatives to the standard Monte-Carlo to address the following problems: Sign problem (finite chemical potential, fermions etc.) Large-N extrapolation (AdS/CFT, AdS/QCD) SUSY on the lattice? Elimination of finite-volume effects Diagrammatic Methods

Motivation: Diagrammatic MC, Worm Algorithm, ... Standard Monte-Carlo: directly evaluate the path integral Diagrammatic Monte-Carlo: stochastically sum all the terms in the perturbative expansion

Motivation: Diagrammatic MC, Worm Algorithm, ... Worm Algorithm [Prokof’ev, Svistunov]: Directly sample Green functions, Dedicated simulations!!! Example: Ising model X, Y – head and tail of the worm Applications: Discrete symmetry groups a-la Ising [Prokof’ev, Svistunov] O(N)/CP(N) lattice theories [Wolff] – so far quite complicated

Difficulties with “worm’’ DiagMC Typical problems: Nonconvergence of perturbative expansion (non-compact variables) [Prokof’ev et al., 1006.4519] Explicit knowledge of the structure of perturbative series required (difficult for SU(N) see e.g. [Gattringer, 1104.2503]) Finite convergence radius for strong coupling Algorithm complexity grows with N Weak-coupling expansion (=lattice perturbation theory): complicated, volume-dependent...

DiagMC based on SD equations Basic idea: Schwinger-Dyson (SD) equations: infinite hierarchy of linear equations for correlators G(x1, …, xn) Solve SD equations: interpret them as steady-state equations for some random process Space of states: sequences of coordinates {x1, …, xn} Extension of the “worm” algorithm: multiple “heads” and “tails” but no “bodies” Main advantages: No truncation of SD equations required No explicit knowledge of perturbative series required Easy to take large-N limit

Example: SD equations in φ4 theory

SD equations for φ4 theory: stochastic interpretation Steady-state equations for Markov processes: Space of states: sequences of momenta {p1, …, pn} Possible transitions: Add pair of momenta {p, -p} at positions 1, A = 2 … n + 1 Add up three first momenta (merge) Start with {p, -p} Probability for new momenta:

Example: sunset diagram…

Normalizing the transition probabilities Problem: probability of “Add momenta” grows as (n+1), rescaling G(p1, … , pn) – does not help. Manifestation of series divergence!!! Solution: explicitly count diagram order m. Transition probabilities depend on m Extended state space: {p1, … , pn} and m – diagram order Field correlators: wm(p1, …, pn) – probability to encounter m-th order diagram with momenta {p1, …, pn} on external legs

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

Diagrammatic interpretation Histories between “Restarts”: unique Feynman diagrams Measurements of connected, 1PI, 2PI correlators are possible!!! In practice: label connected legs Kinematical factor for each diagram: qi are independent momenta, Qj – depend on qi Monte-Carlo integration over independent momenta

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 (~ ΛIRD)

Resummation Integral representation of Cn,m = Γ(n/2 + m + 1/2) x-(n-2) y-m: Pade-Borel resummation. Borel image of correlators!!! Poles of Borel image: exponentials in wn,m Pade approximants are unstable Poles can be found by fitting Special fitting procedure using SVD of Hankel matrices

Resummation: fits by multiple exponents

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

Test: triviality of φ4 theory Renormalized mass: Renormalized coupling: CPU time: several hrs/point (2GHz core) Compare [Wolff 1101.3452] Several core-months (!!!)

Conclusions: DiagMC from SD eq-s Advantages: Implicit construction of perturbation theory No critical slow-down Naturally treats divergent series Easy to take large-N limit [Buividovich 1009.4033] No truncation of SD eq-s Disadvantages: No “strong-coupling” expansions (so far?) Large statistics in IR region Requires some external resummation procedure Extensions? Spontaneous symmetry breaking (1/λ – terms???) Non-Abelian LGT: loop equations [Migdal, Makeenko, 1980] Strong-coupling expansion: seems quite easy Weak-coupling expansion: more adequate, but not easy Supersymmetry and M(atrix)-models

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