Diagrammatic MonteCarlo for nonAbelian field theories Pavel Buividovich

Diagrammatic Monte-Carlo for non-Abelian field theories? Pavel Buividovich (Regensburg University) DIMOCA workshop, September 2017, Mainz

Diagrammatic Monte-Carlo for dense QCD and sign problem So far lattice strong-coupling expansion: (leading order or few lowest orders) • Worldlines of quarks/mesons/baryons • Confining strings Very good approximation! Physical degrees of freedom! ü Phase diagram, tri-critical X Hadron spectrum, potentials X Bosonic part is most difficult …

Strong- vs weak-coupling expansions • Confinement • Dynamical mass gap generation ARE NATURAL for SC, BUT… Continuum physics is at weak-coupling! At N->∞, phase transition between strong and weak coupling, in any volume

In this talk So far, the strategy is often this Start with the action Sign problem? Find alternative representation (duality, Feynman rules, …) Use Monte-Carlo sampling on dual variables … so far Abelian, SU(2), Abelian color cycles… Both strong and weak-coupling expansions for non-Abelian SU(N) gauge theories (or sigmamodels) are so far difficult

In this talk • Anyway, to go to continuum we need high orders of strong- or weakcoupling expansions • How to generate complicated bosonic series automatically /by Monte-Carlo?

In this talk • Alternative Monte-Carlo approach based on Schwinger-Dyson equations • Unlike FUNctional methods, it is potentially exact • Best suitable for large-N limit • Avoids explicit duality transforms/action manipulations • Unifies expansion parameterization with sampling algorithm

Starter: finite-N matrix model Consider 0 D matrix field theory (finite-N matrix model) • Path integral unbounded (Monte-Carlo impossible!!!) • BUT exists in the framework of 1/N expansion • Leading order: planar diagrams (can be drawn on sphere), each vertex ~ λ • Higher-genus surfaces@1/N 2 g Can we devise e. g. worm algorithm for such diagrams?

Schwinger-Dyson equations Let’s write down Schwinger-Dyson equations Full set of observables: multi-trace correlators Factorized solution in the large-N limit:

Schwinger-Dyson equations • Diagrams with L legs and M vertices • Diagrams with L’<=L+2 legs and M-1 vertices • Recursive solution yields perturbative series

Stochastic solution of linear equations • Schwinger-Dyson equations can be always cast in linear form (although they might admit equivalent and simpler non-linear forms) • Formal series solution gives rise to perturbation theory • Let’s explicitly write the solution as • A(X | Y) matrix is very sparse Evaluate the sum over n and X 0 … Xn using Monte-Carlo!!!

Stochastic solution of linear equations Assume: A(X|Y), b(X) are positive, |eigenvalues| < 1 Solution using the Metropolis algorithm: Sample sequences {Xn, …, X 0} with the weight Basic transitions: • Add new index Xn+1 , • Remove index • Restart

Stochastic solution of linear equations • With probability p+: Add index step • With probability (1 -p+): Remove index/Restart Ergodicity: any sequence can be reached (unless A(X|Y) has some block-diagonal structure) Acceptance probabilities (no detailed balance, Metropolis-Hastings) • Parameter p+ can be tuned to reach optimal acceptance • Probability distribution of N(X) is crucial to asses convergence Finally: make histogram of the last element Xn in the sequence Solution φ(X) , normalization factor

Practical implementation • Keeping the whole sequence{Xn, …, X 0} in memory is not practical (size of X can be quite large) • Use the sparseness of A(X | Y) , remember the sequence of transitions Xn→ Xn+1 • Every transition is a summand in a symbolic representation of SD equations • Every transition is a “drawing” of some element of diagrammatic expansion (either weak- or strong-coupling one) Save: • current diagram • history of drawing Need DO and UNDO operations for every diagram element Construction of algorithms is almost automatic and can be nicely combined with symbolic calculus software (e. g. Mathematica)

Sign problem and reweighting • • Now lift the assumptions A(X | Y) > 0 , b(X)>0 Use the absolute value of weight for the Metropolis sampling • Sign of each configuration: • Define • Effectively, we solve the system • The expansion convergence has smaller radius of • Reweighting fails if the system approaches the critical point (one of eigenvalues approach 1) One can only be saved by a suitable reformulation of equations which makes the sign problem milder

Diag. MC for φ4 matrix model Basic operations of “diagram drawing”: • Insert line • Merge singlet operators • Create vertex

MC history of diagram order: ϕ 4 matrix model Large autocorrelation time and large fluctuations near the phase transition

Numerical results [From Marino, Schiappa, Weiss 0711. 1954]

Intermediate resume The algorithm is ideal for models involving planar (fixed genus) diagrams with positive weights: • Planar surfaces on the lattice (link with meson scattering amplitudes and Veneziano formula) [Makeenko, Olesen, Armoni, PB, …] • Planar φ4 theory

Strong-coupling expansion: U(N) matrix model • Diagram-counting interpretation not straightforward and not universal • Diagram weights non-positive, sign problem • Some kind of “sign blessing” also happens here

Approaching QCD: SU(N) principal chiral model Non-Abelian theory Asymptotic freedom Classical action is scale invariant Dynamical non-perturbative mass gap generation • Admits large-N limit • Perturbative series • •

Perturbative expansion it should be first-principle and automatic • • • Take N->∞ to reduce diagram space Small fluctuations of SU(N) fields Map SU(N) to Hermitian matrices Cayley map

“Perturbative” action Expand action and Jacobian in φ Infinitely many interaction vertices

SU(N) principal chiral model Power series in t’Hooft λ? Factorial growth even at large N due to IR renormalons … [Bali, Pineda] Can be sampled, but resummation difficult …Bare mass term ~λ from Jacobian? ? ? [a-la Fujikawa for axial anomaly] ü Massive planar fields ü Suitable for Diag. MC ? How to expand in λ? Count vertices !? ?

Counting powers of λ Consider a planar diagram: • f faces = loop momenta • v vertices ~ λ ~ m 02 • l bare propagators = lines • In planar limit, f – l + v = 2 • • Standard power counting Wk can only contain: Λ 2 UV, m 20, m 40/ Λ 2 UV , …. … Times probably logs !!!? ? ?

Minimal working example: 2 D O(N) sigma model @ large N Non-perturbative mass gap Jacobian reads Cayley map Again, bare mass term from the Jacobian… [PB, 1510. 06568]

O(N) sigma model @ large N Full action in new coordinates We blindly do perturbation theory [with A. Davody] Only cactus diagrams @ large N

Trans-series and Resurgence From our perturbative expansion we get Same for PCM!!! Resurgent trans-series [Écalle, 81] PT Zero modes Classical solutions [Argyres, Dunne, Unsal, …, 2011 -present]

O(N) sigma model @ large N Relative error of mass vs. order M Numerical evidence of convergence!!!

O(N) sigma model @ large N Relative error of mass vs. order M Numerical evidence of convergence!!!

O(N) sigma model @ large N Convergence rate: first extremum in ε(M) Has physical scaling!!! Similar to critical slowing -down in Monte-Carlo

Back to SU(N): Schwinger-Dyson eqs * * * Scalar field theory with infinitely many interaction vertices (momentumdependent) Some interaction vertices have negative weight reweighting + sign problem? Sign cancellations also in observables

Diag. MC for matrix field theory model Basic operations of “diagram drawing”: • Insert line Merge singlet operators • Create vertex • The two new momenta are {p, -p} • Probability distributions proportional to bare propagators

Sign problem at high orders * * * Mean sign decays exponentially with order Limits practical simulations to orders ~ 10 Sign problem depends on spacing, not volume

Restoration of SU(N)x. SU(N) symmetry * * * Perturbative vacuum not SU(N)x. SU(N) symm. Symmetry seems to be restored at high orders Restoration is rather slow

Mean link vs expansion order * * * Good agreement with N->∞ extrapolation Convergence slower than for standard PT MC Data from [Vicari, Rossi, Campostrini’ 94 -95]

Mean link vs expansion order * Wrong or no convergence after large-N phase transition (λ > λc = 3. 27 ) [hep-lat/9412102]

Long-distance correlators * * Constant values at large distances, consistent with <1/N tr gx> > 0 Converges slower than standart PT, but IR-finite

Finite temperature (phase) transition? Weak enhancement of correlations at L 0 ~ 35 -40 [P. B. , Valgushev, 1706. 08954]

Resume Weak-coupling Diag. MC in the large-N limit: + IR-finite, convergent series + Volume-independent algorithm - Sign problem vs. Standard MC Slower convergence than standard PT Starts with symmetry-breaking vacuum Finite-density matter: complex propagators

Outlook Resummation of logs: Easy in mean-field-approximation (for O(N) sigma-model just one exponent) Diag. MC with mean-field (Bold Diag. MC)? ? ? … Not easy in non-Abelian case … Matrix-valued Lagrange multipliers

Outlook Diag. MC based on strong-coupling expansion? + Sign problem really reduced + Volume-independent + Correct vacuum from the very beginning + Hadrons/mass gap/confinement are natural - No continuum extrapolation ? In practice, high-order SC expansion can work „reasonably“ well even in the scaling region …

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

Large-N gauge theory in the Veneziano limit • Lattice action: No EK reduction in the large-N limit! Center symmetry broken by fermions. Naive Dirac fermions: Nf is infinite, no need to care about doublers!!! (But axial anomaly physics might be subtle, mind the signature of γ 5) • Basic observables: § Wilson loops = closed string amplitudes § Wilson lines with quarks at the ends = open string amplitudes • Zigzag symmetry for QCD strings!!!

Migdal-Makeenko loop equations Loop equations in the closed string sector: Loop equations in the open string sector: Infinite hierarchy of quadratic equations! [Migdal, Makeenko, Eguchi, Kawai, …, 198 x]

Loop equations illustrated Quadratic term Iterations of loop equations generate hopping/strong-coupling expansion

Main difficulty • In gauge theory, Wilson loops have “zigzag symmetry” [Polyakov, hep-th/9711002] • Along with “useful loops”, we produce a lot of “trash loops” • Their cancellations are important for “straightforward” loop equations • Solution: loop equations in the space of irreducible loops, sort of gauge fixing • Equations become more complicated