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

2. 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!
Phase diagram, tri-critical
X Hadron spectrum, potentials
X Bosonic part is most difficult …

3. 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

4. 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 sigma-models) are so far difficult

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

6. 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

7. Starter: finite-N matrix model
Consider 0D 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
Can we devise e.g. worm algorithm for such diagrams?

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

9. 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

10. 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 X0 … Xn using Monte-Carlo!!!

11. Solution using the Metropolis algorithm:
Stochastic solution of linear equations Assume: A(X|Y), b(X) are positive, |eigenvalues| < 1
Solution using the Metropolis algorithm:
Sample sequences {Xn, …, X0} with the weight
Basic transitions:
Add new index Xn+1 ,
Remove index
Restart

12. 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

13. Practical implementation
Keeping the whole sequence{Xn, …, X0} 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)

14. 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 has smaller radius of convergence
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

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

16. MC history of diagram order: ϕ4 matrix model
Large autocorrelation time and large fluctuations near the phase transition

17. Numerical results
[From Marino,Schiappa,Weiss ]

18. 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

19. 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

20. SU(N) principal chiral model
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

21. 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

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

23. 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]
Massive planar fields
Suitable for DiagMC
? How to expand in λ?
Count vertices !??

24. Counting powers of λ l bare propagators = lines
Consider a planar diagram:
f faces = loop momenta
v vertices ~ λ ~ m02
l bare propagators = lines
In planar limit, f – l + v = 2
Standard power counting
Wk can only contain:
Λ2UV, m20, m40/ Λ2UV , ….
… Times probably logs !!!???

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

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

27. 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]

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

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

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

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

32. DiagMC 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

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

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

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

36. Mean link vs expansion order
Wrong or no convergence after large-N phase transition (λ > λc = 3.27 ) [hep-lat/ ]

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

38. Finite temperature (phase) transition?
Weak enhancement of correlations at L0 ~ [P.B.,Valgushev, ]

39. Resume - Sign problem vs. Standard MC
Weak-coupling DiagMC 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

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

41. Outlook - No continuum extrapolation + Sign problem really reduced
DiagMC 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 …

42. 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.

43. 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 γ5)
Basic observables:
Wilson loops = closed string amplitudes
Wilson lines with quarks at the ends = open string amplitudes
Zigzag symmetry for QCD strings!!!

44. Infinite hierarchy of quadratic equations!
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,…, 198x]

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

46. Main difficulty In gauge theory, Wilson loops have
“zigzag symmetry” [Polyakov, hep-th/ ]
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