Numerical Stochastic Perturbation Theory (NSPT) Francesco Di Renzo - PowerPoint PPT Presentation

Numerical Stochastic Perturbation Theory (NSPT) Francesco Di Renzo Giuseppe Marchesini Memorial Conference GGI Firenze, 19-05-2017

A (by now) quite long story started with Pino and Enrico as supervisors In early 90’s Parma was provided an APE100 prototype (tubo). Only a few Gflops computing power, but something very intriguing at that time (well… by now a piece of cake…) QUESTION: what to do? From my point of view: I was starting my PhD Pino and Enrico were very keen on numerically implementing Stochastic Perturbation Theory. From my point of view: the start of a career in research, under the supervision of people who have always been firm reference points.

An invitation Perturbation Theory (PT) is nothing less than ubiquitous in Field Theory. In principle the lattice is a regulator among the others ... in practice it is a dreadful one so that when it comes to compute something in Lattice Perturbation Theory (LPT) you will probably start to get nervous ... In particular for LGT: lot of vertices (not given once and for all) Sums and/or integrals ... a lot of trigonometrics ... A variety of actions (both for glue and for quarks) and as an extra bonus ... often bad convergence properties r A � p , q � �� gT A � i � � cos � 2 � � r sin � 2 � � , p � � q � p � � q � 2 � A � p , q � �� gT A c SW V c 1 � � � � V 1 � � � cos � 2 � sin � p � � q � � , p � � q � 4 � � � � � 4 cos � 2 � cos � 2 � cos � 2 � a r k 1 � k 2 � q � � p � 2 g 2 1 a AB � p , q , k 1 , k 2 � �� 2 g 2 if ABC T C c SW AB � p , q � � V c 2 � � 2 � T A , T B � � � � V 2 � � � cos � 2 � � 2 cos � 2 � cos � 2 � � � � � � � � � � sin � 2 � � sin � k 2 � � � sin � k 1 � �� � , q � � p � k 1 � k 2 � q � � p � � � i � � sin � 2 � � r cos � 2 � � , p � � q � p � � q � � 1 � p � a 2 � � q a 2 6 g 3 1 ABC � p , q � � 6 � T A � T B , T C � � T B � T C , T A � V 3 � � � � T C � T A , T B � � � � � � �� � i � � cos � 2 � p � � q � � r sin � 2 � � , p � � q � ABC � p , q , k 1 , k 2 , k 3 � �� 3 ig 3 a 2 6 c SW r � T A T B T C � � � � �� � i � � � � � 6 cos � 2 � sin � q � � p � � 1 q � � p � V c 3 � � � � � cos � 2 � cos � 2 � cos � 2 � sin � 2 � � � q � � p � q � � p � k 3 � � k 1 � k 2 � 1 2 � T A T B T C � T C T B T A � i � � � � � � � 2 cos � 2 � cos � 2 � cos � � sin � 2 � q � � p � q � � p � k 3 � � k 2 � k 1 � 2 � � � � sin � 2 � cos � 2 � k 2 � � � � �� sin � � cos � 2 � cos � 2 � , k 3 � � k 2 � k 1 � k 1 � � 2 k 2 � � k 3 � q � � p � k 3 � � k 1 � FIG. 1. Momentum assignments for the quark-antiquark-gluon 2 vertices.

Despite this ...

Agenda - Basics of Stochastic Quantization and Stochastic Perturbation Theory - From Stochastic Perturbation Theory to NSPT (moving straight to LGT) - Stochastic Gauge Fixing - Fermionic loops in NSPT - A few different frameworks for NSPT (i.e. a few handles to possibly improve it) - A canonical application: renormalization constants - Something maybe more field-theoretic (numerics stumbles on fundamental QFT…) a. Renormalons b. Resurgence? (*) Of course I had to make a selection, with Pino on my mind - Conclusions

Basics of Stochastic Quantization and Stochastic Perturbation Theory

Basics of Stochastic Quantization and Stochastic Perturbation Theory D φ O [ φ ] e − S [ φ ] R You start with a field theory you want to solve h O [ φ ] i = R D φ e − S [ φ ] Parisi-Wu, Sci. Sinica 24 (1981) 35, Damgaard-Huffel, Phys Rept 152 (1987) 227 You now want an extra degree of freedom which you will think of as a stochastic time in which an evolution takes place according to the Langevin equation d φ η ( x ; t ) ∂ S [ φ ] = − ∂φ η ( x ; t ) + η ( x ; t ) φ ( x ) 7! φ η ( x ; t ) dt The drift term is given by the equations of motion... ... but beware! This is a stochastic differential equation due to the presence of the gaussian noise h η ( x, t ) η ( x 0 , t 0 ) i η = 2 δ ( x � x 0 ) δ ( t � t 0 ) η ( x ; t ) : D η ( z, τ ) . . . e − 1 dzd τη 2 ( z, τ ) R R 4 Noise expectation values are now naturally defined h . . . i η = D η ( z, τ ) e − 1 R dzd τη 2 ( z, τ ) R 4 The key assertion of Stochastic Quantization can be now simply stated h O [ φ η ( x 1 ; t ) . . . φ η ( x n ; t )] i η ! t →∞ h O [ φ ( x 1 ) . . . φ ( x n )] i

A conceptually simple proof comes from the Fokker Planck equation formalism D η O [ φ η ( t )] e − 1 dzd τη 2 ( z, τ ) R R Z 4 h O [ φ η ( t )] i η = = D φ O [ φ ] P [ φ , t ] D η e − 1 R dzd τη 2 ( z, τ ) R 4 ✓ δ S [ φ ] ◆ Z δ δ ˙ P [ φ , t ] = δφ ( x ) + P [ φ , t ] dx δφ ( x ) δφ ( x ) Floratos-Iliopoulos, Nucl.Phys. B 214 (1983) 392 for the solution of which we can introduce a perturbative expansion which generates a hierarchy of equations X g k P k [ φ , t ] P [ φ , t ] = k =0 Leading order is easy to solve and admits an infinite time (equilibrium) limit such that 0 [ φ ] = e − S 0 [ φ ] P 0 [ φ , t ] → t →∞ P eq Z 0 P k [ φ , t ] → t →∞ P eq In a convenient weak sense at every order one gets equilibrium k [ φ ] in terms of quantities which are interelated by a set of relations in which one recognizes the Schwinger-Dyson equations ... i.e. we are done! We want to go via another expansion, i.e. the expansion of the solution of Langevin equation in power of the coupling constant Parisi-Wu, Damgaard-Huffel φ η ( x ; t ) = φ (0) X g n φ ( n ) η ( x ; t ) + η ( x ; t ) n> 0

∂ η ( k, t ) = − ( k 2 + m 2 ) φ (0) ∂ t φ (0) Langevin equation for the free scalar field (momentum space) η ( k, t ) + η ( k, t ) Z t ∂ ∂ tG (0) ( k, t ) = − ( k 2 + m 2 ) G (0) ( k, t ) + δ ( t ) φ ( k, t ) = d τ G ( k, t − τ ) η ( k, τ ) Look for (propagator) 0 G (0) ( k, t ) = θ ( t ) exp ( − ( k 2 + m 2 ) t ) i.e. Z t φ (0) ( k, t ) = φ (0) ( k, 0) exp ( − ( k 2 + m 2 ) t ) + d τ exp ( − ( k 2 + m 2 )( t − τ )) η ( k, τ ) 0 Interacting case (cubic interaction in the following) is solved by superposition ... Z t  � η ( k, τ ) − λ Z dpdq d τ exp − ( k 2 + m 2 )( t − τ ) φ ( k, t ) = (2 π ) 2 n φ ( p, τ ) φ ( q, τ ) δ ( k − p − q ) 2! 0 ... which leaves the solution in a form which is ready for iteration. It is actually also ready for a graphical intepretation and for the formulation of a diagrammatic Stochastic Perturbation Theory Z G η − λ Z Z Z Z φ = G ( G η )( G η ) + . . . x + + + + 3! ‘stochastic diagrams’, The stochastic diagrams one obtains when averaging over the noise (contractions!) reconstruct, in a convenient infinite time + + limit, the contributions of the (topologically) ~ Q + Q correspondent Feynman diagrams ... . )< + (3.42) but we do not want to go this way ... x +

From Stochastic Perturbation Theory to NSPT (directly for LGT…)

Stochastic Quantization for LGT Batrouni et al (Cornell group) PRD 32 (1985) S G = − β ⇣ ⌘ X U P + U † We now start with the Wilson action Tr P 2 N c P We now deal with a theory formulated in terms of group variables and Langevin equation reads ∂ U µx = e A µ ( x ) ∂ tU xµ ( t ; η ) = ( � i r xµ S G [ U ] � i η xµ ( t )) U xµ ( t ; η ) where the Lie derivative is in place 1 ⇣ ⌘ e i α T a V r xµ = T a r a xµ = T a r a r a V f ( V ) = lim α ( f � f ( V )) U xµ α → 0 This is again a stochastic differential equation with (gaussian) noise averages satisfying t →∞ h O [ U ( t ; η )] i η = 1 Z DU e − S G [ U ] O [ U ] lim Z In order to proceed we now need a (numerical) integration scheme to simulate, e.g. Euler F xµ [ U, ⌘ ] = ✏ r xµ S G [ U ] + p ✏ ⌘ xµ U xµ ( n + 1; η ) = e − F xµ [ U, η ] U xµ ( n ; η ) ⇣ − 1 ⌘� F xµ [ U, ⌘ ] = ✏� ⌘ ⇣ + √ ✏ ⌘ xµ X U P − U † U P − U † Tr P P 4 N c N c U P ⊃ U xµ  δ il δ km � 1 � h η i,k ( z ) η l,m ( w ) i η = δ ik δ lm δ zw N c

NSPT (directly in the LGT case) Di Renzo, Marchesini, Onofri 94 Now we look for a solution in the form of a perturbative expansion X β − k/ 2 U ( k ) U xµ ( t ; η ) → 1 + xµ ( t ; η ) k =1 then we plug it into the (numerical scheme!) Langevin equation and get a hierarchy of equations! U (1) 0 = U (1) − F (1) U (2) 0 = U (2) − F (2) + 1 2 F (1) 2 − F (1) U (1) U (3) 0 = U (3) − F (3) + 1 2( F (2) F (1) + F (1) F (2) ) − 1 3! F (1) 3 − ( F (2) − 1 2 F (1) 2 ) U (1) − F (1) U (2) . . . In practice: we do not look closely at the (underlying) Stochastic Perturbation Theory because the computer is going to (numerically) take care of it and all that you are interested in are the observables, for which T X g k φ ( k ) X X g k h O k ( t ) i η h O [ η ( t )] i η = t →∞ h O k ( t ) i η = lim lim T →∞ 1 /T O k ( jn ) k k j =1 Beware! Lattice PT is (always!) a decompactification of lattice formulation, so that ultimately one should be able to make contact with the continuum Langevin equation, i.e. ∂ ∂ tA a µ ( η , x ; t ) = D ab ν F b ν µ ( η , x ; t ) + η a Where has this gone? µ ( x ; t )