recent applications of numerical stochastic perturbation theory

Recent Application of NSPTRecent Application of NSPTFrancesco Di RenzoFrancesco Di Renzo

Bielefeld - October 13, 2006Bielefeld - October 13, 2006QCD on Teraflops ComputersQCD on Teraflops Computers

Recent ApplicationsRecent Applicationsofof

Numerical Stochastic Perturbation Numerical Stochastic Perturbation TheoryTheory

F. Di RenzoF. Di Renzo (1)(1), V. Miccio , V. Miccio (2)(2), L. Scorzato, L. Scorzato (3)(3) andand C. Torrero C. Torrero ((44))

together with together with M. Laine M. Laine (4)(4) andand Y. Schr Y. Schröder öder (4)(4)

(1) (1) Università di Parma Università di Parma andand INFN, Parma, Italy INFN, Parma, Italy(2) (2) INFN Milano Bicocca, ItalyINFN Milano Bicocca, Italy

(3)(3) ECT* Trento, Italy ECT* Trento, Italy (4)(4) University of Bielefeld, Germany University of Bielefeld, Germany



Lattice Perturbation TheoryLattice Perturbation Theory and its difficulties and its difficultiesDespite the fact that in PT the Lattice is in principle a regulator like any other, it is in practice a Despite the fact that in PT the Lattice is in principle a regulator like any other, it is in practice a very ugly one… As a matter of fact, the Lattice is mainly intended as a non-perturbative regulator. very ugly one… As a matter of fact, the Lattice is mainly intended as a non-perturbative regulator. Still, LPT is something you can not actually live without!Still, LPT is something you can not actually live without!

>>>> Often (large) use is made of Often (large) use is made of Boosted PTBoosted PT ( (ParisiParisi, , Lepage & MackenzieLepage & Mackenzie).).

>>>> On top of all this, On top of all this, LPT converges badly!LPT converges badly!

>>>> We can compute to We can compute to HIGHHIGH LOOPSLOOPS! And, if needed, we can use BPT! And, if needed, we can use BPT to assess to assess convergence properties and truncation errorsconvergence properties and truncation errors of the series. of the series.

>> In many (traditional) playgrounds >> In many (traditional) playgrounds LPT LPT has often been replaced byhas often been replaced by non-perturbat. non-perturbat. methodsmethods: renormalization constants, : renormalization constants, SymanzikSymanzik improvement coefficients, ... improvement coefficients, ...

>>>> In practice: In practice: LPT LPT is reallyis really cumbersome cumbersome and usually computations are and usually computations are 1 LOOP1 LOOP;; 2 LOOPS are really hard and 3 LOOPS almost unfeasible.2 LOOPS are really hard and 3 LOOPS almost unfeasible.

>>>> The key point: The key point: LPT LPT is substantiallyis substantially more involved more involved than other (perturb.) regulators. than other (perturb.) regulators. The main difficulty for The main difficulty for gauge theoriesgauge theories is a big is a big proliferation of verticesproliferation of vertices! ! Also, momentum space is not better than configuration space and manyAlso, momentum space is not better than configuration space and many continuum techniques do not apply. continuum techniques do not apply.



OutlineOutline

• We saw some motivations ...We saw some motivations ...

• Some technical details : just a flavour of what Some technical details : just a flavour of what NSPTNSPT is and what the is and what the computational demandscomputational demands are (and/or can be) are (and/or can be)

• A first application: A first application: renormalization constantsrenormalization constants for Lattice QCD (quarks for Lattice QCD (quarks bilinearsbilinears).).

• Applications in Applications in Finite Temperature QCDFinite Temperature QCD (the (the QCD pressureQCD pressure by dimensional by dimensional reduction and all that).reduction and all that).

• An application to come: what about expansions in the (imaginary) An application to come: what about expansions in the (imaginary) chemical chemical potential potential ??



From From Stochastic QuantizationStochastic Quantization to to NSPTNSPTActually Actually NSPT NSPT comes almost for free from the framework of comes almost for free from the framework of Stochastic QuantizationStochastic Quantization ( (Parisi and Wu, Parisi and Wu, 19801980). From the latter originally both a non-perturbative alternative to standard Monte Carlo and a ). From the latter originally both a non-perturbative alternative to standard Monte Carlo and a new version of Perturbation Theory were developed. NSPT in a sense interpolates between the two. new version of Perturbation Theory were developed. NSPT in a sense interpolates between the two.

Now, the main assertion is very simply stated: asymptotically Now, the main assertion is very simply stated: asymptotically

Stochastic QuantizationStochastic Quantization

In the previous formula, In the previous formula, is a gaussian noise, from which the stochastic nature of the is a gaussian noise, from which the stochastic nature of the equation originates.equation originates.

Given a field theory, Stochastic Quantization basically amounts to giving to the field an Given a field theory, Stochastic Quantization basically amounts to giving to the field an extra degree of freedom, to be thought of as a extra degree of freedom, to be thought of as a stochastic timestochastic time in which an evolution in which an evolution takes place according to the takes place according to the Langevin equationLangevin equation



((NumericalNumerical)) Stochastic Perturbation Theory Stochastic Perturbation Theory

Since the solution of Langevin equation will depend on the coupling constant of the Since the solution of Langevin equation will depend on the coupling constant of the theory, look for the solution as a theory, look for the solution as a power expansionpower expansion

If you insert the previous expansion in the Langevin equation, the latter gets translated If you insert the previous expansion in the Langevin equation, the latter gets translated into a into a hierarchy of equationshierarchy of equations, each for each order, each dependent on lower orders., each for each order, each dependent on lower orders.

Now, also Now, also observablesobservables are expanded are expanded

and we get power expansions from Stochastic Quantization’s main assertion, e.g.and we get power expansions from Stochastic Quantization’s main assertion, e.g.

Just to gain some insight (bosonic theory with quartic interaction): you can solve by iteration!Just to gain some insight (bosonic theory with quartic interaction): you can solve by iteration! Diagrammatically ... Diagrammatically ...

+ + λλ + + λλ22 ( ( + + ... ) + O(... ) + O(λλ33 ))

+ 3 + 3 λλ ( ( ++ ) + O() + O(λλ22))... And this is a propagator ...... And this is a propagator ...



NSPTNSPT ( (Di Renzo, Marchesini, Onofri 94Di Renzo, Marchesini, Onofri 94) simply amounts to the ) simply amounts to the numerical integrationnumerical integration of of these equations on a computer!these equations on a computer!

Numerical Stochastic Perturbation TheoryNumerical Stochastic Perturbation Theory

• From fields to From fields to collections of fieldscollections of fields order norder n

• From scalar operations to From scalar operations to order by order operationsorder by order operations order norder n22

• Not too bad from the parallelism point of view!Not too bad from the parallelism point of view!

• ’’94-’00 - 94-’00 - APE100APE100 - - Quenched LQCDQuenched LQCD (Now on (Now on PCPC’s! Now also with Fadeev-Popov, ’s! Now also with Fadeev-Popov, but no ghosts!).but no ghosts!).

• ’’00-now - 00-now - APEmilleAPEmille - - Unquenched LQCDUnquenched LQCD: Dirac matrix easy to invert (in PT!): Dirac matrix easy to invert (in PT!) (of course happy with (of course happy with apeNEXTapeNEXT!)!)

• ’’06-... - 06-... - apeNEXTapeNEXT - we have resources to undertake also - we have resources to undertake also something newsomething new ... ...



One wants to work at One wants to work at zero quark masszero quark mass in order to get a in order to get a mass-independent mass-independent schemescheme..

project on the tree level structureproject on the tree level structure

We work in the We work in the RI’-MOMRI’-MOM scheme: compute quark bilinears operators between scheme: compute quark bilinears operators between (off-shell p) quark states and then amputate to get G functions (off-shell p) quark states and then amputate to get G functions

where the field renormalization constant is defined viawhere the field renormalization constant is defined via

Renormalization conditions read Renormalization conditions read

Renormalization constantsRenormalization constants and and LPTLPTDespite the fact that there is no theoretical obstacle to computing log-div RC in PT, on the lattice Despite the fact that there is no theoretical obstacle to computing log-div RC in PT, on the lattice one tries to compute them NP. Popular (intermediate) schemes are one tries to compute them NP. Popular (intermediate) schemes are RI’-MOMRI’-MOM ( (Rome groupRome group) and ) and SFSF ((alpha Collalpha Coll).).



Computation ofComputation of Renormalization Constants Renormalization ConstantsWe compute everything in PT. Usually divergent parts (anomalous dimensions) are “easy”, while We compute everything in PT. Usually divergent parts (anomalous dimensions) are “easy”, while fixing finite parts is hard. In our approach it is just the other way around! fixing finite parts is hard. In our approach it is just the other way around!

RI’-MOMRI’-MOM is an is an infinite-volume schemeinfinite-volume scheme, while we have to perform , while we have to perform finite V finite V computationscomputations! Care will be taken of this (crucial) aspect.! Care will be taken of this (crucial) aspect.

We actually take the We actually take the ’s for granted. See ’s for granted. See J.GraceyJ.Gracey ( (20032003): 3 loops!): 3 loops!

We take small values for (lattice) momentum and look for We take small values for (lattice) momentum and look for “hypercubic symmetric” “hypercubic symmetric” Taylor expansionsTaylor expansions to fit the finite parts we want to get. to fit the finite parts we want to get.

We know which form we have to expect for a generic coefficient (at loop L)We know which form we have to expect for a generic coefficient (at loop L)

- Wilson gauge – Wilson fermion (- Wilson gauge – Wilson fermion (WWWW) action on ) action on 323244 and and 161644 lattices. lattices.

- Gauge fixed toGauge fixed to Landau Landau (no anomalous dimension for the quark field at 1 loop level).(no anomalous dimension for the quark field at 1 loop level).

- - nnff = 0 = 0 (both (both 323244 and and 161644); ); 22 , , 33,, 4 4 ( (323244). ).

- Relevant Relevant mass countertemmass countertem (Wilson fermions) plugged in (in order to stay at zero quark mass). (Wilson fermions) plugged in (in order to stay at zero quark mass).



Let’s talk about the Let’s talk about the care needed when dealing with anomalous dimensionscare needed when dealing with anomalous dimensions. We . We recall the situation for the recall the situation for the scalar currentscalar current (1 loop): from the master formula (1 loop): from the master formula

i.e. you have to subtract a logi.e. you have to subtract a log

On the right the right thing to do: On the right the right thing to do: subtract a “tamed log”subtract a “tamed log” ( (finite volumefinite volume!) We are !) We are working out corrections at 2 and 3 loops: hard, but precious!working out corrections at 2 and 3 loops: hard, but precious!

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(pa)2

It is a It is a Finite Volume effectFinite Volume effect! Top: 16! Top: 1644 and 32 and 3244 signals for O signals for Opp (pseudoscalar). Bottom: (pseudoscalar). Bottom:

161644 and 32 and 3244 signals for O signals for Oss (scalar). Middle: the same for the ratio O (scalar). Middle: the same for the ratio Oss/O/Opp (finite). (finite).



CertainCertain ratios ratios are are finite finite andand safe to compute! safe to compute!

- 4.86(3)- 1.40(1)- 0.487(1)4

- 5.13(3)- 1.43(1)- 0.487(1)3

- 5.35(3)- 1.46(1)- 0.487(1)2

- 5.72(3)- 1.50(1)- 0.487(1)0

O(-3)O(-2)O(-1)nf

- 4.86(3)- 1.40(1)- 0.487(1)4

- 5.13(3)- 1.43(1)- 0.487(1)3

- 5.35(3)- 1.46(1)- 0.487(1)2

- 5.72(3)- 1.50(1)- 0.487(1)0

O(-3)O(-2)O(-1)nf

- 2.57(2)- 0.732(6)- 0.244(1)4

- 2.72(2)- 0.744(6)- 0.244(1)3

- 2.83(2)- 0.759(5)- 0.244(1)2

- 3.02(2)- 0.780(5)- 0.244(1)0

O(-3)O(-2)O(-1)nf

- 2.57(2)- 0.732(6)- 0.244(1)4

- 2.72(2)- 0.744(6)- 0.244(1)3

- 2.83(2)- 0.759(5)- 0.244(1)2

- 3.02(2)- 0.780(5)- 0.244(1)0

O(-3)O(-2)O(-1)nf

ZZpp/Z/Zss ZZvv/Z/Zaa

Good Good

nnff dependence dependence



ZZaa andand ZZvv

- 3.50(8)- 1.31(3)- 0.800(2)2

- 4.04(4)- 1.39(3)- 0.800(2)0

O(-3)O(-2)O(-1)nf

- 3.50(8)- 1.31(3)- 0.800(2)2

- 4.04(4)- 1.39(3)- 0.800(2)0

O(-3)O(-2)O(-1)nf

- 5.42(8)- 1.88(3)- 1.044(2)2

- 6.10(8)- 1.98(3)- 1.044(2)0

O(-3)O(-2)O(-1)nf

- 5.42(8)- 1.88(3)- 1.044(2)2

- 6.10(8)- 1.98(3)- 1.044(2)0

O(-3)O(-2)O(-1)nf



Resumming ZResumming Zaa andand Z Zv v (to 4 loops!)(to 4 loops!) One can compare to NP results from One can compare to NP results from SPQSPQCDCDRR

We can now have numbers for Za and Zv. We resum (@ b=5.8) using different coupling definitions:

ZZaa = 0.79(1) = 0.79(1) ZZvv = 0.70(1) = 0.70(1)

The missing ingredient is the second loop of cThe missing ingredient is the second loop of cSWSW. Of course we will try to compute it. Still, there is something we are already . Of course we will try to compute it. Still, there is something we are already

doing to give estimates at three loops, i.e. making use of the Alpha Collaboration parametrization for their non-perturbative doing to give estimates at three loops, i.e. making use of the Alpha Collaboration parametrization for their non-perturbative determination (below the ndetermination (below the n ff=2 formula)=2 formula)

Just a comment onJust a comment on Clover fermions Clover fermions



The The QCD pressureQCD pressure is a key observable in Finite Temperature QCD: it is both a is a key observable in Finite Temperature QCD: it is both a phenomenologically relevant quantity and a beautiful phenomenologically relevant quantity and a beautiful theoretical laboratory.

An application in FT QCD: theAn application in FT QCD: the Pressure Pressure fromfrom Dimensional Dimensional ReductionReduction

All the All the reduction procedurereduction procedure can be formulated in ( can be formulated in (continuumcontinuum) ) PTPT, but one wants to get , but one wants to get the genuine the genuine non-perturbative contributionnon-perturbative contribution form the form the latticelattice: a : a matchingmatching is needed, which is needed, which can be safely computed in (can be safely computed in (LatticeLattice) ) PTPT (we are in 3d) (we are in 3d).

Unfortunately, the computation is a tough one ... And here NSPT comes in place.

a decomposition which comes from the Effective Field Theories approacha decomposition which comes from the Effective Field Theories approach

which in terms of fields meanswhich in terms of fields means

A beautiful setting to undertake the computation is Dimensional Reduction (M. Laine, Y. Schröder et al):



It is not only an It is not only an high high ((44thth)) order order computation; at the highest order it is computation; at the highest order it is IR divergentIR divergent!

We know what the log-divergence is and since we want to match to a continuum computation, se make use of the same IR regulator: a mass.

This breaks Gauge Invariance, i.e. we need to fix a gauge in the same way as in the continuum: Fadeev-Popov procedure without ghosts! (C. Torrero)

Dealing with a Dealing with a mass regulator mass regulator ((JHEP0607:026JHEP0607:026))

Beware: first extrapolate to infinite volume, then subtract the log-divergence and finally remove the regulator!



An expansion in theAn expansion in the Chemical Potential Chemical Potential ? ?Dealing with a non-zero density is a big challenge: experimentalists are urging the Lattice Dealing with a non-zero density is a big challenge: experimentalists are urging the Lattice community, but we have to circumvent the sign problem …community, but we have to circumvent the sign problem …

>>>> (Very) preliminary steps taken (in the framework of a collaboration with (Very) preliminary steps taken (in the framework of a collaboration with M.D’EliaM.D’Elia and and MP. LombardoMP. Lombardo))

>>>> This framework is in a sense closer to original work by This framework is in a sense closer to original work by ParisiParisi. . BewareBeware! This time! This time Dirac matrix inversionDirac matrix inversion is is notnot a a perturbativeperturbative one! one!

>>>> Why don’t we make use of Why don’t we make use of NSPTNSPT ( (imaginaryimaginary) ) expansions expansions??

>> A way to deal with a chemical potential on the lattice comes from >> A way to deal with a chemical potential on the lattice comes from imaginary imaginary simulations (simulations (D’Elia-LombardoD’Elia-Lombardo, , Philipsen-DeForcrandPhilipsen-DeForcrand).).

>> Analytic continuation relies (also) on Taylor expansions. Coefficients of the>> Analytic continuation relies (also) on Taylor expansions. Coefficients of the expansions can themselves be computed and employed to study the phaseexpansions can themselves be computed and employed to study the phase diagram (diagram (Bielefeld-SwanseaBielefeld-Swansea).).



ConclusionsConclusions

• NSPTNSPT is by now a is by now a mature techniquemature technique. Computations in many different . Computations in many different frameworks can be (and are actually) undertaken.frameworks can be (and are actually) undertaken.

• More results are to come (e.g. More results are to come (e.g. different actionsdifferent actions for renormalization constants; for renormalization constants; the pressure computations go on in the the pressure computations go on in the EQCDEQCD sector; collaboration with sector; collaboration with G. G.

BaliBali on on HQ potentialsHQ potentials))

• Other developments are possible ...Other developments are possible ...

• ... Stay tuned!... Stay tuned!

recent applications of numerical stochastic perturbation theory

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