NSPT@apeNEXT Francesco Di Renzo GGI - Firenze - February 9, 2007 apeNEXT workshop NSPT@apeNEXT F. Di Renzo (1) and L. Scorzato (2) in collaboration with

NSPT@apeNEXTNSPT@apeNEXTFrancesco Di RenzoFrancesco Di Renzo

GGI - Firenze - February 9, 2007GGI - Firenze - February 9, 2007apeNEXT workshopapeNEXT workshop

NSPT@apeNEXTNSPT@apeNEXTF. Di RenzoF. Di Renzo (1) (1) andand L. ScorzatoL. Scorzato (2)(2)

in collaboration within collaboration with C. TorreroC. Torrero (3)(3)

(1)(1) University of University of Parma. INFN Parma – Parma. INFN Parma – MI11MI11 (2)(2) ECT* Trento. INFN Parma – ECT* Trento. INFN Parma – MI11 MI11 (3)(3) University of Bielefeld University of Bielefeld



NSPTNSPT: a tool for: a tool for getting more fromgetting more from Lattice Perturbation TheoryLattice Perturbation Theory

Despite 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!

>> 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 In many (traditional) playgrounds LPT LPT has often been replaced byhas often been replaced by non-perturbat.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, ...

>>>> On top of all this, On top of all this, LPT converges badly LPT converges badly and one often tries to make use of and one often tries to make use of Boosted PTBoosted PT ( (ParisiParisi, , Lepage & MackenzieLepage & Mackenzie). This should be carefully assessed.). This should be carefully assessed.

>>>> The key point: The key point: LPT LPT is substantiallyis substantially more involved more involved than other (perturb.) regulators. than other (perturb.) regulators. LPT LPT is reallyis really cumbersome cumbersome and usually (diagrammatic) computations are and usually (diagrammatic) 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.

>>>> With With NSPTNSPT we can compute to we can compute to HIGHHIGH LOOPSLOOPS! We can ! We can assessassess convergence convergence properties properties andand truncation errors truncation errors of the series. With this respect we think LPT of the series. With this respect we think LPT should not be necessarily regarded as a second choice.should not be necessarily regarded as a second choice.

>> In the following we will mainly focus on (>> In the following we will mainly focus on (quark bilinearsquark bilinears) ) renormalization renormalization constantsconstants..



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. are.

• A little bit on the status of A little bit on the status of renormalization constantsrenormalization constants for Lattice QCD: quarks for Lattice QCD: quarks bilinears bilinears for the for the WWWW (Wilson gauge and Wilson fermions) (Wilson gauge and Wilson fermions) actionaction..

• Current Lattice QCD projects can be interested in different combinations of Current Lattice QCD projects can be interested in different combinations of gauge/fermion actions: take into account gauge/fermion actions: take into account Symanzik gaugeSymanzik gauge action and action and Clover Clover fermionsfermions as well! as well!

• apeNEXT can do the jobapeNEXT can do the job (first configurations production just started) (first configurations production just started)



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! For LGT this means integrating (these equations on a computer! For LGT this means integrating (Batrouni et al 85Batrouni et al 85))

Numerical Stochastic Perturbation TheoryNumerical Stochastic Perturbation Theory

where (Langevin eq. has been formulated in terms of a Lie derivative and Euler scheme)where (Langevin eq. has been formulated in terms of a Lie derivative and Euler scheme)

and everything should be intended as a series expansion, i.e. one has to plug in and everything should be intended as a series expansion, i.e. one has to plug in



NSPTNSPT is not so bad to put on a computer! In particular on a parallel one (APE family) ... is not so bad to put on a computer! In particular on a parallel one (APE family) ...

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 (WW action): Dirac matrix easy to invert (WW action): Dirac matrix easy to invert (it is PT, after all!)(it is PT, after all!)

• ’’07-... - 07-... - apeNEXTapeNEXT - we have resources to undertake - we have resources to undertake sistematic investigation sistematic investigation ofof different actionsdifferent actions ... ...



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: our state of the art: our state of the art

Despite 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 must be taken of this (crucial) aspect when dealing with log’s.! Care must be taken of this (crucial) aspect when dealing with log’s.

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).



Computation ofComputation of Renormalization Constants Renormalization ConstantsAlways keep in mind the master Always keep in mind the master formula! formula!

... and the definition of Z... and the definition of Zqq

• The The O(p)O(p) are the quantities to be actually computed. They are made out of convenient are the quantities to be actually computed. They are made out of convenient inversions of the Dirac operator on sources (we work out everything in mom space!)inversions of the Dirac operator on sources (we work out everything in mom space!)

• If one computes ratios of If one computes ratios of OO’s one obtains ratios of ’s one obtains ratios of ZZ’s, in which in particular ’s, in which in particular ZZqq cancels cancels

out. Convenient ratios are finite.out. Convenient ratios are finite.

• ZZvv ( (ZZaa) can be computed by taking convenient ratios of ) can be computed by taking convenient ratios of OOvv ( (OOaa) and ) and SS-1-1, thus eliminating , thus eliminating

ZZqq. They can also be computed taking ratios of . They can also be computed taking ratios of OOvv ( (OOaa) and the corresponding conserv. ) and the corresponding conserv.

currents.currents.

• ZZss ( (ZZpp) requires to subtract log’s in order to obtain finite quantities. This needs care.) requires to subtract log’s in order to obtain finite quantities. This needs care.

• Once one is left with finite quantities one can extrapolate to zero the irrelevant terms Once one is left with finite quantities one can extrapolate to zero the irrelevant terms which go away with powers of which go away with powers of papa (these powers comply to hypercubic simmetry) (these powers comply to hypercubic simmetry)



Ratios Ratios of bilinears Z’s are of bilinears Z’s 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 Zpp/Z/Zs s (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 nf=2 results @ β=5.8 using different coupling definitions:

ZZpp/Z/Zss = 0.77(1) = 0.77(1)

• Results less and less dependent on the order at fixed scheme and less and less dependent Results less and less dependent on the order at fixed scheme and less and less dependent on the scheme at higher and higher order. Zon the scheme at higher and higher order. Zpp/Z/Zss and Z and Zss/Z/Zpp quite well inverse of each other. quite well inverse of each other.

• Compare to Compare to SPQSPQCDCDR R result Zresult Zpp/Z/Zss = 0.75(3) = 0.75(3)



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

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

• SPQSPQCDCDR R result Zresult Za a = 0.76(1) and Z= 0.76(1) and Zvv = 0.66(2) = 0.66(2)

• Keep in mind chiral extrapolation!Keep in mind chiral extrapolation!



• Remember:Remember: n nff enters as a parameter and you would like to fit the n enters as a parameter and you would like to fit the nff dependence. dependence.

• APEmilleAPEmille (some work started on Clover) (some work started on Clover) is not enough (is not enough (~~10 months for a given n10 months for a given nff))

Renormalization constantsRenormalization constants: what next?: what next?

In these days Lattice QCD has great opportunities in really performing first principle computations. In these days Lattice QCD has great opportunities in really performing first principle computations. There are nevertheless a variety of options as for the choice of the actionThere are nevertheless a variety of options as for the choice of the action

On top of On top of Wils/WilsWils/Wils (Wilson-gauge/Wilson-fermion) action we want to take into (Wilson-gauge/Wilson-fermion) action we want to take into account the possible combinations ofaccount the possible combinations of

• Wilson gaugeWilson gauge action action

• tree-level Symanzik gaugetree-level Symanzik gauge action action

• (unimproved) (unimproved) Wilson fermionWilson fermion action action

• (Wilson improved)(Wilson improved) Clover Clover fermionfermion action action

Results will also apply to twisted mass (renormalization conditions in the massless Results will also apply to twisted mass (renormalization conditions in the massless limit)limit)



Why was in the endWhy was in the end NSPT NSPT quite efficient? quite efficient?

apeNEXT can do the job!apeNEXT can do the job!

• The APEmille code was not so brilliant on apeNEXT ...The APEmille code was not so brilliant on apeNEXT ...

• ... but we can optimize a little bit. For example we can make use of ... but we can optimize a little bit. For example we can make use of prefetching prefetching queuesqueues. We also have . We also have sofansofan at hand. Ok, then the at hand. Ok, then the cost for one ncost for one nff is is ~~2 months2 months..

• You do not have to store fields, but You do not have to store fields, but collections of fieldscollections of fields, on which the most , on which the most intensive FPU operations are intensive FPU operations are order-by-order multiplicationsorder-by-order multiplications (remember the (remember the observation on parallelism!)observation on parallelism!)

• This is a situation in which in there is a reasonable hope to perform well on a This is a situation in which in there is a reasonable hope to perform well on a disegned-to-number-crunch machinedisegned-to-number-crunch machine ... ...Keep register file and pipelines busyKeep register file and pipelines busy! !

(in Parma they would say (in Parma they would say “fitto come il rudo”“fitto come il rudo” ... this is ... this is packed rubbishpacked rubbish ...) ...)

• This was traditionally quite easy on APE100 and APEmille (program memory and This was traditionally quite easy on APE100 and APEmille (program memory and data memory are not the same)data memory are not the same)



ff7db6 | break pipeff7db6 | break pipe | | | !! first staple down| !! first staple down | | | Faux0[0] = Umu[link_c+o4]| Faux0[0] = Umu[link_c+o4] | Faux = Faux0[o4bis]^+ | Faux = Faux0[o4bis]^+ | Faux0[0] = Umu[link_c+o5]^+| Faux0[0] = Umu[link_c+o5]^+ | Faux = AUmultU11(Faux,Faux0[0])| Faux = AUmultU11(Faux,Faux0[0]) | Faux = AUmultU11(Faux,Umu[link_c+o6])| Faux = AUmultU11(Faux,Umu[link_c+o6]) | F = F + Faux| F = F + Faux | | 1831 74 % C: 1343 F: 0 M: 0 X: 2 L: 0 I: 4 IQO: 4/4 275/17 110/21831 74 % C: 1343 F: 0 M: 0 X: 2 L: 0 I: 4 IQO: 4/4 275/17 110/2

ff84dd | break pipeff84dd | break pipe | | | !! second staple up| !! second staple up | | | Faux0[0] = Umu[link_c+o8]^+| Faux0[0] = Umu[link_c+o8]^+ | Faux = AUmultU11(Umu[link_c+o7],Faux0[0])| Faux = AUmultU11(Umu[link_c+o7],Faux0[0]) | Faux0[0] = Umu[link_c+o9]^+| Faux0[0] = Umu[link_c+o9]^+ | Faux = AUmultU11(Faux,Faux0[0])| Faux = AUmultU11(Faux,Faux0[0]) | F = F + Faux| F = F + Faux | | 1807 75 % C: 1353 F: 0 M: 0 X: 2 L: 0 I: 4 IQO: 3/3 275/17 110/21807 75 % C: 1353 F: 0 M: 0 X: 2 L: 0 I: 4 IQO: 3/3 275/17 110/2

......

ffc75d | do i = 0, Sp_Vmffc75d | do i = 0, Sp_Vm | Faux = logU(Umu[j1]) | Faux = logU(Umu[j1]) | Faux = Faux - MomNull[Dir]| Faux = Faux - MomNull[Dir] | Faux = stricToA(Faux) | Faux = stricToA(Faux) | Umu[j1] = expA(Faux)| Umu[j1] = expA(Faux) | j1 = j1 + 1| j1 = j1 + 1 3172 92 % C: 2855 F: 0 M: 0 X: 31 L: 10 I: 32 IQO: 2/2 381/39 111/93172 92 % C: 2855 F: 0 M: 0 X: 31 L: 10 I: 32 IQO: 2/2 381/39 111/9

......

ffb599 | break pipeffb599 | break pipe | | | !! Enforcing unitary constraint ... | !! Enforcing unitary constraint ... | | | Faux = logU(Faux)| Faux = logU(Faux) | Faux = stricToA(Faux) | Faux = stricToA(Faux) | MomNull[Dir] = MomNull[Dir] + Faux | MomNull[Dir] = MomNull[Dir] + Faux | | | Umu[link_c] = expA(Faux)| Umu[link_c] = expA(Faux) | | 3240 90 % C: 2856 F: 0 M: 0 X: 31 L: 10 I: 31 IQO: 1/1 325/37 230/183240 90 % C: 2856 F: 0 M: 0 X: 31 L: 10 I: 31 IQO: 1/1 325/37 230/18



/AUseries_fact -> "expA" "(" AUseries_expr^a ")" {/AUseries_fact -> "expA" "(" AUseries_expr^a ")" {

temporary AUseries res, aux, auxRtemporary AUseries res, aux, auxR temporary su3 a_temporary su3 a_ temporary complex jnk, jnk1temporary complex jnk, jnk1

res = SetUpAres = SetUpAaux = aaux = ares = res + auxres = res + auxjnk = complex_1jnk = complex_1jnk1 = complex_1jnk1 = complex_1

/for n = 2 to ordine {/for n = 2 to ordine {

auxR = SetUpAauxR = SetUpA

queue = a.AU.[0]queue = a.AU.[0] /for i = 1 to (ordine-n) {/for i = 1 to (ordine-n) { a_ = queuea_ = queue /for j = (n-1) to (ordine-i) {/for j = (n-1) to (ordine-i) { auxR.AU.[i+j-1] = auxR.AU.[i+j-1] + a_ * aux.AU.[j-1]auxR.AU.[i+j-1] = auxR.AU.[i+j-1] + a_ * aux.AU.[j-1] }} queue = a.AU.[i]queue = a.AU.[i] } } a_ = queuea_ = queue auxR.AU.[ordine-1] = auxR.AU.[ordine-1] + a_ * aux.AU.[n-2]auxR.AU.[ordine-1] = auxR.AU.[ordine-1] + a_ * aux.AU.[n-2]

jnk1 = jnk1 + complex_1jnk1 = jnk1 + complex_1 jnk = jnk/jnk1jnk = jnk/jnk1 /for i = (n-1) to (ordine-1) {/for i = (n-1) to (ordine-1) { res.AU.[i] = res.AU.[i] + jnk * auxR.AU.[i]res.AU.[i] = res.AU.[i] + jnk * auxR.AU.[i] }} aux = auxRaux = auxR

}}

res.U0 = (1.0,0.0)res.U0 = (1.0,0.0)

rreturn resrreturn res

}}

Here is an example taken from bulk computations (going from a power-expanded AHere is an example taken from bulk computations (going from a power-expanded Aμμ field to field to

a power-expanded Ua power-expanded Uμμ field) field)



ConclusionsConclusions

• NSPTNSPT is by now quite a is by now quite 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 for renormalization constantsMore results for renormalization constants are to come for are to come for different actionsdifferent actions: : on top of on top of Wils-WilsWils-Wils, also , also Wils-ClovWils-Clov, , Sym-WilsSym-Wils and and Sym-ClovSym-Clov. apeNEXT can . apeNEXT can manage the job!manage the job!

• Other developments are possible ... (expansions in the chemical potential?)Other developments are possible ... (expansions in the chemical potential?)

• So, if you want ... stay tuned!So, if you want ... stay tuned!

Documents

NSPT@apeNEXT Francesco Di Renzo GGI - Firenze - February 9, 2007 apeNEXT workshop NSPT@apeNEXT F. Di Renzo (1) and L. Scorzato (2) in collaboration with