Diapositiva 1

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(1) University of Parma. INFN Parma MI11
(2) ECT Trento. INFN Parma MI11 (3)
University of Bielefeld
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NSPT a tool for getting more from Lattice
Perturbation Theory
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. Still, LPT is
something you can not actually live without!
gtgt In many (traditional) playgrounds LPT has
often been replaced by non-perturbat.
methods renormalization constants, Symanzik
improvement coefficients, ...
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Outline

• We saw some motivations ...
• Some technical details just a flavour of what
  NSPT is and what the computational demands are.
• A little bit on the status of renormalization
  constants for Lattice QCD quarks bilinears for
  the WW (Wilson gauge and Wilson fermions) action.
• Current Lattice QCD projects can be interested in
  different combinations of gauge/fermion actions
  take into account Symanzik gauge action and
  Clover fermions as well!
• apeNEXT can do the job (first configurations
  production just started)

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From Stochastic Quantization to NSPT
Actually NSPT comes almost for free from the
framework of Stochastic Quantization (Parisi and
Wu, 1980). 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.
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(Numerical) Stochastic Perturbation Theory
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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
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Numerical Stochastic Perturbation Theory
NSPT is not so bad to put on a computer! In
particular on a parallel one (APE family) ...

• 94-00 - APE100 - Quenched LQCD (Now on PCs!
  Now also with Fadeev-Popov,
• but no ghosts!).
• 00-now - APEmille - Unquenched LQCD (WW action)
  Dirac matrix easy to invert (it is PT, after
  all!)
• 07-... - apeNEXT - we have resources to
  undertake sistematic investigation of different
  actions ...

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One wants to work at zero quark mass in order to
get a mass-independent scheme.
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RI-MOM is an infinite-volume scheme, while we
have to perform finite V computations! Care must
be taken of this (crucial) aspect when dealing
with logs.
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... and the definition of Zq

• The O(p) 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!)
• If one computes ratios of Os one obtains ratios
  of Zs, in which in particular Zq cancels
• out. Convenient ratios are finite.
• Zv (Za) can be computed by taking convenient
  ratios of Ov (Oa) and S-1, thus eliminating
• Zq. They can also be computed taking ratios of
  Ov (Oa) and the corresponding conserv.
• currents.
• Zs (Zp) requires to subtract logs in order to
  obtain finite quantities. This needs care.
• Once one is left with finite quantities one can
  extrapolate to zero the irrelevant terms
• which go away with powers of pa (these powers
  comply to hypercubic simmetry)

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Ratios of bilinears Zs are finite and safe to
compute!
Good nf dependence
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Za and Zv
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Resumming Zp/Zs (to 4 loops!) One can
compare to NP results from SPQCDR
We can now have numbers for Za and Zv. We resum
nf2 results _at_ ß5.8 using different coupling
definitions
Zp/Zs 0.77(1)

• Results less and less dependent on the order at
  fixed scheme and less and less dependent on the
  scheme at higher and higher order. Zp/Zs and
  Zs/Zp quite well inverse of each other.
• Compare to SPQCDR result Zp/Zs 0.75(3)

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Resumming Za and Zv (to 4 loops!) One can
compare to NP results from SPQCDR
Za 0.79(1)
Zv 0.70(1)

• SPQCDR result Za 0.76(1) and Zv 0.66(2)
• Keep in mind chiral extrapolation!

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• On top of Wils/Wils (Wilson-gauge/Wilson-fermion)
  action we want to take into account the possible
  combinations of
• Wilson gauge action
• tree-level Symanzik gauge action
• (unimproved) Wilson fermion action
• (Wilson improved) Clover fermion action
• Results will also apply to twisted mass
  (renormalization conditions in the massless limit)

• Remember nf enters as a parameter and you would
  like to fit the nf dependence.
• APEmille (some work started on Clover) is not
  enough (10 months for a given nf)

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apeNEXT can do the job!
Why was in the end NSPT quite efficient?

• You do not have to store fields, but collections
  of fields, on which the most intensive FPU
  operations are order-by-order multiplications
  (remember the observation on parallelism!)
• This is a situation in which in there is a
  reasonable hope to perform well on a
  disegned-to-number-crunch machine ...Keep
  register file and pipelines busy!
• (in Parma they would say fitto come il
  rudo ... this is packed rubbish ...)
• This was traditionally quite easy on APE100 and
  APEmille (program memory and data memory are not
  the same)

• The APEmille code was not so brilliant on apeNEXT
  ...
• ... but we can optimize a little bit. For example
  we can make use of prefetching queues. We also
  have sofan at hand. Ok, then the cost for one nf
  is 2 months.

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Here is an example taken from bulk computations
(going from a power-expanded Aµ field to a
power-expanded Uµ field)
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Conclusions

• NSPT is by now quite a mature technique.
  Computations in many different frameworks can be
  (and are actually) undertaken.
• More results for renormalization constants are to
  come for different actions on top of Wils-Wils,
  also Wils-Clov, Sym-Wils and Sym-Clov. apeNEXT
  can manage the job!

• Other developments are possible ... (expansions
  in the chemical potential?)
• So, if you want ... stay tuned!