Diapositiva 1

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3 (or 4!) loops renormalization constants for
lattice QCD
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Outline
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gtgt LPT converges badly and usually computations
are 1 LOOP (analytic 2 LOOP on their way).
gtgt Often (large) use is made of Boosted PT
(Parisi, Lepage Mackenzie).
gtgt We can compute to 3 (or even 4) LOOPS!
gtgt We make use of the idea of BPT and we are
able to assess convergence properties and
truncation errors of the series.
gtgt We want to assess consistency with NP
determinations (if available). This is
the case we will focus on Zp/Zs (see Tarantino
_at_LAT05).
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NSPT comes as an application of Stochastic
Quantization (Parisi Wu) the field is given an
extra degree of freedom, to be thought of as a
stochastic time, in which an evolution takes
place according to the Langevin equation
Both the Langevin equation and the main assertion
get translated in a tower of relations ...
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Renormalization scheme (definitions) Martinelli
al NP 445 (1995) 81
One wants to work at zero quark mass in order to
get a mass-independent scheme.
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Renormalization scheme (comments)
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!
We actually take the gs for granted. See
J.Gracey (2003) 3 loops!
We take small values for (lattice) momentum and
look for hypercubic symmetric Taylor expansions
to fit the finite parts we want to get.
RI-MOM is an infinite-volume scheme, while we
have to perform finite V computations! Care will
be taken of this (crucial) aspect.
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Computational setup
Configurations (some hundreds) up to 3 (4...)
LOOPs have been generated and stored in order to
perform many computations.
- Wilson gauge Wilson fermion (WW) action on
324 and 164 lattices.

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

- nf 0 (both 324 and 164) 2 , 3, 4 (324). We
will focus on nf 2.

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

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1Loop example (Zq)
Easy example (no log in Landau gauge) what do we
expect for the inverse quark propagator?
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A first less trivial example would be 1Loop for
the scalar current
Just be patient for a few minutes there is
something more direct ...
The perfect quantity to compute is the ratio
Zp/Zs (or Zs/Zp)
- quark field renormalization drops out in the
ratio
- no anomalous dimension around

• as an extra bonus, from the point of view of the
  signals the two quantities are independent.
  Therefore, one can verify that the series are
  inverse of each other.

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• Series are actually inverse of each other and
  finite V effects are under
• control. Irrelevant effects are taken into
  account by the hyp-expans!

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Resummation at fixed order (blue1,green2,red3)
vs value of the couplings (x-axis) from left to
right x0, x1, x2, x3.

• Remember, for this quantity we do not need to
  know an anomalous dimension. Its tantalizing, so
  ... go for 4 loops!
• Notice we know the critical mass counterterm!

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There is a clean signal!
... and as a byproduct you get the critical
mass to 4 loop.
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• At fixed coupling milder and milder variations
  with the order.

• At fixed order milder and milder variations
  changing the coupling.

• Resumming at this order the series are almost
  inverse of each other.

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What do quote as a result? This is our sistematic
(truncation) error.
Take the phenomenologists attitude (deviations
from previous order) Zp/Zs .77(1).
This is also consistent with sort of scaling of
deviations from previous order.
Compare to NP (see Tarantino _at_LAT05) Zp/Zs
.75(1).
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About scaling of deviations from previous order
...
(This of course should not be taken too seriously
...)
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A caveat on BOOSTED PERTURBATION THEORY! (a
trivial one)
We now exaggerate the boosting of coupling x0,
x1, x2, x3, , xi b/Ph,
The bottom line is obvious there is no free
lunch in BPT ...
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We now go back to Zs (1 LOOP)
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Much the same holds for Zp, so apparently there
is a common problem.
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It is a finite volume effect!
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Much the same holds at 2 LOOP ...
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Remember apparently the log is tamed by finite
volume.
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This is a way of drawing which is closer to what
we saw log (diamonds) and tamed-log (circles)
on the finite size we are interested in.
so take this signal for the tamed-log and
plug it into our subtraction!
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It works! Here is the signal for Zs
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... and here comes Zp
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PRELIMINARY!
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Conclusions and perspectives
gtgt NSPT can give you a valuable tool for
computation of Z s.
gtgt The effect of Boosted PT can be carefully
assessed and convergence properties (which can be
not so bad!) can be inspected.
gtgt Care should be taken for finite volume when
logs are in place.
gtgt Configurations are there many computations
are possible (also in other fermionic schemes).
Moreover, improvement is on the way ...