Playing around with different vacua

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Playing around with different vacua a heretical
(perturbative) way to FT Lattice QCD
F. Di Renzo
Università di Parma and INFN, Parma, Italy
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A disclaimer
Despite the fact that I have been working (also)
in Finite Temperature for some time, I still
regard myself as an ousider in the field.
Much of what I know comes from collaborations
with experts in the field (M. Laine. Y.
Schroeder, M.P. Lombardo, M. DElia)
in what follows errors and naiveness are of my
own
My own expertise has been for quite a long time
in a (non diagrammatic) way of doing Lattice
Perturbation Theory. While LPT has never been
regarded as such a useful tool in FT Lattice QCD
(even harder than at T0!), I will try to
elaborate on a proposal aiming at gaining some
information from it. No results will be given.
This is really the discussion of a proposal.
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Outline

• Preludio Finite Temperature Perturbation Theory
• vs Finite Temperature non-perturbative
  Lattice QCD.
• A naive computation in LPT Polyakov loop to two
  loop.
• A skecth of the technique by which computations
  were made (NSPT)
• from Stochastic Quantization to Stochastic
  Perturbation Theory
• from SPT to Numerical SPT
• An How-To for Lattice Gauge Theories and why we
  mention different vacua.
• The proposal (an even less standard LPT)
• Can we learn anything from convergence
  properties of FT series?
• Z3 sectors are obvious different vacua for
  Perturbative Lattice QCD
• and an interesting computation could be the
  Dirac operator spectrum!

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FT Perturbation Theory vs FT Lattice QCD
A simple-minded comparison

• Finite Temperature PT is simply derived by
  compactifying one dimension, but this results in
  quite delicate issues. Simply keep in mind
• T and g are both parameters to deal with!
• IR problems, resummations needed, different
  scales (2pT, gT, g2T)

• In non-perturbative Lattice QCD simulations life
  appears a bit easier with some respects
• Basic ingredient is a NtNs3 lattice (Nt lt Ns)
• There is no explicit reference to T b (i.e. the
  coupling) is determining it, once Nt is fixed b
  is the only parameter you explicitely deal with!

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A naive Lattice PT computation
And an even more ingenuous curiosity
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(No Transcript)
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From Stochastic Quantization to NSPT
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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To understand, take the standard example f4
theory ...
The free case is easy to solve in term of a
propagator ...
... and for the interacting case you can always
trade the differential equation for an integral
one ...
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If you insert the previous expansion in the
Langevin equation, the latter gets translated
into a hierarchy of equations, each for each
order, each dependent on lower orders.
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Numerical Stochastic Perturbation Theory
NSPT (Di Renzo, Marchesini, Onofri 94) simply
amounts to the numerical integration of SPT
equations on a computer! Lets take again the f4
theory, but notice that this time we are dealing
with a LATTICE regularization in x-space and the
time evolution has of course been discretized ...
These equation are now put on a computer. A
measurement is now obtained by constructing
composite operators, i.e.
Remember the main result of Stochastic
Quantization the expectation values are now
traded for temporal averages over the stochastic
evolution ...
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NSPT for Lattice Gauge Theories (JHEP0410073)
Langevin equation for LGT goes back to the 80s
(Cornell Group 84) the main point is to
formulate a stochastic process in the group
manifold.
Then one has to implement a finite difference
integration scheme (i.e. Euler)
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NSPT around non trivial vacua
1 is not the only trivial order for our
expansion! Other vacua are viable choices as well!
Uxm(0)(th)
Since dynamics is dictated by the equations of
motion, any classical solution is good eneugh!
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Fermionic observables are then constructed by
inverting (maybe several times) the Dirac matrix
on convenient sources. The Dirac matrix in turn
is a function of the gluonic field, and because
of that is expressed as a series as well
The good point is that free part is diagonal in
p-space, while interactions are diagonal in
x-space go back and forth via FFT! This is also
crucial in taking into account fermions in the
evolution.
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The proposal
A heretical approach to Finite Temperature (PT)
(Lattice QCD)
In the Polyakov loop computation we were sitting
on a given lattice size (4243) and started
computing ... No reference to temperature T was
made from the beginning. We now would like to
have a FT strategy to implement.

• We do not want to have a standard FT perturbative
  approach! We would rather go for the attitude of
  standard non-perturbative FT Lattice QCD let b
  be our only parameter and let us keep on
  expanding in b-1.
• Take a NtNs3 lattice and compute observables
  as series in b-1.
• Take Ns be bigger and bigger (one would like a
  limit to infinity ) at fixed Nt , i.e. try an
  infinite volume extrapolation in order to get the
  series you are aiming at.
• Your analysys of the series could suggest a
  (quasi?) singular behavior in b(Nt).
• Convert to a temperature. This should be done in
  terms of (asymptotic) scaling and knowledge of
  Lattice L parameter.
• Repeat for bigger and bigger Nt aiming at a
  continuum limit.

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• Once again, some comments are in order. One could
  ask So, what? This really looks like what one
  does in the non-perturbative framework Well
• Convergencies properties can be quite precise in
  describing singular points.
• One does not need to scan a region in b and
  could save resources to pin down a better
  continuum limit.
• It could be that subtleties of standard FT
  Perturbation Theory are avoided only the
  coupling in place (this required to commit to a
  finite number of points ...)
• One needs to revert to (asymptotic) scaling to
  translate to a physical temperature (but remember
  that the L parameter is by now quite well known).
• Fermions are easily treated in NSPT.
• The idea of different vacua is quite intriguing
  in this framework
• Different Z3 sectors are natural candidate to
  investigate.

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Actually the Polyakov loop was measured in the
background not of 1, but of z1. As a check, one
could verify that multiplying by z one goes back
to a real result.

• A useful (I think) computation to undertake the
  eig-problem for the Dirac operator in the
  background of different Z3 sectors. See C.
  Gattringer PRL 97 (06) 032003.
• Notice that computing corrections to a spectrum
  (the perturbative, field-independent, free field
  fermionic spectrum) is a text-book excercise.
  Only some caveats
• Degenerate case of Perturbation Theory.
• The Wilson Dirac operator (the first to
  undertake) is not hermitian, but (only)
  g5-hermitian. Go for Overlap as well!
• I would have liked to give some preliminary
  results Unfortunately I cant

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Conclusions

• I only discussed some idea that are at the moment
  a proposal.

• The NSPT Dirac operator spectrum computation will
  be undertaken for sure.

• These were only ideas, so that I suspect a
  possible comment could be Wheres the beef?
  Ok, you cant eat, but maybe I was able to let
  you smell it!