playing around with different vacua francesco di renzo frascati - august 6, 2007 xqcd07 playing...

Playing around with different vacuaPlaying around with different vacuaFrancesco Di RenzoFrancesco Di Renzo

Frascati - August 6, 2007Frascati - August 6, 2007xQCD07xQCD07

Playing around with different Playing around with different vacuavacua

a heretical (perturbative) way to FT Lattice QCDa heretical (perturbative) way to FT Lattice QCDF. Di RenzoF. Di Renzo

Università di Parma Università di Parma andand INFN, Parma, Italy INFN, Parma, Italy



A disclaimer …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. D’Elia) …

… 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 T=0!), 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.



OutlineOutline

Preludio: Preludio: Finite Temperature Perturbation Theory Finite Temperature Perturbation Theory vs vs Finite Temperature non-perturbative Lattice QCDFinite Temperature non-perturbative Lattice QCD..

A naive computation in LPT: A naive computation in LPT: Polyakov loop to two loopPolyakov loop to two loop..

A skecth of the technique by which computations were made (A skecth of the technique by which computations were made (NSPTNSPT):):

from Stochastic Quantization to Stochastic Perturbation Theoryfrom Stochastic Quantization to Stochastic Perturbation Theory from SPT to Numerical SPTfrom SPT to Numerical SPT

An An How-ToHow-To for Lattice Gauge Theories and why we mention for Lattice Gauge Theories and why we mention different vacuadifferent vacua..

The The proposalproposal (an even less standard LPT): (an even less standard LPT):Can we learn anything from convergence properties of “Can we learn anything from convergence properties of “FT seriesFT series”?”?

Z3 sectors Z3 sectors are obvious are obvious different vacua different vacua for Perturbative Lattice QCD …for Perturbative Lattice QCD …

… … and an interesting computation could be the and an interesting computation could be the Dirac operator spectrumDirac operator spectrum!!



FT Perturbation Theory FT Perturbation Theory vsvs FT Lattice QCD FT Lattice QCDA simple-minded comparison …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 (2T, gT, g2T) …

In non-perturbative Lattice QCD simulations life appears a bit easier with some respects:

• Basic ingredient is a Nt*Ns3 lattice (Nt < Ns)

• There is no explicit reference to T: (i.e. the coupling) is determining it, once Nt is fixed … is the only parameter you explicitely deal with!



A naive Lattice PT computationA naive Lattice PT computationAnd an even more ingenuous curiosity …And an even more ingenuous curiosity …

The Polyakov loop is one the most important quantities in FT Lattice QCD. For a one loop computation (on finite lattices) see Heller, Karsch NPB 251 (85) 254.

I took a 4*243 lattice and computed it to two loop (going higher would be quite easy)

The series does not appear to be much convergent …

Much the same I could inspect in the computation of Ps - Pt



Some comments are in order:

• Inferring convergencies properties from a two loop computation is crazy …• Both quantity (as considered) are not so well defined (Polyakov loop is dominated by the linearly divergent HQ self-energy; the difference of the plaquettes is not the properly defined energy density).

• LPT is not so celebrated as for convergence properties (still, many Z’s are fine).• Having said all that, I was nevertheless quite impressed: even the (in)famous 10 loop plaquette appears by far more convergent …

A question comes to your mind (at least if you are as naive as I am): Can the behavior be a signature of the critical temperature? i.e. Can one learn anything in FT Lattice QCD from the convergence properties of the series?

This is quite common in statistical mechanics, e.g. power-law singularities traced by



From From Stochastic QuantizationStochastic Quantization to to NSPTNSPTNSPT NSPT comes almost for free from the framework of comes almost for free from the framework of Stochastic QuantizationStochastic Quantization ( (Parisi and WuParisi and Wu,, 1980 1980). ). From the latter originally both a non-perturbative alternative to standard Monte Carlo and a new 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. version of Perturbation Theory were developed. NSPT in a sense interpolates between the two.

Now, the Now, the main assertionmain assertion is very simply stated: is very simply stated: asymptoticallyasymptotically

Stochastic QuantizationStochastic Quantization

In the previous formula, In the previous formula, is a is a gaussian noisegaussian noise, from which the stochastic nature of the , 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



To understand, take the standard example: To understand, take the standard example: 44 theory ... theory ...

The free case is easy to solve in term of a propagator ...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 ... ... and for the interacting case you can always trade the differential equation for an integral one ...



If you insert the previous expansion in the Langevin equation, the latter gets translated into a If you insert the previous expansion in the Langevin equation, the latter gets translated 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.

Stochastic Perturbation TheoryStochastic Perturbation Theory

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

Observation: we can get power expansions from Stochastic Quantization’s main assertion, e.g.Observation: we can get power expansions from Stochastic Quantization’s main assertion, e.g.

We already know the solutions for We already know the solutions for 44 theory: theory: Diagrammatically ... Diagrammatically ...

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

Now, also Now, also observablesobservables are expanded are expanded

+ 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 SPT of SPT equations on a computer! Let’s take again the equations on a computer! Let’s take again the φφ44 theory, but notice that this time we are dealing theory, but notice that this time we are dealing with a with a LATTICELATTICE regularization in x-space and the time evolution has of course been discretized ... regularization in x-space and the time evolution has of course been discretized ...

Numerical Stochastic Perturbation TheoryNumerical Stochastic Perturbation Theory

These equation are now put on a computer. A measurement is now obtained by constructing These equation are now put on a computer. A measurement is now obtained by constructing composite operators, i.e.composite operators, i.e.

Remember the main result of Stochastic Quantization: the expectation values are now traded for Remember the main result of Stochastic Quantization: the expectation values are now traded for temporal averages over the stochastic evolution ...temporal averages over the stochastic evolution ...



Langevin equation for LGT Langevin equation for LGT goes back to the 80’s (goes back to the 80’s (Cornell Group 84Cornell Group 84): the main point is to formulate ): the main point is to formulate a a stochastic process in the group manifoldstochastic process in the group manifold..

NSPT for Lattice Gauge Theories NSPT for Lattice Gauge Theories ((JHEP0410:073JHEP0410:073))

Then one has to implement a finite difference integration scheme (i.e. Euler) Then one has to implement a finite difference integration scheme (i.e. Euler)



1 1 is not the only trivial order for our expansion! Other is not the only trivial order for our expansion! Other vacua vacua are viable choices as well!are viable choices as well!

NSPT around non trivial vacuaNSPT around non trivial vacua

Since dynamics is dictated by the equations of motion, any Since dynamics is dictated by the equations of motion, any classical solution classical solution is good eneugh!is good eneugh!

UUxx(t;(t;))



Fermionic observables Fermionic observables are then constructed by are then constructed by invertinginverting (maybe several times) the (maybe several times) the Dirac matrixDirac matrix on on convenient sources. The Dirac matrix in turn is a function of the gluonic field, and because of that is convenient sources. The Dirac matrix in turn is a function of the gluonic field, and because of that is expressed as a expressed as a seriesseries as well as well

The good point is that The good point is that free partfree part is diagonal in is diagonal in p-spacep-space, while , while interactionsinteractions are diagonal in are diagonal in x-spacex-space: go : go back and forth via back and forth via FFTFFT! This is also crucial in taking into account fermions in the evolution.! This is also crucial in taking into account fermions in the evolution.



TheThe proposal proposalA heretical approach to Finite Temperature (PT) (Lattice QCD)A heretical approach to Finite Temperature (PT) (Lattice QCD)

In the Polyakov loop computation we were sitting on a given lattice size (4*243) 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 be our only parameter and let us keep on expanding in .

• Take a Nt*Ns3 lattice and compute observables as series in .

• 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 Nt•Convert to a temperature. This should be done in terms of (asymptotic) scaling and knowledge of Lattice parameter.

• Repeat for bigger and bigger Nt aiming at a continuum limit.



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 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 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.



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) 5-hermitian. Go for Overlap as well!

I would have liked to give some preliminary results … Unfortunately I can’t …



ConclusionsConclusions

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

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

• These were only ideas, so that I suspect a possible comment could be: These were only ideas, so that I suspect a possible comment could be: “Where’s the beef?” Ok, you can’t eat, but maybe I was able to let you smell “Where’s the beef?” Ok, you can’t eat, but maybe I was able to let you smell it!it!

playing around with different vacua francesco di renzo frascati - august 6, 2007 xqcd07 playing...

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