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Playing around with different vacua Francesco Di Renzo Frascati - August 6, 2007 xQCD07 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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Playing around with different vacua Francesco Di Renzo Frascati - August 6, 2007 xQCD07 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 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.

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Playing around with different vacua Francesco Di Renzo Frascati - August 6, 2007 xQCD07 Outline Preludio: Finite Temperature Perturbation Theory Preludio: Finite Temperature Perturbation Theory vs Finite Temperature non-perturbative Lattice QCD. vs Finite Temperature non-perturbative Lattice QCD. A naive computation in LPT: Polyakov loop to two loop. A naive computation in LPT: Polyakov loop to two loop. A skecth of the technique by which computations were made (NSPT): A skecth of the technique by which computations were made (NSPT): from Stochastic Quantization to Stochastic Perturbation Theory from Stochastic Quantization to Stochastic Perturbation Theory from SPT to Numerical SPT from SPT to Numerical SPT An How-To for Lattice Gauge Theories and why we mention different vacua. An How-To for Lattice Gauge Theories and why we mention different vacua. The proposal (an even less standard LPT): 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 … Z3 sectors are obvious different vacua for Perturbative Lattice QCD … … and an interesting computation could be the Dirac operator spectrum! … and an interesting computation could be the Dirac operator spectrum!

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Playing around with different vacua Francesco Di Renzo Frascati - August 6, 2007 xQCD07 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 (2 T, gT, g 2 T) … In non-perturbative Lattice QCD simulations life appears a bit easier with some respects: Basic ingredient is a N t * N s 3 lattice (N t < N s ) There is no explicit reference to T: (i.e. the coupling) is determining it, once N t is fixed … is the only parameter you explicitely deal with!

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Playing around with different vacua Francesco Di Renzo Frascati - August 6, 2007 xQCD07 A naive Lattice PT computation 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 * 24 3 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 P s - P t

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Playing around with different vacua Francesco Di Renzo Frascati - August 6, 2007 xQCD07 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 Zs 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

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Playing around with different vacua Francesco Di Renzo Frascati - August 6, 2007 xQCD07 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. Now, the main assertion is very simply stated: asymptotically Stochastic Quantization In the previous formula, is a gaussian noise, from which the stochastic nature of the equation originates. Given a field theory, Stochastic Quantization basically amounts to giving to the field an extra degree of freedom, to be thought of as a stochastic time in which an evolution takes place according to the Langevin equation

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Playing around with different vacua Francesco Di Renzo Frascati - August 6, 2007 xQCD07 To understand, take the standard example: 4 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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Playing around with different vacua Francesco Di Renzo Frascati - August 6, 2007 xQCD07 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. Stochastic Perturbation Theory Since the solution of Langevin equation will depend on the coupling constant of the theory, look for the solution as a power expansion Observation: we can get power expansions from Stochastic Quantizations main assertion, e.g. We already know the solutions for 4 theory: Diagrammatically... + λ + λ 2 ( +... ) + O(λ 3 ) Now, also observables are expanded + 3 λ ( + ) + O(λ 2 )... and this is a propagator...

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Playing around with different vacua Francesco Di Renzo Frascati - August 6, 2007 xQCD07 NSPT (Di Renzo, Marchesini, Onofri 94) simply amounts to the numerical integration of SPT equations on a computer! Lets take again the φ 4 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... Numerical Stochastic Perturbation Theory 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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Playing around with different vacua Francesco Di Renzo Frascati - August 6, 2007 xQCD07 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. NSPT for Lattice Gauge Theories (JHEP0410:073) Then one has to implement a finite difference integration scheme (i.e. Euler)

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Playing around with different vacua Francesco Di Renzo Frascati - August 6, 2007 xQCD07 1 is not the only trivial order for our expansion! Other vacua are viable choices as well! NSPT around non trivial vacua Since dynamics is dictated by the equations of motion, any classical solution is good eneugh! U x (t; )

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Playing around with different vacua Francesco Di Renzo Frascati - August 6, 2007 xQCD07 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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Playing around with different vacua Francesco Di Renzo Frascati - August 6, 2007 xQCD07 The proposal A heretical approach to Finite Temperature (PT) (Lattice QCD) In the Polyakov loop computation we were sitting on a given lattice size (4 * 24 3 ) 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 N t * N s 3 lattice and compute observables as series in. Take N s be bigger and bigger (one would like a limit to infinity …) at fixed N t, 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 N t Convert to a temperature. This should be done in terms of (asymptotic) scaling and knowledge of Lattice parameter. Repeat for bigger and bigger N t aiming at a continuum limit.

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Playing around with different vacua Francesco Di Renzo Frascati - August 6, 2007 xQCD07 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.

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Playing around with different vacua Francesco Di Renzo Frascati - August 6, 2007 xQCD07 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 cant …

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Playing around with different vacua Francesco Di Renzo Frascati - August 6, 2007 xQCD07 Conclusions 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: Wheres the beef? Ok, you cant eat, but maybe I was able to let you smell it!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!

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