Recent Applications of Numerical Stochastic Perturbation Theory

1. Recent Applications of Numerical Stochastic Perturbation Theory F. Di Renzo(1), V. Miccio (2), L. Scorzato(3)and C. Torrero(4) together with M. Laine (4)and Y. Schröder (4) (1) Università di Parma and INFN, Parma, Italy(2) INFN Milano Bicocca, Italy (3) ECT* Trento, Italy (4) University of Bielefeld, Germany

2. >> The key point: LPT is substantially more involved than other (perturb.) regulators. The main difficulty for gauge theories is a big proliferation of vertices! Also, momentum space is not better than configuration space and many continuum techniques do not apply. >> In practice: LPT is really cumbersome and usually computations are 1 LOOP; 2 LOOPS are really hard and 3 LOOPS almost unfeasible. >> On top of all this, LPT converges badly! >> Often (large) use is made of Boosted PT (Parisi, Lepage & Mackenzie). Lattice Perturbation Theory and its difficulties 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! >> In many (traditional) playgrounds LPT has often been replaced by non-perturbat. methods: renormalization constants, Symanzik improvement coefficients, ... >>We can compute to HIGHLOOPS! And, if needed, we can use BPT to assess convergence properties and truncation errors of the series.

3. Outline • We saw some motivations ... • Some technical details : just a flavour of what NSPT is and what the computational demands are (and/or can be) • A first application: renormalization constants for Lattice QCD (quarks bilinears). • Applications in Finite Temperature QCD (the QCD pressure by dimensional reduction and all that). • An application to come: what about expansions in the (imaginary) chemical potential ?

4. Stochastic Quantization 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 In the previous formula, h is a gaussian noise, from which the stochastic nature of the equation originates. Now, the main assertion is very simply stated: asymptotically 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.

5. Since the solution of Langevin equation will depend on the coupling constant of the theory, look for the solution as a power expansion 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. Now, also observables are expanded and we get power expansions from Stochastic Quantization’s main assertion, e.g. Just to gain some insight (bosonic theory with quartic interaction): you can solve by iteration! Diagrammatically ... ... And this is a propagator ... + λ + λ2 ( + ... ) + O(λ3) ) + O(λ2) + 3 λ ( + (Numerical) Stochastic Perturbation Theory

6. From fields to collections of fieldsorder n • From scalar operations to order by order operationsorder n2 • Not too bad from the parallelism point of view! Numerical Stochastic Perturbation Theory NSPT (Di Renzo, Marchesini, Onofri 94) simply amounts to the numerical integration of these equations on a computer! • ’94-’00 - APE100 - Quenched LQCD (Now on PC’s! Now also with Fadeev-Popov, • but no ghosts!). • ’00-now - APEmille - Unquenched LQCD: Dirac matrix easy to invert (in PT!) • (of course happy with apeNEXT!) • ’06-... - apeNEXT - we have resources to undertake also something new ...

7. Renormalization constants and LPT Despite the fact that there is no theoretical obstacle to computing log-div RC in PT, on the lattice one tries to compute them NP. Popular (intermediate) schemes are RI’-MOM (Rome group) and SF (alpha Coll). We work in the RI’-MOM scheme: compute quark bilinears operators between (off-shell p) quark states and then amputate to get G functions project on the tree level structure Renormalization conditions read where the field renormalization constant is defined via One wants to work at zero quark mass in order to get a mass-independent scheme.

8. Computation of Renormalization Constants 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 g’s for granted. See J.Gracey (2003): 3 loops! We know which form we have to expect for a generic coefficient (at loop L) We take small values for (lattice) momentum and look for “hypercubic symmetric” Taylor expansions to fit the finite parts we want to get. - 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). • Relevant mass countertem (Wilson fermions) plugged in (in order to stay at zero quark mass). RI’-MOM is an infinite-volume scheme, while we have to perform finite V computations! Care will be taken of this (crucial) aspect.

9. On the right the right thing to do: subtract a “tamed log” (finite volume!) We are working out corrections at 2 and 3 loops: hard, but precious! It is a Finite Volume effect! Top: 164 and 324 signals for Op (pseudoscalar). Bottom: 164 and 324 signals for Os (scalar). Middle: the same for the ratio Os/Op (finite). Let’s talk about the care needed when dealing with anomalous dimensions. We recall the situation for the scalar current (1 loop): from the master formula i.e. you have to subtract a log

10. Zp/Zs Zv/Za Certain ratios are finite and safe to compute! Good nf dependence

11. ZaandZv

12. Resumming Zaand Zv (to 4 loops!)One can compare to NP results from SPQCDR We can now have numbers for Za and Zv. We resum (@ b=5.8) using different coupling definitions: Zv = 0.70(1) Za = 0.79(1) Just a comment on Clover fermions The missing ingredient is the second loop of cSW. Of course we will try to compute it. Still, there is something we are already doing to give estimates at three loops, i.e. making use of the Alpha Collaboration parametrization for their non-perturbative determination (below the nf=2 formula)

13. A beautiful setting to undertake the computation is Dimensional Reduction (M. Laine, Y. Schröder et al): a decomposition which comes from the Effective Field Theories approach which in terms of fields means An application in FT QCD: the Pressure from Dimensional Reduction The QCD pressure is a key observable in Finite Temperature QCD: it is both a phenomenologically relevant quantity and a beautiful theoretical laboratory. All the reduction procedure can be formulated in (continuum) PT, but one wants to get the genuine non-perturbative contribution form the lattice: a matching is needed, which can be safely computed in (Lattice) PT (we are in 3d). Unfortunately, the computation is a tough one ... And here NSPT comes in place.

14. Dealing with a mass regulator (JHEP0607:026) It is not only an high (4th) order computation; at the highest order it is IR divergent! We know what the log-divergence is and since we want to match to a continuum computation, se make use of the same IR regulator: a mass. This breaks Gauge Invariance, i.e. we need to fix a gauge in the same way as in the continuum: Fadeev-Popov procedure without ghosts! (C. Torrero) Beware: first extrapolate to infinite volume, then subtract the log-divergence and finally remove the regulator!

15. >> A way to deal with a chemical potential on the lattice comes from imaginary m simulations (D’Elia-Lombardo, Philipsen-DeForcrand). >> Analytic continuation relies (also) on Taylor expansions. Coefficients of the expansions can themselves be computed and employed to study the phase diagram (Bielefeld-Swansea). >> Why don’t we make use of NSPT (imaginary) m expansions? >> (Very) preliminary steps taken (in the framework of a collaboration with M.D’Elia and MP. Lombardo) >>This framework is in a sense closer to original work by Parisi. Beware! This time Dirac matrix inversion is not a perturbative one! An expansion in the Chemical Potential ? Dealing with a non-zero density is a big challenge: experimentalists are urging the Lattice community, but we have to circumvent the sign problem …

16. Conclusions • NSPT is by now a mature technique. Computations in many different frameworks can be (and are actually) undertaken. • More results are to come (e.g. different actions for renormalization constants; the pressure computations go on in the EQCDsector; collaboration with G. Bali on HQ potentials) • Other developments are possible ... • ... Stay tuned!