On the lattice searching for the gluon condensate is difficult because a
large perturbative contribution to the expectation value of the action has to
be subtracted before looking for a small contribution from a possible gluon
condensate. The perturbative calculation therefore has to be very precise. We
use a modified version of stochastic perturbation theory to calculate a
perturbative series in a boosted coupling, which converges more rapidly than
the series with the usual lattice coupling, reducing the uncertainties in our
results. We do not see any condensate of dimension two, as suggested by some
earlier lattice studies, but we do find a contribution from a dimension four
condensate. The value of this condensate is approximately 0.04(1) GeV^4, but
with large uncertainties.Comment: 6 pages, 4 figures, contribution to Lattice2005(Theoretical
  developments

Rakow, P. E. L.

English

arXiv

Paul E.L. Rakow

Crossref

Stochastic Perturbation Theory and the Gluon Condensate

On the lattice searching for the gluon condensate is difficult because a large perturbative contribution to the expectation value of the action has to be subtracted before looking for a small contribution from a possible gluon condensate. The perturbative calculation therefore has to be very precise. We use a modified version of stochastic perturbation theory to calculate a perturbative series in a boosted coupling, which converges more rapidly than the series with the usual lattice coupling, reducing the uncertainties in our results. We do not see any condensate of dimension two, as suggested by some earlier lattice studies, but we do find a contribution from a dimension four condensate. The value of this condensate is approximately 0.04(1) GeV^4, but with large uncertainties

Rakow, Paul EL

University of Liverpool Repository

PoS(LAT2005)284Stochastic Perturbation Theory and the GluonCondensatePaul E. L. Rakow Theoretical Physics Division, Department of Mathematical Sciences, University of LiverpoolLiverpool, L69 3BX, UKE-mail: rakow@amtp.liv.ac.ukOn the lattice searching for the gluon condensate is difcult because a large perturbative con-tribution to the expectation value of the action has to be subtracted before looking for a smallcontribution from a possible gluon condensate. The perturbative calculation therefore has to bevery precise. We use a modied version of stochastic perturbation theory to calculate a pertur-bative series in a boosted coupling, which converges more rapidly than the series with the usuallattice coupling, reducing the uncertainties in our results. We do not see any condensate of di-mension two, as suggested by some earlier lattice studies, but we do nd a contribution from adimension four condensate. The value of this condensate is approximately 0  041  GeV4, butwith large uncertainties.XXIIIrd International Symposium on Lattice Field Theory25-30 July 2005Trinity College, Dublin, IrelandSpeaker.cCopyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(LAT2005)284Stochastic Perturbation Theory and the Gluon Condensate Paul E. L. Rakow1. IntroductionThere is a long history of attempts to determine a gluon condensate on the lattice [1, 2] bylooking at the expectation value of the gluonic action, which in the case of the Wilson action issimply the plaquette expectation valueP 1NcTr  U 	 (1.1)This is difficult, because the signal from any non-perturbative condensate will be much smallerthan the background caused by perturbative gluon fluctuations, dominated by the contribution fromgluons with momenta near the ultra-violet cut-off. The relationship between ∆P, the differencebetween the perturbative plaquette and the non-perturbative value measured in a Monte Carlo cal-culation, and the gluon condensate introduced in [3] is given by∆P  g2  Ppert  g2  PMC  g2 a4pi212Nc b0g3β  g αpiGG  (1.2)where a is the lattice spacing, β  g  the β -function, and  b0g3 the first term in the expansion ofβ  g  .In eq.(1.2) we make the conventional assumption that the condensate term has dimension four,so that ∆P will scale like a4. There has however been much interest in the possibility, suggestedin [4], that the dominant correction to perturbation theory scales like a2 instead.In this work we concentrate on quenched SU(3) with the Wilson gauge action. Most of ourresults are based on 12 loop series found on a 124 lattice, though we will also present some ex-ploratory 16 loop perturbation theory results from 84 lattices.A long perturbation series can also be used to investigate other interesting issues. For example,how long does the series need to be before we see that we have an asymptotic series rather than aconvergent one?2. The Perturbative SeriesIn the introduction we have argued that a long perturbation series is needed to answer the ques-tions that interest us. In this section we discuss a number of methods of finding pn, the coefficientsin the series for the plaquette,Ppert  g2 1 ﬀ ∑n ﬁ 1png2n  (2.1)2.1 Diagrammatic Lattice Perturbation TheoryThe first method that suggests itself is to use diagrammatic methods, similar to those used inthe continuum. On the lattice the Feynman rules are more complicated than in the continuum, thereare extra vertices, and the integrands in the loop integrals are much more complicated than in thecontinuum. Nevertheless the first three coefficients in the series have been found by conventionaltechniques [2, 5]. The advantage of this technique is that the coefficients can be found with veryhigh accuracy, for example in [6] p2 is given to 9 significant figures, and p3 to 6, which is muchbetter than what can currently be achieved with stochastic methods. The disadvantage is that thedifficulty of the calculation grows very rapidly with the number of loops, and it will probably be along time before we gain many more terms in the series.284 / 2PoS(LAT2005)284Stochastic Perturbation Theory and the Gluon Condensate Paul E. L. RakowFigure 1: The rst 16 coefcients in the perturbation series for the plaquette. Filled symbols show positivecoefcients, empty symbols show negative coefcients. Errors are smaller than the symbols.2.2 Stochastic Perturbation TheoryThe solution to this problem is stochastic perturbation theory, a mechanised way to do latticeperturbation theory introduced in [7]. Each field of lattice gauge theory is replaced by a series in g,and the result of a simulation is a Monte Carlo estimate for each pn. The precision not as large asthat found by doing integrals, but the effort needed grows as a polynomial of loop number, so wecan go much farther in n. In [7] the coefficients up to p8 were found, [8] extended this to the 10loop level, and in Fig.1 we present exploratory results up to 16 loops, calculated on an 84 lattice.Unfortunately the series converges rather slowly, and simply truncating the sum at 16 loopsdoes not give a good estimate of the complete sum. As can be seen from the graph, the dependenceof the coefficients on n is quite smooth, so one possible approach is to fit a curve to the knowncoefficients and use this to estimate the pn for higher n. This was the approach we used in [9].The new coefficients agree very well with the predictions made in [9] when only 10 loops wereavailable. Even at 16 loops we do not yet see signs of the series diverging, it still looks like a serieswith a finite radius of convergence. This suggests that the crossover region between the strongcoupling and weak coupling regions is still having a strong influence on the series, as discussedin [9]. Motivated by this behaviour in the perturbative series, the possibility that there is a third-order phase transition in this region was recently examined in [10].2.3 Boosted Perturbation TheoryIt is well-known that perturbation theory in the lattice coupling g2 converges slowly. The ex-planation is that the ΛQCD parameter of the bare lattice coupling g2 is surprisingly small, Λlat 0  035ΛMSso g2 is not a very good expansion parameter. The remedy is to use a “boosted” cou-pling which has a more reasonable ΛQCD. The most popular choice is g2	  g2 ﬂ P which hasΛlat  0  38ΛMS  The boosted coupling is larger than g2, but we hope that the coefficients willnow decrease faster, and that we will find a reliable answer with fewer terms in the series. If we284 / 3PoS(LAT2005)284Stochastic Perturbation Theory and the Gluon Condensate Paul E. L. RakowFigure 2: The boosted perturbation series (black curve, square symbols) compared with the series in thelattice coupling (blue curve, round symbols). Filled symbols show positive coefcients, open symbols shownegative.had the conventional series to very high accuracy, transforming to the new coupling would be easy.However if we only have limited accuracy in the coefficients the transformation is numerically dif-ficult as we are looking for small differences between large quantities. The solution to this problemis to modify stochastic perturbation theory so that it works directly with a boosted coupling.In Fig.2 we show the boosted perturbation series (from a 124 lattice) and compare it with theconventional series. Clearly the boosted series falls off much faster. Even though we are interestedin regions where the boosted coupling is 50% to 100% larger the change is worth while.3. Perturbation Theory and DataWe are now ready to compare perturbation theory with Monte Carlo data, to see if there is anyremainder which we could attribute to a condensate. If there is, how does the condensate scale, likea4 or a2?In Fig.3 we show the result of summing the ordinary and the boosted perturbation series tovarious orders. We can see that boosted perturbation theory has a much better behaviour, it hasalmost ‘converged’ before we get to 12 loops, while the series in the original coupling is stillchanging visibly. Note that neither series shows any signs of diverging, though it is possible that ifwe go much further in loop number we will at some point see terms growing.284 / 4PoS(LAT2005)284Stochastic Perturbation Theory and the Gluon Condensate Paul E. L. RakowFigure 3: Convergence of the two expansions of the plaquette at β ﬃ 6  20. The series in the bare latticecoupling is shown with blue circles, the series in the boosted coupling with black squares. It can be seen thatthe boosted series has much better apparent convergence.The quantity which interests us here is the difference between perturbation theory and latticeMonte Carlo data, ∆P  Ppert  PMC. The horizontal red line in Fig.3 shows the value of theplaquette as measured in lattice simulations. Looking at the boosted perturbation series it doesseem that Ppert is measurably larger than PMC. It would not be as easy to estimate the gap using theunimproved series, because the result would depend heavily on extrapolated values of the pn.To see whether ∆P scales like a2 or a4 we plot ∆P ﬂ a2 against a2 in Fig.4. Most of the plaquettevalues are from [11], with some additional data calculated by QCDSF. If the non-perturbativeFigure 4: ∆P  a2 plotted against a2, with both quantities measured in physical units. If ∆P scales like a2 thedata will be constant, if ∆P scales like a4 the data will lie on a straight line passing through the origin.284 / 5PoS(LAT2005)284Stochastic Perturbation Theory and the Gluon Condensate Paul E. L. Rakowcontribution scales like a2, the ∆P ﬂ a2 would be a constant, but if the non-perturbative contributionscales like a4 then the data should lie on a straight line passing through the origin. We see that thedata looks more like a dimension four condensate.We can use this graph to set a bound on any dimension-two condensate. Around a2  0  05GeV ! 2we see that ""∆P ﬂ a2 ""$#0  0015GeV2 % 40MeV  2. Since the natural scale for a dimension-two con-densate is Λ2QCD, this is a rather tight bound.Using eq. (1.2) we can find the dimension-four condensate from the slope of this graph,αpiGG & 0  04  1  GeV4 ' 450 MeV  4  (3.1)This is about the right order of magnitude, though somewhat larger than phenomenological esti-mates [3, 12] which tend to lie in the range 0  012 GeV4 to 0  024 GeV4.4. ConclusionsStochastic Perturbation theory gives long series for some simple quantities, allows us to testideas that we could not look at with conventional perturbation theory. We see, for example, thatboosted perturbation theory does dramatically accelerate series convergence. Although there aregood reasons for expecting that perturbation theory diverges, we see no sign of this yet. Both theconventional and boosted perturbation series still look well-behaved.The gluon condensate, found from the difference Ppert  PMC, seems to be dimension four, notdimension two. The value of the gluon condensate is approximately 0  04  1  GeV4, but with largeuncertainties.References[1] T. Banks, R. Horsley, H. R. Rubinstein and U. Wolff, Nucl. Phys. B 190 (1981) 692.[2] A. Di Giacomo and G. C. Rossi, Phys. Lett. B 100 (1981) 481.[3] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 147 (1979) 385; 448.[4] G. Burgio, F. Di Renzo, G. Marchesini and E. Onofri, Phys. Lett. B 422 (1998) 219.[5] B. Alles, M. Campostrini, A. Feo and H. Panagopoulos, Phys. Lett. B 324 (1994) 433.[6] A. Athenodorou, H. Panagopoulos and A. Tsapalis, Nucl. Phys. Proc. Suppl. 140 (2005) 794.[7] F. Di Renzo, E. Onofri and G. Marchesini, Nucl. Phys. B 457 (1995) 202.[8] F. Di Renzo and L. Scorzato, arXiv:hep-lat/0011067.[9] R. Horsley, P. E. L. Rakow and G. Schierholz, Nucl. Phys. Proc. Suppl. 106 (2002) 870.[10] L. Li and Y. Meurice, arXiv:hep-lat/0507034.[11] G. Boyd et al., Nucl. Phys. B 469 (1996) 419.[12] S. Narison, Phys. Lett. B 387 (1996) 162.284 / 6

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