Instantons and the ⟨A2⟩ condensate

Instantons and the ŠA2‹condensate Ph. Boucaud,1J. P. Leroy,1A. Le Yaouanc,1J. Micheli,1O. Pe `ne,1F. De Soto,2A. Donini,3H. Moutarde,4 and J. Rodrı ´guez-Quintero5 1Laboratoire de Physique The ´orique, Universite ´de Paris XI, Ba ˆtiment 210, 91405 Orsay Cedex, France 2Departamento de Fı ´sica Ato ´mica, Molecular y Nuclear, Universidad de Sevilla, Apdo. 1065, 41080 Sevilla, Spain 3I.N.F.N., Roma I and Dip. Fisica, Universita `di Roma ‘‘La Sapienza,’’ P.le A. Moro 2, 00185 Rome, Italy 4Centre de Physique The ´orique Ecole Polytechnique, 91128 Palaiseau Cedex, France 5Departamento de Fı ´sica Aplicada, E.P.S. La Ra ´bida, Universidad de Huelva, 21819 Palos de la fra., Spain 共Received 16 March 2002; published 19 August 2002兲 We argue that the 具 AOPE 2 典 condensate found in the Landau gauge on lattices, when an operator product expansion of Green’s functions is performed, might be explained by instantons. We use cooling to estimate the instanton contribution and extrapolate back the result to the thermalized configuration. The resulting 具 Ainst 2 典 is similar to 具 AOPE 2 典 . DOI: 10.1103/PhysRevD.66.034504 PACS number共s兲: 12.38.Aw, 11.15.Ha, 12.38.Cy, 12.38.Gc I. INTRODUCTION Lattice calculations of the gluon propagator and threepoint Green’s function in the Landau gauge indicate that the expected perturbative behavior at large momentum phas to be corrected by a O(1/p2) contribution sizeable up to 10 GeV 关1–5兴. An understanding of this contribution as the effect of an A2⬅A ␮ aAa ␮ condensate 关in the Landau gauge A2is the only mass dimension-two operator liable to have a vacuum expectation value 共VEV兲兴 has been gained by verifying that two independent Green’s functions could be described by the perturbative contribution corrected by the effect of one common value of 具 AOPE 2 典 , as expected from the operator product expansion 共OPE兲. The physical origin of this condensate is an important question, possibly involving the non-trivial topology of the QCD vacuum. In particular, instantons provide an interesting insight into a wide range of low energy QCD properties 共关6兴and references therein兲. They have been put into evidence on the lattice using different cooling procedures. In this paper we claim that instantons provide for 具 A2 典 a value close to what is needed for the OPE fit to Green’s functions. We propose a method to identify instantons from the cooled gauge configuration, count them and measure their radii; we also check that these results are compatible with the instanton number deduced from the two-point correlation function of an instanton. We then estimate 具 Ainst 2 典 , the contribution of the instantons to 具 A2 典 in cooled configurations, extrapolate back to the thermalized configurations 共zero cooling sweeps兲and, finally, compare the outcome with the OPE estimate. II. COOLING AND INSTANTON COUNTING BY SHAPE RECOGNITION A. Cooling In order to study the influence of the underlying classical properties of a given lattice configuration, the first step will be to isolate these structures from UV modes. The method we use is due to Teper 关7兴; it consists of replacing each link by a unitary matrix proportional to the staple. A cooling sweep is performed after replacing all the links of the lattice. This procedure introduces largely discussed biases, such as UV instanton disappearance and instanton–anti-instanton pair annihilations, that increase with the number of cooling sweeps; alternative cooling methods have been proposed 共see for example 关8兴and references therein兲to cure these diseases. We will try to reduce them by identifying the instantons after a few cooling sweeps and extrapolating back to the thermalized gauge configuration. B. Instantons Instantons 共anti-instantons兲are classical solutions 关9兴of the equations of motion. We work in the Landau gauge which is defined on the lattice by minimizing 兺xA ␮ (x)2. For an instanton solution this prescription leads to the singular Landau gauge, where the gauge field is 关9兴 A ␮ (I)a⫽2 ␩ ¯ ␮ ␯ ax ␯ ␳ 2 x2共x2⫹ ␳ 2兲,共1兲 ␳ being the instanton radius. The instanton chosen has been centered at the origin with a conventional color orientation. These solutions have Q⫽⫾1 topological charge, Q⬅g2 32 ␲ 2 冕 d4xFa ␮ ␯ F ˜ ␮ ␯ a⫽⫾ g2 32 ␲ 2 冕 d4xFa ␮ ␯ F ␮ ␯ a,共2兲 where F ˜ ␮ ␯ ⫽1 2 ⑀ ␮ ␯␳␴ F ␳␴ . From Eq. 共1兲the topological charge density is Q ␳ 共x兲⫽⫾ 6 ␲ 2 ␳ 4 冉 ␳ 2 x2⫹ ␳ 2 冊 4 .共3兲 On the lattice, the topological charge density will be computed as Qlatt共x兲⫽1 29 ␲ 2兺 ⫾1 ⫾4 ⑀ ˜ ␮ ␯␳␴ Tr关⌸ ␮ ␯ 共x兲⌸ ␳␴ 共x兲兴,共4兲 PHYSICAL REVIEW D 66, 034504 共2002兲 0556-2821/2002/66共3兲/034504共5兲/$20.00 ©2002 The American Physical Society66 034504-1 with ⌸ ␮ ␯ (x)⫽U ␮ (x)U ␯ (x⫹ ␮ a ˆ)U ␮ †(x⫹ ␮ a ˆ⫹ ␯ a ˆ)U ␯ †(x ⫹ ␯ a ˆ), and ⑀ ˜ ␮ ␯␳␴ the antisymmetric tensor, with an extra minus sign for each negative index. C. Identification of instantons A common belief is that an instanton liquid gives a fair description of important features of the QCD vacuum. Along this line we will try a description of our cooled gauge configuration as an ensemble of non-interacting instantons with random positions and color orientations. We hence also neglect the interaction-induced instanton deformations and correlations. Although the instanton ensemble for the QCD vacuum cannot be considered as a dilute gas 关6–12兴, this crude assumption allows a qualitatively reasonable picture, especially near the instanton center. Many enlightening works have studied the instanton properties from lattice gauge configurations,1among which are 关13–16兴. As for us, we start by searching regions where the topological charge density looks like that of Eq. 共3兲. Starting from each local maximum or minimum of Qlatt(x) we integrate over all neighboring points with 兩 Qlatt(x) 兩 ⭓ ␣ 兩 Qlatt(xmax) 兩 , for different values of ␣ ranging from 0.8 to 0.4. A local extremum is accepted to be an 共anti-兲instanton if the ratio ⑀ between the lattice integral and its theoretical counterpart, Q ␳ (x), ⑀ ⫽共1⫺3 ␣ 1/2⫹2 ␣ 3/4兲⫺1 冕 x/ 兩 Q(x) 兩 ⭓ ␣ 兩 Q(0) 兩 d4xQ latt共x兲 共5兲 shows a plateau when ␣ is varied. Indeed for a theoretical 共anti-兲instanton ⑀ ⫽1( ⑀ ⫽⫺1) for any ␣ 苸关0,1兴. As a crosscheck of self-duality, this instanton shape recognition 共ISR兲 procedure is applied on the lattice to both expressions for Q introduced in Eq. 共2兲. D. Instanton numbers and radii The ISR method resolves the semiclassical structure on the lattice with only ⬃5 cooling sweeps. This early recognition reduces the possible cooling-induced bias. Whenever this method succeeds, we measure the radius of the accepted instanton in two separate ways: from the maximum value of Q, written Qlatt max , at the center of the instanton, and from the number N ␣ of lattice points in the volume 兰 x/ 兩 Q(x) 兩 ⭓ ␣ 兩 Q(0) 兩 d4x. We find ␳ /a⫽ 冑 46 ␲ 2Qlatt max ⫽1 冑 ␣ ⫺1/4⫺1 冑 42N ␣ ␲ 2.共6兲 These two measures of the radius agree within 10%. III. GLUON PROPAGATOR A. The instanton gauge-field correlation function The classical gauge-field two-point correlation function verifies, for any position and color orientation, 1 8兺 aG ¯ ␮ ␯ aa⬅1 8V兺 a„A ␮ (I)a共k兲A ␯ (I)a共⫺k兲… ⫽G ¯ (2)共k2兲 冉 ␦ ␮ ␯ ⫺k ␮ k ␯ k2 冊 ,共7兲 where Vis the volume in the Euclidean four-dimensional space and A ␮ (I)a(k) is the Fourier transform of Eq. 共1兲. The resulting scalar form factor is G ¯ (2)共k2兲⫽32 ␲ 4 Vk6 冉 1⫺共k ␳ 兲2 2K2共k ␳ 兲 冊 2 ,共8兲 K2being a Bessel function 关17兴. Equation 共8兲equally applies to instantons and anti-instantons.2 In a perfect gas approximation for an ensemble of nI(nA) 共anti-兲instantons of radius ␳ , the classical gauge-field correlation function is simply given by Eq. 共8兲times the number of instantons and anti-instantons, nI⫹nA. This correlation function is the contribution of the background field to the gluon propagator. We expect this formula to describe the behavior of the lattice gluon propagator once the effect of quantum UV fluctuations is removed by the cooling procedure 关7兴. The effect of instanton interactions is known 关10,11兴to modify the instanton shape far from its center, in the IR region. But the large k2behavior should be appropriately given by Eq. 共8兲i.e. ⬀1/k6. This is shown in Fig. 1共a兲 for one generic lattice gauge field configuration. The theoretical lines in that plot3are generated by Eq. 共8兲 using the average radius ␳ and nI⫹nAcomputed from the ISR method. Note that the matching improves with the number of cooling sweeps. This agrees with the expectation that decreasing the instanton density reduces the instanton deformation and that quantum fluctuations are damped by cooling. Reversely, if we know the average radius from the ISR method, we can compute nI⫹nAfrom the fit to the measured propagator. B. The hard gluon propagator Let us now consider a hard gluon of momentum p ␮ propagating in an instanton gas background. The gluon interacts with the instanton gauge field. This can be computed with Feynman graphs and it is easy to see that when the instanton modes k ␮ verifies k ␮ ⬃1/ ␳ Ⰶp ␮ , the dominant contribution is an O(1/p2) correction to the perturbative gluon. This correction is equal to the standard OPE Wilson 1A detailed comparison of our results to theirs will appear in a forthcoming extensive work. 2A similar analysis is being performed parallel, in other context, by Broniowski and Dorokhov 关18兴. 3We multiply by k2to compute a dimensionless object and perform the matching. PH. BOUCAUD et al. PHYSICAL REVIEW D 66, 034504 共2002兲 034504-2 coefficient 关2,3兴times 具 Ainst 2(0) 典 ⬅1/V 兰 d4x(A(I))2(x). We will now proceed to estimate this instanton-induced condensate. IV. ŠA2‹CONDENSATE A. ŠA2‹in instantons From Eq. 共1兲we get 具 Ainst 2共nc兲 典 ⬅nI⫹nA V 冕 d4x兺 ␮ ,aA ␮ (I)aA ␮ (I)a ⫽12 ␲ 2 ␳ 2nI⫹nA V,共9兲 where ␳ is the average instanton radius in the considered cooled configuration and ncis the number of cooling sweeps. We use an ensemble of 10 independent gauge configurations4at ␤ ⫽6.0 on a 244lattice. Each configuration has been cooled and after 5, 7, 10, 15, 30 and 100 cooling sweeps transformed into the Landau gauge. Using the ISR method on each gauge configuration, we obtain the results of Table I. In this table we also present the number of 共anti-兲instantons and the corresponding value for 具 A2(nc) 典 computed by a correlation function fit 共CFF兲i.e.afitofthe lattice propagators to the instanton correlation function, Eq. 共8兲. The CFF method is expected to be affected differently from the ISR method by systematic uncertainties: instanton interactions, deformations and quantum fluctuations 关as we can see in Fig. 1共a兲, at low momentum兴, and we therefore consider as quite encouraging the qualitative agreement, becoming quantitative at large nc, between ISR and CFF results. We have, for simplicity, translated our lattice results into physical units using, for all values of nc, the nc⫽0 inverse lattice spacing, a⫺1(nc⫽0)⫽1.996 GeV 共at ␤ ⫽6.0). This simple recipe overlooks the effect of cooling on the lattice spacing 共see Ref. 关15兴and references therein兲but this simplification becomes harmless after extrapolating back our results to nc⫽0. B. ŠAinst 2„nc…‹ at zero cooling The instanton number depends on the number of cooling sweeps. This result may imply that the cooling procedure destroys not only quantum UV fluctuations but something else from the semiclassical background of gauge fields. To lessen this problem we take advantage of the early recognition of the instanton content in a gauge configuration ensured by the ISR method and perform an extrapolation 关19兴to nc ⫽0 of the ISR results for 具 Ainst 2(nc) 典 in the table. We then obtain 关see Fig. 1共b兲兴: 具 Ainst 2共nc⫽0兲 典 ⫽1.76共23兲GeV2.共10兲 We have used a form a/(b⫹nc) to fit and extrapolate. We have also varied a little this functional form to check the stability of the extrapolation. We take this result as indicative of the non-perturbative instanton contribution to the 具 A2 典 4Considering the present size of our systematic uncertainties we did not consider it worthwhile to further increase the statistics. TABLE I. Estimates of 具 Ainst 2(nc) 典 . ␳ (fm) nI⫹nA 具 A2 典 (GeV2) ncISR ISR CFF ISR CFF 5 0.329共2兲87共2兲93共10兲1.38共3兲0.9共8兲 7 0.361共2兲74共2兲59共1兲1.42共5兲1.12共2兲 10 0.394共4兲60共1兲38共1兲1.36共4兲0.86共2兲 15 0.417共5兲43共1兲28共1兲1.11共4兲0.72共2兲 30 0.452共9兲26共1兲19共1兲0.80共6兲0.57共3兲 100 0.53共1兲10共1兲9共1兲0.43共3兲0.37共3兲 FIG. 1. In Fig. 1共a兲共left兲we present the lattice gluon propagators after several number of cooling sweeps 共points兲and the corresponding theoretical instanton gauge-field correlation functions 共lines兲in the perfect instanton gas approximation, Eq. 共8兲, plotted as a function of n2 (n ␮ ⬅L/(2 ␲ )k ␮ ,Lbeing the lattice length兲. In Fig. 1共b兲共right兲, we show the extrapolation at zero cooling of 具 A2(nc) 典 in physical units 共for three different trial functions兲. INSTANTONS AND THE 具 A2 典 CONDENSATE PHYSICAL REVIEW D 66, 034504 共2002兲 034504-3 condensate. If we applied other lattice estimates of instanton gas parameters taken from the available literature to Eq. 共9兲, the value of 具 A(I)2 典 would range5from 1 to 2 GeV2.Onthe other hand, parameters from instanton liquid based phenomenology 关6兴yield estimates of the order of 0.5 GeV2. As the quoted error in Eq. 共10兲is only statistical, this last range somehow estimates a certain systematic uncertainty. C. Comparison with ŠA2‹from OPE Our instanton estimate of 具 A2 典 is a semiclassical one, deprived of the necessary UV fluctuations, and therefore not directly comparable with 关2兴 具 AOPE 2 典 (10 GeV) ⫽2.4(5) GeV2. There is of course no exact recipe to compare both estimates, since the separation between the semiclassical nonperturbative domain and the perturbative one cannot be exact. We may appeal to the fact that at the renormalization point ␮ , the radiative corrections are minimized; therefore a semiclassical estimate must best correspond to 具 AOPE 2 典 at some reasonable ␮ . In the example of ␾ 2vacuum expectation in the spontaneously broken ␾ 4model given in 关21兴, one finds indeed that it equals the classical estimate for ␮ around the spontaneously generated mass. In our problem, one could guess that the corresponding scale should typically be around 1/ ␳ ⯝0.7 GeV, or some gluonic mass, a very low scale anyway. We cannot run 具 AOPE 2 典 (10 GeV) down to such a low scale 关2兴, AR, ␮ 2⫽AR, ␮ 0 2 冉 1⫹35 44ln ␮ ␮ 0 冒 ln ␮ ⌳MOM 冊 .共11兲 We therefore stop arbitrarily ␮ ⬃2.6 GeV, where the OPE corrected perturbative running of the Green’s functions fails to correctly describe their behavior. This scale of 2.6 GeV turns out to be of the same order 关2兴as the critical mass 关22兴 for the gluon propagator. At this scale, we obtain 具 AOPE 2共2.6 GeV兲 典 ⫽1.4共3兲共3兲GeV2,共12兲 where the first quoted error just propagates the uncertainty from the OPE determination of 具 A2 典 and the second one takes into account, in the way proposed in Ref. 关4兴, higher orders6in ␣ sfor running. V. DISCUSSION AND CONCLUSIONS We are aware that our method of comparison of 具 Ainst 2(nc⫽0) 典 and 具 AOPE 2 典 suffers from a lot of arbitrariness and approximations 共such as the perfect gas approximation, possible errors in the instanton identification, the uncertainty in the extrapolation to zero cooling sweeps, etc.兲. We have taken care to cross check our estimates by comparing different methods at each step of the computation, in particular the ISR and CFF 关see Fig. 1共a兲and Table I兲. A comparison with direct ‘‘measurements’’ of the 具 A2 典 condensate from cooled lattice configurations could be thought of as an additional crosscheck. Qualitative agreement is found for a large enough number of cooling sweeps, but this agreement is manifestly destroyed by UV fluctuations already for nc⬃30. Of course, by using ISR and instanton gas approximation we sharply separate UV fluctuations from the semiclassical background. All these imprecisions seem inherent to the subject anyway. With this in mind, we nevertheless take the fair agreement between Eqs. 共10兲and 共12兲as a convincing indication that the A2condensate receives a significant instantonic contribution. In other words, the instanton liquid picture might yield the explanation for the 1/p2corrections to the perturbative behavior of Green functions computed with thermalized configurations on the lattice. ACKNOWLEDGMENTS These calculations were performed on the Orsay APEmille purchased thanks to funding from the Ministe `re de l’Education Nationale and the CNRS. We are indebted to Bartolome Alles, Carlos Pena and Michele Pepe for illuminating discussions. A.D. acknowledges the M.U.R.S.T. for financial support through Decreto 1833/2001 共short term visit program兲and F.S. acknowledges the Fundacio ´nCa ´mara for financial support. A.D. wishes to thank the LPT-Orsay for its warm hospitality. This work was supported in part by the European Union Human Potential Program under contract No. HPRN-CT-2000-00145, Hadrons/Lattice QCD. 关1兴Ph. Boucaud et al., J. High Energy Phys. 04, 006 共2000兲. 关2兴Ph. Boucaud, A. Le Yaouanc, J.P. Leroy, J. Micheli, O. Pene, and J. Rodriguez-Quintero, Phys. Lett. B 493, 315 共2000兲. 关3兴Ph. Boucaud, A. Le Yaouanc, J.P. Leroy, J. Micheli, O. Pene, and J. Rodriguez-Quintero, Phys. Rev. D 63, 114003 共2001兲. 关4兴D. Becirevic, Ph. Boucaud, J.P. Leroy, J. Micheli, O. Pene, J. Rodriguez-Quintero, and C. Roiesnel, Phys. Rev. D 60, 094509 共1999兲;61, 114508 共2000兲. 关5兴F. De Soto and J. Rodriguez-Quintero, Phys. Rev. D 64, 114003 共2001兲. 关6兴T. Schafer and E.V. Shuryak, Rev. Mod. Phys. 70, 323 共1998兲. 关7兴M. Teper, Phys. Lett. 162B, 357 共1985兲; Phys. Lett. B 171,86 共1986兲. 关8兴M. Garcia Perez, O. Philipsen, and I.O. Stamatescu, Nucl. Phys. B551, 293 共1999兲. 关9兴G. ’t Hooft, Phys. Rev. D 14, 3432 共1976兲;18, 2199共E兲共1976兲. 5We use the parameters obtained in 关20兴for simulations on different lattices and ␤ ’s with a cooling improved to let scale invariant instanton solutions exist for large enough instanton sizes. We only quote anyway the results where the packing rate, ␲ 2/2( ␳ /L)4(nI ⫹nA), is as much as 1, since our method to estimate 具 Ainst 2(nc) 典 assumes limited overlap between instantons. 6We re-write Eq. 共11兲in terms of the coupling constant renormalized in other schemes, such as MS. PH. BOUCAUD et al. PHYSICAL REVIEW D 66, 034504 共2002兲 034504-4 关10兴D. Diakonov and V.Y. Petrov, Nucl. Phys. B245, 259 共1984兲. 关11兴J.J. Verbaarschot, Nucl. Phys. B362,33共1991兲;B386, 236共E兲 共1991兲. 关12兴M. Hutter, hep-ph/0107098. 关13兴M.C. Chu, J.M. Grandy, S. Huang, and J.W. Negele, Phys. Rev. D 49, 6039 共1994兲. 关14兴P. de Forcrand, M. Garcia Perez, and I.O. Stamatescu, Nucl. Phys. B499, 409 共1997兲. 关15兴UKQCD Collaboration, D.A. Smith, and M.J. Teper, Phys. Rev. D 58, 014505 共1998兲. 关16兴C. Michael and P.S. Spencer, Phys. Rev. D 52, 4691 共1995兲. 关17兴I. S. Gradshteyn and I. M. Ryzhiz, Table of Integrals, Series, and Products, 5th ed., edited by Alan Jeffre 共Academic, New York, 1994兲. 关18兴W. Broniowski 共private communication兲. 关19兴J.W. Negele, Nucl. Phys. B 共Proc. Suppl.兲73,92共1999兲. 关20兴P. de Forcrand, M. Garcı ´aPe ´rez, J. E. Hetrick, and I.-O. Stamatescu, in Proceedings of the Theory of Elementary Particles: 31st International Symposium Ahrenshoop, 1997, Buclow, Germany, edited by H. Dorn, D. Lust, and G. Weigt 共Wiley, New York, 1998兲, hep-lat/9802017. 关21兴V.A. Novikov, M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov, Nucl. Phys. B249, 445 共1985兲. 关22兴V.A. Novikov, M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov, Nucl. Phys. B191, 301 共1981兲. INSTANTONS AND THE 具 A2 典 CONDENSATE PHYSICAL REVIEW D 66, 034504 共2002兲 034504-5