Gluon two- and three-point Green Functions computed in Landau gauge from the
lattice show the existence of power corrections to the purely perturbative
expressions, that can be explained through an Operator Product Expansion as the
influence of a non gauge invariant mass dimension two condensate. The
relationship of this condensate with topological properties of QCD, namely
instantons, will be studied, giving a first estimate of the contribution of
instantons to this condensate based in the direct lattice measure, after a
cooling process, of the instanton liquid properties.Comment: Lattice2002(topology) contribution, 3 pages, 2 figure

't Hooft

A. Donini

A. Le Yaouanc

Alles

Becirevic

Boucaud

Capitani

De Soto

F. De Soto

Garcia Perez

Gubarev

H. Moutarde

J. Micheli

J. Rodri´guez-Quintero

J.P. Leroy

Kondo

O. Pe`ne

Ph. Boucaud

Schafer

Teper

English

arXiv

Crossref

An instanton picture O.P.E.〈A2〉 condensate?

Gluon two- and three-point Green Functions computed in Landau gauge from the lattice show the existence of power corrections to the purely perturbative expressions, that can be explained through an Operator Product Expansion as the influence of a non gauge invariant mass dimension two condensate. The relationship of this condensate with topological properties of QCD, namely instantons, will be studied, giving a first estimate of the contribution of instantons to this condensate based in the direct lattice measure, after a cooling process, of the instanton liquid properties.Universidad Pablo de Olavide. Departamento de Sistemas Físicos, Químicos y NaturalesPostprin

de Soto, Feliciano

Boucaud, P.

Donini, A.

Leroy, J. P.

Le Yaouanc, A.

Micheli, J.

Moutarde, H.

Pène, O.

Rodríguez-Quintero, J.

RIO: Repositorio Institucional Olavide

1An Instanton Picture O.P.E.〈A2〉Condensate?Ph. Boucauda, F. De Sotob, A. Doninic, J.P. Leroya, A. Le Yaouanca, J. Michelia, H. Moutarded, O.Pènea, J. Rodŕıguez-QuinteroeaLaboratoire de Physique Théorique, Université de Paris XI, Batiment 210, 91405 Orsay-Cedex, France.bDpto. de F́ısica Atómica Molecular y Nuclear, Universidad de Sevilla. Avda. Reina Mercedes s/n,41012 Sevilla, Spain.cI.N.F.N., Roma I and Dip. Fisica, Universitá di Roma ”La Sapienza”, P. le A. Moro 2, 00185 Rome,ItalydCentre de Physique Thèorique Ecole Polytechnique, 91128 Palaiseau Cedex, FranceeDpto. de F.A., E.P.S. La Rábida, Universidad de Huelva, 21819 Palos de la Fra., Spain.Gluon two- and three-point Green Functions computed in Landau gauge from the lattice show the existenceof power corrections to the purely perturbative expressions, that can be explained through an Operator ProductExpansion as the influence of a non gauge invariant mass dimension two condensate. The relationship of thiscondensate with topological properties of QCD, namely instantons, will be studied, giving a first estimate of thecontribution of instantons to this condensate based in the direct lattice measure, after a cooling process, of theinstanton liquid properties.1. QCD coupling constant, O.P.E. and〈A2〉condensate.Lattice calculations of the QCD coupling con-stant and gluon propagator based in the GreenFunctions Method [1], suggest the necessity toadd power corrections to the purely perturbativeexpressions to correctly describe their running [3].An Operator Product Expansion (O.P.E.) analy-sis of the Green functions in Landau gauge1 re-lates this power corrections to the existence ofa non-perturbative〈A2〉condensate [4], throughexpressions:G(2)O.P.E.(p2) = G(2)Pert.(p2) + c〈A2〉R,µp2,αO.P.Es (p2) = αPert.s (p2) + c′〈A2〉R,µp2, (1)1In the lattice we will work in the minimum A2 Landaugauge, ∂µAµ = 0, so all gauge dependent quantities, willbe expressed in this particular gauge.where perturbative expressions are developed atthree loops, and the functions c and c′ includethe Wilson coefficient of the expansion, and theanomalous dimension of the condensate at leadinglogarithm.By performing a combined fit of lattice re-sults to expressions in (1), in two different MOMschemes, a value of ΛMS is extracted, in fairlygood agreement with the one obtained by the AL-PHA collaboration [2], by a completely differentmethod. A value of the condensate comes outfrom the analysis. The physical meaning of thiscondensate is still an open question, and a lotof work is being devoted to its study during lastyears, for example, in relation to confinement [5].The aim in this work will be to study the possi-ble semiclassical contribution to this condensatecoming from instantons, and whether they mightexplain the presence of power corrections in GreenFunctions.22. The role of instantons.Instantons have been extensively studied as apossible description of the QCD vacuum (See [7]for a general overview), and so as a major sourceof QCD properties at low energies. In relationwith the aim of this work, an ensemble of non-interacting instantons (I) and antiinstantons (I)in Landau gauge would give a contribution to the〈A2〉condensate;〈A2〉inst≈NV∫d4xAaµ(x)Aaµ(x) = 12π2ρ2n, (2)where Aaµ(x) is the standard ’t Hooft Polyakovinstanton gauge field [6], ρ the average radius,and n =NI+NIV the density.If we accept the phenomenological values as-signed to n and ρ by the Instanton Liquid Model[7] (n ∼ 0.5fm−4 and ρ ∼ 1/3fm), the instan-tonic contribution will be〈A2〉Inst.∼ 0.5GeV2.We will perform, however, our own analysis, thustesting the latter approach.2.1. Cooling.In principle, a direct measure of A2 in the lat-tice should be possible, but the presence of theUV divergent part is hardly separable from thesoft, instantonic one. The other possibility is toperform a cooling procedure, that will allow usto compute the number and size of instantons,giving an indirect measure of the A2 through (2).We will use the traditional cooling method [8],even if it introduces a number of known biases, asI− I annihilation, and a modification of instantonsizes and lattice spacing. The approach proposedhere is to compute instanton properties for dif-ferent number of cooling sweeps, and extrapolateback to the thermalised situation in order to re-cover their physical meaning2.2.2. Shape Recognition.Instantons will be localised in cooled latticesvia a geometrical method (Described in [10].)2 The use of improved cooling methods, as the one devel-oped in [9], could improve this approach, as radii evolutionis minimised, but I − I annihilation is unavoidable, so theextrapolation will be anyway necessary.that accepts a topological charge lump as an in-stanton when the ratio of the integral over a givenfraction of the topological charge at the maxi-mum, α, and its theoretical counterpart, ǫ, is ∼ 1,for a range of values of α:ǫ =∫x/|Qρ(x)||Qρ(0)|≥αd4xQρ(x)1 − 3α1/2 + 2α3/4(3)Once the lump has been identified as an instan-ton, the radius will be computed from the size ofthe cluster where the integral has been developed.2.3. A naive model of annihilation.With the method outlined above, we computethe density and size of instantons in a lattice, fordifferent numbers of cooling sweeps, nc, obtain-ing values with a strong dependence on nc (Seefigure), that avoids to obtain any physical infor-mation at fixed nc.As a first approach to the understanding of thisevolution, we will make a simple model, whereinstantons annihilate with antiinstantons (Beingso ∆N = NI − NI a constant) proportionally totheir packing ratio, and to the number of antiin-stantons, so that the equation for the evolutionof N = NI + NI is:∂N∂nc= −λ2Vρ4(nc)(N(nc)2− ∆N2). (4)If we assume ρ(nc) = cte, the solution of Eq. (4)will give N(nc) ∼N(0)1+λnc, the expression used in[10], as a first order approach, but our coolingprocedure modifies instanton’s size (See figure),in a way thah we phenomenologically parametriseas:ρ(nc) = ρ(0)(1 + a ln(1 + nc)) . (5)We will include (5) in equation (4), with ρ(0) theextrapolated radius at the thermalised situationand a a constant to determine.After performing a combined fit of our latticeresults to the expressions (5) and the one comingfrom the integration of (4), we can fix the initialvalues of the density, n(0) and the radius ρ(0),3and the two constants that govern the evolution,λ and a.0 50 100 150 200nc00.511.5A2  (GeV2 )1 10 100nc0.30.40.50.60.7ρ (fm)Figure 1. Results of the combined fit for the instan-ton density and radius as a function of the number ofcooling sweeps for a 244 lattice at β=6.0.3. Results and conclussion.The result of the combined fit gives a valueof the instantonic contribution to〈A2Ins〉∼0.4GeV2, however the result of the extrapolationis highly dependent on the value of ρ, which dueto the logarithmic behaviour is hardly reliable.We therefore prefer the value at the maximum,1.12(11) GeV2, as a crude estimation of〈A2Ins〉.This semiclassical evaluation of〈A2〉, whichdoes not run with the scale, is difficult to re-late to that appearing in the O.P.E. expansion,which does depend on the renormalisation schemeand scale. The typical scale of instantons isρ−1 ∼ 0.7GeV. Unluckly it is not possible to runthe〈A2O.P.E.〉to such a low energy, where per-tubative QCD is not valid. The lowest reachebleenergy scale is 2.6GeV [4,10];〈A2O.P.E.(2.6GeV)〉= 1.4(3)(3)GeV2, (6)the first error coming from the OPE determina-tion of the condensate renormalised at 10 GeV,and the second from higher orders in the running.Keeping in mind the level of uncertainty ofthese calculations, we can nevertheless claim arather encouraging agreement between the in-stantonic contribution to the condensate and theone computed from the running of the GreenFunctions.Acknowledgements.F. de Soto wants to thank L.P.T. for itswarm hospitality and Fundación Cámara and theMCYT (Contract BFM2002-03315) for financialsupport.REFERENCES1. B. Alles et al., Nucl. Phys. B502 (1997) 3252. S. Capitani et al., NPPS 63 (1998) 153; NPB544 (1999) 669.3. Ph. Boucaud et al. JHEP 10 (98) 017; D. Be-cirevic et al. PRD 60 (1999) 094509.4. Ph. Boucaud et al. PLB 493, 315 (2000), F.De Soto and J. Rodŕıguez-Quintero PRD 64(2001) 114003.5. K. I. Kondo, et al. PRD 65, 085034 (2002),F.V. Gubarev, V.I. Zakharov, PLB 501 28(2001).6. G. ’t Hooft, Phys. Rev. D 14 (1976) 3432[Erratum-ibid. D 18 (1976) 2199].7. T. Schafer and E. V. Shuryak, Rev. Mod.Phys. 70 (1998) 323.8. M. Teper, Phys. Lett. B 162, 357 (1985);9. M. Garcia Perez, O. Philipsen and I. O. Sta-matescu, Nucl. Phys. B 551, 293 (1999)10. Ph. Boucaud et al., Phys. Rev. D 66 (2002)034504.

Nuclear Physics B - Proceedings Supplements

An instanton picture O.P.E.<A2> condensate?

Gluon two- and three-point Green Functions computed in Landau gauge from the lattice show the existence of power corrections to the purely perturbative expressions, that can be explained through an Operator Product Expansion as the influence of a non gauge invariant mass dimension two condensate. The relationship of this condensate with topological properties of QCD, namely instantons, will be studied, giving a first estimate of the contribution of instantons to this condensate based in the direct lattice measure, after a cooling process, of the instanton liquid properties.Ciencias Integrada

Boucaud, Ph.

De Soto, F.

Rodríguez Quintero, José

Arias Montano: Institutional Repository of the University of Huelva

Nuclear Physics B

An Instanton Picture O.P.E. (A2) Condensate?

Gluon two- and three-point Green Functions computed in Landau gauge from the lattice show the existence of power corrections to the purely perturbative expressions, that can be explained through an Operator Product Expansion as the influence of a non gauge invariant mass dimension two condensate. The relationship of this condensate with topological properties of QCD, namely instantons, will be studied, giving a first estimate of the contribution of instantons to this condensate based in the direct lattice measure, after a cooling process, of the instanton liquid properties.MCYT BFM2002-0331

Pène, Olivier

Soto Borrero, Feliciano Carlos de

idUS. Depósito de Investigación Universidad de Sevilla

arXiv:hep-ph/0209031v1  3 Sep 20021An Instanton Picture O.P.E.〈A2〉Condensate?Ph. Boucauda, F. De Sotob, A. Doninic, J.P. Leroya, A. Le Yaouanca, J. Michelia, H. Moutarded, O.Pènea, J. Rodŕıguez-QuinteroeaLaboratoire de Physique Théorique, Université de Paris XI, Batiment 210, 91405 Orsay-Cedex, France.bDpto. de F́ısica Atómica Molecular y Nuclear, Universidad de Sevilla. Avda. Reina Mercedes s/n,41012 Sevilla, Spain.cI.N.F.N., Roma I and Dip. Fisica, Universitá di Roma ”La Sapienza”, P. le A. Moro 2, 00185 Rome,ItalydCentre de Physique Thèorique Ecole Polytechnique, 91128 Palaiseau Cedex, FranceeDpto. de F.A., E.P.S. La Rábida, Universidad de Huelva, 21819 Palos de la Fra., Spain.Gluon two- and three-point Green Functions computed in Landau gauge from the lattice show the existenceof power corrections to the purely perturbative expressions, that can be explained through an Operator ProductExpansion as the influence of a non gauge invariant mass dimension two condensate. The relationship of thiscondensate with topological properties of QCD, namely instantons, will be studied, giving a first estimate of thecontribution of instantons to this condensate based in the direct lattice measure, after a cooling process, of theinstanton liquid properties.1. QCD coupling constant, O.P.E. and〈A2〉condensate.Lattice calculations of the QCD coupling con-stant and gluon propagator based in the GreenFunctions Method [1], suggest the necessity toadd power corrections to the purely perturbativeexpressions to correctly describe their running [3].An Operator Product Expansion (O.P.E.) analy-sis of the Green functions in Landau gauge1 re-lates this power corrections to the existence ofa non-perturbative〈A2〉condensate [4], throughexpressions:G(2)O.P.E.(p2) = G(2)Pert.(p2) + c〈A2〉R,µp2,αO.P.Es (p2) = αPert.s (p2) + c′〈A2〉R,µp2, (1)1In the lattice we will work in the minimum A2 Landaugauge, ∂µAµ = 0, so all gauge dependent quantities, willbe expressed in this particular gauge.where perturbative expressions are developed atthree loops, and the functions c and c′ includethe Wilson coefficient of the expansion, and theanomalous dimension of the condensate at leadinglogarithm.By performing a combined fit of lattice re-sults to expressions in (1), in two different MOMschemes, a value of ΛMS is extracted, in fairlygood agreement with the one obtained by the AL-PHA collaboration [2], by a completely differentmethod. A value of the condensate comes outfrom the analysis. The physical meaning of thiscondensate is still an open question, and a lotof work is being devoted to its study during lastyears, for example, in relation to confinement [5].The aim in this work will be to study the possi-ble semiclassical contribution to this condensatecoming from instantons, and whether they mightexplain the presence of power corrections in GreenFunctions.22. The role of instantons.Instantons have been extensively studied as apossible description of the QCD vacuum (See [7]for a general overview), and so as a major sourceof QCD properties at low energies. In relationwith the aim of this work, an ensemble of non-interacting instantons (I) and antiinstantons (I)in Landau gauge would give a contribution to the〈A2〉condensate;〈A2〉inst≈NV∫d4xAaµ(x)Aaµ(x) = 12π2ρ2n, (2)where Aaµ(x) is the standard ’t Hooft Polyakovinstanton gauge field [6], ρ the average radius,and n =NI+NIV the density.If we accept the phenomenological values as-signed to n and ρ by the Instanton Liquid Model[7] (n ∼ 0.5fm−4 and ρ ∼ 1/3fm), the instan-tonic contribution will be〈A2〉Inst.∼ 0.5GeV2.We will perform, however, our own analysis, thustesting the latter approach.2.1. Cooling.In principle, a direct measure of A2 in the lat-tice should be possible, but the presence of theUV divergent part is hardly separable from thesoft, instantonic one. The other possibility is toperform a cooling procedure, that will allow usto compute the number and size of instantons,giving an indirect measure of the A2 through (2).We will use the traditional cooling method [8],even if it introduces a number of known biases, asI− I annihilation, and a modification of instantonsizes and lattice spacing. The approach proposedhere is to compute instanton properties for dif-ferent number of cooling sweeps, and extrapolateback to the thermalised situation in order to re-cover their physical meaning2.2.2. Shape Recognition.Instantons will be localised in cooled latticesvia a geometrical method (Described in [10].)2 The use of improved cooling methods, as the one devel-oped in [9], could improve this approach, as radii evolutionis minimised, but I− I annihilation is unavoidable, so theextrapolation will be anyway necessary.that accepts a topological charge lump as an in-stanton when the ratio of the integral over a givenfraction of the topological charge at the maxi-mum, α, and its theoretical counterpart, ǫ, is ∼ 1,for a range of values of α:ǫ =∫x/|Qρ(x)||Qρ(0)|≥αd4xQρ(x)1− 3α1/2 + 2α3/4(3)Once the lump has been identified as an instan-ton, the radius will be computed from the size ofthe cluster where the integral has been developed.2.3. A naive model of annihilation.With the method outlined above, we computethe density and size of instantons in a lattice, fordifferent numbers of cooling sweeps, nc, obtain-ing values with a strong dependence on nc (Seefigure), that avoids to obtain any physical infor-mation at fixed nc.As a first approach to the understanding of thisevolution, we will make a simple model, whereinstantons annihilate with antiinstantons (Beingso ∆N = NI −NI a constant) proportionally totheir packing ratio, and to the number of antiin-stantons, so that the equation for the evolutionof N = NI +NI is:∂N∂nc= −λ2Vρ4(nc)(N(nc)2−∆N2). (4)If we assume ρ(nc) = cte, the solution of Eq. (4)will give N(nc) ∼N(0)1+λnc, the expression used in[10], as a first order approach, but our coolingprocedure modifies instanton’s size (See figure),in a way thah we phenomenologically parametriseas:ρ(nc) = ρ(0)(1 + a ln(1 + nc)) . (5)We will include (5) in equation (4), with ρ(0) theextrapolated radius at the thermalised situationand a a constant to determine.After performing a combined fit of our latticeresults to the expressions (5) and the one comingfrom the integration of (4), we can fix the initialvalues of the density, n(0) and the radius ρ(0),3and the two constants that govern the evolution,λ and a.0 50 100 150 200nc00.511.5A2  (GeV2 )1 10 100nc0.30.40.50.60.7ρ (fm)Figure 1. Results of the combined fit for the instan-ton density and radius as a function of the number ofcooling sweeps for a 244 lattice at β=6.0.3. Results and conclussion.The result of the combined fit gives a valueof the instantonic contribution to〈A2Ins〉∼0.4GeV2, however the result of the extrapolationis highly dependent on the value of ρ, which dueto the logarithmic behaviour is hardly reliable.We therefore prefer the value at the maximum,1.12(11) GeV2, as a crude estimation of〈A2Ins〉.This semiclassical evaluation of〈A2〉, whichdoes not run with the scale, is difficult to re-late to that appearing in the O.P.E. expansion,which does depend on the renormalisation schemeand scale. The typical scale of instantons isρ−1 ∼ 0.7GeV. Unluckly it is not possible to runthe〈A2O.P.E.〉to such a low energy, where per-tubative QCD is not valid. The lowest reachebleenergy scale is 2.6GeV [4,10];〈A2O.P.E.(2.6GeV)〉= 1.4(3)(3)GeV2, (6)the first error coming from the OPE determina-tion of the condensate renormalised at 10 GeV,and the second from higher orders in the running.Keeping in mind the level of uncertainty ofthese calculations, we can nevertheless claim arather encouraging agreement between the in-stantonic contribution to the condensate and theone computed from the running of the GreenFunctions.Acknowledgements.F. de Soto wants to thank L.P.T. for itswarm hospitality and Fundación Cámara and theMCYT (Contract BFM2002-03315) for financialsupport.REFERENCES1. B. Alles et al., Nucl. Phys. B502 (1997) 3252. S. Capitani et al., NPPS 63 (1998) 153; NPB544 (1999) 669.3. Ph. Boucaud et al. JHEP 10 (98) 017; D. Be-cirevic et al. PRD 60 (1999) 094509.4. Ph. Boucaud et al. PLB 493, 315 (2000), F.De Soto and J. Rodŕıguez-Quintero PRD 64(2001) 114003.5. K. I. Kondo, et al. PRD 65, 085034 (2002),F.V. Gubarev, V.I. Zakharov, PLB 501 28(2001).6. G. ’t Hooft, Phys. Rev. D 14 (1976) 3432[Erratum-ibid. D 18 (1976) 2199].7. T. Schafer and E. V. Shuryak, Rev. Mod.Phys. 70 (1998) 323.8. M. Teper, Phys. Lett. B 162, 357 (1985);9. M. Garcia Perez, O. Philipsen and I. O. Sta-matescu, Nucl. Phys. B 551, 293 (1999)10. Ph. Boucaud et al., Phys. Rev. D 66 (2002)034504.

An instanton picture O.P.E. (A2) condensate?

An Instanton Picture O.P.E. <A^2> Condensate?

An Instanton Picture O.P.E. <A^2> Condensate?

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RIO: Repositorio Institucional Olavide

Arias Montano: Institutional Repository of the University of Huelva

idUS. Depósito de Investigación Universidad de Sevilla