Instantons and the ⟨A2⟩ condensate

Ins an ons and he ŠA2‹condensa e Ph. Boucaud,1J. P. Le oy,1A. Le Yaouanc,1J. Micheli,1O. Pe `ne,1F. De So o,2A. Donini,3H. Mou a de,4 and J. Rod ı ´guez-Quin e o5 1Labo a oi e de Physique The ´o ique, Uni e si e ´de Pa is XI, Ba ˆ imen 210, 91405 O say Cedex, F ance 2Depa amen o de Fı ´sica A o ´mica, Molecula y Nuclea , Uni e sidad de Se illa, Apdo. 1065, 41080 Se illa, Spain 3I.N.F.N., Roma I and Dip. Fisica, Uni e si a `di Roma ‘‘La Sapienza,’’ P.le A. Mo o 2, 00185 Rome, I aly 4Cen e de Physique The ´o ique Ecole Poly echnique, 91128 Palaiseau Cedex, F ance 5Depa amen o de Fı ´sica Aplicada, E.P.S. La Ra ´bida, Uni e sidad de Huel a, 21819 Palos de la a., Spain 共Recei ed 16 Ma ch 2002; published 19 Augus 2002兲 We a gue ha he 具 AOPE 2 典 condensa e ound in he Landau gauge on la ices, when an ope a o p oduc expansion o G een’s unc ions is pe o med, migh be explained by ins an ons. We use cooling o es ima e he ins an on con ibu ion and ex apola e back he esul o he he malized con igu a ion. The esul ing 具 Ains 2 典 is simila o 具 AOPE 2 典 . DOI: 10.1103/PhysRe D.66.034504 PACS numbe 共s兲: 12.38.Aw, 11.15.Ha, 12.38.Cy, 12.38.Gc I. INTRODUCTION La ice calcula ions o he gluon p opaga o and h ee- poin G een’s unc ion in he Landau gauge indica e ha he expec ed pe u ba i e beha io a la ge momen um phas o be co ec ed by a O(1/p2) con ibu ion sizeable up o 10 GeV 关1–5兴. An unde s anding o his con ibu ion as he e - ec o an A2⬅A ␮ aAa ␮ condensa e 关in he Landau gauge A2is he only mass dimension- wo ope a o liable o ha e a acuum expec a ion alue 共VEV兲兴 has been gained by e i- ying ha wo independen G een’s unc ions could be de- sc ibed by he pe u ba i e con ibu ion co ec ed by he e - ec o one common alue o 具 AOPE 2 典 , as expec ed om he ope a o p oduc expansion 共OPE兲. The physical o igin o his condensa e is an impo an ques ion, possibly in ol ing he non- i ial opology o he QCD acuum. In pa icula , ins an ons p o ide an in e es ing insigh in o a wide ange o low ene gy QCD p ope ies 共关6兴and e e ences he ein兲. They ha e been pu in o e idence on he la ice using di e - en cooling p ocedu es. In his pape we claim ha ins an ons p o ide o 具 A2 典 a alue close o wha is needed o he OPE i o G een’s unc ions. We p opose a me hod o iden i y ins an ons om he cooled gauge con igu a ion, coun hem and measu e hei adii; we also check ha hese esul s a e compa ible wi h he ins an on numbe deduced om he wo-poin co ela ion unc ion o an ins an on. We hen es ima e 具 Ains 2 典 , he con i- bu ion o he ins an ons o 具 A2 典 in cooled con igu a ions, ex apola e back o he he malized con igu a ions 共ze o cooling sweeps兲and, inally, compa e he ou come wi h he OPE es ima e. II. COOLING AND INSTANTON COUNTING BY SHAPE RECOGNITION A. Cooling In o de o s udy he in luence o he unde lying classical p ope ies o a gi en la ice con igu a ion, he i s s ep will be o isola e hese s uc u es om UV modes. The me hod we use is due o Tepe 关7兴; i consis s o eplacing each link by a uni a y ma ix p opo ional o he s aple. A cooling sweep is pe o med a e eplacing all he links o he la ice. This p ocedu e in oduces la gely discussed biases, such as UV ins an on disappea ance and ins an on–an i-ins an on pai annihila ions, ha inc ease wi h he numbe o cooling sweeps; al e na i e cooling me hods ha e been p oposed 共see o example 关8兴and e e ences he ein兲 o cu e hese dis- eases. We will y o educe hem by iden i ying he ins an- ons a e a ew cooling sweeps and ex apola ing back o he he malized gauge con igu a ion. B. Ins an ons Ins an ons 共an i-ins an ons兲a e classical solu ions 关9兴o he equa ions o mo ion. We wo k in he Landau gauge which is de ined on he la ice by minimizing 兺xA ␮ (x)2. Fo an ins an on solu ion his p esc ip ion leads o he singula Landau gauge, whe e he gauge ield is 关9兴 A ␮ (I)a⫽2 ␩ ¯ ␮ ␯ ax ␯ ␳ 2 x2共x2⫹ ␳ 2兲,共1兲 ␳ being he ins an on adius. The ins an on chosen has been cen e ed a he o igin wi h a con en ional colo o ien a ion. These solu ions ha e Q⫽⫾1 opological cha ge, Q⬅g2 32 ␲ 2 冕 d4xFa ␮ ␯ F ˜ ␮ ␯ a⫽⫾ g2 32 ␲ 2 冕 d4xFa ␮ ␯ F ␮ ␯ a,共2兲 whe e F ˜ ␮ ␯ ⫽1 2 ⑀ ␮ ␯␳␴ F ␳␴ . F om Eq. 共1兲 he opological cha ge densi y is Q ␳ 共x兲⫽⫾ 6 ␲ 2 ␳ 4 冉 ␳ 2 x2⫹ ␳ 2 冊 4 .共3兲 On he la ice, he opological cha ge densi y will be com- pu ed as Qla 共x兲⫽1 29 ␲ 2兺 ⫾1 ⫾4 ⑀ ˜ ␮ ␯␳␴ T 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 Ame ican Physical Socie y66 034504-1 wi h ⌸ ␮ ␯ (x)⫽U ␮ (x)U ␯ (x⫹ ␮ a ˆ)U ␮ †(x⫹ ␮ a ˆ⫹ ␯ a ˆ)U ␯ †(x ⫹ ␯ a ˆ), and ⑀ ˜ ␮ ␯␳␴ he an isymme ic enso , wi h an ex a minus sign o each nega i e index. C. Iden i ica ion o ins an ons A common belie is ha an ins an on liquid gi es a ai desc ip ion o impo an ea u es o he QCD acuum. Along his line we will y a desc ip ion o ou cooled gauge con- igu a ion as an ensemble o non-in e ac ing ins an ons wi h andom posi ions and colo o ien a ions. We hence also ne- glec he in e ac ion-induced ins an on de o ma ions and co - ela ions. Al hough he ins an on ensemble o he QCD acuum canno be conside ed as a dilu e gas 关6–12兴, his c ude assump ion allows a quali a i ely easonable pic u e, especially nea he ins an on cen e . Many enligh ening wo ks ha e s udied he ins an on p op- e ies om la ice gauge con igu a ions,1among which a e 关13–16兴. As o us, we s a by sea ching egions whe e he opological cha ge densi y looks like ha o Eq. 共3兲. S a ing om each local maximum o minimum o Qla (x) we in e- g a e o e all neighbo ing poin s wi h 兩 Qla (x) 兩 ⭓ ␣ 兩 Qla (xmax) 兩 , o di e en alues o ␣ anging om 0.8 o 0.4. A local ex emum is accep ed o be an 共an i-兲ins an on i he a io ⑀ be ween he la ice in eg al and i s heo e ical coun e pa , Q ␳ (x), ⑀ ⫽共1⫺3 ␣ 1/2⫹2 ␣ 3/4兲⫺1 冕 x/ 兩 Q(x) 兩 ⭓ ␣ 兩 Q(0) 兩 d4xQ la 共x兲 共5兲 shows a pla eau when ␣ is a ied. Indeed o a heo e ical 共an i-兲ins an on ⑀ ⫽1( ⑀ ⫽⫺1) o any ␣ 苸关0,1兴. As a c oss- check o sel -duali y, his ins an on shape ecogni ion 共ISR兲 p ocedu e is applied on he la ice o bo h exp essions o Q in oduced in Eq. 共2兲. D. Ins an on numbe s and adii The ISR me hod esol es he semiclassical s uc u e on he la ice wi h only ⬃5 cooling sweeps. This ea ly ecogni- ion educes he possible cooling-induced bias. Whene e his me hod succeeds, we measu e he adius o he accep ed ins an on in wo sepa a e ways: om he maxi- mum alue o Q, w i en Qla max , a he cen e o he ins an on, and om he numbe N ␣ o la ice poin s in he olume 兰 x/ 兩 Q(x) 兩 ⭓ ␣ 兩 Q(0) 兩 d4x. We ind ␳ /a⫽ 冑 46 ␲ 2Qla max ⫽1 冑 ␣ ⫺1/4⫺1 冑 42N ␣ ␲ 2.共6兲 These wo measu es o he adius ag ee wi hin 10%. III. GLUON PROPAGATOR A. The ins an on gauge- ield co ela ion unc ion The classical gauge- ield wo-poin co ela ion unc ion e i ies, o any posi ion and colo o ien a ion, 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 whe e Vis he olume in he Euclidean ou -dimensional space and A ␮ (I)a(k) is he Fou ie ans o m o Eq. 共1兲. The esul ing scala o m ac o 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 unc ion 关17兴. Equa ion 共8兲equally applies o ins an ons and an i-ins an ons.2 In a pe ec gas app oxima ion o an ensemble o nI(nA) 共an i-兲ins an ons o adius ␳ , he classical gauge- ield co e- la ion unc ion is simply gi en by Eq. 共8兲 imes he numbe o ins an ons and an i-ins an ons, nI⫹nA. This co ela ion unc ion is he con ibu ion o he backg ound ield o he gluon p opaga o . We expec his o mula o desc ibe he beha io o he la ice gluon p opaga o once he e ec o quan um UV luc ua ions is emo ed by he cooling p oce- du e 关7兴. The e ec o ins an on in e ac ions is known 关10,11兴 o modi y he ins an on shape a om i s cen e , in he IR egion. Bu he la ge k2beha io should be app op i- a ely gi en by Eq. 共8兲i.e. ⬀1/k6. This is shown in Fig. 1共a兲 o one gene ic la ice gauge ield con igu a ion. The heo e ical lines in ha plo 3a e gene a ed by Eq. 共8兲 using he a e age adius ␳ and nI⫹nAcompu ed om he ISR me hod. No e ha he ma ching imp o es wi h he num- be o cooling sweeps. This ag ees wi h he expec a ion ha dec easing he ins an on densi y educes he ins an on de o - ma ion and ha quan um luc ua ions a e damped by cooling. Re e sely, i we know he a e age adius om he ISR me hod, we can compu e nI⫹nA om he i o he measu ed p opaga o . B. The ha d gluon p opaga o Le us now conside a ha d gluon o momen um p ␮ p opaga ing in an ins an on gas backg ound. The gluon in e - ac s wi h he ins an on gauge ield. This can be compu ed wi h Feynman g aphs and i is easy o see ha when he ins an on modes k ␮ e i ies k ␮ ⬃1/ ␳ Ⰶp ␮ , he dominan con ibu ion is an O(1/p2) co ec ion o he pe u ba i e gluon. This co ec ion is equal o he s anda d OPE Wilson 1A de ailed compa ison o ou esul s o hei s will appea in a o hcoming ex ensi e wo k. 2A simila analysis is being pe o med pa allel, in o he con ex , by B oniowski and Do okho 关18兴. 3We mul iply by k2 o compu e a dimensionless objec and pe - o m he ma ching. PH. BOUCAUD e al. PHYSICAL REVIEW D 66, 034504 共2002兲 034504-2 coe icien 关2,3兴 imes 具 Ains 2(0) 典 ⬅1/V 兰 d4x(A(I))2(x). We will now p oceed o es ima e his ins an on-induced conden- sa e. IV. ŠA2‹CONDENSATE A. ŠA2‹in ins an ons F om Eq. 共1兲we ge 具 Ains 2共nc兲 典 ⬅nI⫹nA V 冕 d4x兺 ␮ ,aA ␮ (I)aA ␮ (I)a ⫽12 ␲ 2 ␳ 2nI⫹nA V,共9兲 whe e ␳ is he a e age ins an on adius in he conside ed cooled con igu a ion and ncis he numbe o cooling sweeps. We use an ensemble o 10 independen gauge con igu a ions4a ␤ ⫽6.0 on a 244la ice. Each con igu a ion has been cooled and a e 5, 7, 10, 15, 30 and 100 cooling sweeps ans o med in o he Landau gauge. Using he ISR me hod on each gauge con igu a ion, we ob ain he esul s o Table I. In his able we also p esen he numbe o 共an i-兲ins an ons and he co esponding alue o 具 A2(nc) 典 compu ed by a co ela ion unc ion i 共CFF兲i.e.a i o he la ice p opaga o s o he ins an on co ela ion unc ion, Eq. 共8兲. The CFF me hod is expec ed o be a ec ed di e en ly om he ISR me hod by sys ema ic unce ain ies: ins an on in e ac ions, de o ma ions and quan um luc ua ions 关as we can see in Fig. 1共a兲, a low momen um兴, and we he e o e conside as qui e encou aging he quali a i e ag eemen , be- coming quan i a i e a la ge nc, be ween ISR and CFF e- sul s. We ha e, o simplici y, ansla ed ou la ice esul s in o physical uni s using, o all alues o nc, he nc⫽0 in e se la ice spacing, a⫺1(nc⫽0)⫽1.996 GeV 共a ␤ ⫽6.0). This simple ecipe o e looks he e ec o cooling on he la ice spacing 共see Re . 关15兴and e e ences he ein兲bu his simpli ica ion becomes ha mless a e ex apola ing back ou esul s o nc⫽0. B. ŠAins 2„nc…‹ a ze o cooling The ins an on numbe depends on he numbe o cooling sweeps. This esul may imply ha he cooling p ocedu e des oys no only quan um UV luc ua ions bu some hing else om he semiclassical backg ound o gauge ields. To lessen his p oblem we ake ad an age o he ea ly ecogni- ion o he ins an on con en in a gauge con igu a ion ensu ed by he ISR me hod and pe o m an ex apola ion 关19兴 o nc ⫽0 o he ISR esul s o 具 Ains 2(nc) 典 in he able. We hen ob ain 关see Fig. 1共b兲兴: 具 Ains 2共nc⫽0兲 典 ⫽1.76共23兲GeV2.共10兲 We ha e used a o m a/(b⫹nc) o i and ex apola e. We ha e also a ied a li le his unc ional o m o check he s abili y o he ex apola ion. We ake his esul as indica i e o he non-pe u ba i e ins an on con ibu ion o he 具 A2 典 4Conside ing he p esen size o ou sys ema ic unce ain ies we did no conside i wo hwhile o u he inc ease he s a is ics. TABLE I. Es ima es o 具 Ains 2(nc) 典 . ␳ ( m) 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le 兲we p esen he la ice gluon p opaga o s a e se e al numbe o cooling sweeps 共poin s兲and he co esponding heo e ical ins an on gauge- ield co ela ion unc ions 共lines兲in he pe ec ins an on gas app oxima ion, Eq. 共8兲, plo ed as a unc ion o n2 (n ␮ ⬅L/(2 ␲ )k ␮ ,Lbeing he la ice leng h兲. In Fig. 1共b兲共 igh 兲, we show he ex apola ion a ze o cooling o 具 A2(nc) 典 in physical uni s 共 o h ee di e en ial unc ions兲. INSTANTONS AND THE 具 A2 典 CONDENSATE PHYSICAL REVIEW D 66, 034504 共2002兲 034504-3 condensa e. I we applied o he la ice es ima es o ins an on gas pa ame e s aken om he a ailable li e a u e o Eq. 共9兲, he alue o 具 A(I)2 典 would ange5 om 1 o 2 GeV2.On he o he hand, pa ame e s om ins an on liquid based phenom- enology 关6兴yield es ima es o he o de o 0.5 GeV2. As he quo ed e o in Eq. 共10兲is only s a is ical, his las ange somehow es ima es a ce ain sys ema ic unce ain y. C. Compa ison wi h ŠA2‹ om OPE Ou ins an on es ima e o 具 A2 典 is a semiclassical one, de- p i ed o he necessa y UV luc ua ions, and he e o e no di ec ly compa able wi h 关2兴 具 AOPE 2 典 (10 GeV) ⫽2.4(5) GeV2. The e is o cou se no exac ecipe o com- pa e bo h es ima es, since he sepa a ion be ween he semi- classical nonpe u ba i e domain and he pe u ba i e one canno be exac . We may appeal o he ac ha a he eno - maliza ion poin ␮ , he adia i e co ec ions a e minimized; he e o e a semiclassical es ima e mus bes co espond o 具 AOPE 2 典 a some easonable ␮ . In he example o ␾ 2 acuum expec a ion in he spon aneously b oken ␾ 4model gi en in 关21兴, one inds indeed ha i equals he classical es ima e o ␮ a ound he spon aneously gene a ed mass. In ou p oblem, one could guess ha he co esponding scale should ypically be a ound 1/ ␳ ⯝0.7 GeV, o some gluonic mass, a e y low scale anyway. We canno un 具 AOPE 2 典 (10 GeV) down o 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 he e o e s op a bi a ily ␮ ⬃2.6 GeV, whe e he OPE co ec ed pe u ba i e unning o he G een’s unc ions ails o co ec ly desc ibe hei beha io . This scale o 2.6 GeV u ns ou o be o he same o de 关2兴as he c i ical mass 关22兴 o he gluon p opaga o . A his scale, we ob ain 具 AOPE 2共2.6 GeV兲 典 ⫽1.4共3兲共3兲GeV2,共12兲 whe e he i s quo ed e o jus p opaga es he unce ain y om he OPE de e mina ion o 具 A2 典 and he second one akes in o accoun , in he way p oposed in Re . 关4兴, highe o de s6in ␣ s o unning. V. DISCUSSION AND CONCLUSIONS We a e awa e ha ou me hod o compa ison o 具 Ains 2(nc⫽0) 典 and 具 AOPE 2 典 su e s om a lo o a bi a iness and app oxima ions 共such as he pe ec gas app oxima ion, possible e o s in he ins an on iden i ica ion, he unce ain y in he ex apola ion o ze o cooling sweeps, e c.兲. We ha e aken ca e o c oss check ou es ima es by compa ing di e - en me hods a each s ep o he compu a ion, in pa icula he ISR and CFF 关see Fig. 1共a兲and Table I兲. A compa ison wi h di ec ‘‘measu emen s’’ o he 具 A2 典 condensa e om cooled la ice con igu a ions could be hough o as an addi ional c osscheck. Quali a i e ag eemen is ound o a la ge enough numbe o cooling sweeps, bu his ag eemen is mani es ly des oyed by UV luc ua ions al eady o nc⬃30. O cou se, by using ISR and ins an on gas app oxima ion we sha ply sepa a e UV luc ua ions om he semiclassical backg ound. All hese imp ecisions seem inhe en o he sub- jec anyway. Wi h his in mind, we ne e heless ake he ai ag eemen be ween Eqs. 共10兲and 共12兲as a con incing indica ion ha he A2condensa e ecei es a signi ican ins an onic con ibu- ion. In o he wo ds, he ins an on liquid pic u e migh yield he explana ion o he 1/p2co ec ions o he pe u ba i e beha io o G een unc ions compu ed wi h he malized con- igu a ions on he la ice. ACKNOWLEDGMENTS These calcula ions we e pe o med on he O say APEmille pu chased hanks o unding om he Minis e ` e de l’Educa ion Na ionale and he CNRS. We a e indeb ed o Ba olome Alles, Ca los Pena and Michele Pepe o illumi- na ing discussions. A.D. acknowledges he M.U.R.S.T. o inancial suppo h ough Dec e o 1833/2001 共sho e m isi p og am兲and F.S. acknowledges he Fundacio ´nCa ´ma a o inancial suppo . A.D. wishes o hank he LPT-O say o i s wa m hospi ali y. 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