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. This wo k was suppo ed in pa by he
Eu opean Union Human Po en ial P og am unde con ac
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INSTANTONS AND THE
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