Spectral function for overoccupied gluodynamics from classical lattice simulations

Vol. 50 (2019) Ac a Physica Polonica B No 6
SPECTRAL FUNCTION FOR OVEROCCUPIED
GLUODYNAMICS FROM CLASSICAL LATTICE
SIMULATIONS∗
K. Bogusla skia,b, A. Ku kelac,d, T. Lappib,e, J. Peu on
aIns i u ü Theo e ische Physik, Technische Uni e si ä Wien
1040 Vienna, Aus ia
bDepa men o Physics, P.O. Box 35, 40014 Uni e si y o Jy äskylä, Finland
cTheo e ical Physics Depa men , CERN, 1211 Gene a, Swi ze land
dFacul y o Science and Technology, Uni e si y o S a ange
4036 S a ange , No way
eHelsinki Ins i u e o Physics, P.O. Box 64, 00014 Uni e si y o Helsinki, Finland
Eu opean Cen e o Theo e ical S udies in Nuclea Physics
and Rela ed A eas (ECT*) and Fondazione B uno Kessle
S ada delle Taba elle 286, 38123 Villazzano (TN), I aly
(Recei ed Ap il 2, 2019)
We s udy he spec al p ope ies o an o e occupied gluonic sys em
a om equilib ium. Using classical Yang–Mills simula ions and linea
esponse heo y, we de e mine he s a is ical and spec al unc ions. We
measu e dispe sion ela ions and damping a es o ans e sally and longi-
udinally pola ized exci a ions in he gluonic plasma, and also s udy u he
s uc u es in he spec al unc ion.
DOI:10.5506/APhysPolB.50.1105
1. In oduc ion
The main pu pose o he p og am o ul a ela i is ic hea y-ion colli-
sions is o c ea e and s udy he p ope ies o decon ined QCD ma e in he
labo a o y. Ou pu pose he e is o s udy s ongly o e occupied colo ield
con igu a ions ha a e ele an o se e al o he di e en aspec s o he col-
lision p ocess. In he e y ini ial p e-equilib ium s age a e he collision, he
dynamics is domina ed by gluon sa u a ion. The cha ac e is ic aspec o his
egime is he exis ence o a semiha d dominan ans e se momen um scale,
he sa u a ion scale QsΛQCD, gene a ed by nonlinea in e ac ions o he
∗P esen ed a he C acow Epiphany Con e ence on Ad ances in Hea y Ion Physics,
K aków, Poland, Janua y 8–11 2019.
(1105)
1106 K. Bogusla ski e al.
dense gluonic sys em. A he sa u a ion scale, he gluon ield is nonpe u -
ba i ely s ong. This means ha he gluon ield s eng hs, o occupa ion
numbe s o gluonic s a es, a e pa ame ically p opo ional o he in e se o
he coupling AµAµ∼1/αs. La e in he e olu ion, i is gene ally belie ed
ha he plasma eaches a s a e close o local he mal equilib ium. In such a
he mal sys em, a pa o he deg ees o eedom, namely he so ields wi h
momen a p.gT a e, simila ly, nonpe u ba i ely la ge. These so ields
a e impo an o many p ope ies o QCD ma e . Thus, in bo h cases,
we a e in a si ua ion whe e we wan o unde s and he eal- ime beha io
o QCD sys ems wi h bo h a pe u ba i e momen um scale (gene ically de-
no ed he e by QΛQCD) and, he e o e, weak coupling cons an αs1,
bu also gluon ields (a leas o some impo an momen um modes) in
an o e occupied s a e. In his egime, he app oxima ion o classical ields
p o ides a powe ul nonpe u ba i e ool.
The s anda d me hod o unde s anding eal- ime QCD dynamics in
weak coupling is p o ided by he ha d ( he mal) loop (HTL) app oach. He e,
one de elops a pe u ba ion heo y based on a sepa a ion o wo di e en
momen um scales. The deg ees o eedom a he ha d scale Qcan be
hough o as (classical) pa icles, and in e ac wi h so (∼mD) modes
ha can be hough o as (classical) ields. In an equilib ium plasma, he
small coupling cons an p o ides such a scale sepa a ion, bu his app oach
can also be gene alized o nonequilib ium sys ems. In addi ion o analy ical
calcula ions, he e a e also many nume ical implemen a ions (see e.g. [1–3])
o his idea, based on explici ly di e en desc ip ions o pa icle- and ield-
like deg ees o eedom. Such calcula ions ha e been used o unde s and
e.g. sphale on ansi ions in he mal sys ems and plasma ins abili ies in
aniso opic ones.
In a hea y-ion collision he sys em s a s om a con igu a ion, a e y
ea ly imes τ∼1/Qs, whe e he e is only one scale Q∼mD∼Qs. Equi-
lib a ion is hen he p ocess whe e he wo scales de elop h ough a ious
s ages o become pa ame ically di e en , mDT. Following such a sce-
na io, especially a i s ea lies s ages, is p oblema ic wi h a quali a i ely
di e en desc ip ion o ha d and so modes. In p ac ical e ms, i he so
modes a e desc ibed as ields on a la ice wi h la ice spacing a, he la ice
needs o be ine enough o ep esen hem: mD1/a. Howe e , a pa icle-
like physical pic u e o he ha d modes implies ha hey a e localized a he
scale o he la ice spacing, equi ing a1/Q, since he de B oglie wa e-
leng h o he ha d pa icles is ∼1/Q. In such a desc ip ion, i is he e o e
no possible o ha e mD∼Q.
The app oach ha we a e ollowing he e a oids his p oblem. We ea
all deg ees o eedom on he same la ice, subjec o he same la ice UV
cu o 1/a. Thus, we do no need o ha e a la ge scale sepa a ion. On he
Spec al Func ion o O e occupied Gluodynamics om Classical La ice . . . 1107
o he hand, classical la ice simula ions being ela i ely inexpensi e, one
can i in a a he la ge sepa a ion be ween he ha d and so scales, e en
while main aining a con ollable con inuum limi Q1/a. We ea e en
he ha d modes as classical ields, no pa icles. In he HTL limi , he
classical ea men o he ha d scales should no ma e o he ha d+so
in e ac ions, since he only impo an hing abou he ha d modes a e hei
colo cu en s, no whe he hese cu en s a e made o pa icles o ields. A
d awback o he classical la ice app oach is ha he in e ac ions be ween
ha d modes a e ea ed inco ec ly (as classical ields ins ead o pa icles
as hey should), al hough in he o e occupied egime he e o is o highe
o de in αs. Thus, ou sys em would, ul ima ely, he malize o an unphysical
classical equilib ium. The in e ac ions be ween he ha d modes a e, howe e ,
much slowe , and o en neglec ed in he pa icle+ ield simula ions in any
case. We e e he eade o e.g. Re s. [4–6] o a discussion on he alidi y
o he classical app oxima ion. One can see ou calcula ional se up as an
ex ension o HTL se up o si ua ions whe e he scale sepa a ion mD/Q can
be a ied smoo hly up o la ge alues.
This pape e iews he ecen esul s p esen ed in mo e de ail in Re . [7].
In he ollowing, we shall b ie ly desc ibe ou nume ical me hod o linea ized
luc ua ions, based on he algo i hm de eloped in Re . [8] and he obse ables
ha we measu e. We hen in oduce ou es case sys em, he iso opic sel -
simila UV cascade o gluons. We will hen e iew some o ou nume ical
esul s ob ained so a and discuss in e es ing p ospec s o u u e p ojec s.
2. Me hods
Fo he ime e olu ion o classical Yang–Mills ields, we use he s an-
da d Hamil onian la ice [9,10] o mula ion o gauge heo y in eal ime.
He e, ins ead o he gauge po en ial Aiand he co a ian de i a i e Di=
∂i+ig[Ai,·], one wo ks wi h link ma ices connec ing la ice si es Ui(x) =
eiagAi(x)and co a ian ini e di e ences ob ained using he links. The canon-
ical conjuga e a iable o he gauge po en ial is he ch omoelec ic ield
Ei=∂ Ai. The Hamil onian se up is o mula ed in he empo al gauge
A0= 0, whe e one mus ake ca e o sa is y also he Gauss’ law cons ain
[Di, Ei] = 0.
Ou i s measu able is he “s a is ical unc ion”, de ined as a symme ic
wo-poin unc ion o he ield ope a o
Fab
jk x, x0=1
2Dnˆ
Aa
j(x),ˆ
Ab
kx0oE.(1)
The s a is ical unc ion measu es “ he mal” luc ua ions in he ield, and is
ela ed o he phase-space densi y o pa icles in sys em (p). In he classical
1108 K. Bogusla ski e al.
app oxima ion, he commu a o is supp essed by a powe o g(because Ai∼
1/g ). Thus, he e Fis jus a wo-poin unc ion o he classical ield
Fab
jk x, x0=DAb
j(x)Ab
kx0Ecl .(2)
The s a is ical unc ion a equal ime is an o en measu ed quan i y in CYM
calcula ions, bu in Re . [7], we also measu e i a unequal ime.
In addi ion o he s a is ical unc ion, he o he in gene al-independen
co ela o in quan um ield heo y is he “spec al unc ion”
ρab
jk x, x0=iDhˆ
Aa
j(x),ˆ
Ab
kx0iE.(3)
I is a genuinely “quan um” obse able and p opo ional o ∼~due o he
ield ope a o commu a ion ela ions. We can ne e heless measu e i in he
CYM simula ions by using i s ela ion o he e a ded p opaga o as
Gab
R,jk  , 0, p=θ( − 0)ρab
jk  , 0, p.(4)
The e a ded p opaga o measu es he linea esponse o he sys em o an
ex e nal pe u ba ion, and we measu e i wi h he algo i hm de eloped in
Re . [8]. We spli he gauge ield in o a backg ound ield and a luc ua ion
ˆ
Aa
i(x)→ˆ
Aa
i(x) + ˆaa
i(x),(5)
whe e he luc ua ion gene a ed by an ex e nal in ini esimal cu en is gi en
by he e a ded p opaga o
Dˆab
i(x)E=Zd4x0Gbc
R,ik x, x0jk
cx0.(6)
This leads o ou algo i hm o calcula e he s a is ical unc ion. We i s , a
ime 0, pe u b he sys em wi h a cu en ha exis s only o one imes ep:
jk
c(x)∼eik·xδ( − 0). We hen ollow he linea ized equa ions o mo ion o
he luc ua ion and i s ime de i a i e aa
i(x) = hˆaa
i(x)i, ei
a(x). Finally, a
a la e ime > 0, we measu e he co ela ion o he ield aa
i( )wi h he
cu en ji
a( 0)and use his o ex ac he momen um space spec al unc ion
ρ(p, ). A simila p ocedu e wi h he elec ic ield luc ua ion can be used
o ob ain ime de i a i es o he spec al unc ion.
3. O e occupied cascade
Le us hen mo e on o discuss ou es case o e occupied nonequilib ium
sys em, he iso opic sel -simila cascade. This sys em has been ex ensi ely
Spec al Func ion o O e occupied Gluodynamics om Classical La ice . . . 1109
s udied by many g oups (see e.g. [4,6,11]), and i has been ound ha
a kine ic heo y o mula ion can desc ibe he basic p ope ies seen in he
nume ical s udies. In his sys em, one s a s om an iso opic ield con igu-
a ion whe e only he modes up o some cha ac e is ic ha d momen um a e
occupied
(p)∼n0
g2θ(p0−p).(7)
In p ac ical calcula ions, one ypically eplaces a s ic he a unc ion wi h
a smoo he Gaussian momen um dis ibu ion, bu his de ail, o he ini ial
occupa ion numbe n0/g2∼1/g2o momen um scale p0ma e li le o
he la e sel -simila ime e olu ion o he sys em. This is ins ead ully
de e mined by he conse ed o al ene gy densi y, which de ines he eal
cha ac e is ic ha d scale o he p oblem as ε∼Q4/g2. In wha ollows,
we speci ically de ine Q=4
q10π3/2g2ε/ √2(N2
c−1), scale dimension ul
quan i ies wi h Q o dimensionless ones and, unless o he wise s a ed, plo
quan i ies measu ed a Q = 1500.
Du ing he ime e olu ion o he sys em, he ene gy cascades owa ds
UV modes in such a way ha all he momen um modes up o pmax ∼ 1/7
a e occupied. The ypical occupa ion numbe (e.g. a he ha d scale pmax)
dec eases wi h ime as
(pmax)∼ −4/7.(8)
The scaling beha io
( , p) = −4/7 sp/ 1/7(9)
becomes e iden in he nume ical esul when we scale he momen um dis-
ibu ion o gluons by 4/7, and plo i in e ms o he scaled momen um
−1/7p, as shown in Fig. 1(le ).
In he HTL o kine ic heo y, one es ima es he Debye o plasmon scale
om he (ha d) pa icle dis ibu ion as
m2= 2NcZd3p
(2π)3
(p)
p.(10)
In he scaling egime (9), his gi es an es ima e
m∼ −1/7(11)
o he ime dependence o he so scale. This scaling is e i ied nume ically
in Fig. 1( igh ). This leads us o an impo an aspec o his sel -simila
a ac o sys em, namely ha he scale sepa a ion pmax/m g ows wi h ime.

1110 K. Bogusla ski e al.
0.001
0.01
0.1
1
0.1 1
Dis ibu ion: ( / 0)4/7 g2
Momen um: ( / 0)-1/7 p / Q
Q = 250
Q = 400
Q = 800
Q = 1500
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 1000 2000 3000 4000 5000 6000
Mass om HTL: (Q )1/7 m / Q
Time: Q
n0 = 3.2
n0 = 2
n0 = 1
n0 = 0.5
n0 = 0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 1000 2000 3000
(p0 )1/7 m / p0
p0
Fig. 1. Le : Scaled momen um dis ibu ion in e ms o he escaled momen um,
demons a ing he sel -simila na u e o he cascade. Righ : Scaling beha io o
he Debye scale, wi h he inse showing he unscaled alues. The dis ibu ion (p)
is es ima ed using he equal ime elec ic ield co ela o .
By looking a di e en imes, one can he e o e smoo hly u n on o o he
pa ame e ha de e mines he alidi y o he HTL app oxima ion. In he
ollowing, we will be mos ly wo king a a he la e imes whe e his scale
sepa a ion is clea , in o de o compa e ou calcula ion o HTL in a egime
whe e i should be a good desc ip ion o he sys em.
4. Spec al unc ion and dispe sion ela ions
We s a his discussion wi h he s a is ical and spec al unc ions o
ans e sely pola ized quasipa icles. Rele an ques ions he e a e whe he
hey bo h exhibi he same peak s uc u e, and wha a e he loca ions and
wid hs o he peaks. A small equencies, one would also expec o see an
addi ional s uc u e, he “Landau cu ”. The spec al unc ion is na u ally
no malized o uni y (ac ually ~), whe eas he s a is ical unc ion is p opo -
ional o he numbe o pa icles in he sys em. In o de o compa e he
peak s uc u e in he wo co ela o s, i is con enien o di ide he s a is-
ical unc ion by i s alue a ∆ = 0− = 0, which is wha we will do in
he ollowing plo s. Figu e 2shows he spec al unc ion o he ans e se
pola iza ion. Fi s o all, we see ha he e is a e y nice ag eemen be ween
he wo unc ions, which could be in e p e ed as a gene alized luc ua ion–
dissipa ion heo em. In equency space, one can see a e y nice Lo en zian
shape, and ex ac he quasipa icle wid h (plasmon damping a e). One
can e en clea ly see he Landau cu s uc u e a small equencies, in ough
ag eemen wi h he HTL heo y cu e ha uses only he alue o mex ac ed
nume ically om he da a as an inpu pa ame e .
Spec al Func ion o O e occupied Gluodynamics om Classical La ice . . . 1111
-0.5
0
0.5
1p = 0.09 Q p = 0.55 Q
F
..T / F
..T( , )
ρ
.T
-0.5
0
0.5
1p = 0.15 Q
Co ela ion unc ions
p = 0.70 Q
-1
-0.5
0
0.5
1
0 50 100 150
p = 0.30 Q
Time: Q ∆
0 50 100 150 200
p = 0.90 Q
0
20
40
60
80
100 p = 0.09 Q p = 0.55 Q
F
..T / F
..T( , )
ρ
.T
0
20
40
60
80
100 p = 0.15 Q
Co ela ion unc ions
p = 0.70 Q
0
20
40
60
80
100
0 0.1 0.2 0.3 0.4 0.5
p = 0.30 Q
F equency: ω / Q
0.5 0.6 0.7 0.8 0.9 1
p = 0.90 Q
-6
-4
-2
0
2
4
6 p = 0.05 Q
Spec al unc ion: ρT
0
400
800
-6
-4
-2
0
2
4
6p = 0.12 Q
0
200
400
-4
-2
0
2
4
0 50 100 150 200
p = 0.22 Q
Time: Q ∆
0
100
200
0 0.1 0.2 0.3 0.4
F equency: ω / Q
Fig. 2. Le : Spec al and s a is ical unc ions in he ime domain o ans e se
pola iza ion. Cen e : The same unc ions in he equency domain, co esponding
o ωρ(ω). He e, he s a is ical unc ion has been di ided by i s equal ime alue,
which makes he no maliza ion he same a ze o ime sepa a ion ∆ = 0. Righ :
The spec al unc ion ρin he ime and equency domain. Wi h one powe o
equency less, he s uc u es a small ωa e now mo e isible. The dashed black
line is he LO HTL unc ional o m, exhibi ing a ze o-wid h peak and a “Landau
cu ” egion a small ω.
Fo he longi udinal pola iza ion mode, as shown in Fig. 3, he s o y is
e y simila . The e is a good ag eemen be ween he s a is ical and spec al
unc ions, when he o me is no malized o he equal ime alue. Fo he
longi udinal pola iza ion s a e, he measu emen is ha de , since a high
momen a, he quasipa icle peak ge s weake and me ges wi h he Landau
cu , as could ha e been expec ed om HTL.
-1
-0.5
0
0.5
1p = 0.05 Q
Longi udinal spec al unc ion: ρ
.L
0
50
100
150
-0.5
0
0.5 p = 0.12 Q
0
20
40
60
-0.4
-0.2
0
0.2
0.4 p = 0.22 Q
0
5
10
15
-0.2
0
0.2
0 50 100 150 200 250
p = 0.29 Q
Time: Q ∆
0
2
4
6
8
0 0.1 0.2 0.3 0.4
F equency: ω / Q
Fig. 3. (Colo online) S a is ical (medium g ay/o ange) and spec al (o he colo s)
unc ion o he longi udinal pola iza ion in he ime and equency domains.
1112 K. Bogusla ski e al.
By ex ac ing he loca ion o he peak as a unc ion o momen um, we can
ex ac a quasipa icle dispe sion ela ion om ou da a as shown in Fig. 4.
O e all, wi hin ou s a is ics, i is no possible o di e en ia e be ween he
HTL unc ional o m and a ela i is ic dispe sion ela ion ω2=m2+p2.
Figu e 4also shows da a o he longi udinal pola iza ion. The di e ence
be ween he wo pola iza ion s a es is quali a i ely as one would expec
om HTL. Quan i a i ely, we can cha ac e ize he dispe sion ela ion by
wo di e en masses om he small and la ge momen um limi s by de ining
he plasmon mass ωpl ≡ω(p→0) and he mass gap a p→ ∞, deno ed by
m∞. Ou nume ical es ima e o he a io o hese scales is
ωpl
m∞
= 0.96 ,(12)
whe e he HTL p edic ion o his quan i y would be
ωpl
m∞
=p2/3≈0.82 ,(13)
and he NLO co ec ion has been calcula ed o be nega i e [12].
2
1
0.2 0.5 21
ω = p
Longi udinal disp. es ima e: ωL / mHTL
Momen um: p / mHTL
1923, Qas = 0.47
2563, Qas = 0.7
ans e se da a
ωLHTL
ωTHTL
0.8
0.9
1
0 1 2 3
Combina ion: (ωT2 - p2)0.5 / mHTL
Momen um: p / mHTL
Fig. 4. (Colo online) Le : Dispe sion ela ion o ans e se and longi udinal
pola iza ions, compa ed o he HTL unc ional o m applying mcalcula ed om
(p)using Eq. (10), and o a ela i is ic dispe sion ela ion ω2=m2+p2. Righ :
pω2−p2/m e sus p o be e illus a e he di e ence be ween a ela i is ic dis-
pe sion ela ion (dashed s aigh line), he HTL unc ional o m (solid ed line)
and he nume ical esul o he ans e se pola iza ion s a e. The g een c osses
a e a simula ion wi h Qa = 0.47 and he blue s a s wi h Qa = 0.7.
5. Fu he obse ables
Le us now discuss a li le mo e he equal ime ield co ela o , i.e. (p),
which so a was used as a no maliza ion ac o o compa e he s a is ical
and spec al unc ions. In he small momen um limi , he expec a ion om
HTL heo y would be o see a classical he mal dis ibu ion
Spec al Func ion o O e occupied Gluodynamics om Classical La ice . . . 1113
(p)≈T∗
ω(p).(14)
The e ec i e empe a u e o he so ields should be gi en by
T∗≡
1
2Rp ( , p) ( ( , p) + 1)
Rp
( ,p)
√m2+p2∼ −3/7,(15)
whe e o classical ields, one neglec s he 1 in ( + 1). Ou esul s o
he equal ime elec ic ield co ela o a e shown in Fig. 5(le ). We see
ha he e is a signi ican enhancemen in he in a ed compa ed o he
expec a ion. We do no cu en ly ha e a compelling in e p e a ion o his
obse a ion, bu ce ainly he ag eemen wi h expec a ions om HTL is no
as good as o he spec al unc ion.
10-2
10-1
100
101
1 10
S a is ical unc ion: F
.. / T*
Momen um: p / mHTL
ans, Q = 250
ans, Q = 1500
long, Q = 250
long, Q = 1500
0
0.005
0.01
0.015
0.02
0 0.2 0.4 0.6 0.8 1
Damping a e: γT,L / Q
Momen um: p / Q
ans., 1923, Qas = 0.47
ans., 2563, Qas = 0.7
long., 1923, Qas = 0.47
long., 2563, Qas = 0.7
γHTL(p=0)
Fig. 5. Le : The equal ime elec ic ield co ela o o ans e se and longi udinal
pola iza ions. The shaded g ay bands a e he expec a ion om HTL heo y. Righ :
The plasmon damping a e. F om he HTL heo y, we only ha e an es ima e o
he alue a p= 0.
In he HTL heo y, he plasmon damping a e is also ela ed o he e ec-
i e empe a u e T∗. Ou nume ical esul o he damping a e, ex ac ed
om i ing he ime domain signal wi h an exponen ially decaying unc-
ion, is shown in Fig. 5( igh ). Ou measu emen s o he ime dependence
o γa e consis en wi h he ime scaling om Eq. (15). Howe e , we a e
able o ex ac a esul in a a he la ge ange o momen a, whe eas om
he HTL heo y, we only ha e poin s a p= 0 (shown in he igu e) and
p=∞[13] (no shown, bu consis en wi hin e o s). Also he e, he ag ee-
men o his T∗-dependen obse able wi h HTL is no as good as o he
spec al unc ion.