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Hea y qua k di usion coe icien du ing hyd odynamiza ion : non-equilib ium s.
equilib ium
© 2024 he Au ho s
Published e sion
Peu on, Ja kko; Bogusla ski, Ki ill; Ku kela, Aleksi; Lappi, Tuomas; Lindenbaue ,
Flo ian
Peu on, J., Bogusla ski, K., Ku kela, A., Lappi, T., & Lindenbaue , F. (2024). Hea y qua k di usion
coe icien du ing hyd odynamiza ion : non-equilib ium s. equilib ium. In Ha dP obes2023:
11 h In e na ional Con e ence on Ha d and Elec omagne ic P obes o High-Ene gy Nuclea
Collisions (A icle 091). Sissa Medialab. POS P oceedings o Science, 438.
h ps://doi.o g/10.22323/1.438.0091
2024
PoS(Ha dP obes2023)091
Hea y qua k di usion coe icien du ing
hyd odynamiza ion - non-equilib ium s. equilib ium
K. Bogusla ski,𝑎A. Ku kela,𝑏T. Lappi,𝑐,𝑑 F. Lindenbaue 𝑎and J. Peu on𝑐,𝑑,𝑒,∗
𝑎Ins i u e o Theo e ical Physics, Technische Uni e si ä Wien, 1040 Vienna, Aus ia
𝑏Facul y o Science and Technology, Uni e si y o S a ange , 4036 S a ange , No way
𝑐Depa men o Physics, P.O. Box 35, 40014 Uni e si y o Jy äskylä, Finland
𝑑Helsinki Ins i u e o Physics, P.O. Box 64, 00014 Uni e si y o Helsinki, Finland
𝑒Dep . o Physics, Lund Uni e si y, Söl ega an 14A, Lund,SE-223 62, Sweden
E-mail: [email p o ec ed],[email p o ec ed],
[email p o ec ed],[email p o ec ed],
[email p o ec ed]
We compu e he hea y qua k momen um di usion coe icien using e ec i e kine ic heo y o
a sys em going h ough bo om-up iso opiza ion un il app oxima e hyd odynamiza ion. We
ind ha when compa ing he non he mal di usion coe icien o he he mal one o he same
ene gy densi y, he obse ed de ia ions h oughou he whole e olu ion a e wi hin 30% om he
he mal alue. Fo he mal sys ems ma ched o o he quan i ies we obse e conside ably la ge
de ia ions. We also obse e ha he di usion coe icien in he ans e se di ec ion domina es
a la ge occupa ion numbe , whe eas o an unde occupied sys em he longi udinal di usion
coe icien domina es. Simila ly, we s udy he je quenching pa ame e , whe e we ob ain a
smoo h e olu ion connec ing he la ge alues o he glasma phase wi h he smalle alues in he
hyd odynamical egime.
Ha dP obes2023
26-31 Ma ch 2023
Ascha enbu g, Ge many
∗Speake
©Copy igh owned by he au ho (s) unde he e ms o he C ea i e Commons
A ibu ion-NonComme cial-NoDe i a i es 4.0 In e na ional License (CC BY-NC-ND 4.0). h ps://pos.sissa.i /
PoS(Ha dP obes2023)091
Hea y qua k di usion coe icien du ing hyd odynamiza ion - non-equilib ium s. equilib ium J. Peu on
10−210−1100
hpλ i/hpi
100
101
102
PT/PL
λ= 0.5
λ= 1
λ= 2
λ= 5
λ= 10
Figu e 1: T ajec o y o he sys em du ing he bo om-up he maliza ion on he occupa ion numbe aniso opy
plane. Solid and dashed cu es co espond o di e en ini ial condi ions. Rep oduced om [13].
1. In oduc ion
Recen s udies on anspo coe icien s ou o equilib ium ha e indica ed ha he glasma s age
can ha e conside able impac on he coe icien s [1–8]. Howe e , he e has been a li e a u e gap
un il e y ecen ly: equilib ium anspo coe icien s a e ela i ely well known bu he e olu ion
o anspo coe icien s du ing hyd odynamiza ion emained poo ly unde s ood. We epo he e o
ou ecen s udies whe e we aimed o close he gap and in es iga ed he hea y qua k momen um
di usion coe icien 𝜅[9] and he je quenching pa ame e ˆ𝑞[10] du ing hyd odynamiza ion using
e ec i e kine ic heo y.
The wo main ques ions hese p oceedings add ess in ol e he magni ude o 𝜅compa ed o
i s equilib ium alue du ing hyd odynamiza ion and he ela i e impo ance o ans e se and
longi udinal di usion coe icien s du ing he hyd odynamiza ion p ocess.
2. Me hod: e ec i e kine ic heo y and bo om-up he maliza ion
We ep oduce he bo om-up he maliza ion [11] scena io using e ec i e kine ic heo y [12],
as in [13]. The e olu ion o he sys em is illus a ed in Fig. 1. In o de o make a connec ion o he
e olu ion o he sys em and o he quan i ies, we ha e placed a ew ma ke s in Fig. 1. The s a ma ke
is placed a 𝑓=1
/𝜆=1
/(4𝜋𝑁𝑐𝛼𝑠),whe e 𝑁𝑐is he numbe o colo s and 𝛼𝑠is he s ong coupling
cons an . Fo weak couplings his also co esponds o maximum aniso opy. The ci cle ma ke is
placed a minimum occupancy, which in he bo om-up he maliza ion pic u e co esponds o 𝛼𝑠.
Finally, he iangle ma ke is placed a 𝑃𝑇/𝑃𝐿=2, co esponding o app oxima e iso opy. The
c osses a he bo om co espond o he expec ed alues a he mal equilib ium.
In e ec i e kine ic heo y he dynamical deg ee o eedom is he gluon phase space densi y
𝑓(𝒑)=1
𝜈𝑔
d𝑁
d3𝑥d3𝒑,whose ime-e olu ion is gi en by he Bol zmann equa ion
−𝜕 𝑓 (𝒑)
𝜕𝜏 =C1↔2[𝑓(𝒑)] + C2↔2[𝑓(𝒑)] − 𝑝𝑧
𝜏
𝜕
𝜕𝑝𝑧
𝑓(𝒑).(1)
2
PoS(Ha dP obes2023)091
Hea y qua k di usion coe icien du ing hyd odynamiza ion - non-equilib ium s. equilib ium J. Peu on
10−410−310−210−1100
τ/τBMSS
0.0
0.2
0.4
0.6
0.8
1.0
1.2
κ/κT∗
eq
λ= 0.5
λ= 1
λ= 2
λ= 5
λ= 10
10−510−410−310−210−1100
τ/τBMSS
0.0
0.2
0.4
0.6
0.8
1.0
1.2
κ/κε
eq
λ= 0.5
λ= 1
λ= 2
λ= 5
λ= 10
10−510−410−310−210−1100101
τ/τBMSS
0
1
2
3
4
5
κ/κmD
eq
λ= 10
λ= 5
λ= 2
λ= 1
λ= 0.5
Figu e 2: Th ee di e en ways o compa e equilib ium and nonequilib ium. Le : compa ing o he same
e ec i e empe a u e o he in a ed modes. Cen e : o he same ene gy densi y. Righ : o he same
sc eening mass.
The dominan con ibu ion o he di usion coe icien 𝜅a ises om sca e ing wi h he medium
gluons ia -channel gluon exchange [14]. The coe icien is gi en by
3𝜅=Δ𝑘2
Δ𝑡=1
2𝑀∫𝒌𝒌′𝒑′
(2𝜋)3𝛿3(𝒑+𝒌−𝒑′−𝒌′)2𝜋𝛿 (𝑘′−𝑘)𝒒2|M𝜅|2𝑓(𝒌)(1+𝑓(𝒌′)),
(2)
whe e 𝑘, 𝑘′a e he ingoing and ou going gluon momen a, 𝑞=𝑘−𝑘′,is he momen um ans-
e and 𝑝, 𝑝′a e he ingoing and ou going hea y qua k momen a. The in eg a ion measu e is
gi en by ∫𝒑=∫d𝑝3/2𝑝0(2𝜋)3. The ma ix elemen co esponding o his p ocess is |M|2
𝜅=
𝑁𝑐𝐶𝐻𝑔416𝑀2𝑘2(1+cos2𝜃𝒌𝒌′)
(𝑞2+𝑚2
𝐷)2.The e ec i e empe a u e o he in a ed modes is
𝑇∗=2𝜆
𝑚𝐷∫d3𝑝/(2𝜋)3𝑓(𝑝)(1+𝑓(𝑝)),whe e he Debye sc eening mass is 𝑚2
𝐷=4∫d3𝑝/(2𝜋)3𝜆 𝑓 (𝑝)/𝑝.
When compa ing equilib ium and nonequilib ium sys ems, we need an es ima e o he empe a u e
o he co esponding equilib ium sys em. This empe a u e is de ined h ough ene gy densi y 𝜀
as 𝑇𝜀=(30 𝜀/𝜋2𝜈𝑔)1/4.In equilib ium he quan i ies abo e (𝑇∗, 𝑚𝐷, 𝑇𝜀)a e compu ed using he
Bose-Eins ein dis ibu ion.
3. Resul s
Since he e is no unambiguous way o compa e equilib ium and nonequilib ium sys ems,
we will y o compa e he equilib ium and nonequilib ium sys ems o he same 𝑚𝐷, 𝑇∗and 𝜀.
The compa ison is done as a unc ion o ime. As a consequence, he co esponding he mal
sys em changes du ing he ime-e olu ion. We escale he ime wi h he he maliza ion imescale
𝜏BMSS =𝛼−13/5
𝑠/𝑄𝑠.
The esul s a e shown in Fig. 2. We obse e ha when ma ching o he same sc eening mass
𝑚𝐷o in a ed empe a u e 𝑇∗ he e a e la ge de ia ions du ing he equilib a ion. Howe e when
ma ching o he same ene gy densi y 𝜀 he de ia ions a e (depending on he coupling) wi hin
app oxima ely ∼30% du ing he e olu ion. Thus ma ching o he same ene gy densi y (Landau
ma ching) is he bes way o compa e equilib ium and nonequilib ium sys ems in his case.
We can also b eak he compa ison down in o ans e se and longi udinal componen s as we
ha e done in Fig. 3. We obse e ha he ans e se (𝜅𝑇) and longi udinal (𝜅𝐿) di usion coe icien s
3
PoS(Ha dP obes2023)091
Hea y qua k di usion coe icien du ing hyd odynamiza ion - non-equilib ium s. equilib ium J. Peu on
10−510−410−310−210−1100
τ/τBMSS
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
κT/κ
eq
λ= 0.5
λ= 1
λ= 2
λ= 5
λ= 10
10−510−410−310−210−1100
τ/τBMSS
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
κz/κ
eq
λ= 0.5
λ= 1
λ= 2
λ= 5
λ= 10
Figu e 3: T ans e se and longi udinal di usion coe icien s compa ed o hei equilib ium alues o he
same ene gy densi y.
10−510−410−310−210−1100
τ/τBMSS
0.0
0.5
1.0
1.5
2.0
2.5
κT/κz
λ= 0.5
λ= 1
λ= 2
λ= 5
λ= 10
Figu e 4: Ra io o he ans e se and longi udinal di usion coe icien s du ing he hyd odynamiza ion
p ocess.
beha e quali a i ely simila ly o he ull coe icien (excep in he case o he longi udinal di usion
coe icien a e y ea ly imes). Fo smalle coupling 𝜆we obse e la ge de ia ions. This is mos
likely due o he ac ha o small coupling he bo om-up he maliza ion is ep oduced mo e
accu a ely.
Fig. 4shows he a io o ans e se and longi udinal di usion coe icien s du ing he e olu ion.
We obse e ha he ans e se di usion coe icien is ini ially enhanced compa ed o he longi udinal
coe icien . When he sys em becomes unde occupied, he hie a chy is in e ed, and he longi udinal
coe icien is enhanced compa ed o he ans e se coe icien . The aniso opy o he coe icien s
can become sizable, o he o de 10 - 40 %, depending on he coupling s eng h. The plo also
shows he eme gence o a limi ing a ac o , which we will discuss elsewhe e in mo e de ail.
Then we p oceed o he je quenching ac o ˆ𝑞de ined as ˆ𝑞𝑖 𝑗 =d⟨𝑞𝑖𝑞𝑗⟩
d𝐿. He e we use he
ollowing con en ion: ˆ𝑥je di ec ion, ˆ𝑧beam di ec ion. The je quenching ac o is gi en by
ˆ𝑞𝑖 𝑗 =1
4𝑑𝑅
lim
|p|→∞ ∫kk′p′
𝑞⊥<Λ⊥
𝑞𝑖
⊥𝑞𝑗
⊥(2𝜋)4𝛿4(𝑃+𝐾−𝑃′−𝐾′)M𝑎𝑔
𝑎𝑔
2
|p|𝑓k(1+𝑓k′),(3)
whe e M𝑎𝑔
𝑎𝑔
2is he ma ix elemen co esponding o elas ic sca e ings o in-medium gluons.
He e we conside a qua k je . Howe e he alue o ˆ𝑞 o a gluon je can be ob ained by scaling
wi h a simple Casimi ac o . The cu es shown in Fig. 5a e ob ained as ollows: We ma ch 𝜀 o
glasma as in [6] o ob ain he alue o 𝑄𝑠a he ini ial condi ion. Then ˆ𝑞is ma ched o he esul
o JETSCAPE [15] a he iangle ma ke o ob ain a alue o he ans e se momen um ans e
4

PoS(Ha dP obes2023)091
Hea y qua k di usion coe icien du ing hyd odynamiza ion - non-equilib ium s. equilib ium J. Peu on
10−1100101
τ( m/c)
0
2
4
6
8
10
ˆ
q(GeV2/ m)
Glasma Kine ic
heo y Hyd odynamics
ˆq om glasma
λ= 10
Qs= 1.4GeV
Eje = 100 GeV
Eje = 20 GeV
Figu e 5: The alue o he je quenching ac o ˆ𝑞compu ed acco ding o he p ocedu e desc ibed in he ex .
cu o Λ⊥a ha ime. The bands co espond o di e en cu o models and ini ial condi ions. We
obse e ha ou esul s ma ch he glasma simula ion a ea ly imes ela i ely well and smoo hly
connec o he hyd odynamic e olu ion.
4. Conclusions
The wo main conclusions o hese p oceedings a e, ha du ing he hyd odynamiza ion 𝜅
is wi hin 30 % om i s equilib ium alue when he equilib ium and nonequilib ium sys ems a e
ma ched o he same ene gy densi y. The second conclusion is ha he e is a clea hie a chy be ween
ans e se and longi udinal di usion coe icien s. Ini ially he ans e se di usion coe icien 𝜅𝑇
domina es. A unde occupa ion 𝜅𝑧is la ge . In bo h cases he de ia ion is oughly a ac o o wo.
Ou esul s may be used o phenomenological desc ip ions o hea y qua k di usion, qua ko-
nium dynamics and je quenching. Ou u u e plans in ol e s udying limi ing a ac o s using 𝜅and
ˆ𝑞as es obse ables.
Acknowledgmen s
The au ho s would like o hank N. B ambilla, M. Escobedo, D.I. Mülle , A. Ro hkop and
M. S ickland o aluable discussions. This wo k is suppo ed by he Eu opean Resea ch Council,
ERC-2018-ADG-835105 Yoc oLHC. This wo k was also suppo ed unde he Eu opean Union’s
Ho izon 2020 esea ch and inno a ion by he STRONG-2020 p ojec (g an ag eemen No. 824093).
The con en o his a icle does no e lec he o icial opinion o he Eu opean Union and espon-
sibili y o he in o ma ion and iews exp essed he ein lies en i ely wi h he au ho s. This wo k
was unded in pa by he Knu and Alice Wallenbe g ounda ion, con ac numbe 2017.0036. TL
and JP ha e been suppo ed by he Academy o Finland, by he Cen e o Excellence in Qua k
Ma e (p ojec 346324) and p ojec 321840. KB and FL would like o hank he Aus ian Science
Fund (FWF) o suppo unde p ojec P 34455, and FL is addi ionally suppo ed by he Doc o al
P og am W1252-N27 Pa icles and In e ac ions. The au ho s wish o acknowledge CSC – IT
Cen e o Science, Finland, o compu a ional esou ces. We acknowledge g an s o compu e
capaci y om he Finnish G id and Cloud In as uc u e (pe sis en iden i ie u n:nbn: i: esea ch-
in as-2016072533 ). The au ho s wish o acknowledge he Vienna Scien i ic Clus e (VSC) p ojec
71444 o compu a ional esou ces.
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Hea y qua k di usion coe icien du ing hyd odynamiza ion - non-equilib ium s. equilib ium J. Peu on
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6