Heavy quark diffusion coefficient during hydrodynamization : non-equilibrium vs. equilibrium

This is a sel -a chi ed e sion o an o iginal a icle. This e sion may di e om he o iginal in pagina ion and ypog aphic de ails. Au ho (s): Ti le: Yea : Ve sion: Copy igh : Righ s: Righ s u l: Please ci e he o iginal e sion: CC BY-NC-ND 4.0 h ps://c ea i ecommons.o g/licenses/by-nc-nd/4.0/ 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 ). 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