PoS(LATTICE2019)170 8. We = τ N . https://pos.sissa.it/ c T † ∗ [email protected] [email protected] [email protected] [email protected] [email protected] Speaker. This research is carried out under the project with contract number "SHuSs 2017/28", with the approval and support present preliminary results for distances up to 2.5 fm and temperatures up to 1.09 We explore the profiles of theSU(3) flux tube lattice connecting gauge a theory quark in and closethis an vicinity work, antiquark to we in the consider high critical the temperature temperature morethe of realistic previous the case study phase to of transition. SU(3) the gauge flux In grouping tube and procedure with making for dynamical use quarks, noise of the extending reduction. Gradientstrengths The flow method in profiles in the of smooth- the flux chromoelectric tubethe and have highly chromomagnetic been improved field staggered measured quark from (HISQ) Polyakov action loop-plaquette on a correlations lattice using with temporal extent ∗ † Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). 37th International Symposium on Lattice Field16-22 Theory June - 2019 Lattice2019 Wuhan, China of the Ministry ofTechnology. Education, This work Culture, was Science partly and- supported by TRR Sports the 211. of DFG We Mongolia (German areWe Research and grateful also Foundation) to Mongolian - thank the Foundation project HotQCD Bielefeld number for collaborationacknowledge 315477589 colleagues Science for CCNU providing colleagues for and us for the with theirand so the financial analysis dynamical support called have fermion and been ParallelGPUCode configurations. fruitful performed program discussion at the on of Institute physics. of the Flux Physics Gradient and tube Technology measurements flow of Mongolian method. Academy of Sciences. We Heng-Tong Ding Institute of Particle Physics, Central ChinaE-mail: Normal University, 152 LuoYu Road, Wuhan 430079 Olaf Kaczmarek Fakultät für Physik, Universität Bielefeld, D-33615Institute Bielefeld, of Germany Particle Physics, Central ChinaE-mail: Normal University, 152 LuoYu Road, Wuhan 430079 Battogtokh Purev Department of Theoretical Physics, Institute ofSciences, Physics Peace and Ave. Technology, 54b, Mongolian 13330 Academy Ulaanbaatar, of Mongolia E-mail: Sodbileg Chagdaa Department of Theoretical Physics, Institute ofSciences, Physics Peace and Ave. Technology, 54b, Mongolian 13330 Academy Ulaanbaatar, of Mongolia E-mail: Enkhtuya Galsandorj Department of Theoretical Physics, Institute ofSciences, Physics Peace and Ave. Technology, 54b, Mongolian 13330 Academy Ulaanbaatar, of Mongolia E-mail: Flux tube with dynamical fermions fromtemperature high SU(3) lattice gauge theory PoS(LATTICE2019)170 (2.1) Sodbileg Chagdaa , ) µ , x ( U = 0 = t | ) µ , x ( t U , ) µ , x 1 ( t U } ) t U ( W S . µ c , T x ∂ { at each value of the gauge coupling and the strange to light quark 2 0 s g m − ) = 8 using RHMC algorithm based on the simple molecular dynamics. In order 27. We present some new preliminary results for interquark distances up to µ , × x = 3 ( l t ˙ m U / s m Field configurations with dynamical quarks are taken from HotQCD collaboration. With the The generated configurations have been flowed by the flow equation [11, 12] The strong interaction of matter, one of the four fundamental forces of nature, provides the Mediators of the strong interaction are gluons and due to the non-abelian nature of the gauge In this work we study the profiles of the distribution of chromoelectric and chromomagnetic highly improved staggered quark action with tree level6000 improved - Symanzik 9000 gauge action gauge (HISQ/tree) field configurations werethe generated lattice for of each size value 32 of the five gauge couplings on 2.5 fm and temperatures up to 1.09 2. Gradient flowed configurations to reduce the autocorrelation time, thelected configurations from were the taken data after one 10it configuration sweeps. to from about Further each 1150 we five so col- configurations thaturations a reducing single flowed the GPU by total in means number the of institute of theprogram of Gradient [10]. Ulaanbaatar flow can make method the encoded generated in config- the so called ParallelGPUCode mass ratio is basic force which is responsibleIn for particular, binding quarks elementary particles and together gluonsof to appear color form to confinement, nuclear be matter. which confinedphenomena is in of one ordinary of the matter the strong duemechanism most interaction. is to challenging one the of and the mechanism Reaching not central athey goals yet are of detailed fully relevant experimental for as understanding explained heavy well ion physics of as collision theoretical color experiment high and energy confinement physics for as cosmology. interactions in QCD they formare a numerous narrow accurate tube lattice of simulationsthe chromoelectric that flux flux tubes, have between particularly, made two in considerable quarks. thetemperatures progress cases [1–9] There in all of concerning SU(2) simulation to and of close address SU(3) vicinity the pure to flux gauge the tube theories critical behavior at temperaturelittle at zero of zero progress and the temperature in finite deconfinement as understanding phase well the transition.to as flux But the tube there non-locality behavior has of been in the effective theaction lattice presence action the of for logarithm the dynamical of gluonic fermions gauge themore due fields. non-local determinant for In of fermions the with the effective small gauge fermion mass.quenched matrix approximation, In order but appears to they which avoid can this becomes not problem represent more early the simulations and real used world. field components, energy density and widthsition of in the QCD flux with tube (2+1) aroundmethod flavors, the for using deconfinement noise HISQ/tree reduction. phase action We tran- on work on ais the lattice fixed line exploiting to of the its constant Gradient physics, physical where flow value the strange quark mass QCD flux tube at high temperature 1. Introduction PoS(LATTICE2019)170 − 15, . , we (3.1) (3.2) (2.2) from 0 ) 390 . That ⊥ . R t = x , t k x ( x is chromofield . We measured c Sodbileg Chagdaa µν T )) f , where a Polyakov ρ 09 in the range 6 ) , . gives [10] x 1 R ( ) β ( t †  − + i U c Ω ) L ( T . t ref W − around quark-antiquark pair. is the coupling constant and 8 S x ) 97 . ( 25 of the flow-time value with µ . ρ , √ a x , µν 0 µν f x ∂ = ( −  and Ω ( . called flow-time and the gauge fields 01 − h . ) are space-time components that define β t tr smear τ i r 6 1 , ) 14 f n x . Expression in the right hand side of the from the axis connecting the two sources ~ ( ( ) − i 4 x x in the range 0 and its conjugate ) 16 which give the physical distances ranging ( µν )) U R † ) and ρ 1 T ρ ( τ −  0 , = ) N ( W + 34 ∏ τ x f R L L ( 2 ( ) † , Tr 0 + c Ω 24 ( 1 L f N L ) − h at distance 0 ) ( ≡ ) ρ L ν ) , , h n ~ x µ (  ( in the range 4 U 4 L ( a β Ω a with the staple ( / Tr ) 2 1 R c are the space-space components and define chromomagnetic con- 1 ρ N ) = , x 23 x ) = , f = ( 25 only in this paper. t † . R U ( 0 ( µν W 15 at all temperatures in our case. Indeed our data at the first five values ) is and . µν = W  f 0 ρ S t n ~ , 13 ρ , f x = x ( , ∂ t ) numerically for link smearing. 2 0 U 12 g f 2.1 ) = ρ is a SU(3)-valued differential operator, calculation of , x µ , ( ) have been constructed from the link variables on the lattice and it makes it convenient to x ∂ Ω 2.2 Simulations have been performed at five values of the gauge coupling The gauge field configurations are flowed in the range 0 In our flux tube simulation program Polyakov loop-plaquette correlation operators [13, 14] If 500 which corresponds to physical temperatures . loop at some location measure the chromofield components from the Gradient flowed configurations. chromoelectric contributions to the chromofield strength. even numbers of lattice distance from 0.5 fm to 2.5errors. fm. The Jackknife method has been used to get the estimates of the statistical strength where tributions to the chromofield strength, while lattice spacing respectively. The timeeach propagation other of is a quark represented and by an the antiquark Polyakov at loop distance 6 correlates with the two Polyakov loops andare thus able varying the to position scan of distribution theOne plaquette, surface can of then the investigate various chromofield profiles strength with of respect the to distribution the surface axis by parallel as changingpair. well profile as coordinate perpendicular to the plane containing the quark-antiquark Eq. ( The plaquette variable solve the Eq. ( QCD flux tube at high temperature which turned out to belattice a numerical very simulation. efficient It computational introducesalong strategy extra the to coordinate flow increase become signal-to-noise smoother with ratio the in smearing radius where 3. Flux tube measurement the step of 0.01. And wetime have increases done all the the measurements at fourpoint the components seems eight are to values reaching of be their them.of plateau When the the values flow-time flow- at gave the some bad point0.20 signal-to-noise of ratio, and while 0.25 those gave at the theshow good last the three one. results which from are As this holds for all temperatures and distances, we choose to PoS(LATTICE2019)170 data (4.1) (5.1) (5.2) (4.2) (4.3) K a f calculated with presents energy Sodbileg Chagdaa ε 1 14 f + )) 2 fm. The second plot shows 0 34 b , f = ) 20 R β + ( ( ) . / . 3 fm 24 ) β f · f ) β c ( ) β τ 2 β − 2MeV ( from below. In the deconfinement phase = β f ( ( N / K ) c · K √ E 2MeV T ) β / . a f exp 10 a f 3MeV / 1 √ β ( . ) . ( 2 0 / K 2 b M ≡ 10 K 1 c 2 . ( ( and 197 ) 3671 as updated by 2018. Here one fits 156 + ) / K 2 a f τ 1 3 · τ d b ) + = N E ) 156 N of those profiles as functions of the interquark dis- ) , − β , + β c K 2 ) ( = 23 ( )) β 1 β d f f 0 scaling as determined in Appendix B of [15] with the ( ( K ε ) = β K 0 c τ T / K = + c T a f f N , ⊥ 10 13 = f ( x β 0 , ) = ) = ( c b 2 + T T β β T / ( ( 12 R 46049 and a K f ( ) = ( = = a f − β k K 2 ( x c f = ( M 49415, . 7 = are the coefficients of the universal two-loop beta functions. Then we can express K 0 1 c ) at several values of interquark distance and temperature. The Fig. b 5.2 and 0 The six chromofield strength components compose energy density that gives the physical flux To set the lattice spacing we used b tance and temperature. The midpointsources. belongs to The the first central plot region shows of thatclearly our the decreases interest energy with density between increasing in the distance the two and middle vanishes region at between about the two sources tube. In order tophase investigate transition the it behavior of is the worthdependence. real to Total magnetic physical consider and flux various electric profiles tube chromofield of across strengths the the which are, energy deconfinement respectively, density and their parameter where and lattice spacing in fm 5. Results 5.1 Energy density in the flux tube define the total energy density as We were able to obtain longitudinal and transverse profiles of the energydensity density value at the point its temperature dependence, from that oneof can the see flux that tube in decreases the when confinement temperaturethe phase approaches the energy energy density density further decreases with temperature at all distances. We see from the plot that and temperature in MeV Now one can obtain estimates of theat corresponding each temperatures gauge in coupling units by of the the critical ratio temperature the Eq. ( coefficients QCD flux tube at high temperature 4. Scaling function to the Ansatz PoS(LATTICE2019)170 β in c T (5.3) (5.4) . Thus 09 c . T 09 R=0.56 fm R=0.60 fm R=0.90 fm R=1.20 fm R=1.40 fm R=1.50 fm R=1.90 fm R=2.20 fm R=2.40 fm . Sodbileg Chagdaa in physical units as 1.08 ) and due to its finite 2 8. 3.1 × c 1.04 3 T/T 1 between a quark and an antiquark as ) ) 0 ) K of the Eq. ( K = − ref ⊥ ) − K 0.96 x x ) ⊥ , + x ⊥ 2 ( ⊥ x / 0.02 0.00 0.10 0.08 0.06 0.04 f -0.02 -0.04 x (

R 2 2 fm in the region with temperature 1

f (

a .

] (R/2, 0) [GeV 0) (R/2, 16 for each gauge coupling. We choose those

ε ( 4 2 ⊥ , is displayed in the Fig. , 2 = − is an unphysical parameter which is due to the ε x ⊥ e k 4 x 1 14 ⊥ x = D 2 , ( . Width of the flux tube increases with interquark K a x 2 R d T . 12 d R 2 , ) = R a ⊥ 2.4 10 x , c c c c c = ( 8 2 f , and a 6 T=0.97T T=1.00T T=1.03T T=1.06T T=1.09T 1 / , 2 2 ε a 4 D = a / 1.6 and temperature R R [fm] R to be of an even number in order to fit the transverse profiles on the middle has been computed via N 1.2 ε 9 fm but to suppress it with further separation which could be an indication of D 2 fm. . 0 ≈ 0.8 = R 2 between the two sources. R / R 0.4 The energy density value at the midpoint = distances with k 0.06 0.04 0.02 0.00 0.10 0.08 -0.04 -0.02 x

We define physical width of the flux tube as width of its energy density and estimate it quan- The resulting width of the energy density, Na

(R/2, 0) [GeV 0) (R/2, ]

ε 4 can be similar to its zero temperature case studied in [8] where no string breaking observed up = c from that the width titatively by fitting the middle transversetion distribution [16] of the energy density to an exponential func- functions of the interquark distance and temperature from the lattice of size 32 the flux tubes with length of less than 2.2 fmfull do QCD. still This survive might for be temperaturesT suggesting up that to flux 1 tube into full the QCD distance at high temperature but just above 5.2 Width of the flux tube distance until the distanceincreasing reaches distance. around The 0.9 presence fm, of dynamicalup after quarks to which seems around to it widen strongly thehadronization. decreases flux This in with looks a in further small contrast distance with thea case monotone of or SU(3) pure a gauge weak theory logarithmic [17] where spreading one of finds the flux tube as a function of interquark distance. we find string breaking distance to be around distance from the centerand of seven the lattice distances flux tube [17]. The fit is done for each five gauge coupling functions of the distance Figure 1: point QCD flux tube at high temperature in terms of thestatistical fitting fluctuations parameters of the fields at the reference point R PoS(LATTICE2019)170 1.1 1.08 R=0.56 fm R=0.60 fm R=0.83 fm R=0.90 fm R=1.10 fm R=1.20 fm R=1.50 fm R=2.00 fm R=2.20 fm Sodbileg Chagdaa 1.06 c 1.04 T/T 1.02 and flux tubes with length of c . We see that the temperature c T 1 T 06 . 09 . 2 fm in the deconfinement phase. . 0.98 2 = 0.96 R 3 2 1 0

0.5 3.5 2.5 1.5

ε [fm] D 5 9 fm after which it is strongly suppressed due to . 9 fm could indicate strong localization of the QCD . 0 0 2.2 = = 2

. These are in qualitative agreement with the conclusion R c c c c c R c T 1.8 T=0.97T T=1.00T T=1.03T T=1.06T T=1.09T 09 . 1.6 . In the deconfinement phase it tends to decrease with temperature c T 1.4 R [fm] 1.2 for various distances. We see that the width of the flux tube increases as a 2 1 0.8 0.6 8. 0.4 × 0 2 1 3 The width of the flux tube as functions of the interquark distance and temperature from the lattice 3 2.5 1.5 0.5 3.5

suggest that flux tube structure melts eventually beyond certain distance and temperature ε [fm] D c We find that the width of the flux tube does not always increase as a function of distance in The decreases of the both energy density and width of the flux tube with temperature above Also our results confirm the earlier result of very first study of the breaking of the flux tube T both confined and deconfined phasesit when increases the dynamical with fermions distance are up presented. to In around both phases value in the deconfined phase.tube structures Our with study length in of the 2 fm presence persists of up dynamical to fermions temperature shows 1 that flux 6. Conclusion the yet shows non-vanishing values at our largest temperature. the dynamical fermions. This resultresult supports that is [18]’s in result the that deconfined is phase.work in Our [18] present the study with confined has phase advantages the asdynamical and it modern fermions. [17]’s is computational updating the strategies pioneer and differing from [17] in accounting for 1.1 fm persists up to temperature 1 in full QCD bytance. [18] Difference in is that that our both resultsconfinement our show that and and this deconfinement, hadronization their takes at study place in least showing the hadronization up both two beyond to phases, certain temperature dis- 1 function of temperature up to Figure 2: of size 32 The strong decrease with distance after QCD flux tube at high temperature string yet we observed no string breaking up to around accelerates the rate of the hadronizationthe probably dynamical causing fermions. an Temperature additional dependence polarizationsecond of effects plot the due of width to the at Fig. a fixed distance is shown in the in [7] that the fluxthat tube in structure presence survives of to dynamical the fermions deconfinement for transition a and sufficiently conclusion large in distance [19] between sources the flux PoS(LATTICE2019)170 , Phys. 293 Nucl. , , B180 , (2016) 033, Sodbileg Chagdaa ]. Nucl. Phys. 06 , (2010) 071, JHEP (2010) 063, 08 , Commun. Math. Phys. , 08 (1987) 673-691. ]. JHEP , Master’s thesis, Bielefeld , ]. B291 ]. JHEP , (2017) 025002. hep-lat/9406021 ]. 45 ]. Width and string tension of the flux tube in ]. Flux tubes of two- and three-quark system in Nucl. Phys. , Distribution of the color fields around static J. Phys. G 6 , hep-lat/0212036 hep-lat/9409005 Observing long color flux tubes in SU(2) lattice gauge ]. (1996) 389-403, [ Flux tubes in QCD with (2+1) HISQ fermions Flux tubes at finite temperature ]. Pure gauge QCD flux tubes and their widths at finite Pure gauge flux tubes and their widths at finite temperature How thick are chromo-electric flux tubes? hep-lat/1701.03371 D53 hep-lat/0812.0284 hep-lat/0903.4155 hep-lat/1207.6523 Flux tube profiles with dynamical quarks and finite temperatures (2016), [ (2003) 899-902, [ ]. Phys.Rev. The distribution of chromoelectric flux in SU(2) lattice gauge theory ]. ]. On the intrinsic width of the chromoelectric flux tube in finite temperature On the linear increase of the flux tube thickness near the deconfinement , (1995) 5165-5198, [ A721 (2009) 073, [ D51 The QCD transition temperature: results with physical masses in the continuum (2019) 88-112. (2009) 088, [ hep-lat/0907.5491 01 (2012) 174, [ 06 hep-lat/1702.03454 11 Properties and uses of the Wilson flow in lattice QCD Trivializing maps, the Wilson flow and the HMC algorithm Chromoflux distribution in lattice QCD (1992) 299-309. ,[ B940 Flux tube delocalization at the deconfinement point (1983) 374-378. JHEP Nucl. Phys. Applications of the Gradient flow method in lattice QCD , , JHEP Phys. Rev. , JHEP , B370 , 132B hep-lat/1006.4518 hep-lat/1004.3875 hep-lat/1511.01783 (2010) 899-919, [ Nucl. Phys. Phys. limit II temperature University (2017). [ Lett. SU(2) lattice gauge theory at high temperature LGTs [ [ full QCD transition [FS2] (1980) 1-12. PoS(LATTICE2016)344 theory quarks: Flux tube profiles [1] G. S. Bali, K. Schilling and Ch. Schlichter, [9] P. Bicudo, N. Cardoso and M. Cardoso, [6] M. Caselle and P. Grinza, [8] H. Ichie, V. Bornyakov, T. Streuer and G. Schierholz, [2] R. W. Haymaker, V. Singh, Y. Peng and J. Wosiek, [3] M. Lüscher, G. Münster and P. Weisz, [7] P. Cea, L. Cosmai, F. Cuteri and A. Papa, [4] A. Allais and M. Caselle, [5] M. Caselle, [18] W. Feilmair and H. Markum, [19] L. Cosmai, P. Cea, F. Cuteri and A. Papa, [13] M. Fukugita and T. Niuya, [16] R. Sommer, QCD flux tube at high temperature tube structure disappears. In futurerange, study improve we the plan fit to function increase and the compute statistics, relevant extend observables the of temperature interest. References [10] L. Mazur, [11] M. Lüscher, [12] M. Lüscher, [14] S. Chagdaa, E. Galsandorj, E. Laermann and B. Purev, [15] A. Bazavov et al., [17] P. Bicudo, N. Cardoso and M. Cardoso,