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PHYSICAL REVIEW D 98, 014504 (2018) Localization of topological charge density near Tc in quenched QCD with Wilson flow † ‡ You-Hao Zou,* Jian-Bo Zhang, and Guang-Yi Xiong Department of Physics, Zhejiang University, Zhejiang 310027, People’s Republic of China (Received 21 May 2018; published 10 July 2018) We smear quenched lattice QCD ensembles with lattice volume 323 × 8 by using Wilson flow. Six ensembles at temperature near the critical temperature Tc corresponding to the critical inverse coupling βc ¼ 6.06173ð49Þ are used to investigate the localization of topological charge density. If the effective smearing radius of Wilson flow is large enough, the density, size and peak of Harrington-Shepard (HS) caloron-like topological lumps of ensembles are stable when β ≤ 6.050, but start to change significantly when β ≥ 6.055. The inverse participation ratio (IPR) of topological charge density shows similar results, it begins to increase when β ≥ 6.055 and is stable when β ≤ 6.050. The pseudoscalar glueball mass is extracted from the topological charge density correlator (TCDC) of ensembles at T ¼ 1.19Tc,and1.36Tc, the masses are 1.915(98) and 1.829(123) GeV respectively, they are consistent with results from conventional methods. DOI: 10.1103/PhysRevD.98.014504 I. INTRODUCTION Since the topological structure is connected with chiral Topological properties of the QCD vacuum are believed symmetry breaking and confinement, we are interested in to play an important role in QCD. For example, the the behavior of topological structures when the temperature topological susceptibility has the famous Witten- is near the critical temperature Tc. The temperature in Veneziano relation, which can explain the U(1) anomaly lattice QCD is given by: 0 and the large mass of the η meson [1–3]. The topological 1 structure of the QCD vacuum is related to chiral symmetry T ¼ ; ð1Þ N a breaking and may be also related to confinement [4,5]. t t A usual way to study the topological structure is in which at is the lattice spacing in the temporal direction, investigating the localization of topological charge density, and Nt is the temporal lattice size. Therefore we can such as Belavin-Polyakov-Schwartz-Tyupkin (BPST) change Nt or at to vary the temperature T. If we change instanton-like localized topological lumps at zero temper- Nt,becauseNt cannot be too large the temperature will be ature. BPST instanton is a semi-classical solution of the changed coarsely. Thus we cannot get different ensembles QCD Lagrangian in Euclidean space [6]. Isolated instan- with small variation of temperature near Tc. Therefore we tons are zero modes of the Dirac operator. When these will vary the temperature by changing at, which means modes mix with each other they will shift away from zero that we will generate different temperature ensembles by modes [5]. The way how they mix is important, since it is slightly varying the inverse coupling β. The conventional the topological structure of the QCD vacuum. When we use UV filters like cooling, smoothing and smearing [12–16] the gluonic definition for the topological charge density lead to different smearing effects when the ensembles qðxÞ to investigate the topological localized structures, have different lattice spacings, even though the parameters such as instantons, a UV filter is needed to remove the are set to be the same. So we will use the gradient flow, short-ranged topological fluctuations and preserve the which provides a general energy scale. Its effective – pffiffiffiffi long-ranged topological structures [7 11]. smearing radius λ ¼ 8t [17], where t is the flow time. Recent works [18–20] show that the gradient flow is *[email protected] consistent with standard cooling, therefore like using † [email protected] cooling we can also use the gradient flow to study ‡ [email protected] topological structures. Then we can compare the topo- logical structure of different ensembles and avoid the Published by the American Physical Society under the terms of different smearing effects. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to In our work, we used the Harrington-Shepard (HS) the author(s) and the published article’s title, journal citation, caloron solutions [21] to filter the localized topological and DOI. Funded by SCOAP3. lumps, which is the generalized form of BPST instantons at 2470-0010=2018=98(1)=014504(8) 014504-1 Published by the American Physical Society ZOU, ZHANG, and XIONG PHYS. REV. D 98, 014504 (2018) finite temperature with periodic boundary condition at the TABLE I. The quenched ensembles of Wilson action in this temporal direction. We also used the inverse participation work. The lattice size is 323 × 8. 10000 sweeps were done before ratio (IPR) [22] to investigate the topological localization. thermalization. Each configuration is separated by 10 sweeps. The IPR is defined by: Each sweep includes 5 times quasi heat-bath and 5 steps of P leapfrog. jqðxÞj2 IPR ¼ V Px ; ð Þ β χ ð j ð ÞjÞ2 2 Ncnfg P a x q x 6.045 2000 3.02ð38Þ × 10−4 0.0863 fm in which qðxÞ is the topological charge density. In this work 6.050 2000 4.92ð14Þ × 10−4 0.0856 fm we use the gluonic definition for qðxÞ: 6.055 2000 7.67ð23Þ × 10−4 0.0849 fm 9 36ð47Þ 10−4 1 6.060 2000 . × 0.0842 fm ð Þ¼ ϵ ½ ð Þ ð Þ ð Þ 6.065 2000 8 15ð19Þ 10−4 0.0835 fm q x 2 μνρσTrC Fμν x Fρσ x ; 3 . × 32π −4 6.070 2000 5.82ð18Þ × 10 0.0828 fm in which ϵμνρσ is the Levi-Civita symbol, TrC is the trace running over the color space, and the field tensor Fμν is Θ ð3Þ defined by: in which is the Z rotated Polyakov loop: 8 i † < ReP exp½−2iπ=3; arg P ∈ ½π=3; πÞ; FμνðxÞ¼− ðCμνðxÞ − CμνðxÞÞ 2 Θ ¼ ∈ ½−π 3 π 3Þ ð Þ : ReP; arg P = ; = ; 6 1 i − − ð ð Þ − † ð ÞÞ ð Þ ReP exp½2iπ=3; arg P ∈ ½−π; −π=3Þ; 3 ReTrC 2 Cμν x Cμν x ; 4 ð Þ where P is the usual Polyakov loop of each configuration. in which Cμν x is the average of the four plaquettes on the β μ − ν In Table I the 6 ensembles we used to find c are listed. plane. When all topological charges focus on one 323 8 lattice site IPR ¼ V, IPR would decrease if the topological The lattice size is × . We expect that the finite volume charge density becomes more delocalized. Finally it will effects are negligible. The lattice spacing a is found by equal to 1 when the topological charge density distributes using [25] uniformly. 2 The topological charge density correlator (TCDC) of a ¼ r0 expð−1.6804 − 1.7331ðβ − 6Þþ0.7849ðβ − 6Þ quenched QCD can be used to extract pseudoscalar glue- − 0 4428ðβ − 6Þ3Þ ð Þ ball masses at zero temperature with Wilson flow [23].In . ; 7 our work, we extracted the pseudoscalar glueball mass from TCDC at finite temperature with Wilson flow. The results where r0 is set to be 0.5 fm from Ref. [26]. Obviously are compared with those from Ref. [24]. Unlike conven- Table I shows that βc is near 6.060. The critical inverse tional methods, this method does not need large lattice size coupling βc is obtained by interpolating to the location in the temporal direction to do fitting, which is hard to be where χP is maximum. We use a B-spline interpolation satisfied in ensembles at finite temperature especially at and obtain βc ¼ 6.06173ð49Þ, which is compatible with high temperatures. βc ¼ 6.06239ð38Þ in Ref. [27]. II. LOCATING THE HS CALORON-LIKE B. HS caloron-like topological lumps TOPOLOGICAL LUMPS In this paper we use the HS caloron solutions to filter the β A. Find the critical inverse coupling c localized topological charge density lumps. The localized First, we need to find the critical temperature Tc. In other topological lumps are defined by sites that have maximum β 4 words we need to determine the critical inverse coupling c. absolute value of qðxcÞ in a 3 hypercube centered at site 323 8 We use pure gauge ensembles that have lattice size × xc. The center xc is also mentioned as peak. After applying in our work. We use the susceptibility χP of the Polyakov the HS caloron filters in the following, we can get calorons- loop to find βc. χP is defined as like topological lumps. In SU(2) gauge theory at temperature T, HS caloron χ ¼hΘ2i − hΘi2; ð Þ P 5 solution of gauge field AμðxÞ has the exact form as [21] a a a AμðxÞ¼AμðxÞT ;Tis the generators for SUð2Þ; πρ2 ð2 πj⃗− ⃗jÞ a ðÞ sinh T x xc AμðxÞ¼ηaμν∂ν ln ΦðxÞ; ΦðxÞ¼1 þ ; ð8Þ jx⃗− x⃗cj=T coshð2Tπjx⃗− x⃗cjÞ − cosð2Tπðx4 − xc4ÞÞ 014504-2 LOCALIZATION OF TOPOLOGICAL CHARGE DENSITY … PHYS. REV. D 98, 014504 (2018) ρ ð Þ¼ where xc is the center of a HS caloron, is the size of whereP the normalized action density s x a HS caloron. It satisfies the (anti-)self-dual condition a4 2 ð Þ 8π2 8π2 μ<νtrCFμν x , the normalization factor ¼˜ ˜ ¼ 1 ϵ ηðÞ ’ Fμν Fμν, Fμν 2 μνρσFρσ, aμν is the t Hooft symbol: comes from the actionR of a single HS caloron g2 4 S ¼ 2 jQj with Q ¼ d xqðxÞ. ηðÞ ¼ ϵ μ ν ¼ 1 2 3 π aμν aμν; ; ; ; ; (iii) To avoid double countings of two peaks of a single ðÞ ðÞ ðÞ 0 η ¼ −η ¼δ η ¼ 0 ð Þ but distorted HS caloron, we filter peak xc by a4ν aν4 aν; a44 : 9 j − 0 j ϵρð Þ ð Þ When the temperature T → 0, it approaches the BPST if xc xc < xc : 16 Φð Þ → 1 þ ρ2 instanton solution x ð − Þ2 [6].

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