Axial U(1) symmetry and mesonic correlators at high temperature in N f =2 lattice QCD
Kei Suzuki, Sinya Aoki, Yasumichi Aoki, Guido Cossu, Hidenori Fukaya, Shoji Hashimoto, Christian Rohrhofer
OOU-HET-1039
Axial U(1) symmetry and mesonic correlators athigh temperature in N f = lattice QCD Kei Suzuki ∗ a , Sinya Aoki b , Yasumichi Aoki c , Guido Cossu d , Hidenori Fukaya e ,Shoji Hashimoto f , g , Christian Rohrhofer e (JLQCD Collaboration) a Advanced Science Research Center, Japan Atomic Energy Agency (JAEA), Tokai 319-1195,Japan b Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto 606-8502,Japan c RIKEN Center for Computational Science, Kobe 650-0047, Japan d School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3JZ, UnitedKingdom e Department of Physics, Osaka University, Toyonaka 560-0043, Japan f KEK Theory Center, High Energy Accelerator Research Organization (KEK), Tsukuba305-0801, Japan g School of High Energy Accelerator Science, The Graduate University for Advanced Studies(Sokendai), Tsukuba 305-0801, Japan
We investigate the high-temperature phase of QCD using lattice QCD simulations with N f = U ( ) symmetry, overlap-Dirac spectra, screening masses from mesonic correlators, and topologicalsusceptibility. We find that some of the observables are quite sensitive to lattice artifacts due to asmall violation of the chiral symmetry. For those observables, we reweight the Möbius domain-wall fermion determinant by that of the overlap fermion. We also check the volume dependenceof observables. Our data near the chiral limit indicates a strong suppression of the axial U ( ) anomaly at temperatures ≥
220 MeV. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] J a n xial U(1) symmetry and mesonic correlators at high temperature in N f = lattice QCD Kei Suzuki
1. Introduction
In the high-temperature region of quantum chromodynamics (QCD), one of open questions isthe fate of the U ( ) A symmetry. In the low-temperature phase, the U ( ) A symmetry is known tobe broken by a quantum anomaly which is related to topological excitations of gluon fields, e.g,instantons. In the high-temperature region with restored chiral symmetry (in other words, abovethe critical temperature, T > T c ), the restoration or violation of the U ( ) A symmetry is still a long-standing problem not only in theoretical approaches [1, 2, 3] but also in lattice QCD simulations at N f = N f = + U ( ) A symmetry breaking above thecritical temperature. However, many studies applied the staggered-type fermions, where chiralsymmetry is explicitly broken, and it was difficult to precisely measure how much of U ( ) A sym-metry breaking is due to lattice artifacts. Recently, chiral fermions were employed to simulatelattice QCD at high temperature [4, 5, 7, 9, 10, 12, 13] (in Refs. [12, 13], only for valence quarksector). JLQCD Collaboration studied with N f = Q = U ( ) A anomaly (at around 1 . T c ) was also reported in simulationswith N f = U ( ) A symmetry is good at 1 . T c but not near T c .In these proceedings, we report on our recent results of the observables at T =
220 MeVsuch as the Dirac spectrum, U ( ) A susceptibility, screening masses from mesonic correlators, and Table 1:
Numerical parameters of lattice simulations. L × L t and m are the lattice size and quark mass,respectively. ¯ ∆ ov π − δ and χ t are our results of the U ( ) A susceptibility and topological susceptibility from thefermionic definition, respectively. L × L t am ¯ ∆ ov π − δ a on OV χ t a ×
12 0.001 1.5(0.6) × − ≈ ×
12 0.0025 3.6(1.3) × − × − ×
12 0.00375 0.00017(7) 2.3(0.7) × − ×
12 0.005 0.00091(42) 9.0(2.0) × − ×
12 0.01 0.00389(92) 1.7(0.2) × − ×
12 0.001 1.8(1.4) × − × − ×
12 0.0025 0.00017(6) 3.5(3.0) × − ×
12 0.00375 0.00026(8) 7.9(3.0) × − ×
12 0.005 0.00291(188) 9.3(1.9) × − ×
12 0.01 0.01358(263) 2.9(0.4) × − ×
12 0.005 0.00785(178) 5.4(0.6) × − ×
12 0.01 0.01162(140) 2.0(0.2) × − ×
12 0.001 2.2(0.9) × − × − ×
12 0.0025 0.00012(4) 4.9(4.4) × − ×
12 0.00375 0.00032(12) 1.5(0.7) × − ×
12 0.005 0.00135(63) 2.9(1.1) × − xial U(1) symmetry and mesonic correlators at high temperature in N f = lattice QCD Kei Suzuki ρ ( | λ | ) [ G e V ] Overlap Dirac eigenvalue | λ | [MeV]32 x12, β =4.3, T=220MeV, m=0.001(2.64MeV)on DWon OV 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 50 100 150 200 250 ρ ( | λ | ) [ G e V ] Overlap Dirac eigenvalue | λ | [MeV]32 x12, β =4.3, T=220MeV, m=0.01(26.4MeV)on DWon OV Figure 1:
Spectral density ρ ( | λ | ) for overlap-Dirac eigenvalues λ at T =
220 MeV. Upper panel: m = .
64 MeV. Lower panel: m = . topological susceptibility in N f = β = .
30 and the latticespacing 1 / a = .
64 GeV ( a ∼ .
075 fm), which is finer than that of configurations used in theprevious works [4, 7]. We simulate lattice volumes L = , , ,
48, and the length of the fifthdimension in the MDW fermion formulation is L s =
16. The physical quark mass (as the average ofup and down quark masses) is estimated to be am = . ( ) (3 . ( ) MeV). Some of our resultswere already reported in previous proceedings [18, 19, 20, 21].
2. Overlap Dirac spectrum
In Fig. 1, we plot spectral density of overlap Dirac eigenvalues, ρ ( λ ) = ( / V ) (cid:104) ∑ λ (cid:48) δ ( λ − λ (cid:48) ) (cid:105) for two typical ensembles. The blue and magenta bins denote the spectra on the MDW fermionsensembles (DW) and reweighted overlap fermion ensembles (OV), respectively. At m = .
64 MeVfor the OV ensembles, we find a suppression of both low eigenmodes and chiral zero modes.The suppression of the low eigenmodes is related to the U ( ) A symmetry restoration in the lightquark mass region. The disappearance of the chiral zero modes is related to the suppression of thetopological susceptibility. At m = . U ( ) A symmetry breaking. U ( ) A susceptibility The U ( ) A susceptibility ∆ π − δ is an order parameter of the U ( ) A symmetry breaking. Thisis defined from a spacetime integral of the difference between two-point correlators of isovector-pseudoscalar ( π a ≡ i ¯ ψτ a γ ψ ) and isovector-scalar ( δ a ≡ ¯ ψτ a ψ ) operators: ∆ π − δ ≡ χ π − χ δ ≡ (cid:90) d x (cid:104) π a ( x ) π a ( ) − δ a ( x ) δ a ( ) (cid:105) , (3.1)where a is an isospin index in N f = U ( ) A susceptibility in the lattice theory is definedby a summation of low-lying eigenvalues of the overlap Dirac operator, λ ( ov , m ) i [22]: ∆ ov π − δ = V ( − m ) (cid:42) ∑ i m ( − λ ( ov , m ) i ) λ ( ov , m ) i (cid:43) , (3.2)2 xial U(1) symmetry and mesonic correlators at high temperature in N f = lattice QCD Kei Suzuki N t =12, β =4.3T=220MeV U ( ) A s u sc ep t i b ili t y ∆ π - δ [ G e V ] Quark mass m [MeV] ∆ - ov on OV (UV subt.) 24x12 ∆ - ov on OV (UV subt.) 32x12 ∆ - ov on OV (UV subt.) 40x12 ∆ - ov on OV (UV subt.) 48x12 0 0.001 0.002 0.003 2 4 6 8 10 Figure 2: U ( ) A susceptibilities, ¯ ∆ ov π − δ (3.2), from the eigenvalue density of the overlap-Dirac operators at T =
220 MeV. where we set the lattice spacing a =
1. This summation is truncated at the lowest 40 eigenvalues. In Fig. 2, we show the U ( ) A susceptibility at T =
220 MeV. In the light quark mass region,we find strong suppression of the ∆ ov π − δ . For example, at the lowest quark mass and L =
32, theratio of ∆ ov π − δ to temperature is (cid:112) ∆ ov π − δ / T ≈ L = L =
24, whoseaspect ratio against temperature is L / L t =
4. Screening mass difference from spatial mesonic correlators
The screening mass is defined by the exponential decay of spatial correlators", which may beused to measure a violation of U ( ) A symmetry. We investigate the difference between the effectivescreening masses ∆ m scr ( z ) = | m PSscr ( z ) − m Sscr ( z ) | , (4.1)where m PSscr ( z ) and m Sscr ( z ) are the effective screening masses at a spatial coordinate z for isovector-pseudoscalar ( π a ≡ i ¯ ψτ a γ ψ ) and isovector-scalar ( δ a ≡ ¯ ψτ a ψ ) operators, respectively.In Fig. 3, we show the difference between the effective screening masses measured by theMDW operator (without reweighting), where the horizontal axis is a dimensionless spatial distance( zT = ( n z a / N t a ) = n z / N t ). For the screening masses with light quark mass, we find a small valueof ∆ m scr ( zT ) , which indicate the restoration of the U ( ) A symmetry and it is consistent with theresults of the U ( ) A susceptibility ¯ ∆ ov π − δ . For heavy quark masses, the mass difference becomeslarge, which implies the U ( ) A symmetry breaking.
5. Topological susceptibility
The topological susceptibility χ t is defined as a gauge ensemble average of the topologicalcharge Q t : χ t = (cid:104) Q t (cid:105) V , (5.1) From this definition, we further apply two types of subtractions: a subtraction of the contributions from chiral zeromodes and an ultraviolet divergence (or lattice cutoff). For a justification of the zero mode subtraction, see Ref. [2, 7].For the parametrization scheme of the lattice cutoff contribution by different valence quark masses, see Ref. [20, 21]. xial U(1) symmetry and mesonic correlators at high temperature in N f = lattice QCD Kei Suzuki Δ m e ff ( z T ) / M e V zTm ud =0.001m ud =0.0025m ud =0.00375m ud =0.005m ud =0.01 Figure 3:
Difference between effective screening masses (4.1) from spatial mesonic collerators for U ( ) A partners at T =
220 MeV and L =
32. The horizontal axis is defined as a dimensionless spatial distance zT = ( n z a / N t a ) = n z / N t . For the topological charge Q t , we employ two definitions. As a fermionic definition, Q t is definedthrough the index theorem for the overlap Dirac operator: Q t = n + − n − , (5.2)where n ± are the numbers of chiral zero modes with positive or negative chirality, respectively. Asa gluonic definition, Q t is defined as a summation over spacetime x at a flow time t : Q t ( t ) = π ∑ x ε µνρσ Tr F µν ( x , t ) F ρσ ( x , t ) , (5.3)where F µν ( x , t ) is the clover-type discretization of the field strength tensor [23]. In Fig. 4, we plot the topological susceptibility χ t at T =
220 MeV. We show the results fromthe fermionic definition (5.2) on the OV ensembles and the gluonic definition (5.3) on the MDWensembles, respectively. In the light quark mass region, χ t is strongly suppressed with both thedefinitions. Furthermore, the volume dependence between L =
24 and 48 is small. In the heavyquark mass region, the value of χ t becomes nonzero, which is in agreement with the peak structureof the Dirac spectra in the lower panel of Fig. 1.
6. Summary and discussion
In these proceedings, we studied the high-temperature phase of QCD at T =
220 MeV byusing N f = U ( ) A susceptibility (3.2) and the difference of mesonic screening masses (4.1) in lightquark mass region, m (cid:46)
10 MeV, which indicates the U ( ) A symmetry restoration in the chirallimit ( m → This definition is usually not an integer, but we find a well-discretized distribution of Q t ( t ) at t = xial U(1) symmetry and mesonic correlators at high temperature in N f = lattice QCD Kei Suzuki
0 5 10 15 20 25 30 c t [ M e V ] m [MeV]T=220MeV L=24 (1.8fm)T=220MeV L=32 (2.4fm)T=220MeV L=48 (3.6fm)T=220MeV L=40 (3.0fm)gluonic Figure 4:
Topological susceptibilities χ t at T =
220 MeV. Colored points: χ t from the fermionic defini-tion (5.2) on reweighted OV ensembles. Uncolored points: χ t from the gluonic defnition (5.3) on MDWensembles. Acknowledgment
Numerical simulations are performed on IBM System Blue Gene Solution at KEK under asupport of its Large Scale Simulation Program (No. 16/17-14) and Oakforest-PACS at JCAHPCunder a support of the HPCI System Research Projects (Project IDs: hp170061, hp180061 andhp190090) and Multidisciplinary Cooperative Research Program in CCS, University of Tsukuba(Project IDs: xg17i032 and xg18i023). This work is supported in part by the Japanese Grant-in-Aid for Scientific Research (No. JP26247043, JP18H01216 and JP18H04484), and by MEXT as“Priority Issue on Post-K computer" (Elucidation of the Fundamental Laws and Evolution of theUniverse) and by Joint Institute for Computational Fundamental Science (JICFuS).
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