PoS(LATTICE2019)133 + 2 = f N https://pos.sissa.it/ lattice with lattice 4 gauge ensemble for all topological sectors are sampled e topological susceptibility and the ogy (Nos. 107-2119-M-003-008, 108-2112-M-003- ipei, Taiwan 11677, R.O.C. ice2019 aiwan 10617, R.O.C. 9, R.O.C. lattice QCD with physical 3). The salient feature of this gauge ensemble is 1 domain-wall quarks, on the 64 > ) + L c π , s 1 M , d / + u ive Commons ( 2 = onal License (CC BY-NC-ND 4.0). 4 fm, and f > L N (TWQCD collaboration) 064 fm ( . ∗† 0 ∼ a 1 lattice QCD with physical [email protected] + Using 10-16 units of Nvidia DGX-1, we have generated the first spacing Speaker. This work is supported by the Ministry of Science and Technol that the chiral symmetry is preserved to a high precision and ergodically. In this paper,ground-state we hadron present mass spectra. the first results of th 1 † ∗ Copyright owned by the author(s) under the terms of the Creat c E-mail: Physics Department, National Taiwan University, Taipei, T Ting-Wai Chiu Physics Department, National Taiwan Normal University, Ta First study of Institute of Physics, Academia Sinica, Taipei, Taiwan 1152 domain-wall quarks 37th International Symposium on Lattice Field16-22 Theory June - 2019 Latt Wuhan, China Attribution-NonCommercial-NoDerivatives 4.0 Internati 005) PoS(LATTICE2019)133 1 + 1 + requires 2 8 months. D = ∼ f Ting-Wai Chiu N he QCD vacuum simulation on the n-wall fermion [5], (HMC) simulation of 128 GB. In general, to 2050 HMC trajectories. = ∼ nsists of eight V100 GPUs ale lattice QCD simulations. 1, running for l point, with sufficiently large xact chiral symmetry at finite ch heavier than the sum of the 16 GB ermion action is expressed as the massless quarks is spontaneously metry in the massless limit. At the mising approach, discretizing the f × first principles of QCD, it requires can use the exact one-flavor pseud- ting physical observables by Monte ce memory. Moreover, using EOFA N uting the pseudofermion force in the an Dirac operator [7], thus avoiding main-wall quarks on a 5-dimensional these gauge ensembles. Nevertheless, arameters have been given in Ref. [4]. y there are only 3 lattice QCD groups dependent HMC simulation, we obtain of 6 units of Nvidia DGX-1, each running aditional lattice fermions (e.g., Wilson, ce action used for the simulation, which e 4-dimensional lattice [2, 3]. Neverthe- pontaneous chiral symmetry breaking as constitutes the origin of hadron masses, ) 018 GeV for 400 gauge configurations. . f i 0 ugh domain-wall fermion (DWF) on the 5- N ( ¯ qq ± R h SU lattice with any lattice Dirac operator 188 1 . × 4 3 ) f = N ( 1 10 months, resulting a total of , requiring a large number of long vectors (proportional L − Φ − a 2 6 16 in the fifth dimension, following our first SU / 1 ∼ = − ) s N D † D ( † Φ 64 lattice [6]. It is interesting to point out that the entire 1 lattice QCD on the 64 × + 3 , due to the strong interaction between quarks and gluons in t 1 ) f + N 2 ( 1 lattice QCD at the physical point [4] with the optimal domai lattice QCD with physcial domain-wall quarks V = 1 + f lattice with the extent + SU 1 4 N 1 + + 16 lattice can be fitted into one unit of Nvidia DGX-1, which co 2 2 The holy grail of lattice QCD is to simulate QCD at the physica Since quarks are Dirac fermions, they possess the chiral sym The thermalization is performed with one unit of Nvidia DGX- Since 2018, TWQCD has been performing the hybrid Monte Carlo However, in lattice QCD, formulating lattice fermion with e × = = 4 f f N 1. Introduction volume and fine lattice spacing, thenthe to characteristics extract of physics a from gauge ensemble dependsin on turn the has latti significant impacts on the physics outcome. zero temperature, the chiral symmetry memory much larger than 128 GB,rational since approximation each of one-flavor pseudof to the number of poles in the rational approximation) in comp molecular dynamics. However, for domain-wallofermion fermion, action one (EOFA) with athe positive-definite rational and approximation Hermiti and savingalso a enhances large the amount HMC of efficiency significantly. devi Then the thermalized configurations are distributedan to independent 10-1 HMC simulation for broken to By sampling one configuration every 5 trajectories400 in each gauge in configurations. The lattice setupNow and simulation the p lattice spacing is updated to (with non-trivial topology). The quark condensate 64 and resolves the puzzles suchbare as quark masses “why of the its mass constituents". of To investigate a the proton s is mu well as the hadron physics (e.g., the mass spectrum) from the dimensional lattice [1] and the overlap-Dirac fermion on th nonperturbative methods. So far,continuum space-time lattice on QCD a is 4-dimensionalCarlo the lattice, simulation. most and pro compu lattice spacing is rather nontrivial. This is realized thro less, the computational requirement forlattice lattice is QCD 10-100 with times do morestaggered, than and their their counterparts variants). with tr Thisworldwide is (RBC/UKQCD, one JLQCD, TWQCD) of using the DWF for reasons large-sc wh N simulate interconnected by the NVLink, with the total device memory 8 on the 64 simulation on the 32 PoS(LATTICE2019)133 ) 2. 1. = 2 / π p (2.2) (2.1) , ≫ ab i 32 δ 2 p ( a 2 t / / quarks. [MeV] Q t )] th. Never- )= h c x b ( res 0.045(4) 0.006(1) 0.178(5) T Ting-Wai Chiu a λσ , and F T s ) ( , 5 5 x symmetry, and re- 5 ( d − − − / ) µν u . However, to project 1 10 10 d, t 10 ( F [ a m A Q × × × tr U ) ) res ) , which hardly changes for 4 )] 12 17 )= pological charge ( t ( ( ssless overlap-Dirac operator , ( he topological fluctuations of lattice is prohibitively expen- µνλσ ov 21 ε 45 77 . 4 D . . x , ( e configurations for a sufficiently ∑ gical susceptibility and the quark )] clover x aking the Q e 64 pherical region of root-mean-square ( = [ is obtained from the four plaquettes behaves like a constant for a gauge ensemble without the proper ned by the Wilson flow [8, 9], which is ) λσ ) ght and heavy quarks. In the following, a m x t F mass spectrum (or any physical observ- q ( ( ) clover ty and the hadron mass spectra. m x The residual masses of 0.550 0 0.040 1 µν ( Q 0.00125 5 F clover µν t gets larger, the gauge configuration becomes F χ Q t [ tr d 2 s c / x Table 1: u 4 4-dimensional volume quark d is much heavier than other non-singlet (approximate) = ′ Z Ω η 2 π ) , c µνλσ , i 32 ε s 2 t computed with , Ω Q ) d = t h / ( t t u MeV. For Q = χ ( MeV, and is the flow-time. As quark, the t ) t χ ) 5 ) plane], unless the gauge configuration is sufficiently smoo ( 4 d ˆ ν ( / , converges to its nearest integer, round 6% of its bare u ˆ . µ ) 178 t . 4 045 ( . is the matrix-valued field tensor, with the normalization tr , where ∼ is the (integer-valued) topological charge of the gauge fiel t a µν t 8 clover F Q on the ( a √ Q lattice QCD with physcial domain-wall quarks , satisfying the Atiyah-Singer index theorem, index t x 1 = gT Q MeV respectively. This + quarks, the residual masses = 1 rms 1. Consequently, the ) c , where R + 1 µν ( 2 ≫ In lattice QCD with exact chiral symmetry, the index of the ma The vacuum of QCD has a non-trivial topological structure. T F ··· , 2 = and 2 a 006 f . , / Among all moments, the topological susceptibility smoother, and mass, amounting to 0 N Also, the residual masses of residual mass is sive. On the other hand, the clover topological charge theless, the smoothness of a gauge configurationa can continuous-smearing be process attai to average gaugeradius field over a s and solves the puzzle why the flavor-singlet the zero modes of the massless overlap-Dirac operator for th quarks are updated inconfigurations. Table 1 for 400 For s is not reliable [where the matrix-valued field tensor Goldstone bosons. In general, it can be shown that the topolo condensates are closely related, whichtopological susceptibility in cannot turn give implies the that correct hardon is equal to able) from the first principles of QCD. In other words, applying the Wilson flow to an ensemble of gaug 0 demonstrates that the optimal DWF can preserve the chiral symmetry to a high precision, for both li we present the first results of the topological susceptibili 2. Topological Susceptibility the QCD vacuum can be measured in terms of the moments of the to surrounding are even smaller, 0 1 is the most crucial one, which plays the important role in bre t PoS(LATTICE2019)133 t 2 is χ a 01. / . t (2.4) (2.3) 0 . The t clover = χ Q t for fitting ∆ t 0 [10]. Thus 10. Fitting Ting-Wai Chiu = > t 256. (right panel) ge of 2 a = / 2 256 is plotted on the t a in computing / 2 = t computed with . The horizontal line is the t/a )] 2 2 0 to 256 with t versus the flow-time t at a a χ ( ) / / = t t t s spectrum of QCD nonper- ( t at clover clover ) Q , Q t om clover [ ( of Q tra from the time-correlation func- p-Dirac operator at the experimental data. To this end, 7 aryon interpolators respectively, and MeV − tic value of clover ) iral symmetry can be obtained from the Q with 10 milar to the gauge fields in the continuum 78 0 50 100 150 200 250 . × ematic one. -5 -6 -7 0 )
10 10 10
t a
±
42 4 versus the flow-time . ) 0 30 t 3 . ( 1 ± becomes almost a constant for in the Wilson flow. t ± 66 χ . clover 7 29 Q . clover = ( 94 Q 4 =256 a
2 = ( t
2 0.15 0.13 0.10 0.08 0.05 0.03 0.00
χ In this study, the flow equation is numerically integrated fr One of the main objectives of lattice QCD is to extract the mas P(Q) = f is plotted on the right panel. Evidently, the left panel, while the topological susceptibility computed where the first/second uncertainty is the statistical/syst In Fig. 1, the histogram of the probability distribution of Figure 1: theory. Moreover, it has been demonstrated that the asympto N long flow-time can let them fall into topological sectors, si the topological susceptibility in lattice QCDasymptotic value with of exact ch to a constant for 30 in good agreement with that computed with the index of overla with final result of The topological susceptibility computed with constant fit for 30 as well as by replacing 3. Hardon Mass Spectrum turbatively from the firstthe principles, first and to step compare istions it to of with extract quark-antiquark the meson ground-state interpolators hadron and mass 3-quark spec b PoS(LATTICE2019)133 , , ), d = = ¯ sc / A u , Γ m min ¯ ss λ , / ¯ uc , max ¯ λ us , where Ting-Wai Chiu c , 3, Γ . ¯ ¯ ud c 1 and the proximity ) ), pseudovector ( = C 0 ( V j P respectively. Then the , and m J 1 c ) Γ kij ¯ s pseudovector 16, (1170) , c 1 0 1 1 s1 1 s h oretical state is incompatible h = K b 3097 D D Γ ( flavor contents Experiment (PDG) Experiment work This ), vector ( s ¯ s P ψ N , the conventional quark-antiquark / 55) of the sea quarks, where j c perators are measured with point- es as a prediction of lattice QCD. . J 1 Γ rs ( 0 ¯ state. Thus the conventional quark- ll components of this exotic hadron u l value. In this case, further theoreti- h the experimental mass spectra. If a of interpolators consisting of 3-quark, = 0 , 1 1 te by the same s1 and 1 c1 quark-antiquark-gluon operators; while s 1 a f K a D D ation matrix of interpolators consisting of ) Γ c ¯ e of this exotic hadron state. Theoretically, u m , , d quarks are predictions from the first principles j 1020 04 ( Γ . 1 c ¯ 0 u φ ), pseudoscalar ( , then there could be two possibilities. It could * vector pseudovector s ) 4 S K* , D* C D = ) ( J/ and P a s s J , 140 m d ( , , ± 0 u * π scalar * 0 s0 * c0 / 0 (980) 0 D D 0 f K f 00125 . 0 5 = 0 a s d c K / D D u : P pseudoscalar m J , corresponding to scalar ( 4 3 2 1 0 }
k
γ Mass[GeV] j γ i jk lattice QCD with physcial domain-wall quarks ε , 1 i are fixed by the masses of γ + 5 The ground-state masses of the quark-antiquark mesons with c γ 1 , m i 05) and masses ( . + γ , versus the experimental states [11]. 0 , 2 In the following, the time-correlation functions of local o The meson interpolators in this study are: 5 / ¯ cc γ and = , 20 s f 1I . { Another possibility is thatantiquark the or targeted the hadron 3-quark isand interpolator an gives cannot exotic a theoretical overlap mass withcal/experimental different studies a from are required the to experimenta clarifyfor the the natur exotic meson state, it requiresquark-antiquark, to meson-meson, analyze the diquark-antidiquark, correl and for the exotic baryon state,meson-baryon, to and study diquark-diquark-antiquark operators. the correlation matrix be a state to be observed in the future experiments, thus serv to-point quark propagators computed with the same paramete of mass, then wemeson can or infer 3-quark that baryon. thiswith hadron On any state the experimental behaves state other like with hand, the if same the mass of a the and 3.1 Meson mass spectrum masses of any other hadrons containing Figure 2: N to check whether the theoretical results are compatible wit theoretical state can be identified with an experimental sta of QCD. 6 m PoS(LATTICE2019)133 , , , , ) ) q )] )] s ) i k γ and γ 15 11 j 5 , and ± γ γ )( )( ) 2317 2460 ¯ s 2 1400 ( ( C / , ( i jk 62 ( 54 0 1 s 1 ( ∗ s s 1 ( ε P i c ¯ q γ J K D D = 5 , γ P ) 5138 ¯ 4017 u Ting-Wai Chiu J )[ , )[ 3/2 c , and cc − d − , and ccc ) i ) 2 2 1170 , while γ c c Experiment (PDG) Experiment work This / / ( 5 3 1 1 γ ( ( ++ ¯ h u 2300 1 cc 2420 ( ccc ( ∗ 0 Ω 1 = Ω , D , D )] l counterparts. This is also , )] 3/2 , 6 PC ) cc ccc ) J )( 11 c c c c )( 980 12 ith the error bar (statistical and coming paper. ( ( he first and the third columns in c 0 1285 22 f ( ( orrelation fucntions of these meson 1 , g all states, the ground-state masses f pseudovector meson operators with ) 3624 , )[ nd-state masses of the scalar operators 4760 ) c l counterpart with the same es in high energy experiments. It turns 1/2 cc + 700 )[ 2 ( c ial-vector operators c + / ∗ 0 2 1 1270 K / ( transforms like ( c , 3 cc 1 cc ( ) flavor combinations, versus the experimental states q i K Ω ) γ ccc , , c 5 5 500 ) , Ω γ )] ( s , 1/2 ¯ 5 , q 0 )] d c c c σ )( cc 5 / / 1260 u 0 )( ( : 26 ( f P 1 ( J 26 a ( The ground-state masses of 3-quark baryons with 5 4 3 2 1 respectively.
3040
) Mass [GeV] Mass 4139 , for all )[ , )[ ± − )] 2 − 2 / 2 / 2536 10 Figure 3: 3 [11]. ( / 3 1 3 ( )( s ( c D cc 34 Ω ( , Ω ± , )] 2 fla- )] 7 also agree well with ) respectively. Note that / 3 , and 1 2983 ) T )( also agree well with ) )( ¯ sc )[ c = c 32 , − 10 i ( s ( . P 2 γ , 2430 lattice QCD with physcial domain-wall quarks 5 J / − ( (in the last column of Fig. 2), they are also compatible with d , and γ 3 1 are plotted, 1 + ¯ s / ( k 1 ¯ operators are in very good agreement with their experimenta c 3009 3721 uc u D + γ j ( , Σ , )[ ± )[ c 1 = γ ) 2 + Γ − ¯ ss + / ¯ 2 c 2 , , and i jk PC / 2 c / ε The details of all mesons in Fig. 2 will be presented in a forth The construction of In Fig. 2, the ground-state masses extracted from the time-c i J 3 ¯ us 1 γ ( 1235 = ( , 5 ( = c cc f γ 1 ¯ ¯ b 3.2 Baryon mass spectrum u Ω respectively. Similarly, the ground-state masses of the ax N and pseudovector ( respectively. Likewise, forΓ the ground-state masses of the and 3 Ω 3-quark baryontor opera- andof the ground-state extraction masses from the time-correlation function follow the pre- scriptions invious our studies pre- In [12, Fig. 6]. 3,state masses the of ground- 3-quark baryons with like for all operators are plotted, versus theout corresponding that meson for stat each theoretical state, there is an experimenta vor combinations. They are all inment with good their counter- agree- parts in the experiments. Also, wedictions have for 9 the pre- (in masses unitsthe of following MeV)baryons: charmed of ud the case for theFig. pseudoscalar 2. and Moreover, vector it mesons, is as interesting to shown point in out t that the grou the theoretical mass is insystematic good combined) less agreement than with 2% the ofof PDG its the central mass, value. w Amon PoS(LATTICE2019)133 and ) − 2 / threshold, threshold. 1 ( K c ∗ Ting-Wai Chiu DK Ω D tibility (2.4) and extracted GC) and National Center for 1 lattice QCD with physical predicted states for charmed + , 193 (2017) , 55 (2014) facilities. This work is supported hcoming paper. , 95 (2015) 1 12-M-003-005, 107-2119-M-003- 767 738 is 41 MeV below the + 744 ) B B 2 is 32 MeV above the ve like the conventional meson/baryon ) interacting through the gluons with the , 092 (2014)] , no. 3, 030001 (2018) and 2019 update. if any) to meson-meson/meson-baryon one. Here we identify , 040 (2018) = resting to see that our theoretical hadron n Figs. 2-3) couple predominantly to the states observed by LHCb [13] in 2017, , no. 18, 182001 (2017) f 2317 98 c ( rk-diquark-antiquark interpolators. N 0 1403 2536 Ω 118 ∗ s 2018 ( , 305 (1995) 1 D s D 443 6 , 471 (2005) , 064 (2006) 755 of the five respectively. 0603 ) − 2 threshold, and / quarks in the sea. Note that some hadron states in Figs. 2-3 3050 K ( , 141 (1998) , 342 (1992) ∗ ) c , 071601 (2003); Phys. Lett. B c D , Ω and 3 90 417 s 288 , − d 2 , and , 071 (2010) Erratum: [JHEP / u ( ) [Particle Data Group], Phys. Rev. D 1008 3000 to be 1 ( c P et al. [LHCb Collaboration], Phys. Rev. Lett. J Ω lattice QCD with physcial domain-wall quarks 1 et al. is 44 MeV below the with domain-wall quarks, and determined the topological suscep + ) ) ) 1 c − , + 2 s 2 / , We have generated the first gauge ensemble for The details of all baryons in Fig. 3 will be presented in a fort We are grateful to Academia Sinica Grid Computing Center (AS 2460 3 d ( = ( 1 / c s f [1] D. B. Kaplan, Phys. Lett. B [2] H. Neuberger, Phys. Lett. B [3] R. Narayanan and H. Neuberger, Nucl. Phys. B [4] T. W. Chiu [TWQCD Collaboration], PoS LATTICE [5] T. W. Chiu, Phys. Rev. Lett. [6] Y. C. Chen, T. W. Chiu [TWQCD Collaboration], Phys. Lett. [9] M. Luscher, JHEP [7] Y. C. Chen, T. W. Chiu [TWQCD Collaboration], Phys. Lett. [8] R. Narayanan and H. Neuberger, JHEP u N where the first/second error is the statistical/systematic Ω and predict their 4. Concluding Remark ( the ground-state hadron mass spectra in Figs. 2-3. It is inte mass spectra are in good agreement with the PDG masses, plus 9 baryons. This implies that thesecomposed ground-state of hadrons valence beha quark-antiquark/quark-quark-quark, quantum fluctuations of are close to the threshold of strong decays, e.g., D This also suggests that the ground-statequark-antiquark/3-quark mesons/baryons (i interpolators, and onlyinterpolators, weakly not to ( mention diquark-antidiquark/diqua Acknowledgement High Performance Computing (NCHC) for the computerby time and the Ministry of Science008). and Technology (Grant Nos. 108-21 References [10] T. W. Chiu and T. H.[11] Hsieh, arXiv:1908.01676 [hep-lat]. M. Tanabashi [12] T. W. Chiu and T. H. Hsieh, Nucl. Phys. A [13] R. Aaij