First study of N f =2+1+1 lattice QCD with physical domain-wall quarks
aa r X i v : . [ h e p - l a t ] F e b First study of N f = + + lattice QCD with physicaldomain-wall quarks Ting-Wai Chiu ∗† (TWQCD collaboration) Physics Department, National Taiwan Normal University, Taipei, Taiwan 11677, R.O.C.Institute of Physics, Academia Sinica, Taipei, Taiwan 11529, R.O.C.Physics Department, National Taiwan University, Taipei, Taiwan 10617, R.O.C.E-mail: [email protected]
Using 10-16 units of Nvidia DGX-1, we have generated the first gauge ensemble for N f = + + ( u / d , s , c ) domain-wall quarks, on the 64 lattice with latticespacing a ∼ .
064 fm ( L > M π L > ∗ Speaker. † This work is supported by the Ministry of Science and Technology (Nos. 107-2119-M-003-008, 108-2112-M-003-005) 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/ f = + + lattice QCD with physcial domain-wall quarks Ting-Wai Chiu
1. Introduction
The holy grail of lattice QCD is to simulate QCD at the physical point, with sufficiently largevolume and fine lattice spacing, then to extract physics from these gauge ensembles. Nevertheless,the characteristics of a gauge ensemble depends on the lattice action used for the simulation, whichin turn has significant impacts on the physics outcome.Since quarks are Dirac fermions, they possess the chiral symmetry in the massless limit. At thezero temperature, the chiral symmetry SU L ( N f ) × SU R ( N f ) of N f massless quarks is spontaneouslybroken to SU V ( N f ) , due to the strong interaction between quarks and gluons in the QCD vacuum(with non-trivial topology). The quark condensate h ¯ qq i constitutes the origin of hadron masses,and resolves the puzzles such as “why the mass of a proton is much heavier than the sum of thebare quark masses of its constituents". To investigate the spontaneous chiral symmetry breaking aswell as the hadron physics (e.g., the mass spectrum) from the first principles of QCD, it requiresnonperturbative methods. So far, lattice QCD is the most promising approach, discretizing thecontinuum space-time on a 4-dimensional lattice, and computing physical observables by MonteCarlo simulation.However, in lattice QCD, formulating lattice fermion with exact chiral symmetry at finitelattice spacing is rather nontrivial. This is realized through domain-wall fermion (DWF) on the 5-dimensional lattice [1] and the overlap-Dirac fermion on the 4-dimensional lattice [2, 3]. Neverthe-less, the computational requirement for lattice QCD with domain-wall quarks on a 5-dimensionallattice is 10-100 times more than their counterparts with traditional lattice fermions (e.g., Wilson,staggered, and their variants). This is one of the reasons why there are only 3 lattice QCD groupsworldwide (RBC/UKQCD, JLQCD, TWQCD) using DWF for large-scale lattice QCD simulations.Since 2018, TWQCD has been performing the hybrid Monte Carlo (HMC) simulation of N f = + + lattice with the extent N s =
16 in the fifth dimension, following our first N f = + + ×
64 lattice [6]. It is interesting to point out that the entire simulation on the64 ×
16 lattice can be fitted into one unit of Nvidia DGX-1, which consists of eight V100 GPUsinterconnected by the NVLink, with the total device memory 8 ×
16 GB =
128 GB. In general, tosimulate N f = + + lattice with any lattice Dirac operator D requiresmemory much larger than 128 GB, since each one-flavor pseudofermion action is expressed as therational approximation of Φ † ( D † D ) − / Φ , requiring a large number of long vectors (proportionalto the number of poles in the rational approximation) in computing the pseudofermion force in themolecular dynamics. However, for domain-wall fermion, one can use the exact one-flavor pseud-ofermion action (EOFA) with a positive-definite and Hermitian Dirac operator [7], thus avoidingthe rational approximation and saving a large amount of device memory. Moreover, using EOFAalso enhances the HMC efficiency significantly.The thermalization is performed with one unit of Nvidia DGX-1, running for ∼ ∼ −
10 months, resulting a total of ∼ a − = . ± .
018 GeV for 400 gauge configurations.1 f = + + lattice QCD with physcial domain-wall quarks Ting-Wai Chiu
Table 1:
The residual masses of u / d , s , and c quarks. quark m q a m res a m res [MeV] u / d . ( ) × − s . ( ) × − c . ( ) × − ( u / d , s , c ) quarks are updated in Table 1 for 400configurations. For u / d quark, theresidual mass is ∼ .
6% of its baremass, amounting to 0 . ( ) MeV. For s and c quarks, the residual massesare even smaller, 0 . ( ) MeV, and0 . ( ) MeV respectively. Thisdemonstrates that the optimal DWF canpreserve the chiral symmetry to a high precision, for both light and heavy quarks. In the following,we present the first results of the topological susceptibility and the hadron mass spectra.
2. Topological Susceptibility
The vacuum of QCD has a non-trivial topological structure. The topological fluctuations ofthe QCD vacuum can be measured in terms of the moments of the topological charge h Q pt i , p = , , · · · , where Q t is the (integer-valued) topological charge of the gauge field, Q t = ε µνλσ π Z d x tr [ F µν ( x ) F λσ ( x )] , (2.1)and F µν = gT a F a µν is the matrix-valued field tensor, with the normalization tr ( T a T b ) = δ ab / χ t χ t = h Q t i Ω , Ω = , (2.2)is the most crucial one, which plays the important role in breaking the U A ( ) symmetry, and re-solves the puzzle why the flavor-singlet η ′ is much heavier than other non-singlet (approximate)Goldstone bosons. In general, it can be shown that the topological susceptibility and the quarkcondensates are closely related, which in turn implies that a gauge ensemble without the propertopological susceptibility cannot give the correct hardon mass spectrum (or any physical observ-able) from the first principles of QCD. In lattice QCD with exact chiral symmetry, the index of the massless overlap-Dirac operatoris equal to Q t , satisfying the Atiyah-Singer index theorem, index ( D ov ) = Q t . However, to projectthe zero modes of the massless overlap-Dirac operator for the 64 lattice is prohibitively expen-sive. On the other hand, the clover topological charge Q clover = ∑ x ε µνλσ tr [ F µν ( x ) F λσ ( x )] / ( π ) is not reliable [where the matrix-valued field tensor F µν ( x ) is obtained from the four plaquettessurrounding x on the ( ˆ µ , ˆ ν ) plane], unless the gauge configuration is sufficiently smooth. Never-theless, the smoothness of a gauge configuration can be attained by the Wilson flow [8, 9], which isa continuous-smearing process to average gauge field over a spherical region of root-mean-squareradius R rms = √ t , where t is the flow-time. As t gets larger, the gauge configuration becomessmoother, and Q clover ( t ) converges to its nearest integer, round [ Q clover ( t )] , which hardly changes for t / a ≫
1. Consequently, the χ t ( t ) computed with Q clover ( t ) behaves like a constant for t / a ≫ f = + + lattice QCD with physcial domain-wall quarks Ting-Wai Chiu long flow-time can let them fall into topological sectors, similar to the gauge fields in the continuumtheory. Moreover, it has been demonstrated that the asymptotic value of χ t computed with Q clover isin good agreement with that computed with the index of overlap-Dirac operator at t = χ t computed with Q clover in the Wilson flow.In this study, the flow equation is numerically integrated from t = ∆ t = . Q clover ( t ) at t / a =
256 is plotted on theleft panel, while the topological susceptibility computed with Q clover ( t ) versus the flow-time t / a is plotted on the right panel. Evidently, the χ t becomes almost a constant for t / a >
10. Fitting χ t to a constant for 30 ≤ t / a ≤
256 gives χ t a = ( . ± . ) × − (2.3)with χ / d.o.f. = .
11. The systematic error can be estimated by changing the range of t for fittingas well as by replacing Q clover ( t ) with its nearest integer round [ Q clover ( t )] in computing χ t . Thefinal result of χ t in the energy units is χ / t = ( . ± . ± . ) MeV , (2.4)where the first/second uncertainty is the statistical/systematic one. Q -10 -5 0 5 10 P ( Q ) t/a = 256t
3. Hardon Mass Spectrum
One of the main objectives of lattice QCD is to extract the mass spectrum of QCD nonper-turbatively from the first principles, and to compare it with the experimental data. To this end,the first step is to extract the ground-state hadron mass spectra from the time-correlation func-tions of quark-antiquark meson interpolators and 3-quark baryon interpolators respectively, and3 f = + + lattice QCD with physcial domain-wall quarks Ting-Wai Chiu to check whether the theoretical results are compatible with the experimental mass spectra. If atheoretical state can be identified with an experimental state by the same J P ( C ) and the proximityof mass, then we can infer that this hadron state behaves like the conventional quark-antiquarkmeson or 3-quark baryon. On the other hand, if the mass of a theoretical state is incompatiblewith any experimental state with the same J P ( C ) , then there could be two possibilities. It couldbe a state to be observed in the future experiments, thus serves as a prediction of lattice QCD.Another possibility is that the targeted hadron is an exotic state. Thus the conventional quark-antiquark or the 3-quark interpolator cannot overlap with all components of this exotic hadronand gives a theoretical mass different from the experimental value. In this case, further theoreti-cal/experimental studies are required to clarify the nature of this exotic hadron state. Theoretically,for the exotic meson state, it requires to analyze the correlation matrix of interpolators consisting ofquark-antiquark, meson-meson, diquark-antidiquark, and quark-antiquark-gluon operators; whilefor the exotic baryon state, to study the correlation matrix of interpolators consisting of 3-quark,meson-baryon, and diquark-diquark-antiquark operators.In the following, the time-correlation functions of local operators are measured with point-to-point quark propagators computed with the same parameters ( N s = m = . λ max / λ min = . / .
05) and masses ( m u / d a = . , m s a = . , m c a = .
55) of the sea quarks, where m u / d , m s and m c are fixed by the masses of π ± ( ) , φ ( ) and J / ψ ( ) respectively. Then themasses of any other hadrons containing u , d , s and c quarks are predictions from the first principlesof QCD. M a ss [ G e V ] Experiment (PDG)This work J P : KD c J/ Meson operator: q, Q,q = {u/d, s, c} j j kij j D s D*D s * pseudoscalar scalar vector pseudovector pseudovector K* c0 c1 h c D s1 D s1 K K D D D s0 *D * K *f / a h (1170) f b f (980) Figure 2:
The ground-state masses of the quark-antiquark mesons with flavor contents ¯ ud , ¯ us , ¯ uc , ¯ ss , ¯ sc ,and ¯ cc , versus the experimental states [11]. The meson interpolators in this study are: ¯ u Γ d , ¯ u Γ s , ¯ u Γ c , ¯ s Γ s , ¯ s Γ c , and ¯ c Γ c , where Γ = { , γ , γ i , γ γ i , ε i jk γ j γ k } , corresponding to scalar ( S ), pseudoscalar ( P ), vector ( V ), pseudovector ( A ),4 f = + + lattice QCD with physcial domain-wall quarks Ting-Wai Chiu and pseudovector ( T ) respectively. Note that ¯ q γ γ i q transforms like J PC = ++ , while ¯ q ε i jk γ j γ k q like J PC = + − .In Fig. 2, the ground-state masses extracted from the time-correlation fucntions of these mesonoperators are plotted, versus the corresponding meson states in high energy experiments. It turnsout that for each theoretical state, there is an experimental counterpart with the same J P ( C ) , andthe theoretical mass is in good agreement with the PDG mass, with the error bar (statistical andsystematic combined) less than 2% of its central value. Among all states, the ground-state massesof the ¯ c Γ c operators are in very good agreement with their experimental counterparts. This is alsothe case for the pseudoscalar and vector mesons, as shown in the first and the third columns inFig. 2. Moreover, it is interesting to point out that the ground-state masses of the scalar operators¯ ud , ¯ us , ¯ ss , ¯ uc , and ¯ sc also agree well with f / σ ( ) , K ∗ ( ) , f ( ) , D ∗ ( ) , and D ∗ s ( ) respectively. Similarly, the ground-state masses of the axial-vector operators ¯ u γ γ i d , ¯ u γ γ i s , ¯ s γ γ i s ,¯ u γ γ i c , and ¯ s γ γ i c also agree well with a ( ) , K ( ) , f ( ) , D ( ) , and D s ( ) respectively. Likewise, for the ground-state masses of the pseudovector meson operators with Γ = ε i jk γ j γ k (in the last column of Fig. 2), they are also compatible with h ( ) , K ( ) , b ( ) , D ( ) , and D s ( ) respectively.The details of all mesons in Fig. 2 will be presented in a forthcoming paper. M a ss [ G e V ] Experiment (PDG)
This work J P : c c c c cc c c c c c c c c c c ccc ccc cc cc cc cc c c Figure 3:
The ground-state masses of 3-quark baryons with J P = / ± and3 / ± , for all ( u / d , s , c ) flavor combinations, versus the experimental states[11]. The construction of3-quark baryon opera-tor and the extractionof ground-state massesfrom the time-correlationfunction follow the pre-scriptions in our pre-vious studies [12, 6].In Fig. 3, the ground-state masses of 3-quarkbaryons with J P = / ± and 3 / ± are plotted,for all ( u / d , s , c ) fla-vor combinations. Theyare all in good agree-ment with their counter-parts in the experiments.Also, we have 9 pre-dictions for the masses(in units of MeV) ofthe following charmedbaryons: Σ c ( / − )[ ( )( )] , Ω c ( / − )[ ( )( )] , Ω c ( / − )[ ( )( )] , Ω cc ( / + )[ ( )( )] , Ω cc ( / − )[ ( )( )] , Ω cc ( / + )[ ( )( )] , Ω cc ( / − )[ ( )( )] , Ω ccc ( / + )[ ( )( )] , Ω ccc ( / − )[ ( )( )] ,5 f = + + lattice QCD with physcial domain-wall quarks Ting-Wai Chiu where the first/second error is the statistical/systematic one. Here we identify Ω c ( / − ) and Ω c ( / − ) with Ω c ( ) and Ω c ( ) of the five Ω c states observed by LHCb [13] in 2017,and predict their J P to be 1 / − and 3 / − respectively.The details of all baryons in Fig. 3 will be presented in a forthcoming paper.
4. Concluding Remark
We have generated the first gauge ensemble for N f = + + ( u / d , s , c ) domain-wall quarks, and determined the topological susceptibility (2.4) and extractedthe ground-state hadron mass spectra in Figs. 2-3. It is interesting to see that our theoretical hadronmass spectra are in good agreement with the PDG masses, plus 9 predicted states for charmedbaryons. This implies that these ground-state hadrons behave like the conventional meson/baryoncomposed of valence quark-antiquark/quark-quark-quark, interacting through the gluons with thequantum fluctuations of ( u , d , s , c ) quarks in the sea. Note that some hadron states in Figs. 2-3are close to the threshold of strong decays, e.g., D s ( ) is 32 MeV above the D ∗ K threshold, D s ( ) is 44 MeV below the D ∗ K threshold, and D ∗ s ( ) is 41 MeV below the DK threshold.This also suggests that the ground-state mesons/baryons (in Figs. 2-3) couple predominantly to thequark-antiquark/3-quark interpolators, and only weakly (if any) to meson-meson/meson-baryoninterpolators, not to mention diquark-antidiquark/diquark-diquark-antiquark interpolators. Acknowledgement
We are grateful to Academia Sinica Grid Computing Center (ASGC) and National Center forHigh Performance Computing (NCHC) for the computer time and facilities. This work is supportedby the Ministry of Science and Technology (Grant Nos. 108-2112-M-003-005, 107-2119-M-003-008).
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