Pseudoscalar Meson in Two Flavors QCD with the Optimal Domain-Wall Fermion
aa r X i v : . [ h e p - l a t ] S e p NTUTH-11-505E
Pseudoscalar Meson in Two Flavors QCD with the OptimalDomain-Wall Fermion
Ting-Wai Chiu,
1, 2, 3
Tung-Han Hsieh, and Yao-Yuan Mao (TWQCD Collaboration) Physics Department, National Taiwan University, Taipei 10617, Taiwan Center for Quantum Science and Engineering,National Taiwan University, Taipei 10617, Taiwan Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan Research Center for Applied Sciences, Academia Sinica, Taipei 115, Taiwan
Abstract
We perform hybrid Monte Carlo (HMC) simulations of two flavors QCD with the optimal domain-wall fermion (ODWF), on the 16 ×
32 lattice (with lattice spacing a ∼ . l and ¯ l , the pion decay constant, the chiral condensate, and the average up anddown quark mass. { ω s , s = 1 , · · · , N s } , onefor each layer in the fifth dimension [3]. Thus the artifacts due to the chiral symmetrybreaking with finite N s can be reduced to the minimum, especially in the chiral regime. The4-dimensional effective Dirac operator of massless ODWF is D = m [1 + γ S opt ( H w )] ,S opt ( H w ) = 1 − Q N s s =1 T s Q N s s =1 T s , T s = 1 − ω s H w ω s H w , which is exactly equal to the Zolotarev optimal rational approximation of the overlap Diracoperator. That is, S opt ( H w ) = H w R Z ( H w ), where R Z ( H w ) is the optimal rational approxi-mation of ( H w ) − / [5, 6].Recently we have demonstrated that it is feasible to perform a large-scale unquenchedQCD simulation which not only preserves the chiral symmetry to a good precision, but alsosamples all topological sectors ergodically [7]. To recap, we perform HMC simulations of 2flavors QCD on a 16 ×
32 lattice, with ODWF at N s = 16, and plaquette gauge action at β = 5 .
95. Then we compute the low-lying eigenmodes of the overlap Dirac operator, anduse its index to obtain the topological charge of each gauge configuration, and from whichwe compute the topological susceptibility for 8 sea-quark masses, each of 300 configurations.Our result of the topological susceptibility agrees with the sea-quark mass dependence pre-dicted by the NLO ChPT [8], and provides the first determination of both the pion decayconstant and the chiral condensate simultaneously from the topological susceptibility.In this paper, we perform further simulations and increase the ensemble of each sea-quarkmass from 300 to 500 configurations. That is, for each sea-quark mass, we generate 5000trajectories after thermalization, and sample one configuration every 10 trajectories. Thenwe compute the valence quark propagators and the time-correlation function of the pseu-doscalar meson operator, and from which we extract the mass M π and the decay constant2 π of the pseudoscalar meson. We compare our results of M π and F π with the NLO ChPT[9], and find that our results are in good agreement with the sea-quark mass dependencepredicted by NLO ChPT, and from which we obtain the low-energy constants F , Σ, ¯ l and¯ l . With the low-energy constants, we determine the average up and down quark mass m MS ud (2 GeV), and the chiral condensate Σ MS (2 GeV).First, we outline our HMC simulation of 2 flavors QCD with ODWF. Starting from theODWF action S = ¯Ψ D Ψ [3] on the 5D lattice, we separate the even and the odd sites (theso-called even-odd preconditioning) on the 4D lattice, and rewrite D as D ( m q ) = S − M D OE w C M D EO w S − , where m q denotes the bare quark mass, D w denotes the standard Wilson Dirac operatorplus a negative parameter − m (Here m = 1 . D EO/OE w denotes thepart of D w with gauge links pointing from odd/even sites to even/odd sites, and M = (cid:2) (4 − m ) + ω − / (1 − L )(1 + L ) − ω − / (cid:3) − , ( ω ) ss ′ = ω s δ ss ′ ,L = P + L + + P − L − , P ± = (1 ± γ ) / , L − = ( L + ) T , ( L + ) ss ′ = δ s − ,s ′ , < s ≤ N s − ( m q / m ) δ N s ,s ′ , s = 1 ; S = M ω − / , S = (1 + L ) − ω − / ,C = 1 − M D OE w M D EO w . Since det D = det S − · det C · det S − , and S and S do not depend on the gauge field,we can just use C for the HMC simulation. After including the Pauli-Villars fields (with m q = 2 m ), the pseudo-fermion action for 2 flavors QCD ( m u = m d ) can be written as S pf = φ † C † P V ( CC † ) − C P V φ, C
P V ≡ C (2 m ) . (1)In the HMC simulation [10], we first generate random noise vector ξ with Gaussiandistribution, then we obtain φ = C − P V Cξ using the conjugate gradient (CG). With fixed φ , the system is evolved under a fictituous Hamiltonian dynamics, the so-called moleculardynamics (MD). In the MD, we use the Omelyan integrator [11], and the Sexton-Weingartenmultiple-time scale method [12]. The most time-consuming part in the MD is to compute3he vector η = ( CC † ) − C P V φ with CG, which is required for the evaluation of the fermionforce in the equation of motion for the conjugate momentum of the gauge field. Here wetake advantage of the remarkable floating-point capability of the Nvidia GPU, and performthe CG with mixed precision [13]. Moreover, the computations of the gauge force and thefermion force, and the update of the gauge field are also ported to the GPU. In other words,almost the entire HMC simulation is performed within a single GPU.Furthermore, we introduce an auxillary heavy fermion field with mass m H ( m q ≪ m H < m ), similar to the case of the Wilson fermion [14]. For two flavors QCD, the pseudofermionaction (with C H ≡ C ( m H )) becomes, S Hpf = φ † C † H ( CC † ) − C H φ + φ † H C † P V ( C H C † H ) − C P V φ H , which gives exactly the same fermion determinant of (1). Nevertheless, the presence of theheavy fermion field plays a crucial role in reducing the light fermion force and its fluctuation,thus diminishes the change of the Hamiltonian in the MD trajactory, and enhances theacceptance rate. A detailed description of our HMC simulations will be presented in aforthcoming paper [15].We determine the lattice spacing by heavy quark potential which is extracted from Wil-son loops of size ( R , R , T ), where R , R and T are the sizes in spatial and temporaldirections. The spatial distance between the heavy quark and antiquark is R = p R + R .We measure all planar and non-planar Wilson loops W with a ≤ R ≤ a and a ≤ T ≤ a .Fitting the data of W ( R, T ) to the formula h W i = C exp( − T V ( R )), we obtain the heavyquark potential V ( R ) as a function of R . Here we have used all 5000 trajectories afterthermalization, and we estimate the error of V ( R ) using the jackknife method with the binsize of which the statistical error saturates. Then we fit our data of V to the formula V ( R ) = A + BR + σR, (2)to obtain A , B , and σ . We summarize our results in Table I.Using the empirical formula deduced by Sommer [16], F ( r ) r = 1 . , F ( r ) ≡ ddr V ( r ) = − Br + σ, (3)and setting the Sommer parameter r = 0 .
49 fm, we obtain the lattice spacing a = r r σ .
65 +
B , (4)4 q a A B σ χ /dof a [fm]0.01 0.7777(57) -0.3814(70) 0.0577(10) 0.0329 0.1045(13)0.02 0.7827(46) -0.3818(41) 0.0584(9) 0.0275 0.1051(10)0.03 0.7792(54) -0.3789(62) 0.0595(9) 0.0368 0.1060(12)0.04 0.7916(71) -0.3995(78) 0.0598(13) 0.0440 0.1071(16)0.05 0.7797(73) -0.3798(72) 0.0615(13) 0.0456 0.1078(16)0.06 0.7762(50) -0.3785(44) 0.0628(11) 0.0458 0.1089(11)0.07 0.7783(47) -0.3855(53) 0.0633(8) 0.0255 0.1097(10)0.08 0.7719(69) -0.3744(64) 0.0649(12) 0.0569 0.1105(14)TABLE I: The parameters of A , B , and σ obtained by fitting our data of heavy quark potential V ( R ) to Eq. (2), together with the χ /dof of the fit. The lattice spacing in the last column isobtained by (4). where the results are given in the last column of Table I. Using the linear fit, we obtainthe lattice spacing in the chiral limit, a = 0 . χ /dof = 0.10, where thesystematic error is estimated by varying the number of sea-quark masses. This gives theinverse lattice spacing a − = 1 . D ( m q ) | Y i = D (2 m ) B − | source vector i , (5)where B − x,s ; x ′ ,s ′ = δ x,x ′ ( P − δ s,s ′ + P + δ s +1 ,s ′ ) with periodic boundary conditions in the fifthdimension. Then the solution of (5) gives the valence quark propagator( D c + m q ) − x,x ′ = (2 m − m q ) − [( BY ) x, x ′ , − δ x,x ′ ] . To measure the chiral symmetry breaking due to finite N s , we compute the residual masswith the formula [17] m res = * tr( D c + m q ) − , tr[( D † c + m q )( D c + m q )] − , + { U } − m q , (6)5here ( D c + m q ) − denotes the valence quark propagator with m q equal to the sea-quarkmass, tr denotes the trace running over the color and Dirac indices, and the subscript { U } denotes averaging over an ensemble of gauge configurations. In Table II, we list the residualmasses for eight sea quark masses, together with those obtained by setting ω s = 1 (polarapproximation of the sign function of H w ) in the valance quark propagator. In the lattercase, even though the chiral symmetry of the valence quarks is different from that of the seaquarks, it may serve as an estimate of the residual mass in the unitary limit with ω s = 1.We see that turning on { ω s } with λ min /λ max = 0 . / .
40, the residual mass is decreasedby a factor of 25-40, while the cost of computing quark propagators is increased by a factorof 2-5. Moreover, for m q a = 0 .
01, we also computed the residual mass with N s = 32 and ω s = 1, and obtained m res = 0 . { ω s } with N s = 16 and λ min /λ max = 0 . / .
40, while the cost is almost the same in bothcases. This suggests that ODWF is a viable way to preserve the chiral symmetry on thelattice, without increasing N s . For ODWF, using the linear fit, we obtain the residual massin the chiral limit, m res a = 0 . m q is corrected by its residualmass, i.e., m q → m q + m res . m q a m res (ODWF) m res ( ω s = 1) ratio0.01 0.000418(31) 0.01064(17) 0.039(3)0.02 0.000380(29) 0.01139(15) 0.033(3)0.03 0.000269(40) 0.01047(13) 0.026(4)0.04 0.000259(43) 0.01043(12) 0.025(4)0.05 0.000269(41) 0.01000(13) 0.027(4)0.06 0.000357(47) 0.01029(11) 0.035(4)0.07 0.000248(45) 0.00988(15) 0.025(6)0.08 0.000219(38) 0.00991(13) 0.022(4)TABLE II: The residual mass (second column) versus the sea quark mass for two flavors QCDwith ODWF. The third column is the residual mass obtained by setting ω s = 1 in the valencequark propagator. The last column is the ratio m res (ODWF)/ m res ( ω s = 1). Using the valence quark propagator with quark mass equal to the sea-quark mass, we6 x 32 x 16, β = 5.95, m val =m sea t C ( t ) -4 -3 -2 -1 m = 0.01 m = 0.02 m = 0.03 m = 0.04 m = 0.05 m = 0.06 m = 0.07 m = 0.08 Pseudoscalar Meson t m e ff m = 0.01 m = 0.02 m = 0.03 m = 0.04 m = 0.05 m = 0.06 m = 0.07 m = 0.08 x 32 x 16, β = 5.95, m val =m sea Pseudoscalar Meson (a) (b)FIG. 1: (color online) (a) The time-correlation function of the pseudoscalar meson for eight seaquark masses. (b) The effective mass of (a). The dashed line connecting the data points of thesame sea-quark mass is for guiding the eyes. compute the time-correlation function of the pseudoscalar interpolator C ( t ) = X ~x tr { γ ( D c + m q ) − ,x γ ( D c + m q ) − x, } , where the trace runs over the Dirac and color space. In Fig. 1, we plot C ( t ) and its effectivemass m eff ( t ) = cosh − { [ C ( t + 1) + C ( t − / (2 C ( t )) } for eight sea quark masses respectively.Then h C ( t ) i is fitted to the formula Z [ e − M π t + e − M π ( T − t ) ] / (2 M π ) to extract the pion mass M π and the decay constant F π = m q √ Z/M π , where the excited states have been neglected.Here we have chosen the fitting range [ t , t ] in which the effective mass attaining a plateau,and we estimate the errors of M π and F π using the jackknife method with the bin size of 15configurations of which the statistical error saturates.We make the correction for the finite volume effect using the estimate within ChPTcalculated up to O ( M π / (4 πF π ) ) [18]. In Table III, we give the values of M π and F π (withfinite volume corrections), together with their finite volume correction factors computedusing the formulas given in [18]. In Fig. 2, we plot M π /m q and F π versus m q respectively.For the lighest pion, M π L ≃ .
0, the formulas for finite volume correction may be unreliable,according to Ref. [18]. Thus, we perform the ChPT fit with the lightest pion excluded. Thenwe will check whether the lightest pion falls on the curve of the ChPT fit.Taking into account of the correlation between M π /m q and F π for the same sea-quark7 q a [ t , t ] χ /dof M π [GeV] F π [GeV] 1 + R M π R F π M π and F π . The second column is the range [ t , t ] of thetime-correlation function used for fitting, the third column is the χ /dof of the fit, and the lasttwo columns are finite volume corrections for M π and F π respectively. m q [GeV] M π / m q [ G e V ] N f = 2 x 32 x 16, β = 5.95, m val =m sea m q [GeV] F π [ G e V ] N f = 2 x 32 x16, β = 5.95, m val =m sea (a) (b)FIG. 2: Physical results of 2 flavors QCD with ODWF (a) M π /m q , and (b) F π . The solid linesare the simultaneous fits to the NLO ChPT, for seven sea-quark masses ( m q a = 0 . − . M π m q = 2Σ F (cid:20) (cid:18) Σ m q π F (cid:19) ln (cid:18) m q F Λ (cid:19)(cid:21) , (7) F π = F (cid:20) − (cid:18) Σ m q π F (cid:19) ln (cid:18) m q F Λ (cid:19)(cid:21) , (8)where Λ and Λ are related to the low energy constants ¯ l and ¯ l as follows.¯ l = ln (cid:18) Λ m π ± (cid:19) , ¯ l = ln (cid:18) Λ m π ± (cid:19) , m π ± = 0 .
140 GeV . The strategy of our data fitting is to search for the values of the parameters Σ, F , Λ and Λ such that they minimize χ = X i V Ti C − i V i , V i = ( M π /m q ) i − ( M π /m q ) ChPT i ( F π ) i − ( F π ) ChPT i , where C i is the 2 × M π /m q and F π with the same sea-quark mass.For seven sea-quark masses corresponding to pion masses in the range 309 −
565 MeV,our fit gives Σ = [0 . , (9) F = 0 . , (10)¯ l = 4 . , (11)¯ l = 4 . , (12)with χ /dof = 0.07, where the systematic errors are estimated by varying the number ofdata points from 7 ( M π ≤
565 MeV) to 4 ( M π ≤
459 MeV). In Fig. (2), we see that the datapoints of the lightest pion also fall on the curves of NLO ChPT fit. This seems to suggestthat the finite volume corrections for the lightest pion (with M π L ≃ .
0) may be correct.To obtain the physical bare quark mass, we use the physical ratio ( M π /F π ) phys =0 . / .
093 = 1 .
45 as the input, and solve the equation M π ( m q ) /F π ( m q ) = 1 .
45 to ob-tain the physical bare quark mass m physq = 0 . F π = 0 . , (13) M π = 0 . . (14)9ince we have used the physical ratio 1.45 as the input, in principle, we can only regardeither (13) or (14) as our predicted physical result.In order to convert the chiral condensate Σ and the average m u and m d to those in theMS scheme, we calculate the renormalization factor Z MS s (2 GeV) using the non-perturbativerenormalization technique through the RI/MOM scheme [19], and our result is [20] Z MS s (2 GeV) = 1 . . (15)Then the values of Σ and the average of m u and m d are transcribed toΣ MS (2 GeV) = [230(4)(6) MeV] , (16) m MS ud (2 GeV) = 4 . , (17)where the systematic errors follow from those in Eqs. (9) and (15).Since our calculation is done at a single lattice spacing the discretization error cannot bequantified reliably, but we do not expect much larger error because our lattice action is freefrom O ( a ) discretization effects.We also investigated to what extent our results of the low-energy constants dependingon the chiral symmetry of the valence quark propagators. We repeated above analysis withvalence quark propagators computed with N s = 32 and λ min /λ max = 0 . / .
4, which hasthe residual mass m res a = 0 . SU (2) low-energy constants, the chiral condensate, and theaverage up and down quark mass are compatible with those obtained by other lattice groupsusing unitary dynamical quarks with N f = 2, e.g., Ref. [21]. A detailed comparison withall lattice results [22] is beyond the scope of this paper.To conclude, our results of the mass and the decay constant of the pseudoscalar mesonare in good agreement with the sea-quark mass dependence predicted by the next-to-leadingorder (NLO) ChPT, and provide a determination of the low-energy constants ¯ l and ¯ l ,the pion decay constant, the chiral condensate, and the average up and down quark mass.Together with our recent result of the topological susceptibility [7], these suggest that thenonperturbative chiral dynamics of the sea quarks are well under control in our HMC simu-lations. Moreover, this study also shows that it is feasible to perform large-scale simulations10f unquenched lattice QCD, which not only preserve the chiral symmetry to a good pre-cision, but also sample all topological sectors ergodically. This provides a new strategy totackle QCD nonperturbatively from the first principles.This work is supported in part by the National Science Council (Nos. NSC99-2112-M-002-012-MY3, NSC99-2112-M-001-014-MY3) and NTU-CQSE (No. 10R80914-4). We alsothank NCHC and NTU-CC for providing facilities to perform part of our calculations. [1] D. B. Kaplan, Phys. Lett. B , 342 (1992); Nucl. Phys. Proc. Suppl. , 597 (1993).[2] H. Neuberger, Phys. Lett. B , 141 (1998); R. Narayanan and H. Neuberger, Nucl. Phys.B , 305 (1995).[3] T. W. Chiu, Phys. Rev. Lett. , 071601 (2003); Phys. Lett. B , 97 (2003); hep-lat/0303008[4] T.W. Chiu et al. [TWQCD Collaboration], PoS LATTICE2009 , 034 (2009). [arXiv:0911.5029[hep-lat]][5] N. I. Akhiezer, ”Theory of approximation”, Reprint of 1956 English translation, Dover, NewYork, 1992.[6] T. W. Chiu, T. H. Hsieh, C. H. Huang and T. R. Huang, Phys. Rev. D , 114502 (2002).[7] T. W. Chiu, T. H. Hsieh and Y. Y. Mao, Phys. Lett. B , 131 (2011). [arXiv:1105.4414[hep-lat]][8] Y. Y. Mao and T. W. Chiu [TWQCD Collaboration], Phys. Rev. D , 034502 (2009).[9] J. Gasser and H. Leutwyler, Nucl. Phys. B , 465 (1985).[10] S. Duane, A. D. Kennedy, B. J. Pendleton and D. Roweth, Phys. Lett. B , 216 (1987).[11] T. Takaishi and P. de Forcrand, Phys. Rev. E , 036706 (2006).[12] J. C. Sexton and D. H. Weingarten, Nucl. Phys. B , 665 (1992).[13] T. W. Chiu et al. [ TWQCD Collaboration ], PoS LATTICE2010 , 030 (2010).[arXiv:1101.0423 [hep-lat]], and references therein.[14] M. Hasenbusch, Phys. Lett. B , 177 (2001).[15] T. W. Chiu et al. [TWQCD Collaboration], “Monte Carlo simulation of lattice QCD with theoptimal domain-wall fermion”, in preparation.[16] R. Sommer, Nucl. Phys. B , 839 (1994)[17] Y. C. Chen, T .W. Chiu [TWQCD Collaboration] arXiv:1205.6151 [hep-lat].
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