-
In our simulations, we computed the correlation functions of the H-dibaryon and Λ. We also calculated the correlation function of N, Σ, and Ξ. The generic form of a correlation function is
G(→x,τ)=<O(→x,τ)O+(0)>,
(1) and for the H-dibaryon interpolating operator, we chose the local operator. The starting point is the following operator notation for the different combinations of six quarks [30]:
[abcdef]= ϵijkϵlmn(biCγ5P+cj)×(elCγ5P+fm)(akCγ5P+dn)(→x,t),
(2) where
a,b,…,f denote generic quark flavors, andP+=(1+γ0)/2 projects the quark fields to positive parity. We chose the operatorOH as the H-dibaryon interpolating operator which transforms under the singlet irreducible representation of flavor SU(3) [17, 51–53],OH=148([sudsud]−[udusds]−[dudsus]),
(3) and the H-dibaryon correlation function can be obtained from the formulae in Ref. [53].
For the baryon Λ interpolating operator, we chose the standard definition (see, for example, Refs. [54–56]),
OΛ(x)=1√6ϵabc{2(uTa(x) Cγ5 db(x))sc(x)+ (uTa(x) Cγ5 sb(x))dc(x)− (dTa(x) Cγ5 sb(x))uc(x)} ,
(4) and for the correlation function of the baryon Λ, we also chose the standard definition (see, for example, Refs. [54–56]).
For the baryons N, Σ, and Ξ, we selected the corresponding standard definitions of the interpolating operator and correlator (see Refs. [54–56]).
After obtaining the correlation function, the mass can be obtained by fitting the exponential ansatz:
G(τ)=A+e−m+τ+A−e−m−(1/T−τ),
(5) where
m+ is the mass of the particle of interest, and τ takes values in the interval0≤τ<1/T on a lattice at finite temperature T.To obtain the ground state energy of the particle concerned, the best approach is to choose a large time extent lattice. However, at finite temperature, if the simulation is conducted on such a lattice, the lattice spacing must be small. Therefore, performing lattice simulations at finite temperature constitutes a dilemma presently. To obtain the ground state energy as accurately as possible on a relatively small time extent lattice, it is suitable for our procedure to incorporate an extrapolation method to obtain the ground state mass. We fit Eq. (5) to correlators in a series of time range
[τ1,τ2] , whereτ2 is fixed to the whole time extent andτ1 is swept across several values:τ1=1,2,3,... Thus, we obtained a series of mass values corresponding to the suppression of different early Euclidean time slices. Subsequently, we plotted the mass values obtained in different time intervals[τ1,τ2] against1/τ1 and fit a linear expression to those mass values. Finally, we extrapolated the linear expression toτ1→∞ . -
Hadron properties are encoded in spectral functions, which can provide important information on hadrons. Two approaches and their variants are usually adopted to reconstruct the spectral function. The first is the maximum entropy method and its variants [57–59]. The second is the Backus-Gilbert method and its variants [60–63] (reviews on the spectral function in lattice QCD can be found in Refs. [64, 65] and references therein). Recently, a new method based on the Backus-Gilbert method was presented in Ref. [50]. This method allows for choosing a smearing function at the beginning of the reconstruction procedure. To render this paper self-contained, we briefly present the method proposed in Ref. [50] in this section. In the following, the notations and symbols are almost the same as those used in Ref. [50].
The correlation function can be expressed as
G(τ)=∫∞0dEρL(E)b(τ,E),
(6) where
ρL(E) is the spectral function. We choose the basis function asb(τ,E)=e−τ+e−(1/T−τ),
(7) and
ρL(E⋆) can be approximated byˉρL(E⋆) , whereˉρL(E⋆) is evaluated byˉρL(E⋆)=τm∑τ=0gτ(E⋆)G(τ+1),
(8) once the coefficients
gτ(E⋆) are determined.The coefficients
gτ(E⋆) are determined by minimizing the linear combinationW[λ,g] of the deterministic functionalA[g] and error functionalB[g] W[λ,g]=(1−λ)A[g]+λB[g]G(0)2,
(9) under the unit area constraint
∫∞0dEˉΔσ(E,E⋆)=1,
(10) where
A[g] is defined asA[g]=∫∞E0dE|ˉΔσ(E,E⋆)−Δσ(E,E⋆)|2,
(11) where
ˉΔσ(E,E⋆) andΔσ(E,E⋆) are the smearing and target smearing functions, respectively. These two functions are expressed asˉΔσ(E,E⋆)=τm∑0gτ(λ,E⋆)b(τ+1,E),
(12) and
Δσ(E,E⋆)=e−(E−E⋆)22σ2∫∞0dEe−(E−E⋆)22σ2,
(13) respectively.
B[g] is expressed asB[g]=gTCovg,
(14) where
Cov is the covariance matrix of the correlation functionG(τ) . Further details are given in Ref. [50]. -
Before presenting the simulation results, we describe the computation details. The simulations were performed on
Nf=2+1 Generation2 (Gen2) FASTSUM ensembles [43]; the ensembles at the lowest temperatures were those provided by the HadSpec collaboration [66, 67]. The computation setup was the same as that used in Ref. [43]. We summarize the simulation details in Tables 1, 2, and 3 from Ref. [43].Gauge coupling (fixed-scale approach) β=1.5 tree-level coefficients c0=5/3,c1=−1/12 bare gauge, fermion anisotropy γg=4.3 ,γf=3.399 ratio of bare anisotropies ν=γg/γf=1.265 spatial tadpole (without, with smeared links) us=0.733566 ,˜us=0.92674 temporal tadpole (without, with smeared links) uτ=1 ,˜uτ=1 spatial, temporal clover coefficient cs=1.5893 ,cτ=0.90278 stout smearing for spatial links ρ=0.14 , isotropic, 2 stepsbare light quark mass for Gen2 ˆm0,light=−0.0840 bare strange quark mass ˆm0,strange=−0.0743 light quark hopping parameter for Gen2 κlight=0.2780 strange quark hopping parameter κstrange=0.2765 Table 1. Parameters in the lattice action. This table is recompiled from Ref. [43].
aτ /fm0.0350(2) a−1τ /GeV5.63(4) ξ=as/aτ 3.444(6) as /fm0.1205(8) Ns 24 mπ /MeV384(4) mπL 5.63 Table 2. Parameters such as lattice spacing and pion mass from Ref. [43] for Generation 2 ensemble.
Ns Nτ T/MeV T/Tc Ncfg aτmN aτmΣ aτmΞ aτmΛ aτmH 24 128 44 0.24 304 0.2133(24)(6) 0.2349(21)(5) 0.2459(18)(5) 0.2299(23)(6) 0.457(27)(5) 32 48 117 0.63 601 0.208(2)(3) 0.231(1)(2) 0.243(1)(2) 0.226(2)(3) 0.448(17)(4) 24 40 141 0.76 502 0.203(2)(5) 0.228(2)(4) 0.239(2)(4) 0.221(2)(4) 0.437(14)(9) 24 36 156 0.84 501 0.196(2)(6) 0.221(2)(5) 0.231(2)(5) 0.214(2)(5) 0.42(1)(2) 24 32 176 0.95 1000 0.181(2)(9) 0.204(2)(8) 0.215(2)(7) 0.199(2)(8) 0.393(7)(21) 24 28 201 1.09 1001 0.179(2)(12) 0.191(2)(12) 0.201(2)(11) 0.190(2)(11) 0.38(1)(2) 24 24 235 1.27 1001 0.172(3)(15) 0.179(3)(15) 0.191(3)(14) 0.182(3)(14) 0.36(1)(3) 24 20 281 1.52 1000 0.159(4)(18) 0.164(4)(18) 0.176(4)(17) 0.169(4)(17) 0.33(1)(4) 24 16 352 1.90 1000 0.154(6)(24) 0.158(6)(24) 0.171(6)(23) 0.164(6)(23) 0.31(2)(4) Table 3. Spatial and temporal extent, temperature in MeV, number of configurations, and masses of N, Σ, Ξ, Λ, and H-dibaryon. Masses of baryons and H-dibaryon are obtained by the extrapolation method. Estimates of statistical and systematic errors are contained in the first and second brackets, respectively. The errors of the fitting parameters of linear extrapolation are the systematic errors of the masses. The ensembles at the lowest temperatures are those provided by HadSpec [66, 67] (Gen2).
The ensembles were generated with a Symanzik-improved gauge action and a tadpole-improved clover fermion action with stout-smeared links. The details of the action are given in Ref. [43]. The parameters in the lattice action are recompiled in Table 1. The
Nf=2+1 Gen2 ensembles correspond to a physical strange quark mass and a bare light quark mass ofaτml=−0.0840 , yielding a pion mass ofmπ=384(4) MeV (see Table 2).The ensemble details are listed in Table 3, which is recompiled from Ref. [43] with a slight difference on the ensemble
N3s×Nτ=323×48 . The corresponding physical parameters, such as lattice spacing and pion mass, are listed in Table 2.The quark propagators were computed using the deflation-accelerated algorithm [68, 69]. When computing the propagator, the spatial links were stout smeared [70] with two steps of smearing, using the weight
ρ=0.14 . For the sources and sinks, we used Gaussian smearing [71],η′=C(1+κH)nη,
(15) where H is the spatial hopping part of the Dirac operator, and C is an appropriate normalization factor [49].
The correlators of the Λ and H-dibaryon are presented in Figs. 1 and 2, respectively. For the correlators of the Λ and H-dibaryon, we found a similar behavior to that displayed in Fig. 1 in Ref. [49] for N. For the correlator of Λ on large
Nτ and relatively smallNs lattice, especially the243×128 lattice, some correlator data points take negative values. These points are not depicted in the plot because the vertical axis is rescaled logarithmically. At some points, the error bar looks strange. It is because, at these points, the errors reach the magnitude of the correlator value, and the vertical axis is rescaled. For the plot of the H-dibaryon correlator, the same observation can be made.Figure 1. (color online) Euclidean correlator
G(τ)/G(0) of Λ as a function ofτT at different temperatures. At the lowest temperatureT/Tc=0.24 , the correlators at some points are not displayed because they take negative values.Figure 2. (color online) Euclidean correlator
G(τ)/G(0) of H-dibaryon as a function ofτT at different temperatures. At the lowest temperatureT/Tc=0.24 , the correlators at some points are not displayed because they take negative values.We used the extrapolation method to extract the ground state masses for N, Ξ, Σ, Λ, and H-dibaryon. We first fit Eq. (5) to the correlator by suppressing different early time slices to obtain a series of mass values. We present the results of nucleon and H-dibaryon on lattice
Nτ=128 in Fig. 3. After obtaining a series of mass values with different early time slices suppressed, we extrapolated the mass values linearly according to the scenario described in the last paragraph in Sec. II. We present the results of linear extrapolation for the nucleon and H-dibaryon on latticeNτ=128 in Fig. 4. In the extrapolation procedure, we used one portion of the data presented in Fig. 3.Figure 3. (color online) Mass values of nucleon and H-dibaryon obtained by fitting Eq. (5) to correlators on
Nτ=128 ensembles. The mass values are the fitting parameterm+ in Eq. (5) extracted by the fitting procedure in different intervals[τ1,τ2] . The horizontal axis labelτ1 represents different number of time slices suppressed corresponding to the lower bound of the interval[τ1,τ2] .Figure 4. (color online) Linear extrapolation of mass values for nucleon and H-dibaryon on
Nτ=128 ensembles. The horizontal axis represents inverse values of time slices suppressed.The results are listed in Table 3. Note that the masses decrease when the temperature increases. We compare our results of N and Λ below
Tc with those in Refs. [45, 49]. The results are consistent within errors.We also calculated the spectral density
ˉρL(E⋆) of the correlation function of N, Σ, Ξ, Λ, and H-dibaryon using a public computer program [72].We present the spectral density for
σ= 0.02, 0.04, 0.06, and 0.08 and N, Ξ, and H-dibaryon at three temperatures in Fig. 5. The upper panel in Fig. 5 for N atT/Tc=0.24 indicates that too large σ values may skip the peak structure of spectral density. Note from the upper panel in Fig. 5 that the spectral density distribution obtained forσ=0.08 has only one position whereˉρL(E⋆) takes a local maximum value. This position is approximately atE⋆=0.37 . AtT/Tc=0.24 , the time extentNτ=128 is large enough to extract the ground state energy.Figure 5. (color online) Spectral density computed with different values of the parameter σ for the target smearing function
Δσ(E,E⋆) for H-dibaryon, Ξ, and N at different temperatures.However, even if we do not suppress any early Euclidean time slices in the fitting procedure based on Eq. (5), we cannot obtain a mass value
aτmN larger than0.30 . The largest value ofaτmN that we obtained by suppressing different numbers of early time slices was approximately 0.25, which is smaller than 0.30, as can be clearly seen from Fig. 3. The mass value of 0.30 is somewhat an arbitrary value between the two peak positions of 0.17 and 0.38 for the spectral density presented in Table 4. Thus, we conclude that setting a large σ value may lead to missing some peak structures. By contrast, the spectral densityˉρL(E⋆) obtained forσ=0.02 in the upper panel of Fig. 5 has a peak position atE⋆≈0.05 with a small peak value. This peak structure may be due to lattice artifacts.Nτ E⋆ E⋆ E⋆ N 128 0.05 0.17 0.38 48 0.06 0.35 – 40 0.09 0.42 – 36 0.10 0.52 – 32 0.11 0.56 – 28 0.18 0.57 – 24 0.26 – – 20 0.24 – – 16 0.43 – – Table 4. For different
Nτ lattices, peak positionE⋆ of the spectral density for N.Nτ E⋆ E⋆ E⋆ Σ 128 0.05 0.17 0.37 48 0.06 0.35 – 40 0.08 0.41 – 36 0.09 0.49 – 32 0.11 0.56 – 28 0.17 0.57 – 24 0.20 – – 20 0.22 – – 16 0.39 – – Table 5. For different
Nτ lattices, peak positionE⋆ of the spectral density for Σ.Nτ E⋆ E⋆ Ξ 128 0.05 0.33 48 0.06 0.34 40 0.08 0.41 36 0.09 0.48 32 0.10 0.55 28 0.13 0.63 24 0.17 0.71 20 0.20 – 16 0.32 – Table 6. For different
Nτ lattices, peak positionE⋆ of the spectral density for Ξ.Nτ E⋆ E⋆ E⋆ Λ 128 0.05 0.18 0.38 48 0.06 0.35 – 40 0.08 0.42 – 36 0.09 0.49 – 32 0.11 0.56 – 28 0.14 0.61 – 24 0.18 0.64 – 20 0.22 – – 16 0.35 – – Table 7. For different
Nτ lattices, peak positionE⋆ of the spectral density for Λ.Nτ E⋆ E⋆ E⋆ E⋆ H-dibaryon 128 0.10 0.19 0.35 0.69 48 0.05 0.24 0.64 – 40 0.06 0.32 0.78 – 36 0.07 0.35 – – 32 0.08 0.45 – – 28 0.10 0.52 – – 24 0.11 – – – 20 0.14 – – – 16 0.17 – – – Table 8. For different
Nτ lattices, peak positionE⋆ of the spectral density for H-dibaryon.The middle panel in Fig. 5 for Ξ at
T/Tc=0.95 shows that smaller σ values can lead to a more pronounced peak structure of the spectral density in smallE⋆ regions. The lower panel for the H-dibaryon atT/Tc=1.90 suggests that setting different σ values has little effect on the computation of spectral density at high temperature. Therefore, we only present the spectral density results computed forσ=0.020 in the following.The spectral density
ˉρL(E⋆) of Ξ, Λ, and H-dibaryon is presented in Figs. 6, 7, and 8, respectively. The spectral densityˉρL(E⋆) of N and Σ has a similar behaviour to that of Λ. Note from Figs. 6, 7, and 8 that the spectral densityˉρL(E⋆) of Ξ and Λ has a similar behaviour, whileˉρL(E⋆) of the H-dibaryon is slightly different. All the peak positions ofˉρL(E⋆) are provided in Tables 4−8.Note from Figs. 6, 7, and 8 that, at the lowest temperature
T/Tc=0.24 , the spectral density for Ξ, Λ, and H-dibaryon has a rich peak structure. Despite the two peaks approximately located betweenE⋆=0.20 andE⋆=0.40 , the spectral densityˉρL(E⋆) of Ξ and Λ in the range ofE⋆ from0.20 to0.40 are approximately the same. The mass valuesaτmΞ=0.2459 andaτmΛ=0.2299 obtained by the extrapolation method lie in that range ofE⋆ .However,
aτmH=0.44 atT/Tc=0.24 for the H-dibaryon is in the neighborhood of the peak positionE⋆=0.35 , where theˉρL(E⋆) value is not very large. Note thataτmH=0.44 atT/Tc=0.24 is obtained by suppressing more early Euclidean time slices. The upper panel of Fig. 8 shows that more high frequency components of the spectral density should be suppressed in the extrapolation procedure.When temperature increases, the multi-peak structure of the spectral density distribution turns into a two-peak structure for Ξ and Λ, and at high temperatures, i.e.,
T/Tc= 1.27, 1.52, and 1.90, the spectral density distribution presents one peak.At intermediate temperatures, the spectral density
ˉρL(E⋆) exhibits a two-peak structure. If we take the smaller values ofE⋆ at peak positions as the ground state energies of the corresponding particle, then these mass values obtained by the peak position ofˉρL(E⋆) are smaller than those mass values obtained in Refs. [49] and [45]. The mass values ofaτmΞ ,aτmΛ , andaτmH presented in Table 3 are not consistent with the peak positions of the corresponding spectral density. Note that the mass values obtained by the extrapolation method are affected by the two-peak structure of the spectral density.At high temperatures, i.e.,
T/Tc= 1.27, 1.52, and 1.90, the spectral density distribution for N exhibits one peak structure, and the peak position shifts towards large values with increasing temperature. Note that the peak broadens and becomes smooth. It means that in the mass spectrum structure of the nucleon, there is no δ function structure contributing to the correlation function, as shown in Fig. 9 for N. A similar behaviour can be found for Σ, Ξ, and Λ. The smooth distribution of the spectral density implies that a one-particle state does not exist at high temperature.Figure 9. (color online) Spectral density distribution of N at different temperatures:
T/Tc= 0.84, 1.27, 1.52, and 1.90. AtT/Tc=0.84, Nτ=36 , the spectral density of N exhibits two peaks. AtT/Tc= 1.27, 1.52, and 1.90, the spectral density distribution becomes approximately smooth.This is not the case for the H-dibaryon. Its spectral density distribution at
T/Tc=0 , 1.27, 1.52, and 1.9 is presented in Fig. 10. Note that, atT/Tc= 1.27 and 1.52, the spectral density distribution still exhibits a one peak structure; atT/Tc=1.90 , the spectral density distribution broadens and becomes smooth. This observation may imply that, at temperaturesT/Tc= 1.27 and 1.52, the H-dibaryon still exhibits a one-particle state.Figure 10. (color online) Spectral density distribution of the H-dibaryon at different temperatures:
T/Tc=0.84,1.27,1.52, and1.90 . AtT/Tc=0.84,Nτ=36 , the spectral density of the H-dibaryon exhibits two peaks. AtT/Tc=1.27,1.52 , the spectral density distribution has one peak. AtT/Tc=1.90 , the spectral density distribution becomes approximately smooth.
Gauge coupling (fixed-scale approach) | |
tree-level coefficients | |
bare gauge, fermion anisotropy | |
ratio of bare anisotropies | |
spatial tadpole (without, with smeared links) | |
temporal tadpole (without, with smeared links) | |
spatial, temporal clover coefficient | |
stout smearing for spatial links | |
bare light quark mass for Gen2 | |
bare strange quark mass | |
light quark hopping parameter for Gen2 | |
strange quark hopping parameter |