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In our simulation, we compute the correlation functions of H-dibaryon and Λ, we also calculate the correlation function of N, Σ and Ξ. The generic form of correlation function is:
$ G(\vec{x},\tau) = < O(\vec{x},\tau)O^+(0)>, $
(1) For the H-dibaryon interpolating operator, we choose the local operator. The starting point is the following operator notation for the different six quark combination [30]:
$ \begin{aligned}[b] [abcdef] = \ & \epsilon_{ijk} \epsilon_{lmn} \Big( b^i C\gamma_5 P_+ c^j \Big) \\ & \times\Big( e^l C\gamma_5 P_+ f^m \Big) \Big( a^k C\gamma_5 P_+ d^n \Big) ({\vec{x}}, t)\,, \end{aligned} $
(2) where
$ a, b,\ldots,f $ denote generic quark flavors, and$ P_+=(1+\gamma_0)/2 $ projects the quark fields to positive parity. We choose the operator$ O_{\bf{H}} $ as the H-dibaryon interpolating operator which transforms under the singlet irreducible representation of flavor SU(3) [17, 51–53]:$ O_{\bf{H}} = \frac{1}{48}\Big( [sudsud] - [udusds] - [dudsus] \Big), $
(3) The H-dibaryon correlation function can be obtained based on the formulae in Ref. [53].
For the baryon Λ interpolating operator, we choose the standard definition which is given by (see, for example, Refs. [54–56]):
$ \begin{aligned}[b] O_{\Lambda}(x) =\;& \frac{1}{\sqrt{6}} \epsilon_{abc} \left\{ 2 \left( u^T_a(x)\ C \gamma_5\ d_b(x) \right) s_c(x) \right. \\ & +\ \left( u^T_a(x)\ C \gamma_5\ s_b(x) \right) d_c(x) \\ & \left. -\ \left( d^T_a(x)\ C \gamma_5\ s_b(x) \right) u_c(x) \right\}\ , \end{aligned} $
(4) For the baryon Λ correlation function, we choose the standard definition (see, for example, Refs. [54–56]).
For the baryon N, Σ and Ξ, we take the standard definition of interpolating operator, and the corresponding definition of correlator (also, see, Refs. [54–56]).
After we get the correlation function, the mass can be obtained by fitting the exponential ansatz:
$ G(\tau) = A_+ e^{-m_+\tau} + A_- e^{-m_-(1/T-\tau)}, $
(5) with
$ m_{+} $ being the mass of the particle of interest, and τ in the interval$ 0 \leq \tau<1/T $ on a lattice at finite temperature T.In order to get the ground state energy of the particle concerned, it is best to choose large time extent lattice. However, at finite temperature, if we make simulation on large time extent lattice, the lattice spacing must be chosen to be small. Therefore, it is a dilemma for us to make lattice simulation at finite temperature presently. In order to get the ground state energy as possible as we can on a relatively small time extent lattice, it is expedient for us to take an extrapolation method in our procedure to get the ground state mass. We fit equation (5) to correlators in a series of time range
$ [\tau_1,\tau_2] $ where$ \tau_2 $ is fixed to the whole time extent, and$ \tau_1 $ runs over several values from$ \tau_1=1, 2, 3, 4,... $ , then we can get a series of mass values which correspond to different early Euclidean time slices suppression. After that, we plot the mass values obtained in different time interval$ [\tau_1,\tau_2] $ against$ 1/\tau_1 $ , and fit a linear expression to those mass values, then, extrapolate the linear expression to$ \tau_1 \to \infty $ . -
Hadron properties are encoded in spectral functions which can provide us important information on hadrons. Two approaches and their variants are adopted to reconstruct spectral function. The first is the maximum entropy method and their variants [57–59]. The second is the Backus-Gilbert method and their variants [60–63](reviews on the spectral function in lattice QCD can be found in Ref. [64, 65] and references therein). Recently, based on the Backus-Gilbert method, a new 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 which was designed 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 written as:
$ G(\tau) = \int_0^\infty dE\rho_L(E)b(\tau,E), $
(6) with
$ \rho_L(E) $ being the spectral function. We choose the basis function as:$ b(\tau,E) = e^{-\tau} + e^{-(1/T-\tau)}, $
(7) We can approximate
$ \rho_L(E_\star) $ by$ \bar{\rho}_L(E_\star) $ , where$ \bar{\rho}_L(E_\star) $ can be evaluated by$ \bar{\rho}_L(E_\star) = \sum\limits_{\tau=0}^{\tau_{m}} g_\tau(E_\star) G(\tau+1), $
(8) after those coefficients
$ g_\tau(E_\star) $ are determined.The coefficients
$ g_\tau(E_\star) $ are determined by minimizing the linear combination$ W[\lambda,g] $ of the deterministic functional$ A[g] $ and error functional$ B[g] $ $ W[\lambda,g] = (1-\lambda) A[g] + \lambda \frac{B[g]}{G(0)^2}, $
(9) under the unit area constraint
$ \int_0^\infty dE \bar\Delta_\sigma (E,E_\star) =1, $
(10) where
$ A[g] $ is defined as:$ A[g] = \int_{E_0}^\infty dE | \bar\Delta_\sigma (E,E_\star) - \Delta_\sigma (E,E_\star)|^2, $
(11) with
$ \bar\Delta_\sigma (E,E_\star) $ and$ \Delta_\sigma (E,E_\star) $ being the smearing function and target smearing function, respectively. These two functions are given by$ \bar\Delta_\sigma (E,E_\star) = \sum\limits_0^{\tau_m} g_\tau (\lambda, E_\star) b(\tau +1,E), $
(12) and
$ \Delta_\sigma (E,E_\star) = \frac{e^{-\frac{(E-E_\star)^2}{2\sigma^2}}} {\int_0^\infty dE e^{-\frac{(E-E_\star)^2}{2\sigma^2}}} , $
(13) respectively.
$ B[g] $ is written as:$ B[g]= g^T {{\rm{Cov}}} g, $
(14) with
$ { {\rm{Cov}}} $ is the covariance matrix of the correlation function$ G(\tau) $ . More details are given in Ref. [50]. -
Before presenting the simulation results, we describe the computation details. The simulations are carried out on
$ N_f=2+1 $ Generation2 (Gen2) FASTSUM ensembles [43] of which the ensembles at the lowest temperature are provided by the HadSpec collaboration [66, 67], so the computation details are the same as those used in Ref. [43]. We recompile the simulation details in the following three Tables 1, 2, and 3 from Ref. [43].gauge coupling (fixed-scale approach) $ \beta = 1.5 $ tree-level coefficients $ c_0=5/3,\,c_1=-1/12 $ bare gauge, fermion anisotropy $ \gamma_g = 4.3 $ ,$ \gamma_f = 3.399 $ ratio of bare anisotropies $ \nu = \gamma_g / \gamma_f = 1.265 $ spatial tadpole (without, with smeared links) $ u_s = 0.733566 $ ,$ \tilde{u}_s = 0.92674 $ temporal tadpole (without, with smeared links) $ u_\tau = 1 $ ,$ \tilde u_\tau = 1 $ spatial, temporal clover coefficient $ c_s = 1.5893 $ ,$ c_\tau = 0.90278 $ stout smearing for spatial links $ \rho = 0.14 $ , isotropic, 2 stepsbare light quark mass for Gen2 $ \hat m_{0, {\rm{light}}} = -0.0840 $ bare strange quark mass $ \hat m_{0, {\rm{strange}}} = -0.0743 $ light quark hopping parameter for Gen2 $ \kappa_{{\rm{light}}} = 0.2780 $ strange quark hopping parameter $ \kappa_{{\rm{strange}}} = 0.2765 $ Table 1. Parameters in the lattice action. This table is recompiled from Ref. [43].
$ a_\tau $ [fm]0.0350(2) $ a_\tau^{-1} $ [GeV]5.63(4) $ \xi=a_s/a_\tau $ 3.444(6) $ a_s $ [fm]0.1205(8) $ N_s $ 24 $ m_\pi $ [MeV]384(4) $ m_\pi L $ 5.63 Table 2. Parameters such as lattice spacing, pion mass etc collected from Ref. [43] for Generation 2 ensemble.
$ N_s $ $ N_\tau $ $ T {{\rm{[MeV]}}} $ $ T/T_c $ $ N_{{\rm{cfg}}} $ $ a_\tau m_N $ $ a_\tau m_\Sigma $ $ a_\tau m_\Xi $ $ a_\tau m_{\Lambda} $ $ a_\tau m_H $ 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, mass of N, Σ, Ξ, Λ and H-dibaryon. Masses of baryons and H-dibaryon are obtained by extrapolation method. Estimates of statistical and systematic errors are contained in the first and second brackets, respectively. The errors of fitting parameters of linear extrapolation are taken to be the systemtatic errors of the masses. The ensembles at the lowest temperatures were provided by HadSpec [66, 67] (Gen2).
The ensembles are 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
$ N_f=2+1 $ Gen2 ensembles correspond to a physical strange quark mass and a bare light quark mass of$ a_\tau m_l=-0.0840 $ , yielding a pion mass of$ m_\pi=384(4) $ MeV (see Table 2).The ensemble detail is listed in Table 3 which is recompiled from Ref. [43] with a slight difference on ensemble
$ N_s^3\times N_\tau = 32^3\times 48 $ . The corresponding physical parameters such as the lattice spacing, and the pion mass etc are collected in Table 2.The quark propagators are computed by using the deflation-accelerated algorithm [68, 69]. When computing the propagator, The spatial links are stout smeared [71] with two steps of smearing, using the weight
$ \rho = 0.14 $ . For the sources and sinks, we use the Gaussian smearing [70]$ \eta' = C\left(1+\kappa H\right)^n\eta, $
(15) where H is the spatial hopping part of the Dirac operator and C an appropriate normalisation [49].
The correlators of Λ and H-dibaryon are presented in Fig. 1 and Fig. 2, respectively. For the correlators of Λ and H-dibaryon, we find similar behavior which was displayed in Fig. 1 in Ref. [49] for N. For the correlator of Λ on large
$ N_\tau $ and relatively small$ N_s $ lattice, especially$ 24^3\times 128 $ lattice, some correlator data points are negative, and these points are not displayed on 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 are the magnitude of the correlator value, and the vertical axis is rescaled. For the plot of H-dibaryon correlator, the same observation can be observed.Figure 1. (color online) Euclidean correlator
$ G(\tau)/G(0) $ of Λ as a function of$ \tau T $ at different temperatures. At the lowest temperature$ T/T_c =0.24 $ , the correlators at some points are not displayed due to minus values.Figure 2. (color online) Euclidean correlator
$ G(\tau)/G(0) $ of H-dibaryon as a function of$ \tau T $ at different temperatures. At the lowest temperature$ T/T_c =0.24 $ , the correlators at some points are not displayed due to minus values.We use the extrapolation method to extract the ground state masses for N, Ξ, Σ, Λ, and H-dibaryon. We first fit equation (5) to correlator by suppressing different early time slices to get a series of mass values. We present the results of nucleon and H-dibaryon on lattice
$ N_\tau=128 $ in Fig. 3. After we get a series of mass values with different early time slices suppressed, we extrapolate the mass values linearly with the scenario described in the last paragraph in Sec. II. We present the results of linear extrapolation for nucleon and H-dibaryon on lattice$ N_\tau=128 $ in Fig. 4. In the extrapolation procedure, we use one portion of the data presented in Fig. 3.Figure 3. (color online) Mass values of nucleon and H-dibaryon obtained by fitting equation (5) to correlators on
$ N_\tau=128 $ ensembles. The mass values are the fitting parameter$ m_+ $ in equation 5 extracted by fitting procedure in different interval$ [\tau_1, \tau_2] $ . Horizontal axis label$ \tau_1 $ represents different number of time slices suppressed which corresponds the lower bound of interval$ [\tau_1, \tau_2] $ .Figure 4. (color online) Linear extrapolation of mass values for nucleon and H-dibaryon on
$ N_\tau=128 $ ensembles. Horizontal axis represents inverse values of time slices suppressed.The results are listed in Table 3. We can find that the masses decrease when temperature increases. We compare our results of N and Λ below
$ T_c $ with those in Refs. [45, 49]. The results are consistent within errors.We also calculate the spectral density
$ \bar\rho_L(E_\star) $ of the correlation function of N, Σ, Ξ, Λ and H-dibaryon by using the public computer program [72].We present the spectral density with different
$ \sigma = 0.02, \ 0.04, \ 0.06,\ 0.08 $ for N, Ξ and H-dibaryon at three temperatures in Fig. 5. The upper panel in Fig. 5 for N at$ T/T_c = 0.24 $ indicates that too large σ value may skip peak structure of spectral density. From the upper panel in Fig. 5, we can find that the spectral density distribution obtained by using$ \sigma = 0.08 $ has just one position where$ \bar\rho_L(E_\star) $ takes locally maximum value. The position is about at$ E_\star =0.37 $ . At$ T/T_c =0.24 $ , the time extent$ N_\tau=128 $ is large enough to extract the ground state energy.Figure 5. (color online) Spectral density computed at different parameter σ for the target smearing function
$ \Delta_\sigma (E,E_\star) $ 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 with equation (5), we cannot get a mass value
$ a_\tau m_N $ which is larger than$ 0.30 $ . The largest value of$ a_\tau m_N $ we get by suppressing different number of early time slices is about 0.25 which is smaller than 0.30. It can be seen clearly 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 from Table 4. So we think taking large σ may lead to missing some peak structure. On the other hand, spectral density$ \bar\rho_L(E_\star) $ obtained by using$ \sigma = 0.02 $ in the upper panel of Fig. 5 has a peak position at$ E_\star \approx 0.05 $ with small peak value. This peak structure may be due to the lattice artefact.$ N_\tau $ $ E_\star $ $ E_\star $ $ E_\star $ 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. On different
$ N_\tau $ lattice, peak position$ E_\star $ of spectral density for N.$ N_\tau $ $ E_\star $ $ E_\star $ $ E_\star $ 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. On different
$ N_\tau $ lattice, peak position$ E_\star $ of spectral density for Σ.$ N_\tau $ $ E_\star $ $ E_\star $ 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. On different
$ N_\tau $ lattice, peak position$ E_\star $ of spectral density for Ξ.$ N_\tau $ $ E_\star $ $ E_\star $ $ E_\star $ 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. On different
$ N_\tau $ lattice, peak position$ E_\star $ of spectral density for Λ.$ N_\tau $ $ E_\star $ $ E_\star $ $ E_\star $ $ E_\star $ 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. On different
$ N_\tau $ lattice, peak position$ E_\star $ of spectral density for H-dibaryon.The middle panel in Fig. 5 for Ξ at
$ T/T_c = 0.95 $ shows that smaller σ value can make peak structure of spectral density in small$ E_\star $ region more pronounced. The lower panel for H-dibaryon at$ T/T_c = 1.90 $ suggests that different σ value has little effect on the computation of spectral density at high temperature. So, we just present the spectral density results computed with$ \sigma =0.020 $ in the following.The spectral density
$ \bar\rho_L(E_\star) $ of Ξ, Λ and H-dibaryon are given in Fig. 6, 7 and 8, respectively. The spectral density$ \bar\rho_L(E_\star) $ of N and Σ has similar behaviour to that of Λ. From Fig. 6, 7 and 8, we can find that the spectral density$ \bar\rho_L(E_\star) $ of Ξ and Λ has similar behaviour, while$ \bar\rho_L(E_\star) $ of H-dibaryon is slightly different. All the peak positions of$ \bar\rho_L(E_\star) $ are collected in Table 4-8.From Fig. 6, 7 and 8, we can find that at the lowest temperature
$ T/T_c = 0.24 $ , the spectral density for Ξ, Λ and H-dibaryon has rich peak structure. Despite there are two peaks approximately between$ E_\star=0.20 $ and$ E_\star=0.40 $ , the spectral density$ \bar\rho_L(E_\star) $ of Ξ and Λ in the range of$ E_\star $ from$ 0.20 $ to$ 0.40 $ are almost the same. The mass values$ a_\tau m_\Xi = 0.2459 $ ,$ a_\tau m_\Lambda = 0.2299 $ obtained by extrapolation method are in that range of$ E_\star $ .However,
$ a_\tau m_H = 0.44 $ at$ T/T_c = 0.24 $ for H-dibaryon is in the neighbour of peak position$ E_\star = 0.35 $ where the$ \bar\rho_L(E_\star) $ value is not very large. Obtaining$ a_\tau m_H = 0.44 $ at$ T/T_c = 0.24 $ is just by suppressing more early Euclidean time slices. From the upper panel of Fig. 8, more high frequency components of the spectral density should be suppressed in the extrapolation procedure.When temperature increases, the multi-peak structure of spectral density distribution turns into two-peak structure for Ξ and Λ until at high temperature
$ T/T_c = 1.27, 1.52, 1.90 $ , the spectral density distribution has one peak.At the intermediate temperatures, the spectral density
$ \bar\rho_L(E_\star) $ has a two-peak structure. If we take the smaller values of$ E_\star $ at peak positions as the ground state energies of corresponding particle, then these mass values obtained by the peak position of$ \bar\rho_L(E_\star) $ are smaller than those mass values obtained in Ref. [49] and Ref. [45]. Mass values of$ a_\tau m_\Xi $ ,$ a_\tau m_\Lambda $ and$ a_\tau m_H $ presented in Table 3 are not consistent with the peak positions of corresponding spectral density. This observation shows that the mass values obtained by extrapolation method are affected by the two-peak structure of spectral density.At high temperature
$ T/T_c = 1.27, 1.52, 1.90 $ , for the spectral density distribution for N, the spectral density exhibits one peak structure, and the peak position shifts towards large value with increasing temperature. In the meantime, we can find the peak broadens and becomes smooth. It means that in the mass spectrum structure of nucleon, there is no δ function structure contributing to the correlation function. We can find this observation from Fig. 9 for N. Similar behaviour can be found for Σ, Ξ and Λ. We think the smooth distribution of spectral density implies that there does not exist one-particle state at high temperature.Figure 9. (color online) Spectral density distribution of N at different temperature
$ T/T_c = 0.84, 1.27, 1.52,1.90 $ . At$ T/T_c = 0.84, N_\tau = 36 $ , the spectral density of N exhibits two peaks. At$ T/T_c = 1.27, 1.52,1.90 $ , the spectral density distribution almost becomes smooth.This is not the case for H-dibaryon. The spectral density distribution for H-dibaryon at
$ T/T_c = 1.27, 1.52, 1.90 $ is presented in Fig. 10 from which we can find that at$ T/T_c = 1.27, 1.52 $ , the spectral density distribution still exhibits one peak structure until at$ T/T_c = 1.90 $ , the spectral density distribution broadens and becomes smooth. This observation may imply that at temperature$ T/T_c = 1.27, 1.52 $ , H-dibaryon still remains as a one-particle state.Figure 10. (color online) Spectral density distribution of H-dibaryon at different temperature
$ T/T_c = 0.84, 1.27, 1.52,1.90 $ . At$ T/T_c = 0.84, N_\tau = 36 $ , the spectral density of H-dibaryon exhibits two peaks. At$ T/T_c = 1.27, 1.52 $ , the spectral density distribution has one peak. At$ T/T_c = 1.90 $ , the spectral density distribution almost becomes smooth.
Masses of the conjectured H-dibaryon at different temperatures
- Received Date: 2024-03-06
- Available Online: 2024-08-01
Abstract: We present a lattice QCD determination of masses of the conjectured H-dibaryon