Open Boundary Condition, Wilson Flow and the Scalar Glueball Mass

A major problem with periodic boundary condition on the gauge fields used in current lattice gauge theory simulations is the trapping of topological charge in a particular sector as the continuum limit is approached. To overcome this problem open boundary condition in the temporal direction has been proposed recently. One may ask whether open boundary condition can reproduce the observables calculated with periodic boundary condition. In this work we find that the extracted lowest glueball mass using open and periodic boundary conditions at the same lattice volume and lattice spacing agree for the range of lattice scales explored in the range 3 GeV $\leq $ 1/a $\leq$ 5 GeV. The problem of trapping is overcome to a large extent with open boundary and we are able to extract the glueball mass at even larger lattice scale $\approx$ 5.7 GeV. To smoothen the gauge fields and to reduce the cut off artifacts recently proposed Wilson flow is used. The extracted glueball mass shows remarkable insensitivity to the lattice spacings in the range explored in this work, 3 GeV $\leq$ 1/a $\leq$ 5.7 GeV.


Motivation
Even though lattice QCD continues to make remarkable progress in confronting experimental data, certain problems have persisted. For example, the spanning of the gauge configurations over different topological sectors become progressively difficult as the continuum limit is approached. This is partly intimately related to the use of periodic boundary condition on the gauge field in the temporal direction of the lattice. As a consequence, in the continuum limit the different topological sectors are disconnected from each other. Thus at smaller and smaller lattice spacings the generated gauge configurations trend to get trapped in a particular topological sector for a very long computer simulation time resulting in very long autocorrelations. This may sometime even invalidate the result of the simulation. Open boundary condition on the gauge field in the temporal direction has been recently proposed to overcome this problem [1][2][3]. Advantages of using open boundary conditions have also been studied in the context of SU (2) lattice gauge theory at weak coupling [4].
In the context of spanning of different topological sectors, an important quantity to study is the topological susceptibility (χ) in pure Yang-Mills lattice theory which is related to the η mass by the Witten-Veneziano formula [5][6][7]. For recent high precision calculations of χ with periodic boundary condition see, for example, Refs. [8][9][10]. In Ref. [11] we have addressed the question whether open boundary condition in the temporal direction can yield the expected value of χ. We have shown that with the open boundary it is possible to get the expected value of χ and the result agrees with our own numerical simulation employing periodic boundary condition.
In this work we continue our exploration of open boundary condition in the context of extraction of lowest glueball mass from the temporal decay of correlators. Extraction of glueball masses compared to hadron masses is much more difficult due to the presence of large vacuum fluctuations present in the correlators of gluonic observables. Moreover the computation of low lying glueball masses which are much higher than the masses of hadronic ground states, in principle requires relatively small lattice spacings. To overcome these problems, anisotropic lattices together with improved actions and operators have been employed [12][13][14] successfully to obtain accurate glueball masses. On the other hand, the calculation of glueball masses with isotropic lattice has a long history (see for example, the reviews, Refs. [15,16] a −1 = 3.73(6) GeV [17,18]. One would like to continue these calculations to even higher lattice scale which however eventually will face the problem of efficient spanning of the space of gauge configurations. Such trapping has been already demonstrated [11]. It is interesting to investigate whether the open boundary condition can reproduce the glueball masses extracted with periodic boundary condition at reasonably small lattice spacings achieved so far and whether the former can be extended to even smaller lattice spacings. Our main objective in this paper is to address these issues. An important ingredient in the extraction of masses is the smearing of gauge field which is necessary both to suppress unwanted fluctuations due to lattice artifacts and to increase the ground state overlap [19]. In the past various techniques have been proposed towards smearing the gauge fields [20][21][22]. Recently proposed Wilson flow [23][24][25] puts the technique of smearing on a solid mathematical footing. The same idea is referred to in the mathematical literature by the name of gradient flow [26][27][28]. Another motivation of the present work is the study of the effectiveness of Wilson flow in the extraction of masses.

Simulation details
We have generated gauge configurations with open boundary condition (denoted by O) in SU (3) lattice gauge theory at different lattice volumes and gauge couplings using the openQCD program [29]. Gauge configurations using periodic boundary conditions (denoted by P) also have been generated for several of the same lattice parameters (necessary changes to implement periodic boundary condition in temporal direction were made in the openQCD package for pure Yang-Mills case). Details of the simulation parameters are summarized in table 1.
To extract the scalar glueball mass, in this initial study we use the correlator of E which is the average of the action density over spatial volume at a particular time slice given in Ref. [2]. Since the action is a sum over the plaquettes, this is similar to the use of plaquette-plaquette correlators which have been used in the literature [30,31]. As in the latter case, there is room for operator improvement. One may use simple four link plaquette (unimproved) or one may use the clover definition of the field strength in the action (improved). Correlator is measured over N cnfg number of configurations with two successive ones separated by 32 and thus making the total length of simulation time to be N cnfg × 32. The lattice spacings quoted in table 1 are determined using the results from Refs. [32,33]. To smooth the gauge configurations, Wilson flow [23][24][25] is used and the reference flow time t 0 is determined through the implicit equation where t is the Wilson flow time and T is the temporal extent of the lattice. Through this equation, the reference flow time provides a reference scale to calculate the physical quantities from lattice data. An alternative to the t 0 scale is the w 0 scale proposed in Ref. [34]. However, we don't see any significant difference in our results using the two different scales. Since we extract the scalar glueball mass from the temporal decay of the correlator of E(x 0 ) where x 0 denotes the particular temporal slice, we first look at the effect of open boundary on the E(x 0 ) . In figure 1 we plot E(x 0 ) versus x 0 at flow time t = t 0 at β = 6.21 and lattice volume 24 3 × 48 for ensemble O 1 . Breaking of translational invariance due to open boundary condition in the temporal direction is clearly visible in the plot. To calculate the correlator we need to pick the sink and source points from the region free from boundary artifacts, which can be identified from such plot. To facilitate the identification better, we also plot E(x 0 ) for periodic boundary condition in the temporal direction for the same lattice volume and lattice spacing (ensemble P 1 ). Preservation of translation invariance is evident in this case. Clearly, for open boundary condition, source and sink points need to be chosen from the region where E(x 0 ) is almost flat. We note that for both open and periodic cases the central region E(x 0 ) is not perfectly flat but exhibits an oscillatory behaviour on a fine scale. Next we discuss the extraction of glueball mass. As already discussed in section 2, one may use the unimproved (naive plaquette) or improved (clover) version of the operator E(x 0 ). In general we expect improved operator to be preferable over unimproved one. However, for the extraction of masses Wilson flow is essential and this may diminish the difference between the results using them. In this work we have used Wilson flow in all the four directions as originally conceived. Due to the smearing in the temporal direction we should expect to get glueball mass for separation between source and sink which are larger than twice the smearing radius (≈ 2 × √ 8t). However a successful extraction of glueball mass in this case requires reasonably small statistical error at such large temporal separation. In figure 3 we plot glueball effective mass am e f f (0 ++ ) versus the temporal difference x 0 (x 0 = x source 0 − x sink 0 ) at Wilson flow time √ 8t = 0.28 fm, β = 6.42 and lattice volume 32 3 × 64 for ensemble P 2 for improved and unimproved choices of operators (from here onwards, we denote the temporal difference by x 0 ). As expected the plateau appears for relatively larger temporal separation and presumably thanks to Wilson flow the statistical error is reasonably small. We have verified that the results are very similar at all other Wilson flow times under consideration. Even though we find that there is no noticeable difference between them, we employed the improved operator for the rest of calculation in this paper.

Numerical Results
We extract the effective mass for the glueball (0 ++ ) state from the temporal decay of the correlator E(  figure 4 we plot the lowest glueball effective mass am e f f (0 ++ ) versus x 0 at four Wilson flow times t, β = 6.42 and lattice volume 32 3 × 64 for ensemble P 2 . We find that the effective mass is sensitive to Wilson time for initial temporal differences x 0 but becomes independent of different Wilson flow times in the plateau region within statistical error. Note that as expected, the plateau region moves to the right as Wilson flow time increases. Also shown in the figure is the fit to the plateau region of the data for √ 8t = 0.35 fm. The fit nevertheless passes through the plateau regions of data sets corresponding to other Wilson flow times.
For comparison with traditional methods to smoothen the gauge field configurations, in figure  4 we plot the lowest glueball effective mass am e f f (0 ++ ) versus x 0 at five smearing levels for four dimensional HYP smearing [21] at β = 6.42 and lattice volume 32 3 × 64 for ensemble P 2 . We find that the effective mass for different smear levels converge in a very narrow window where we can identify the plateau region and extract the mass. This behaviour is to be contrasted with that in the case of Wison flow discussed in the previous paragraph. Also shown in the figure is the fit to the plateau region of the data for smear level 18. In physical units the fitted mass is found to be 1409 (59) MeV which has a marginal overlap with the same [1510 (52) Mev] obtained with Wilson flow. We have observed from our studies with all the β values, the results obtained with HYP smearing are systematically lower than those obtained with Wilson flow. We note that the latter value is closer to the range of glueball mass quoted by other collaborations. The works presented in the rest   With open boundary condition the translational invariance in the temporal direction is broken and hence we can not average over all the source points to improve statistical accuracy as we have done in the case of periodic boundary condition. Nevertheless, we can average over few source points chosen far away from the boundary. In figure 6 we plot the lowest glueball effective mass am e f f (0 ++ ) versus x 0 at Wilson flow time ( √ 8t = 0.35 fm), β = 6.42 and lattice volume 32 3 × 64 for both open and periodic boundary conditions (ensembles O 2 and P 2 ). We find that effective mass agree for the two choices of the boundary conditions but as expected statistical error is larger for open boundary data.    In table 2 we have shown the fit range used to extract and the extracted lattice glueball mass for the ensembles studied in this paper. A linear fit is used to extract the mass.

Conclusions
In lattice Yang-Mills theory, we have shown that the open boundary condition on the gauge fields in the temporal direction of the lattice can reproduce the lowest scalar glueball mass extracted with periodic boundary condition at reasonably large lattice scales investigated in the range 3 GeV ≤ 1 a ≤ 5 GeV. With open boundary condition we are able to overcome, to a large extent, the problem of trapping and performed simulation and extract the glueball mass at even larger lattice scale ≈ 5.7 GeV. The use of recently proposed Wilson flow is used to smoothen the gauge fields and to reduce the cut off artifacts on the extracted glueball mass. The extracted glueball mass shows remarkable insensitivity to the lattice spacings in the range explored in this work 3 GeV ≤ 1 a ≤ 5.7 GeV. Conventionally, due to various theoretical reasons, in the calculation of masses from correlators, smearing of gauge field is carried out only in spatial directions. In this work, however, Wilson flow is carried out in all the four directions and our results show that one can indeed extract mass with relatively small statistical error at relatively large temporal separations. A critical evaluation of the strengths and weaknesses of the four-dimensional vesrus three-dimensional smoothening of the gauge field in the calculation of masses is beyond the scope of the present work.