- Email: [email protected]
Properties of the FLIC overlap quark propagator
Nuclear Physics B (Proc. Suppl.) 141 (2005) 217–222 www.elsevierphysics.com
Properties of the FLIC overlap quark propagator W. Kamleha , P.O. Bowmanb, D.B. Leinwebera , A.G. Williamsa , J-B. Zhanga a
Special Research Centre for the Subatomic Structure of Matter (CSSM) and Department of Physics, University of Adelaide, SA 5005, Australia. b
Nuclear Theory Center, Indiana University, Bloomington IN 47408, USA.
FLIC overlap fermions are a type of smeared link action in which the FLIC (Fat Link Irrelevant Clover) fermion action is used as the overlap kernel instead of the standard Wilson fermion action. We study the (quenched) quark propagator in momentum space using the FLIC overlap fermion action, and compare it to that of the Wilson overlap. We find that the two actions have different discretization errors which decrease with the quark mass. We also examine the FLIC overlap quark propagator within the dynamical kernel approximation, where the FLIC Overlap action is used for the valence quarks and the FLIC fermion action is used for the quark sea. Our results indicate that the quark propagator is relatively insensitive to the fermion sea.
1. INTRODUCTION
fermionic Green’s function,
The quark propagator in momentum space is a fundamental quantity in QCD. Its infrared structure gives us insight into the dynamical generation of mass due to the spontaneous breaking of chiral symmetry within QCD. Its ultraviolet behaviour allows us to obtain the running quark mass [1–3] and OPE condensates [4]. Lattice QCD allows a direct probe of the nonperturbative quark propagator. The momentum space quark propagator has been studied previously using different gauge fixing and fermion actions [3,5–10], including studies of the (standard) Wilson overlap in Landau gauge [3,9,10]. The tree-level (Aµ (x) = 0) quark propagator is identified with the (Euclidean space) fermionic Greens function, (∂/ + m0 )∆f (x, y) = δ 4 (x − y),
(1)
where m0 is the bare quark mass. In momentum space this equation is solved straightforwardly, 1 ˜ f (p) = ∆ . ip / + m0
(2)
˜ f (p) to be the tree-level propDenote S (0) (p) ≡ ∆ agator in momentum space. Then in the presence of gauge field interactions, define Sbare (p) to be the Fourier transform of the (interacting) 0920-5632/$ – see front matter © 2004 Published by Elsevier B.V. doi:10.1016/j.nuclphysbps.2004.12.033
(x, y) = δ 4 (x − y). (D / + m0 )∆bare f
(3)
We define the mass function M (p) and the bare renormalisation function Z(p) such that the bare quark propagator has the form Sbare (p) =
Z(p) . ip / + M (p)
(4)
Then for the renormalisation point ζ, the renormalised quark propagator is given by Sζ (p) =
Zζ (p) = Z2 (ζ, a)Sbare (p), (5) ip / + M (p)
where Zζ (p) is the (ζ-dependent) renormalisation function, and Z2 (ζ, a) is the wave function renormalisation constant, which depends on ζ and the regulator parameter a. The a-dependence of Sbare is implicit. Z2 (ζ, a) is chosen such that Zζ (ζ) = 1.
(6)
As Sζ (p) is multiplicatively renormalisable, all of the ζ-dependence is contained within Zζ (p), that is, the mass function is ζ-independent. Overlap fermions [11–15] are a lattice fermion action which possess an exact lattice chiral symmetry which reduces to the continuum chiral
218
W. Kamleh et al. / Nuclear Physics B (Proc. Suppl.) 141 (2005) 217–222
Table 1 Parameters for the four different lattices. All use a tadpole-improved Luscher-Weisz gluon action. Lattice spacings are set by the string tension. Shown are the lattice volume, gauge coupling β, sea quark hopping parameter κsea , lattice spacing, sea quark mass (in MeV), the mean link and the physical volume. Volume 83 × 16 123 × 24 83 × 16 123 × 24
β 4.286 4.60 3.70 4.00
κsea 0.0000 0.0000 0.1340 0.1318
a (fm) 0.200 0.120 0.208 0.120
symmetry in the continuum limit. The overlap formalism is constructed through the use of a non-chiral lattice action, the overlap kernel, which represents a single fermion species of large negative mass. Typically, the standard Wilson fermion action is used as the kernel [16]. However, the use of the FLIC (Fat Link Irrelevant Clover) fermion action [17,18] has been shown to speedup the evaluation of the overlap by a factor of two [19]. The FLIC fermion action is a cloverimproved fermion action which makes use of APE smeared links in the irrelevant terms. In this work, we are interested in comparing the quark propagator in momentum space calculated using the Wilson-overlap and FLICoverlap actions in quenched QCD. The mass function and renormalisation function on the lattice are defined analogously to their continuum versions, for the details of their construction see Ref. [9,10,20]. Further, we study the effects of dynamical fermions on the structure of the FLIC overlap propagator. For computational feasability, we use FLIC overlap fermions for the valence quark action, and FLIC fermions (the overlap kernel) for the sea quarks [21]. We refer to this combination of valence and sea quark fermion actions as the dynamical kernel approximation. 2. SIMULATION DETAILS While FLIC overlap and Wilson overlap are both free of O(a) errors, the actions may in general differ at O(a2 ), and as such there may be some difference in their properties at finite a. We conduct a comparison between the two actions by calculating the FLIC overlap propagator on the same set of quenched lattices as in the Wil-
msea ∞ ∞ 120 162
u0 0.8721 0.8888 0.8625 0.8338
Phys. Vol. (fm4 ) 1.63 × 3.04 1.443 × 2.952 1.6643 × 3.312 1.443 × 2.976
son overlap studies [9,10]. Furthermore, the FLIC overlap propagator with a dynamical kernel is calculated on two lattices to perform an exploratory study of the dynamical sector. All the lattices have approximately the same volume, and the dynamical-fermion lattice spacings approximate those of the quenched spacings. The details of all five lattices are given in Table 1. Landau gauge is chosen for the gauge fixing. An improved gauge fixing scheme [22] is used, and a Conjugate Gradient Fourier Acceleration [23] algorithm is chosen to perform the gauge fixing. We use periodic boundary conditions in the spatial directions and anti-periodic in the time direction. At tree level, the FLIC overlap and Wilson overlap have identical behaviour (although the choice of mw = 1.4 for the FLIC overlap was slightly different from that chosen for the Wilson overlap). The tree level propagator is calculated directly by setting the links and the mean link to unity. The kinematical lattice momentum q is obtained numerically from the tree level propagator, although it could equally well have been obtained from the analytic form for q derived in Ref. [9]. Each lattice ensemble consists of 50 configurations. The lattice Green’s function is obtained by inverting the FLIC overlap Dirac operator on each configuration using a multi-mass CG inverter [24]. The Fourier transform is taken to move to momentum space, and the bare quark propagator is obtained from the ensemble average. The quark propagator is calculated for 15 different masses. The details of the FLIC overlap parameters used are presented in Table 2.
W. Kamleh et al. / Nuclear Physics B (Proc. Suppl.) 141 (2005) 217–222
219
Table 2 FLIC overlap parameters. For each gauge coupling β, displayed are the regulator parameter mw , the number of APE smearing sweeps used in the FLIC kernel at a smearing fraction of 0.7, the number of low eigenmodes projected out, and the 15 lattice masses µ. The final column shows the bare quark mass in MeV, which is approximately the same for each lattice. β mw nape nlow µ1 µ2 µ3 µ4 µ5 µ6 µ7 µ8 µ9 µ10 µ11 µ12 µ13 µ14 µ15
4.286 1.4 7 20 0.00622 0.01244 0.01866 0.02488 0.03110 0.03732 0.04354 0.04976 0.06220 0.07464 0.09330 0.12439 0.15549 0.18659 0.21769
4.60 1.4 4 15 0.00400 0.00800 0.01200 0.01600 0.02000 0.02400 0.02800 0.03200 0.04000 0.04800 0.06000 0.08000 0.10000 0.12000 0.14000
3. RESULTS We now turn to the results of our investigation. The mass function M (p) and the renormalisation function Z(p) are calculated on each of the lattices. A cylinder cut [9] is applied to all the data, to reduce the effects of rotational symmetry violation. While the quark masses obtained from the FLIC overlap and Wilson overlap calculations are similar, they are not matched. In particular, the lightest five FLIC overlap masses are significantly lighter than the corresponding Wilson overlap masses. In order to compare the two actions, for each fixed momenta value we perform a cubic fit of both M (p) and Z(p) as a function of m0 , excluding the lightest mass (for finite volume reasons). Having obtained the fit functions M (p, m0 ) and Z(p, m0 ) we then choose m0 = 0.0, 0.05, 0.1, 0.2, 0.4, 0.6 GeV and compare the functions at these values of the bare mass. In this way we can compare the two actions with
3.70 1.4 14 30 0.00622 0.01244 0.01866 0.02488 0.03110 0.03732 0.04354 0.04976 0.06220 0.07464 0.09330 0.12439 0.15549 0.18659 0.21769
4.00 1.4 6 30 0.00400 0.00800 0.01200 0.01600 0.02000 0.02400 0.02800 0.03200 0.04000 0.04800 0.06000 0.08000 0.10000 0.12000 0.14000
m0
18 36 54 72 90 108 126 144 181 217 271 362 452 543 633
matched input masses. Results are shown in Fig. 1, plotted against the kinematical momenta q. First we examine the mass function. We see that the actions agree well in the infrared, and are both flat in the ultraviolet. At the lighter quark masses the two actions agree across the whole momentum spectrum, but for the heavier quark masses they have different asymptotic values of the quark mass. The difference becomes more pronounced as the quark mass increases. What this shows is that with a matched input bare mass, the low energy properties of the two lattice actions are the same. That is, where the lattice approximation is good, the actions agree. There are well known difficulties associated with putting heavy quarks on the lattice, as the Compton wavelength of the quarks becomes small and the lattice approximation becomes poor. It is in this area that we are likely to see sensitivity to the discretisation errors, and this is shown by the
220
W. Kamleh et al. / Nuclear Physics B (Proc. Suppl.) 141 (2005) 217–222
Figure 1. The mass function M (p) (left) and the renormalisation function Z(p) (right), on three different lattices at a = 0.200 (top), a = 0.120 (bottom), plotted against the kinematical momentum q. The curves for the two actions results from a cubic fit to the data, and then interpolating to six common values of the bare mass, including the chiral limit. The different Z(p) have been offset vertically for clarity, the actual asymptotic values approximate unity. data. In the very infrared the two actions agree for all masses, but at heavy quark mass as the momenta increases they begin to differ, with the coarser lattice showing a more rapid divergence (as expected). Given the relatively strange behaviour of the Wilson overlap mass function on the coarsest lattice at the heaviest masses, we can also conclude that the FLIC overlap inherits the benefits of the improved ultraviolet behaviour of the FLIC kernel, due to the APE smearing of the irrelevant operators.
The comparison of the renormalisation function results between the two actions leads to similar conclusions. Two things should be noted however. The first is that the two actions seem to disagree in the ultraviolet regardless of the quark mass. This suggests that Z(p) is more sensitive to the cutoff than M (p). Secondly, we note that we are comparing Z(p) and not Zζ (p). There is a good reason for this, as by fixing the behaviour of Zζ (p) at high momenta and then comparing the two actions, we would have been led to be-
W. Kamleh et al. / Nuclear Physics B (Proc. Suppl.) 141 (2005) 217–222
221
Figure 2. The mass function M (p) (left) and the renormalisation function Z(p) (right), on two different lattices at a = 0.200 (top) and a = 0.120 (bottom), plotted against the kinematical momentum q. The curves for the two actions results from a cubic fit to the data, and then interpolating to six common values of the bare mass, including the chiral limit. The quenched and dynamical lattices are approximately matched. The different Z(p) have been offset vertically for clarity, the actual asymptotic values approximate unity. lieve (spuriously) that the actions agreed in the ultraviolet but not in the infrared. The results comparing the quenched and dynamical results on the two (approximately) matched lattices are displayed in Fig. 2. We used the same technique to perform this comparison, constructing the fit functions and choosing common input masses. The dynamical lattices have a sea quark mass of roughly the strange quark mass. We see that there is little difference be-
tween the quenched and dynamical results for both the mass function and the renormalisation function, although in the intermediate momentum regime the difference seems to be more significant (the lightest mass and heaviest masses on the 83 × 16 lattices shows some difference in the low momentum regime, but the finer lattice reveals these to be lattice spacing artifacts). Our findings are consistent with predictions from the Dyson-Schwinger equations [25].
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W. Kamleh et al. / Nuclear Physics B (Proc. Suppl.) 141 (2005) 217–222
4. CONCLUSION The quark propagator in momentum space is calculated using FLIC overlap fermions, and compared with previous studies of the Wilson overlap propagator. We find that in the infrared where the lattice approximation is good that the physical properties of the actions are the same, but in the ultraviolet the actions have different discretisation errors, but the differences disappear as one moves to smaller lattice spacing and lighter quark mass. A calculation of the FLIC overlap quark propagator in the dynamical kernel approximation finds limited unquenching effects. The results of these studies will be presented in more detail elsewhere [20,26]. ACKNOWLEDGEMENTS We thank the Australian Partnership for Advanced Computing (APAC) and the Australian National Computing Facility for Lattice Gauge Theory for generous grants of supercomputer time which have enabled this project. This work is supported by the Australian Research Council. REFERENCES 1. JLQCD, S. Aoki et al., Phys. Rev. Lett. 82 (1999) 4392, hep-lat/9901019. 2. D. Becirevic et al., Phys. Rev. D61 (2000) 114507, hep-lat/9909082. 3. P. O. Bowman, et al., Nucl. Phys. Proc. Suppl. 128 (2004) 23 [arXiv:heplat/0403002]. 4. E. R. Arriola, P. O. Bowman and W. Broniowski, arXiv:hep-ph/0408309. 5. J.I. Skullerud and A.G. Williams, Phys. Rev. D63 (2001) 054508, hep-lat/0007028. 6. J. Skullerud, D.B. Leinweber and A.G. Williams, Phys. Rev. D64 (2001) 074508, hep-lat/0102013. 7. P.O. Bowman, U.M. Heller and A.G. Williams, Nucl. Phys. Proc. Suppl. 106 (2002) 820, hep-lat/0110081. 8. T. Blum et al., Phys. Rev. D66 (2002) 014504, hep-lat/0102005. 9. CSSM Lattice, F.D.R. Bonnet et al., Phys. Rev. D65 (2002) 114503, hep-lat/0202003.
10. J.B. Zhang et al., Nucl. Phys. Proc. Suppl. 119 (2003) 831-833, hep-lat/0208037. 11. R. Narayanan and H. Neuberger, Phys. Lett. B302 (1993) 62, hep-lat/9212019. 12. R. Narayanan and H. Neuberger, Nucl. Phys. B412 (1994) 574, hep-lat/9307006. 13. R. Narayanan and H. Neuberger, Phys. Rev. Lett. 71 (1993) 3251, hep-lat/9308011. 14. R. Narayanan and H. Neuberger, Nucl. Phys. B443 (1995) 305, hep-th/9411108. 15. H. Neuberger, Phys. Lett. B417 (1998) 141, hep-lat/9707022. 16. R.G. Edwards, U.M. Heller and R. Narayanan, Nucl. Phys. B540 (1999) 457, hep-lat/9807017. 17. CSSM Lattice, J.M. Zanotti et al., Phys. Rev. D65 (2002) 074507, hep-lat/0110216. 18. W. Kamleh, (2002), hep-lat/0209154. 19. W. Kamleh et al., Phys. Rev. D66 (2002) 014501, hep-lat/0112041. 20. W. Kamleh et al., in preparation. 21. W. Kamleh, D.B. Leinweber and A.G. Williams, Phys. Rev. D70 (2004) 014502, hep-lat/0403019. 22. F.D.R. Bonnet et al., Austral. J. Phys. 52 (1999) 939, hep-lat/9905006. 23. A. Cucchieri and T. Mendes, Phys. Rev. D57 (1998) 3822, hep-lat/9711047. 24. A. Frommer et al., Int. J. Mod. Phys. C6 (1995) 627, hep-lat/9504020. 25. C.S. Fischer and R. Alkofer, Phys. Rev. D67 (2003) 094020, hep-ph/0301094. 26. W. Kamleh et al., in Preparation.
Properties of the FLIC overlap quark propagator W. Kamleha , P.O. Bowmanb, D.B. Leinwebera , A.G. Williamsa , J-B. Zhanga a
Special Research Centre for the Subatomic Structure of Matter (CSSM) and Department of Physics, University of Adelaide, SA 5005, Australia. b
Nuclear Theory Center, Indiana University, Bloomington IN 47408, USA.
FLIC overlap fermions are a type of smeared link action in which the FLIC (Fat Link Irrelevant Clover) fermion action is used as the overlap kernel instead of the standard Wilson fermion action. We study the (quenched) quark propagator in momentum space using the FLIC overlap fermion action, and compare it to that of the Wilson overlap. We find that the two actions have different discretization errors which decrease with the quark mass. We also examine the FLIC overlap quark propagator within the dynamical kernel approximation, where the FLIC Overlap action is used for the valence quarks and the FLIC fermion action is used for the quark sea. Our results indicate that the quark propagator is relatively insensitive to the fermion sea.
1. INTRODUCTION
fermionic Green’s function,
The quark propagator in momentum space is a fundamental quantity in QCD. Its infrared structure gives us insight into the dynamical generation of mass due to the spontaneous breaking of chiral symmetry within QCD. Its ultraviolet behaviour allows us to obtain the running quark mass [1–3] and OPE condensates [4]. Lattice QCD allows a direct probe of the nonperturbative quark propagator. The momentum space quark propagator has been studied previously using different gauge fixing and fermion actions [3,5–10], including studies of the (standard) Wilson overlap in Landau gauge [3,9,10]. The tree-level (Aµ (x) = 0) quark propagator is identified with the (Euclidean space) fermionic Greens function, (∂/ + m0 )∆f (x, y) = δ 4 (x − y),
(1)
where m0 is the bare quark mass. In momentum space this equation is solved straightforwardly, 1 ˜ f (p) = ∆ . ip / + m0
(2)
˜ f (p) to be the tree-level propDenote S (0) (p) ≡ ∆ agator in momentum space. Then in the presence of gauge field interactions, define Sbare (p) to be the Fourier transform of the (interacting) 0920-5632/$ – see front matter © 2004 Published by Elsevier B.V. doi:10.1016/j.nuclphysbps.2004.12.033
(x, y) = δ 4 (x − y). (D / + m0 )∆bare f
(3)
We define the mass function M (p) and the bare renormalisation function Z(p) such that the bare quark propagator has the form Sbare (p) =
Z(p) . ip / + M (p)
(4)
Then for the renormalisation point ζ, the renormalised quark propagator is given by Sζ (p) =
Zζ (p) = Z2 (ζ, a)Sbare (p), (5) ip / + M (p)
where Zζ (p) is the (ζ-dependent) renormalisation function, and Z2 (ζ, a) is the wave function renormalisation constant, which depends on ζ and the regulator parameter a. The a-dependence of Sbare is implicit. Z2 (ζ, a) is chosen such that Zζ (ζ) = 1.
(6)
As Sζ (p) is multiplicatively renormalisable, all of the ζ-dependence is contained within Zζ (p), that is, the mass function is ζ-independent. Overlap fermions [11–15] are a lattice fermion action which possess an exact lattice chiral symmetry which reduces to the continuum chiral
218
W. Kamleh et al. / Nuclear Physics B (Proc. Suppl.) 141 (2005) 217–222
Table 1 Parameters for the four different lattices. All use a tadpole-improved Luscher-Weisz gluon action. Lattice spacings are set by the string tension. Shown are the lattice volume, gauge coupling β, sea quark hopping parameter κsea , lattice spacing, sea quark mass (in MeV), the mean link and the physical volume. Volume 83 × 16 123 × 24 83 × 16 123 × 24
β 4.286 4.60 3.70 4.00
κsea 0.0000 0.0000 0.1340 0.1318
a (fm) 0.200 0.120 0.208 0.120
symmetry in the continuum limit. The overlap formalism is constructed through the use of a non-chiral lattice action, the overlap kernel, which represents a single fermion species of large negative mass. Typically, the standard Wilson fermion action is used as the kernel [16]. However, the use of the FLIC (Fat Link Irrelevant Clover) fermion action [17,18] has been shown to speedup the evaluation of the overlap by a factor of two [19]. The FLIC fermion action is a cloverimproved fermion action which makes use of APE smeared links in the irrelevant terms. In this work, we are interested in comparing the quark propagator in momentum space calculated using the Wilson-overlap and FLICoverlap actions in quenched QCD. The mass function and renormalisation function on the lattice are defined analogously to their continuum versions, for the details of their construction see Ref. [9,10,20]. Further, we study the effects of dynamical fermions on the structure of the FLIC overlap propagator. For computational feasability, we use FLIC overlap fermions for the valence quark action, and FLIC fermions (the overlap kernel) for the sea quarks [21]. We refer to this combination of valence and sea quark fermion actions as the dynamical kernel approximation. 2. SIMULATION DETAILS While FLIC overlap and Wilson overlap are both free of O(a) errors, the actions may in general differ at O(a2 ), and as such there may be some difference in their properties at finite a. We conduct a comparison between the two actions by calculating the FLIC overlap propagator on the same set of quenched lattices as in the Wil-
msea ∞ ∞ 120 162
u0 0.8721 0.8888 0.8625 0.8338
Phys. Vol. (fm4 ) 1.63 × 3.04 1.443 × 2.952 1.6643 × 3.312 1.443 × 2.976
son overlap studies [9,10]. Furthermore, the FLIC overlap propagator with a dynamical kernel is calculated on two lattices to perform an exploratory study of the dynamical sector. All the lattices have approximately the same volume, and the dynamical-fermion lattice spacings approximate those of the quenched spacings. The details of all five lattices are given in Table 1. Landau gauge is chosen for the gauge fixing. An improved gauge fixing scheme [22] is used, and a Conjugate Gradient Fourier Acceleration [23] algorithm is chosen to perform the gauge fixing. We use periodic boundary conditions in the spatial directions and anti-periodic in the time direction. At tree level, the FLIC overlap and Wilson overlap have identical behaviour (although the choice of mw = 1.4 for the FLIC overlap was slightly different from that chosen for the Wilson overlap). The tree level propagator is calculated directly by setting the links and the mean link to unity. The kinematical lattice momentum q is obtained numerically from the tree level propagator, although it could equally well have been obtained from the analytic form for q derived in Ref. [9]. Each lattice ensemble consists of 50 configurations. The lattice Green’s function is obtained by inverting the FLIC overlap Dirac operator on each configuration using a multi-mass CG inverter [24]. The Fourier transform is taken to move to momentum space, and the bare quark propagator is obtained from the ensemble average. The quark propagator is calculated for 15 different masses. The details of the FLIC overlap parameters used are presented in Table 2.
W. Kamleh et al. / Nuclear Physics B (Proc. Suppl.) 141 (2005) 217–222
219
Table 2 FLIC overlap parameters. For each gauge coupling β, displayed are the regulator parameter mw , the number of APE smearing sweeps used in the FLIC kernel at a smearing fraction of 0.7, the number of low eigenmodes projected out, and the 15 lattice masses µ. The final column shows the bare quark mass in MeV, which is approximately the same for each lattice. β mw nape nlow µ1 µ2 µ3 µ4 µ5 µ6 µ7 µ8 µ9 µ10 µ11 µ12 µ13 µ14 µ15
4.286 1.4 7 20 0.00622 0.01244 0.01866 0.02488 0.03110 0.03732 0.04354 0.04976 0.06220 0.07464 0.09330 0.12439 0.15549 0.18659 0.21769
4.60 1.4 4 15 0.00400 0.00800 0.01200 0.01600 0.02000 0.02400 0.02800 0.03200 0.04000 0.04800 0.06000 0.08000 0.10000 0.12000 0.14000
3. RESULTS We now turn to the results of our investigation. The mass function M (p) and the renormalisation function Z(p) are calculated on each of the lattices. A cylinder cut [9] is applied to all the data, to reduce the effects of rotational symmetry violation. While the quark masses obtained from the FLIC overlap and Wilson overlap calculations are similar, they are not matched. In particular, the lightest five FLIC overlap masses are significantly lighter than the corresponding Wilson overlap masses. In order to compare the two actions, for each fixed momenta value we perform a cubic fit of both M (p) and Z(p) as a function of m0 , excluding the lightest mass (for finite volume reasons). Having obtained the fit functions M (p, m0 ) and Z(p, m0 ) we then choose m0 = 0.0, 0.05, 0.1, 0.2, 0.4, 0.6 GeV and compare the functions at these values of the bare mass. In this way we can compare the two actions with
3.70 1.4 14 30 0.00622 0.01244 0.01866 0.02488 0.03110 0.03732 0.04354 0.04976 0.06220 0.07464 0.09330 0.12439 0.15549 0.18659 0.21769
4.00 1.4 6 30 0.00400 0.00800 0.01200 0.01600 0.02000 0.02400 0.02800 0.03200 0.04000 0.04800 0.06000 0.08000 0.10000 0.12000 0.14000
m0
18 36 54 72 90 108 126 144 181 217 271 362 452 543 633
matched input masses. Results are shown in Fig. 1, plotted against the kinematical momenta q. First we examine the mass function. We see that the actions agree well in the infrared, and are both flat in the ultraviolet. At the lighter quark masses the two actions agree across the whole momentum spectrum, but for the heavier quark masses they have different asymptotic values of the quark mass. The difference becomes more pronounced as the quark mass increases. What this shows is that with a matched input bare mass, the low energy properties of the two lattice actions are the same. That is, where the lattice approximation is good, the actions agree. There are well known difficulties associated with putting heavy quarks on the lattice, as the Compton wavelength of the quarks becomes small and the lattice approximation becomes poor. It is in this area that we are likely to see sensitivity to the discretisation errors, and this is shown by the
220
W. Kamleh et al. / Nuclear Physics B (Proc. Suppl.) 141 (2005) 217–222
Figure 1. The mass function M (p) (left) and the renormalisation function Z(p) (right), on three different lattices at a = 0.200 (top), a = 0.120 (bottom), plotted against the kinematical momentum q. The curves for the two actions results from a cubic fit to the data, and then interpolating to six common values of the bare mass, including the chiral limit. The different Z(p) have been offset vertically for clarity, the actual asymptotic values approximate unity. data. In the very infrared the two actions agree for all masses, but at heavy quark mass as the momenta increases they begin to differ, with the coarser lattice showing a more rapid divergence (as expected). Given the relatively strange behaviour of the Wilson overlap mass function on the coarsest lattice at the heaviest masses, we can also conclude that the FLIC overlap inherits the benefits of the improved ultraviolet behaviour of the FLIC kernel, due to the APE smearing of the irrelevant operators.
The comparison of the renormalisation function results between the two actions leads to similar conclusions. Two things should be noted however. The first is that the two actions seem to disagree in the ultraviolet regardless of the quark mass. This suggests that Z(p) is more sensitive to the cutoff than M (p). Secondly, we note that we are comparing Z(p) and not Zζ (p). There is a good reason for this, as by fixing the behaviour of Zζ (p) at high momenta and then comparing the two actions, we would have been led to be-
W. Kamleh et al. / Nuclear Physics B (Proc. Suppl.) 141 (2005) 217–222
221
Figure 2. The mass function M (p) (left) and the renormalisation function Z(p) (right), on two different lattices at a = 0.200 (top) and a = 0.120 (bottom), plotted against the kinematical momentum q. The curves for the two actions results from a cubic fit to the data, and then interpolating to six common values of the bare mass, including the chiral limit. The quenched and dynamical lattices are approximately matched. The different Z(p) have been offset vertically for clarity, the actual asymptotic values approximate unity. lieve (spuriously) that the actions agreed in the ultraviolet but not in the infrared. The results comparing the quenched and dynamical results on the two (approximately) matched lattices are displayed in Fig. 2. We used the same technique to perform this comparison, constructing the fit functions and choosing common input masses. The dynamical lattices have a sea quark mass of roughly the strange quark mass. We see that there is little difference be-
tween the quenched and dynamical results for both the mass function and the renormalisation function, although in the intermediate momentum regime the difference seems to be more significant (the lightest mass and heaviest masses on the 83 × 16 lattices shows some difference in the low momentum regime, but the finer lattice reveals these to be lattice spacing artifacts). Our findings are consistent with predictions from the Dyson-Schwinger equations [25].
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4. CONCLUSION The quark propagator in momentum space is calculated using FLIC overlap fermions, and compared with previous studies of the Wilson overlap propagator. We find that in the infrared where the lattice approximation is good that the physical properties of the actions are the same, but in the ultraviolet the actions have different discretisation errors, but the differences disappear as one moves to smaller lattice spacing and lighter quark mass. A calculation of the FLIC overlap quark propagator in the dynamical kernel approximation finds limited unquenching effects. The results of these studies will be presented in more detail elsewhere [20,26]. ACKNOWLEDGEMENTS We thank the Australian Partnership for Advanced Computing (APAC) and the Australian National Computing Facility for Lattice Gauge Theory for generous grants of supercomputer time which have enabled this project. This work is supported by the Australian Research Council. REFERENCES 1. JLQCD, S. Aoki et al., Phys. Rev. Lett. 82 (1999) 4392, hep-lat/9901019. 2. D. Becirevic et al., Phys. Rev. D61 (2000) 114507, hep-lat/9909082. 3. P. O. Bowman, et al., Nucl. Phys. Proc. Suppl. 128 (2004) 23 [arXiv:heplat/0403002]. 4. E. R. Arriola, P. O. Bowman and W. Broniowski, arXiv:hep-ph/0408309. 5. J.I. Skullerud and A.G. Williams, Phys. Rev. D63 (2001) 054508, hep-lat/0007028. 6. J. Skullerud, D.B. Leinweber and A.G. Williams, Phys. Rev. D64 (2001) 074508, hep-lat/0102013. 7. P.O. Bowman, U.M. Heller and A.G. Williams, Nucl. Phys. Proc. Suppl. 106 (2002) 820, hep-lat/0110081. 8. T. Blum et al., Phys. Rev. D66 (2002) 014504, hep-lat/0102005. 9. CSSM Lattice, F.D.R. Bonnet et al., Phys. Rev. D65 (2002) 114503, hep-lat/0202003.
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