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Dynamical quark effects on glueballs and topology in lattice QCD
Nuclear Physics B (Proc. Suppl .) 26 (1992) 275-277 North-Holland
DYNAMICAL QUARK EFFECTS ON GLUEBALLS AND TOPOLOGY IN LATTICE QCD Y. Kuramashia, M . Fukugita6, H . Mino`, M. Okawad, and A. Ukawae a Department of Physics, University of Tokyo, Tokyo 118, Japan b Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606,Japan c Faculty of Engineering, Yamanashi University, Kofu .¢00, Japan d National Laboratory for High Energy Physics(KEK), Ibaraki 805,Japan e Institute of Physics, University of Tsukuba, Ibaraki 805, Japan
The 0++ and 2++ glueball masses are calculated for gauge configurations with two flavors of dynamical staggered quarks . The 0++ glueball mass exhibits a large decrease with the quark mass suggesting the possibility of a substantial mixing effect with the 4q state. The effect of dynamical quarks on the topological susceptibility is also discussed.
1 . INTRODUCTION In the last few years a number of full QCD simulations including dynamical quarks have been performed for the flavor non-singlet hadron mass spectrum.[1] These calculations, however, are yet to uncover a clear deviation of the spectrum from the quenched one. The situation could be different for glueballs; possible mixings of these states with flavor singlet qq states might lead to a glueball spectrum substantially different from that of the pure gauge theory . Another quantity for which dynamical quarks should manifest their influence is topological fluctuations of the gauge field . Configurations with a non-vanishing Pontryagin number are suppressed for small quark masses, and indeed the topological susceptibility X is expected to satisfy the relation[2] 1 m2a f2a +v(mq) . 2t t~ f
Motivated by these considerations we have measured the glueball mass and the topological *
PRESENTED BY Y. KURAMASHI
0920-5632/92/$05 .00 0 1992- Elsevier Science Publishers B.V
susceptibility [3] on the gauge configurations recently generated with two flavors of dynamical Kogut-Susskind staggerd quarks at Q = 5.7 on a 20' lattice.[4] The number of configurations is 140 (separated by 5 time units) for m, a = 0 01 and 100 for mq,a = 0.02 . We should remark that the analyses presented here is of preliminary nature, since a reliable extraction of glueball mass may require statistics much higher than we have at hand from our run of over a year . 2. GLUEBALL MASS SPECTRUM For the glueball mass measurement we employed the blocking method of Michael and Teper[5] and use the operator Re(Tr(Uxy -1- Uyt + U,zx)) for the 0++ state and Re(Tr(-Uyz +U~x)) for 2++ . Here U=j denotes the plaquette in the ij-plane constructed from the blocked links. Our best results to be quoted below are obtained at the third blocking, i. e., with U=j containing up to 8 x 8 loops on the original lattice. In fig. 1 we show the 0++ propagator at m4 = 0 .01 . The solid curve represents a fit to a single hyperbolic cosine over the range I = I - 4 . Our estimates fer the glueball masses obtained by a All rights reserved .
shows xthe 1quarks the conspicuous corresponds gauge value its while that = errors on 104 closely 0++ to may us smaller that fit with bin quenched 0by This amass the of aof sweeps result ßlff now single glueball of are Q theory renormalize lighter estimate itthe size the calculated qq right for the and suggests, = isvalue, 1examine summarized for This propagator hyperbolic 2++ mixing states small 0++ the 5high of of feature pure 0to propagator case obtained depicts full value Füuranuishi the 5the mass agives 0++ 2Interpolating mass which the momentum configurations for gauge unit QCD as by effect albeit The heat the the magnitude in even (m9a cosine state 2++ plaquette compared as the the feff 3in fig on in coupling for X2 will et decrease bath question the istheory is=within al jackknife contrast 0++ in fig a 2fit 0with large mqa we lower Mynaniical drive 204 4isthe quark for fluctuations algorithm 52athe mass otiçerve of in =to tat lattice for the substantial to the The itself absence =of The than a0full renormal5mass to aan the available 1both the method glueball for -shifted propasizable mixing aquark error open effecQCD 4Solid with is orig0++ that first the flat tfor dean ofeffects on glueóalls of 0 into 2is measured pion due m(0++)a the mqa that still to us mass due Glueball important of while We with In in value decrease esect an"' channels a) because solid the two to the energy make off at 0++ the at exceeds fig the and to remark topology dependence the possibility 0mqa sea for pions qq This 114-ma an the 30++ glueball value causes absence mass flavor a=0++ large we skates for quark with of increase the speculation 0as = slope 5as m(0++),/2 in tbr estimate that mass illustrate the two 0in atindicated at mqa lattice singlet valence the errors The at the these 0mass, the that of masses could mqa former dynamical the expected of =with Qeff Barring is the of two QCD minimum 5right-most 0++ the 0++ the decreased qq falls quark the = on by the also lead quark mixing lines the (see, towards in meson 0(circles) mixing latter state dashed what renormalization for the quark data 5the slightly decreases to mixing masses, to ethe mass, circle momentum figure but such effect uncertaincannot (branch pattern should from as therefore effect rearrange aflavors and lines glueball [5]) smaller an higher below istaken effect asince 02++ SupThe bethe the we esdebebe of of b)
Y
276 10',
2.0 C
.01)
0"
ma
10-2
9" 0.8
1e
0.4 v 0
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t
i
i .01 .
.
gator for circle pure 3 .5 A decrease behavior creases . errors . state Let more that for . expected ; sea tively mass mixing . We ization :matching wiwh coupling . mqa .01 .02 inal .7.
-1L 0.02
0
Fig. . line similar bars
0"
.
.
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.
.
.
. .
Fig. . (squares) mass quenched
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.7.
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.
.
. .
Y. Fj.rwnashi etal /Dynarnical quark effects on glueballs andtopology in lattice QCD
277
X (x10 -a )
0.01
i
0.02
0.03
0.04
Fig. 3. Speculative mixing pattern of the 0++ glueball and the flavor singlet q7 state.
pose nom, that crossing of the broken lines occurs slightly below 7nga = 0.02. The quark mass dependence of our data for the 0+-' state is qualitatively consistent with that of the lower solid line, especially if the fact is taken into account that mass estimates at m9a = 0.02 could be influenced toward a larger value due to the presence of the nearby upper solid line . In this interpretation the 0++ state we measured at mqa = 0.01 would be mostly made of qq, and might be identified with 0,*(075) rather than a glueball state. We emphasize that much more statistics is perhaps needed to conrrm the speculative mass pattern discussed here . 3. TOPOLOGICAL SUSCEPTIBILITY We have measured the topological charge applying the cooling method[6] to the operator Q
0.01
m qa
327r2 sites
and calculated the topological susceptibility defined by X =C Q2 > / V with V the space-time lattice volume . Our result for X ~~ shown in fig . 4 . The solid line represents the first term on the right hand-side of eq. (1) calculated with the aid of m = 5.36(20)m q and f,r = 0.0403(12) at 1T'tq -° 0 obtained in ref.[4á . The measured X is quite close to the line' suppcr ting the validity of
0 .02
mq a
Fig. 4. Comparison of the measured value (circles) of the topological susceptibility at P = 5 .7 with Nf = 2 and the prediction from eq. (1) .
the relation (1) to the leading order in mq . Alternatively one may fit the data for X with the function a - inl . This gives y = 1 .45(34) which is a little larger than y = 1 expected for small mq . 4. SUMMARY We have explored possible manifestations of the effect of dynamical quarks in glueball states and topology . Especially interesting, though requiring further studies, is the indication that the 0++ glueball state mixes largely with the flavor singlet 4q state . References D. Toussaint, these proceedings . E. Witten, Nucl. Phys. B156 (1979) 269 ; G. Veneziano, Nucl. Phys. B159 (1979) 213 . For a previous calculation, see, K . M. Bitar et at ., Phys . Rev. D44 (1991) 2090. [4] M . Fukugita, H. Mino, M . Okawa and A. Ukawa, preprint KEK-TH-302(1991) (to appear in Phys. Rev . U tt.); H. Mino, these proceedings . C. iichael and M. Teper, Nucl . Phys. B314 (1999) 347 . [6] See, e.g., a . Hoek, M. Teper and J. Waterhouse, Nucl. Phys. B238 (1967) 589.
DYNAMICAL QUARK EFFECTS ON GLUEBALLS AND TOPOLOGY IN LATTICE QCD Y. Kuramashia, M . Fukugita6, H . Mino`, M. Okawad, and A. Ukawae a Department of Physics, University of Tokyo, Tokyo 118, Japan b Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606,Japan c Faculty of Engineering, Yamanashi University, Kofu .¢00, Japan d National Laboratory for High Energy Physics(KEK), Ibaraki 805,Japan e Institute of Physics, University of Tsukuba, Ibaraki 805, Japan
The 0++ and 2++ glueball masses are calculated for gauge configurations with two flavors of dynamical staggered quarks . The 0++ glueball mass exhibits a large decrease with the quark mass suggesting the possibility of a substantial mixing effect with the 4q state. The effect of dynamical quarks on the topological susceptibility is also discussed.
1 . INTRODUCTION In the last few years a number of full QCD simulations including dynamical quarks have been performed for the flavor non-singlet hadron mass spectrum.[1] These calculations, however, are yet to uncover a clear deviation of the spectrum from the quenched one. The situation could be different for glueballs; possible mixings of these states with flavor singlet qq states might lead to a glueball spectrum substantially different from that of the pure gauge theory . Another quantity for which dynamical quarks should manifest their influence is topological fluctuations of the gauge field . Configurations with a non-vanishing Pontryagin number are suppressed for small quark masses, and indeed the topological susceptibility X is expected to satisfy the relation[2] 1 m2a f2a +v(mq) . 2t t~ f
Motivated by these considerations we have measured the glueball mass and the topological *
PRESENTED BY Y. KURAMASHI
0920-5632/92/$05 .00 0 1992- Elsevier Science Publishers B.V
susceptibility [3] on the gauge configurations recently generated with two flavors of dynamical Kogut-Susskind staggerd quarks at Q = 5.7 on a 20' lattice.[4] The number of configurations is 140 (separated by 5 time units) for m, a = 0 01 and 100 for mq,a = 0.02 . We should remark that the analyses presented here is of preliminary nature, since a reliable extraction of glueball mass may require statistics much higher than we have at hand from our run of over a year . 2. GLUEBALL MASS SPECTRUM For the glueball mass measurement we employed the blocking method of Michael and Teper[5] and use the operator Re(Tr(Uxy -1- Uyt + U,zx)) for the 0++ state and Re(Tr(-Uyz +U~x)) for 2++ . Here U=j denotes the plaquette in the ij-plane constructed from the blocked links. Our best results to be quoted below are obtained at the third blocking, i. e., with U=j containing up to 8 x 8 loops on the original lattice. In fig. 1 we show the 0++ propagator at m4 = 0 .01 . The solid curve represents a fit to a single hyperbolic cosine over the range I = I - 4 . Our estimates fer the glueball masses obtained by a All rights reserved .
shows xthe 1quarks the conspicuous corresponds gauge value its while that = errors on 104 closely 0++ to may us smaller that fit with bin quenched 0by This amass the of aof sweeps result ßlff now single glueball of are Q theory renormalize lighter estimate itthe size the calculated qq right for the and suggests, = isvalue, 1examine summarized for This propagator hyperbolic 2++ mixing states small 0++ the 5high of of feature pure 0to propagator case obtained depicts full value Füuranuishi the 5the mass agives 0++ 2Interpolating mass which the momentum configurations for gauge unit QCD as by effect albeit The heat the the magnitude in even (m9a cosine state 2++ plaquette compared as the the feff 3in fig on in coupling for X2 will et decrease bath question the istheory is=within al jackknife contrast 0++ in fig a 2fit 0with large mqa we lower Mynaniical drive 204 4isthe quark for fluctuations algorithm 52athe mass otiçerve of in =to tat lattice for the substantial to the The itself absence =of The than a0full renormal5mass to aan the available 1both the method glueball for -shifted propasizable mixing aquark error open effecQCD 4Solid with is orig0++ that first the flat tfor dean ofeffects on glueóalls of 0 into 2is measured pion due m(0++)a the mqa that still to us mass due Glueball important of while We with In in value decrease esect an"' channels a) because solid the two to the energy make off at 0++ the at exceeds fig the and to remark topology dependence the possibility 0mqa sea for pions qq This 114-ma an the 30++ glueball value causes absence mass flavor a=0++ large we skates for quark with of increase the speculation 0as = slope 5as m(0++),/2 in tbr estimate that mass illustrate the two 0in atindicated at mqa lattice singlet valence the errors The at the these 0mass, the that of masses could mqa former dynamical the expected of =with Qeff Barring is the of two QCD minimum 5right-most 0++ the 0++ the decreased qq falls quark the = on by the also lead quark mixing lines the (see, towards in meson 0(circles) mixing latter state dashed what renormalization for the quark data 5the slightly decreases to mixing masses, to ethe mass, circle momentum figure but such effect uncertaincannot (branch pattern should from as therefore effect rearrange aflavors and lines glueball [5]) smaller an higher below istaken effect asince 02++ SupThe bethe the we esdebebe of of b)
Y
276 10',
2.0 C
.01)
0"
ma
10-2
9" 0.8
1e
0.4 v 0
v
t
i
i .01 .
.
gator for circle pure 3 .5 A decrease behavior creases . errors . state Let more that for . expected ; sea tively mass mixing . We ization :matching wiwh coupling . mqa .01 .02 inal .7.
-1L 0.02
0
Fig. . line similar bars
0"
.
.
quenched find timate switched
.
.
.
. .
Fig. . (squares) mass quenched
quark (branch masses modest mq effect, steeper equal mixing into
.7.
;z_-
.82 .
the than ties suggests comes to .01 . cay the 27r/,L Let mixing havior .
.86
.7.
.g., .86
.02,
.01 . ;
.02
.
.
.
. .
Y. Fj.rwnashi etal /Dynarnical quark effects on glueballs andtopology in lattice QCD
277
X (x10 -a )
0.01
i
0.02
0.03
0.04
Fig. 3. Speculative mixing pattern of the 0++ glueball and the flavor singlet q7 state.
pose nom, that crossing of the broken lines occurs slightly below 7nga = 0.02. The quark mass dependence of our data for the 0+-' state is qualitatively consistent with that of the lower solid line, especially if the fact is taken into account that mass estimates at m9a = 0.02 could be influenced toward a larger value due to the presence of the nearby upper solid line . In this interpretation the 0++ state we measured at mqa = 0.01 would be mostly made of qq, and might be identified with 0,*(075) rather than a glueball state. We emphasize that much more statistics is perhaps needed to conrrm the speculative mass pattern discussed here . 3. TOPOLOGICAL SUSCEPTIBILITY We have measured the topological charge applying the cooling method[6] to the operator Q
0.01
m qa
327r2 sites
and calculated the topological susceptibility defined by X =C Q2 > / V with V the space-time lattice volume . Our result for X ~~ shown in fig . 4 . The solid line represents the first term on the right hand-side of eq. (1) calculated with the aid of m = 5.36(20)m q and f,r = 0.0403(12) at 1T'tq -° 0 obtained in ref.[4á . The measured X is quite close to the line' suppcr ting the validity of
0 .02
mq a
Fig. 4. Comparison of the measured value (circles) of the topological susceptibility at P = 5 .7 with Nf = 2 and the prediction from eq. (1) .
the relation (1) to the leading order in mq . Alternatively one may fit the data for X with the function a - inl . This gives y = 1 .45(34) which is a little larger than y = 1 expected for small mq . 4. SUMMARY We have explored possible manifestations of the effect of dynamical quarks in glueball states and topology . Especially interesting, though requiring further studies, is the indication that the 0++ glueball state mixes largely with the flavor singlet 4q state . References D. Toussaint, these proceedings . E. Witten, Nucl. Phys. B156 (1979) 269 ; G. Veneziano, Nucl. Phys. B159 (1979) 213 . For a previous calculation, see, K . M. Bitar et at ., Phys . Rev. D44 (1991) 2090. [4] M . Fukugita, H. Mino, M . Okawa and A. Ukawa, preprint KEK-TH-302(1991) (to appear in Phys. Rev . U tt.); H. Mino, these proceedings . C. iichael and M. Teper, Nucl . Phys. B314 (1999) 347 . [6] See, e.g., a . Hoek, M. Teper and J. Waterhouse, Nucl. Phys. B238 (1967) 589.