- Email: [email protected]
Heavy flavor meson decay constants with Wilson fermions at β = 6.4
Nuclear Physics B (Proc. Suppl.) 26 (1992) 344-346 North-Holland
HEAVY FLAVOR MESON DECAY CONSTANTS WITH WILSON FERMIONS AT ,Q = 6.4 A .Abada l ,C.R.Allton2 ,Ph.Boucaud l , D.B.Carpenter3,M.Crisafulli4 ,J .Galand 1 ,S.GÜsken', G. artinelli4 ,O.Pènes ,C.T .Sachrajda2,R.Sarno4 , K .Schilling',R.Sommeró , i LPTHE, Orsay, France; 2 Dept. of Physics, The University, Southampton S09 5NH, UK ; 3 Dept . of Electronics and Computer Science, The University, Southampton S09 5NH, UK; a Dip . di Fisica, University di Roma `La Sapienza', I-00185 Roma,Italy and INFN, Sezione di Roma, Italy ; s Physics Department, University of Wuppertal, D-5600 Wuppertal 1, Fed .Rep . Germany ; 6 CERN Theory Division, CH-1211 Geneva 28, Switzerland .
presented by O .Pène . We present an extensive lattice study of the physical properties of mesons, composed of a heavy, H, and a light, q, quark, at ß = 6.4 on a 243 x 60 lattice, using the Wilson action in the quenched approximation . We have studied
the mass spectrum and the decay constants of vector and pseudoscalar mesons . We find significant violations of the mass scaling law fpm = const. ( ti 50% for D-mesons and -" 20% for B-mesons) . The results using quenched but propagating quarks are remarkably consistent with the static results when the scale is taken from the pion decay constant. Combining the results obtained by several cálaUat:ous of the pseudoscalar decay constants as a function of the meson mass, at different values of the lattice spacing, we obtain by extrapolation fBV as = (220 f 40) MeV (fB = (205 ±40) MeV and BB = 1 .16±0.07), where Fag is the renormalization group invariant B-parameter relevant
for Bd- Bd mixing. We also find fB. BB, /fBd BB, = 1 .19± 0 .10 . This last result is relevant in experimental studies of Ba-19a mixing . The vector-pseudoscalar mass splittings do not follow the predicted behaviour, My - Mp - const ., which is expected ( and found experimentally) in the limit of large heavy quark masses (i .e. when mQ > AQCD)-
We present results [1] obtained from a Monte Carlo study performed at ,Q = 6.4 using the standard Wilson :wtion for the gauge fields and the quark propagators [2], in the quenched approximation. We have generated 15 independent gauge field configurations on a 243 x 30 lattice, separated by 500-1600 sweeps, using the overrelaxed algorithm. On each configuration we have computed the quark propagators for 7 different values of the Wilson hopping parameter KW, corresponding to "heavy" quarks, KH = 0 .1275, 0 .1325, 0.1375, 0.1425, and "light" quarks, Kq = 0.1465, 0.1490 and 0.1495 . We have extracted Kct., the critical value of KW, corresponding to the point at which the pseudoscalar meson becomes the massless Gold0-5632íR405.
stone boson of QCD Kcr = 0.1506(2) For the value of the inverse lattice spacing a-1 we get from the p meson mass a -1 (GeV) = (3 .7 f 0 .2),
(2)
while we find from the pseudoscalar decay constant a-1 = (3.3 f 0 .6)GeV . For meson spectroscopy, we will use the scale derived from the p mass, eq .(2), (this includes the determination of the bare lattice quark masses, i .e. the KW corresponding to strange and charm
0 1992 - Elsevier Science Publishers B.V All rights reserved .
A. Abada et at. /Heavy flavor meson
decay constants with Wilsonfermions at ß = 64
quarks),which we call calibration "b" . For the determination of the pseudoscalar decay constants we prefer calibration "a" based on eq .(3), since this method reduces the systematic uncertainties. We refer the reader to [1] for a detailed account of our results concerning masses . Let us simply stress the following : we confirm a trend already noticed [3,4,6] that the vector-pseudoscalar mass differences are predicted smaller than experiment by the quenched lattices with Wilson action . The discrepancy increases with increasing meson mass, the prediction for the J/1@-% mass difference being around one third of the experimental value. We see no sign of improvement with increasing Q . Turning now to meson decay constants, we obtain with a-1 = 3.66 GeV (cf. eq . 2) . f = (145 ± 30)MeV
(4)
We also get fKlf, - 1 = 0.16±0.07
(5)
Our main goal was to study the heavy flavor meson decay constants. We have used a-1 = 3 .3 GeV, Is', = 0.1495 (from the kaon mass) and I{charm = 0 .1383 (from the D meson mass) . The ratio fp(KL = K, )l fp (KL = K,,.) is compatible with a constant [1] when the heavy quark mass mH varies, leading to fD, IfDd = fB o/fBd = 1 .06 ± 0.04
(6)
in agreement with static quark calculations [6,5] at ,Q = 6.0 that give respectively 1 .09 ± 0.04 and 1 .08 ± 0.04. The Heavy Quark Effective Theory predicts that, in the limit mH --" oo, the vector and pseudoscalar decay constants scale with the mass of the heavy quark, 711H, as : 1iÎ = fp = ~ a'IAO(M) fV
fpv N"P
1/MP
345
1/MP
' a ',
fpv N"P Ge V312 »6 "
Ge V-1 "b
0.20(3)
0.60
0.31(3)
0.44 0.57
Ge V312
Ge V-1
6.0
0.56(9)
0.0
ref.5
a "
0.86(10)
0.0
0.18(3)
0.76
0.28(3)
ref.6
0.55(6)
0.0
0.85(8)
0.0
ref.3
0.23(3)
0.73
0.35(4)
0.55
6.2
0.33(3)
0.38
0.28(2)
0.42
ref.4
0.32(3)
0.43
0.27(2)
0.48
0.30(3)
0.50
0.26(2)
0.56
0.27(3)
0.61
0.23(2)
0.68
0.23(3)
0.80
0.20(2)
0.90
ref.3
0.29(4)
0.61
0.37(4)
0.51
6.4
0.33(6)
0.39
0.38(3)
0.35
ref.1
0.31(5)
0.46
0.36(3)
0.42
0.28(5)
0.57
0.33(3)
0.51
0.24
0.76
0.28(3)
0.69 E
Table 1 fp VfWp and 1/Mp in physical units with the two different calibrations, "a and "b", of the lattice spacing. The static results, obtained at P = 6, have been multiplied by a factor (a,(MB)/a,(l/a))-6/33 .
where M=Mp=MV=MH(MpandMv being respectively the pseudoscalar and vector meson masses) . Let us define U(M) = fvfP1M
(8)
which should be one in the asymptotic limit, eq.(7) . We perform a linear fit in 1/M, with M = (Mp + My )/2, that gives U(M = oo) = 0 .96±0.04, U(Á1 = (MB+MB .)/2) = 0.90±0.04 and U(M = (MD + MD. )l2) = 0.80 ± 0 .04. To study the second part of eq .(7) we have reported in table 1 our results [1] as well as the results from refs .[3],[4],[5], where the decay constants fp were computed with moving heavy quarks, and, from refs .[5,6], where fp was computed with static quarks, i.e. at lowest order in Eichten's expansion [7]. Since lattice artefacts are expected to become important for large
A. Abada et aL /Heavy~nwson decay cmstants with Wilson fermions at ,6 = 64
masses, we have used only points corresponding to aMH < 0.7. Let us define (MP) =
(CIS
(MP)Iaa(MB))s/33, t(MP)
(9)
which is finite in the infinite mass limit. The values for 6(Mp) corresponding to table 1 have been plotted in fig. 1 in the case of calibration "a" (i.e. the scale taken from fA ) . Fig. 1 shows i) a satisfactory scaling when the lattice spacing is varied, i.e. an agreement between points computed for different values of ,c3; ii) the asymptotic beheaviour (7), clearly visible, with rather large 1/Mp corrections, since not only do the moving quark points (1/Mp > 0) show a linear beheaviour in 1/Mp, but they agree well with the static points (1/Mp = 0). (p) (Gev)
T . 1 . . . - . 1 - . .. . . .
-
8.6 8.s
c a~ aArcu ®t aL e eaeds et aL A dut= st aL X this werk
ryingly large X2 we estimate the errors on fB and fD by allowing a variation of - 5 on X2 in the case of calibration "b" # 1 , and we use the difference between the two calibrations to estimate the systematic errors . Due to the slope in 1/Mp we see large violations to the asymptotic scaling relation fp MP constant for charmed mesons (- 40 -100%) and sizeable ones (- 15 - 30%) for B-mesons. From our fits we finally obtain fB =
(205 f 40)MeV
(10)
where the systematic error has been included . For fD we only need the moving quarks since they cover the region of the physical charm. We quote: ƒD = (210 f 15)MeV
(11)
We have also studied the "B-parameters" , and we simply quote the most relevant results for phenomenology (Bil -Bil and B,-B, mixings) fBd
BBd =
(220 f 40)MeV ,
(12)
where the scale invariant B-paramreter has been used, and Rid = ƒB. BB .I .fB, BBd = 1 .19 10-10 .
(13)
References Figure 1 .
1/Mp (GeV)-1
These features are confirmed by the linear (quadratic) fit of dP(Mp) that we have performed in 1/Mp. It gives a X 2 per degree of freedom, 2 ,/d.n.f. = 0.9 (X 2 /d .o.f. = 0.7) . However the same exercise performed with calibration "b" (i .e. the scale taken from Mp) gives X 2 /d .o.f. _ i;.C (X 2 /d.o.f. = 2 .5) for a linear (quadratic) fit, although showing the same qualitative trend as the previous fit. To take into account these wor-
A .Abada et al .Roma Preprint n.823, CERN-TH. 6271/91, LPTHE-ORSAY 91/36, SHEP 91/92-3. [2] K.G .Wilson, in "New Phenomena in Sub-nuclear Physics", ed . A.Zichichi, Plenum, New York (1977) . [3] M.B .Gavela et al ., Phys.Lett. 206B (1988) 113. [4] C .R .Allton et al ., Nucl .Phys. B(Proc .Supp1 .20) (1991) 504. [5] C .Alexandrou et al ., Phys .Lett. 256B (1991) 60. [6] C .R .Allton et al ., Nucl .Phys. B349 (1991) 598. [7] E.Eichten, Nucl .Phys. B(Proc .Suppl .)4 (1988) 170. A variation of one on X2 li ave been used to estimate the errors in the case of calibration 'W' .
HEAVY FLAVOR MESON DECAY CONSTANTS WITH WILSON FERMIONS AT ,Q = 6.4 A .Abada l ,C.R.Allton2 ,Ph.Boucaud l , D.B.Carpenter3,M.Crisafulli4 ,J .Galand 1 ,S.GÜsken', G. artinelli4 ,O.Pènes ,C.T .Sachrajda2,R.Sarno4 , K .Schilling',R.Sommeró , i LPTHE, Orsay, France; 2 Dept. of Physics, The University, Southampton S09 5NH, UK ; 3 Dept . of Electronics and Computer Science, The University, Southampton S09 5NH, UK; a Dip . di Fisica, University di Roma `La Sapienza', I-00185 Roma,Italy and INFN, Sezione di Roma, Italy ; s Physics Department, University of Wuppertal, D-5600 Wuppertal 1, Fed .Rep . Germany ; 6 CERN Theory Division, CH-1211 Geneva 28, Switzerland .
presented by O .Pène . We present an extensive lattice study of the physical properties of mesons, composed of a heavy, H, and a light, q, quark, at ß = 6.4 on a 243 x 60 lattice, using the Wilson action in the quenched approximation . We have studied
the mass spectrum and the decay constants of vector and pseudoscalar mesons . We find significant violations of the mass scaling law fpm = const. ( ti 50% for D-mesons and -" 20% for B-mesons) . The results using quenched but propagating quarks are remarkably consistent with the static results when the scale is taken from the pion decay constant. Combining the results obtained by several cálaUat:ous of the pseudoscalar decay constants as a function of the meson mass, at different values of the lattice spacing, we obtain by extrapolation fBV as = (220 f 40) MeV (fB = (205 ±40) MeV and BB = 1 .16±0.07), where Fag is the renormalization group invariant B-parameter relevant
for Bd- Bd mixing. We also find fB. BB, /fBd BB, = 1 .19± 0 .10 . This last result is relevant in experimental studies of Ba-19a mixing . The vector-pseudoscalar mass splittings do not follow the predicted behaviour, My - Mp - const ., which is expected ( and found experimentally) in the limit of large heavy quark masses (i .e. when mQ > AQCD)-
We present results [1] obtained from a Monte Carlo study performed at ,Q = 6.4 using the standard Wilson :wtion for the gauge fields and the quark propagators [2], in the quenched approximation. We have generated 15 independent gauge field configurations on a 243 x 30 lattice, separated by 500-1600 sweeps, using the overrelaxed algorithm. On each configuration we have computed the quark propagators for 7 different values of the Wilson hopping parameter KW, corresponding to "heavy" quarks, KH = 0 .1275, 0 .1325, 0.1375, 0.1425, and "light" quarks, Kq = 0.1465, 0.1490 and 0.1495 . We have extracted Kct., the critical value of KW, corresponding to the point at which the pseudoscalar meson becomes the massless Gold0-5632íR405.
stone boson of QCD Kcr = 0.1506(2) For the value of the inverse lattice spacing a-1 we get from the p meson mass a -1 (GeV) = (3 .7 f 0 .2),
(2)
while we find from the pseudoscalar decay constant a-1 = (3.3 f 0 .6)GeV . For meson spectroscopy, we will use the scale derived from the p mass, eq .(2), (this includes the determination of the bare lattice quark masses, i .e. the KW corresponding to strange and charm
0 1992 - Elsevier Science Publishers B.V All rights reserved .
A. Abada et at. /Heavy flavor meson
decay constants with Wilsonfermions at ß = 64
quarks),which we call calibration "b" . For the determination of the pseudoscalar decay constants we prefer calibration "a" based on eq .(3), since this method reduces the systematic uncertainties. We refer the reader to [1] for a detailed account of our results concerning masses . Let us simply stress the following : we confirm a trend already noticed [3,4,6] that the vector-pseudoscalar mass differences are predicted smaller than experiment by the quenched lattices with Wilson action . The discrepancy increases with increasing meson mass, the prediction for the J/1@-% mass difference being around one third of the experimental value. We see no sign of improvement with increasing Q . Turning now to meson decay constants, we obtain with a-1 = 3.66 GeV (cf. eq . 2) . f = (145 ± 30)MeV
(4)
We also get fKlf, - 1 = 0.16±0.07
(5)
Our main goal was to study the heavy flavor meson decay constants. We have used a-1 = 3 .3 GeV, Is', = 0.1495 (from the kaon mass) and I{charm = 0 .1383 (from the D meson mass) . The ratio fp(KL = K, )l fp (KL = K,,.) is compatible with a constant [1] when the heavy quark mass mH varies, leading to fD, IfDd = fB o/fBd = 1 .06 ± 0.04
(6)
in agreement with static quark calculations [6,5] at ,Q = 6.0 that give respectively 1 .09 ± 0.04 and 1 .08 ± 0.04. The Heavy Quark Effective Theory predicts that, in the limit mH --" oo, the vector and pseudoscalar decay constants scale with the mass of the heavy quark, 711H, as : 1iÎ = fp = ~ a'IAO(M) fV
fpv N"P
1/MP
345
1/MP
' a ',
fpv N"P Ge V312 »6 "
Ge V-1 "b
0.20(3)
0.60
0.31(3)
0.44 0.57
Ge V312
Ge V-1
6.0
0.56(9)
0.0
ref.5
a "
0.86(10)
0.0
0.18(3)
0.76
0.28(3)
ref.6
0.55(6)
0.0
0.85(8)
0.0
ref.3
0.23(3)
0.73
0.35(4)
0.55
6.2
0.33(3)
0.38
0.28(2)
0.42
ref.4
0.32(3)
0.43
0.27(2)
0.48
0.30(3)
0.50
0.26(2)
0.56
0.27(3)
0.61
0.23(2)
0.68
0.23(3)
0.80
0.20(2)
0.90
ref.3
0.29(4)
0.61
0.37(4)
0.51
6.4
0.33(6)
0.39
0.38(3)
0.35
ref.1
0.31(5)
0.46
0.36(3)
0.42
0.28(5)
0.57
0.33(3)
0.51
0.24
0.76
0.28(3)
0.69 E
Table 1 fp VfWp and 1/Mp in physical units with the two different calibrations, "a and "b", of the lattice spacing. The static results, obtained at P = 6, have been multiplied by a factor (a,(MB)/a,(l/a))-6/33 .
where M=Mp=MV=MH(MpandMv being respectively the pseudoscalar and vector meson masses) . Let us define U(M) = fvfP1M
(8)
which should be one in the asymptotic limit, eq.(7) . We perform a linear fit in 1/M, with M = (Mp + My )/2, that gives U(M = oo) = 0 .96±0.04, U(Á1 = (MB+MB .)/2) = 0.90±0.04 and U(M = (MD + MD. )l2) = 0.80 ± 0 .04. To study the second part of eq .(7) we have reported in table 1 our results [1] as well as the results from refs .[3],[4],[5], where the decay constants fp were computed with moving heavy quarks, and, from refs .[5,6], where fp was computed with static quarks, i.e. at lowest order in Eichten's expansion [7]. Since lattice artefacts are expected to become important for large
A. Abada et aL /Heavy~nwson decay cmstants with Wilson fermions at ,6 = 64
masses, we have used only points corresponding to aMH < 0.7. Let us define (MP) =
(CIS
(MP)Iaa(MB))s/33, t(MP)
(9)
which is finite in the infinite mass limit. The values for 6(Mp) corresponding to table 1 have been plotted in fig. 1 in the case of calibration "a" (i.e. the scale taken from fA ) . Fig. 1 shows i) a satisfactory scaling when the lattice spacing is varied, i.e. an agreement between points computed for different values of ,c3; ii) the asymptotic beheaviour (7), clearly visible, with rather large 1/Mp corrections, since not only do the moving quark points (1/Mp > 0) show a linear beheaviour in 1/Mp, but they agree well with the static points (1/Mp = 0). (p) (Gev)
T . 1 . . . - . 1 - . .. . . .
-
8.6 8.s
c a~ aArcu ®t aL e eaeds et aL A dut= st aL X this werk
ryingly large X2 we estimate the errors on fB and fD by allowing a variation of - 5 on X2 in the case of calibration "b" # 1 , and we use the difference between the two calibrations to estimate the systematic errors . Due to the slope in 1/Mp we see large violations to the asymptotic scaling relation fp MP constant for charmed mesons (- 40 -100%) and sizeable ones (- 15 - 30%) for B-mesons. From our fits we finally obtain fB =
(205 f 40)MeV
(10)
where the systematic error has been included . For fD we only need the moving quarks since they cover the region of the physical charm. We quote: ƒD = (210 f 15)MeV
(11)
We have also studied the "B-parameters" , and we simply quote the most relevant results for phenomenology (Bil -Bil and B,-B, mixings) fBd
BBd =
(220 f 40)MeV ,
(12)
where the scale invariant B-paramreter has been used, and Rid = ƒB. BB .I .fB, BBd = 1 .19 10-10 .
(13)
References Figure 1 .
1/Mp (GeV)-1
These features are confirmed by the linear (quadratic) fit of dP(Mp) that we have performed in 1/Mp. It gives a X 2 per degree of freedom, 2 ,/d.n.f. = 0.9 (X 2 /d .o.f. = 0.7) . However the same exercise performed with calibration "b" (i .e. the scale taken from Mp) gives X 2 /d .o.f. _ i;.C (X 2 /d.o.f. = 2 .5) for a linear (quadratic) fit, although showing the same qualitative trend as the previous fit. To take into account these wor-
A .Abada et al .Roma Preprint n.823, CERN-TH. 6271/91, LPTHE-ORSAY 91/36, SHEP 91/92-3. [2] K.G .Wilson, in "New Phenomena in Sub-nuclear Physics", ed . A.Zichichi, Plenum, New York (1977) . [3] M.B .Gavela et al ., Phys.Lett. 206B (1988) 113. [4] C .R .Allton et al ., Nucl .Phys. B(Proc .Supp1 .20) (1991) 504. [5] C .Alexandrou et al ., Phys .Lett. 256B (1991) 60. [6] C .R .Allton et al ., Nucl .Phys. B349 (1991) 598. [7] E.Eichten, Nucl .Phys. B(Proc .Suppl .)4 (1988) 170. A variation of one on X2 li ave been used to estimate the errors in the case of calibration 'W' .