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Dependence of the decay constant of a heavy-light pseudoscalar meson on the heavy quark mas
Physics Letters B 276 (1992) 191-195 North-Holland
PHYSICS LETTERS B
Dependence of the decay constant of a heavy-light pseudoscalar meson on the heavy quark mass V. Eletsky 1TEP. SU- 117 259 Moscow, USSR
and E. Shuryak Physics Department, SUNY at Stony Brook. Stony Brook, NY 11794, USA Received 12 November 1991
We use the QCD sum rules for an evaluation of the masses of light-and-heavy mesons and their decay constants in the limit of a large heavy quark mass mo, finding both the leading terms and the O( 1/ma) corrections. The results are summarised and compared to those of other authors in fig. 2.
A general interest to mesons m a d e o f heavy and light quarks ( d e n o t e d Qq in what follows) is related to the fact that they are a kind o f " h y d r o g e n a t o m s " o f the hadronic world, with only one light quark participating in the non-perturbative d y n a m i c s o f the Q C D vacuum. As noted in ref. [ 1 ], a family o f such mesons with heavy quarks (c,b,t, etc. ) has very similar properties, like hydrogen atoms with p,d,t nuclei. Studies o f their asymptotic properties, as well as O( 1 / m o ) corrections, have therefore attracted much attention in the last few years (see e.g. ref. [2] ). The particular p a r a m e t e r s discussed below are the energy difference between the meson and the heavy quark masses Er = m p - m o and, especially, the decay constant fe o f the pseudoscalar mesons P o f this kind, defined in a standard way as
( 0 [Oy~, Y5q [meson ) = ife p~,.
( 1)
At large heavy quark mass m o it scales as follows:
f Zpmo=Co( l + G / m o
+... ) ,
Er = E o + E l / m 0 + ... ,
(2)
(3)
where Co, G , Eo, El are constants to be calculated below. Note, that the p a r a m e t e r Co has a simple physical meaning: it is p r o p o r t i o n a l to the light quark density at the center. The constant fe was evaluated previously in a n u m b e r o f potential models [ 3], by means o f Q C D sum rules [ 1,4-6,8,7,9-11 ] and the instanton liquid model [ 12 ], the QCD-string approach [ 13 ] and recently in a n u m b e r o f lattice works, both with a d y n a m i c a l heavy quark [ 14-16 ] and with a static one [ 17-19 ]. A m o n g the m o t i v a t i o n s for the present study was the observation that these two types o f lattice calculations do not match well, so the question about the magnitude o f O ( 1 / m o) corrections to this quantity was raised. The results o f a d y n a m i c a l a p p r o a c h forfD agree with practically all sum rule calculations in the case o f the D meson. Being extrapolated to the B meson, they result i n f s = 110-140 MeV, consistent (within the uncertain0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
] 91
Volume 276, number
PHYSICS
1,2
LETTERS
B
6 February
1992
ties) with the QCD sum rule results obtained in refs. [ 1,4,5,8], although some authors [ 6,7,9,11] suggest somewhat larger numbers, fB= 170-230 MeV. The static lattice approach yields very large numbers in the heavy quark limit, fstat= 300-400 MeV, and the question is whether it can be compatible with other results after taking into account 0 ( 1/m,) corrections. Considering the correlation function P( Q’) of two pseudoscalar currents made of light and heavy quarks at euclidean momentum Q2 > 0, one gets the following expression [ 41:
(4)
in which we have neglected the non-perturbative effects due to the gluonic condensate and the four-fermion operators, which are negligible [ 41. The last term corresponds to the mixed quark-gluon operator, (qGq) = rni (Qq) where rni =0.8 GeV ’ [ 201. We account for the anomalous dimension of operator (44) with L= log( mu/A) /log(p/A), where we take the normalization point ,ii= 0.5 GeV and ,4 = 0.2 GeV. Making a number of standard steps: (i) writing the physical spectral density in the delta-plus-theta function form
(5) with three parametersf,, mP, s,; (ii) subtracting the contribution of the non-resonance states from both sides; (iii) making the Bore1 transformation [ 2 1 ] ; and (iv) dividing by exp( - m;/A4’), one finally arrives at the following sum rules at finite quark mass [ 41:
”
(S--~)2exp(-slM2)[l+O(a,(m~))lds
_cqqjL4,9_
s
w
(I-m&/2M2)
Taking the limit mQ+Go and defining then non-relativistic variables E,, EC, ,D through mp-mQ =E,, so= ( mQ + EC)2,M2 = 2mQp,one obtains the following sum rule for the constant C, defined above:
~F,(E~I~)a;“d-~~q)L4’9+m6(44)/16~2 and, after taking the logarithmic the resonance EO:
derivative
>
,
(7)
with respect to ,u, one has the following sum rule for the position
of
(8) These sum rules contain only one free parameter, the continuum threshold E,. Here F, and F2 are two standard functions appearing from the dispersion integral related to the contribution of the quark loop and continuum:
(9)
F,(E,/~)=l-(1+E,/~+E~/2~2)exp(-E,/~), F,(EJ,u) 192
= 1 - (1 +E,/P+E,~/~P~+E?~P~)
exp( -EC/p)
(10)
Volume 276, number 1,2
PHYSICS LETTERS B
6 February 1992
a n d a~"d a n d a~ao are r a d i a t i v e c o r r e c t i o n factors ~"
a [ad= l + ot~(m~ )(2.8 +21og( mQ/ 4/t ) ) ,
(11)
a~d= l +c~( m~ )(2.8 + 21og( mo/6/t ) ) .
(12)
T h e s e e x p r e s s i o n s for Eo a n d Co c o i n c i d e with t h o s e d e r i v e d in ref. [ 1 ]; they h a v e a rather s i m p l e n o n - r e l a t i v istic m e a n i n g . In particular, in this l i m i t the Borel p a r a m e t e r / t = l / r , w h e r e r is the euclidean time or d i s t a n c e o f the m e s o n p r o p a g a t i o n , a n d these e x p r e s s i o n s can be d e r i v e d in a m u c h s i m p l e r way directly f r o m the correlator in the c o o r d i n a t e r e p r e s e n t a t i o n . Let us r e m i n d the r e a d e r o f the m e a n i n g o f these two relations: t h e y are w r i t t e n in such a way, that t h e i r R H S s are s o m e f u n c t i o n s o f the Borel p a r a m e t e r / t , while the L H S s are j u s t constants. T h i s is possible only if some particular values of the parameters are chosen, and o n l y inside s o m e " / t - w i n d o w " , w h e r e the expessions used are reliable. In particular, the w i n d o w is restricted by the f o l l o w i n g two c o n d i t i o n s : ( i ) T h e n o n - p e r t u r b a t i v e c o r r e c t i o n s to the c o r r e l a t o r should be e v a l u a t e d reliably. In practice this m e a n s that /t is large enough, so that the last i n c l u d e d t e r m o f the O P E is not too large. ( i i ) T h e c o n t r i b u t i o n o f the n o n - r e s o n a n c e ( c o n t i n u u m ) states should not be too large, so selection o f its p a r t i c u l a r p a r a m e t e r i z a t i o n is not really v e r y i m p o r t a n t . F o r definiteness, we do not plot the c u r v e s in the region w h e r e b o t h the c o n t r i b u t i o n o f the ( ( I G q ) t e r m a n d that o f the c o n t i n u u m e x c e e d h a l f o f the total result. T h e last c o n d i t i o n ( n o w for the the fitted p a r a m e t e r s ) is: (iii) T h e R H S s o f eqs. ( 7 ) , ( 8 ) d o not c h a n g e inside the w i n d o w by m o r e than, say, 10%. W i t h these c o n d i t i o n s i m p o s e d , we h a v e fig. l a at w h i c h a set o f lines for the R H S o f eq. ( 8 ) are shown, ~J The argument of the log is changed according to the position of the maximum of the corresponding dispersion integral. Let us also comment, that inside the logs we simply substitute mQ=rob,so, stricktly speaking, we do not really go to the limit mQ~,~, as far as the log of this quantity is concerned. We do not discuss the academic question of summation of these logs, which can affect our curves in fig. 2 very close to the point 1/me= O.
.8 .6 .4 .2
71'''1'''1'''1'''1'' ~--:
0
llllllll .2
.3
.2 I .4
ll~
~- -_:_---:_-. -
o .5
.2
illllll,llll~ .3
.4
.5
t
zI
I
IrllllllllllllJllllll
.15
C
.2 .2
lllllllr .3
I I I
.4
-3
0
I~ -4 .5
lllllll] .2
.3
I I I ~
.4
.2
.4 mp
.6
.8
1
.5
Fig. 1. Four quantities E0(a), C0(b), El(c), G ( d ) defined in (2), (3) evaluated from the corresponding sum rules (7), (8), ( 13 ), (14) as functions of the non-relativistic Borel parameter #. The units are in GeV, according to dimension. The solid, dashed and dash-doned lines correspond to Ec= 1.0, 1. l and 1.2 GeV, respectively. The lines end where the conditions described in the text are not satisfied, so one can see only the "working window".
Fig. 2. The combination fpm~/2 (in GeV 3 / 2 ) v e r s u s inverse meson mass (in GeV ~). The crosses are lattice data compiled in ref. [ 19 ], the round and square points correspond to ref. [ 1] and ref. [4], respectively. The solid, dashed and dash-dotted lines correspond to our results with Ec= 1.0, 1.1 and 1.2 GeV, respectively (as in fig. 1). The dotted line corresponds to the results of ref. [11]. 193
Volume 276, number 1,2
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6 February 1992
corresponding to different values of the continuum threshold, E c = 1.0, 1.1 and 1.2 GeV. It shows that for this region of Ec "stability" (independence on #) is indeed good enough. The corresponding values of Eo can then be read offoffig, la, and used below, e.g. in fig. lb for Co. Now we make the next step and evaluate the O( 1/mQ) corrections, expanding the relativistic sum rules (2) up to the next order. After a little algebra, one obtains the following sum rules for parameters El and C1 defined above: 36 El = -- ~Eo - Con2 exp(Eo/#)#4( 4 # - E o ) F z ( E c / # ) a ~ ad 1
2
-
-
3 2Con2exp[(Eo-E~)/#]E4[(Eo-Ec)a[aa-4#a~aa]-
1 -~oexp(Eo/#)(1-Eo/#)mz(glq)
,
(13)
and
C1 = - 4 E o +
Eo2 + 2E~ 2~
36 Con 2 exp(E°/#)#4Fz(Ec/#)a~ad
3 + 2~on2 exp[ (Eo - E ~ ) / # ] E c a , 4
rad
_
1 4#Co e x p ( E o / # ) m ~ ((tq) •
_
_
(14)
The RHSs of these equations are shown in figs. 1c, 1d for the set of parameters defined above. Finally, we rewrite the expansion equation (2) in terms of the meson mass, me, and compare our results presented as a set of lines with a set of data points corresponding to various lattice calculations as compiled in ref. [ 19 ] (see fig. 2 ). We conclude that our results are quite consistent with lattice ones obtained with dynamical heavy quarks, but are significantly lower than those done with static ones (the point at 1 / m p = 0). The origin of this deviation remains unclear to us. A few comments about the discrepancy on the value off/~ within the Q C D sum rule method should be made. One source of this discrepancy is due to different values of the b quark mass used (or Er in the non-relativistic approach) which enter in the exponent. For instance, changing mb from 4.8 GeV to 4.6 GeV increasesfB from 130 to 180 MeV. A consistent way to determine the input value of mb is to obtain it from the sum rule itself, as we did above, but one may of course use some additional information as well. If one uses the mass of the B meson, from our Eo the value of the b quark mass appears to be m b = 4 . 7 - 4 . 8 GeV, which is consistent with the sum rules for the Y" mesons. Another source of disagreement is due to different criteria in the fitting procedure. E.g., in refs. [ 9,11 ] the author is not just looking for a smooth behavior of the fitting parameters in a certain interval of the Borel parameter, but rather looks for a m i n i m u m and demands this m i n i m u m to be stable against the variation of the continuum threshold. This pushes the continuum threshold to very high values. In our opinion, this exclusive treatment of one of the three parameters (the position of the resonance pole, its residue and the threshold of the continuum) is not justified. The dotted curve in fig. 2 corresponds to the result obtained in ref. [ 11 ]; it seems to be in contradiction with both our calculation and the lattice data. In summary, we have performed a 1/mQ expansion of the relativistic sum rules for the decay constantfp, and have studied its dependence on the heavy quark mass. The results are summarised in fig. 2, where they are compared with the results in literature and also with lattice data. We are thankful to G. Martinelli and A. Soni for clarifying comments on the lattice results and to M.A. Shifman, who has encouraged us to perform this calculation.
194
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References [ 1 ] E.V. Shuryak, Nucl. Phys. B 198 (1982) 83. [2] H. Georgi, Phys. Lett. B 240 (1990) 447; B. Grinstein, Nucl. Phys. B 339 (1990) 253; N. Isgur and M.B. Wise, Phys. Lett. B 232 (1989) 113; B 237 (1990) 527. [3] M. Suzuki, Phys. Lett. B 162 (1985) 392; D. Silverman and H. Yao, Phys. Rev. D 38 (1988) 214; S. Capstick and S. Godfrey, Phys. Rev. D 41 (1990) 2856. [4] T.M. Aliev and V.L. Eletsky, Sov. J. Nucl. Phys. 38 (1983) 936. [5] A.R. Zhitnitski, I.R. Zhitnitski and V.L. Chernyak, Sov. J. Nucl. Phys. 38 (1983) 773. [6] V.S. Mathur and M.T. Yamawaki, Phys. Rev. D 29 (1984) 2057. [7] L.J. Reinders, Phys. Rev. D 38 (1988) 947. [8] C.A. Dominguez and N. Paver, Phys. Lett. B 197 (1987) 423. [9] S. Narison, Phys. Lett. B 198 (1987) 104; B 218 (1989) 238. [ 10] C.A. Dominguez and N. Paver, Phys. Lett. B 246 (1990) 493. [ 11 ] S. Narison, CERN preprint CERN-TH.6112/91. [ 12] E.V. Shuryak, Nucl. Phys. B 328 (1989) 85. [ 13 ] Yu.A. Simonov, Heidelberg preprint ( 1991 ). [ 14] M.B. Gavela et al., Nucl. Phys. B 305 (1988) 677. [ 15 ] T.A. De Grand and D.R. Loft, Phys. Rev. D 38 (1988) 954. [ 16] C. Bernard, T. Draper, G. Hockney and A. Soni, Phys. Rev. D 38 (1988) 3540. [ 17] C.R. Allton et al., Nucl. Phys. B 349 ( 1991 ) 598. [ 18 ] C. Alexandrou, S. Giisken, F. Jegerlehner, K. Schilling and R. Sommer, Phys. Lett. B 256 ( 1991 ) 60. [ 19] L. Maini, University of Rome "La Sapienza", preprint no. 798 ( 1991 ); G. Martinelli, University of Rome "La Sapienza" preprint no. 799 ( 1991 ). [20] V.M. Belyaev and B.L. loffe, Zh. Eksp. Teor. Fiz. 83 (1982) 876. [21 ] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 147 (1979) 385,448.
195
PHYSICS LETTERS B
Dependence of the decay constant of a heavy-light pseudoscalar meson on the heavy quark mass V. Eletsky 1TEP. SU- 117 259 Moscow, USSR
and E. Shuryak Physics Department, SUNY at Stony Brook. Stony Brook, NY 11794, USA Received 12 November 1991
We use the QCD sum rules for an evaluation of the masses of light-and-heavy mesons and their decay constants in the limit of a large heavy quark mass mo, finding both the leading terms and the O( 1/ma) corrections. The results are summarised and compared to those of other authors in fig. 2.
A general interest to mesons m a d e o f heavy and light quarks ( d e n o t e d Qq in what follows) is related to the fact that they are a kind o f " h y d r o g e n a t o m s " o f the hadronic world, with only one light quark participating in the non-perturbative d y n a m i c s o f the Q C D vacuum. As noted in ref. [ 1 ], a family o f such mesons with heavy quarks (c,b,t, etc. ) has very similar properties, like hydrogen atoms with p,d,t nuclei. Studies o f their asymptotic properties, as well as O( 1 / m o ) corrections, have therefore attracted much attention in the last few years (see e.g. ref. [2] ). The particular p a r a m e t e r s discussed below are the energy difference between the meson and the heavy quark masses Er = m p - m o and, especially, the decay constant fe o f the pseudoscalar mesons P o f this kind, defined in a standard way as
( 0 [Oy~, Y5q [meson ) = ife p~,.
( 1)
At large heavy quark mass m o it scales as follows:
f Zpmo=Co( l + G / m o
+... ) ,
Er = E o + E l / m 0 + ... ,
(2)
(3)
where Co, G , Eo, El are constants to be calculated below. Note, that the p a r a m e t e r Co has a simple physical meaning: it is p r o p o r t i o n a l to the light quark density at the center. The constant fe was evaluated previously in a n u m b e r o f potential models [ 3], by means o f Q C D sum rules [ 1,4-6,8,7,9-11 ] and the instanton liquid model [ 12 ], the QCD-string approach [ 13 ] and recently in a n u m b e r o f lattice works, both with a d y n a m i c a l heavy quark [ 14-16 ] and with a static one [ 17-19 ]. A m o n g the m o t i v a t i o n s for the present study was the observation that these two types o f lattice calculations do not match well, so the question about the magnitude o f O ( 1 / m o) corrections to this quantity was raised. The results o f a d y n a m i c a l a p p r o a c h forfD agree with practically all sum rule calculations in the case o f the D meson. Being extrapolated to the B meson, they result i n f s = 110-140 MeV, consistent (within the uncertain0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
] 91
Volume 276, number
PHYSICS
1,2
LETTERS
B
6 February
1992
ties) with the QCD sum rule results obtained in refs. [ 1,4,5,8], although some authors [ 6,7,9,11] suggest somewhat larger numbers, fB= 170-230 MeV. The static lattice approach yields very large numbers in the heavy quark limit, fstat= 300-400 MeV, and the question is whether it can be compatible with other results after taking into account 0 ( 1/m,) corrections. Considering the correlation function P( Q’) of two pseudoscalar currents made of light and heavy quarks at euclidean momentum Q2 > 0, one gets the following expression [ 41:
(4)
in which we have neglected the non-perturbative effects due to the gluonic condensate and the four-fermion operators, which are negligible [ 41. The last term corresponds to the mixed quark-gluon operator, (qGq) = rni (Qq) where rni =0.8 GeV ’ [ 201. We account for the anomalous dimension of operator (44) with L= log( mu/A) /log(p/A), where we take the normalization point ,ii= 0.5 GeV and ,4 = 0.2 GeV. Making a number of standard steps: (i) writing the physical spectral density in the delta-plus-theta function form
(5) with three parametersf,, mP, s,; (ii) subtracting the contribution of the non-resonance states from both sides; (iii) making the Bore1 transformation [ 2 1 ] ; and (iv) dividing by exp( - m;/A4’), one finally arrives at the following sum rules at finite quark mass [ 41:
”
(S--~)2exp(-slM2)[l+O(a,(m~))lds
_cqqjL4,9_
s
w
(I-m&/2M2)
Taking the limit mQ+Go and defining then non-relativistic variables E,, EC, ,D through mp-mQ =E,, so= ( mQ + EC)2,M2 = 2mQp,one obtains the following sum rule for the constant C, defined above:
~F,(E~I~)a;“d-~~q)L4’9+m6(44)/16~2 and, after taking the logarithmic the resonance EO:
derivative
>
,
(7)
with respect to ,u, one has the following sum rule for the position
of
(8) These sum rules contain only one free parameter, the continuum threshold E,. Here F, and F2 are two standard functions appearing from the dispersion integral related to the contribution of the quark loop and continuum:
(9)
F,(E,/~)=l-(1+E,/~+E~/2~2)exp(-E,/~), F,(EJ,u) 192
= 1 - (1 +E,/P+E,~/~P~+E?~P~)
exp( -EC/p)
(10)
Volume 276, number 1,2
PHYSICS LETTERS B
6 February 1992
a n d a~"d a n d a~ao are r a d i a t i v e c o r r e c t i o n factors ~"
a [ad= l + ot~(m~ )(2.8 +21og( mQ/ 4/t ) ) ,
(11)
a~d= l +c~( m~ )(2.8 + 21og( mo/6/t ) ) .
(12)
T h e s e e x p r e s s i o n s for Eo a n d Co c o i n c i d e with t h o s e d e r i v e d in ref. [ 1 ]; they h a v e a rather s i m p l e n o n - r e l a t i v istic m e a n i n g . In particular, in this l i m i t the Borel p a r a m e t e r / t = l / r , w h e r e r is the euclidean time or d i s t a n c e o f the m e s o n p r o p a g a t i o n , a n d these e x p r e s s i o n s can be d e r i v e d in a m u c h s i m p l e r way directly f r o m the correlator in the c o o r d i n a t e r e p r e s e n t a t i o n . Let us r e m i n d the r e a d e r o f the m e a n i n g o f these two relations: t h e y are w r i t t e n in such a way, that t h e i r R H S s are s o m e f u n c t i o n s o f the Borel p a r a m e t e r / t , while the L H S s are j u s t constants. T h i s is possible only if some particular values of the parameters are chosen, and o n l y inside s o m e " / t - w i n d o w " , w h e r e the expessions used are reliable. In particular, the w i n d o w is restricted by the f o l l o w i n g two c o n d i t i o n s : ( i ) T h e n o n - p e r t u r b a t i v e c o r r e c t i o n s to the c o r r e l a t o r should be e v a l u a t e d reliably. In practice this m e a n s that /t is large enough, so that the last i n c l u d e d t e r m o f the O P E is not too large. ( i i ) T h e c o n t r i b u t i o n o f the n o n - r e s o n a n c e ( c o n t i n u u m ) states should not be too large, so selection o f its p a r t i c u l a r p a r a m e t e r i z a t i o n is not really v e r y i m p o r t a n t . F o r definiteness, we do not plot the c u r v e s in the region w h e r e b o t h the c o n t r i b u t i o n o f the ( ( I G q ) t e r m a n d that o f the c o n t i n u u m e x c e e d h a l f o f the total result. T h e last c o n d i t i o n ( n o w for the the fitted p a r a m e t e r s ) is: (iii) T h e R H S s o f eqs. ( 7 ) , ( 8 ) d o not c h a n g e inside the w i n d o w by m o r e than, say, 10%. W i t h these c o n d i t i o n s i m p o s e d , we h a v e fig. l a at w h i c h a set o f lines for the R H S o f eq. ( 8 ) are shown, ~J The argument of the log is changed according to the position of the maximum of the corresponding dispersion integral. Let us also comment, that inside the logs we simply substitute mQ=rob,so, stricktly speaking, we do not really go to the limit mQ~,~, as far as the log of this quantity is concerned. We do not discuss the academic question of summation of these logs, which can affect our curves in fig. 2 very close to the point 1/me= O.
.8 .6 .4 .2
71'''1'''1'''1'''1'' ~--:
0
llllllll .2
.3
.2 I .4
ll~
~- -_:_---:_-. -
o .5
.2
illllll,llll~ .3
.4
.5
t
zI
I
IrllllllllllllJllllll
.15
C
.2 .2
lllllllr .3
I I I
.4
-3
0
I~ -4 .5
lllllll] .2
.3
I I I ~
.4
.2
.4 mp
.6
.8
1
.5
Fig. 1. Four quantities E0(a), C0(b), El(c), G ( d ) defined in (2), (3) evaluated from the corresponding sum rules (7), (8), ( 13 ), (14) as functions of the non-relativistic Borel parameter #. The units are in GeV, according to dimension. The solid, dashed and dash-doned lines correspond to Ec= 1.0, 1. l and 1.2 GeV, respectively. The lines end where the conditions described in the text are not satisfied, so one can see only the "working window".
Fig. 2. The combination fpm~/2 (in GeV 3 / 2 ) v e r s u s inverse meson mass (in GeV ~). The crosses are lattice data compiled in ref. [ 19 ], the round and square points correspond to ref. [ 1] and ref. [4], respectively. The solid, dashed and dash-dotted lines correspond to our results with Ec= 1.0, 1.1 and 1.2 GeV, respectively (as in fig. 1). The dotted line corresponds to the results of ref. [11]. 193
Volume 276, number 1,2
PHYSICS LETTERS B
6 February 1992
corresponding to different values of the continuum threshold, E c = 1.0, 1.1 and 1.2 GeV. It shows that for this region of Ec "stability" (independence on #) is indeed good enough. The corresponding values of Eo can then be read offoffig, la, and used below, e.g. in fig. lb for Co. Now we make the next step and evaluate the O( 1/mQ) corrections, expanding the relativistic sum rules (2) up to the next order. After a little algebra, one obtains the following sum rules for parameters El and C1 defined above: 36 El = -- ~Eo - Con2 exp(Eo/#)#4( 4 # - E o ) F z ( E c / # ) a ~ ad 1
2
-
-
3 2Con2exp[(Eo-E~)/#]E4[(Eo-Ec)a[aa-4#a~aa]-
1 -~oexp(Eo/#)(1-Eo/#)mz(glq)
,
(13)
and
C1 = - 4 E o +
Eo2 + 2E~ 2~
36 Con 2 exp(E°/#)#4Fz(Ec/#)a~ad
3 + 2~on2 exp[ (Eo - E ~ ) / # ] E c a , 4
rad
_
1 4#Co e x p ( E o / # ) m ~ ((tq) •
_
_
(14)
The RHSs of these equations are shown in figs. 1c, 1d for the set of parameters defined above. Finally, we rewrite the expansion equation (2) in terms of the meson mass, me, and compare our results presented as a set of lines with a set of data points corresponding to various lattice calculations as compiled in ref. [ 19 ] (see fig. 2 ). We conclude that our results are quite consistent with lattice ones obtained with dynamical heavy quarks, but are significantly lower than those done with static ones (the point at 1 / m p = 0). The origin of this deviation remains unclear to us. A few comments about the discrepancy on the value off/~ within the Q C D sum rule method should be made. One source of this discrepancy is due to different values of the b quark mass used (or Er in the non-relativistic approach) which enter in the exponent. For instance, changing mb from 4.8 GeV to 4.6 GeV increasesfB from 130 to 180 MeV. A consistent way to determine the input value of mb is to obtain it from the sum rule itself, as we did above, but one may of course use some additional information as well. If one uses the mass of the B meson, from our Eo the value of the b quark mass appears to be m b = 4 . 7 - 4 . 8 GeV, which is consistent with the sum rules for the Y" mesons. Another source of disagreement is due to different criteria in the fitting procedure. E.g., in refs. [ 9,11 ] the author is not just looking for a smooth behavior of the fitting parameters in a certain interval of the Borel parameter, but rather looks for a m i n i m u m and demands this m i n i m u m to be stable against the variation of the continuum threshold. This pushes the continuum threshold to very high values. In our opinion, this exclusive treatment of one of the three parameters (the position of the resonance pole, its residue and the threshold of the continuum) is not justified. The dotted curve in fig. 2 corresponds to the result obtained in ref. [ 11 ]; it seems to be in contradiction with both our calculation and the lattice data. In summary, we have performed a 1/mQ expansion of the relativistic sum rules for the decay constantfp, and have studied its dependence on the heavy quark mass. The results are summarised in fig. 2, where they are compared with the results in literature and also with lattice data. We are thankful to G. Martinelli and A. Soni for clarifying comments on the lattice results and to M.A. Shifman, who has encouraged us to perform this calculation.
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Volume 276, number 1,2
PHYSICS LETTERS B
6 February 1992
References [ 1 ] E.V. Shuryak, Nucl. Phys. B 198 (1982) 83. [2] H. Georgi, Phys. Lett. B 240 (1990) 447; B. Grinstein, Nucl. Phys. B 339 (1990) 253; N. Isgur and M.B. Wise, Phys. Lett. B 232 (1989) 113; B 237 (1990) 527. [3] M. Suzuki, Phys. Lett. B 162 (1985) 392; D. Silverman and H. Yao, Phys. Rev. D 38 (1988) 214; S. Capstick and S. Godfrey, Phys. Rev. D 41 (1990) 2856. [4] T.M. Aliev and V.L. Eletsky, Sov. J. Nucl. Phys. 38 (1983) 936. [5] A.R. Zhitnitski, I.R. Zhitnitski and V.L. Chernyak, Sov. J. Nucl. Phys. 38 (1983) 773. [6] V.S. Mathur and M.T. Yamawaki, Phys. Rev. D 29 (1984) 2057. [7] L.J. Reinders, Phys. Rev. D 38 (1988) 947. [8] C.A. Dominguez and N. Paver, Phys. Lett. B 197 (1987) 423. [9] S. Narison, Phys. Lett. B 198 (1987) 104; B 218 (1989) 238. [ 10] C.A. Dominguez and N. Paver, Phys. Lett. B 246 (1990) 493. [ 11 ] S. Narison, CERN preprint CERN-TH.6112/91. [ 12] E.V. Shuryak, Nucl. Phys. B 328 (1989) 85. [ 13 ] Yu.A. Simonov, Heidelberg preprint ( 1991 ). [ 14] M.B. Gavela et al., Nucl. Phys. B 305 (1988) 677. [ 15 ] T.A. De Grand and D.R. Loft, Phys. Rev. D 38 (1988) 954. [ 16] C. Bernard, T. Draper, G. Hockney and A. Soni, Phys. Rev. D 38 (1988) 3540. [ 17] C.R. Allton et al., Nucl. Phys. B 349 ( 1991 ) 598. [ 18 ] C. Alexandrou, S. Giisken, F. Jegerlehner, K. Schilling and R. Sommer, Phys. Lett. B 256 ( 1991 ) 60. [ 19] L. Maini, University of Rome "La Sapienza", preprint no. 798 ( 1991 ); G. Martinelli, University of Rome "La Sapienza" preprint no. 799 ( 1991 ). [20] V.M. Belyaev and B.L. loffe, Zh. Eksp. Teor. Fiz. 83 (1982) 876. [21 ] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 147 (1979) 385,448.
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