- Email: [email protected]
Charm effects in the MS bottom quark mass from ϒ mesons
15 June 2000
Physics Letters B 483 Ž2000. 94–98
Charm effects in the MS bottom quark mass from F mesons A.H. Hoang a , A.V. Manohar b a
b
Theory DiÕision, CERN, CH-1211 GeneÕa 23, Switzerland Department of Physics, UniÕersity of California at San Diego, 9500 Gilman DriÕe, La Jolla, CA 92093-0319, USA Received 30 November 1999; received in revised form 21 March 2000; accepted 10 May 2000 Editor: R. Gatto
Abstract We study the shift in the F mass due to a non-zero charm quark mass. This shift affects the value of the MS b-quark mass extracted from the F system by about y20 MeV, due to an incomplete cancellation of terms that are non-analytic in the charm quark mass. The precise size of the shift depends on unknown higher order corrections, and might have a considerable uncertainty if they are large. q 2000 Elsevier Science B.V. All rights reserved.
The bottom quark mass is an important parameter for the theoretical description of B meson decays and b jet production cross sections in collider experiments. In continuum QCD, the most precise determinations of the bottom quark mass parameter have been obtained from data on the spectrum and the electronic partial widths of the F mesons. Recently, a number of MS bottom quark determinations have been carried out, which were based on F meson sum rules at next-to-next-to-leading order ŽNNLO. in the non-relativistic expansion, and which consistently eliminated all linear sensitivity to small momenta w1–3x. The latter is mandatory to reduce the systematic uncertainty in the bottom quark mass below the typical hadronization scale LQCD w4,5x. The analyses mentioned above, however, have treated all quarks other than the b quark as massless. This treatment is justified for light quarks that have masses much smaller than the inverse Bohr radius 1r² r : of the non-relativistic bottom–antibottom system i.e. for up, down and strange quarks, because in this case the theoretical expressions describing the bottom–antibottom dynamics and the conversion to the MS bottom mass definition can be expanded in the light quark masses. Like the contributions that are linearly sensitive to small momenta, the terms linear Žand non-analytic. in these light quark masses cancel out in the analysis. At NLO in the non-relativistic expansion this can be seen explicitly by considering the effects of a light virtual quark to the static energy of a bottom–antibottom quark pair with spatial distance r,1 Estat s 2 Mb q Vstat Ž r . ,
Ž 1.
where Mb is the bottom quark pole mass and Vstat the potential energy of the non-relativistic bottom–antibottom
E-mail address: [email protected] ŽA.H. Hoang.. The dominant terms that are linearly sensitive to low momenta in the Schrodinger equation that describes the bottom–antibottom ¨ dynamics are all contained in the static energy. w4,5x 1
0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 5 9 5 - 5
A.H. Hoang, A.V. Manoharr Physics Letters B 483 (2000) 94–98
95
quark system. At order a s2 the correction coming from the finite mass of a light quark q to the pole mass contribution reads w6x 4 a s2 mq d Mbq s Mb D , Ž 2. 2 3 p Mb
ž /
p2 DŽ r . s
ry
8
3 4
r2q
p2 8
r3y
`
y
ž
1 4
ln2 r y
13 24
p2 ln r q
151 q
24
288
/
r4
Ý Ž 2 F Ž n . ln r q F Ž n . . r 2 n , X
Ž 3.
ns3
3 Ž n y 1.
F Ž n. '
, Ž 4. 4 n Ž n y 2. Ž 2 n y 1. Ž 2 n y 3. where d Mbq is the shift in the b-quark pole mass keeping the b-quark MS mass fixed, m q is the mass of the light quark, and F X Ž n. ' EEn F Ž n.. In the limit m qrMb 0, Eq. Ž2. reduces to d Mbq s 16 a s2 m c . Ž 5. q q The shift d Mb is non-analytic in the quark masses, and should be regarded as being of the form d Mb s a s2 m2crMb2 Ž Mbr6 . , so that it is explicitly proportional to Mb , which breaks the b-quark chiral symmetry. The limiting value Eq. Ž5. can be easily computed using heavy quark effective theory. At the scale m s m b , one matches to a theory in which the b-quark is treated as a static field, with the residual mass term equal to zero, so that the propagator is irŽ k P Õ .. At the lower scale m s m c , one integrates out the charm quark. The matching condition at this scale induces a residual mass term for the b-quark, which is given by computing the graph in Fig. 1 The shift in the bottom–antibottom quark potential energy due to a light quark q is
™
(
q d Vstat Ž r . sy
R q q Ž m q , l. s
4 as 3 r
(
as
ž p/ ½ 6
4 m2q
ž
m2q
ž /
ž /
1q
m2
d l2
`
q
2 m2q
H4 m
2 q
l2
R q q Ž m q , l . exp Ž yl r .
5
,
/
Ž 6.
Ž 7. l2 l2 where R q q is the qq pair production cross section for the center-of-mass energy l in eq ey-annihilation normalized to the massless cross section. For m q < 1r² r : f Mb a s the static potential energy can be expanded in m q r: 4 as as a2 q d Vstat Ž r. Ž 8. ln Ž m r . q g q 56 4 y 13 a s2 m q q O s m2q r , 3 r 3p p and the contribution linear in m q cancels in the total static energy, Eq. Ž1. between Mb and V Ž r .. The dominant light quark correction is of order Ž a srp . 2 m 2qrMb and can be neglected for all practical purposes. ŽFor the charm case this would amount to a correction of several MeV..
™
1y
ln
ž
/
Fig. 1. Diagram contributing to the residual mass term of the b quark in HQET.
A.H. Hoang, A.V. Manoharr Physics Letters B 483 (2000) 94–98
96
On the other hand, for the case that a light quark mass is comparable in size to the inverse Bohr radius, i.e. q has to be taken into account. In this case the cancellation for the charm quark, the full m q-dependence of d Vstat of the linear charm mass term between the pole mass and potential contributions to the static energy, Eq. Ž2., is incomplete. In this letter we show that this feature can lead to sizeable effects in the MS bottom quark mass determination, compared to when the charm quark is treated as massless. There are two reasons why the incomplete cancellation of the linear charm mass contribution can lead to a sizeable effect. First, the linear charm mass term is not suppressed by a factor p 2 , as one might expect from a contribution coming from a loop integration Žsee Eq. Ž2... This is because the linear charm mass contribution represents a correction generated by a non-analytic linear dependence on infrared momenta. A similar effect is known in chiral perturbation theory where non-analytic mp3 corrections from loop diagrams are also not suppressed by powers of p . Second, the scale of the strong coupling of the linear charm mass term in Eq. Ž2. is of order the charm mass rather than the bottom quark mass. This can be understood from the fact that the charm mass serves as an infrared cutoff for the non-analytic linear dependence on infrared momenta just mentioned before. Thus, the linear charm mass term is generated at momenta of order m c . The effective field theory computation of Fig. 1 also indicates that one should use a s Ž m c .. These arguments are supported by an explicit calculation of the order a s3 BLM corrections to the linear charm mass term in Eq. Ž2.. For this calculation it is sufficient to consider the static limit, i.e. we only need to determine the linear charm mass corrections to the bottom quark self-energy due to the chromostatic Coulomb field. The order a s3 BLM linear charm mass contributions in the bottom quark pole mass can be obtained from the formula
d Mbc,stat s
as
d l2
`
H 6p 4 M
2 c
l
2
R q q Ž m c , l.
d3p
H Ž 2p .
2 4 p as 3
2
3 p ql
2
as
1q2
4p
m2
5
žž / /
b 0 ln
p
2
q
3
,
Ž 9.
where we assumed that m c is the charm pole mass, and b 0 s 11 y 23 n l for n l s 3 light quark flavors. The factor of two in front of the b 0 term arises because the charm quark loop can be inserted on both sides of the massless fermion loops. We have also included the chromostatic self-energy contribution at order a s2 . The term linear in m c contained in Eq. Ž9. reads
d Mbc,linear s
a s2 Ž m . 6
mc 1 q 2
as
ž p/ 4
m2
žž /
b 0 ln
m 2c
y 4ln2 q
14 3
/
,
Ž 10 .
pole which corresponds to the BLM scale m BLM s 0.388 m c . We emphasize again that expression Ž10. represents a non-analytic contribution Žfrom momenta p 2 Q m2c . that cannot be obtained from a naive Taylor expansion of the integrand in Eq. Ž9. with respect to m c . Obviously, the BLM calculation indicates a very low renormalization scale for the strong coupling governing the linear charm quark mass contribution. For a s Ž m c . s 0.36 and m c s 1.5 GeV the order a s2 and the a s3 terms each amount to 32 MeV. At this point we note that the linear charm mass approximation in Eq. Ž10. indeed dominates d Mbc. At high orders of perturbation theory it is true because at high orders d Mbc is dominated by momenta p 2 < m2c , leading to a suppression of the contributions that are not contained in the linear mass term. At low orders it can be checked explicitly: at order a s2 the linear charm mass term is within 15% of the result containing the full charm mass dependence displayed in Eq. Ž2. for Ž m crMb . 2 s 0.1. At order a s3 only a comparison to the full charm mass result in the large-b 0 approximation is possible Žsee e.g. w7x.. Using the same input parameters used to produce the numbers shown in Table 4 of Ref. w7x we find that the a s3 term is within 10% of the large-b 0 result containing the full charm mass dependence. To illustrate the size of the corrections caused by the incomplete cancellation of the linear charm quark terms let us examine the difference in the MS bottom quark mass m b Ž m b ., determined from the mass of the F Ž1S . meson, MF Ž1 S . s 9.460 GeV, for the two cases that the finite charm mass effects are either taken into account or neglected. For simplicity, we only consider an extraction of the MS bottom quark mass at NLO in the non-relativistic expansion. A more thorough analysis using full NNLO expressions and including also a sum rule analysis based on data for all known F mesons will be carried out elsewhere w8x.
A.H. Hoang, A.V. Manoharr Physics Letters B 483 (2000) 94–98
97
Including the effects of the charm quark mass from Eq. Ž7. properly in first order time-independent perturbation theory, the F Ž1S . meson mass at NLO in the non-relativistic expansion reads w9,10x Ž a s ' a sŽ n fs4 . Ž m ..
½
MF Ž1 S . s 2 Mb 1 y
CF2 a 2s
y
CF2 a 2s
8
8
as
žp/
a1
b0 Ž L q 1 . q
2
2 q 3
mc
žž / ž ln
qh
m
2 mc Mb CF a s
// 5
,
Ž 11 . where w10x
°Ž2 y x y 4 x . 2
4
2 'x 2 y 1 h Ž x . ' y 116 y 2 x 2 q
3 xp 4
~ Ž2yx
qx3 pq
ž
m CF a s Mb
/
y4 x4.
2
¢ L s ln
2
Ž2yx2 y4 x4 . 4 '1 y x 2
tany1 Ž 'x 2 y 1 . ,
x)1
,
xs1 ,
ln
ž
1 q '1 y x 2 1 y '1 y x 2
/
Ž 12 .
x-1
,
,
Ž 13 .
™
™
and a1 s 313 y 109 n l for n l s 3 massless quark flavors w11,12x. For x 0 the function h has the limiting behavior hŽ x 0. s y11r6q lnŽ2rx . q 3p xr4 q O Ž x 2 .. Using the upsilon expansion up to order e 2 w13x and the expressions given in Eqs. Ž2. and Ž11. we arrive at the following formula for the shift in the MS bottom quark mass Ž CF s 4r3.:
Dmb s
MF Ž1 S . 2 =
½
CF2 a s2 8
as
žp/
2 3
h
ž
4 mc MF Ž 1 S . CF a s
/
11 q 6
y ln
ž
MF Ž 1 S . CF a s 2 mc
/
1 y 3
a s2
mc MF Ž1 S .
5
,
Ž 14 .
where we have taken into account only the term linear in the charm quark mass from Eq. Ž2..2 The scale of the strong coupling contained in the first line of Eq. Ž14. is of order the inverse Bohr radius. We identify this scale with the one of the linear charm mass term. In deriving Eq. Ž14., the terms y11r6q lnŽ2rx . in hŽ x 0. have been absorbed into Eq. Ž11., with the replacement n l n l q 1 s 4. This gives the usual relation between the 1S and MS masses neglecting the charm quark mass, so these terms in hŽ x . do not contribute to the shift D m b . In Fig. 2 we have displayed yD m b as a function of a s . a s has to be evaluated at a scale of order the charm mass as discussed above. The solid, dashed, dash-dotted and dotted lines correspond to the choices m c s 1.7,1.5,1.3 and 1.1 GeV, respectively, for the charm quark mass. For a s f 0.4, which corresponds to a choice of the renormalization scale equal to the charm quark mass, we find that the shift is between y10 and y20 MeV. If a smaller renormalization scale is assumed, the shift can amount to more than y50 MeV for larger choices of the charm quark mass. The spread in the curves displayed in Fig. 2 shows that the shift in the MS bottom quark
™
™
2 It is sufficient to only include the term linear in m c in Eq. Ž2., since the higher order terms are suppressed by powers of m c r Mb which is small. Higher order terms in Eq. Ž11. depend on powers of 2 Mc rŽ Mb CF a s ., which is not small, and it is necessary to include the full functional dependence in hŽ x ..
A.H. Hoang, A.V. Manoharr Physics Letters B 483 (2000) 94–98
98
Fig. 2. The function y D m b as a function of a s . The solid, dashed, dash-dotted and dotted lines correspond to the choices m c s1.7, 1.5, 1.3 and 1.1 GeV, respectively, for the charm quark mass.
mass can be of order several tens of MeV. The exact value, however, contains a considerable uncertainty, which is amplified by the large value of the strong coupling governing the D m b . A complete NNLO analysis for the MS bottom quark extraction from F sum rules will be indispensable to accurately determine the effect of a nonzero charm mass. However, if the scale governing the strong coupling constant in D m b is indeed as low as the BLM scale estimate carried out above indicates, a considerable uncertainty might persist. For a moderate choice of the strong coupling in D m b between 0.4 and 0.6 the charm mass shift is smaller than the uncertainties of 60–80 MeV in the value of m b Ž m b . obtained in recent NNLO analyses of the F sum rules w1–3x, where charm mass effects have been neglected. However, the inclusion of the charm mass effects will be essential for a future extraction of m b Ž m b . at NNNLO.
Acknowledgements A.H. would like thank the members of the UCSD high energy theory group for their hospitality, and for the pleasant time at the UCSD Physics Department, where this work was finalized. A.H. is supported in part by the EU Fourth Framework Program ‘‘Training and Mobility of Researchers’’, Network ‘‘Quantum Chromodynamics and Deep Structure of Elementary Particles’’, contract FMRX-CT98-0194 ŽDG12-MIHT.. A.M. is supported in part by the US Department of Energy under contract DOE DE-FG03-97ER40546.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x
K. Melnikov, A. Yelkhovky, Phys. Rev. D 59 Ž1999. 114009. A.H. Hoang, Phys. Rev. D 61 Ž2000. 034005. M. Beneke, A. Signer, Phys. Lett. B 471 Ž1999. 233. A.H. Hoang, M.C. Smith, T. Stelzer, S.S. Willenbrock, Phys. Rev. D 59 Ž1999. 114014. M. Beneke, Phys. Lett. B 434 Ž1998. 115. N. Gray, D.J. Broadhurst, W. Grafe, K. Schilcher, Z. Phys. C 48 Ž1990. 673. P. Ball, M. Beneke, V.M. Braun, Nucl. Phys. B 452 Ž1995. 563. A.H. Hoang, M. Melles, in preparation. S. Titard, F.J. Yndurain, Phys. Rev. D 49 Ž1994. 6007. D. Eiras, J. Soto, hep-phr9905543. W. Fischler, Nucl. Phys. B 129 Ž1977. 157. A. Billoire, Phys. Lett. B 92 Ž1980. 343. A.H. Hoang, Z. Ligeti, A.V. Manohar, Phys. Rev. Lett. 82 Ž1999. 277; Phys. Rev. D 59 Ž1999. 074017.
Physics Letters B 483 Ž2000. 94–98
Charm effects in the MS bottom quark mass from F mesons A.H. Hoang a , A.V. Manohar b a
b
Theory DiÕision, CERN, CH-1211 GeneÕa 23, Switzerland Department of Physics, UniÕersity of California at San Diego, 9500 Gilman DriÕe, La Jolla, CA 92093-0319, USA Received 30 November 1999; received in revised form 21 March 2000; accepted 10 May 2000 Editor: R. Gatto
Abstract We study the shift in the F mass due to a non-zero charm quark mass. This shift affects the value of the MS b-quark mass extracted from the F system by about y20 MeV, due to an incomplete cancellation of terms that are non-analytic in the charm quark mass. The precise size of the shift depends on unknown higher order corrections, and might have a considerable uncertainty if they are large. q 2000 Elsevier Science B.V. All rights reserved.
The bottom quark mass is an important parameter for the theoretical description of B meson decays and b jet production cross sections in collider experiments. In continuum QCD, the most precise determinations of the bottom quark mass parameter have been obtained from data on the spectrum and the electronic partial widths of the F mesons. Recently, a number of MS bottom quark determinations have been carried out, which were based on F meson sum rules at next-to-next-to-leading order ŽNNLO. in the non-relativistic expansion, and which consistently eliminated all linear sensitivity to small momenta w1–3x. The latter is mandatory to reduce the systematic uncertainty in the bottom quark mass below the typical hadronization scale LQCD w4,5x. The analyses mentioned above, however, have treated all quarks other than the b quark as massless. This treatment is justified for light quarks that have masses much smaller than the inverse Bohr radius 1r² r : of the non-relativistic bottom–antibottom system i.e. for up, down and strange quarks, because in this case the theoretical expressions describing the bottom–antibottom dynamics and the conversion to the MS bottom mass definition can be expanded in the light quark masses. Like the contributions that are linearly sensitive to small momenta, the terms linear Žand non-analytic. in these light quark masses cancel out in the analysis. At NLO in the non-relativistic expansion this can be seen explicitly by considering the effects of a light virtual quark to the static energy of a bottom–antibottom quark pair with spatial distance r,1 Estat s 2 Mb q Vstat Ž r . ,
Ž 1.
where Mb is the bottom quark pole mass and Vstat the potential energy of the non-relativistic bottom–antibottom
E-mail address: [email protected] ŽA.H. Hoang.. The dominant terms that are linearly sensitive to low momenta in the Schrodinger equation that describes the bottom–antibottom ¨ dynamics are all contained in the static energy. w4,5x 1
0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 5 9 5 - 5
A.H. Hoang, A.V. Manoharr Physics Letters B 483 (2000) 94–98
95
quark system. At order a s2 the correction coming from the finite mass of a light quark q to the pole mass contribution reads w6x 4 a s2 mq d Mbq s Mb D , Ž 2. 2 3 p Mb
ž /
p2 DŽ r . s
ry
8
3 4
r2q
p2 8
r3y
`
y
ž
1 4
ln2 r y
13 24
p2 ln r q
151 q
24
288
/
r4
Ý Ž 2 F Ž n . ln r q F Ž n . . r 2 n , X
Ž 3.
ns3
3 Ž n y 1.
F Ž n. '
, Ž 4. 4 n Ž n y 2. Ž 2 n y 1. Ž 2 n y 3. where d Mbq is the shift in the b-quark pole mass keeping the b-quark MS mass fixed, m q is the mass of the light quark, and F X Ž n. ' EEn F Ž n.. In the limit m qrMb 0, Eq. Ž2. reduces to d Mbq s 16 a s2 m c . Ž 5. q q The shift d Mb is non-analytic in the quark masses, and should be regarded as being of the form d Mb s a s2 m2crMb2 Ž Mbr6 . , so that it is explicitly proportional to Mb , which breaks the b-quark chiral symmetry. The limiting value Eq. Ž5. can be easily computed using heavy quark effective theory. At the scale m s m b , one matches to a theory in which the b-quark is treated as a static field, with the residual mass term equal to zero, so that the propagator is irŽ k P Õ .. At the lower scale m s m c , one integrates out the charm quark. The matching condition at this scale induces a residual mass term for the b-quark, which is given by computing the graph in Fig. 1 The shift in the bottom–antibottom quark potential energy due to a light quark q is
™
(
q d Vstat Ž r . sy
R q q Ž m q , l. s
4 as 3 r
(
as
ž p/ ½ 6
4 m2q
ž
m2q
ž /
ž /
1q
m2
d l2
`
q
2 m2q
H4 m
2 q
l2
R q q Ž m q , l . exp Ž yl r .
5
,
/
Ž 6.
Ž 7. l2 l2 where R q q is the qq pair production cross section for the center-of-mass energy l in eq ey-annihilation normalized to the massless cross section. For m q < 1r² r : f Mb a s the static potential energy can be expanded in m q r: 4 as as a2 q d Vstat Ž r. Ž 8. ln Ž m r . q g q 56 4 y 13 a s2 m q q O s m2q r , 3 r 3p p and the contribution linear in m q cancels in the total static energy, Eq. Ž1. between Mb and V Ž r .. The dominant light quark correction is of order Ž a srp . 2 m 2qrMb and can be neglected for all practical purposes. ŽFor the charm case this would amount to a correction of several MeV..
™
1y
ln
ž
/
Fig. 1. Diagram contributing to the residual mass term of the b quark in HQET.
A.H. Hoang, A.V. Manoharr Physics Letters B 483 (2000) 94–98
96
On the other hand, for the case that a light quark mass is comparable in size to the inverse Bohr radius, i.e. q has to be taken into account. In this case the cancellation for the charm quark, the full m q-dependence of d Vstat of the linear charm mass term between the pole mass and potential contributions to the static energy, Eq. Ž2., is incomplete. In this letter we show that this feature can lead to sizeable effects in the MS bottom quark mass determination, compared to when the charm quark is treated as massless. There are two reasons why the incomplete cancellation of the linear charm mass contribution can lead to a sizeable effect. First, the linear charm mass term is not suppressed by a factor p 2 , as one might expect from a contribution coming from a loop integration Žsee Eq. Ž2... This is because the linear charm mass contribution represents a correction generated by a non-analytic linear dependence on infrared momenta. A similar effect is known in chiral perturbation theory where non-analytic mp3 corrections from loop diagrams are also not suppressed by powers of p . Second, the scale of the strong coupling of the linear charm mass term in Eq. Ž2. is of order the charm mass rather than the bottom quark mass. This can be understood from the fact that the charm mass serves as an infrared cutoff for the non-analytic linear dependence on infrared momenta just mentioned before. Thus, the linear charm mass term is generated at momenta of order m c . The effective field theory computation of Fig. 1 also indicates that one should use a s Ž m c .. These arguments are supported by an explicit calculation of the order a s3 BLM corrections to the linear charm mass term in Eq. Ž2.. For this calculation it is sufficient to consider the static limit, i.e. we only need to determine the linear charm mass corrections to the bottom quark self-energy due to the chromostatic Coulomb field. The order a s3 BLM linear charm mass contributions in the bottom quark pole mass can be obtained from the formula
d Mbc,stat s
as
d l2
`
H 6p 4 M
2 c
l
2
R q q Ž m c , l.
d3p
H Ž 2p .
2 4 p as 3
2
3 p ql
2
as
1q2
4p
m2
5
žž / /
b 0 ln
p
2
q
3
,
Ž 9.
where we assumed that m c is the charm pole mass, and b 0 s 11 y 23 n l for n l s 3 light quark flavors. The factor of two in front of the b 0 term arises because the charm quark loop can be inserted on both sides of the massless fermion loops. We have also included the chromostatic self-energy contribution at order a s2 . The term linear in m c contained in Eq. Ž9. reads
d Mbc,linear s
a s2 Ž m . 6
mc 1 q 2
as
ž p/ 4
m2
žž /
b 0 ln
m 2c
y 4ln2 q
14 3
/
,
Ž 10 .
pole which corresponds to the BLM scale m BLM s 0.388 m c . We emphasize again that expression Ž10. represents a non-analytic contribution Žfrom momenta p 2 Q m2c . that cannot be obtained from a naive Taylor expansion of the integrand in Eq. Ž9. with respect to m c . Obviously, the BLM calculation indicates a very low renormalization scale for the strong coupling governing the linear charm quark mass contribution. For a s Ž m c . s 0.36 and m c s 1.5 GeV the order a s2 and the a s3 terms each amount to 32 MeV. At this point we note that the linear charm mass approximation in Eq. Ž10. indeed dominates d Mbc. At high orders of perturbation theory it is true because at high orders d Mbc is dominated by momenta p 2 < m2c , leading to a suppression of the contributions that are not contained in the linear mass term. At low orders it can be checked explicitly: at order a s2 the linear charm mass term is within 15% of the result containing the full charm mass dependence displayed in Eq. Ž2. for Ž m crMb . 2 s 0.1. At order a s3 only a comparison to the full charm mass result in the large-b 0 approximation is possible Žsee e.g. w7x.. Using the same input parameters used to produce the numbers shown in Table 4 of Ref. w7x we find that the a s3 term is within 10% of the large-b 0 result containing the full charm mass dependence. To illustrate the size of the corrections caused by the incomplete cancellation of the linear charm quark terms let us examine the difference in the MS bottom quark mass m b Ž m b ., determined from the mass of the F Ž1S . meson, MF Ž1 S . s 9.460 GeV, for the two cases that the finite charm mass effects are either taken into account or neglected. For simplicity, we only consider an extraction of the MS bottom quark mass at NLO in the non-relativistic expansion. A more thorough analysis using full NNLO expressions and including also a sum rule analysis based on data for all known F mesons will be carried out elsewhere w8x.
A.H. Hoang, A.V. Manoharr Physics Letters B 483 (2000) 94–98
97
Including the effects of the charm quark mass from Eq. Ž7. properly in first order time-independent perturbation theory, the F Ž1S . meson mass at NLO in the non-relativistic expansion reads w9,10x Ž a s ' a sŽ n fs4 . Ž m ..
½
MF Ž1 S . s 2 Mb 1 y
CF2 a 2s
y
CF2 a 2s
8
8
as
žp/
a1
b0 Ž L q 1 . q
2
2 q 3
mc
žž / ž ln
qh
m
2 mc Mb CF a s
// 5
,
Ž 11 . where w10x
°Ž2 y x y 4 x . 2
4
2 'x 2 y 1 h Ž x . ' y 116 y 2 x 2 q
3 xp 4
~ Ž2yx
qx3 pq
ž
m CF a s Mb
/
y4 x4.
2
¢ L s ln
2
Ž2yx2 y4 x4 . 4 '1 y x 2
tany1 Ž 'x 2 y 1 . ,
x)1
,
xs1 ,
ln
ž
1 q '1 y x 2 1 y '1 y x 2
/
Ž 12 .
x-1
,
,
Ž 13 .
™
™
and a1 s 313 y 109 n l for n l s 3 massless quark flavors w11,12x. For x 0 the function h has the limiting behavior hŽ x 0. s y11r6q lnŽ2rx . q 3p xr4 q O Ž x 2 .. Using the upsilon expansion up to order e 2 w13x and the expressions given in Eqs. Ž2. and Ž11. we arrive at the following formula for the shift in the MS bottom quark mass Ž CF s 4r3.:
Dmb s
MF Ž1 S . 2 =
½
CF2 a s2 8
as
žp/
2 3
h
ž
4 mc MF Ž 1 S . CF a s
/
11 q 6
y ln
ž
MF Ž 1 S . CF a s 2 mc
/
1 y 3
a s2
mc MF Ž1 S .
5
,
Ž 14 .
where we have taken into account only the term linear in the charm quark mass from Eq. Ž2..2 The scale of the strong coupling contained in the first line of Eq. Ž14. is of order the inverse Bohr radius. We identify this scale with the one of the linear charm mass term. In deriving Eq. Ž14., the terms y11r6q lnŽ2rx . in hŽ x 0. have been absorbed into Eq. Ž11., with the replacement n l n l q 1 s 4. This gives the usual relation between the 1S and MS masses neglecting the charm quark mass, so these terms in hŽ x . do not contribute to the shift D m b . In Fig. 2 we have displayed yD m b as a function of a s . a s has to be evaluated at a scale of order the charm mass as discussed above. The solid, dashed, dash-dotted and dotted lines correspond to the choices m c s 1.7,1.5,1.3 and 1.1 GeV, respectively, for the charm quark mass. For a s f 0.4, which corresponds to a choice of the renormalization scale equal to the charm quark mass, we find that the shift is between y10 and y20 MeV. If a smaller renormalization scale is assumed, the shift can amount to more than y50 MeV for larger choices of the charm quark mass. The spread in the curves displayed in Fig. 2 shows that the shift in the MS bottom quark
™
™
2 It is sufficient to only include the term linear in m c in Eq. Ž2., since the higher order terms are suppressed by powers of m c r Mb which is small. Higher order terms in Eq. Ž11. depend on powers of 2 Mc rŽ Mb CF a s ., which is not small, and it is necessary to include the full functional dependence in hŽ x ..
A.H. Hoang, A.V. Manoharr Physics Letters B 483 (2000) 94–98
98
Fig. 2. The function y D m b as a function of a s . The solid, dashed, dash-dotted and dotted lines correspond to the choices m c s1.7, 1.5, 1.3 and 1.1 GeV, respectively, for the charm quark mass.
mass can be of order several tens of MeV. The exact value, however, contains a considerable uncertainty, which is amplified by the large value of the strong coupling governing the D m b . A complete NNLO analysis for the MS bottom quark extraction from F sum rules will be indispensable to accurately determine the effect of a nonzero charm mass. However, if the scale governing the strong coupling constant in D m b is indeed as low as the BLM scale estimate carried out above indicates, a considerable uncertainty might persist. For a moderate choice of the strong coupling in D m b between 0.4 and 0.6 the charm mass shift is smaller than the uncertainties of 60–80 MeV in the value of m b Ž m b . obtained in recent NNLO analyses of the F sum rules w1–3x, where charm mass effects have been neglected. However, the inclusion of the charm mass effects will be essential for a future extraction of m b Ž m b . at NNNLO.
Acknowledgements A.H. would like thank the members of the UCSD high energy theory group for their hospitality, and for the pleasant time at the UCSD Physics Department, where this work was finalized. A.H. is supported in part by the EU Fourth Framework Program ‘‘Training and Mobility of Researchers’’, Network ‘‘Quantum Chromodynamics and Deep Structure of Elementary Particles’’, contract FMRX-CT98-0194 ŽDG12-MIHT.. A.M. is supported in part by the US Department of Energy under contract DOE DE-FG03-97ER40546.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x
K. Melnikov, A. Yelkhovky, Phys. Rev. D 59 Ž1999. 114009. A.H. Hoang, Phys. Rev. D 61 Ž2000. 034005. M. Beneke, A. Signer, Phys. Lett. B 471 Ž1999. 233. A.H. Hoang, M.C. Smith, T. Stelzer, S.S. Willenbrock, Phys. Rev. D 59 Ž1999. 114014. M. Beneke, Phys. Lett. B 434 Ž1998. 115. N. Gray, D.J. Broadhurst, W. Grafe, K. Schilcher, Z. Phys. C 48 Ž1990. 673. P. Ball, M. Beneke, V.M. Braun, Nucl. Phys. B 452 Ž1995. 563. A.H. Hoang, M. Melles, in preparation. S. Titard, F.J. Yndurain, Phys. Rev. D 49 Ž1994. 6007. D. Eiras, J. Soto, hep-phr9905543. W. Fischler, Nucl. Phys. B 129 Ž1977. 157. A. Billoire, Phys. Lett. B 92 Ž1980. 343. A.H. Hoang, Z. Ligeti, A.V. Manohar, Phys. Rev. Lett. 82 Ž1999. 277; Phys. Rev. D 59 Ž1999. 074017.