- Email: [email protected]
Estimating quenching errors in ƒB and BB
UCLEAR PHYSIC~
ELSEVIER
Nuclear PhysicsB (Proc. Suppl.) 47 (1996) 441-444
PROCEEDINGS SUPPLEMENTS
Estimating Quenching Errors in fB and BB Stephen R. Sharpe a* and Yan Zhang a ~Physics Department, University of Washington, Seattle, WA 98195, USA We use quenched chiral perturbation theory to estimate the quenching errors in fB, fBs, BB and BBs.
1. I N T R O D U C T I O N The interpretation of experimental results for B-meson mixing requires knowledge of 8-2 2 (ff¢lbT"(1-75)dbT,(1-75)dlB °) -- 5IbmBBB .(1) The right hand side is a parameterization in terms of the usual decay constant (normalized so that f~ = 132 MeV) ifBp~, = (Ol~Tu75blB-). At present, the only way to calculate such matrix elements from first-principles is by numerical simulation of lattice QCD. In fact, lack of computer power has limited most simulations to the quenched approximation to QCD (QQCD). Such simulations have, however, progressed to the point that the error due to quenching is comparable to, or larger than, that due to statistics or systematics other than quenching [1]. Furthermore, preliminary work suggests substantial differences in fB upon the inclusion of quark loops [2]. Thus it is important to estimate quenching errors in mixing matrix elements such as (1). We have made such estimates using quenched chiral perturbation theory [3,4]. We have extended the formalism of Ref. [3] to include the couplings of "heavy-light" mesons to pseudoGoldstone bosons (PGB's) in QQCD. Our results are valid for mb --* oo, but we expect the limb corrections to be small. Technical details can be found in Ref. [5], and we attempt here a complementary discussion focusing on the physical origin of quenching errors, and the major results. Ref. [5] also contains our results for "partially quenched" theories--dynamical quarks are included, but have different masses from the valence quarks. *Speaker 1996 Elsevier Science B.V. PII: S0920-5632(96)00091-6
Similar work for QQCD has been done by Booth [6], and he has also extended the results to include 1~rob corrections [7]. Where our works overlap, the results are in agreement. 2. B A S I C I D E A The essential point is that quenching alters the clouds of PGBs surrounding B mesons. These clouds contribute to the mixing matrix elements, and, in a chiral expansion, the leading contributions can be expressed in terms of a few parameters. The results for QQCD and QCD differ substantially, and this difference can be used to estimate the quenching errors. To illustrate these comments, we show, in Figs. 1 and 2, quark flow diagrams corresponding to the one-loop chiral corrections to lB. The thick and thin lines represent heavy and light quarks, respectively. The solid box represents the axial current. We have indicated the various coupling constants that appear; e.g. g is the B-B*-rr coupling in QCD. Clearly, quenched loops arise from a different source than those in QCD. (Strictly speaking the diagrams of Fig. 1 are present also in QQCD, but they give zero contribution because of the exact cancelation between quark and ghost-quark loops.) All of the QQCD contributions are proportional to a coupling constant (m02, a or g') present only in the quenched Lagrangian. Here m02 is the "hairpin" vertex which iterates to give the rf a mass in QCD, but must be treated as an interaction in QQCD [3,4], while a is the corresponding momentum dependent interaction which renormalizes the y' field in QCD. g' is the "disconnected" part of the B-B*-Tf vertex, irrelevant in QCD where the I/' is not a PGB, but
442
S.R. Sharpe, Y Zhang/Nuclear Physics B (Proc. Suppl.) 47 (1996) 441-444
> B
B*
>
B
B
B*
B
B
>
>
B
m
B*
B
Figure 1. Chiral loops in QCD needed in QQCD.
B
3. R E S U L T S F O R C H I R A L L O G S It turns out to be advantageous in lattice work to separately calculate the quantities fB and BB, so we consider corrections to these two quantities individually. We use the notation exemplified by s
=
I+(ASB s)qcD
(2)
+ cl(A)(m,, + md + ms) + c2(h)m, + . . . ] fO,QCD is the decay constant in the chiral limit, A f is the chiral logarithmic correction, c1,~ are constants parameterizing the analytic corrections (arising from higher order terms in the chiral Lagrangian and the weak current) and the ellipsis represents terms of size mq2 ,,, m~. The chiral logs for QCD have been calculated in Ref. [8]; the result in this example is (assuming mu = ma) (A/Bs)Qc D _
(l(4~rf) 2 +3g ~) (m~- log m~.A 2
+ -~m, ,og ~ - ) .
(z)
>
< m
Figure 2. Chiral loops in QQCD The A dependence in the chiral log is unphysical; it is canceled by that in c1,~. Note that both the chiral log and the analytic corrections vanish in the chiral limit. The corresponding result in QQCD is [6,5]
1
[
, 2
(A/Bs)QQCD _ (4~rf) 2 t - 3gg m , , + (1 +3g~)(2am28 - m02)] log A s .
(4)
Here ms, is the mass of the quenched meson composed of s and ~, which is the only meson appearing in loops, as can be seen from Fig. 2. The A dependence is canceled by analytic terms of the form c• (A)m02 + c• (A)mq. We note 1. There is no remnant of the QCD result.
S.R. Sharpe, Y. Zhang/Nuclear Physics B (Proc. Suppl.) 47 (1996) 441-444
2. The result diverges as m , , --~ 0, indicating a sickness of the chiral limit of QQCD. Its origin is the same as the divergences found previously in m~/mq, fK, (¢¢)[3] and SKI4]. The divergence is weak, however, being significant numerically only for m,, g m~, well below masses simulated at present. 3. For masses used in present simulations, the chiral log can be numerically important, ranging in size from - 0 . 1 to 0.3. In present simulations, the result for fBs is obtained by using an s quark on the lattice, while that for fB is obtained by linear extrapolation from quark masses in the range 0.5ma-m,. We mimic this procedure by defining fR in QQCD using linear extrapolation from m, and 0.5m, to mq -- 0. The analytic result can he deduced from that for fBs given above. For the B-parameters we find (ABBs) QQCD -- ( 4 - f ) 2 6gg'm2,,+ ( 3 g ~ - 1)(m~,
2
2
1
(5) ~
- 5 a m , , + 5too)
.
m,, A2 ,
to be compared with (ABBs)QC D
=
2(392 - 1) 2 3(47rf)2 m,7 log -A-~'m'
(6)
The divergence is still present in QQCD, but there is also a remnant of the QCD result in the m~s term. This is because some of the quarkline diagrams for QCD do not involve additional quark loops. The situation is not as favorable as for BK with degenerate quarks, however, where the QCD and QQCD expressions coincide [4]. 4. Q U A N T I T A T I V E ESTIMATES QUENCHING E R R O R S
OF
At first sight, our results are not very useful because of the unknown constants. To estimate fQQCD/fQCD, for example, we clearly need to know something about ¢0,QQCD/¢0,QCD JB /JB . The cleanest way of proceeding is to devise quantities which cancel as many of the unknown constants as possible [3]. To do this we consider theories
443
with three non-degenerate quarks, labeled u, d and s, and satisfying mu < md < m,. We then construct Of
=
\~BD,] m,
X
--
-
\ f B o ,] mu
' m,
2(md-m~,)'
Y-
-
mu
2(m,--md)'
(8)
and the similar ratio QB involving the Bparameters. To the order that we work in chiral perturbation theory, the quark masses appearing in QI can be replaced by the corresponding squared meson-masses, e.g. m, --* m~,. Q! and QB are designed to vanish if chiral corrections depend linearly on the quark masses, and are small. Thus all dependence o n f~,QCD, on the analytic terms cl and c2, and on A, cancels. The expressions for QQCD can be deduced from the results given above. To calculate the Q's in QCD requires the inclusion of r ° - rI mixing; the result is given in Ref. [5]. We content ourselves here with numerical examples. p h y s , and mu : We consider the choice rn~ = ~s,~, md: m, = 1 : 2 : 3, which is realizable in present quenched simulations. With these masses we have QI = fBv f B s / f ~ o - 1. To quote results we need values for various parameters. For QCD we take g2 = 0.4, the estimate used in Ref. [8]. For QQCD we use the values which are representative of the range considered in Ref. [5]: gO = g, g~ = 0, ~ = 0.7 and ~ = 2r,0V(3(4rf) 2) = 0.05(0.15). The two choices for m0~ roughly cover the range of values suggested by the numerical evidence for quenched chiral logs [9]. We then find Q?CD
=
0.009,
QQ,°"
=
0,
Q?QCD ----0.025 (0.041); (9)
Q QQcD = 0 . 0 0 2 ( - 0 . 0 0 1 ) . ( 1 0 )
The vanishing of QQCD appears to be accidental; there are corrections of O(m~) to all quantities. There is a clear, though small (2 - 3%), difference between QQCD and QCD results for QI. Aside from higher order corrections, this is a firm result for an error due to quenching, and is large enough that it might be measurable in present simulations. We view this as a rough lower bound on the quenching errors in decay constants themselves, because there are likely to be cancelations
444
S.R. Sharpe, Y Zhang/Nuclear Physics B (Proc. Suppl.) 47 (1996) 441-444
in the errors when forming the ratio in Qf. It is not a precise bound because the theory with dynamical quarks that we are calling "QCD" has different quark masses from real QCD. The small values for QB suggest that the quenching error for B-parameters is smaller than that for decay constants. The quenching error grows as the mass ratios are increased--for m, = _I1~$ p h y s , and mu : m d : m8 = 1 : 2 : 10, the error in Q! is ,~ 20%. This suggests that one should not use small quark masses in QQCD.
0.4
,
[
BB
0.2
0.0
-0.2
-.
fB .....................
-
°.....-'"
-0.4 -0.6
"l
I
-0.5
5. Q U A L I T A T I V E ESTIMATES QUENCHING ERRORS
OF
Further assumptions are required to estimate quenching errors in individual decay constants and B-parameters. We assume that the dominant quenching error is due to the chiral log, i.e. that the fractional quenching error is A(AfB)(A) m (AfB)QQCD(A)- (AfB)QCD(A), where QCD now refers to the theory in which dynamical and valence quarks have their physical masses. This is equivalent to assuming that QQCD and QCD agree except for the contributions from (the logarithmically divergent part of) chiral loops with momenta in the range 0 < ]k I < A. This is, at best, an assumption valid on average over several quantities for a reasonable choice of A. It is quite possible that the differences in the loop integrals are partly compensated by the analytic parameters. So we view these estimates as rough upper bounds on quenching errors. They might indicate in which quantities the error is likely to be larger. We show these estimates in Fig. 3, taking A = 1GeV, g~ = g~ = 0.4, a = 0.7 and 6 = 0.15, and the conservative range -gQ < g~ < gq. The estimated error can be substantiM, but is poorly determined. We have found, however, that there are no reasonable values of the parameters for which the estimates are simultaneously small (< 10%) for all four quantities. It is thus possible that there are substantial quenching errors in one or more of the quantities.
i
i
i
l
,,,,I
i
0.0
g'
0.5
Figure 3. Estimates of quenching errors, A ( A X ) , for the four quantities of interest. If we restrict ourselves to [g'/gQ[ < 1/3, which we consider to be the most likely range, then we can draw further conclusions. • Quenching errors in B-parameters are < 10%. • The quenching error in f B s / f B is negative and in the range - 0 . 1 to - 0 . 3 . The latter estimate is surprisingly large, but is likely to be more reliable than those for individual decay constants, since the dependence on f~'0 Q Q C D /f~'0 Q C D cancels. Taken literally, it implies that f B s / f B is 10-30% larger in QCD than in QQCD. More conservatively, it suggests that the deviation of this ratio from unity, which is ( f B s / f s - 1) ~ 0.1 - 0.2 in present QQCD simulations, has a quenching error of 100% or more. REFERENCES 1. C. Allton, these proceedings. 2. C. Bernard, these proceedings. 3. C. Bernard and M. Golterman, Phys. Rev. D 4 6 , 8 5 3 (1992). 4. S. Sharpe, Phys. Rev. D46, 3146 (1992). 5. S. Sharpe and Y. Zhang, UW/PT-95-10 6. M. Booth, Phys. Rev. D51, 2338 (1995). 7. M. Booth, hep-ph/9412228. 8. B. Grinstein el al., Nucl. Phys. B 3 8 0 (1992) 369. 9. R. Gupta, Nucl. Phys. B ( P r o c . S u p p l . ) 42 (1995) 85.
ELSEVIER
Nuclear PhysicsB (Proc. Suppl.) 47 (1996) 441-444
PROCEEDINGS SUPPLEMENTS
Estimating Quenching Errors in fB and BB Stephen R. Sharpe a* and Yan Zhang a ~Physics Department, University of Washington, Seattle, WA 98195, USA We use quenched chiral perturbation theory to estimate the quenching errors in fB, fBs, BB and BBs.
1. I N T R O D U C T I O N The interpretation of experimental results for B-meson mixing requires knowledge of 8-2 2 (ff¢lbT"(1-75)dbT,(1-75)dlB °) -- 5IbmBBB .(1) The right hand side is a parameterization in terms of the usual decay constant (normalized so that f~ = 132 MeV) ifBp~, = (Ol~Tu75blB-). At present, the only way to calculate such matrix elements from first-principles is by numerical simulation of lattice QCD. In fact, lack of computer power has limited most simulations to the quenched approximation to QCD (QQCD). Such simulations have, however, progressed to the point that the error due to quenching is comparable to, or larger than, that due to statistics or systematics other than quenching [1]. Furthermore, preliminary work suggests substantial differences in fB upon the inclusion of quark loops [2]. Thus it is important to estimate quenching errors in mixing matrix elements such as (1). We have made such estimates using quenched chiral perturbation theory [3,4]. We have extended the formalism of Ref. [3] to include the couplings of "heavy-light" mesons to pseudoGoldstone bosons (PGB's) in QQCD. Our results are valid for mb --* oo, but we expect the limb corrections to be small. Technical details can be found in Ref. [5], and we attempt here a complementary discussion focusing on the physical origin of quenching errors, and the major results. Ref. [5] also contains our results for "partially quenched" theories--dynamical quarks are included, but have different masses from the valence quarks. *Speaker 1996 Elsevier Science B.V. PII: S0920-5632(96)00091-6
Similar work for QQCD has been done by Booth [6], and he has also extended the results to include 1~rob corrections [7]. Where our works overlap, the results are in agreement. 2. B A S I C I D E A The essential point is that quenching alters the clouds of PGBs surrounding B mesons. These clouds contribute to the mixing matrix elements, and, in a chiral expansion, the leading contributions can be expressed in terms of a few parameters. The results for QQCD and QCD differ substantially, and this difference can be used to estimate the quenching errors. To illustrate these comments, we show, in Figs. 1 and 2, quark flow diagrams corresponding to the one-loop chiral corrections to lB. The thick and thin lines represent heavy and light quarks, respectively. The solid box represents the axial current. We have indicated the various coupling constants that appear; e.g. g is the B-B*-rr coupling in QCD. Clearly, quenched loops arise from a different source than those in QCD. (Strictly speaking the diagrams of Fig. 1 are present also in QQCD, but they give zero contribution because of the exact cancelation between quark and ghost-quark loops.) All of the QQCD contributions are proportional to a coupling constant (m02, a or g') present only in the quenched Lagrangian. Here m02 is the "hairpin" vertex which iterates to give the rf a mass in QCD, but must be treated as an interaction in QQCD [3,4], while a is the corresponding momentum dependent interaction which renormalizes the y' field in QCD. g' is the "disconnected" part of the B-B*-Tf vertex, irrelevant in QCD where the I/' is not a PGB, but
442
S.R. Sharpe, Y Zhang/Nuclear Physics B (Proc. Suppl.) 47 (1996) 441-444
> B
B*
>
B
B
B*
B
B
>
>
B
m
B*
B
Figure 1. Chiral loops in QCD needed in QQCD.
B
3. R E S U L T S F O R C H I R A L L O G S It turns out to be advantageous in lattice work to separately calculate the quantities fB and BB, so we consider corrections to these two quantities individually. We use the notation exemplified by s
=
I+(ASB s)qcD
(2)
+ cl(A)(m,, + md + ms) + c2(h)m, + . . . ] fO,QCD is the decay constant in the chiral limit, A f is the chiral logarithmic correction, c1,~ are constants parameterizing the analytic corrections (arising from higher order terms in the chiral Lagrangian and the weak current) and the ellipsis represents terms of size mq2 ,,, m~. The chiral logs for QCD have been calculated in Ref. [8]; the result in this example is (assuming mu = ma) (A/Bs)Qc D _
(l(4~rf) 2 +3g ~) (m~- log m~.A 2
+ -~m, ,og ~ - ) .
(z)
>
< m
Figure 2. Chiral loops in QQCD The A dependence in the chiral log is unphysical; it is canceled by that in c1,~. Note that both the chiral log and the analytic corrections vanish in the chiral limit. The corresponding result in QQCD is [6,5]
1
[
, 2
(A/Bs)QQCD _ (4~rf) 2 t - 3gg m , , + (1 +3g~)(2am28 - m02)] log A s .
(4)
Here ms, is the mass of the quenched meson composed of s and ~, which is the only meson appearing in loops, as can be seen from Fig. 2. The A dependence is canceled by analytic terms of the form c• (A)m02 + c• (A)mq. We note 1. There is no remnant of the QCD result.
S.R. Sharpe, Y. Zhang/Nuclear Physics B (Proc. Suppl.) 47 (1996) 441-444
2. The result diverges as m , , --~ 0, indicating a sickness of the chiral limit of QQCD. Its origin is the same as the divergences found previously in m~/mq, fK, (¢¢)[3] and SKI4]. The divergence is weak, however, being significant numerically only for m,, g m~, well below masses simulated at present. 3. For masses used in present simulations, the chiral log can be numerically important, ranging in size from - 0 . 1 to 0.3. In present simulations, the result for fBs is obtained by using an s quark on the lattice, while that for fB is obtained by linear extrapolation from quark masses in the range 0.5ma-m,. We mimic this procedure by defining fR in QQCD using linear extrapolation from m, and 0.5m, to mq -- 0. The analytic result can he deduced from that for fBs given above. For the B-parameters we find (ABBs) QQCD -- ( 4 - f ) 2 6gg'm2,,+ ( 3 g ~ - 1)(m~,
2
2
1
(5) ~
- 5 a m , , + 5too)
.
m,, A2 ,
to be compared with (ABBs)QC D
=
2(392 - 1) 2 3(47rf)2 m,7 log -A-~'m'
(6)
The divergence is still present in QQCD, but there is also a remnant of the QCD result in the m~s term. This is because some of the quarkline diagrams for QCD do not involve additional quark loops. The situation is not as favorable as for BK with degenerate quarks, however, where the QCD and QQCD expressions coincide [4]. 4. Q U A N T I T A T I V E ESTIMATES QUENCHING E R R O R S
OF
At first sight, our results are not very useful because of the unknown constants. To estimate fQQCD/fQCD, for example, we clearly need to know something about ¢0,QQCD/¢0,QCD JB /JB . The cleanest way of proceeding is to devise quantities which cancel as many of the unknown constants as possible [3]. To do this we consider theories
443
with three non-degenerate quarks, labeled u, d and s, and satisfying mu < md < m,. We then construct Of
=
\~BD,] m,
X
--
-
\ f B o ,] mu
' m,
2(md-m~,)'
Y-
-
mu
2(m,--md)'
(8)
and the similar ratio QB involving the Bparameters. To the order that we work in chiral perturbation theory, the quark masses appearing in QI can be replaced by the corresponding squared meson-masses, e.g. m, --* m~,. Q! and QB are designed to vanish if chiral corrections depend linearly on the quark masses, and are small. Thus all dependence o n f~,QCD, on the analytic terms cl and c2, and on A, cancels. The expressions for QQCD can be deduced from the results given above. To calculate the Q's in QCD requires the inclusion of r ° - rI mixing; the result is given in Ref. [5]. We content ourselves here with numerical examples. p h y s , and mu : We consider the choice rn~ = ~s,~, md: m, = 1 : 2 : 3, which is realizable in present quenched simulations. With these masses we have QI = fBv f B s / f ~ o - 1. To quote results we need values for various parameters. For QCD we take g2 = 0.4, the estimate used in Ref. [8]. For QQCD we use the values which are representative of the range considered in Ref. [5]: gO = g, g~ = 0, ~ = 0.7 and ~ = 2r,0V(3(4rf) 2) = 0.05(0.15). The two choices for m0~ roughly cover the range of values suggested by the numerical evidence for quenched chiral logs [9]. We then find Q?CD
=
0.009,
QQ,°"
=
0,
Q?QCD ----0.025 (0.041); (9)
Q QQcD = 0 . 0 0 2 ( - 0 . 0 0 1 ) . ( 1 0 )
The vanishing of QQCD appears to be accidental; there are corrections of O(m~) to all quantities. There is a clear, though small (2 - 3%), difference between QQCD and QCD results for QI. Aside from higher order corrections, this is a firm result for an error due to quenching, and is large enough that it might be measurable in present simulations. We view this as a rough lower bound on the quenching errors in decay constants themselves, because there are likely to be cancelations
444
S.R. Sharpe, Y Zhang/Nuclear Physics B (Proc. Suppl.) 47 (1996) 441-444
in the errors when forming the ratio in Qf. It is not a precise bound because the theory with dynamical quarks that we are calling "QCD" has different quark masses from real QCD. The small values for QB suggest that the quenching error for B-parameters is smaller than that for decay constants. The quenching error grows as the mass ratios are increased--for m, = _I1~$ p h y s , and mu : m d : m8 = 1 : 2 : 10, the error in Q! is ,~ 20%. This suggests that one should not use small quark masses in QQCD.
0.4
,
[
BB
0.2
0.0
-0.2
-.
fB .....................
-
°.....-'"
-0.4 -0.6
"l
I
-0.5
5. Q U A L I T A T I V E ESTIMATES QUENCHING ERRORS
OF
Further assumptions are required to estimate quenching errors in individual decay constants and B-parameters. We assume that the dominant quenching error is due to the chiral log, i.e. that the fractional quenching error is A(AfB)(A) m (AfB)QQCD(A)- (AfB)QCD(A), where QCD now refers to the theory in which dynamical and valence quarks have their physical masses. This is equivalent to assuming that QQCD and QCD agree except for the contributions from (the logarithmically divergent part of) chiral loops with momenta in the range 0 < ]k I < A. This is, at best, an assumption valid on average over several quantities for a reasonable choice of A. It is quite possible that the differences in the loop integrals are partly compensated by the analytic parameters. So we view these estimates as rough upper bounds on quenching errors. They might indicate in which quantities the error is likely to be larger. We show these estimates in Fig. 3, taking A = 1GeV, g~ = g~ = 0.4, a = 0.7 and 6 = 0.15, and the conservative range -gQ < g~ < gq. The estimated error can be substantiM, but is poorly determined. We have found, however, that there are no reasonable values of the parameters for which the estimates are simultaneously small (< 10%) for all four quantities. It is thus possible that there are substantial quenching errors in one or more of the quantities.
i
i
i
l
,,,,I
i
0.0
g'
0.5
Figure 3. Estimates of quenching errors, A ( A X ) , for the four quantities of interest. If we restrict ourselves to [g'/gQ[ < 1/3, which we consider to be the most likely range, then we can draw further conclusions. • Quenching errors in B-parameters are < 10%. • The quenching error in f B s / f B is negative and in the range - 0 . 1 to - 0 . 3 . The latter estimate is surprisingly large, but is likely to be more reliable than those for individual decay constants, since the dependence on f~'0 Q Q C D /f~'0 Q C D cancels. Taken literally, it implies that f B s / f B is 10-30% larger in QCD than in QQCD. More conservatively, it suggests that the deviation of this ratio from unity, which is ( f B s / f s - 1) ~ 0.1 - 0.2 in present QQCD simulations, has a quenching error of 100% or more. REFERENCES 1. C. Allton, these proceedings. 2. C. Bernard, these proceedings. 3. C. Bernard and M. Golterman, Phys. Rev. D 4 6 , 8 5 3 (1992). 4. S. Sharpe, Phys. Rev. D46, 3146 (1992). 5. S. Sharpe and Y. Zhang, UW/PT-95-10 6. M. Booth, Phys. Rev. D51, 2338 (1995). 7. M. Booth, hep-ph/9412228. 8. B. Grinstein el al., Nucl. Phys. B 3 8 0 (1992) 369. 9. R. Gupta, Nucl. Phys. B ( P r o c . S u p p l . ) 42 (1995) 85.