Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://qfthep.sinp.msu.ru/talks2013/Talk_Melikhov_QFTHEP2013.pdf Äàòà èçìåíåíèÿ: Fri Jun 21 22:32:53 2013 Äàòà èíäåêñèðîâàíèÿ: Thu Feb 27 20:31:35 2014 Êîäèðîâêà:

Accurate bottom quark mass from sum rules for decay constants of B

mesons

Wolfgang Lucha, Dmitri Melikhov, Silvano Simula
HEPHY, Vienna, Austria & SINP, Moscow State University, Russia & INFN, Uni Roma Tre, Italy,

We show that Borel QCD sum rules for heavy­light currents yield very strong correlations between the b-quark mass mb and the B-meson decay constant fB: fB/ fB -8 mb/mb. This fact opens the possibility of an accurate extraction of mb from QCD sum rules using fB as input. Combining precise lattice QCD determinations of fB with our sum-rule analysis based on heavy­light correlation function leads to mb(mb) = (4.247 ± 0.027 ± 0.011) GeV


2

Precise knowledge of the b-quark mass is highly desirable [mb(µ) mb(µ), mb mb(mb)]
4.6

m (GeV)

4.4
Hoang et al. LMS

b

4.2

Chet et al.

QCD-SR
4.0

LQCD (N =2)
f

¯ Moment SRs for bb two-point functions in pQCD with 4-loop accuracy vs experimental data: low-n moments (Chetyrkin et al): mb = 4.163 ± 0.016 GeV large-n moments (Hoang et al): mb = 4.235 ± 0.055
(pert)

± 0.03

(exp)

GeV

We report that Borel QCD sum rules for heavy­light correlators provide the possibility to extract mb with comparable accuracy if a precise value for fB is used as input.


3

Explore the sensitivity of fB to the precise value of the b-quark mass: In a nonrelativistic potential model the following relationship between the ground-state wave function at the origin, (r = 0), and the ground-state binding energy : |(r = 0)| 3/2. The decay constant fB (r = 0); in the heavy-quark limit fB 1/ mQ: f
B

MB = ( MB - mQ)3/2.

Dependence of fB on small variations mQ of the heavy-quark (pole) mass near some mQ: MB = 5.27 GeV; fB 200 MeV for mQ 4.6 Â 4.7 GeV 0.9 Â 1.0 and fB -0.5 mQ.

The sensitivity of fB to the precise value of the heavy-quark mass should be very high: mQ fB -(11 Â 12) . fB mQ


4

Recent QCD sum-rule results for fB based on 3-loop two-point function have been presented: Narison'2001 Jamin'2002 LMS`2009 Narison'2012 mb (GeV) 4.05 ± 0.06 4.21 ± 0.05 4.245 ± 0.025 4.236 ± 0.069 fB (MeV) 203 ± 23 210 ± 19 193 ± 15 206 ± 7 Not all the results here are equally trustable. Recall: the values of the ground-state parameters are strongly influenced by · the reliable perturbative expansion of the two-point function; · the way of fixing the auxiliary parameters of the sum-rule approach, particularly, the effective continuum threshold. For fixed inputs of the correlator (condensates, s, etc.) we find ( ) mb - 4.247 GeV fB(mb) = 192.0 - 37 ± 3(syst) MeV. 0.1 GeV Strong correlation between fB and mb! Combining our sum-rule analysis with fB from lattice QCD f mb = (4.247 ± 0.027(OPE
+fB ) LQCD B

= (191.5 ± 7.3) MeV leads to

± 0.011(syst) GeV.


5

Correlation function , OPE, and heavy

quark mass

The basic object is T -product of 2 pseudoscalar currents, j5( x) = (mb + m) q( x)i5b( x), ¯ ( ) 2 4 i px ( p ) = i d x e 0 T j5( x) j5(0) 0 and its Borel image
24 () = fB MBe-
2 MB

+
sphys

ds e

- s


hadr

( s) =
(mb +m)2

ds e

- s

pert( s, µ) +

power

(, µ).

here sphys = ( MB + MP)2, and fB is the decay constant defined by
2 (mb + m)0|qi5b| B = fB MB. ¯

To exclude the excited-state contributions, one adopts the duality Ansatz: all contributions of excited states are counterbalanced by the perturbative contribution above an effective continuum threshold, seff () which differs from the physical continuum threshold.


6

Applying the duality assumption yields:
24 fB MBe-
2 MB

seff ()

=
(mb +m)2

ds e

- s



pert

( s, µ ) +

power

(, µ)

dual

(, seff ()).

The rhs is the dual correlator

dual

(, seff ()).

Even if the QCD inputs pert( s, µ) and power(, µ) are known, the extraction of the decay constant requires, in addition, a criterion for determining seff (). As first step, we need a reasonably convergent OPE for both correlator and dual correlator. The best-known 3-loop calculations of the perturbative spectral density have been performed in form of an expansion in terms of the MS strong coupling s(µ) and the pole mass Mb: ( )2 s(µ) s(µ) (1) 2 2 2 ( s, Mb ) + (2)( s, Mb , µ) + · · · . pert( s, µ) = (0)( s, Mb ) + An alternative option is to reorganize the perturbative expansion in terms of the running MS 2 mass, mb(), by substituting Mb in the spectral densities (i)( s, Mb ) via its perturbative expansion in terms of the running mass mb() ( )2 1 + s() r + s() r + . . . . Mb = mb() 2 1


7

OPE in terms of b-quark pole (left) and MS mass (right): SR results for fB; seff = 35 GeV2.
f
2 dual

,s 0.06 0.05 0.04 0.03 0.02 0.01

eff

GeV2

f

2

dual

,s 0.06 0.05

eff

Gev2 total

total

0.04 O1 0.03 0.02

O1 O O 2 0.1 0.15 0.2

power
0.01 O 0.1 0.15

power O 2 0.2 Gev 0.25
2

GeV 0.25

2

0

Lessons: 1. For dual correlator calculated through the heavy-quark pole mass, perturbative expansion exhibits no sign of convergence; the O(1), O( s), and O(2) terms are of the same magnitude. s In pole-mass scheme one cannot expect higher orders to give smaller contributions. 2. Formulating the perturbative series in terms of the heavy-quark MS mass yields a clear hierarchy of contributions. We employ the MS-mass OPE in our SR analysis. 3. fB extracted from the pole-mass truncated OPE ( fB = 188 MeV) is substantially smaller than that from the MS-mass OPE truncated at the same order ( fB = 220 MeV). However, both decay constants exhibit stability over a wide range of . Borel stability does not guarantee the reliability.


8

Extraction of the decay constant
According to the standard procedures of QCD sum rules, one executes the following steps: 1. The Borel window The working -window is chosen such that the OPE gives an accurate description of the exact correlator (i.e., all higher-order radiative and power corrections are under control) and at the same time the ground state gives a "sizable" contribution to the correlator. Our -window for the B( s) mesons is 0.05 (GeV-2) 0.175. 2. The effective continuum threshold To find seff (), we employ a previously developed algorithm which provides a reliable extraction of the ground-state parameters in quantum-mechanics and of the charmed-meson decay constants in QCD. We introduce the dual invariant mass Mdual and the dual decay constant fdual
2 Mdual() -

d log d

dual

(, seff ()),

f

2 dual

( ) M

2 -4 M B e B



dual

(, seff ()).

The dual mass should reproduce the true ground-state mass MB; the deviation of Mdual from MB measures the contamination of the dual correlator by excited states. Starting from an Ansatz for seff () and requiring a minimum deviation of Mdual from MB in the -window generates a variational solution for seff (). With the latter at our disposal, fdual() yields the desired decay-constant estimate. We consider polynomials in , including also a -independent constant: s () =
(n) eff n j=0

s(jn) j.


9

2 2 We obtain s(jn) by minimizing the squared difference between Mdual and MB in the -window: N ] 1 [ 2 22 Mdual(i) - MB . N i=1 2

Uncertainties in the extracted decay constant

The resulting fB is sensitive to the input values of the OPE parameters -- which determines what we call the OPE-related error -- and to the details of the adopted prescription for fixing the behaviour of the effective continuum threshold seff () -- the systematic error.
OPE related error

We estimate the size of the OPE-related error by perform a bootstrap analysis, assuming Gaussian distributions for all OPE parameters but the renormalization scales. For the latter, we assume uniform distributions in the range 3 µ, (GeV) 6. The resulting distribution of the decay constant turns out to be close to Gaussian shape. Hence, the quoted OPE-related error is a Gaussian error. Systematic error The systematic error, related to the limited intrinsic accuracy of the method of sum rules, is a subtle point. In quantum mechanics, we observed that considering polynomial parameterizations of the effective continuum threshold seff (), the band of results obtained from linear, quadratic, and ¨ cubic Ansatze for seff (), encompasses the true value of the decay constant. Thus, the half-width of this band may be regarded as a realistic estimate for the systematic uncertainty of the prediction.


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Decay constant of B

meson

For mb mb(mb) = 4.247 GeV, µ = = mb, and central values of the other relevant parameters:
Mdual MB 1.01 1.005 1 n 1n 2 0.995 31.5 GeV 0.05 0.075 0.1 0.125 0.15 0.175 0.2
2

s

eff

GeV2 34 n1 n3 n2 n0 GeV 0.05 0.075 0.1 0.125 0.15 0.175 0.2
2

f

dual

MeV n1 n3 n2

33.5 n0 n3 32.5 32 33

194 192 190 188 186 184 182

n0 GeV 0.05 0.075 0.1 0.125 0.15 0.175 0.2
2

fB MeV 240 220 200 n0 180 160 4.2 4.25 4.3 mb GeV 4.35 n3 n2 n1

Our results for fB may be parameterized by (for fixed values of other OPE parameters) [ ( ) ( ) mb - 4.247 GeV |qq|1/3 - 0.269 GeV ¯ dual fB (mb, µ = = mb, qq) = 192.0 - 37 ¯ +4 ±3 0.1 GeV 0.01 GeV

]
(syst)

MeV,


11

Performing the bootstrap analysis we find ( fB = 192.0 ± 14.3

(OPE)

± 3.0(syst) MeV.

)

A similar procedure yields for the Bs meson ( ) fBs = 228.0 ± 19.4(OPE) ± 4(syst) MeV. and fBs / fB = 1.184 ± 0.023(OPE) ± 0.007(syst).
240
m =4.163(16) GeV
b

290 270

m =4.163(16) GeV
b

220

f (MeV)

(MeV) f
Bs m =4.247(29) GeV
b

250 230 210
m =4.247(29) GeV
b

200

B

180
N =2
f

N = 2+1
f

N =2
f

N = 2+1
f

160
-2.00

QCD-SR
0.00 2.00 4.00

LQCD
6.00 8.00 10.00

190
-2.00

QCD-SR
0.00 2.00 4.00

LQCD
6.00 8.00 10.00

The main contributions to the OPE uncertainty in the extracted fB and f the renormalization-scale dependence and the errors in mb.

Bs

arise from


12

Extraction of the bottom
4.6
f
LQCD B

quark mass

Using lattice average fB = (191.5 ± 7.3) MeV and applying our algorithms yields:
constant
= 191.5 ± 7.3 MeV

4.5

linear quadratic cubic

m (GeV)

4.4

b

4.3
Hoang et al.

4.2
Chetyrkin et al.

4.1 LO NLO NNLO

perturbative order

m m m

b b b

LO NLO NNLO

4.38 0.1 0.020syst GeV 4.27 0.04 0.015syst GeV 4.247 0.027 0.011syst GeV

Moving from LO to NLO of the perturbative expansion (i) decreases sizeably mb and reduces OPE-error. The extracted values of mb exhibit a nice "convergence" depending on the accuracy of the perturbative correlation function. The major contribution to the OPE-error in mNNLO (60%) due to scale-variations. The N3LO b correction is not known. Nevertheless, we do not expect a sizeable shift of the central value of mb, but expect a reduction of the OPE-error.


13
60
Hoang et al. Chetyrkin et al.

50

40

Count

30

f

LQCD B

= 191.5 ± 7.3 MeV

20

10

0 4.1 4.2
b

4.3

4.4

m (GeV)

Distribution of mb range fB = (191.5 with the associated distributions in the

as obtained by the bootstrap analysis: Gaussian distributions for fB in the ± 7.3) MeV and for all the OPE parameters (except for the scales µ and ) uncertainties are employed. For the independent parameters µ and , uniform range 3 GeV < µ, < 6 GeV are assumed.


14

Summary
We presented a detailed QCD sum-rule analysis of the B- and Bs-meson decay constants, with particular emphasis on the study of the errors in the extracted decay-constant values: the OPE uncertainty due to the errors of the QCD parameters and the intrinsic error of the sum-rule approach due to the limited accuracy of the extraction procedure. Our main findings are: · The extraction of hadronic properties is improved by allowing a Borel-parameter dependence for the effective continuum threshold, which increases the accuracy of the duality approxima¨ tion. Considering suitably optimized polynomial Ansatze for the effective continuum threshold provides an estimate of the intrinsic uncertainty of the method of QCD sum rules. · For beauty mesons, a strong correlation between mb and the sum-rule result for fB is reported: fB mb -8 . fB mb Combining our sum-rule analysis with the latest results for fB and f
mb 4.247 0.025
OPE f
B

Bs

from lattice QCD yields

0.011syst GeV

The main contribution to the first error (about 60%) is due to the variation of the renormalization scales. Good news is that the systematic uncertainty of the sum-rule method, estimated from the ¨ spread of the results for different Ansatze of the effective continuum threshold remains under control.


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Our value of mb is extracted from the heavy­light correlator known to O(2) accuracy. Since the s value of mb is changing only marginally when moving from the O( s) to O(2) accuracy of the s 3 correlator, we do not expect the inclusion of the presently unknown O( s ) correction to lead to a substantial change in the extracted value of mb. Our result compares well with mb = (4.209 ± 0.050) GeV found from moment sum rules for heavy­heavy correlators evaluated to the same O(2) accuracy s as in our analysis. We observe an excellent agreement with the prediction of the sum rule mb = (4.235 ± 0.055(pert) ± 0.003
(exp)

) GeV.

We see, however, a pronounced tension with the prediction mb = (4.163 ± 0.016) GeV, based on moment sum rules for heavy­heavy correlators calculated to O(3) accuracy. The origin s of this disagreement requires further considerations. We conclude by emphasizing that the properly formulated Borel QCD sum rules for heavy­light correlators provide a competitive tool for the reliable calculation of heavy-meson properties and for the extraction of basic QCD parameters by making use of the results from lattice QCD and the experimental data.


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The OPE parameters: md (2 GeV) = (3.5 ± 0.5) MeV, m s(2 GeV) = (95 ± 5) MeV, s( MZ ) = 0.1184 ± 0.0007, 3 s qq(2 GeV) = -((269 ± 17) MeV) , s s(2 GeV)/qq(2 GeV) = 0.8 ± 0.3, ¯ ¯ ¯ = GG (0.024 ± 0.012) GeV4.