Документ взят из кэша поисковой машины. Адрес оригинального документа : http://qfthep.sinp.msu.ru/talks/Melikhov_Talk.pdf Дата изменения: Tue Sep 14 11:02:40 2010 Дата индексирования: Mon Oct 1 19:40:37 2012 Кодировка: Поисковые слова: m 35

OPE, heavy quark mass, and decay constants of heavy mesons from QCD sum rules
Dmitri Melikhov
SINP, Moscow State University, Moscow, Russia

A new extraction of the decay constants of D, D s, B, and Bs mesons from the two-point function of heavy-light pseudoscalar currents is presented. The main emphasis of this talk is laid on the uncertainties in these quantites, both related to the OPE for the relevant correlators and to the extraction procedures of the method of sum rules.

Based on W.Lucha, D.Melikhov, S.Simula "Decay constants of heavy pseudoscalar mesons from QCD sum rules" arXiv:1008.2698


2

A QCD sum-rule calculation of hadron parameters involves two steps: I. Calculating the operator product expansion (OPE) series for a relevant correlator For heavy-light currents, one observes a very strong dependence of the OPE for the correlator (and, consequently, of the extracted decay constant) on the heavy-quark mass used, i.e., on-shell (pole), or running M S mass. We make use of the three-loop OPE for the correlator by Chetyrkin et al, reorganized in terms of M S mass, in which case OPE exhibits a reasonable convergence.

II. Extracting the parameters of the ground state by a numerical procedure
NEW :

(a) Make use of the new more accurate duality relation based on Borel-parameter-dependent threshold. Allows a more accurate extraction of the decay constants and provides realistic estimates of the intrinsic (systematic) errors -- those related to the limited accuracy of sum-rule extraction procedures. (b) Study the sensitivity of the extracted value of fP to the OPE parameters (quark masses, condensates,. . . ). The corresponding error is referred to as OPE uncertainty, or statistical error.


3

Basic object: OPE for ( p2) = i d xe and its Borel transform ( p2 ).
Quark
2 MQ

i px

0|T j5( x) j(0) |0 , 5

j5( x) = (mQ + m)qi5 Q( x) Ї

hadron duality assumption :

24 fQ MQe-

=

se ff e- spert (mQ +mu )2

( s, , mQ, µ) d s +

power

(, mQ, µ) dual(, µ, seff )

In order the l.h.s. and the r.h.s. have the same -behavior
seff is a function of and : seff ,

2 The "dual" mass: Mdual() = -

d d

log

dual

(, seff ()).

If quark-hadron duality is implemented "perfectly", then Mdual should be equal to MQ; The deviation of Mdual from the actual meson mass MQ measures the contamination of the dual correlator by excited states. Better reproduction of MQ more accurate extraction of fQ. Taking into account -dependence of seff improves the accuracy of the duality approximation. Obviously, in order to predict fQ, we need to fix seff . How to fix seff ?


4

Our new algorithm for extracting ground

state parameters when MQ is known

For a given trial function seff () there exists a variational solution which minimizes the deviation of the dual mass from the actual meson mass in the -"window" (only a few lowest-dimension power corrections are known, work at mQ 1). (i) Consider a set of Polynomial -dependent Ansaetze for seff : s(n)() = eff
n j=0

s(jn)() j.

2 2 (ii) Minimize the squared difference between the "dual" mass Mdual and the known value MQ in the -window. This gives us the parameters of the effective continuum threshold.

(iii) Making use of the obtained thresholds, calculate the decay constant. (iv) Take the band of values provided by the results corresponding to linear, quadratic, and cubic effective thresholds as the characteristic of the intrinsic uncertainty of the extraction procedure. Illustration: D-meson
Mdual M 1.02 1.01 1 0.99 GeV 0.1 0.2 0.3 0.4 0.5 0.6
2 D

n0 n1 n 2n 3

fdual MeV 230 220 210 200 190 180 170 0.1

n2 n3 n1 n0
2

GeV 0.2 0.3 0.4 0.5 0.6


5

Extraction of f
100

D

mc(mc) = 1.279 ± 0.013 GeV, µ = 1 - 3 GeV.
230
+0.013

m = 1.279
c

-0.013

GeV

220 210

80

Count

60

f (MeV)

200 190 180 170
m = 1.279(13) GeV

40

D

20

160 150

c

N =2
f

N =3
f

QCD-SR
-dependent constant

LATTICE

PDG

0 0.18 0.19 0.20
D

0.21

0.22

0.23

0.24

f (GeV)

f

D

206.2

7.3OPE

5.1syst MeV

f

D

const

181.3

7.4OPE MeV

The effect of -dependent threshold is visible!


6

Extraction of f
100

Ds

mc(mc) = 1.279 ± 0.013 GeV, µ = 1 - 3 GeV.
300
+0.013

m = 1.279
c

-0.013

GeV

80

Count

60

40

f
200
m = 1.279(13) GeV
c

Ds

(MeV)

250

20
150

N =2
f

N =3
f

QCD-SR

LATTICE

PDG

0 0.20 0.25 0.30

f

Ds

(GeV)

f

Ds

246.5

15.7OPE

5syst MeV

-dependent

constant

f

Ds

const

218.8

16.1OPE MeV


7

Extraction of fB: a very strong sensitivity to mb(mb)
fB MeV 240 220 200 180 160 4.2 4.25 4.3 mb GeV 4.35 n 0n 1n 2n 3

-dependent effective threshold: f
dual B

qq 1/3 - 0.267 GeV Ї mb - 4.245 GeV +4 (mb, qq , µ = mb) = 206.5 ± 4 - 37 Ї 0.1 GeV 0.01 GeV 37 MeV on fB!

M eV,

± 10 MeV on mb


8

The prediction for fB is not feasible without a very precise knowledge of mb:
80
+0.17

200
m = 4.20
b -0.07

200
m = 4.163 ± 0.016 GeV
b

70 60 50

GeV

m = 4.245 ± 0.025 GeV
b

150

150

Count

Count

40 30 20 10 0 0.05 0.10 0.15
B

100

Count
0.10 0.15
B

100

50

50

0 0.20 0.25 0.30 0.05 0.20 0.25 0.30

0 0.05 0.10 0.15
B

0.20

0.25

0.30

f (GeV)
240
m =4.163(16) GeV
b

f (GeV)

f (GeV)

220

f (MeV)

200

B

180
m =4.245(25) GeV
b

N =2
f

N =3
f

160

QCD-SR

LATTICE

Our estimate : mb mb
4.3syst MeV
fB const

4.245
184

0.025 GeV
13OPE MeV

f

B

193.4

12.3OPE


9

Extraction of f
80 70 60 50
b

Bs
300

m = 4.245 ± 0.025 GeV

280

m = 4.163(16) GeV
b

Count

260

40 30 20

(MeV) f
Bs

240

220
10 0 0.15 0.20 0.25
Bs
m = 4.245(25) GeV
b

200
0.30

N =2
f

N =3
f

QCD-SR
180

LATTICE

f

(GeV)

f

Bs

232.5

18.6OPE

2.4syst MeV

fBs const

218

18OPE MeV


10

Conclusions
The effective continuum threshold seff is an important ingredient of the method which determines to a large extent the numerical values of the extracted hadron parameters. Finding a criterion for fixing seff poses a problem in the method of sum rules. · -dependence of seff emerges naturally when trying to make quark-hadron duality more accurate. For those cases where the ground-state mass MQ is known, we proposed a new algorithm for fixing seff . We have tested that our algorithm leads to the extraction of more accurate values of bound-state parameters than the standard algorithms used in the context of sum rules before. · -dependent seff is a useful concept as it allows one to probe realistic intrinsic uncertainties of the extracted parameters of the bound states. · We obtained predictions for the decay constants of heavy mesons fQ which along with the "statistical" errors related to the uncertainties in the QCD parameters, for the first time include realistic "systematic" errors related to the uncertainty of the extraction procedure of the method of QCD sum rules. · Matching our SR estimate to the average of the recent lattice results for fB allowed us to obtain a rather accurate estimate mb(mb) = 4.245 ± 0.025 GeV. Interestingly, this range does not overlap with a very accurate range reported by Chetyrkin et al. Why?


11

OPE : heavy
To 2 -accuracy, mb s
, pole

quark pole mass or running mass ?

= 4.83 GeV mb (mb ) = 4.20 GeV:

Spectral densities

(mb , s , s) (mb , s , ) (mb (mb , s ), s , ) (mb , s , ) (mb , s , s)
i i s 5 4 Pole mass OPE 3 2 1 0 i 0 i 1 i 2 Full Pert 20 25 30 35 s 40 i i s, mb 5 4 3 2 1 0 20 25 30 MS scheme

i 0 i 1 i 2 Full Pert 35 s 40

· In pole mass scheme poor convergence of perturbative expansion · In M S scheme the pert. spectral density has negative regions higher orders NOT negligible

Extracted decay constant
f 2 dual ,s 0.05 0.04 0.03 0.02 0.01 0 0.01 0.1 0.12 0.14 0.16 0.18 O 1 O O 2 power total
0

f 2 dual ,s 0.05 0.04 0.03 0.02 0.01 0 0.01

0

O 1 O O 2 power total

0.1

0.12

0.14

0.16

0.18

· Decay constant in pole mass shows NO hierarchy of perturbative contributions · Decay constant in M S -scheme shows such hierarchy. Numerically, fP using pole mass fP using M S mass.


12

OPE Aliev 1983 Narison 2001 Jamin 2001 Our results O O 2 O 2 O 2

mQ, GeV pole : 4.8 pole : 4.7 pole : 4.83 MS : 4.05 MS : 4.21 MS : 4.25 0.05 0.025

fB, MeV 130 203 215 193 20 % 23OPE 19OPE 13OPE 4syst