Heavy - quark masses and heavy - meson decay constants from Borel sum rules in QCD
Wolfgang Lucha1, Dmitri Melikhov1,2, Silvano Simula3
1 - HEPHY, Vienna; 2- SINP, Moscow State University; 3 - INFN, Roma tre, Roma
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. Our main emphasis 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 “Heavy-meson decay constants from QCD sum rules” arXiv:1008.3129
“OPE, charm-quark mass, and decay constants of D and D s mesons from QCD sum rules”
arXiv:1101.5986, Phys. Lett. B 701, in press
2
A QCD sum-rule calculation of hadron parameters involves two steps:
I. Calculating the operator product expansion (OPE) series for a relevant correlator
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 MS mass.
We make use of the three-loop OPE for the correlator by Chetyrkin et al, reshuffled in terms of MS
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
2
R
ipx
Basic object: OPE for Π(p ) = i dxe h0|T
j5(x) j†5(0)
|0i,
j5(x) = (mQ + m)q̄iγ5 Q(x)
and its Borel transform (p2 → τ).
Quark - hadron duality assumption :
2
fQ2 MQ4 e−MQτ
=
R
seff
e−sτρpert (s, α, mQ, µ) ds
(mQ +mu )2
+ Πpower(τ, mQ, µ) ≡ Πdual(τ, µ, seff )
In order the l.h.s. and the r.h.s. have the same τ-behavior
seff is a function of Τ Hand ΜL : seff HΤ, ΜL
d
2
(τ) = − dτ
log Πdual(τ, seff (τ)).
The “dual” mass: Mdual
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 τ-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 (τ) there exists a variational solution which minimizes the deviation of
the dual mass from the actual meson mass in the τ-“window” (only a few lowest-dimension power
corrections are known, work at τmQ ≤ 1).
n
P
(n)
j
(i) Consider a set of Polynomial τ-dependent Ansaetze for seff : s(n)
(τ)
=
s
j (τ) .
eff
j=0
2
and the known value MQ2 in
(ii) Minimize the squared difference between the “dual” mass Mdual
the τ-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MD
1.02
1.01
fdual@MeVD
230
220
210
200
190
180
170
n=0
n=1
1
n=2 n=3
0.99
0.1
0.2
0.3
0.4
0.5
0.6
Τ@GeV-2 D
0.1
n=2
n=3
n=1
n=0
0.2
0.3
0.4
0.5
0.6
Τ@GeV-2 D
5
Extraction of fD
mc(mc) = 1.279 ± 0.013 GeV, µ = 1 − 3 GeV.
230
100
+0.013
m = 1.279
c
-0.013
220
GeV
210
f (MeV)
60
200
190
D
Count
80
40
180
170
m = 1.279(13) GeV
160
20
c
N =2
f
QCD-SR
N =3
f
LATTICE
PDG
0.18
0.19
0.20
0.21
0.22
0.23
0.24
f (GeV)
D
!-dependent
0
constant
150
fD = 206.2 ± 7.3OPE ± 5.1syst MeV
The effect of τ-dependent threshold is visible!
fD HconstL = 181.3 ± 7.4OPE MeV
6
Extraction of fDs
mc(mc) = 1.279 ± 0.013 GeV, µ = 1 − 3 GeV.
100
300
+0.013
m = 1.279
c
-0.013
GeV
(MeV)
40
250
Ds
60
f
Count
80
200
m = 1.279(13) GeV
20
c
QCD-SR
N =2
f
N =3
f
LATTICE
PDG
0.20
0.25
f
Ds
(GeV)
0.30
!-dependent
0
constant
150
fDs = 246.5 ± 15.7OPE ± 5syst MeV
fDs HconstL = 218.8 ± 16.1OPE MeV
7
Extraction of fB: a very strong sensitivity to mb(mb)
fB@MeVD
240
220
n=0 n=1 n=2 n=3
200
180
160
4.2
4.25
4.3
mb @GeVD
4.35
τ-dependent effective threshold:
!
!#
"
1/3
h
q̄qi
−
0.267
GeV
m
−
4.245
GeV
b
+4
MeV,
fBdual(mb, hq̄qi, µ = mb) = 206.5 ± 4 − 37
0.1 GeV
0.01 GeV
± 10 MeV on mb → ∓ 37 MeV on fB!
8
The prediction for fB is not feasible without a very precise knowledge of mb:
200
80
200
+0.17
70
m = 4.20
b
GeV
m = 4.163 ± 0.016 GeV
m = 4.245 ± 0.025 GeV
b
-0.07
b
150
60
150
40
Count
Count
Count
50
100
100
30
50
20
50
10
0
0
0.05
0.10
0.15
0.20
0.25
0.30
0
0.05
0.10
0.15
f (GeV)
f (GeV)
B
240
0.20
B
0.25
0.30
0.05
0.10
0.15
0.20
0.25
0.30
f (GeV)
B
m =4.163(16) GeV
b
200
B
f (MeV)
220
180
m =4.245(25) GeV
b
N =2
f
160
QCD-SR
N =3
f
LATTICE
 
Our estimate : mb Hmb L = 4.245 ± 0.025 GeV
fB = 193.4 ± 12.3OPE ± 4.3syst MeV
fB HconstL = 184 ± 13OPE MeV
9
Extraction of fBs
80
70
300
m = 4.245 ± 0.025 GeV
b
280
60
b
260
(MeV)
Count
50
m = 4.163(16) GeV
40
240
f
Bs
30
20
220
10
m = 4.245(25) GeV
b
0
0.15
N =2
200
0.20
0.25
f
Bs
(GeV)
0.30
f
QCD-SR
N =3
f
LATTICE
180
fBs = 232.5 ± 18.6OPE ± 2.4syst MeV
fBs HconstL = 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.
• τ-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.
• τ-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 - quark pole mass or running mass ?
To α2s -accuracy, mb,pole = 4.83 GeV ↔ mb (mb ) = 4.20 GeV:
Spectral densities
ρ(mb , α s , s) → Π(mb , α s , τ) → Π(mb (mb , α s ), α s , τ) → Π(mb , α s , τ) → ρ(mb , α s , s)
 L
HΑ Π Li ΡiHs, Μ =m
b
5
4
3
2
1
0
HΑ Π Li Ρi HsL
5
4 Pole mass OPE
3
2
1
0
i=0 i=1 i=2 Full Pert
20
25
30
35

MS-scheme
i=0 i=1 i=2 Full Pert
s
40
20
25
30
35
s
40
• In pole mass scheme poor convergence of perturbative expansion
• In MS scheme the pert. spectral density has negative regions → higher orders NOT negligible

Extracted
decay constant

f 2 dualHΤ,s0 L
0.05
f 2 dualHΤ,s0 L
0.05
0.04
0.04
0.03
OH1L OHΑL OHΑ2 L power total
0.03
0.02
0.02
0.01
0.01
0
0
-0.01
0.1
0.12
0.14
0.16
Τ
0.18
OH1L OHΑL OHΑ2 L power total
-0.01
0.1
0.12
0.14
0.16
Τ
0.18
• Decay constant in pole mass shows NO hierarchy of perturbative contributions
• Decay constant in MS -scheme shows such hierarchy. Numerically, fP using pole mass fP using MS mass.
12
OPE
Aliev @1983 D
O HΑL
mQ, GeV
pole : 4.8
Narison @2001 D O IΑ2 M
pole : 4.7
O IΑ2 M
pole : 4.83
Jamin @2001 D
Our results
O IΑ2 M

MS : 4.05

MS : 4.21 ± 0.05

MS : 4.25 ± 0.025
fB, MeV
130 H± 20 %L
203 ± 23OPE
215 ± 19OPE
193 ± 13OPE ± 4syst