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Recent progress in QCD sum rule calculations of heavy meson properties
Dmitri Melikhov
SINP, Moscow

I discuss the details of calculating hadron properties from the OPE for correlators of quark currents in QCD, which constitutes the basis of the method of QCD sum rules. The main emphasis is laid on gaining control over the systematic uncertainties of the hadron parameters obtained within this method. We start with examples from quantum mechanics, where bound-state propè erties may be calculated independently in two ways: exactly, by solving the Schrodinger equation, and approximately, by the method of sum rules. Knowing the exact solution allows us to control each step of the sum-rule extraction procedure. On the basis of this analysis, we formulate several improvements of the method of sum rules. We then apply these modifications to the analysis of the decay constants of heavy charm and beauty mesons.


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 MS mass. OPE reorganized in terms of MS mass exhibits a reasonable convergence of the perturbative expansion for hadron observables.

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

1. Basic object in QCD: ( ) 2 i px ( p ) = i d xe 0|T j5( x) j5(0) |0, and its Borel transform (), p2 .

j5( x) = (mQ + m)qi5 Q( x) ï

Analogue in quantum mechanics: Polarization operator (E ) is defined through the full Green function G(E ): (E ) = f = 0| r and its Borel transform E T , 1 |i = 0. r H-E

1 exp(-H T ) H-E which leads to the evolution operator in imaginary time T (T = 1/MBorel): (T ) = f = 0| exp(-H T )|i = 0. r r


4

Exact and Feynman propagators of a confined particle

Feynman propagator of a NR particle with mass m DF ( E , 2 ) = k 1 2 - 2mE - i0 k
exact

.

The plot compares the values at E = 0 of DF and D
DNR k 2 GeV 0.5 0.4 0.3
2

for HO potential m2r2/2.

DF

1 k2

0.2 Dexact 0.1 0.1 0.2 0.3 0.4 k 0.5
2

GeV2

As soon as "soft" momenta in Feynman diagrams are essential, nonperturbative effects in propagators are essential. At large k2, one finds D
exact

2 4 (k ) = DF (k ) + # 4 + # 6 + . . . k k
2 2


5

Correlator in a realistic potential model : confinement

Coulomb

Polarization operator (E ) = f = 0| r

1 r H -E |i

= 0 . (r) - . r

k2 H= +V 2m Expansion of (E ) in powers of the interaction:

conf

+ 0 + 1 + 2 0


+ 0 + 1 + 2

2

+

+

+

+ ...


6

§ Analogue of the OPE for the Borel image (T ): For the case V
OPE pert conf

(r) =

m2 r2 2

an explicit double expansion in powers of and powers of T (T ),



power

(T ) = pert(T ) + ( m )3/2 [ (T ) = 2T ( m )3/2 [ (T ) = 2T

] 122 1 + 2mT + m T , 3 ( ( ) )] 122 11 1541 19 4 4 - T 1+ T 1+ 2mT + 2mT 4 12 480 1824
power 3/2





pert

(m) (T ) = 2


0

[ ] z 3/2m2 dz exp(-zT ) 2 + 2m + 3z

§ The "phenomenological" representation for (T ) í in the basis of hadron eigenstates: (T ) = f = 0| exp(-H T )|i = 0 = r r En - energy of the n-th bound state, Rn = |n( = 0)|2. r
n=0

Rn exp(-EnT ),


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How to calculate E
T

n=0

and R

n=0

of the ground state from (T ) known numerically?
T exp E0 T
5

log
2.2 2

T

E

0

R0

4 1.8 3 1.6 1.4 1.2 0.5 1 1.5 2 2.5 3 3.5 4 2

T

1 1 1.5 2 2.5 3 3.5 4

T

Black - exact (T ); Red - OPE with 4 power corrections, Green - OPE with 100 power corrections.

With a few power corrections the plateau cannot be reached. Some other concept: "Quark-hadron duality" assumption will be used.


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Sum rule: OPE side:
OPE OPE

(T ) = phys(T ).
-zT

(T ) =
0

dze

pert(z) +

power

(T ).

Physical side with the "Standard Ansatz" for the excited states: (T ) = dze-zT phys(z), phys(z) = R0(z - E0) + (z - zeff )pert(z).
0

This gives R0 e
-E0 T

z eff

=

dual

(T , zeff )
0

dze-zT
0

pert

(z) +

power

(T ). (T , zeff ), E
dual

From this sum rule one obtains estimates for R -dT log (T , zeff ). These depend on unphysical parameters T and zeff . How to fix these parameters?

and E0, R

dual

(T , zeff )

=

choose z

eff

The standard and almost obvious criterion is "maximal stability": such that the dependence of Edual and Rdual on T in the T -"window" is minimal.


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How this works in quantum mechanics for H =
Edual T , z Eg 1 0.9 0.8 0.7 0.6 0.6 0.7 0.8 0.9 1 T GeV
1 c

k2 2m

+

m2 r2 2

- . r

zc from maximal stability of Edual

f

dual

T ,zc ,Edual z 0.1 0.08 0.06

c

GeV Exact

f

dual

T , zc , E 0.1

g

GeV

0.08

exact zc from maximal stability of f
dual

using exact E

g

0.06

Extracted 0.04 T GeV
1

0.04
1

0.6

0.7

0.8

0.9

1

T GeV 0.6 0.7 0.8 0.9 1

What is bad? § A model for the excited states is oversimplified. § Within the assumption zeff = con st, maximal stability does not automatically lead to a success.


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Exact effective threshold: Use the exact E0 and R0 obtained from Schroedinger equation and solve the relation R0 e
-E0 T z e
ff

=
0

dze

-zT

pert(z) +

power

(T )

with respect to zeff . The obtained "exact threshold" is a slightly rising function of T , zeff (T ).
zexact T 1.175 1.15 1.125 1.1 1.075 1.05 1.025 T 0.4 0.6 0.8 1
dual

We have neglected this dependence when calculating E

= -dT log dual(T , zeff (T )).


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Correlator in QCD
( p ) = i
2



) d xe |T j5( x) (0) |,
i px

(

j 5

j5( x) = (mQ + m)qi5 Q( x) ï

Physical QCD vacuum | is complicated and differs from perturbative QCD vacuum |0. Wilsonian OPE: ) 2 ^ ^ T j5( x) j (0) = C0( x , ²)1 + Cn( x2, ²) : O(0, ²) : (
5 n

Condensates í nonzero expectation values of gauge-invariant operators over physical vacuum: ^ | : O(0, ²) : | 0. Borel transform ( p2 ): Green functions in Minkowski space evolution operator in Euclidean space () =
(mQ +mu )2

e- spert( s, , mQ, ²) d s + § §
pert (0)

power

(, mQ, ²), ( s) + ( ) s (²) 2 (2)( s) + § § §

( s, ² ) = ( s) +



power

(, ²) í power expansion in in terms of the condensates: { [ ( ) ( )] } 2 2 m2 m2 m0 Q Q 1 F (, ² = mQ) = (mQ + m)2e-mQ -mQ qq 1 + 2C s 1 - 2 + 2 1 - 2 ï + 12 s GG .
power



s (²) (1)

Sum rule:

OPE

() =

hadron

()


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Sum rule:

OPE

() =

hadron

()

Duality concept: where pQCD calculations may be applied in hadron physics?


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Spectral densities of the polarization operator (OPE vs hadron language):
xx

Im (s)

+

f

B

theoretical physical

seff
2 m2 MB b

s
*

scont =(MB +m )2
hadron duality assumption :

Quark



seff

d s exp(- s)pert( s) =

sphys.cont.

d s exp(- s)hadr ( s).


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With the help of the duality assumption, the contribution of the excited states cancels against the high-energy region of the perturbative contribution, and from we come to
2 4 - MQ e Q

OPE

() =

hadron

()

fM

2 Q

=

s e

ff

e- s

pert

( s, , mQ, ²) d s +

power

(, mQ , ²)

dual

(, ², seff )

(mQ +mu )2

Note: nonperturbative contributions are all referred to the ground state. Extraction of bound-state parameters is possible only if we fix seff by some "external" criterion.

For heavy-meson observables one faces two problems: 1. How to reliably calculate the truncated OPE for the correlator? 2. How to fix s
eff

and estimate the errors in the extracted value of fQ?


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OPE : heavy Spectral densities

quark pole mass or running mass ?

(mb, s, s) (mb, s, ) (mb(mb, s), s, ) (mb, s, ) (mb, s, s)
To 2 -accuracy, m 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
b, pole

= 4.83 GeV mb (mb ) = 4.20 GeV

i 0 i 1 i 2 Full Pert 35 s 40

§ In pole mass scheme poor convergence of perturbative expansion § In MS scheme the perturbative spectral density has negative region

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 power total
2 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 MS-scheme shows such hierarchy. Numerically, fP using pole mass fP using MS mass.


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Quark

hadron duality assumption :

fM

2 Q

2 4 - MQ Qe

=



seff e- spert (mQ +mu )2

( s, , mQ, ²) d s +

power

(, mQ, ²)

dual

(, ², 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 ? § 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".


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Our new algorithm for extracting ground

state parameters when MQ is known
n j=0

(i) Consider a set of Polynomial -dependent Ansaetze for seff : s(n)() = eff

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


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Extraction of fP: QCD vs potential model Potential Model (HO + Coulomb)
Edual T , z Eg 1.04 n0 1.02 1 0.98 0.96 0.2 0.4 0.6 0.8 1 T GeV
1 eff

QCD ( fB for mb(mb) = 4.20 GeV) ïï
Mdual M 1.02 1.01 1
B

T

n0

n2 n1

0.99

n 2n 3 n1 GeV 0.05 0.075 0.1 0.125 0.15 0.175 0.2
2

f

dual

T ,zeff T 0.082 0.08 0.078 0.076 0.074 0.072

GeV n1 exact n2 n0 T GeV
1

fdual MeV 220 215 210 205 200 n1 n3 n2

n0 GeV 0.05 0.075 0.1 0.125 0.15 0.175 0.2
2

0.2

0.4

0.6

0.8

1

Surprising? No: As soon as quark-hadron duality is implemented as a cut on the perturbative correlator, the extraction of the ground-state parameters in QCD and in potential model are very similar.


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Extraction of f
100

D

mc(mc) = 1.279 ‘ 0.013 GeV, ² = 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!


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Extraction of f
100

Ds

mc(mc) = 1.279 ‘ 0.013 GeV, ² = 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


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Extraction of fB. Problem 1: 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: [ ( ) ( )] mb - 4.245 GeV qq1/3 - 0.267 GeV ï dual fB (mb, qq, ² = mb) = 206.5 ‘ 4 - 37 ï +4 M eV, 0.1 GeV 0.01 GeV ‘ 10 MeV on mb 37 MeV on fB!


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Problem 2. The dependence on the renormalization scale ². Even with NNLO corrections to the correlator, the sensitivity to the choice of ² is rather large. This signals that NNNLO (4 loops) are non-negligible. Often, the contribution of the omitted higher orders is probed by the variation of the scale ². "Standard" in B-physics: mb/2 < ² < 2mb. But why?
i i s, 5 4 mb 2 3 2 1 0 1 i 0i 1i 2 24
f
2 dual

i i s, 5 4 mb 3 2 1 0 1 i 0i 1i 2 28 29 s 30
f
2 dual

i i s, 5 4 2mb 3 2 1 0 1 i 0i 1i 2 24 26 28 s 30
f
2 dual

25

26

27

18
,s0 , mb 0.08

20

22

s 15 17.5 20 22.5 25 27.5 30
,s0 , 2mb 0.08 0.06 0.04 0.02 0 0.02

,s0 , mb 2

0.08 0.06 0.04 0.02 0 0.02 i 0 i 1 i 2 power total 0.04 0.1 0.12 0.14 0.16 0.18

0.06 0.04 0.02 0 0.02

i 0 i 1 i 2 power total 0.04 0.1 0.12 0.14 0.16 0.18 0.04

i 0 i 1 i 2 power total 0.1 0.12 0.14 0.16 0.18

What is the relevant range of the ²-variation to probe higher-order contributions?


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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


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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


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Conclusions
The effective continuum threshold seff is an important ingredient of the method of dispersive sum rules which determines to a (very) large extent the numerical values of the extracted hadron parameter. Finding a criterion for fixing seff poses a problem in the method of sum rules. § seff depends on the external kinematical variables (e.g., momentum transfer in sum rules for 3point correlators and light-cone sum rules) and "unphysical" parameters (renormalization scale ², Borel parameter ). Borel-parameter -dependence of seff emerges naturally when trying to make quark-hadron duality more accurate. § We proposed a new algorithm for fixing -dependent seff , for those problems where the groundstate mass MQ is known. We have tested that our algorithm leads to more accurate values of ground-state parameters than the "standard" algorithms used in the context of dispersive sum rules before. Moreover, our algorithm allows one to probe "systematic" ( "intrinsic") uncertainties related to the limited accuracy of the extraction procedure in the method of QCD sum rules. § We reported the decay constants of D, D s, B, Bs mesons which along with the "statistical" errors related to the uncertainties in the QCD parameters, for the first time include realistic "systematic" errors. Many other results are to come.