[PPT] - OPE, heavy quark mass, and decayconstants of heavymesons from QCD PowerPoint Presentation

SLIDE 1

OPE, heavy quark mass, and decayconstants of heavymesons from QCD sum rules

Dmitri Melikhov

SINP, Moscow State University, Moscow, Russia

A new extraction of the decay constants of D, Ds, B, and Bs mesons from the two-point function

f 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

SLIDE 2

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. We make use of the three-loop OPE for the correlator by Chetyrkin et al, reorganized 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, conden- sates,.. . ). The corresponding error is referred to as OPE uncertainty, or statistical error.

SLIDE 3

3

1. Basic object:

Π(p2) = i

dxeipx0|T
j5(x)j†

5(0)

|0,

j5(x) = (mQ + m)¯ qiγ5Q(x)

2. Calculate OPE series and its Borel transform (p2 → τ):

Green functions in Minkowski space → evolution operator in Euclidean space, τ related to Euclidean time

Π(τ) =

∞

(mQ+mu)2

e−sτρpert(s, α, mQ, µ) ds + Πpower(τ, mQ, µ),

ρpert(s, µ) = ρ(0)(s) + αs(µ)

π ρ(1)(s) +

αs(µ)

π

2 ρ(2)(s) + · · ·

Πpower(τ, µ) – power expansion in τ in terms of condensates.

Quark hadron dualityassumption :

f 2

QM4 Qe−M2

Qτ =

seff

(mQ+mu)2

e−sτρpert(s, α, mQ, µ) ds + Πpower(τ, mQ, µ) ≡ Πdual(τ, µ, seff)

SLIDE 4

4

OPE : heavy quark pole mass or running mass? Spectral densities

ρ(mb, αs, s) → Π(mb, αs, τ) → Π(mb(mb, αs), αs, τ) → Π(mb, αs, τ) → ρ(mb, αs, s)

To α2

s-accuracy, mb,pole = 4.83 GeV ↔ mb(mb) = 4.20 GeV 20 25 30 35 40 s 1 2 3 4 5 ΑΠ i Ρis i0 i1 i2 Full Pert Pole mass OPE 20 25 30 35 40 s 1 2 3 4 5 ΑΠi Ρis,Μm

b

i0 i1 i2 Full Pert MS scheme

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

Extracted decay constant

0.1 0.12 0.14 0.16 0.18 Τ 0.01 0.01 0.02 0.03 0.04 0.05 f2

dualΤ,s0

O1 OΑ OΑ2 power total 0.1 0.12 0.14 0.16 0.18 Τ 0.01 0.01 0.02 0.03 0.04 0.05 f2

dualΤ,s0

O1 OΑ OΑ2 power total
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.

SLIDE 5

5

OPE mQ, GeV fB, MeV Aliev 1983 OΑ pole : 4.8 130 20% Narison 2001 OΑ2 pole : 4.7 MS

: 4.05

203 23OPE Jamin 2001 OΑ2 pole : 4.83 MS

: 4.21 0.05

215 19OPE Our results OΑ2 MS

: 4.25 0.025

193 13OPE 4syst

SLIDE 6

6

Quark hadron dualityassumption :

f 2

QM4 Qe−M2

Qτ =

seff

(mQ+mu)2 e−sτρpert(s, α, mQ, µ) ds + 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 and Μ : seff Τ, Μ

The “dual” mass: M2

dual(τ) = − d dτ log Πdual(τ, seff(τ)).

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?

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”.

SLIDE 7

7

Our new algorithm for extracting ground state parameters when MQ is known

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

eff(τ) = n

j=0

s(n)

j (τ)j.

(ii) Minimize the squared difference between the “dual” mass M2

dual and the known value M2 Q in

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

0.1 0.2 0.3 0.4 0.5 0.6 ΤGeV2 0.99 1 1.01 1.02 MdualMD n1 n0 n2 n3 0.1 0.2 0.3 0.4 0.5 0.6 ΤGeV2 170 180 190 200 210 220 230 fdualMeV n0 n1 n2 n3

SLIDE 8

8

Extraction of fD mc(mc) = 1.279 ± 0.013 GeV, µ = 1 − 3 GeV.

! "! #! $! %! &!! !'&% !'&( !'"! !'"& !'"" !'") !'"#

*+,-. /

0123456

7

8191&'":(111111111345 ;!'!&) <!'!&)

!"# !$# !%# !&# !'# (## (!# ((# ()#

*

+,-./01

23+456 7899:3;

<=>?@A>@ !4B/C/>B/>@

D

*,E,(

D

*,E,)

F

<,E,!G(%'-!)1,H/0

I+H

fD 206.2 7.3OPE 5.1syst MeV

fDconst 181.3 7.4OPE MeV

The effect of τ-dependent threshold is visible!

SLIDE 9

9

Extraction of fDs mc(mc) = 1.279 ± 0.013 GeV, µ = 1 − 3 GeV.

! "! #! $! %! &!! !'"! !'"( !')!

*+,-. /

01234567

8

92:2&'";<222222222456 =!'!&) >!'!&)

!"# $## $"# %##

&

'()*+,-.

/0'123 4566708

9:;(<=;< !1>,?,;>,;<

@

&)A)$

@

&)A)%

B

9)A)!C$DE*!%.)F,-

G'F

fDs 246.5 15.7OPE 5syst MeV

fDsconst 218.8 16.1OPE MeV

SLIDE 10

10

Extraction of fB: a very strong sensitivity to mb(mb) 4.2 4.25 4.3 4.35 mbGeV 160 180 200 220 240 fBMeV n0 n1 n2 n3 τ-dependent effective threshold: f dual

B

(mb, ¯ qq, µ = mb) =

206.5 ± 4 − 37

mb − 4.245 GeV 0.1 GeV

+ 4

¯ qq1/3 − 0.267 GeV 0.01 GeV

MeV,

± 10 MeV on mb → ∓ 37 MeV on fB!

SLIDE 11

11

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

! "! #! $! %! &! '! (! )! !*!& !*"! !*"& !*#! !*#& !*$!

+,-./

1234567

8

92:2%*#!2222222456 ;!*"( <!*!(

! "! #!! #"! $!! !%!" !%#! !%#" !%$! !%$" !%&!

'()*+ ,

./0123

4

5.6.7%#8&.9.!%!#8.012

! "! #!! #"! $!! !%!" !%#! !%#" !%$! !%$" !%&!

'()*+ ,

./0123

4

5.6.7%$7".8.!%!$".012

!"# !$# %## %%# %&#

'

()*+,-.

/01234 5677809

:

');)%

:

');)<

=

>;&?!"<*!".)@,-

=

>;&?%&A*%A.)@,-

Our estimate : m

bm
b 4.245 0.025 GeV

fB 193.4 12.3OPE 4.3syst MeV

fBconst 184 13OPE MeV

SLIDE 12

12

Extraction of fBs

! "! #! $! %! &! '! (! )! !*"& !*#! !*#& !*$!

+,-./

12345678

9

:3;3%*#%&3<3!*!#&3567

!"# $## $$# $%# $&# $"# '##

(

)*+,-./0

123456 7899:2;

<

(+=+$

<

(+=+'

>

?+=+%@!&',!&0+A./

>

?+=+%@$%B,$B0+A./

fBs 232.5 18.6OPE 2.4syst MeV

fBsconst 218 18OPE MeV

SLIDE 13

13

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 accu-
rate. 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

f 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?

OPE, heavy quark mass, and decayconstants of heavymesons from QCD sum rules

Dmitri Melikhov

SINP, Moscow State University, Moscow, Russia

A new extraction of the decay constants of D, Ds, B, and Bs mesons from the two-point function

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

A QCD sum-rule calculation of hadron parameters involves two steps:

Π(p2) = i

j5(x) = (mQ + m)¯ qiγ5Q(x)

Green functions in Minkowski space → evolution operator in Euclidean space, τ related to Euclidean time

Π(τ) =

e−sτρpert(s, α, mQ, µ) ds + Πpower(τ, mQ, µ),

αs(µ)

2 ρ(2)(s) + · · ·

Quark hadron dualityassumption :

f 2

e−sτρpert(s, α, mQ, µ) ds + Πpower(τ, mQ, µ) ≡ Πdual(τ, µ, seff)

OPE : heavy quark pole mass or running mass? Spectral densities

ρ(mb, αs, s) → Π(mb, αs, τ) → Π(mb(mb, αs), αs, τ) → Π(mb, αs, τ) → ρ(mb, αs, s)

To α2

Extracted decay constant

OPE mQ, GeV fB, MeV Aliev 1983 OΑ pole : 4.8 130 20% Narison 2001 OΑ2 pole : 4.7 MS

203 23OPE Jamin 2001 OΑ2 pole : 4.83 MS

215 19OPE Our results OΑ2 MS

193 13OPE 4syst

Quark hadron dualityassumption :

f 2

seff

In order the l.h.s. and the r.h.s. have the same τ-behavior

seff is a function of Τ and Μ : seff Τ, Μ

The “dual” mass: M2

the dual mass from the actual meson mass in the τ-“window”.

Our new algorithm for extracting ground state parameters when MQ is known

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

s(n)

(ii) Minimize the squared difference between the “dual” mass M2

Extraction of fD mc(mc) = 1.279 ± 0.013 GeV, µ = 1 − 3 GeV.

fD 206.2 7.3OPE 5.1syst MeV

fDconst 181.3 7.4OPE MeV

The effect of τ-dependent threshold is visible!

Extraction of fDs mc(mc) = 1.279 ± 0.013 GeV, µ = 1 − 3 GeV.

fDs 246.5 15.7OPE 5syst MeV

fDsconst 218.8 16.1OPE MeV

Extraction of fB: a very strong sensitivity to mb(mb) 4.2 4.25 4.3 4.35 mbGeV 160 180 200 220 240 fBMeV n0 n1 n2 n3 τ-dependent effective threshold: f dual

(mb, ¯ qq, µ = mb) =

mb − 4.245 GeV 0.1 GeV

¯ qq1/3 − 0.267 GeV 0.01 GeV

± 10 MeV on mb → ∓ 37 MeV on fB!

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

Our estimate : m

fB 193.4 12.3OPE 4.3syst MeV

fBconst 184 13OPE MeV

Extraction of fBs

fBs 232.5 18.6OPE 2.4syst MeV

fBsconst 218 18OPE MeV

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.

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.

the extracted parameters of the bound states.

“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

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?

eff

fB

+ x x

M

2 B 2 b

m

2 π B

cont

s =(M +m )

Im (s) Π s

theoretical physical

s