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

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 , D s , B , and B s 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 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 f P to the OPE parameters (quark masses, conden- sates,.. . ). The corresponding error is referred to as OPE uncertainty, or statistical error.

3 1. Basic object: � � � j 5 ( x ) j † Π ( p 2 ) = i dxe ipx � 0 | T 5 (0) | 0 � , j 5 ( x ) = ( m Q + m )¯ qi γ 5 Q ( x ) 2. Calculate OPE series and its Borel transform ( p 2 → τ ): Green functions in Minkowski space → evolution operator in Euclidean space, τ related to Euclidean time ∞ � e − s τ ρ pert ( s , α, m Q , µ ) ds + Π power ( τ, m Q , µ ) , Π ( τ ) = ( m Q + m u ) 2 � 2 ρ (2) ( s ) + · · · ρ pert ( s , µ ) = ρ (0) ( s ) + α s ( µ ) � α s ( µ ) π ρ (1) ( s ) + • π Π power ( τ, µ ) – power expansion in τ in terms of condensates. • Quark � hadron dualityassumption : s e ff Q τ = Q e − M 2 � f 2 Q M 4 e − s τ ρ pert ( s , α, m Q , µ ) ds + Π power ( τ, m Q , µ ) ≡ Π dual ( τ, µ, s e ff ) ( m Q + m u ) 2

4 OPE : heavy � quark pole mass or running mass? Spectral densities ρ ( m b , α s , s ) → Π ( m b , α s , τ ) → Π ( m b ( m b , α s ) , α s , τ ) → Π ( m b , α s , τ ) → ρ ( m b , α s , s ) To α 2 s -accuracy, m b , pole = 4 . 83 GeV ↔ m b ( m b ) = 4 . 20 GeV � Α � Π � i Ρ i � s � � Α � Π � i Ρ i � s, Μ� m �� � b � 5 5 ������ scheme Pole mass OPE MS 4 4 3 3 2 2 1 1 0 0 i � 0 i � 1 i � 2 Full Pert i � 0 i � 1 i � 2 Full Pert s s 20 25 30 35 40 20 25 30 35 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 f 2 dual � Τ ,s 0 � dual � Τ ,s 0 0.05 0.05 0.04 0.04 O � 1 � O � Α � O � Α 2 � power total 0.03 0.03 O � 1 � O � Α � O � Α 2 � power total 0.02 0.02 0.01 0.01 0 0 � 0.01 � 0.01 Τ Τ 0.1 0.12 0.14 0.16 0.18 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, f P using pole mass ≪ f P using MS mass.

5 OPE � mQ, GeV � fB, MeV Aliev � 1983 � O � � Α � 130 � � 20 � % � pole : 4.8 � : 4.05 ���� Narison � 2001 � � O � � Α 2 � � pole : 4.7 MS 203 � 23 OPE � : 4.21 � 0.05 ���� O � � Α 2 � Jamin � 2001 � pole : 4.83 MS 215 � 19 OPE � : 4.25 � 0.025 ���� O � � Α 2 � Our results MS 193 � 13 OPE � 4 syst

6 Quark � hadron dualityassumption : � s e ff Q τ = Q e − M 2 f 2 Q M 4 ( m Q + m u ) 2 e − s τ ρ pert ( s , α, m Q , µ ) ds + Π power ( τ, m Q , µ ) ≡ Π dual ( τ, µ, s e ff ) In order the l.h.s. and the r.h.s. have the same τ -behavior s eff is a function of Τ � and Μ � : s eff � Τ , Μ � dual ( τ ) = − d The “dual” mass: M 2 d τ log Π dual ( τ, s e ff ( τ )) . If quark-hadron duality is implemented “perfectly”, then M dual should be equal to M Q ; The deviation of M dual from the actual meson mass M Q measures the contamination of the dual correlator by excited states. Better reproduction of M Q → more accurate extraction of f Q . Taking into account τ -dependence of s e ff improves the accuracy of the duality approximation. Obviously, in order to predict f Q , we need to fix s e ff . How to fix s e ff ? • For a given trial function s e ff ( τ ) there exists a variational solution which minimizes the deviation of the dual mass from the actual meson mass in the τ -“window”.

7 Our new algorithm for extracting ground � state parameters when M Q is known n (i) Consider a set of Polynomial τ -dependent Ansaetze for s e ff : s ( n ) s ( n ) j ( τ ) j . � e ff ( τ ) = j = 0 (ii) Minimize the squared di ff erence between the “dual” mass M 2 dual and the known value M 2 Q in the τ -window. This gives us the parameters of the e ff ective 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 e ff ective thresholds as the characteristic of the intrinsic uncertainty of the extraction procedure. Illustration: D-meson M dual � M D f dual � MeV � 1.02 230 220 n � 2 n � 0 1.01 210 n � 1 n � 3 200 n � 1 1 n � 2 n � 3 190 n � 0 180 0.99 170 Τ � GeV � 2 � Τ � GeV � 2 � 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6

8 Extraction of f D m c ( m c ) = 1 . 279 ± 0 . 013 GeV , µ = 1 − 3 GeV . ()# &!! ;!'!&) ((# 7 8 191&'":(111111111345 <!'!&) %! (!# (## + ,-./01 $! !'# *+,-. !&# * #! !%# F < ,E,!G(%'-!)1,H/0 D * ,E,( D * ,E,) !$# "! 23+456 7899:3; I+H !"# <=>?@A>@ ! 4B/C/>B/>@ ! !'&% !'&( !'"! !'"& !'"" !'") !'"# / 0 123456 f D � 206.2 � 7.3 OPE � 5.1 syst MeV f D � � const � � 181.3 � 7 . 4 OPE MeV The e ff ect of τ -dependent threshold is visible!

9 Extraction of f Ds m c ( m c ) = 1 . 279 ± 0 . 013 GeV , µ = 1 − 3 GeV . &!! %## =!'!&) 8 9 2:2&'";<222222222456 >!'!&) %! $"# '( )*+,-. $! *+,-. #! & $## B 9 )A)!C$DE*!%.)F,- @ & )A)$ @ & )A)% "! /0'123 4566708 G'F !"# ! 9:;(<=;< ! 1>,?,;>,;< !'"! !'"( !')! / 01 234567 f D s � 246.5 � 15.7 OPE � 5 syst MeV f Ds � � const � � 218.8 � 16.1 OPE MeV

10 Extraction of f B : a very strong sensitivity to m b ( m b ) f B � MeV � 240 220 n � 0 n � 1 n � 2 n � 3 200 180 160 m b � GeV � 4.2 4.25 4.3 4.35 τ -dependent e ff ective threshold: qq � 1 / 3 − 0.267 GeV � � m b − 4.245 GeV � � � ¯ �� f dual ( m b , � ¯ qq � , µ = m b ) = 206 . 5 ± 4 − 37 + 4 MeV , B 0.1 GeV 0.01 GeV ± 10 MeV on m b → ∓ 37 MeV on f B !

11 The prediction for f B is not feasible without a very precise knowledge of m b : )! $!! $!! ;!*"( 8 9 2:2%*#!2222222456 4 5 .6.7%#8&.9.!%!#8.012 4 5 .6.7%$7".8.!%!$".012 (! <!*!( '! #"! #"! &! +,-./ '()*+ '()*+ %! #!! #!! $! #! "! "! "! ! ! ! !*!& !*"! !*"& !*#! !*#& !*$! !%!" !%#! !%#" !%$! !%$" !%&! !%!" !%#! !%#" !%$! !%$" !%&! 0 1 234567 , - ./0123 , - ./0123 = > ;&?!"<*!".)@,- %&# %%# ( )*+,-. %## ' !$# = > ;&?%&A*%A.)@,- : ' );)% : ' );)< �� � �� � 5677809 Our estimate : m b � � m b � � 4.245 � 0.025 GeV !"# /01234 f B � 193.4 � 12.3 OPE � 4.3 syst MeV f B � � const � � 184 � 13 OPE MeV

12 Extraction of f Bs )! '## 9 : 3;3%*#%&3<3!*!#&3567 (! > ? +=+%@!&',!&0+A./ $"# '! &! $&# +,-./ )* +,-./0 %! $%# $! ( #! $$# "! > ? +=+%@$%B,$B0+A./ < ( +=+$ < ( +=+' ! $## !*"& !*#! !*#& !*$! 123456 7899:2; 0 12 345678 !"# f Bs � 232.5 � 18.6 OPE � 2.4 syst MeV f Bs � � const � � 218 � 18 OPE MeV