Heavy quark masses and heavy meson decayconstants from Borel sum - PowerPoint PPT Presentation

Heavy � quark masses and heavy � meson decayconstants from Borel sum rules in QCD Wolfgang Lucha 1 , Dmitri Melikhov 1 , 2 , Silvano Simula 3 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 B s 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 ex- tracted 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, reshu ffl ed 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 � � � j 5 ( x ) j † Basic object: OPE for Π ( p 2 ) = i dxe ipx � 0 | T 5 (0) | 0 � , j 5 ( x ) = ( m Q + m )¯ qi γ 5 Q ( x ) and its Borel transform ( p 2 → τ ). 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 ?

4 Our new algorithm for extracting ground � state parameters when M Q is known 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” (only a few lowest-dimension power corrections are known, work at τ m Q ≤ 1 ). 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

5 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!

6 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

7 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 !

8 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

9 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

10 Conclusions The e ff ective continuum threshold s e ff 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 s e ff poses a problem in the method of sum rules. • τ -dependence of s e ff emerges naturally when trying to make quark-hadron duality more accu- rate. For those cases where the ground-state mass M Q is known, we proposed a new algorithm for fixing s e ff . 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 s e ff 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 f Q 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 f B allowed us to obtain a rather accurate estimate m b ( m b ) = 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 α 2 s -accuracy, m b , pole = 4 . 83 GeV ↔ m b ( m b ) = 4 . 20 GeV: 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 ) � 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 pert. spectral density has negative regions → higher orders NOT negligible 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.