QCD Sum Rules: from Quark Masses
to Hadronic Form Factors
Alexander Khodjamirian
UNIVERSITÄT
SIEGEN
“Approaches to Quantum Chromodynamics”,
Oberwölz, Austria, 8-12 Sept. 2008
QCD sum rules (SVZ)
[M.Shifman, A.Vainshtein and V.Zakharov (1979)]
•
Vacuum correlator of quark currents = hadronic sum
Z
d 4 x eiqx h0|T {j1 (x)j2 (0)}|0i =
X h0|j1 |hihh|j2 |0i
h
•
mh2 − q 2
Local operator product expansion (OPE)
p
|q 2 | Λ2QCD , x ∼ 1/
|q 2 | → 0, j1 (x)j2 (0) →
P
Cd (x 2 )Od (0)
d
Correlator ⊕ OPE → C0 (q 2 , mq , αs ) +
P
Cd (q 2 , mq , αs )h0|Od |0i
d=3,4,..
Cd - calculable , h0|Od |0i- vacuum condensates
• a rigorous dispersion relation,
in practice: truncate
P
d=3,4,5,6
Alexander Khodjamirian
and approximate
P
h
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
2 / 54
Light-cone sum rules (LCSR)
[I.Balitsky,V.Braun et al (1989); V.Chernyak, I.Zhitnisky (1989) ]
•
Z
a different type of correlator:
d 4 x eiqx h0|T {j1 (x)j2 (0)}|H(p)i =
(p2 = mH2 )
X h0|j1 |hihh|j2 |Hi
h
•
mh2 − (p − q)2
OPE near the light-cone,
|q 2 | ∼ |(p − q)2 |Λ2QCD , x 2 → 0, j1 (x)j2 (0) →
P
t
Ct (x 2 )Ot (x, 0),
R1
du exp(iupx)ϕH
t (u)
R
P
Correlator ⊕ OPE →
du Ct (q 2 , (p − q)2 , mq , αs , u)ϕH
t (u)
h0|Ot (x, 0)|H(p)i →
0
t
Ct - calculable, ϕH
t (u) - light-cone distribution amplitudes of H
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
3 / 54
Twofold use of QCD sum rules:
I. hadronic sum from experiment
⇒ QCD/OPE parameters:
mq , condensates, DA’s
II. correlator from OPE
⇒ hadronic matrix elements:
h0|j1 |hi, hh|j2 |Hi
B, D decay constants, B → π, ρ, D (∗) form factors
needed for flavour physics: VCKM determination
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
4 / 54
Some recent applications
Quark mass determination: ms with 5-loop accuracy
Decay constant of Ds -meson: is there a puzzle ?
B → π form factor from light-cone sum rule:
|Vub | determination
Sum rules with B-meson distribution amplitudes:
B → D (∗) form factors at large recoil
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
5 / 54
Quark mass determination
•
Light quark masses
mq ≡ mq (2 GeV), q = u, d, s
• less accurate than the other SM parameters:
in PDG ∼ 25% accuracy for ms ;
compared: ∼ 10% for mc and ∼ 2% for mb
• the reason: ΛQCD ∼ ms mu , md :
small influence of mu,d,s on hadronic observables
(exception: mπ,K ,η )
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
6 / 54
Light quark masses
• Chiral Perturbation Theory:
R=
ms
= 24.4 ± 1.5,
m̂
Q2 =
ms2 − m̂2
= (22.7 ± 0.8)2
md2 − mu2
[ Leutwyler, 1996 ]
m̂ = 12 (mu + md )
⇒
determine ms , obtain mu,d “for free “:
md =
Alexander Khodjamirian
ms ms R − 1
R − 1
,
m
=
1+
1
−
u
R
R
4Q2
4Q2
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
7 / 54
“Nonlattice” methods of mq determination
•
based on quark-current correlators and OPE:
positivity bounds
QCD (SVZ) sum rules
Finite-energy sum rules (FESR)
inclusive τ → sūντ decays
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
8 / 54
ms from QCD sum rules
• Correlator with scalar (pseudoscalar) currents:
jS(P) = ∂ µ s̄γµ (γ5 )q = (ms (+)
− mq )s̄(γ5 )q,
Π(P) (q 2 ) = i
•
R
(q = u, d)
n
o
d 4 x eiq·x h0|T jP (x)jP† (0) |0i
Dispersion relation (doubly differentiated)
for Π(P) (q 2 ) at Q 2 = −q 2 Λ2QCD :
[Π
(P)00
Z∞
2
(q )]OPE = 2
ds
ρ(P) (s)
,
(s − q 2 )3
0
P
ρ(P) (s) = h0|jP |K (q)ihK (q)|jP |0i
P
K
= {kaon ⊕ excitations} ⊕ quark-hadron duality
K
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
9 / 54
Diagrams contributing to OPE
#
q"
u
#
!
s
#
"
!
#
"
!
"
!
4
3
⊕O(α2
s ) ⊕ O(αs )⊕O(αs )
x
x
x
"
x
#
!
x x x
"
hq̄qi
!
hGGi
"
x x xx
!
hq̄Gqi
"
!
hq̄qi2
⊕O(αs )
O(αs4 ) calculated
Alexander Khodjamirian
[Baikov, Chetyrkin, Kühn (2005) ]
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
10 / 54
OPE for the pseudoscalar correlator
expansion in 1/(Q)d+2 , d = 0, 2, 4, 6
(
4
α i
2
X
3(m
+
m
)
s
s
u
2
C
[Π
1
+
(Q )]OPE =
0,i
2
2
π
8π Q
i=1


)
α i
X
ms2 
{d
=
4}
{d
=
6}
s 
−2 2 1 +
C2,i
+
+
π
Q
Q4
Q6
(P)00
i=1,2
{d = 4} ∼ {ms hq̄qi, hG2 i, O(ms4 ) } (1 ⊕ O(αs ))
{d = 6} ∼ ms hq̄Gqi, hq̄qi2
•
vacuum condensate densities:
O3 = q̄q, O4 =
Alexander Khodjamirian
a ,
Gaµν Gµν
h...i = h0|...|0i
O5 = q̄σµν (λa /2)Gaµν q, O6 = q̄Γa q q̄Γa q
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
11 / 54
Hierarchy in αs and d
00
•
Relative contributions to [Π(P) (M 2 )]OPE
(after Borel transformation Q 2 → M 2 )
00
(d)
rn (M 2 )
(d=0,2)
rn
=
(d)
{Π(P) (M 2 )}O(αns )
Π(P)00 (M 2 )
(2.5 GeV2 ) = 52.4%, 28.3%, 14.4%, 4.0%, −0.3%
(n = 0, 1, 2, 3, 4)
r (d=4,6) (2.5
2
GeV ) = 1.2%.
•
power suppressed corrections very small:
uncertainties of vacuum condensate densities inessential
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
12 / 54
The hadronic sum
• {K , K 2π, K ∗ π, ρK , ...}
→ 3-resonance ansatz {K , K1 (1460), K2 (1830))}
mK1 = 1460 MeV, ΓK1 = 260 MeV;
mK2 = 1830 MeV, ΓK2 = 250 MeV [PDG]
(P)
fK2 mK4 δ(mK2 − s)
ρhadr (s) =
+
X
i=1,2
fK2i mK4 i
1
π
ΓKi mKi
2
(s − mK )2 + (ΓKi mKi )2
!
•
decay constants: h0|jP |K (q)i = fK mK2 ,
fK = 159.8 MeV , fK1 ,K2 fK (ChPT)
fitted combining various moments of sum rules and/or FESR
[Kambor,Maltman ’03]
• use of quark-hadron duality for the continuum: negligible
contribution of states at s > mK2 2
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
13 / 54
Pseudoscalar sum rule to O(αs4 )
[Chetyrkin, A.K.,(2006)]
•
The result:
ms (2 GeV) = 105 ± 6
OPE
•
•
± 7
hadr
MeV ,
if the O(α4s ) terms are removed: ' 2 MeV increase of the central value
The scale dependence:
solid- O(α4s ), dashed- O(α3s )
116
114
ms(2 GeV)[MeV]
112
110
108
106
104
102
100
98
1.5
2
2.5
3
3.5
4
4.5
5
µ2[GeV2]
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
14 / 54
Recent ms determinations
Method, accuracy
OPE bound,
O(αs4 )
QCD SR (P), O(αs4 )
QCD SR (P), O(αs4 )
QCD SR (S), O(αs4 )
FESR (P), O(αs4 )
Lattice QCD, 2 ⊕ 1
ms (2GeV) [MeV]
> 76
Lattice QCD, 2 ⊕ 1
105 ± 6 ± 7
97+11
−8
88+9
−7
102± 8
87 ± 4
±4
90 ± 5 ± 4
Lattice QCD,2 ⊕ 1
+14.6
91.1−6.2
Lattice QCD,2 ⊕ 1
107.3 ± 4.4
± 9.7 ± 4.9
Alexander Khodjamirian
References
Baikov,Chetyrkin,
Kühn ’05
A.K., Chetyrkin ’05
Jamin, Oller, Pich ’06
Jamin, Oller, Pich ’06
Dominguez et al.’08
HPQCD’06
Mason et al.
MILC ’06
Bernard et al.
CP-PACS/JLQCD ’07
Ishikawa et al.
RBC /UKQCD ’08
Allton et al,
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
15 / 54
Comments
• all O(αs4 ) sum rule intervals agree with
the 2 ⊕ 1 lattice determinations ,
uncertainty in ms somewhat smaller than in PDG’08:
ms (2GeV ) = 104+26
−34 MeV
•
•
all ms determinations obey the OPE bound,
P, S sum rules:
no further need for improvement of OPE for Π(P,S) (q 2 )
• The hadronic sum in P -channel:
need more data on excited kaon (J P = 0− ) states
(resonances in K ππ and K ∗ π, ρK ),
accessible, e.g. in τ → K1 ντ , D → K1 lνl , B → K1,2 D (∗)
• scalar sum rules, a more complicated hadronic sum:
nonres. K π (J P = 0+ ) states important
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
16 / 54
c, b-quark masses from sum rules
•
recent determinations:
first moments of quarkonium sum rules (SVZ, full QCD),
O(αs3 ) accuracy achieved:
•
m̄c (m̄c )
[GeV]
m̄b (m̄b )
[GeV]
Reference
1.286± 0.013
4.164± 0.025
1.295± 0.015
4.205± 0.058
Kühn,Steinhauser,
Sturm ‘06
Boughezal,Czakon,
Schutzmeier ’06
PDG’08 : m̄c (m̄c ) = 1.27+0.07
−0.11 GeV,
Alexander Khodjamirian
+0.17
m̄b (m̄b ) = 4.20−0.07
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
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fDs : a Puzzle ?
l+
s
Ds
c
ν̄l
the hadronic matrix element:
2
(mc + ms )h0|s̄iγ5 c|Ds i = fDs mD
s
•
Experiment BR(Ds → µ+ νµ , τ + ντ ) :
fDs = 273 ± 10 MeV
[ world average, J.Rosner, S.Stone , review for PDG08 ]
fD = 205 ± 8.5 ± 2.5 MeV, fDs = 267.9 ± 8.2 ± 3.9 MeV
[ recent CLEO data, 0806.3921[hep-ex]], assuming Vcd = Vus
•
the latest lattice QCD results: [see also talk by Follana ]
fD = 208 ± 4 MeV , fDs = 241 ± 3 MeV
[E.Follana et al., HPQCD and UKQCD,2007]
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
18 / 54
QCD (SVZ) sum rule for fDs
• heavy meson decay constants, a well known application
[V. Novikov et al. Phys. Rep.(1978), T. Aliev,V.Eletsky (1983), Broadhurst(1981)]
•
Correlation function of two charmed-strange currents:
j(x) = (mc + ms )s̄(x)iγ5 c(x)
2
Z
Π(q ) = i
d 4 xeiqx h0 | T {j(x)j † (0)} | 0i
=
h0 | j|hihh|j † | 0i
mh2 − q 2
,D ∗ K ,...
X
h=Ds
s → d, Ds → D,
(c → b, D(s) → B(s) )
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
19 / 54
OPE diagrams
#
q
"
#
s
!
c
x
"
!
d=0
x
#
x
"
!
d=3
x
x x x
"
!
d=4
"
x x xx
!
d=5
"
!
d=6
• inputs: m̄c (see quark mass determination),
hq̄qi (GMOR), hs̄si, hG2 i (J/ψ SR )
[condensates, see for a review B.L.Ioffe, hep/ph 0502148 ]
⊕ quark-hadron duality for continuum and excited states
R∞
{D ∗ K , Ds0 ,...} → π1 ds ImΠOPE (s)exp(−s/M 2 )
s0
(Ds∗ π isospin violating)
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
20 / 54
QCD sum rule predictions for fD(s)
• improvement of OPE: O(αs2 ) correction to heavy-light correlator
[K.Chetyrkin, M. Steinhauser (2000)]
• fD = 195 ± 20 MeV
O(α2s ) QCD SR in HQET [ A.Penin,M.Steinhauser ‘01]
• sum rule analysis for fB , fBs ,
O(α2s ), MS mass [M.Jamin, B.Lange ‘01]
-didn’t quote any number for fD(s)
• fD = 203 ± 20 MeV , fDs = 235 ± 24 MeV
[S.Narison ‘02]
• fD = 177± 21 MeV , fDs = 205 ± 22 MeV
[FESR, O(α2s ), J.Bordes,J.Penarrocha, K.Schilcher (2005) ]
•
predicted fDs > fD , and fDs lower than exp.
f /fD = 1.15 ± 0.05 rather robust
Ds
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
21 / 54
a rigorous upper bound for fD(s)
•
from the same correlator/OPE :
2
2
4 e−mD /M + .... = Π(M 2 ; m , m , α , cond., µ, )
fD2 mD
c
s
s
• the correlator has a positive definite spectral density
q
2 /M 2
4 e−mD
fD < Π(M 2 )/(mD
)
⇒
• preliminary result
with O(α2s ), M > 1.0 GeV2 and µ > 1.5 GeV, OPE convergence
fD < 230 MeV , fDs < 260 MeV
• exp. result for fDs looks indeed unexpectedly large ...
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
22 / 54
Comments
•
Ds → lνγ with low Eγ < Emax,CLEO , a possible background ?
•
extra αem , ∼ ml2 suppression removed, involves two
long-distance form factors
•
LCSR estimates for D → lνγ form factors
at (pl + pν )2 ∼ 0 (large Eγ )
[A.K. G. Stoll, D. Wyler, 1995 ]
pole model at large (pl + pν )2 (small Eγ )
[G.Burdman, T.Goldman,D.Wyler ’94 ]
→ the effect is ∼ 1% for D, cannot be much larger for Ds
(study in progress)
•
semileptonic Ds decays with low energy photons and large lepton pair inv. mass
has to be taken under scrutiny
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
23 / 54
Light-Cone Sum Rules for B → π form factors
•
the correlator:
π
π
u
d
+
b
p+q
+...
b
q
Z
Alexander Khodjamirian
+
b
q 2 , (p+q)2 << mb2 ,
Fλ (q, p) = i
π
b-quark highly virtual ⇒ x 2 ∼ 0
d 4 xeiqx hπ(p) | T {ū(x)γλ b(x), b̄(0)iγ5 d(0)} | 0i
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
24 / 54
OPE near the light-cone
Z
F (q, p) = i
Z
+
0
1
(
d 4 xeiqx
h
i
S0 (x 2 , mb2 , µ) + αs S1 (x 2 , mb2 , µ)
⊗hπ(p) | ū(x)Γd(0) | 0i|µ
)
dv S̃(x 2 , mb2 , µ, v ) ⊗ hπ(p) | ū(x)G(vx)Γ̃d(0)} | 0i|µ
+ ...
• S0,1 , S̃ - perturbative amplitudes, (b-quark propagators)
• vacuum-pion matrix elements - expanded near x 2 = 0
⇒ universal distribution amplitudes of π :
Z
hπ(q)|ū(x)[x, 0]γµ γ5 d(0)|0ix 2 =0 = −iqµ fπ
1
du eiuqx ϕπ (u)+O(x 2 ) .
0
• the expansion goes over twists
√
• terms ∼ S̃ suppressed by powers of 1/ mb Λ;
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
25 / 54
The OPE result
2
2
F (q , (p+q) ) =
X
Z
du T (t) (q 2 , (p+q)2 , mb2 , αs , u, µ) ϕ(t)
π (u, µ)
t=2,3,4,..
hard scattering amplitudes ⊗ pion light-cone DA
- LO twist 2,3,4 q q̄ and q̄qG terms:
[V.Belyaev, A.K., R.Rückl (1993); V.Braun, V.Belyaev, A.K., R.Rückl (1996)]
-NLO O(αs ) twist 2, (collinear factorization)
[A.K., R.Rückl, S.Weinzierl, O. Yakovlev (1997); E.Bagan, P.Ball, V.Braun (1997);]
-NLO O(αs ) twist 3 (coll.factorization for asympt. DA)
[P. Ball, R. Zwicky (2001); G.Duplancic,A.K.,B.Melic, Th.Mannel,N.Offen (2007) ]
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
26 / 54
Distribution amplitudes (DA’s) of the pion
•
twist 2 DA: normalized with fπ ,
expansion in Gegenbauer polynomials

ϕπ (u, µ) = 6u(1 − u) 1 +

X
3/2
anπ (µ)Cn (2u − 1) ,
n=2,4,..
π
a2n
(µ) ∼ [Log(µ/ΛQCD )]−γ2n → 0
at µ → ∞
[Efremov-Radyushkin-Brodsky-Lepage evolution]
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
27 / 54
Gegenbauer moments at low scale
•
π (µ ), determined from:
essential parameters: a2,4
0
• matching exp. pion form factors to LCSR,
• two-point QCD sum rules,
• lattice QCD
•
a2π = 0.25 ± 0.15 (average. of recent determinations)
a2π + a4π = 0.1 ± 0.1 (pion-photon form factor)
•
remaining tw 3,4 DA parameters:
normalization constants and first moments,
determined mainly from two-point sum rules
[P. Ball, V.Braun, A.Lenz (2006) ]
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
28 / 54
Derivation of LCSR
•
Hadronic dispersion relation in the variable (p + q)2 :
π
F (q 2, (p + q)2) =
π
u
B
P
p+q
b
Bh
q
b
u
Bh
h
b
B
+
b
q
p+q
+
(q 2)
fB fBπ
P
Bh
→ duality (sB
0)
(q 2 , (p + q)2 mb2 )
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
29 / 54
Derivation of LCSR
•
matching OPE with disp. relation and using quark-hadron
duality
2
2
[F ((p + q) , q )]OPE
•
•
m2 fB f + (q 2 )
+
= 2B Bπ
mB − (p + q)2
Z∞
ds
[ImF (s, q 2 )]OPE
s − (p + q)2
s0B
(t)
inputs: mb , αs , ϕπ (u), t=2,3,4;
fB - determined from two-point (SVZ) sum rule;
uncertainties due to:
• variation of (universal) input parameters,
• quark-hadron duality
(suppressed with Borel transformation, controlled by the mB calculation)
•
•
LCSR contains both “soft” and “hard” contributions to fBπ
the method is used at finite mb
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
30 / 54
+
Recent update of LCSR for fBπ
(q 2 )
[ G. Duplancić, A.K., Th. Mannel, B. Melić, N. Offen, arXiv:0801.1796 [hep-ph],JHEP]
•
•
•
MS b quark mass used
twist-3 O(αs ) corrections recalculated
fitting the q 2 dependence to the measured slope:
a2π (1GeV) = 0.16 ± 0.01; a4π (1GeV) = 0.04 ∓ 0.01
2.4
2.2
2
fBΠ + Hq2 LfBΠ + H0L 1.8
1.6
1.4
1.2
0
2
4
6
8
10
12
q2 HGeV2 L
plot: LCSR vs BK parametrization of the BABAR data (almost indistinguishable):
we “trade” the q 2 6= 0 LCSR calculation for the accuracy
+
of the fBπ
(0) prediction
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
31 / 54
Extracting |Vub |
+
fBπ
(0) =
0.262 ± [0.005]fit ± [0.002]mb
h
i
+0.03
−0.02 m
q
± [0.002]M ± [0.001]µ ± ...
combining all individual uncertainties in quadrature:
+
+0.04
fBπ
(0) = 0.26−0.03
•
using |Vub f + (0)| from P. Ball’s fit of BaBar data:
|Vub | = 3.5[±0.4]th ± [0.2]shape ± [0.1]BR × 10−3 ,
• earlier LCSR result:
(with one-loop pole mass mb = 4.8 ± 0.1 GeV)
= 0.258 ± 0.031
[P.Ball, R.Zwicky(2004)]
+
(0)
fBπ
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
32 / 54
Recent |Vub | determinations from B → πlνl
[ref.]
Okamoto et al.
HPQCD
Arnesen et al.
Becher,Hill
Flynn et al
Bourrely,Caprini,
Lellouch
Ball, Zwicky
this work
Alexander Khodjamirian
+
fBπ
(q 2 )
calculation
lattice
(nf = 3)
lattice
(nf = 3)
-
LCSR
LCSR
+
fBπ
(q 2 )
input
-
3.78±0.25±0.52
-
3.55±0.25±0.50
lattice⊕SCET
lattice
lattice ⊕ LCSR
lattice
3.54 ± 0.17 ± 0.44
3.7 ± 0.2 ± 0.1
3.47 ± 0.29 ± 0.03
3.36 ± 0.23
-
3.5 ± 0.4 ± 0.1
3.5 ± 0.4 ± 0.2 ± 0.1
|Vub | × 103
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
33 / 54
LCSR with B-meson distribution amplitudes
[ A.K., T. Mannel, N.Offen,2005 ]
d
B
p
u
b
q
(a)
B
B
(b)
•
(c)
a similar approach: LCSR for B → π in SCET
[F.De Fazio, Th. Feldmann and T. Hurth, (2005)]
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
34 / 54
B-meson DA’s
h0|q̄2α (x)[x, 0]hv β (0)|B̄v i
"
(
) #
Z∞
φB+ (ω) − φB− (ω)
ifB mB
−iωv ·x
B
dωe
(1 + /
v ) φ+ (ω) −
/x γ5
=−
4
2v · x
0
•
defined in HQET; key input parameter: the inverse moment
Z ∞
φB (ω, µ)
1
=
dω +
λB (µ)
ω
0
•
QCD sum rules in HQET: λB (1 GeV) = 460 ± 110 MeV
βα
[V.Braun, D.Ivanov, G.Korchemsky,2004 ]
•
•
all B → π, K (∗) , ρ form factors calculated
so far only tree-level calculations, 2,3-particle DA’s
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
35 / 54
Form factors from LCSR with B-meson DA’s
form factor
this work
+
fBπ
(0)
0.25±0.05
0.258±0.031, (0.26+0.04
−0.03 )
+
fBK
(0)
0.31±0.04
0.301±0.041±0.008
T (0)
fBπ
0.21±0.04
0.253±0.028
T (0)
fBK
0.27±0.04
0.321±0.037±0.009
V Bρ (0)
0.32±0.10
0.323±0.029
∗
V BK (0)
0.39±0.11
0.411±0.033±0.031
ABρ
1 (0)
0.24±0.08
0.242±0.024
∗
ABK
(0)
1
0.30±0.08
0.292±0.028±0.023
ABρ
2 (0)
0.21±0.09
0.221±0.023
∗
ABK
(0)
2
0.26±0.08
0.259±0.027±0.022
T1Bρ (0)
0.28±0.09
0.267±0.021
∗
T1BK (0)
0.33±0.10
0.333±0.028±0.024
Alexander Khodjamirian
LCSR with light-meson DA’s
[P.Ball and R.Zwicky] ([Duplancic et al])
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
36 / 54
LCSR for B → D (∗) form factors
[S.Faller, A.K.,Ch.Klein, Th.Mannel, [hepph]]
d
B
b
(a)
p
c
B
q
(b)
•
virtual c quark in the correlator with B-meson DA
•
B → D, B → D ∗ form factors near maximal recoil
(not directly accessible in HQET)
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
37 / 54
B → D form factors
hD(p)|c̄γµ b|B̄(p + q)i
= (v + v 0 )µ h+ (w) + (v − v 0 )µ h− (w)
√
mB mD
2
2
mB2 + mD
(∗) − q
w =v ·v =
,
2 mB mD (∗)
0
G2 |Vcb |2
dΓ(B̄ → Dl ν̄l )
3
= F 3 (mB + mD )2 mD
(w 2 − 1)3/2 |G(w)|2 .
dw
48π
the two form factors h± are combined within a single function:
G(w) = h+ (w) −
Alexander Khodjamirian
1−r
h− (w) .
1+r
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
38 / 54
Result for B → D form factors
1.2
G(w)
1
0.8
0.6
0.4
0.2
1
1.1
1.2
1.3
1.4
1.5
1.6
w
LCSR prediction at w ∼ wmax compared with BaBar(2008) data fitted to
Caprini-Lelloch-Neubert-parametrization
•
B → D ∗ form factors calculated in the same region
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
39 / 54
Heavy quark limit of the new sum rules
•
heavy-quark symmetry relations obeyed
h+ (w) = hV (w) = hA1 (w) = hA3 (w) = ξ(w) ,
h− (w) = hA2 (w) = 0 ,
•
(1)
Isgur-Wise function given by the sum rule:
βZ0 /w
dρ exp
ξ(w) =
Λ̄ − ρw
τ
h
1 B i
1 B
φ− (ρ) + 1 −
φ (ρ) ,
2w
2w +
0
• valid at w ∼ wmax
• only tree level, radiative corrections nontrivial
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
40 / 54
Summary on QCD sum rules
•
initially (SVZ):
an analytical method to assess QCD vacuum effects,
(e.g., ∼ 99% of mnucleon is due to quark condensate [ B. Ioffe [1981]])
to understand why hadrons are not alike
•
not intended for very accurate calculations
of hadronic parameters
•
nowadays a useful working tool,
many applications to flavour physics
including new method for hadronic form factors (LCSR)
•
uncertainties will always remain and have to be identified
and estimated case by case
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
41 / 54
How accurate are QCD sum rules
•
two main sources of uncertainties:
(I) OPE: truncated, inputs uncertain
• a reasonable accuracy achieved in 2-point correlators,
due to progress in multiloop calculations,
• αs , quark masses, quark/gluon condensates, DA’s:
accuracy slowly improving
• in LCSR only NLO t ≤ 4 available,
twist expansion demands additional studies
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
42 / 54
(II) hadronic sum approximated
with quark-hadron duality
• more difficult,
the most safe predictions are bounds from OPE
• not easy to estimate the “systematic” error
related with effective threshold s0 :
(fitting s0 by adjusting the hadron mass)
• a better solution: experimental information
on excited states ⇒ the hadronic spectral function
• theoretical information on the spectrum (string-like models)
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
43 / 54
The old problem: Γ(J/ψ → ηc γ)
•
•
quark model: M1 transition (no overlap of wave functions)
QCD sum rules from three-point correlator with c̄c currents
Γ(J/ψ → ηc γ) = 2.2 ± 0.8 KeV (mc = 1.25GeV )
[ A.K. (1980); including gluon condensate correction (1984)]
•
use of disp. relation for ηc → 2γ amplitude:
[M. Shifman, 1979 ]
!
4
mJ/ψ
mη2c
2 Γ(ηc → 2γ)
1− 2
(1 + corr )
αem 3
Γ(J/ψ → ηc γ) =
9 Γ(J/ψ → e+ e− )
mη c
mJ/ψ
= 2.6 ± 1.1KeV (input : Γ(ηc → 2γ) = 6.4 ± 2.8KeV )
•
three-point sum rules including O(αs ) corrections:
Γ(J/ψ → ηc γ) = 2.6 ± 0.5KeV
[V.Beilin, A.Radyushkin (1985)]
•
the only measurement in 1977 [Crystal Ball] → PDG:
Γ(J/ψ → ηc γ) = 1.3 ± 0.4 keV
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
44 / 54
•
for many years considered a serious problem
for QCD sum rule approach
“QCD will not survive if mηc = 2.977 GeV and J/ψ → ηc γ rate is lower than 2 keV...
I expect that the experimental result will turn out close to 2 keV”
[M. Shifman, Z. Phys. (1980)]
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
45 / 54
•
for many years considered a serious problem
for QCD sum rule approach
“QCD will not survive if mηc = 2.977 GeV and J/ψ → ηc γ rate is lower than 2 keV... I
expect that the experimental result will turn out close to 2 keV”
[M. Shifman, Z. Phys. (1980)]
•
the CLEO new data (2008) [0805.0252, hep-ex]
BR(J/ψ → ηc γ) = 1.98 ± 0.09 ± 0.3%
Γ(J/ψ → ηc γ) = 1.84 ± 0.29 keV
no discrepancy with QCD sum rules!
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
46 / 54
BACKUP SLIDES
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
47 / 54
Coefficients multiplying (αs /π)n in d = 0 part: (lQ = log Q 2 /µ2 )
C0,1 =
C0,3 =
11
5071
35
139
17 2
− 2 lQ , C0,2 =
−
ζ3 −
lQ +
l ,
3
144
2
6
4 Q
1995097
π4
65869
715
2720
475
695 2 221 3
−
−
ζ3 +
ζ5 −
lQ +
ζ3 lQ +
l −
l ,
5184
36
216
12
9
4
8 Q
24 Q
most recent:
C0,4 =
−
+
Alexander Khodjamirian
2361295759
2915 4
25214831
192155 2
625
59875
−
π −
ζ3 +
ζ3 +
ζ5 −
ζ6
497664
10368
5184
216
108
48
»
–
52255
43647875
1 4
864685
24025
ζ7 + lQ −
+
π +
ζ3 −
ζ5
256
10368
18
288
48
»
–
»
–
»
–
16785
79333
7735
1778273
lQ2
−
ζ3 + lQ3 −
+ lQ4
,
1152
32
288
384
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
48 / 54
Quark-hadron duality
•
global duality for the hadronic sum (dispersion relation):
strict property based on asymptotic freedom of QCD
X h0|j1 |hihh|j2 |0i
h
mh2
−
q2
Z∞
≡
ρhadr (s)
1
ds
=
2
π
s−q
mh2
Z∞
ds
ImC0 (s)
s − q2
(mq +mq 0 )2
0
h0 - the lowest hadron with flavour {q̄q 0 },
( j1 = q̄Γq 0 , j2 = q¯0 Γq ),
(e.g., π, K , D, B )
•
local duality:
ρhadr (s) '
1
ImC0 (s),
π
an approximation valid at sufficiently large s mh20
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
49 / 54
“Semilocal” duality used in QCD sum rules
h0|j1 |h0 ihh0 |j2 |0i X h0|j1 |hihh|j2 |0i
+
mh20 − q 2
mh2 − q 2
h6=h
0
=
1
π
Zs0
ds
ImC0 (s) 1
+
π
s − q2
(mq1 +mq2 )2
Z∞
ds
ImC0 (s)
s − q2
s0
matching the sum of excited ⊕ continuum h-states
to the integral over ImC0
with s0 , the effective threshold
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
50 / 54
Comments on quark-hadron duality
•
•
“semilocal” duality is a weaker assumption than the local one
•
•
a systematic uncertainty is introduced in the sum rule
works for channels where the hadronic sum is measured
(J/ψ) or dominated by the lowest state (π, K )
Borel transformation
1
(mh2 −q 2 )
→ exp(−mh2 /M 2 )
suppresses the higer-state contributions to the hadronic sum,
the sum rule less sensitive to the duality approximation
•
no standard approach to fix s0 , e.g., calculating
the mass of h0 from the same sum rule by d/d(1/M 2 )
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
51 / 54
The sum rule result for fB
•
inputs: m̄b (see quark mass determination),
hq̄qi (PCAC), hG2 i (J/ψ SR ) ⊕ quark-hadron duality
•
the sum rule result with O(αs2 ) accuracy,
at m̄b (m̄b ) = 4.21 ± 0.05 GeV :
fB = 210 ± 19 MeV, fBs = 244 ± 21 MeV
[M.Jamin,B.O.Lange(2001)]
fB = 206 ± 20 MeV, (HQET)
[A.Penin, M.Steinhauser (2001)]
agree (within still large errors ) with experiment on B → τ νl (Vub - PDG average)
and with the lattice QCD determinations of fB(s)
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
52 / 54
B(s) → K form factors: an update
•
including ms in OPE → kaon DA’s (a1K )
[G. Duplancić, B. Melić , hep-ph 0806]
•
ratios (some uncertainties cancel)
+
fB+s K (0)
fBK
(0)
+0.11
+0.17
=
1.38
= 1.15−0.09
+
+
−0.10
fBπ
(0)
fBπ
(0)
•
relevant for B → Kll,
SU(3)fl relations for B → hh amplitudes:
ξ=
•
+
(mK2 ) mB2 − mπ2
fK fBπ
+0.07
= 1.01−0.15
.
fπ fB+s K (mπ2 ) mB2 s − mK2
close to previous LCSR estimates of SU(3)fl violation
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
53 / 54
B(s) → K form factors: an update
•
including ms in OPE → kaon DA’s (a1K )
[G. Duplancić, B. Melić , hep-ph 0806]
•
ratios (some uncertainties cancel)
+
fB+s K (0)
fBK
(0)
+0.11
+0.17
=
1.38
= 1.15−0.09
+
+
−0.10
fBπ
(0)
fBπ
(0)
•
relevant for B → Kll,
SU(3)fl relations for B → hh amplitudes:
ξ=
•
+
(mK2 ) mB2 − mπ2
fK fBπ
+0.07
= 1.01−0.15
.
fπ fB+s K (mπ2 ) mB2 s − mK2
close to previous LCSR estimates of SU(3)fl violation
Alexander Khodjamirian
QCD Sum Rules: from Quark Masses to Hadronic Form Factors
54 / 54