SVZ Sum Rules: 31 years later
QCD 10 - 25th anniversary
Montpellier 28th june - 2nd july 2010
Stephan Narison
CNRS / IN2P3 - LPTA / LPUM (Montpellier)
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 1/2
Contents
♣ The SVZ sum rules
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 2/2
Contents
♣ The SVZ sum rules
♦ Different forms of the sum rules
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 2/2
Contents
♣ The SVZ sum rules
♦ Different forms of the sum rules
♥ The SVZ-OPE: Adler function in e+ e−
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 2/2
Contents
♣ The SVZ sum rules
♦ Different forms of the sum rules
♥ The SVZ-OPE: Adler function in e+ e−
♠ Theoretical progresses
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 2/2
Contents
♣ The SVZ sum rules
♦ Different forms of the sum rules
♥ The SVZ-OPE: Adler function in e+ e−
♠ Theoretical progresses
♣ Large order PT terms ∆N ≡ ∑n>N cn αns
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 2/2
Contents
♣ The SVZ sum rules
♦ Different forms of the sum rules
♥ The SVZ-OPE: Adler function in e+ e−
♠ Theoretical progresses
♣ Large order PT terms ∆N ≡ ∑n>N cn αns
♦ Traditional Hadron phenomenology
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 2/2
Contents
♣ The SVZ sum rules
♦ Different forms of the sum rules
♥ The SVZ-OPE: Adler function in e+ e−
♠ Theoretical progresses
♣ Large order PT terms ∆N ≡ ∑n>N cn αns
♦ Traditional Hadron phenomenology
♥ PT QCD parameters
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 2/2
Contents
♣ The SVZ sum rules
♦ Different forms of the sum rules
♥ The SVZ-OPE: Adler function in e+ e−
♠ Theoretical progresses
♣ Large order PT terms ∆N ≡ ∑n>N cn αns
♦ Traditional Hadron phenomenology
♥ PT QCD parameters
♠ SVZ power corrections
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 2/2
Contents
♣ The SVZ sum rules
♦ Different forms of the sum rules
♥ The SVZ-OPE: Adler function in e+ e−
♠ Theoretical progresses
♣ Large order PT terms ∆N ≡ ∑n>N cn αns
♦ Traditional Hadron phenomenology
♥ PT QCD parameters
♠ SVZ power corrections
♣ Correlated estimate of 1/Q2
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 2/2
Contents
♣ The SVZ sum rules
♦ Different forms of the sum rules
♥ The SVZ-OPE: Adler function in e+ e−
♠ Theoretical progresses
♣ Large order PT terms ∆N ≡ ∑n>N cn αns
♦ Traditional Hadron phenomenology
♥ PT QCD parameters
♠ SVZ power corrections
♣ Correlated estimate of 1/Q2
♦ Phenomenology of 1/Q2
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 2/2
Contents
♣ The SVZ sum rules
♦ Different forms of the sum rules
♥ The SVZ-OPE: Adler function in e+ e−
♠ Theoretical progresses
♣ Large order PT terms ∆N ≡ ∑n>N cn αns
♦ Traditional Hadron phenomenology
♥ PT QCD parameters
♠ SVZ power corrections
♣ Correlated estimate of 1/Q2
♦ Phenomenology of 1/Q2
♥ Conclusions
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 2/2
The SVZ sum rules
♣ Outstanding Discovery of the 20th century
Shifman Vainshtein Zakharov : Sakurai’s price 1999
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 3/2
The SVZ sum rules
♣ Outstanding Discovery of the 20th century
Shifman Vainshtein Zakharov : Sakurai’s price 1999
♥ Bridge between High AND Low energy QCD regions
ΠH (Q2 ≡ −q2 ) ≡ i
=
Z
Z ∞
t<
QCD OPE
JH (x)
d 4 x h0|T JH (x)JH† (0)|0i
dt
ImΠ(t) + subtraction terms
t + Q2 + iε
= EXP DATA
:
Hadronic current
ψ̄Γψ, ψψψ, αs G2 , gψ̄Gψ, ψ̄Γ1 ψψ̄Γ2 ψ, ...
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 3/2
Different Forms of the Sum Rules
♣ Improvement of the dispersion relation
• Exponential (Borel/Laplace)
Z ∞
SVZ
d
dt exp ImΠ(t) R (τ) ≡ − log L (τ)
L (τ) =
dτ
t<
exponential enhances the low energy contribution.
−tτ
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 4/2
Different Forms of the Sum Rules
♣ Improvement of the dispersion relation
• Exponential (Borel/Laplace)
Z ∞
SVZ
d
dt exp ImΠ(t) R (τ) ≡ − log L (τ)
L (τ) =
dτ
t<
exponential enhances the low energy contribution.
• Moments & FESR
−tτ
Mn =
Z tc
dt t n ImΠ(t)
t<
n≥0
Moments : n ≤ 0 heavy quarks SVZ
FESR : n ≥ 0 light quarks Lugunov-Soloviev-Tavkhelidze 67,
Bramon-Etim-Greco 72, Shankar 77, Floratos-SN-de Rafael 79,
Krasnikov-Pivovarov-Tavkhelidze 83, Bertlmann-Launer-de Rafael (BLR) 85
not appropriate for
extracting (accurately) power corrections : difference of
large numbers
+ Dominguez-Loewe-Perrottet (BDLPR) 88
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 4/2
♦ Some other QCD spectral sum rules
• Gaussian sum rules BLR 85
G(s, σ) =
Z ∞
t<
dt exp
−
(t+s)2
σ
ImΠ(t)
Gaussian centered at s with finite width resolution
√
4πσ
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 5/2
♦ Some other QCD spectral sum rules
• Gaussian sum rules BLR 85
G(s, σ) =
Z ∞
dt exp
t<
−
(t+s)2
σ
ImΠ(t)
Gaussian centered at s with finite width resolution
• τ-like decays
√
4πσ
Braaten - SN - Pich 88, 92, Lediberder - Pich 92
Rnm (Mτ2 ) =
Z M2
τ
0
dt t n 1 −
t
Mτ2
m
ImΠ(t)
Threshold : suppression of contribution near the real axis.
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 5/2
♦ Some other QCD spectral sum rules
• Gaussian sum rules BLR 85
G(s, σ) =
Z ∞
dt exp
t<
−
(t+s)2
σ
ImΠ(t)
Gaussian centered at s with finite width resolution
• τ-like decays
√
4πσ
Braaten - SN - Pich 88, 92, Lediberder - Pich 92
Rnm (Mτ2 ) =
Z M2
τ
0
dt t n 1 −
t
Mτ2
m
ImΠ(t)
Threshold : suppression of contribution near the real axis.
• φ-meson-like sum rule SN
R(tc ) =
Z tc
t<
dt
t
1−2
tc
ImΠ(t)
Suppression of the leading tc -term after integration.
• analytic continuation Mennessier-Causse, Pennarocha et al...
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 5/2
The SVZ-OPE: Adler function in e+e−
♣ D(Q2 ) known in QCD using the SVZ-OPE
C2p h0|O2p |0i
2
2
2 d
Π(Q ) = ∑
D(Q ) ≡ −Q
2
2p
dQ
Q
p=0,1,2,...
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 6/2
The SVZ-OPE: Adler function in e+e−
♣ D(Q2 ) known in QCD using the SVZ-OPE
C2p h0|O2p |0i
2
2
2 d
Π(Q ) = ∑
D(Q ) ≡ −Q
2
2p
dQ
Q
p=0,1,2,...
♦ Anatomy of the OPE in terms of QCD parameters
• p = 0: usual PT series (as ≡ αs /π):
C0 ≡ ∑n cn αns = 1 + as + 1.640 a2s + 6.371 a3s + 49.076 a4s +
∆N ≡ ∑n>N cn αns ??
• p = 1: m̄2q : small corrections
• p = 2: hαs G2 i, mq hψ̄q ψq i: gluon and quark condensates
• p = 3: αs hψ̄q ψq i2 : four-quark condensates
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 6/2
Theoretical Progresses
♣ 1st Step
• Quant. Mec. & Non-Rel.
Bell-Bertlmann (BB), NSVZ 81
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 7/2
Theoretical Progresses
♣ 1st Step
• Quant. Mec. & Non-Rel.
Bell-Bertlmann (BB), NSVZ 81
• Radiative corrections into Exponential SR :
Inverse Laplace NOT Borel transform SN- de Rafael 1981
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 7/2
♣ 2nd Step
• Mixing of operators under renormalization
Tarrach 82, Espriu-Tarrach 82, SN-Tarrach 83,...
• Absorption of Mass singularities into the quark condensate
Becchi-SN-de Rafael-Yndurain (BNRY) 81, Broadhurst-Generalis (BG) 84,
Bagan-Latorre-Pascual 86, BNP 92, Jamin-Munz 95,
Chetyrkin-Steinhauser 01
• High-dimension gluon condensates
Nikolaev-Radyushkin (NR) 83, BG 84, Bagan-Pascual-Tarrach 85,...
• Higher order PT corrections Chetyrkin et al...
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 8/2
♣ 2nd Step
• Mixing of operators under renormalization
Tarrach 82, Espriu-Tarrach 82, SN-Tarrach 83,...
• Absorption of Mass singularities into the quark condensate
Becchi-SN-de Rafael-Yndurain (BNRY) 81, Broadhurst-Generalis (BG) 84,
Bagan-Latorre-Pascual 86, BNP 92, Jamin-Munz 95,
Chetyrkin-Steinhauser 01
• High-dimension gluon condensates
Nikolaev-Radyushkin (NR) 83, BG 84, Bagan-Pascual-Tarrach 85,...
• Higher order PT corrections Chetyrkin et al...
♥ 3rd step
• ChPT constraints on the pion spectral function
Bijnens-Prades-de Rafael 95
• Question on duality violation for spectral function
Shifman 00, Cata-Golterman-Peris 09, G-Alonso-Prades-Pich, de Rafael,
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 8/2
Large order terms: ∆N ≡
n
∑n>N cnαs
♣ IR renormalon chain ?
• cn ∼ n! : usual folklore: eaten by the PT part of hαs G2 i
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 9/2
Large order terms: ∆N ≡
n
∑n>N cnαs
♣ IR renormalon chain ?
• cn ∼ n! : usual folklore: eaten by the PT part of hαs G2 i
♦ UV renormalon (large β approximation) ?
• cn alternate signs : not yet seen here at α4s nor
in the lattice calculation of hαs G2 i to order α16
s ! Rakow
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 9/2
Large order terms: ∆N ≡
n
∑n>N cnαs
♣ IR renormalon chain ?
• cn ∼ n! : usual folklore: eaten by the PT part of hαs G2 i
♦ UV renormalon (large β approximation) ?
• cn alternate signs : not yet seen here at α4s nor
in the lattice calculation of hαs G2 i to order α16
s ! Rakow
♥ Geometricsum instead ? (no sign of IR renormalon !)
• cn ∼ cn−1 cn−1 : 6= pert. calc. to O (α4s ) Chetyrkin et al. ;
cn−2
Lattice calc. hαs G2 i to O (α16
s ) Rakow
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 9/2
Large order terms: ∆N ≡
n
∑n>N cnαs
♣ IR renormalon chain ?
• cn ∼ n! : usual folklore: eaten by the PT part of hαs G2 i
♦ UV renormalon (large β approximation) ?
• cn alternate signs : not yet seen here at α4s nor
in the lattice calculation of hαs G2 i to order α16
s ! Rakow
♥ Geometricsum instead ? (no sign of IR renormalon !)
• cn ∼ cn−1 cn−1 : 6= pert. calc. to O (α4s ) Chetyrkin et al. ;
cn−2
Lattice calc. hαs G2 i to O (α16
s ) Rakow
♠ Dual to 1/Q2 from a tachyonic gluon mass at short distance ?
• ∑n>N cn αns ≃ Cas λ2 /Q2
λ2 ≤ 0 : N ≈ 10 Lattice
Zakharov-Gubarev-Polikarpov, Chetyrkin-SN-Zakharov 95, SN-Zakharov 09
• Short PT series + power corrections ≡ long PT series.
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 9/2
Large order terms: ∆N ≡
n
∑n>N cnαs
♣ IR renormalon chain ?
• cn ∼ n! : usual folklore: eaten by the PT part of hαs G2 i
♦ UV renormalon (large β approximation) ?
• cn alternate signs : not yet seen here at α4s nor
in the lattice calculation of hαs G2 i to order α16
s ! Rakow
♥ Geometricsum instead ? (no sign of IR renormalon !)
• cn ∼ cn−1 cn−1 : 6= pert. calc. to O (α4s ) Chetyrkin et al. ;
cn−2
Lattice calc. hαs G2 i to O (α16
s ) Rakow
♠ Dual to 1/Q2 from a tachyonic gluon mass at short distance ?
• ∑n>N cn αns ≃ Cas λ2 /Q2
λ2 ≤ 0 : N ≈ 10 Lattice
Zakharov-Gubarev-Polikarpov, Chetyrkin-SN-Zakharov 95, SN-Zakharov 09
• Short PT series + power corrections ≡ long PT series.
♣ Similarity with Veneziano duality amplitude !
• A (a + b → c + d) = s channel + t channel exchanges
Only sum on s or t but not on both channels are allowed.
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 9/2
Traditional Hadron Phenomenology
♣ Impressive wide areas of applications: kalashnikov !
• ρ meson, gluon condensate, charm mass since 1979: SVZ
• Meson spectroscopy since Reinders-Rubinstein-Yazaki 81
• Light quark masses since Becchi-SN-de Rafael-Yndurain 81,
SN-Paver-de Rafael-Treleani 83,...
• Corrections to π and K PCAC SN 81,
Dominguez-de Rafael 87, SN book 89, Dominguez
• Heavy quark masses since SVZ 79, SN 87, Chetyrkin et al,
Ioffe-Zyablyuk, SN 10,...,
• Light Baryons since 1981: Ioffe, Dosch et al.
• Heavy Baryons Bagan-Chabab-Dosch-SN 92-93, Albuquerque
• Gluonium since Novikov SVZ 80, SN 83, SN-Veneziano 89
• Light Hybrids Latorre-Pascual-SN 87, Chetyrkin-SN 00, SN 09
• Heavy Hybrids Govaerts-Reinders-Rubinstein-Weyers 85, SN-book 04
• Four-quarks, molecules Latorre-Pascual 85, SN 86;
•
•
•
Sao Paolo : Nielsen et al.; Matheus-SN-Nielsen-Richard 07
Hadronic decays Vertex Navarra, Zanetti Light Cone De Fazio
τ-decay BNP 92, Pich-Lediberder, Prades-Pich, G.-Alonso, Valenzuela
Thermal Hadron Bochkarev-Shaposhnikov 86, Dosch-SN 88, Loewe
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 10/2
Traditional Hadron Phenomenology
♣ Impressive wide areas of applications: kalashnikov !
• ρ meson, gluon condensate, charm mass since 1979: SVZ
• Meson spectroscopy since Reinders-Rubinstein-Yazaki 81
• Light quark masses since Becchi-SN-de Rafael-Yndurain 81,
SN-Paver-de Rafael-Treleani 83,...
• Corrections to π and K PCAC SN 81,
Dominguez-de Rafael 87, SN book 89, Dominguez
• Heavy quark masses since SVZ 79, SN 87, Chetyrkin et al,
Ioffe-Zyablyuk, SN 10,...,
• Light Baryons since 1981: Ioffe, Dosch et al.
• Heavy Baryons Bagan-Chabab-Dosch-SN 92-93, Albuquerque
• Gluonium since Novikov SVZ 80, SN 83, SN-Veneziano 89
• Light Hybrids Latorre-Pascual-SN 87, Chetyrkin-SN 00, SN 09
• Heavy Hybrids Govaerts-Reinders-Rubinstein-Weyers 85, SN-book 04
• Four-quarks, molecules Latorre-Pascual 85, SN 86;
•
•
•
Sao Paolo : Nielsen et al.; Matheus-SN-Nielsen-Richard 07
Hadronic decays Vertex Navarra, Zanetti Light Cone De Fazio
τ-decay BNP 92, Pich-Lediberder, Prades-Pich, G.-Alonso, Valenzuela
Thermal Hadron Bochkarev-Shaposhnikov 86, Dosch-SN 88, Loewe
♥ SN books : WSC 89, Cambridge 04
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 10/2
PT QCD parameters
♣
Param
Values
Sources
0.1192(10)
τ-decay SN 09
QCD coupling
αs (MZ )
[ 0.1191(27)
Z-width]
0.1189(10) average (Bethke, PDG)
Quark masses
mu (2 GeV)
md (2 GeV)
ms (2 GeV)
mc (mc )
mb (mb )
[MeV] O (α3s )
(2.8 ± 0.2)
(5.1 ± 0.2)
(96.1 ± 4.8)
(1260 ± 18)
(4220 ± 17)
average SN09
"
"
J/ψ Q2 = 0 (r2/3 , r3/4 ), 4m2Q (r8/9 , r8/10 )
ϒ 8m2Q (r13/14 , r13/15 ), SN10
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 11/2
SVZ power corrections SNB 04 Chap. 27
♦
d
NPQCD param.
Values [GeV]d
Sources
3
1
¯
2 hūu + ddi(2)
-(0.254 ± .015)3
(pseudo)scal,
GMOR: δπ × 10−2
4
5 ± 0.5
¯
hddi/h
ūui
¯
hs̄si/hddi
1 − 9 × 10−3
hαs G2 i
(7 ± 1)10−2
Lattice
λa
non-norm. ord.(pseudo)scal,
0.74 ± 0.03
µν
5
ghψ̄σµν
ψGa i
6
g3 fabc hGa Gb Gc i
2
1/3β
≡ M02 αs 1 hψ̄ψi
ραs hψ̄ψi2
LSR (less sens αs ) (FESR)
(6.2 ± 1.6)
non-norm. ord.(pseudo)scal
⊕ light & heavy baryons
e+ e− , ϒ − ηb , J/ψ Laplace
Adriano 85, Rakow 09
τ, J/ψ mom : unconclusive
M02 = (0.80 ± 0.02)
Light baryons, B, B∗
(31 ± 13)hαs G2 i
J/ψ-mom
(4.5 ± 0.3)10−4
SN10
ρ = 2.1 ± 0.2
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 12/2
Correlated Estimate of 1/Q2
♣ 1/Q2 −term : tachyonic gluon mass
Channels
e+ e− data
R (τ)
π−sum rule
Lπ
Average
τ-decay : large β − ∑41 PT series
Fixed Order PT
Contour Improved
SN95a, 95b, 09
−(αs /π)λ2 in GeV2 × 102
6.5 ± 0.5
12 ± 6
7±3
2.6 ± 0.8
5.9 ± 0.8
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 13/2
Phenomenology of 1/Q2
♣ Scale for violations of the OPE by power corrections
Chetyrkin-SN-Zakharov 95
• ρ-meson channel :
Π(M 2 ) = (parton
h
model) 1 +
π
2 i−1.05 αs λ2
hα
G
s
π Mρ2
3M 4
q
Mρ2 ≈ 10 π3 hαs G2 i ≈ (0.6 ∼ 0.8) GeV2
i
+ ... =⇒
(Correction ≤ 10%)
• π-meson channel : positivity of spectral function =⇒
r
2 m4
2
2
f
α
2
π π
s λ
Mπ2 ≥ 16π
3 (m +m )2 1 + 4 π M 2 ≈ 1.8 GeV
u
d
• 0++ gluonium channel :
π
h
Π(MG2 ) − Π(0) = (parton model) 1 +
8π
−β1
2
π
αs
hαs G2 i
4
MG
=⇒ MG2 ≈ 20Mρ2 ≈ 15 GeV2
i
h
2
Π(M 2 ) = (parton model) 1−3 Mλ 2 =⇒ MG2 ≈ 15 GeV2
+ ...
i
G
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 14/2
♦ αs from τ-decay SN09
Rτ
≡
Γ(τ → ντ + hadrons|∆S=0 )
Γ(τ → l + ν̄l + ντ )
(2)
= 3|Vud |2 SEW (1 + δ(0) + δ′EW + δm + δsvz + δNST )
Size ×103
Corrections
δsvz = ∑84 δ(D)
(2)
δST ≡ δsvz + δm + δπ + δa0
δinst
δDV
δtach ≡ large β − ∑41 PT
δNST ≡ δinst + δDV + δtach
−(7.8 ± 1.0)
−(10.9 ± 1.1)
−(0.7 ± 2.7)/20
−(15 ± 9) Cata-Golterman-Peris
(17 ± 5) FO
(39 ± 5) CI
(2.0 ± 9.4) FO
(24.0 ± 10.6) CI
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 15/2
Rτ,V +A |exp = 3.479 ± 0.011
=⇒
αs (Mτ ) = 0.3276(34)ex (10)st (85)nst FO
= 0.3221(48)ex (14)st (121)nst CI
=⇒
PT
αs (Mτ )
αs (MZ )
Ref.
FO
0.3276 (34)ex (86)th
0.1195 (4)ex (10)th (2)ev
SN 09
CI
0.3221 (48)ex (122)th
0.1188 (6)ex (15)th (2)ev
hi
0.3249 (29)ex (75)th
0.1192 (4)ex (9)th (2)ev
Z
0.1191 (27)ex (2)th
BCK, DAVIER et al.
hi
0.1189 (10)
BETHKE, PDG
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 16/2
♦ Light Quark Masses Chetyrkin-SN-Zakharov 95, SN 95, 99, 09
• (Pseudo)scalar channel : mq decreases when |λ2 | increases
2
α
L (M 2 ) = 8π32 (m̄u + m̄d )2 1 + 4.8as + 28.9a2s + ... − 4 πs Mλ 2
: 5-6% effects.
• e+ e− I=0 channel : ms increases when |λ2 | increases
h
Q2 ΠD=2
= − 4π1 2 1.05 απs λ2 + 6m̄2s (1 + 2.67as + 24.14a2s + 250a3s )
ss
• τ-decays ∆S = −1 : ms increases when |λ2 | increases
Q2 ΠVD=2
=
h+A
i
− 2π1 2 1.05 απs λ2 + 3m̄2s (1 + 2.33as + 19.58a2s + 202.31a3s )
i
• λ2 6= 0 improves the agreement from various channels and
lattice calculations !
• Notice the geometric sum in the PT series !
NZ 09
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 17/2
ms (2 GeV) to order α3s
Channels
e+ e− data
Refs.
SN05
100 ± 28exp ± 20th
FESR
113 ± 13exp ± 22th
Rτ,φ
∆1φ
99 ± 23exp ± 4th
104.3 ± 15.4
Tentative Average
τ-decay data
FESR
SU(3) breaking SR
Tentative Average
ms (2 GeV) in MeV
SNTAU 05
JGP 05, BCK 05
93 ± 30
81 ± 22
85.2 ± 17.7
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 18/2
ms (2 GeV) to order
3
αs
Pseudoscalar
Pion FESR + ChPT
Pion LSR + ChPT
Pion LSR + ChPT+1/Q2
Pion SR analytic cont.
113.5 ± 22.0
SN99
CNZ99,SNB02
DNRS08
Kaon FESR
KM02
Kaon LSR+1/Q2
SNB02
Kaon LSR
CK05
Kaon LSR
JOP06
Tentative Average
119.6 ± 23.2
BPR95
107.4 ± 22.0
102 ± 8
100 ± 12
119.6 ± 18.4
105 ± 9
97.2 ± 9.6
103.3 ± 4.3
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 19/2
ms (2 GeV) to order α3s
Scalar K0∗
LSR
JOP06
LSR
CFNP97
FESR
M99
Tentative Average
88 ± 8
103.6 ± 15.4
115.2 ± 13.0
96.8 ± 6.2
Chiral condensate
to order αs (not in the average)
N , B∗ − B, D → K ∗ lν
DN98
131 ± 18
Weighted
99.1 ± 3.4
Tentative Global Average
Arithmetic
97.4 ± 10.9
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 20/2
Conclusions
♣ Improvment of some results: inclusion of αs corrections
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 21/2
Conclusions
♣ Improvment of some results: inclusion of αs corrections
♦ Double ratio of moments look appropriate for extracting small
hadron-mass splittings, hαs G2 i,...
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 21/2
Conclusions
♣ Improvment of some results: inclusion of αs corrections
♦ Double ratio of moments look appropriate for extracting small
hadron-mass splittings, hαs G2 i,...
♦ Check of eventual (small) effects due to Duality Violation
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 21/2
Conclusions
♣ Improvment of some results: inclusion of αs corrections
♦ Double ratio of moments look appropriate for extracting small
hadron-mass splittings, hαs G2 i,...
♦ Check of eventual (small) effects due to Duality Violation
♦ Beyond NN....NLO : Improved accuracy needs a better
understanding of the QCD PT and NPT series
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 21/2
Conclusions
♣ Improvment of some results: inclusion of αs corrections
♦ Double ratio of moments look appropriate for extracting small
hadron-mass splittings, hαs G2 i,...
♦ Check of eventual (small) effects due to Duality Violation
♦ Beyond NN....NLO : Improved accuracy needs a better
understanding of the QCD PT and NPT series
♦ What we usually know about IR (n! div) and UV (large β approx: alternate
signs) renormalons can be incorrect: geometric series instead ?
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 21/2
Conclusions
♣ Improvment of some results: inclusion of αs corrections
♦ Double ratio of moments look appropriate for extracting small
hadron-mass splittings, hαs G2 i,...
♦ Check of eventual (small) effects due to Duality Violation
♦ Beyond NN....NLO : Improved accuracy needs a better
understanding of the QCD PT and NPT series
♦ What we usually know about IR (n! div) and UV (large β approx: alternate
signs) renormalons can be incorrect: geometric series instead ?
♥ 1/Q2 term: a possible way to parametrize Large order PT series
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 21/2
Conclusions
♣ Improvment of some results: inclusion of αs corrections
♦ Double ratio of moments look appropriate for extracting small
hadron-mass splittings, hαs G2 i,...
♦ Check of eventual (small) effects due to Duality Violation
♦ Beyond NN....NLO : Improved accuracy needs a better
understanding of the QCD PT and NPT series
♦ What we usually know about IR (n! div) and UV (large β approx: alternate
signs) renormalons can be incorrect: geometric series instead ?
♥ 1/Q2 term: a possible way to parametrize Large order PT series
♠ 1/Q2 term: does not break the OPE and should not be confused
with the hA2 i term often discussed in the literature.
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 21/2
Conclusions
♣ Improvment of some results: inclusion of αs corrections
♦ Double ratio of moments look appropriate for extracting small
hadron-mass splittings, hαs G2 i,...
♦ Check of eventual (small) effects due to Duality Violation
♦ Beyond NN....NLO : Improved accuracy needs a better
understanding of the QCD PT and NPT series
♦ What we usually know about IR (n! div) and UV (large β approx: alternate
signs) renormalons can be incorrect: geometric series instead ?
♥ 1/Q2 term: a possible way to parametrize Large order PT series
♠ 1/Q2 term: does not break the OPE and should not be confused
with the hA2 i term often discussed in the literature.
♣ 1/Q2 term: seems to improve some existing phenomenology
QCD 10 - Montpellier (28th june - 2nd july 2010) – p. 21/2