heavy quark expansion in qcd and b c near zero recoil
TRANSCRIPT
Heavy quark expansion in QCDand b→c near zero recoil
N. Uraltsev
N. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 1 / 21
Introduction; HQE in QCD
The 1/mQ expansion in QCD can be constructed for a few importantcases where we now know quite a bit about its terms
The key information is provided by the Small Velocity heavy quarksum rules, including spin sum rules first established for heavy quarksin QCD
Allowed to predict values of Λ, µ2π, ρ3
D , ... based on the hyperfinemass splitting MB∗−MB≈47 MeV
N. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 2 / 21
A similar analysis has been extended motivated by the formfactorF (0) in B→D∗ `ν near zero recoil Gambino, Mannel, N.U. arXiv:1004.2859 [hep-ph]
Arrived at three apparently isolated, yet linked through the HQE,observations:
Large negative inelastic corrections to F (0) driving it down toF (0)≈0.86
Large value of the nonlocal correlators of Q̄~π2Q and Q̄~σ~BQ inB mesons from the hyperfine splitting ∆M2 in B vs. D
The enhanced negative corrections in F (0) are relatedto the ‘discrepancy’ in the hyperfine splitting ratiobetween charm and beauty mesons
This implies enhanced inclusive yield of radials in b→c `ν
Resolve ‘ 12 >
32 ’ paradox
Account for the missing semileptonic channels
Predict significance of the 32
+‘D-wave’
As a byproduct we find significant corrections to the ground-statefactorization; relevant for precision inclusive decays
N. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 3 / 21
Vcb at zero recoil
dw (B → D∗+`ν̄) ∼ G 2F · |Vcb|2 · |~p | · |FB→D∗(~p )|2
|Vcb| requires FB→D∗ (~p ) – it is shaped by bound-state physics
At ~p =0 ( ~pe =−~pν̄ )almost nothing happened!
Without isotopic effects (in the heavy quark limit) F~p=0 = 1:
Fn/p(0) = 1+0
mc,b+O
(Λ2
QCD
m2c,b
)+O
(Λ3
QCD
m3c,b
)+ ...
No 1/mb,c-corrections(cf. Ademollo-Gatto)
1986 Voloshin, Shifman1990 Luke
N. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 4 / 21
Challenge to theory: corrections to F (0)=1 are driven by 1/mc ,potentially significant!
Originally (before 05/1994) were thought to be only about -0.02 Neubert
In fact, deviations from the symmetry limit in QCD are
considerably larger in QCD Shifman, N.U., Vainshtein
There have been folklore around sum rules for FD∗
often incorrect...
N. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 5 / 21
The QCD approach
T zr(q0) =
∫d3x
∫dx0 e−iq0x0
1
2MB〈B |1
3iT {c̄γkγ5b(x) b̄γkγ5c(0)}|B〉
q0 =MB−MD∗−ε
−2mc
2mb
0
ε
I0(µ) = − 1
2πi
∮|ε|=µ
T zr(ε) dε
T zr(ε) can be calculated in the short-distance expansion at |ε|�ΛQCD,expansion parameters are µhadr
ε, µhadr
2mc+ε, µhadr
2mb−εand αs at the related
scale hence OPE for I0(µ)
Using analytic properties we can shrink the contour onto the physicalcut ε>0, then by optical theorem
I0(µ) =1
π
∫ µ
0
Im T zr(ε) dε = |FD∗|2 +∑ε<µ
|FB→n|2
N. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 6 / 21
The QCD approach
T zr(q0) =
∫d3x
∫dx0 e−iq0x0
1
2MB〈B |1
3iT {c̄γkγ5b(x) b̄γkγ5c(0)}|B〉
q0 =MB−MD∗−ε
−2mc
2mb
0
ε
I0(µ) = − 1
2πi
∮|ε|=µ
T zr(ε) dε
T zr(ε) can be calculated in the short-distance expansion at |ε|�ΛQCD,expansion parameters are µhadr
ε, µhadr
2mc+ε, µhadr
2mb−εand αs at the related
scale hence OPE for I0(µ)
Using analytic properties we can shrink the contour onto the physicalcut ε>0, then by optical theorem
I0(µ) =1
π
∫ µ
0
Im T zr(ε) dε = |FD∗|2 +∑ε<µ
|FB→n|2
N. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 6 / 21
The QCD approach
T zr(q0) =
∫d3x
∫dx0 e−iq0x0
1
2MB〈B |1
3iT {c̄γkγ5b(x) b̄γkγ5c(0)}|B〉
q0 =MB−MD∗−ε
−2mc
2mb
0
ε
I0(µ) = − 1
2πi
∮|ε|=µ
T zr(ε) dε
T zr(ε) can be calculated in the short-distance expansion at |ε|�ΛQCD,expansion parameters are µhadr
ε, µhadr
2mc+ε, µhadr
2mb−εand αs at the related
scale hence OPE for I0(µ)
Using analytic properties we can shrink the contour onto the physicalcut ε>0, then by optical theorem
I0(µ) =1
π
∫ µ
0
Im T zr(ε) dε = |FD∗|2 +∑ε<µ
|FB→n|2
N. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 6 / 21
OPE in QCD
εf<µ∑f 6=D∗
|FB→f|2 ≡ winel(µ)
FD∗ =√
I0(µ)− winel(µ) <√
I0(µ)
OPE:
I0(µ) = ξApert(µ)− ∆ 1
m2Q
− ∆ 1m3
Q
− ∆ 1m4
Q
− ...
∆ 1m2
Q
=µ2
G
3m2c
+µ2π−µ2
G
4
(1
m2c
+ 1m2
b+ 2
3mcmb
)∆ 1
m3Q
=ρ3
D−13ρ3
LS
4m3c
+ρ3
D+ρ3LS
6mcmb
(1
mc− 1
2mb
)...
N. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 7 / 21
Brief synopsis
FD∗ =√ξpertA −∆power − winel√
ξpertA ' 0.98 at µ ' 0.8 GeV, −∆ 1
m2Q
−∆ 1m3
Q
' −0.13
FD∗ ≤ 0.92 – upper bound
It turns out the sum of the excitation probabilities (wavefunctionoverlap deficit) should be quite significant,
N.U. hep-ph/0312001
winel ∼> 0.14 FD∗ ≈ 0.86 – prediction
N. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 8 / 21
Convergence
How well is the OPE under control?
• Perturbative corrections: small numerically√ξA ' 1−0.022+(0.005−0.004)+0.0022−0.0014+...
applies only to Wilsonian ξpertA (µ)!
0.4 0.5 0.6 0.7 0.8 0.9
0.965
0.970
0.975
0.980
0.985
0.990 √ξpertA
εM , GeV
αs = 0.32d order BLMfull O(α2
s )3d order BLM
Assume µ=εM around 0.8 GeV√ξA ' 0.98 at αs(mb)=0.22
uncertainty 1% seems reasonably conservativeN. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 9 / 21
Convergence
• Power corrections to I0:
Take the low-end expectation values µ2π ' 0.4 GeV2, ρ3
D ' 0.15 GeV3:
−∆ 1m2
Q
'−0.095 −∆ 1m3
Q
'−0.028 −∆ 1m4
Q
'0.02 −∆ 1m5
Q
'0.01
• αs-corrections to the coefficients of power-suppressed terms
Calculated for µ2π, correction is extremely small. Expect mild effect
for µ2G ; in 1/m3
Q even a 30% renormalization would not produce asignificant change
√1−∆A ∼< 0.94± 0.01 the upper bound seems safe at 1% level
Would expect formfactor about 0.92 if no overlap deficit were there
N. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 10 / 21
Sum of inelastic probabilities
Wavefunction overlap deficit in the language of Quantum Mechanics
Required to turn the upper bound for FD∗ into an estimate
I1(µ) = − 1
2πi
∮|ε|=µ
T zr(ε) ε dε =∑ε<µ
εn|FB→n|2
winel(µ) =I1(µ)
ε̄(µ)ε̄(µ) – ‘average excitation energy’ (up to µ)
I1(µ) is calculated in the OPE similar to I0(µ):
I1 =−(ρ3
πG +ρ3A)
3m2c
+−2ρ3
ππ−ρ3πG
3mcmb+
ρ3ππ+ρ3
πG +ρ3S+ρ3
A
4
(1
m2c
+ 23mcmb
+ 1m2
b
)+O
(1
m3Q
)︸ ︷︷ ︸ ︸ ︷︷ ︸ ︸ ︷︷ ︸BPS limit (δ BPS)1 (δ BPS)2
N. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 11 / 21
Inelastic piece
I(BPS)1 =
−(ρ3πG +ρ3
A)
3m2c
+O(
1m3
c
)(ρ3πG +ρ3
A)−ρ3LS determines hyperfine splitting to order 1/m2
Q
Extract comparing B and D mesons
δ(2)I1 is positive; δ(1)I1 comes with small coefficient 1/3mcmb and theminimum is very shallow:
I1/I(BPS)1 = 1− (1−ν3/2)
m2c
m2b︸ ︷︷ ︸+ [...]2
<0.07
ν3/2>0
0.8 1.0 1.2 1.4 1.6 1.8 2.00.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
P/G
I1/I(BPS)
1
winel =0.48 GeV3 +κ 0.35 GeV3
3m2c εrad
' 14%
|κ|∼<0.15
A 6% decrease in FD∗
The way to evaluate∑|FB→n|2 through I1 in the ’t Hooft model yields
almost exact numberN. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 12 / 21
Continuum
Excited states should predominantly be resonances (radial excitations)
Continuum is 1/Nc-suppressed and is usually smaller
D(∗)π inelastic contributions can be estimated for soft pions
The complete estimate yields a significant contribution:
0.3 0.4 0.5 0.6 0.7 0.80.00
0.01
0.02
0.03
0.04
0.05
0.06
pπmax
, GeV
Dπ
0.3 0.4 0.5 0.6 0.7 0.80.000
0.005
0.010
0.015
0.020
0.025
pπ
max, GeV
D∗ π
gD∗Dπ =4.9 GeV−1 (ΓD =96 KeV)
gB∗Bπ/gD∗Dπ = 1, 0.8, 0.6 and 0.4 gB∗Bπ/gD∗D∗π = 1, 0.8, 0.6
Altogether we expect D(∗)π piece to yield around 4% in∑|FB→n|2,
about a fourth of the resonance-based estimate δFD∗ ' −2%N. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 13 / 21
FD∗ summary
QCD lower bound:
FD∗ < 0.92 FD∗ < 0.9 including continuum estimate
The unbiased predicted value
FD∗ ∼< 0.86
The central number has about 2% accuracy it may lower if µ2π
turns out larger
Central value goes down for increasing µ2π yet the corrections from higher
power terms also increaseN. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 14 / 21
B → D `ν near zero recoil
Experimentally challenging theoretically advantageousN.U. 2003
〈D(p2)|c̄γνb|B(p1)〉 = f+(p1+p2)ν+f−(p1−p2)νf± ≡ f±(~q 2)
A single amplitude J0 =(MB +MD)f+(0) + (MB−MD)f−(0) at ~q =0
HQ limit: f+ = MB+MD
2√
MBMD, f− = −MB−MD
MB+MDf+
J0
2√
MBMD
= 1− a2
(1
mc− 1
mb
)2
− a3
(1
mc− 1
mb
)2 (1
mc+ 1
mb
)+ ...
Power corrections are well under control and small
Any amplitude with massless leptons depends, however solely on f+,(only the combination of f+ and f− has no 1/m corrections)
F+ ≡ 2√
MBMD
MB+MDf+ has 1/mQ corrections since nothing forbids this in ~J
Not a drawback in the era of dynamicsN. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 15 / 21
B→D
F+ = 1 +(
Λ2−Σ)( 1
mc− 1
mb
)MB−MDMB+MD
−O(
1m2
Q
)Λ = MB−mb, Σ = ...
From inclusive decays and exact sum rules we knowΛ2−Σ (positive, but small ∝ µ2
π−µ2G
3µhadr)
Moreover, we know all power corrections here are smallat small µ2
π−µ2G
N.U. 20032√
MBMD
MB + MDf+(0) = 1.04 ± 0.01 ± 0.01
All orders in 1/m in ‘BPS’, to 1/m2 ·1/BPS2, α1s
The bulk 3% is the perturbative factor, only 1% comes from power terms
N. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 16 / 21
Summary
Comprehensive heavy quark expansion can say much for nonperturbativeeffects in certain cases
Inequalities or positivity properties are essential, require a physicalrenormalization scheme. Conceptually similar to the lattice, yet may differin details
Exploit physical behavior of the correlators in Minkowsky domain
FD∗≈0.86; uncertainty about 2% at known µ2π, ρ3
D plus effect ofhigher-order power terms. Central value goes down for larger µ2
π, yethigher power corrections become significant
Large inelastic contribution decreasing FD∗ by 6%Close to the BPS estimate
Large nonlocal correlators ρ3... from hyperfine splittings
‘Discrepancy’ in the splitting for B and D is settled
Large inclusive yield of ‘radials’, 7÷10% of Γsl
Resolve 12 vs. 3
2 problemMay provide missing semileptonic channels
N. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 17 / 21
Hyperfine splitting in D vs. B
M2B∗−M2
B ' M2D∗−M2
D ' M2K∗−M2
K ' M2ρ−M2
π
If these were exact and if perturbative corrections could be discarded
−(−ρ3LS + ρ3
πG + ρ3A) ' 2Λµ2
G (1+κ)
−(ρ3πG +ρ3
A) ≈ 0.48 GeV3
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.3
0.4
0.5
0.6
0.7
∆M2
π
1/MP
B D K
α2s -runningαs = 0.3αs = 0.22κ≈0
κ>0
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
1.00
1.05
1.10
1.15
1.20 cG (mc )/cG (mb)
mc , GeV
ffµ= 0
αs = 0.22
µ= 0.6 GeV
µ= 0.8 GeV
αs = 0.3
The final outcome is κ≈−0.15 and −(ρ3πG +ρ3
A) ≈ 0.48 GeV3
N. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 19 / 21
Experimental determination
We can take Vcb extracted from inclusive decays and calculate FD∗
Shifman, N.U. 1994
2ρ0 0.5 1 1.5 2
]-3
| [10
cb |V×
F(1)
30
35
40
45
HFAGLP 2009
ALEPH
CLEO
OPAL(part. reco.)
OPAL(excl.)
DELPHI(part. reco.)
BELLE
DELPHI (excl.)
BABAR (excl.)BABAR (D*0)BABAR (Global Fit)
AVERAGE
= 12χ ∆
/dof = 56.9/212χ
%̂2
FD∗ ' 0.810± 0.007± 0.026± δincl BaBar 2008
0.849± 0.011± δincl HFAG Average 2009
0.836± 0.015± δincl HFAG08, CLEO/ ALEPH excluded
(inclusive value taken without error bars)
Finally FD∗ moved practically to the valueobtained in the heavy quark expansion inQCD, 0.86±0.02
N. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 20 / 21
Experimental determination
For B→D `ν:
F+(0) ' 1.021± 0.019± 0.041± δincl
Good agreement with the dynamic heavy quark expansion in QCD
]-3
| [10cb|V×G(1) 10 20 30 40 50
]-3
| [10cb|V×G(1) 10 20 30 40 50
ALEPH 6.20± 11.80 ±38.85
CLEO 3.45± 5.90 ±44.77
BELLE 5.15± 4.40 ±41.11
Average 4.50±42.30
HFAGLP 2007
/dof = 0.26/ 4 (CL = 99 %)2χ
2009
Using |Vcb | from Γsl(B) I predicted
|Vcb|G (1) = 43.7X5· 10−3
There is no tension between inclusiveand exclusive Vcb
rather a remarkable agreement
N. Uraltsev (Siegen ) 1/mQ expansion and FD∗ ECT∗ Trento April 2 2012 21 / 21