quark-gluon correlation inside the b-meson from qcd sum rules based on heavy quark effective theory...
DESCRIPTION
Beneke, Buchalla, Neubert, Sachrajda (’99) Bauer, Pirjol, Stewart (’01) Heavy quark field Exclusive decay of B-meson QCD factorization of exclusive B-decay: B-meson’s LCDA in HQETTRANSCRIPT
![Page 1: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/1.jpg)
Quark-gluon correlation inside the B-meson from QCD sum rules
based on heavy quark effective theory
Tetsuo Nishikawa (Ryotokuji U)Kazuhiro Tanaka (Juntendo U)
![Page 2: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/2.jpg)
Motivation
Exclusive decay of B meson provides important information for understanding CP violation.
In the description of exclusive B-decay based on QCD factorization, a very important role is played by the light cone distribution amplitude (LCDA) of B-meson.
However, surprisingly, little attention to B-meson’s LCDA was received in past. Our poor knowledge about it limits to extract important physics from experimental data.
This work is a part of an attempt to precisely determine B-meson’s LCDA based on QCD.
![Page 3: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/3.jpg)
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄ
ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
, , ,B lππ ργ π ν→ L
Beneke, Buchalla, Neubert, Sachrajda (’99)Bauer, Pirjol, Stewart (’01)
Heavy quark field
Exclusive decay of B-meson
QCD factorization of exclusive B-decay:
B-meson’s LCDA in HQET
bm → ∞
![Page 4: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/4.jpg)
OPE of B meson’s LCDA
dim.3
dim.4
dim.5
{ } { } with up to (Complete OPE re ) and up to dim.5sult ii sC OO a2 2completely represented by HQET p , aram eters ,E Hl lL
B bm mL = -
Kawamura and Tanaka, PLB 673(2009)201
%φB (t, μ ) = Ci (t,μ ) 0 Oi (μ ) B(v)
i∑ L =Lν(itμeγE )
![Page 5: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/5.jpg)
λE and λH: quark-gluon correlation inside the B-meson “Chromo-electric”
“Chromo-magnetic”
λE 、 λH 〜 strength of the color-electric (-magnetic) field inside the B-meson play an important role for the determination of exclusive
B-decay amplitude But, almost unknown (only one estimate by HQET sum r
ule)
(F(μ): B meson’s decay constant)
![Page 6: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/6.jpg)
• NLO perturbative corrections: very large for τ→ 0 and 10-30% level for moderate τ• Nonperturbative corrections (dim. 5 and dim. 4 operators) are important (20-30% level)• Effects from are significant in dim. 5 contributions. , E Hλ λ
“3”
“3+ 4”“3+4+ 5”
LO
-1 GeVt ⎡ ⎤⎣ ⎦
L-N
Behavior of B-meson’s LCDAKawamura and Tanaka, PLB 673(2009)201
![Page 7: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/7.jpg)
Extrapolation to long distance region
In the long distance region, OPE diverges.
For large distances, we must rely on a model (Lee-Neubert’s ansatz is employed here).
smoothly connect the OPE and the model descriptions at certain distance
LCDA for entire distances
OPE up to dim. 5 ops.
Model (Lee-Neubert ansatz)
t c
OPE
L-N ansatz
ct
Kawamura and Tanaka, PLB 673(2009)201
![Page 8: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/8.jpg)
LCDA enters the B-decay amplitude through its inverse moment.
Stable behavior for Switching off λE and λH, stable behavior is not seen.
0.6 GeV−1 à t c à 1 GeV −1
Inverse moment of LCDA
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄ
ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
Kawamura and Tanaka, PLB 673(2009)201
The above results demonstrate the impact ofReliable and precise determinations of is necessary.
λE and λ H
λB
−1(μ ) = dωφ+ (ω, μ )
ω0
∞
∫ = dτ %φ+ (−iτ ,μ )0
τ c
∫ + dτ %φ+ (−iτ , μ )τ c
∞
∫
λE and λ H
![Page 9: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/9.jpg)
Only one HQET sum rule estimate by Grozin and Neubert (1997) is known.
The sum rule analysis for λE and λH is not complete, unless the calculation at NLO accuracy (dim.6 and O(αs) correction to dim.5) is carried out.
Updating the estimate of λE and λH is needed.
Estimate of λE and λH
( ) ( )2 2 2 21 GeV 0.11 0.06 GeV , 1 GeV 0.18 0.07 GeVλ λE H= ± = ±
Ο(α s ) Ο(1)
dominant OPE of hv (x)GG μν (x)q(x),q(0)γ5hv(0)
![Page 10: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/10.jpg)
In a heavy(Q)-light(q) system,
Q is nearly on-shell:
This is equivalent to write
HQET (Heavy Quark Effective Theory)
Light quark cloud
Heavy quark
vμ (veλocity of the πaρeνt μ esoν)
Pμ =μQvμ + kμ , kμ :ρesiduaλ μoμ eνtuμ (kμ = μ Q )
Q =exπ(−iμQv⋅x)hv
Q
![Page 11: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/11.jpg)
Pair creation of QQ cannot occur. The new field hv is constrained to satisfy
(neglect Q degree of freedom)QCD Lagrangian can be simplified to
HQET (Heavy Quark Effective Theory)
LHQET =hviv⋅Dhv+ qiγ ⋅Dq +L
P+hv=hv, P+ =/v+12
extract the physics of heavy-light mesons
![Page 12: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/12.jpg)
■ Current correlation function
■ j(x): “interpolating field” ex. meson:
Basic object of the QCD sum ruleBasic object of the QCD sum rule
P(q) = −i d Dxe−iq⋅x∫ 0 T [ j(x) j†(0)] 0
j =qGq
P(q) =
Interaction between quarks and with vacuum fluctuation
![Page 13: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/13.jpg)
Correlation function at Correlation function at
=
P(q) = −i d4xeiqx 0∫ T [ j(x) j†(0)] 0
Operator ProductExpansion (OPE)
=c01+ c3mq qq + c4
α s
πGμν Gμν +L
q2 → −∞ (x→ 0)
![Page 14: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/14.jpg)
■ :spectral function
■Using analyticity, we can relate and the spectral function as
Imaginary part of the correlation functionImaginary part of the correlation function
(1 /π)Iμ P(q2)
POPE
q2
(1 /π)Iμ P(q2)
Bound state pole
continuum
PΟPE(q2 ) =
1π
dsImΠphenomenology(s)
s − q2 − iη0
∞
∫ (Dispersion relation)
![Page 15: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/15.jpg)
■ Applying “Borel transform” on the dispersion relation, we obtain a sum rule:
■ Physical quantities extracted from the sum rule have mild M-dependence.
∵truncation of OPE, incompleteness of the spectral ansatz choice of a reasonable range of M
QCD (Borel) Sum ruleQCD (Borel) Sum rule
B̂PΟPE(q
2)= ds0
∞
∫ e−s/M2 1πIμ P(s)
approximate
Borel mass (arbitrary parameter)
ansatz
![Page 16: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/16.jpg)
HQET sum rule for λE,H
Non-diagonal correlation function
Representation of Π with hadronic states
B-meson pole at (not mB !) 2-independent Lorentz structures
ω =L =mB − mb
P(ω ) ≡ −i d 4 xe−iωv⋅x 0 T hv (x)ΓGμν (x)q(x),q(0)γ 5hv (0)⎡⎣ ⎤⎦∫ 0
P(ω) =1
ω − Λ − iη−112
F(μ )2
× λ H2 Tr(Γσ μνγ 5P+ ) + (λ H
2 − λ E2 )Tr Γ(vμ vρσ νρ − vν vρσ μρ )γ 5P+⎡⎣ ⎤⎦{ }
+(higher resonnances)
![Page 17: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/17.jpg)
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄ
ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
Dispersion relation for two Lorentz structure
Borel transform
HQET sum rule for λE,H
ωω thL
Spectral ansatzOPE of LHS
HQET sum rulesfor
Pi (ω) =
1π
d ′ωImΠ i ( ′ω )′ω − ω − iη−∞
∞
∫ , (i = 1,2)
F(μ)2λE,H2 d(ω −L)
λE ,H2
![Page 18: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/18.jpg)
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄ
ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
HQET sum rule for λE,H
Sum rules for
Decay constant is independently determined from an HQET sum rule. Neubert, 1992 Bagan, Ball, Braun and Dosch, 1992
up to dim.6 operators, up to O(αs) Wilson coefficients
F(μ)
λH2 and λ H
2 − λ E2
![Page 19: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/19.jpg)
OPE
+ + O(as) coρρectioν to
=
+ ・・・
light quarkheavy quark
This work
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄ
ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
Grozin&Neubert
![Page 20: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/20.jpg)
Renormalization of the interpolating field
O2 =hvγ5q + (couνteρ teρμ )
Counter term =
UV-pole
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
![Page 21: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/21.jpg)
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB + + +
+ + + +UV-pole
O3 =hvγGq + (couνteρ teρμ )
Counter term=
Renormalization of the interpolating field
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅBQuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅBQuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅBQuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
![Page 22: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/22.jpg)
O(αs) correction to dim5 term
UV-divergence is subtracted by counter terms.
Remaining IR-divergence is absorbed into the vacuum condensate.
![Page 23: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/23.jpg)
Results for λH2(μ=1GeV) (preliminary)
ω th = 0.8GeV
ω th = 1.0GeV
ω th = 0.9GeV
ωth:continuum threshold
: Grozin&Neubert: +dim6: +dim6 +O(αs) correction
![Page 24: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/24.jpg)
Results for λH2 - λE
2 (μ=1GeV) (preliminary)
ω th = 0.8GeV
ω th = 1.0GeV
ω th = 0.9GeV
: Grozin&Neubert: +dim6: +dim6 +O(αs) correction
ωth:continuum threshold
![Page 25: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/25.jpg)
Choice of the reasonable M-rangeCriterion for M:
Both of Higher order power corrections in OPEContinuum contribution
should not be large (less than 30-50%).Reasonable range of M
In this range,
λH
2 = 0.12 ± 0.04 GeV2
λ H2 − λ E
2 = 0.045 ± 0.005 GeV2
![Page 26: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/26.jpg)
Summary
λE and λH (quark-gluon correlation inside the B-meson) play important role in B-meson’s LCDA.
HQET sum rule for λE and λH
up to dim.6 operator in OPE radiative correction to the mixed condensate
Small contribution of dim.6 term OPE seems to converge at this order. Radiative correction significantly lowers λE and λH.
Renormalization group improvement etc. Matching the OPE of LCDA Estimation of the inverse moment of LCDA ( )λB
−1
![Page 27: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/27.jpg)
On the results
Contribution of dim.6 is less than 1% OPE seems to converge at this order.O(αs)-correction to dim.5 is significantly lar
ge and tends to suppress λH and λE.After inclusion of O(αs)-correction, stability
of the splitting becomes worse.
![Page 28: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/28.jpg)
Implication to B-meson wave function
![Page 29: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/29.jpg)
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅBQuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅBQuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅBQuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
+ + +
+ + +
+ + +
+ + (counter term)
O(αs) correction to dim5 term
![Page 30: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/30.jpg)
Formulation of B-meson’s HQET sum rule
Correlation function
C.F. evaluated by OPE is related to B-meson’s physical quantities through the dispersion relation
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄ
ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄ
ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
Pi (ω ) =
1π
d ′ωImΠ i ( ′ω )′ω − ω − iη−∞
∞
∫ , (i = 1,2)
![Page 31: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/31.jpg)
Correlation function
Representation of Π with hadronic states
B-meson pole at ω =L =mB − mb
P(ω ) ≡ −i d 4 xe−iωv⋅x 0 T hv (x)ΓGμν (x)q(x),q(0)γ 5hv (0)⎡⎣ ⎤⎦∫ 0
Formulation of HQET sum rule for B-meson
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄ
ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
![Page 32: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/32.jpg)
Matrix elements
Two-body operator
Three body operator
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄ
ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄ
ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
![Page 33: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/33.jpg)
B-meson pole
2-independent Lorentz structuresWrite dispersion relations for
Pi (ω ) =
1π
d ′ωImΠ i ( ′ω )′ω − ω − iη−∞
∞
∫ , (i = 1,2)
P(ω ) =1
ω − Λ − iη−112
F(μ )2
× λ H2 Tr(Γσ μνγ 5P+ ) + (λ H
2 − λ E2 )Tr Γ(vμ vρσ νρ − vν vρσ μρ )γ 5P+⎡⎣ ⎤⎦{ }
+(higher resonnances)
P1 and Π2
![Page 34: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/34.jpg)
Borel transform
QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄ
ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB
HQET sum rule for λE,H
ωω thL
Spectral ansatzOPE of LHS
HQET sum rules
![Page 35: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/35.jpg)
Results for λH2(μ=1GeV) (preliminary)
ωth:continuum threshold
: Grozin&Neubert: +dim6: +dim6 +O(αs) correction
![Page 36: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/36.jpg)
Results for λH2 - λE
2 (μ=1GeV) (preliminary)
: Grozin&Neubert: +dim6: +dim6 +O(αs) correction
ωth:continuum threshold
![Page 37: Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1aed7f8b9ab05997bd0b/html5/thumbnails/37.jpg)
In a heavy(Q)-light(q) system,
Pair creation of QQ cannot occur. The new field hv is constrained to satisfy
QCD Lagrangian can be simplified to
HQET (Heavy Quark Effective Theory)
PQ ; Pmeson
⇓Q =exπ(−iμQv⋅x)hv, (vμ :veλocity of the μ esoν)
LHQET =hviv⋅Dhv+ qiγ ⋅Dq +L
P+hv=hv, P+ =/v+12
Q
Light quark cloud
Heavy quark
vμ