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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)
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.
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, , ,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 → ∞
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 )
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)
• 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
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
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
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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
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)
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
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
■ 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
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)
■ :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)
■ 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
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)
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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
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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
OPE
+ + O(as) coρρectioν to
=
+ ・・・
light quarkheavy quark
This work
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Grozin&Neubert
Renormalization of the interpolating field
O2 =hvγ5q + (couνteρ teρμ )
Counter term =
UV-pole
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+ + + +UV-pole
O3 =hvγGq + (couνteρ teρμ )
Counter term=
Renormalization of the interpolating field
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O(αs) correction to dim5 term
UV-divergence is subtracted by counter terms.
Remaining IR-divergence is absorbed into the vacuum condensate.
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
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
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
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
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.
Implication to B-meson wave function
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+ + +
+ + +
+ + +
+ + (counter term)
O(αs) correction to dim5 term
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
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Pi (ω ) =
1π
d ′ωImΠ i ( ′ω )′ω − ω − iη−∞
∞
∫ , (i = 1,2)
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
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Matrix elements
Two-body operator
Three body operator
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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
Borel transform
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HQET sum rule for λE,H
ωω thL
Spectral ansatzOPE of LHS
HQET sum rules
Results for λH2(μ=1GeV) (preliminary)
ωth:continuum threshold
: Grozin&Neubert: +dim6: +dim6 +O(αs) correction
Results for λH2 - λE
2 (μ=1GeV) (preliminary)
: Grozin&Neubert: +dim6: +dim6 +O(αs) correction
ωth:continuum threshold
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