quark-gluon correlation inside the b-meson from qcd sum rules based on heavy quark effective theory...

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


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




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


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)



Dispersion relation for two Lorentz structure

Borel transform


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




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



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



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









+ + +

+ + +

+ + +

+ + (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





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



Matrix elements

Two-body operator

Three body operator





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




ωω thL

Spectral ansatzOPE of LHS

HQET sum rules

Results for λH2(μ=1GeV) (preliminary)



Results for λH2 - λE

2 (μ=1GeV) (preliminary)



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


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

quark-gluon correlation inside the b-meson from qcd sum rules based on heavy quark effective theory...

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