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.

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

OPE of B meson’s LCDA dim.3 dim.4 dim.5 Kawamura and Tanaka, PLB 673(2009)201

λ E and λ H : quark-gluon correlation inside the B-meson “Chromo-electric” “Chromo-magnetic”  λ E 、 λ H 〜 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 rule) (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. “3” “ ３＋ 4” “ ３＋４＋ 5” LO L-N Behavior of B-meson’s LCDA Kawamura 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) OPE L-N ansatz Kawamura and Tanaka, PLB 673(2009)201

LCDA enters the B-decay amplitude through its inverse moment. Stable behavior for Switching off λ E and λ H, stable behavior is not seen. Inverse moment of LCDA Kawamura and Tanaka, PLB 673(2009)201 The above results demonstrate the impact of Reliable and precise determinations of is necessary.

Only one HQET sum rule estimate by Grozin and Neubert (1997) is known. The sum rule analysis for λ E and λ H is not complete, unless the calculation at NLO accuracy (dim.6 and O(α s ) correction to dim.5) is carried out. Updating the estimate of λ E and λ H is needed. Estimate of λ E and λ H dominant

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 Q

Pair creation of QQ cannot occur. The new field h v is constrained to satisfy (neglect Q degree of freedom) QCD Lagrangian can be simplified to HQET ( Heavy Quark Effective Theory ) extract the physics of heavy-light mesons

■Current correlation function ■j(x): “interpolating field” ex. meson: Basic object of the QCD sum rule Interaction between quarks and with vacuum fluctuation

Correlation function at Correlation function at = Operator Product Expansion (OPE)

■ :spectral function ■Using analyticity, we can relate and the spectral function as Imaginary part of the correlation function Bound state pole continuum (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 rule approximate Borel mass (arbitrary parameter) ansatz

HQET sum rule for λ E,H Non-diagonal correlation function Representation of Π with hadronic states  B-meson pole at (not m B !)  2-independent Lorentz structures

Dispersion relation for two Lorentz structure Borel transform HQET sum rule for λ E,H Spectral ansatz OPE of LHS HQET sum rules for

HQET sum rule for λ E,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(α s ) Wilson coefficients

OPE + + = + ・・・ light quark heavy quark This work Grozin&Neubert

Renormalization of the interpolating field Counter term = UV-pole

UV-pole Counter term= Renormalization of the interpolating field

O(α s ) correction to dim5 term UV-divergence is subtracted by counter terms. Remaining IR- divergence is absorbed into the vacuum condensate.

Results for λ H 2 (μ=1GeV) (preliminary) ω th :continuum threshold : Grozin&Neubert : +dim6 : +dim6 +O(α s ) correction

Results for λ H 2 - λ E 2 (μ=1GeV) (preliminary) : Grozin&Neubert : +dim6 : +dim6 +O(α s ) correction ω th :continuum threshold

Choice of the reasonable M-range Criterion for M: Both of Higher order power corrections in OPE Continuum contribution should not be large (less than 30-50%). Reasonable range of M In this range,

Summary λ E and λ H (quark-gluon correlation inside the B-meson) play important role in B-meson’s LCDA. HQET sum rule for λ E and λ H  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 λ E and λ H. Renormalization group improvement etc. Matching the OPE of LCDA Estimation of the inverse moment of LCDA ( )

On the results  Contribution of dim.6 is less than 1% OPE seems to converge at this order.  O(α s )-correction to dim.5 is significantly large and tends to suppress λ H and λ E.  After inclusion of O(α s )-correction, stability of the splitting becomes worse.

Implication to B-meson wave function

(counter term) O(α s ) 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

Correlation function Representation of Π with hadronic states  B-meson pole at Formulation of HQET sum rule for B-meson

Matrix elements Two-body operator Three body operator

B-meson pole  2-independent Lorentz structures Write dispersion relations for

Borel transform HQET sum rule for λ E,H Spectral ansatz OPE of LHS HQET sum rules

Results for λ H 2 (μ=1GeV) (preliminary) ω th :continuum threshold : Grozin&Neubert : +dim6 : +dim6 +O(α s ) correction

Results for λ H 2 - λ E 2 (μ=1GeV) (preliminary) : Grozin&Neubert : +dim6 : +dim6 +O(α s ) correction ω th :continuum threshold

In a heavy(Q)-light(q) system, Pair creation of QQ cannot occur. The new field h v is constrained to satisfy QCD Lagrangian can be simplified to HQET ( Heavy Quark Effective Theory ) Q Light quark cloud Heavy quark