Light-Cone Sum Rules with B-Meson Distribution Amplitudes

Transcription

1 Light-Cone Sum Rules with B-Meson Distriution Amplitudes Alexander Khodjamirian (University of Siegen) (with Thomas Mannel and Niels Offen) Continuous Advances in QCD, FTPI, Minneapolis, May 11-14, 2006

2 QCD light-cone sum rules (LCSR) Balitsky, Braun, Kolesnichenko; Braun, Filyanov (1989); Chernyak, Zhitnisky(1990) allow to calculate hadronic form factors: several important applications to exclusive B decays ased on: vacuum hadron correlators OPE near the light-cone, inputs: light-cone distriution amplitudes of π, K, ρ,... dispersion relations parton-hadron duality remake of QCD sum rules: Shifman, Vainshtein, Zakharov (1979) ased on vacuum vacuum correlators, OPE in local operators, inputs: quark/gluon condensates

3 LCSR for B π form factor π π π d u p + q q The correlation function: q 2, (p+q) 2 m 2, -quark highly virtual F λ (q, p) = i d 4 xe iqx π(p) T {ū(x)γ λ (x), (0)iγ 5 d(0)} 0 operator-product-expansion (OPE) near the light-cone, x 2 0

4 OPE near the light-cone schematically, F (B) (q, p) = i d 4 xe iqx { [S0 (x 2, m 2, µ) + α s S 1 (x 2, m 2, µ) ] π(p) ū(x)γd(0) 0 µ dv S(x 2, m 2, µ, v) π(p) ū(x)g(vx) Γd(0)} 0 µ } +... * S 0,1, S - perturative amplitudes, (virtual -quark) * universal distriution amplitudes of π (or K, ρ, K ): 1 π(q) ū(x)[x, 0]γ µ γ 5 d(0) 0 x2 =0 = iq µ f π du e iuqx ϕ π (u) + O(x 2 ). * the expansion (light-cone OPE) goes over α s (µ) and powers of 1/µ 2 ; * typical scale µ 2 m Λ, where Λ 1 GeV Λ QCD 0

5 Derivation of LCSR Hadronic dispersion relation in (p + q) 2 : (q 2 m 2 fixed) π π F(q 2,(p + q) 2 ) = B B u q + h B h B h u q p + q p + q f B f + Bπ (q2 ) B h duality (s B 0 ) [f Bπ (q 2 )] LCSR includes oth soft (end-point) and hard ( α s ) contriutions, valid at 0 < q 2 < m 2 B µ2 more details/results: talk y Roman Zwicky

6 New approach: LCSR with B meson DA A.K., T. Mannel, N.Offen PLB(2005), hep-ph/ also (in SCET): F. De Fazio, T. Feldmann and T. Hurth; hep-ph/ The inversed correlator: B meson on-shell, pion interpolated with an axial current duality F µν (B) (p, q) = i d 4 x e ip x 0 T { d(x)γµ γ 5 u(x), ū(0)γ ν (0) } B 0 (p + q). B d u p q B d u B d u q 2 = 0, p 2 < 0, p 2 Λ 2 QCD, u-quark propagates near LC.

7 OPE near the light-cone schematically, { (on-shell B meson with v = p B /m ) [S0 F(q, p) = i d 4 xe iqx (x 2, µ) + α s S 1 (x 2, µ) ] 0 d(x)γ(0) B(v) µ dv S(x 2, m 2, µ, v) 0 d(x)g(vx) Γ(0)} B(v) µ } +... * S 0,1, S - perturative amplitudes, ( virtual u-quark) * universal distriution amplitudes of B(v) : 0 d(x)[x, 0]Γ(0) B(v) x2 =0 first uses of B-meson DA in PQCD factorization for B π. A. Szczepaniak, E. M. Henley and S. J. Brodsky (1990). R. Akhoury, G. Sterman and Y. P. Yao (1994)

8 B-Meson two-particle DA: the definition. A.G.Grozin. M.Neuert (1997) x x x x B 0 (p B ) Light-cone matrix element, consistent with HQET d 0 T { dα (x)[x, 0] β (0) } B 0 (v) x 2 =0 = if { Bm B (1 + /v)γ 5 dωe iωv x φ B 4 +(ω) + φb +(ω) φ B } (ω) /x 2v x 0 [x, 0]-Wilson line, two normalized DA s φ B +(ω) and φ B (ω), variale ω = (l 0 + l 3 ): (l-light spectator momentum in B rest frame) γ 5 βα,

9 Factorization in B γlν l (p l + p ν ) 2 0, E γ m B /2 A(B γlν) dωφ B +(ω)t h (ω) γ T h 1/ω, 1/λ B = dω φb + (ω) 0 ω B u l - the inverse moment ν G. P. Korchemsky, D. Pirjol, T. M. Yan (2000) S. Descotes-Genon, C. T. Sachrajda (2003)] S. W. Bosch, R. J. Hill, B. O. Lange, M. Neuert (2004) factorization in B π, B h 1 h 2 etc.

10 Quark-antiquark-gluon DA s: definition. H. Kawamura, J. Kodaira,. C.F.Qiao and K. Tanaka,(2001) x x x x x B 0 (p B ) d 0 d α (x)g λρ (ux) β (0) B 0 (v) = f Bm B dω dξ e i(ω+uξ)v x [ { ( ) (1 + /v) (v λ γ ρ v ρ γ λ ) Ψ A (ω, ξ) Ψ V (ω, ξ) iσ λρ Ψ V (ω, ξ) ( ) ( ) }] xλ v ρ x ρ v λ xλ γ ρ x ρ γ λ X A (ω, ξ) + Y A (ω, ξ) v x v x βα.

11 What do we know aout B-meson DA s model-independent relations from QCD equation of motion, e.g. Wandzura-Wilczek-type: φ B (ω) = dρ φb +(ρ) + dωdξ{ψ V,A (ω, ξ)} φ B ρ (0) = 1/λ B + {corr.} ω oundary conditions: ω 0: φ B +(ω) ω, φ B (0) = const Evolution of φ B +(ω, µ) calculated in HQET is nontrivial, φ(ω) log(ω/µ)/ω, radiative tail [M. Neuert, B. Lange, (2003)] lim ω no parton interpretation, positive moments divergent, ut λ B (µ) well defined in O(α s ) no prolem for the new sum rules containing integrals over small ω < s 0 /m B

12 models of φ B ±(ω) ased on QCD sum rules in HQET [ A. G. Grozin and M. Neuert (1997)] The correlator for φ B +(ω) : i d 4 xe ik(vx) 0 T{O + (t) h v (x)γ 2 q(x)} 0 = {...}T(t, k). O + (t) = q(tn) n [tn, 0]Γ h v (0), k < 0 - external (Euclidean) momentum variale, k = Λ is B meson pole in HQET, {...} - a trace loop condensate simple ansatz φ B +(ω) = (ω/ω 2 0)e ( ω/ω 0), φ B (ω) = (1/ω 0 )e ( ω/ω 0), A hyrid model for φ(ω, µ) (exponent.ansatz with the radiative tail) S.J. Lee, M. Neuert, (2005)

13 NLO calculation (including radiative corrections) [ V. M. Braun, D. Y. Ivanov and G. P. Korchemsky,(2003)] the sum rule fitted to an explicit ansatz[ for ϕ B +(ω), φ B +(ω, µ = 1 GeV) = 4λ 1 B ω 1 π ω ω (σ B 1) π 2 ] lnω (ω in units of GeV) λ B = (460 ± 110)MeV, σ B = 1.4 ± 0.4 at µ = 1 GeV 0.8, φ B +(ω, µ) [GeV 1 ] ω [GeV] solid (dashed) is the Lee-Neuert (Braun-Ivanov-Korchemsky)

14 Deriving the simplest sum rule OPE result for B π, the LO diagram: only φ B (ω) contriutes F (B) µν = 2if B Hadronic dispersion relation: 0 dω m B ω p 2φB (ω)p µ p ν +..., F (B) µν = 0 dγ µ u π(p) π(p) ūγ ν B(p + q) if π f + Bπ = (0) ρ h (s) p 2 + ds s p 2 p µp ν +..., apply duality in pion channel Borel transformation. s h

15 The relation etween B meson parameters: (using s 0 π m 2 B ): 1 λ B f π f + Bπ (0)m B f B M 2 (1 e sπ 0 /M2 ). inputs: use LCSR for B π form factor (in terms of pion DA s), 2pt sum rule for f B and predict λ B 3-particle B meson DA s, enter 1) soft-gluon diagram 2) indirectly, violation of WW relation estimated - a few %

16 Summary on the inverse moment 1/λ B = dω φb + (ω) 0 ω renorm. scale 1 GeV Method λ B [MeV] Ref. 2pt SR in HQET,LO 350 Grozin,Neuert 2pt SR in HQET, NLO 440 ± 110 Braun, Ivanov,Korchemsky LCSR for B γlν l 600 Ball, Kou inverted LCSR for B π 460 ± 160 A.K.,Mannel, Offen first moments +Ansatz 480 ± 55 Lee, Neuert

17 the new method allows to calculate many different B light form factors in one go, including SU(3) reaking, m q = m s the main advantage: knowledge of pion, K, ρ, K DA s not needed, decay constants, duality thresholds from exp. and/or two-point (SVZ) SR contriutions of 3-particle DA s of B meson suppressed y powers of s 0 /m B (work in progress), preliminary (sample) results for all major heavy-light form factors: inputs: 2-particle DA,s φ B ±(ω), Grozin-Neuert exponential ansatz, λ B = 440 MeV, f B = 180 MeV, M 2 = 1 GeV, m s (1GeV ) = 130MeV more detailed analysis in progress compared with the results of conventional LCSR [BZ] y P. Ball, R. Zwicky, (2005) (B π, K, ρ, K )

18 Form Factor LCSR LCSR Ref. with B DA (prelim) with light-meson DA s f + Bπ (0) 0.267± 0.258± 0.03 [BZ] 0.26± 0.05 [KMMM] 0.25± 0.05 [AGRS] f + BK (0) 0.328± 0.301± [BZ] fbπ(0) T 0.24± 0.253± [BZ] fbk(0) T 0.305± 0.328± 0.04 [BZ] V Bρ (0) 0.382± 0.323± [BZ] V BK (0) 0.442± 0.411± [BZ] A Bρ 1 (0) 0.281± 0.242± [BZ] A BK 1 (0) 0.328± 0.292± [BZ] A Bρ 2 (0) 0.253± 0.221±0.023 [BZ] A BK 2 (0) 0.304± 0.259±0.027 [BZ] T Bρ 1 (0) 0.323± 0.267±0.021 [BZ] T1 BK (0) 0.375± 0.333±0.028 [BZ] *[KMMM] A. K.,T. Mannel, M. Melcher and B. Melic, PRD (2005), hep-ph/ *[AGRS] Arnesen, Grinstein, Rothstein, Stuart, hep-ph/

19 Conclusions new type of LCSR, calculating B light form factors: already leading order has a good agreement with LCSR with light meson DA s quantitative estimates of SU(3) reaking effects; sensitivity to the inverse moment λ B, values < 300MeV disfavored