B γlν and the B meson distribution amplitude

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1 B γlν and the B meson distribution amplitude M. Beneke Institut für Theoretische Teilchenphysik und Kosmologie RWTH Aachen Colour meets Flavour Workshop on Quantumchromodynamics and Quark Flavour Physics On the occasion of Alex Khodjamirian s 60th birthday Siegen, October 2011 Work in progress with J. Rohrwild, and C. Hellmann and J. Rohrwild M. Beneke (RWTH Aachen), B γlν and λ B Colour & Flavour, 14 October /15

2 Introduction Γ(lν) f 2 B Γ(γlν) f 2 B «2 ml m B α em 4π «2 mb No helicity suppression. Simplest, non-trivial, hard-exclusive B decay when 2E γ = O(m b ). Involves B meson light-cone distribution amplitude: λ B 1 λ B (µ) = dω 0 ω Φ B+(ω, µ) Compare» Γ(πlν) ξ Bπ + αsf 2» Bf πφ π Γ(ππ) f πφ π C 2 F Bπ α sf B f 2 πφ π + C 1 m B λ B m B λ B M. Beneke (RWTH Aachen), B γlν and λ B Colour & Flavour, 14 October /15

3 λ B and hadronic B decays 0.4 ΠΡ R R πρ 00 = 2 Γ(B 0 π 0 ρ 0 ) Γ(B 0 π + ρ ) + Γ(B 0 π ρ + ) Λ B GeV (MB, Huber, Li, 2009) Whether QCD factorization for colour-suppressed, non-leptonic, charmless B decays works, depends on the value of λ B. M. Beneke (RWTH Aachen), B γlν and λ B Colour & Flavour, 14 October /15

4 B meson light-cone distribution amplitude (Grozin, Neubert, 1997; BBNS 1999/2000; MB, Feldmann, 2000) if stat(µ)φ B+ (ω, µ) = 1 dt e itω 0 ( q sy s)(tn )n/ γ 5 (Y s h v)(0) B v µ 2π 1 λ B (µ) = dω 0 ω Φ dω B+(ω, µ), σ n(µ) = λ B (µ) µ 0 ω lnn 0 ω Φ B+(ω, µ) Need only inverse, inverse-log moments, since ω Λ QCD. Not related to local operators. 1-loop renormalization known (Lange, Neubert, 2003). Not related to moments, since R 0 dω ω n... does not commute with renormalization. Three-particle amplitudes have been studied (Kawamura et al, 2001 ; Huang et al, 2006; Khodjamirian, Mannel, Offen, 2006; Geyer, Witzel, 2007; Descotes-Genon, Offen, 2009) λ B from QCD sum rules of various forms (Braun, Korchemsky, Ivanov, 2003; Khodjamirian, Mannel, Offen, 2005; Ball, Kou, 2003): λ B (1 GeV) = (460 ± 110) MeV [BKI] Models of the entire distribution amplitude (Lee, Neubert, 2005; Kawamura, Tanaka, 2009/2010) Upper limits on B γlν: λ B > 669 MeV (591 MeV) (BABAR, 2007); λ B > 300 MeV (BABAR, 2009); M. Beneke (RWTH Aachen), B γlν and λ B Colour & Flavour, 14 October /15

5 B γlν amplitude l νγ lγ µ (1 γ 5 )ν uγ µ(1 γ 5 )b B =» = ieɛ ν l ν lγ µ (1 γ 5 )ν) 0 d 4 x e iqx 0 T{j ν em(x)(uγ µ(1 γ 5 )b)(0)} B + d 4 x e iqx l ν T{j ν em(x)( lγ µ(1 γ 5 )ν)(0)} 0 0 uγ µ (1 γ 5 )b B = eɛ ν ū l γ µ(1 γ 5 )u ν T νµ (p, q) ieq l f B ū l /ɛ (1 γ 5 )u ν it νµ(p, q) = iɛ µνρσv ρ q σ F V (E γ) + (g µνv q v νq µ) ˆF A (E γ) + vνvµ v q f Bm B + q ν-terms Contact term fixed by the em Ward identity q νt νµ = if B p µ (Khodjamirian, Wyler, 2001). Replace (g µνv q v νq µ) ˆF A (E γ) + vνvµ v q f Bm B (g µνv q v νq µ) F A (E γ) + g µνf B to cancel the lepton emission term. Then F A = ˆF A + Q lf B E γ M. Beneke (RWTH Aachen), B γlν and λ B Colour & Flavour, 14 October /15

6 B γlν in QCD (Korchemsky, Pirjol, Yan, 1999;...) W W p p q l p q l l q γ q γ Intermediate light-quark propagator has hard-collinear virtuality m b Λ Intermediate heavy-quark propagator has hard virtuality m 2 b i(/q /l ) (q l) 2 = i /q 2q l Λ i(/p /q) + m b (p q) 2 m 2 b = i /q 2p q m b F V = F A = Qum Bf B 2E γλ B Photon inherits the helicity of the energetic up quark. Leading power. F V = F A = Q bm B f B 2E γm b Photon has opposite helicity, because the weak current couples to the spectator quark. Power-suppressed. M. Beneke (RWTH Aachen), B γlν and λ B Colour & Flavour, 14 October /15

7 Factorization at leading power in the heavy-quark mass expansion (Lunghi, Pirjol, Wyler, 2002; Bosch, Hill, Lange, Neubert, 2003; Descotes-Genon, Sachrajda, 2002) Integrate out hard virtualities: it νµ(p, q) = C(E γ, µ) d 4 x e iqx 0 T{[j ν,em(x)] SCET, ( ξw cγ µ(1 γ 5 )h v)(0) µ} B [j ν,em(x)] SCET = ( q sw c γ νξ)(x) h + i d i /n+ yt{ ξ γ ν i /D + i /D γ ν in +D c in +D c 2 ξ(y), ( qsw c i /D ξ)(x)} Integrate out hard-collinear virtualities: F V (E γ) = F A (E γ) = C(E γ, µ) dt J(E γ, t, µ) 0 ( q sy s)(tn )n/ γ 5 (Y s h v)(0) B v µ = Qum Bf B R(Eγ, µ) 2E γλ B (µ) where R can be calculated in perturbation theory and depends on σ i, i 2n at O(α n s ). M. Beneke (RWTH Aachen), B γlν and λ B Colour & Flavour, 14 October /15

8 Radiative corrections (NLL) (Descotes-Genon, Sachrajda, 2002; Lunghi, Pirjol, Wyler, 2002; Bosch, Hill, Lange, Neubert, 2003; MB, Rohrwild, 2011) R(E γ, µ) = C(E γ, µ h1 )K 1 (µ h2 ) U(E γ, µ h1, µ h2, µ) J(E γ, µ) C(E γ, µ) = 1 + αsc F 4π J(E γ, µ) = 1 + αsc F 4π 2 ln 2 2Eγ µ 2Eγ + 5 ln µ 3 2x 1 x ln x 2Li 2(1 x) 6 π 2 12 ln 2 2Eγ µ 0 µ 2 2σ 1 (µ) ln 2Eγ µ 0 µ 2 1 π! σ 2(µ). K(µ) = 1 + αsc F 4π 3 2 ln m! 2 b µ 2 2.! NLL renormalization-group evolution from the hard (m b ) to the hard-collinear scale ((m b Λ) 1/2 ) U(E γ, µ h1, µ h2, µ) = U 1 (E γ, µ h1, µ)u 2 (µ h2, µ) 1 µ d dµ U 1(E γ, µ h, µ) = Γ cusp(α s) ln µ «+ γ(α s) U 1 (E γ, µ h, µ) 2E γ Two-loop anomalous dimension of the heavy-light SCET current from (Bonciani, Ferroglia; Asatrian, Greub, Pecjak; MB, Huber, Li; Bell 2008) M. Beneke (RWTH Aachen), B γlν and λ B Colour & Flavour, 14 October /15

9 Size of radiative corrections (to the amplitude) Photon-energy dependence (and total scale uncertainty) Hard-collinear scale dependence (E γ = 2 GeV) M. Beneke (RWTH Aachen), B γlν and λ B Colour & Flavour, 14 October /15

10 1/m b power corrections Integrate out hard virtualities: it νµ(p, q) = d 4 x e iqx 0 T{[j ν,em(x)] SCET, (ūγ µ(1 γ 5 )b)(0) SCET } B isνµ local B m b Time-ordered products with sub-leading SCET interactions, subleading corrections to the currents, e.g. [ūγ µ(1 γ 5 )Q](0) ds 3X i=1 C (A) i (s)j (A),i µ (s) + ds ds and a local term. At tree level only emission off the heavy quark and i(/q /l ) (q l) 2 = i /q 2q l + i/l 2q l {z } power suppressed = i/n h + 4l 4X i=1 C (B) i (s, s )J (B),i µ (s, s ) il +/n + i/l 4E γl 2E γl {z } SCET time-ordered products + i/n i +, 4E γ {z} local Local operator is S local νµ = q sγ µνn/ ± h v f B. Beyond tree level: photon structure from collinear quark-photon coupling, three-particle B-meson distribution amplitudes, non-factorizable time-ordered products (work in progress) M. Beneke (RWTH Aachen), B γlν and λ B Colour & Flavour, 14 October /15

11 B γ form factors with radiative and power corrections F V (E γ) = F A (E γ) = Qum» Bf B R(Eγ, µ) + ξ(e γ) + Q bm B f B + Qum Bf B 2E γλ B (µ) 2E γm b (2E γ) 2, Qum» Bf B R(Eγ, µ) + ξ(e γ) Q bm B f B Qum Bf B 2E γλ B (µ) 2E γm b (2E γ) 2 + Q lf B. E γ Non-factorizable time-ordered product terms parameterized by ξ(e γ) = ± f B /(2E γ). Irreducible uncertainty in determination of λ B. Drops out in F V F A. 1.5 FA FV FA FV M. Beneke (RWTH Aachen), B γlν and λ B Colour & Flavour, 14 October /15

12 BABAR analysis of B γlν B = τ Bd de γ de l PS-Cuts d 2 Γ de γ de l (BABAR, Phys.Rev. D80 (2009) [ ]) (BABAR, ): E γ (0.45, 2.35) GeV, E l (1.875, 2.64) GeV, cos θ(l, γ) < 0.36 B < 1.7(2.3) 10 6 λ B > 669(591) MeV (BABAR, Phys.Rev. D80 (2009) [ ]): E γ > 1.0 GeV B < λ B > 0.3 GeV Nothing from BELLE!? But both analyses do not include radiative corrections and theoretical uncertainties. Wrong input for power corrections from (Pirjol, Korchemsky, Yan, 1999) M. Beneke (RWTH Aachen), B γlν and λ B Colour & Flavour, 14 October /15

13 Present limits B 10 6 Cuts corresponding the BABAR NLL + power corrections LO NLL B B B λ B > 669 (591) MeV 372 (331) MeV [incl. V ub ] 310 (274) MeV [excl. V ub ] λ B > 300 MeV MeV [excl. V ub ] with data from BABAR2009 But the first line needs to be taken with caution. Unpublished data analysis, E γ cut too low. No stringent constaints yet. M. Beneke (RWTH Aachen), B γlν and λ B Colour & Flavour, 14 October /15

14 Outlook SuperB factories should do this. BELLE? Br B Γ Ν E Γ 1 GeV E Γ 1.7 GeV B Example: Br (B γl ν, E γ > 1.7 GeV) = (2.0 ± 0.4) 10 6 λ B = MeV Dominant theoretical uncertainty about equally from σ 1, σ 2 and ξ. M. Beneke (RWTH Aachen), B γlν and λ B Colour & Flavour, 14 October /15

15 Conclusion B γlν can determine λ B at SuperB factories. Radiative corrections are under control at NLL. Interesting for study of power corrections. F V F A is power-suppressed and depends only on f B (to all orders? work in progress) M. Beneke (RWTH Aachen), B γlν and λ B Colour & Flavour, 14 October /15