B s µ + µ and B X s γ

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1 B s µ + µ and B X s γ Miko laj Misiak (University of Warsaw) 1. Introduction 2. B X s γ: 2a. (Photonic dipole) (four quark) vertex interference at the NNLO for m c = 0 2. Estimating the uncertainties 2c. Update of the SM prediction 3. B s µ + µ : electroweak-scale matching at the NNLO 4. Summary

2 Information on electroweak-scale physics in the sγ transition is encoded in an effective low-energy local interaction: γ s C 7 B ( B 0 or B ) The inclusive B X s γ decay rate is well approximated y the corresponding perturative decay rate of the -quark: Γ( B X s γ) E γ>e 0 = Γ( X p s γ) E γ>e 0 + provided E 0 is large (E 0 m /2) ut not too close to the endpoint (m 2E 0 Λ QCD ). Conventionally, E 0 = 1.6 GeV m /3 is chosen. non-perturative effects (2 ± 5)% Benzke et al., arxiv:

3 The raw photon energy spectra in the inclusive measurements Weights / 100 MeV 40 0 Data Spectator Model CLEO hep-ex/ Photons / 50 MeV a) BELLE arxiv: PRL 87 (2001) PRL 103 (2009) E γ > 2.0 GeV E γ > 1.7 GeV Events/0.1 GeV E (GeV) BB control E γ * (GeV) Continuum control BABAR arxiv: PRL 109 (2012) E γ > 1.8 GeV E c.m.s γ [GeV] The peaks are centered around 1 2 m 2.35 GeV which corresponds to a two-ody sγ decay. Broadening is due to (mainly): perturative gluon remsstrahlung, motion of the quark inside the B meson, motion of the B meson in the Υ(4S) frame.

4 The HFAG average ( ) B( B X s γ) EXP Eγ>1.6 GeV = (3.43 ± 0.21 ± 0.07) 10 4 includes the following measurements: Reference Method # of B B E 0 [GeV] B 10 4 at E 0 CLEO [PRL 87 (2001) ] inclusive ± 0.41 ± 0.26 BABAR [PRL 109 (2012) ] inclusive ± 0.15 ± 0.29 ± ± 0.14 ± 0.19 ± ± 0.12 ± 0.14 ± 0.04 BELLE [PRL 103 (2009) ] inclusive ± 0.15 ± ± 0.13 ± ± 0.11 ± ± 0.10 ± 0.11 BABAR [PRD 77 (2008) ] inclusive with , ± 0.85 ± 0.60 a hadronic tag which gives ± 0.64 ± 0.47 (hadronic ± 0.48 ± 0.35 decay of the tagged ± 0.38 ± 0.27 recoiling B ( B)) events ± 0.30 ± 0.20 BABAR [PRD 86 (2012) ] semi-inclusive ± 0.19 ± 0.48 BELLE [PLB 511 (2001) 151] semi-inclusive ± 0.58 ± 0.46 ± 0.60

5 Comparison of the inclusive measurements of B( B X s γ) y CLEO, BELLE and BABAR for each E 0 separately B 10 4 for each E 0 [GeV] Averages for each E 0 extrapolated to E 0 = 1.6 GeV using the HFAG factors BELLE, arxiv: MB B HFAG BABAR, arxiv: MB B CLEO, hep-ex/ MB B SM, hep-ph/ The HFAG factors Scheme E γ < 1.7 E γ < 1.8 E γ < 1.9 E γ < 2.0 E γ < Kinetic ± ± ± ± ± Neuert SF ± ± ± ± ± Kagan-Neuert ± ± ± ± ± Average ± ± ± ± ± Why do we need to extrapolate to lower E 0? Are the HFAG factors trustworthy?

6 Decoupling of W, Z, t, H 0 effective weak interaction Lagrangian: L weak Σ C i (µ) Q i 8 operators matter in the SM when the higher-order EW and/or CKM-suppressed effects are neglected: c L c L s L s L R γ L R g s L q L q s L Q 1,2 Q 7 Q 8 Q 3,4,5,6 current-current photonic dipole gluonic dipole penguin Γ( B X s γ) Eγ >E 0 = C 7 2 Γ 77 (E 0 ) + (other) Optical theorem: Integrating the amplitude A over E γ : dγ 77 de γ Im{ q B γ 7 X s 7 γ B q } ImA Im E γ E 0 E max γ 1 2 m B Re E γ OPE on the ring Non-perturative corrections to Γ 77(E 0 ) form a series in Λ QCD m and α s that egins with µ 2 π m 2, µ2 G m 2, ρ3 D m 3, ρ3 LS m 3,...; α s µ 2 π (m 2E 0 ) 2, α s µ 2 G m (m 2E 0 ) ;..., where µ π, µ G, ρ D, ρ LS = O(Λ QCD ) are extracted from the semileptonic B X c e ν spectra and the B B mass difference.

7 The relevant perturative quantity: Γ[ X s γ] E γ>e 0 V c /V u 2 Γ[ X u e ν] = Vts V t 2 6αem V c π i,j C i C j K ij Expansions of the Wilson coefficients and K ij : C i (µ ) = C (0) i (µ ) + α s(µ ) 4π C(1) i (µ ) + ( ) αs (µ ) 2 (2) 4π C i (µ ) +... K ij = K (0) ij + α ( ) s(µ ) 4π K(1) ij + αs (µ ) 2 (2) 4π K ij +... Most important at the NNLO: K (2) 77, K(2) 27 and K(2) 17. µ m 2 They depend on µ m, E 0 m and r = m c m.

8 Evaluation of K (2) and K(2) 17 for m c = 0: 27 [M. Czakon, P. Fiedler, T. Huer, M. Misiak, T. Schutzmeier, M. Steinhauser, arxiv:1305.nnnn] c c c c g g g g Q 1,2 s Q 7 Q 1,2 s Q 7 g c c g q c c q g g Q 1,2 s Q 7 Q 1,2 s Q 7 g

9 Master integrals and differential equations: n D n OS n eff n massless 2-particle cuts particle cuts particle cuts Im(z) NI d dz I i(z) = j R ij(z)i j (z), z = p2 m 2. PLE 0 PLE 1 Re(z) Boundary conditions in the vicinity of z = 0: (a) () +

10 Massless integrals for the oundary conditions: 2PCuts 3PCuts 4PCuts 1L2C1 4L4C1 4L4C2 2L2C1 2L3C1 3L2C1 3L3C1 4L4C3 4L4C4 4L2C1 4L2C2 4L3C1 4L3C2 4L3C3 4L4C5 4L4C6 4L2C3 4L2C4 4L3C4 4L3C5 4L3C6 4L2C5 4L2C6 4L3C7 4L3C8 4L3C9 4L4C7 4L4C8

11 The final results: K (2) 27 (r, E 0) = A 2 + F 2 (r,e 0 ) + 3f q (r, E 0 ) + f (r) + f c (r) φ(1) 27 (r, E 0) lnr [ + (4L c x m ) r d ] dr + x d me 0 f NLO (r,e 0 ) de 0 81 x m ( K(1) K(1) K(1) K(1) ) L L2, ) K (2) 17 (r, E 0) = 1 6 K(2) 27 (r,e 0) + A 1 + F 1 (r,e 0 ) + ( K(1) K(1) 78 L L2, where F i (0, 0) 0, A , A from the current calculation. Impact of the unknown F i (r, E 0 ) on the ranching ratio: [ ] 1 { U(r,E 0) C (0)eff 7 (µ ) C (0) 1 (µ )F 1 (r, E 0 ) + B B U(0, 0) = 0, ( C (0) 2 (µ ) 1 6 C(0) 1 (µ ) U(r,E 0 ) at large r: [MM, M. Steinhauser, hep-ph/ , arxiv: ]. ) } F 2 (r, E 0 ).

12 Impact of the unknown U(r,E 0 ) on the ranching ratio: B B U(r,E 0), U(0, 0) = 0, asymptotic ehaviour of U(r,E 0 ) at large r is known r phys U(r,0) r = m / m Dashed line: U(r, 0) = lnr ln 2 r + O ( ) 1 2r c

13 Impact of the unknown U(r,E 0 ) on the ranching ratio: B B U(r,E 0), U(0, 0) = 0, asymptotic ehaviour of U(r,E 0 ) at large r is known r phys U(r,0) r = m / m Dashed line: U(r, 0) = lnr ln 2 r + O ( ) 1 2r Solid line: f q (r, 0) f NLO (r, 0) r d dr f NLO(r, 0) c

14 Impact of the unknown U(r,E 0 ) on the ranching ratio: B B U(r,E 0), U(0, 0) = 0, asymptotic ehaviour of U(r,E 0 ) at large r is known r phys U(r,0) r = m / m Dashed line: U(r, 0) = lnr ln 2 r + O ( ) 1 2r Solid line: f q (r, 0) f NLO (r, 0) r d dr f NLO(r, 0) Estimate at r phys : (0 ± 3)% c

15 Impact of the unknown U(r,E 0 ) on the ranching ratio: B B U(r,E 0), U(0, 0) = 0, asymptotic ehaviour of U(r,E 0 ) at large r is known r phys U(r,0) Small effects of shifting E 0 from 0 to 1.6 GeV: +0.5% in f NLO and +1.5% in f q at r phys, 0.7% in f NLO and 2.4% in f q at r = 0. 0 r = m / m Dashed line: U(r, 0) = lnr ln 2 r + O ( ) 1 2r Solid line: f q (r, 0) f NLO (r, 0) r d dr f NLO(r, 0) Estimate at r phys : (0 ± 3)% c

16 Once U(r, E 0 ) has een neglected, no interpolation in m c is present in the current calculation. Interferences not involving the photonic dipole operator are treated as follows: K 22 : (and analogous K 11 & K 12 ) 2 2 K 28 : (and analogous K 18 ) c c c c c c c c c K 88 : Two-particle cuts are known (just NLO 2 ). Three- and four-particle cuts are known in the BLM approximation only. The NLO+(NNLO BLM) corrections are not ig (+3.8%).

17 Incorporating other perturative contriutions evaluated after the previous phenomenological analysis in hep-ph/ : 1. Four-loop mixing (current-current) (gluonic dipole) M. Czakon, U. Haisch and M. Misiak, JHEP 0703 (2007) 008 [hep-ph/ ] 2. Diagrams with massive quark loops on the gluon lines R. Boughezal, M. Czakon and T. Schutzmeier, JHEP 0709 (2007) 072 [arxiv: ] H. M. Asatrian, T. Ewerth, H. Garielyan and C. Greu, Phys. Lett. B 647 (2007) 173 [hep-ph/ ] T. Ewerth, Phys. Lett. B 669 (2008) 167 [arxiv: ] 3. Complete interference (photonic dipole) (gluonic dipole) H. M. Asatrian, T. Ewerth, A. Ferroglia, C. Greu and G. Ossola, Phys. Rev. D 82 (2010) [arxiv: ] 4. New BLM corrections to contriutions from 3-ody and 4-ody final states for interferences not involving the photonic dipole: A. Ferroglia and U. Haisch, Phys. Rev. D 82 (2010) [arxiv: ] M. Misiak and M. Poradziński, Phys. Rev. D 83 (2011) [arxiv: ] 5. LO contriutions from sγq q, (q = u, d, s) from the four quark operators ( penguin ones or CKM-suppressed ones). M. Kamiński, M. Misiak and M. Poradziński, Phys. Rev. D 86 (2012) [arxiv: ] Taking into account new non-perturative analyses: M. Benzke, S. J. Lee, M. Neuert and G. Paz, JHEP 1008 (2010) 099 [arxiv: ] T. Ewerth, P. Gamino and S. Nandi, Nucl. Phys. B 830 (2010) 278 [arxiv: ] Updating the parameters: P. Gamino, C. Schwanda, to e pulished

18 Update of the SM prediction (preliminary) B( B X s γ) SM Eγ>1.6 GeV = (3.14 ± 0.22) 10 4 Hardly Contriutions to the total TH uncertainty (summed in quadrature): 5% non-perturative, 3% from the unknown U(r,E 0 ) any change w.r.t (3.15±0.23) 10 4 in (several 1 2% corrections cancel y chance). 3% higher order O(α 3 s), 2.2% parametric

19 Update of the SM prediction (preliminary) B( B X s γ) SM Eγ>1.6 GeV = (3.14 ± 0.22) 10 4 Hardly Contriutions to the total TH uncertainty (summed in quadrature): 5% non-perturative, 3% from the unknown U(r,E 0 ) any change w.r.t. (3.15±0.23) 10 4 in (several 1 2% corrections cancel y chance). 3% higher order O(αs), 3 2.2% parametric Mnemotechnics: 1 year π 10 7 s B( B X s γ) π 10 4 B inst (B s µ + µ ) π 10 9 (?)

20 Update of the SM prediction (preliminary) B( B X s γ) SM Eγ>1.6 GeV = (3.14 ± 0.22) 10 4 Hardly Contriutions to the total TH uncertainty (summed in quadrature): 5% non-perturative, 3% from the unknown U(r,E 0 ) any change w.r.t (3.15±0.23) 10 4 in (several 1 2% corrections cancel y chance). 3% higher order O(α 3 s), 2.2% parametric Experimental world average (HFAG, ): B( B X s γ) EXP Eγ>1.6 GeV = (3.43 ± 0.21 ± 0.07) 10 4 Experiment agrees with the SM at etter than 1σ level. Uncertainties: TH 7%, EXP 6.5%.

21 Comparison with the m c -interpolation in hep-ph/ i,j C(0) i C (0) j K (2) ij P (2) 2 = P (2)β P (2)rem (c) () P (2)rem (a) Red line goes to the true oundary at m c = 0. P (2)rem 2 at m c = 0 in the red line case: 77, 78 exact, 17, 27 with no cut on the photon energy, without Q 7 two-ody final states only.

22 Renormalization scale dependence of B( B X s γ) E γ>1.6 GeV B 10 4 B LO LO NLO 3.4 NLO 3.2 NNLO 3.2 NNLO matching scale µ low-energy scale µ B 10 4 NLO LO Central values: µ 0 = 160 GeV µ = 2.5 GeV µ c = 1.5 GeV 3.2 NNLO charm mass renormalization scale µ c

23 B s µ + µ flavour physics highlight of the LHC It is a strongly suppressed, loop-generated process in the SM. Its CP-averaged time-integrated ranching ratio (with final-state photon remsstrahlung included) reads: B SM = (3.67 ± 0.21) 10 9 B inst = (3.42 ± 0.21) 10 9 efore including the NNLO QCD and the full NLO EW matching corrections. It is very sensitive to new physics even in models with Minimal Flavour Violation (MFV). Enhancements y orders of magnitude are possile even when constraints from all the other measurements are taken into account. It has a clear experimental signature: peak in the dimuon invariant mass. First evidence (3.5σ) for its oservation has een recently announced y the LHC Collaoration (arxiv: ): B exp = ( ) 10 9.

24 Error udget for the CP-averaged and time-integrated ranching ratio (κ 1 = 1 τ Bs Γ s /2) K. de Bruyn et al., B(B s µ + µ ) SM = G4 F M4 W m2 µ M Bs 8π 5 Phys. Rev. Lett. 109 (2012) { ( ) ] } Vt V ts 2 κτ Bs fb 2 m 2 2 s [Y t 0 + O(α s ) + O(α em ) + O(α 2 s ) + O(αem ) = M 2 W }{{}}{{}}{{}}{{}}{{} ±3.5% ±1.1% ±( )% ±1.7%(for m t (m t ) = 165(1) GeV) ± 3%(to e removed) (3.67 ± 0.21) 10 9 for f Bs = 225.0(3.0) MeV [HPQCD+, arxiv: ] }{{} 5.7% (4.25 ± 0.39) 10 9 for f Bs = 242.0(9.5) MeV [FNAL/MILC, arxiv: ] }{{} 9.2% The O(α s ) corrections enhance the ranching ratio y around +2.2% when m t (m t ) is used at the leading order. G. Buchalla and A.J. Buras, Nucl. Phys. B 400 (1993) 225, MM and J. Uran, Phys. Lett. B 451 (1999) 161, G. Buchalla and A.J. Buras, Nucl. Phys. B 548 (1999) 309. Logarithmically ( ln ( m 2 t /m2 )) enhanced electromagnetic corrections and the known electroweak corrections suppress the ranching ratio y around 1.7%. G. Buchalla, A. J. Buras, Phys. Rev. D 57 (1998) 216, C. Boeth, P. Gamino, M. Gorahn, U. Haisch, JHEP 0404 (2004) 071, T. Huer, E. Lunghi, MM, D. Wyler, Nucl. Phys. B 740 (2006) 105, A. J. Buras, J. Girrach, D. Guadagnoli, G. Isidori, Eur. Phys. J C72 (2012) Another oservale: (with different NP sensitivity) B(B s µ + µ ) M Bs τ Bs A. J. Buras, Phys. Lett. B 566 (2003) 115..

25 Evaluation of the NNLO QCD matching corrections [T. Hermann, M. Misiak and M. Steinhauser, to e pulished] Conclusion: Comination with the NLO EW calculation is necessary [C. Boeth, M. Gorahn, E. Stamou, to e pulished]

26 Summary The dominant NNLO corrections to B( B X s γ) are now known not only in the large m c limit, ut also at m c = 0. The updated (preliminary) SM prediction reads B( B X s γ) = (3.14 ± 0.22) 10 4 for E 0 = 1.6GeV. Comining the recently calculated NNLO QCD and NLO EW corrections to B( B s µ + µ ) will allow for a significant reduction of the residual perturative uncertainties.

27 BACKUP SLIDES

28 ( ) αs µ The O 2 π (m 2E 0 ) 2 Γ 77 (E 0 ) = Γ tree 77 and O ( α s µ 2 G m (m 2E 0 ) ) corrections { 1 + (pert. corrections) µ2 π 2m 2 [T. Ewerth, P. Gamino and S. Nandi, arxiv: ] 3µ2 G (µ) 2m 2 [ )] 1 + α s (f π 1 (E 0 ) 43 ln µm [ )]} 1 + α s (f π 2 (E 0 ) + 16 ln µm f 1 f E 0 [GeV] -2.5

29 When (m 2E 0 ) Λ Λ QCD, no OPE can e applied. Local operators Non-local operators Non-perturative parameters Non-perturative functions d de γ Γ 77 = N H(E γ ) M B 2Eγ 0 dk P(M B 2E γ k) F(k)+O pert. pert. non-pert. ( Λm) Photon spectra from models of F(k) [Ligeti, Stewart, Tackmann, arxiv: ] (dγs/deγ) / 7 incl (Γ0s C 2 ) [GeV 1 ] c 3 =c 4 =0 c 3 =±0.15, c 4 =0 c 3 =0, c 4 =±0.15 c 3 =±0.1, c 4 =± E γ [GeV] The function F(k) is: perturatively related to the standard shape function S(ω), exponentially suppressed for k Λ, positive definite, constrained y measured moments of the B X c e ν spectrum (local OPE), constrained y measured properties of the B X u e ν and B X s γ spectra (not imposed in the plot).

30 Upgrading the HFAG factors y fitting F(k) to data: The SIMBA Collaoration [arxiv: , arxiv: ] (work in progress) F(k) = 1 λ [ n=0 c nf n ( k λ )] 2, fn asis functions. Truncate and fit. Another way: F(k) = A(k)B(k) and use the SIMBA approach for B(k). ր perfect fit Why do we need to upgrade the HFAG factors? The old models (Kagan-Neuert 1998,...) are not generic enough (too few parameters). ( ) Inclusion of O Λ effects and and taking other operators (Q m i Q 7 ) into account is necessary [Benzke, Lee, Neuert, Paz, arxiv: ]. What aout just fitting C 7 without extrapolation any particular E 0? Fine, ut measurements at low E 0 (even less precise) are still going to e crucial for constraining the parameter space. The fits are going to give the extrapolation factors anyway. Pulishing them is necessary for cross-checks/upgrades y other groups.

31 Non-perturative effects in the presence of other operators (Q i Q 7 ) d de γ Γ( B X s γ) = (Γ 77 -like term) + ÑE 3 γ [Benzke, Lee, Neuert, Paz, arxiv: ]. Re ( C i C j) Fij (E γ ). Remarks: The SCET approach is valid for large E γ only. It is fine for E γ > E m 1.6 GeV. Lower cutoffs are academic anyway. For such E 0, non-perturative effects in the integrated decay rate are estimated to remain within 5%. They scale like: i j Λ2, Λ2 m 2 m 2 c (known), soft u, c Λ Vus V u m V ts V t (negligile), 2 s s 7 Λ, Λ2 Λ, α m m 2 s ut suppressed y tails of suleading shape functions ( 27 ), m α s Λ m to e constrained y future measurements of the isospin asymmetry ( 78 ), α s Λ m ut suppressed y Q 2 d = 1 9 ( 88 ). Extrapolation factors? Tails of suleading functions are less important for them.

32 Perturative expansion of the Wilson coefficients: ( 4π C(1) αs (µ) 4π C i (µ) = C (0) i (µ) + α s(µ) Branching ratio: i (µ) + ) 2 C (2) i (µ) +... B( B X s γ) E γ>e 0 = B( B V X c e ν) ts V t 2 6αem exp V c π C [P(E 0) + N(E 0 )] Γ[ X s γ] E γ>e 0 V c /V u 2 Γ[ X u e ν] = Vts V t 2 6αem V c π P(E 0 ), C = Perturative expansion of K ij : P(E 0 ) = i,j C i C j K ij K ij = K (0) ij + α ( ) s(µ ) 4π K(1) ij + αs (µ ) 2 (2) 4π K ij +... Perturative expansion of P(E 0 ): P = P (0) + α s 4π ( P (1) 1 + P (1) 2 (r) ) + ( ) αs (µ) 2 ( 4π P (2) P (1) 1, P (2) 3 C (0) i C (1) j, P (1) 2, P (2) 2 C (0) i C (0) j, P (2) 1 pert. 1 + P (2) 2 r = m c m Most important at the NNLO: K (2) 77, K(2) 27 non-pert. V u 2 Γ[ B Xc e ν] V c Γ[ B X u e ν] µ m 2 (r) + P (2) 3 (r) ) ( C (0) i C (2) j, C (1) i C (1) j ) and K(2) 17.

33 Perturative evaluation of Γ( X p sγ) at µ m Γ( X p sγ) Eγ > E 0 = G2 F m5 α em 32π 4 LO: G 77 = 1 7 γ s 2 V tsv t 2 γ 8 i,j=1 2. C i (µ )C j (µ )G ij (E 0,µ ) Other LO are small, e.g.,: s NLO: 1996: Quasi-complete G ij { [Greu, Hurth, Wyler, 1996] [Ali, Greu, ] 2002: Complete ( ) G ij { [Buras, Czarnecki, Uran, MM, 2002] [Pott, 1995] ( ) Up to sq qγ channel contriutions involving diagrams similar to the aove LO one. They get suppressed y α s C 3,4,5,6 and phase-space for E 0 m /3. NNLO: We are still on the way to the quasi-complete case: s u,d,s u,d,s [Kamiński, Poradziński, MM, 2012] γ G 77 is fully known: [Blokland et al., 2005] [Melnikov, Mitov, 2005] [Asatrian et al., ] G 78 is fully known: [Asatrian et al., arxiv: ]

34 The most troulesome NNLO contriution to G ij : c c c G 27 : (and analogous G 17 ) m c = 0: Czakon, Fiedler, Huer, Misiak, Schutzmeier Steinhauser, to e pulished] 163 massive 4-loop on-shell master integrals (with cuts). The m c m /2 limit is known [Steinhauser, MM, 2006]. The BLM approximation is known for aritrary m c : { [Bieri, Greu, Steinhauser, 2003], [Ligeti, Luke, Manohar, Wise, 1999]. Towards G 27 at the NNLO for aritrary m c. [M. Czakon, R.N. Lee, A. Rehman, M. Steinhauser, A.V. Smirnov, V.A. Smirnov, MM] in progress. 1. Generation of diagrams and performing the Dirac algera to express everything in terms of four-loop two-scale scalar integrals with unitarity cuts. 2. Reduction to master integrals with the help of Integration By Parts (IBP). Availale C++ codes: FIRE REDUZE [A.V. Smirnov, arxiv: ] (pulic in the Mathematica version only), [C. Studerus, arxiv: ], DiaGen/IdSolver [M. Czakon, unpulished (2004)]. The IBP for 2-particle cuts has just een completed with the help of FIRE: 0.5 TB RAM has een used 1 month at CERN and KIT. Numer of master integrals: around 500.

35 3. Extending the set of master integrals I n so that it closes under differentiation with respect to z = m 2 c/m 2. This way one otains a system of differential equations d dz I n = Σ k w nk (z, ǫ)i k, ( ) where w nk are rational functions of their arguments. 4. Calculating oundary conditions for ( ) using automatized asymptotic expansions at m c m. 5. Calculating three-loop single-scale master integrals for the oundary conditions using dimensional recurrence relations [R.N. Lee, arxiv: ]. 6. Solving the system ( ) numerically [A.C. Hindmarsch, along an ellipse in the complex z plane. Doing so along several different ellipses allows us to estimate the numerical error. This algorithm has already een successfully applied for diagrams with (massless and massive) quark loops on the gluon lines where = 103 master integrals were present. [R. Boughezal, M. Czakon, T. Schutzmeier, arxiv: ]

36 Non-perturative contriutions from the photonic dipole operator alone ( 77 term) are well controlled for E 0 = 1.6 GeV: ) ( O O O [Bauer, 1997], O αs Λ 2 ( α n s Λ vanish, m )n=0,1,2,... ( Λ 2 m 2 ) [Bigi, Blok, Shifman, Uraltsev, Vainshtein, 1992], [Falk, Luke, Savage, 1993], ( Λ 3 m 3 m 2 ) [Ewerth, Gamino, Nandi, 2009]. The dominant non-perturative uncertainty originates from the 27 interference term: [ c B B = 6C 2 C 1 λ 2 + ( ( ) Λ 2 n ) ] m Λ 54C 7 m 2 n O c m 2 n c m 2 c 2 7 λ GeV 2 from B B mass splitting The coefficients n decrease fast with n. [Voloshin, 1996], [Khodjamirian, Rückl, Stoll, Wyler, 1997] [Grant, Morgan, Nussinov, Peccei, 1997] [Ligeti, Randall, Wise, 1997], [Buchalla, Isidori, Rey, 1997] New claims y Benzke, Lee, Neuert and Paz in arxiv: : One cannot really expand in m Λ/m 2 c. All such corrections should e treated as Λ/m ones and estimated using models of suleading shape functions. Dominant contriutions to the estimated ±5% non-perturative uncertainty in B are found this way, with the help of alternating-sign shape functions that undergo weaker suppression at large gluon momenta correction to the aove phase-space suppressed O ( ) αs Λ Main worry in hep-ph/ , and reason for the m ±5% non-perturative uncertainty.

37 The hard contriution to B X s γ J. Chay, H. Georgi, B. Grinstein PLB 247 (1990) 399. A.F. Falk, M. Luke, M. Savage, PRD 49 (1994) Goal: calculate the inclusive sum Σ Xs C 7 (µ ) X s γ O 7 B + C 2 (µ ) X s γ O 2 B The 77 term in this sum is purely hard. It is related via the optical theorem to the imaginary part of the elastic forward scattering amplitude B( p = 0)γ( q) B( p = 0)γ( q): Im{ q B γ 7 7 γ B q } Im A When the photons are soft enough, m 2 X s = m B (m B 2E γ ) Λ 2 Short-distance dominance OPE. However, the B X s γ photon spectrum is dominated y hard photons E γ m /2. Once A(E γ ) is considered as a function of aritrary complex E γ, ImA turns out to e proportional to the discontinuity of A at the physical cut. Consequently, E max γ 1 GeV de γ Im A(E γ ) circle de γ A(E γ ). Im E γ 1 E max γ 1 2 m B Re E γ [GeV] Since the condition m B (m B 2E γ ) Λ 2 is fulfilled along the circle, the OPE coefficients can e calculated perturatively, which gives A(E γ ) [ (j) F polynomial (2E ] γ/m ) circle m n j j (1 2E + O (α γ/m ) k s (µ hard )) B( p = 0) Q (j) j local operator B( p = 0). Thus, contriutions from higher-dimensional operators are suppressed y powers of Λ/m. At (Λ/m ) 0 : B( p) γ µ B( p) = 2p µ Γ( B X s γ) = Γ( Xs parton γ) + O(Λ/m ). At (Λ/m ) 1 : Nothing! All the possile operators vanish y the equations of motion. At (Λ/m ) 2 : B( p) hd µ D µ h B( p) = 2m B λ 1, λ 1 = ( 0.27 ± 0.04)GeV 2 from B Xl ν spectrum. B( p) hσ µν G µν h B( p) ( ) = 6m B λ 2, λ m 2 B m 2 B 0.12 GeV 2. The HQET heavy-quark field h(x) is defined y h(x) = 1 2 (1 + v/)(x) exp(im v x) with v = p/m B.

38 Energetic photon production in charmless decays of the B-meson (E γ > m GeV) [see MM, arxiv: ] A. Without long-distance charm loops: 1. Hard 2. Conversion 3. Collinear 4. Annihilation (q q c c) q s q s s s Dominant, well-controlled. O(α s Λ/m ), ( 1.6 ± 1.2)%. 0.2% or (+0.8 ± 1.1)%. Exp. π 0, η, η, ω sutracted. [Benzke, Lee, Neuert, Paz, 2010] [Kapustin,Ligeti,Politzer, 1995] Perturatively 0.1%. [Benzke, Lee, Neuert, Paz, 2010] B. With long-distance charm loops: 5. Soft 6. Boosted light c c 7. Annihilation of c c in a heavy ( cs)( qc) state gluons state annihilation only (e.g. η c, J/ψ, ψ ) c c c c c c c c s s s s O(Λ 2 /m 2 c ), +3.1%. Exp. J/ψ sutracted (< 1%). O(α s(λ/m) 2 ) O(α s Λ/M) [Voloshin, 1996], [...], Perturatively (including hard): +3.6%. M 2m c, 2E γ, m. [Buchalla, Isidori, Rey, 1997] e.g. B[B D sj (2457) D (2007) 0 ] 1.2%, [Benzke, Lee, Neuert, Paz, 2010]: add (+1.1 ± 2.9)% B[B 0 D (2010) + D (2007) 0 K ] 1.2%.