Non-local 1/m b corrections to B X s γ

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1 Non-local 1/m b corrections to B X s γ Michael Benzke TU München September 16, 2010 In collaboration with S. J. Lee, M. Neubert, G. Paz Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 1 / 26

2 Outline 1 Motivation 2 Theoretical Framework Soft-Collinear Effective Theory Matching Factorization 3 Irreducible Uncertainties 4 Summary Based on arxiv: Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 2 / 26

3 Motivation Radiative decay B X s γ is mediated by FCNC Occurs only at loop level in the Standard Model Comparison of experimental and theoretical ũ ũ branching ratios can strongly constrain new γ physics models Experimental background complicates determination Need cut on photon energy BF( B X s γ)= (3.02 ± 0.10 ± 0.11) 10 4 for E γ > 2 GeV BF( B X s γ)= (3.45 ± 0.15 ± 0.40) 10 4 for E γ > 1.7 GeV Belle Collaboration 09 c) Theoretical analysis must be performed in this endpoint region 0 Photons / 50 MeV c.m.s Eγ [GeV] Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 3 / 26 b W s

4 Motivation The Decay rate is calculated by considering the cut of a B-meson forward matrix element dγ( B X s γ) 1 M B d 3 p γ Disc im( B B) In the endpoint region the Jet X s has a large energy O(m b ) but small invariant mass O( m b Λ QCD ) Large logarithms in partonic calculation The partonic result is known to O(α 2 s ) and for E γ > 1.6 GeV BF( B X s γ)= (3.15 ± 0.23) 10 4 for E γ > 1.6 GeV M. Misiak et al. 06; Becher, Neubert 06 Errors Higher order perturbative uncertainties: 3 % Parametric uncertainties: 3 % m c -interpolation ambiguity: 3 % non perturbative uncertainties: 5 % Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 4 / 26

5 Overview Goal Systematically analyze the non-perturbative corrections in the endpoint region at subleading power in 1/m b Use an effective field theory to resum large logarithms and to generate a systematic expansion in a small parameter The appropriate effective field theories are SCET (hc,hc,s) and HQET C. W. Bauer et al. 01 Light-cone directions n, n Hard-collinear momentum n p hc Λ, n p hc m b In SCET the rate factorizes into perturbatively calculable and hadronic parts Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 5 / 26

6 Theoretical Framework Starting point: Weak effective Lagrangian H eff = G F 2 X p=u,c General procedure: λ p C 1Q p 1 + C2Qp 2 + X i=3,...,10 C i Q i + C 7γQ 7γ + C 8g Q 8g! + h.c. After integrating out hard modes as well as the small components of the hard-collinear fields the SCET Lagrangian only contains (anti-)hard-collinear and soft fields These fields exhibit a certain scaling in powers of λ = Λ m b Match the relevant operators of the weak effective Hamiltonian onto the SCET fields This yields a systematic expansion in powers of λ Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 6 / 26

7 Matching procedure Matching at leading power hc Q 7 s Q 7 ξ hc b h Corresponds to Q 7 = em b 8π 2 sσ µν(1 + γ 5 )F µν b em b n/ 4π 2 e im bv x ξ hc in A/em 2 (x)(1 + γ 5)h(x ) Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 7 / 26

8 Matching procedure At subleading power e.g. hc Q 8 s A ξhc Q 8 q soft A hc b h Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 8 / 26

9 Matching procedure At subleading power e.g. hc Q 8 s A ξhc Q 8 q soft A hc b SCET-Lagrangian insertion suppressed by another λ h Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 8 / 26

10 Matching procedure At subleading power e.g. hc Q 8 s A ξhc Q 8 q soft A hc b SCET-Lagrangian insertion suppressed by another λ h Q 7 A s Q 7 hc A hc ξ hc b h Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 8 / 26

11 Matching procedure At subleading power e.g. hc Q 8 s A ξhc Q 8 q soft A hc b SCET-Lagrangian insertion suppressed by another λ h Q 7 A s Q 7 hc A hc ξ hc b h Additional hard-collinear field suppressed by another λ Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 8 / 26

12 Decay rate Decay rate amplitude squared Leading power contribution to the decay rate: Interference of Q 7 with Q 7 The Rate factorizes at leading power dγ LO H J S h Q 7 hc Korchemsky, Sterman 94; Bauer, Pirjol, Stewart 01 The hard function H and the jet function J parameterize physics at the scales m b and m b Λ respectively and are perturbatively calculable The shape function S is a Fourier transform of a non-local HQET matrix element S(ω) = dt 2π e iωt B(v) h(tn)... h(0) B(v) The leading order shape function ( meson PDF ) is related to the measured photon spectrum ξhc hc Q 7 h Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 9 / 26

13 Calculation at subleading power Subleading contributions originating from the interference Q 7 Q 7 Lee, Stewart 04; Bosch, Neubert, Paz 04; Beneke, Campanario, Mannel, Pecjak 04 But other operators of the weak effective Hamiltonian also contribute Some of them are suppressed by small Wilson coefficients or CKM elements Important Operator combinations Q 1 Q 7, Q 7 Q 8 and Q 8 Q 8 ( ) (Q 1 Q 1 and Q 1 Q 8 only appear at O 1 ) mb 2 Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 10 / 26

14 Calculation at subleading power Distinguish contributions by their factorization properties 1. Direct photon contributions hc A hc hc Q 8 ξ hc Q 8 αs m b in endpoint region h h Factorizes dγ H j S with the subleading jet function j Contributions of this type can be written as a local OPE when integrated to the total rate Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 11 / 26

15 Calculation at subleading power 2. Resolved photon contributions hc hc q soft q soft Λ Q 8 A hc Q 8 m b h h (in this case double resolved) Factorizes dγ res H J s J J The J are new jet functions corresponding to the uncut hard-collinear propagators Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 12 / 26

16 Calculation at subleading power 2. Resolved photon contributions hc hc q soft q soft Λ Q 8 A hc Q 8 m b h h The subleading shape function s is a non-local HQET matrix element It cannot be separately extracted from the photon spectrum It is non-local in two light-cone directions No OPE even in integrated rate Hadronic uncertainty Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 12 / 26

17 Factorization Formula In the endpoint region the rate factorizes dγ H J S + 1 X H ji S + 1 X H J si m b m b + 1 X H J si J + 1 X «1 H J si J J + O m b m b m 2 b Visualize as J J J B H J S H B + B H J S H B + B H J S H B Consider all relevant combinations in turn Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 13 / 26

18 Interference of Q 1 Q 7 hc hc s Non-perturbative contribution to total rate O ( 1 m 2 c dγ res 1 de γ m b g 17 (ω, ω 1 ) = dω δ(ω + p + ) dr 2π e iω1r ) (Voloshin 96) [ ( )] dω 1 m 2 1 F c g 17 (ω, ω 1 ) ω 1 + iε 2E γ ω 1 dt 1 2π e iωt B h(tn)... Gs αβ (r n)... h(0) B M B Expand F for m c m b Reproduce Voloshin term in total rate λ 2 9m 2 c Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 14 / 26

19 Interference of Q 1 Q 7 Estimate the possible contribution to the partial rate Γ(E 0 ) = E 0 de γ dγ de γ PT invariance implies that all considered subleading shape functions are real Moment constraints dω dω1 g 17 (ω, ω 1 ) = 2λ dωω dω1 g 17 (ω, ω 1 ) = ρ 2 Symmetry dωg17 (ω, ω 1 ) = dωg 17 (ω, ω 1 ) Model function e ω 2 1 σ 2 Λ 2 2σ 2 h 17 (ω 1 ) = dωg 17 (ω, ω 1 ) = 2λ 2 ω1 2 Λ2 2π % non-perturbative uncertainty due to Q 1 Q 7 contribution (possible correction to partonic rate) Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 15 / 26

20 Interference of Q 8 Q 8 hc hc hc s s dγ res de γ e2 s 8πα s m b dω δ(ω + p + ) dω 1 ω 1 + iε dω 2 ω 2 iε g 88(ω, ω 1, ω 2 ) g 88 (ω, ω 1, ω 2 ) = 1 M B B h(tn)... s(tn + u n) s(r n)... h(0) B F.T. Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 16 / 26

21 Interference of Q 8 Q 8 hc hc hc s s dγ res de γ e2 s 8πα s m b dω δ(ω + p + ) dω 1 ω 1 + iε dω 2 ω 2 iε g 88(ω, ω 1, ω 2 ) g 88 (ω, ω 1, ω 2 ) = 1 M B B h(tn)... s(tn + u n) s(r n)... h(0) B F.T. Important subtlety Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 16 / 26

22 Interference of Q 8 Q 8 hc hc dγ res de γ e2 s 8πα s m b dω δ(ω + p + ) dω 1 ω 1 + iε dω 2 ω 2 iε g 88(ω, ω 1, ω 2 ) g 88 (ω, ω 1, ω 2 ) = 1 M B B h(tn)... s(tn + u n) s(r n)... h(0) B F.T. Consider scale dependence of direct contribution dγ dir de γ e2 s C F α s 4πm b ( dω 2 ln m ) b(ω + p + ) µ S(ω) Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 16 / 26

23 Interference of Q 8 Q 8 How does the scale dependence cancel? Consider the asymptotic form of g 88 for ω 1,2 Λ QCD g 88(ω, ω 1, ω 2) C F θ(ω 1)ω 1 ɛ Z 1 (4π) 2 ɛ δ(ω1 ω2) dω S(ω )(ω ω) ɛ +... Γ(1 ɛ) ω Convolution integral is UV divergent! Introduce cutoff and consider high and low momentum part of the convolution separately dγ res e2 s 8πα s de γ m b e2 s C F 8πα s 2π 2 m b Λ dω 1 dω δ(ω + p + ) ω 1 + iε ( dω ln Λ(ω + p +) µ Λ dω 2 ω 2 iε g 88(ω, ω 1, ω 2 ) ) S(ω) Scale dependence cancels Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 17 / 26

24 Interference of Q 8 Q 8 - Numerical Estimate This time no moment constraint for g 88 Assume that the convolution of jet and shape function yields a value of O(Λ QCD ) But suppressed by e 2 s % non-perturbative uncertainty due to Q 8 Q 8 contribution Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 18 / 26

25 Interference of Q 7 Q 8 hc s s dγ res αs de γ m b Z» 1 ω 1 + iε + 1 ω 2 iε Z dω δ(ω + p +) dω 1 Z g (i) 1 78 (ω, ω1, ω2) = B h(tn)... h(0) X M B q dω 2 ω 1 ω 2 + iε «g (1) 78 (ω, ω1, ω2) 1 ω 1 + iε 1 ω 2 iε e q q(r n)... q(u n) B F.T. «g (5) 78 (ω, ω1, ω2) Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 19 / 26

26 Interference of Q 7 Q 8 Due to the sum over flavors it is possible to roughly estimate the effects through vacuum insertion approximation Lee, Neubert, Paz 06 Z dω g (i) 78 (ω, ω1, ω2) VIA = f 2 espec B M B 1 1 «φ B ( ω 1)φ B ( ω 2) 8 Nc % non-perturbative uncertainty due to Q 7 Q 8 contribution Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 20 / 26

27 Interference of Q 7 Q 8 Alternatively it is possible to relate the effect to the measured isospin asymmetry Wigner-Eckart Theorem B h(tn)... h(0) q e q q(r n)... q(u n) B = 1 6 Λ 0 ± 1 2 Λ 1 = 1 6 (Λ 0 Λ 1 ) + e spec Λ 1 Isospin asymmetry 0 Λ 1 ; averaged rate Γ avg Λ 0 SU(3) symmetry Λ 0 = Λ 1 M. Misiak % non-perturbative uncertainty for an SU(3) breaking of 30 % Can be reduced by improved measurement of isospin asymmetry Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 21 / 26

28 Summary - The Numbers Operators Effect Q 1 Q % Q 8 Q % Q 7 Q8 VIA % Q 7 Q exp % Therefore the total hadronic uncertainty Γ(E 0) Γ part (E 0 ) Γ part (E 0 ) to the integrated rate is 4.8 % % by VIA 6.4 % % by Isospin Asymmetry (95 % CL) 4.0 % % by Isospin Asymmetry (central value without error) Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 22 / 26

29 Summary In order to estimate the non-perturbative uncertainty in the B X s γ branching ratio the subleading order of the power expansion in 1 m b must be considered At this order the decay rate obeys a new factorization formula dγ H J S + 1 X H ji S + 1 X H J si m b m b + 1 X H J si m J + 1 X «H J si b m J J 1 + O b m 2 b Effects of the new non-local matrix elements can only be estimated A careful consideration of everything we know yields a non-perturbative uncertainty of ±5 % to the partial decay rate Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 23 / 26

30 Outlook What is left to be done? The new soft functions will affect the extraction of heavy-quark parameters from moments of the B X s γ photon spectrum Need models for the subleading shape functions The new jet functions J contain strong phases, that generate contributions to direct CP violation Identify the new contributions and estimate their size Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 24 / 26

31 Outlook What is left to be done? The new soft functions will affect the extraction of heavy-quark parameters from moments of the B X s γ photon spectrum Need models for the subleading shape functions The new jet functions J contain strong phases, that generate contributions to direct CP violation Identify the new contributions and estimate their size Thank you for your attention! Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 24 / 26

32 Bonus Slides Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 25 / 26

33 Kinematics Jet momentum p b = p + p γ p = (m b, 0, 0, 0) (E γ, 0, 0, E γ ) p 2 = m b (m b 2E γ ) Scaling of hard-collinear fields 0 T ξ(x) ξ(y) 0 = n/ n/ n/n/ 0 T ψ(x) ψ(y) d 4 k n/ n/ ik/ n/n/ = (2π) 4 2 k 2 + iɛ 2 e ik(x y) d 4 k i n k = (2π) 4 k 2 + iɛ e ik(x y) λ 2 1 λ = λ ξ λ 1/2 Michael Benzke (JGU) Non-local 1/m b corrections to B X s γ TU München 26 / 26