Global fits to bs ℓℓ data Nazila Mahmoudi Lyon University & CERN TH In collaboration with T. Hurth and S. Neshatpour Rare B decays in 2015 - experiment and theory Higgs Centre for Theoretical Physics, Edinburgh 11-13 May 2015
b → s transitions Inclusive decays B → X s γ Improved theory calculations (Misiak et al. 1503.01789) Excellent agreement with the measurements B → X s ℓ + ℓ − Still waiting for the final words from Belle and Babar! High expectation from Belle II! Exclusive decays B → K ∗ γ First measurements of B s → µ + µ − Angular distributions of B → K ∗ µ + µ − → large variety of experimentally accessible observables Also: B → K µ + µ − and B s → φµ + µ − Issue of hadronic uncertainties in exclusive modes Nazila Mahmoudi 2 / 20 Rare B decays in 2015 - Edinburgh - 13 May 2015
B → V ℓ + ℓ − – Notations Differential decay distribution: d 4 Γ 9 32 π J ( q 2 , θ ℓ , θ V , φ ) dq 2 d cos θ ℓ d cos θ V d φ = J ( q 2 , θ ℓ , θ V , φ ) = � i J i ( q 2 ) f i ( θ ℓ , θ V , φ ) ց angular coefficients J 1 − 9 ց functions of the spin amplitudes A 0 , A � , A ⊥ , A t , and A S Spin amplitudes: functions of Wilson coefficients and form factors Standard Observables: Dilepton invariant mass spectrum: d Γ dq 2 = 3 J 1 − J 2 � � 4 3 Forward backward asymmetry: � d Γ � d Γ d 2 Γ �� 0 � 1 dq 2 = 3 � A FB ( q 2 ) ≡ − 1 − d cos θ l 8 J 6 dq 2 d cos θ l 0 dq 2 0 ≃ − 2 m b m B C eff 9 ( q 2 0 ) Forward backward asymmetry zero-crossing: q 2 + O ( α s , Λ / m b ) C 7 | A 0 | 2 Polarization fraction: F L ( q 2 ) = | A 0 | 2 + | A � | 2 + | A ⊥ | 2 , Nazila Mahmoudi 3 / 20 Rare B decays in 2015 - Edinburgh - 13 May 2015
B → V µ + µ − observables Optimised observables: form factor uncertainties cancel at leading order bin dq 2 [ J 3 + ¯ bin dq 2 [ J 6 s + ¯ � J 3 ] � J 6 s ] � P 1 � bin = 1 � P 2 � bin = 1 bin dq 2 [ J 2 s + ¯ bin dq 2 [ J 2 s + ¯ 2 � 8 � J 2 s ] J 2 s ] 1 � 1 � dq 2 [ J 4 + ¯ dq 2 [ J 5 + ¯ � P ′ � P ′ 4 � bin = J 4 ] 5 � bin = J 5 ] N ′ 2 N ′ bin bin bin bin − 1 − 1 � � � P ′ dq 2 [ J 7 + ¯ � P ′ dq 2 [ J 8 + ¯ 6 � bin = J 7 ] 8 � bin = J 8 ] 2 N ′ N ′ bin bin bin bin with � bin dq 2 [ J 2 s + ¯ bin dq 2 [ J 2 c + ¯ N ′ � � bin = − J 2 s ] J 2 c ] + CP violating clean observables and other combinations U. Egede et al., JHEP 0811 (2008) 032, JHEP 1010 (2010) 056 J. Matias et al., JHEP 1204 (2012) 104 S. Descotes-Genon et al., JHEP 1305 (2013) 137 Or alternatively: S i = J i ( s , c ) + ¯ J i ( s , c ) dq 2 + d ¯ d Γ Γ dq 2 W. Altmannshofer, P. Ball, A. Bharucha, A.J. Buras, D.M. Straub, M. Wick, JHEP 0901 (2009) 019 Nazila Mahmoudi 4 / 20 Rare B decays in 2015 - Edinburgh - 13 May 2015
The LHCb anomalies 3 main LHCb anomalies: P ′ 5 R K BR( B s → φµ + µ − ) × 10 -6 ] 4 c -2 LHCb [GeV 0.1 2 q )/d − µ + µ 0.05 φ → s B ( B d 0 5 10 15 2 2 4 q [GeV / c ] Possible explanations: Statistical fluctuations Theoretical issues New Physics! Nazila Mahmoudi 5 / 20 Rare B decays in 2015 - Edinburgh - 13 May 2015
New Physics interpretation? Global analysis of the latest LHCb data Relevant O perators: O 7 , O 8 , O ( ′ ) 9 µ, e , O ( ′ ) µ P L µ ) ≡ O l and O S − P ∝ (¯ sP R b )(¯ 0 10 µ, e NP manifests itself in the shifts of the individual coefficients with respect to the SM values: C i ( µ ) = C SM ( µ ) + δ C i i → Scans over the values of δ C i → Calculation of flavour observables → Comparison with experimental results → Constraints on the Wilson coefficients C i Nazila Mahmoudi 6 / 20 Rare B decays in 2015 - Edinburgh - 13 May 2015
Global fits Evaluations uncertainties and correlations: Experimental errors and correlations 3 fb − 1 LHCb data for B → K ∗ 0 µ + µ − : provided in LHCb-CONF-2015-002 Theoretical uncertainties and correlations study of more than 100 observables (at a later stage, selection of the relevant observables for each fit) Monte Carlo analysis variation of the “standard” input parameters: masses, scales, CKM, ... for B s → φµ + µ − , mixing effects taken into account decay constants taken from the latest lattice results use for the B ( s ) → V form factors of the lattice+LCSR combinations from 1503.05534, including correlations (Cholesky decomposition method) use for the B → K form factors of the lattice+LCSR combinations from 1411.3161, including correlations two approaches for the exclusive decays: soft form factors, full form factors two sets of hypotheses for the uncertainties associated to the factorisable and non-factorisable power corrections ⇒ Computation of a (theory + exp) correlation matrix Nazila Mahmoudi 7 / 20 Rare B decays in 2015 - Edinburgh - 13 May 2015
Global fits For the exclusive semi-leptonic decays, two approaches and two evaluations of the uncertainties for each decay. At low q 2 : Soft form factor approach Uncertainties of the factorisable and non-factorisable corrections parametrised as q 2 � � A k → A k 1 + a k exp ( i φ k ) + 6 GeV 2 b k exp ( i θ k ) where A k are the helicity amplitudes. a k in [ − 10 % , + 10 %] or [ − 20 % , + 20 %] φ k , θ k in [ − π, + π ] b k in [ − 25 % , + 25 %] or [ − 50 % , + 50 %] Full form factor approach Uncertainties of the non-factorisable power corrections only parametrised in a similar way: a k in [ − 5 % , + 5 %] or [ − 10 % , + 10 %] φ k , θ k in [ − π, + π ] b k in [ − 10 % , + 10 %] or [ − 25 % , + 25 %] At high q 2 , uncertainties parametrised as A k → A k � 1 + a k exp ( i φ k ) � a k in [ − 10 % , + 10 %] or [ − 20 % , + 20 %] φ k in [ − π, + π ] Nazila Mahmoudi 8 / 20 Rare B decays in 2015 - Edinburgh - 13 May 2015
Global fits Global fits of the observables by minimization of � � � � χ 2 = O th − � · (Σ th + Σ exp ) − 1 · O th − � O exp � O exp � (Σ th + Σ exp ) − 1 is the inverse covariance matrix. 58 observables relevant for leptonic and semileptonic decays: BR( B → X s γ ) BR( B → K 0 µ + µ − ) BR( B → K + µ + µ − ) BR( B → X d γ ) ∆ 0 ( B → K ∗ γ ) BR( B → K ∗ e + e − ) BR low ( B → X s µ + µ − ) R K BR high ( B → X s µ + µ − ) B → K ∗ 0 µ + µ − : F L , A FB , S 3 , BR low ( B → X s e + e − ) S 4 , S 5 in five low q 2 and two high BR high ( B → X s e + e − ) q 2 bins BR( B s → µ + µ − ) B s → φµ + µ − : BR, F L BR( B d → µ + µ − ) in three low q 2 and two high BR( B → K ∗ + µ + µ − ) q 2 bins Nazila Mahmoudi 9 / 20 Rare B decays in 2015 - Edinburgh - 13 May 2015
Global fits Statistical approaches: ∆ χ 2 = χ 2 − χ 2 min method Determination of the minimum of χ 2 → best fit point 1 Computation for each point of the scan of the difference of χ 2 with the best fit point 2 Find the 1 − 2 σ regions corresponding to the number of d.o.f. 3 Interpretation: considering the best fit point gives the “real” description, which variations of the parameters are allowed → relative global fit Absolute χ 2 method Computation of the χ 2 for each point 1 Find the 1 − 2 σ regions corresponding to N d.o.f. where N = ( N o observables - n v 2 variables) If an observable is relatively insensitive to the variation of the Wilson coefficients, 3 remove it from the fit Interpretation: global fit assessing if each point is globally in agreement with all the measurements Nazila Mahmoudi 10 / 20 Rare B decays in 2015 - Edinburgh - 13 May 2015
Fit results for two operators: { C 9 , C 10 } → Using soft form factors with 10% power correction errors: ∆ χ 2 method with 20% power correction errors: ∆ χ 2 method Nazila Mahmoudi 11 / 20 Rare B decays in 2015 - Edinburgh - 13 May 2015
Fit results for two operators: { C 9 , C 10 } → Using soft form factors with 10% power correction errors: ∆ χ 2 method Absolute χ 2 method with 20% power correction errors: ∆ χ 2 method Absolute χ 2 method 1 σ agreement for C 9 is possible even in the 2 operator basis! Nazila Mahmoudi 11 / 20 Rare B decays in 2015 - Edinburgh - 13 May 2015
Fit results for two operators: { C 9 , C 10 } → Using full form factors with 5% power correction errors: ∆ χ 2 method with 10% power correction errors: ∆ χ 2 method Nazila Mahmoudi 11 / 20 Rare B decays in 2015 - Edinburgh - 13 May 2015
Fit results for two operators: { C 9 , C 10 } → Using full form factors with 5% power correction errors: ∆ χ 2 method Absolute χ 2 method with 10% power correction errors: ∆ χ 2 method Absolute χ 2 method Using the full form factors, only 2 σ agreement for C 9 could be possible. Nazila Mahmoudi 11 / 20 Rare B decays in 2015 - Edinburgh - 13 May 2015
Fit results for two operators: { C 9 , C ′ 9 } → Using soft form factors with 10% power correction errors: ∆ χ 2 method with 20% power correction errors: ∆ χ 2 method Nazila Mahmoudi 12 / 20 Rare B decays in 2015 - Edinburgh - 13 May 2015
Fit results for two operators: { C 9 , C ′ 9 } → Using soft form factors with 10% power correction errors: ∆ χ 2 method Absolute χ 2 method with 20% power correction errors: ∆ χ 2 method Absolute χ 2 method 1 σ agreement for C 9 is possible even in the 2 operator basis! Nazila Mahmoudi 12 / 20 Rare B decays in 2015 - Edinburgh - 13 May 2015
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