Angular analysis of decay B 0 K 0 µ + µ of LHCb

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1 Angular analysis of decay B 0 K 0 µ + µ of LHCb Linwei Li Peking University, Beijing 0087, China Pro.Dayong Wang Particle and Nuclear Physics lilinwei@pku.edu.cn Aug.22, 206 Introduction. Motivation In 203, LHCb published the results of form factor independent angular analysis of B 0 K 0 µ + µ with fb 7TeV data. There was a 3.7 σ deviation from SM calculations on P5'. In Moriond 205 conference, they have updated the results with 202 data of 2 fb and the discrepancy was conrmed. This is the most pronounced dierence observed so far at LHC with SM, so it raised a lot of interests..2 Denitions and conventions The denition of the angular variables used in this analysis are shown in the sketch shown in Fig.. The angle θ l is dened as the angle between the dirction of the µ + (µ ) and the direction opposite that of the B 0 ( B 0 ) in the dimuon rest frame; the angle θ k is dened as the angle between the dirction of the kaon and the direction opposite that of the B 0 ( B 0 ) in the K 0 rest frame; the angle φ is the angle between the plane containing the µ + and µ and the plane containing the kaon and the pion from the in the B 0 rest frame. 2 Transformation of PDF formulation used in the tting The angular distribution of the decay B 0 K 0 µ + µ can be described by three angles (θ l, θ k and φ) and by the invariant mass squared of the dimuon system (q 2 ). To reduce the number of tting parameters, we could do some transformation of the decay-rate formulation.

2 μ ϕ K + θk θl B 0 d μμ / K* 0 μ + π Figure : A sketch of the denition of the angular variables used in this analysis 2. the decay rate description in full angular variables Using the notation of[2],the dierential angular distribution can be written as: Γ f ull dq 2 d cos θ l d cos θ k dφ = 9 [ 3 32π 4 F T sin 2 θ k + F L cos 2 θ k +( 4 F T sin 2 θ k F L cos 2 θ k ) cos 2θ l + 2 P F T sin 2 θ k sin 2 θ l cos 2φ + F T F L ( 2 P 4 sin 2θ k sin 2θ l cos φ + P 5 sin 2θ k sin θ l cos φ) F T F L (P 6 sin 2θ k sin θ l sin φ 2 P 8 sin 2θ k sin 2θ l sin φ) +2P 2 F T sin 2 θ k cos θ l P 3 F T sin 2 θ k sin 2 θ l sin 2φ ] () where the q 2 dependent observables F L and S i are bilinear combinations of the K 0 decay amplitudes; A F B is the forward-backward asynnetry of the muons; F L is the longitudinal polarization fraction of the K The transformation used Because of the limited number of signal candidates in the data set, we didn't t the data to full dierential distribution of Eq(). Inspired by the derivations in ref [3][4], to reduce the number of parameters in the t, we "fold" the data three times. The "folding" means we divide the decay rate into dierent parts, calculate them separately according to some symmetries and then add them together to obtain the equivalent decay rates. If we take consecutive steps of folding, the similar expansions are used to get the full PDFs. 2

3 Let us take the rst folding as an example. The rst folding is at φ = 0 (for φ < 0, φ φ, the φ's domain is (0,π /2)). To be more clear, we divide the decay rate dγ into two parts corresponding to φ < 0 and φ < 0, i.e. dγ(φ φ > 0), and dγ(φ φ < 0): dˆγ = dγ(φ φ < 0) + dγ(φ φ > 0) = f0(φ φ φ) + f0(φ φ > 0) = f0(cos φ, sin φ) + f0(cos φ, sin φ) (2) According to trigonometric identities cos( φ) = cos φ, sin( φ) = sin φ, we can cancel the terms containing sin φ. The Eq() now reads: Γ full dq 2 d cos θ l d cos θ k dφ = 9 [ 3 6π 4 F T sin 2 θ k + F L cos 2 θ k +( 4 F T sin 2 θ k F L cos 2 θ k ) cos 2θ l + 2 P F T sin 2 θ k sin 2 θ l cos 2φ + F T F L ( 2 P 4 sin 2θ k sin 2θ l cos φ + P 5 sin 2θ k sin θ l cos φ) +2P 2 F T sin 2 θ k cos θ l ] (3) The second folding is performed at θ l = π/2(forθ l > π/2, θ l π θ l ). The domain of θ l is (0,π/2). According to cos(π θ l ) = cos θ l and sin(π θ l ) = sin θ l, we can cancel the terms containing cos θ l. Then the Eq(2) reads: Γ full dq 2 d cos θ l d cos θ k dφ = 9 [ 3 8π 4 F T sin 2 θ k + F L cos 2 θ k +( 4 F T sin 2 θ k F L cos 2 θ k ) cos 2θ l + 2 P F T sin 2 θ k sin 2 θ l cos 2φ + ] F T F L P 5 sin 2θ k sin θ l cos φ The third folding is at φ = π/2 (for φ > π/2, φ φ π/2), the φ's domain is (0,π/2). According cos(φ π/2) = sin φ and sin(φ π/2) = cos φ, we can cancel the terms containing cos φ and sin φ at the same time. Then the Eq(3) now reads: Γ full dq 2 d cos θ l d cos θ k dφ = 9 [ 2( 3 8π 4 F T sin 2 θ k + F L cos 2 θ k + ( 4 F T sin 2 θ k F L cos 2 θ k ) cos 2θ l ) + ] F T F L P 5 sin 2θ k sin θ l (cos φ sin φ) (4) (5) 3

4 2.3 inuence of S-wave interference on the angular distribution Although the K + π invariant mass must be consistent with a K 0, there can be contributions from a spinless (S-wave) K + π combination[4].the presence of a K + π system in an S-wave conguration, due to a nonresonant contribution or to feed through from K + π scalar resonances, results in additional terms in the dierent angular distribution.[4] The dierential angular distribution of S-wave can be written as: Fs sin 2 θ l + A s sin 2 cos θ k + A 4 s sin θ k sin 2θ l cos φ 6π + A 5 s sin θ k sin θ l cos φ + A 7 s sin θ k sin θ l sin φ + A 8 s sin θ k sin 2θ l sin φ ] where F s is the fraction of the S-wave component in the K 0 mass window, and the A i s is the interference terms of the S-wave with the K 0 transversity amplitudes as dened in [4]. For S-wave, we do the same transformation with P-wave, after the rst "folding" Eq(6) now reads: Fs sin 2 θ l + A s sin 2 cos θ k 8π + A 4 s sin θ k sin 2θ l cos 5 s sin θ k sin θ l cos φ ] (7) After the second "folding", Eq(6) reads: Fs sin 2 θ l + A s sin 2 cos θ k + A 5 s sin θ k sin θ l cos φ ] (8) 4π After the third "folding", Eq(6) nally becomes: 2(Fs sin 2 θ l + A s sin 2 cos θ k ) + A 5 s sin θ k sin θ l (cos φ sin φ) ] 4π (9) 2.4 formulas after transformation with both P wave and S-wave Denoting the righthand side of Eq.() by W p and the righthand side of Eq.(6) by W s, the dierential decay rate takes the form ( F s )W p + W s. According to the number of foldings, we can have two dierent decay rate formulation. One option is folding twice, then we can get the dierential decay rate by adding Eq.(4) and Eq.(8); the angular distribution can be written as : (6) 4

5 Γ full dq 2 d cos θ l d cos θ k dφ = 9 { 2 [ (F s + As cos θ k ) ( cos 2 ) ] θ l + A 5 8π 3 s cos 2 θ k cos 2 θ l cos φ [ + ( F s) 2F L cos 2 ( θ k cos 2 ) θ l + 2 ( F L) ( cos 2 ) ( θ k + cos 2 ) θ l + 2 P ( F L )( cos 2 θ k )( cos 2 θ l ) cos 2φ +2P 5 cos θ k FL ( F L ) ]} cos 2 θ k cos 2 θ l cos φ (0) Another formulation corresponds to folding three times, then we can get the dierential decay rate by adding Eq.(5) and Eq.(9); the angular distribution can be written as : Γ full dq 2 d cos θ l d cos θ k dφ = 9 { 2 [ 2(F s + As cos θk ) ( cos 2 ) θ l 8π 3 ] +As 5 cos 2 θ k cos 2 θ l (cos φ sin φ) + ( F s) [ 4F L cos 2 ( θ k cos 2 ) θ l + ( FL ) ( cos 2 ) ( θ k + cos 2 ) θ l +2P 5 cos θ k FL ( F L ) ]} cos 2 θ k cos 2 θ l (cos φ sin φ) () 5

6 3 Summary In this note, the formulation of the PDF used to determine the parameter P 5 B 0 K 0 µ + µ is derived. in the decay References [] Mauro Dinardo and Kevin Stenson. (205). Angular analysis and dierential branching fraction of the decay B 0 K 0 µ + µ, AN-204/29 [2] W.Aitmannshofer,P.Ball,A.Bharucha,A.J.Buras,D.M.Staub, and M.Wick, J.. (2009). High Energy Phys. 09. [3] R.Aaij et al.. (203). (LHCb Collaboration), J.High Energy Phys. 08(203) 3. [4] J.Matias,. (202). Phys.Rev.D 86,094024(202). 6