Prospects of measurements of observables in

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1 Prospects of measurements of observables in B K µ + µ decays Ulrik Egede, Mitesh Patel, Konstantinos A. Petridis Imperial College ondon September 6, 13 Disclaimer: This is part of a large ongoing effort within HCb to define a path for the next round of this analysis. K.A. Petridis (IC) B K µ + µ UK flavour workshop 13 1 / 1

2 Introduction As presented by Mitesh, HCb measurements of b sll have revealed a host of interesting results. This talk focuses on the prospects for the full angular on the 3 fb 1 update of B K µ + µ. Thoughts on how to perform a full angular fit to B K µ + µ decays. Results shown based on realistic 3 fb 1 -equivalent toy data. K.A. Petridis (IC) B K µ + µ UK flavour workshop 13 / 1

3 theoretical uncertainty and a clean sensitivity to characteristic New Physics features.1 + q = s square ofinput the lepton-pair The source of experimental is the differential decay distribution of the 4-body invariant mass. K final state K ( Kπ)#+ #. It is described by four independent kinematic variables, l angle between p~l in l + l rest φ which are traditionally chosen to be: the invariant squared mass q of the lepton pair; the µ frame and dilepton s direction in rest angle θk between the directions of flight of the kaon and the B meson in the rest frame frame of B d θl θk B of the K ; the angle θl between directions of flight of the # and the B meson in ~ in K angle between pthe the K rest K frame and and direction of the K angle in rest the dilepton rest frame; the azimutal φ between the two planes defined by the µ+ frame of B dof B K µ+ µ : Differential decay rate lepton pair and the Kπ system. In terms of these kinematic variables, the differential π+ decay distribution canangle be between writtenthe as planes defined by the two leptons and the K planes.! + Auto noma de Barcelona PRD71 Discussion on (5) the exclusive 949, B! K JHEP (! K )l811:3, l : complete( basis 9 d4 Γ Joaquim Matias Universitat = J1s sin θk + J1c J(q cos, θl K, K+, (J ) =s sin θk + Jc cos θk ) cos θl dq dcos θk dcos θl dφ 3π B K µ µ J1s sin K + J1c cos K + (Js sin K + Jc cos K ) cos l + J3 sin K sin l cos +J sin sin cos + J sin sin cos + (J sin + J cos ) cos l 6c l K K +J3 sin θk sin 4θl coskφ +l J4 sin θ5 K sink θl cos φ + J6s5 sin θ sin θl cos φ K +J7 sin K sin l sin + J8 sin K sin l sin + J9 sin K sin l sin. +(J6s sin θk + J6c cos θk ) cos θl + J7 sin θk sin θl sin φ + J8 sin θk sin θl sin φ " +J9 sin θk sin θl sin φ, (1) The explicit dependence of the coefficients Ji (q ) in terms of transversity amplitudes (Ai ),R observables,r is in Section Thethe point to emphasize hereais,r that only that respect Ji given terms depend. on spin amplitudes,ak,a (ignoring 1 For acontributions representative set ofand references the phenomenology of this decay mode see [1, 11, scalar in mdiscussing ` limit) 1, 13, 14, 15, 16, 17, 18, 19, ]. This definition of the kinematic variables coincides exactly with that of Refs. [9, 1] K.A. Petridis (IC) B K µ+ µ UK flavour workshop 13 3 / 1

4 A R,. An additional complex amplitude A t is required in the massive case, and in the presence of scalar contributions a new amplitude A S must be included. The expressions for these coefficients read, Angular terms J 1s = ( + β l ) 4 [ A + A + A R + A R ] + 4m l q Re ( A A R + A ) A R, J 1c = A + A R + 4m [ l At +Re(A q AR ] ) + β l A S, J s = β [ l A 4 + A + A R + A R ] [, J c = βl A + A R ], J 3 = 1 [ β l A A + A R A R ], J 4 = 1 βl J 5 = β l [Re(A A A R A R m ] l ) q Re(A A S + AR AS ), [ Re(A A + A R A R ] ), [ J 6s =β l Re(A A A R A R ] m l ), J6c =4β l q Re(A A S + A R AS ), J 7 = β l [Im(A A A R A R m ] l )+ q Im(A A S AR AS )), J 8 = 1 [ βl Im(A A + A R A R ] ), J9 = β l [ Im(A A + A R A R ) ], (3) K.A. Petridis (IC) B K µ + µ UK flavour workshop 13 4 / 1

5 B! K µ µ decay amplitudes Amplitudes I At leading [JHEP 91(9)19] order Altmannshofer et al. A (R)? = N p (C e 9 + Ce 9 ) (Ce 1 + Ce 1 ) V(q ) + m b m B + m K q (Ce 7 + Ce 7 )T 1(q ) A (R) k = N p (C (mb mk ) e 9 C e 9 ) (Ce 1 C e 1 ) A 1(q ) A (R) = A t = N m K p q + m b q (Ce 7 C e 7 )T (q ) m B m K (C e 9 C e 9 ) (Ce 1 C e 1 ) (mb mk q )(m B + m K )A 1 (q ) +m b (C e 7 C e 7 ) (m B +3m K q )T (q ) N p q p (C e 1 C e 1 A S = N p (C S C S )A (q ) q )+ (C e P m µ m B C e P ) A (q ) m K T 3 (q ) A (q ) m B + m K C i are Wilson coe cients that we want to measure (they depend on Cthe eff i heavy are thedegrees Wilsonof coefficients freedom). (including 4-quark operator contributions) A, A 1, A, T 1, T and V are form-factors A(these i, T i and are e ectively V i, are formnuisance factorsparameters). typically treated as nuisance parameters T. Blake B! K µ + µ 16 / 3 K.A. Petridis (IC) B K µ + µ UK flavour workshop 13 5 / 1

6 Using Eqs. (7), this relation translates precisely into the relation forthej i given in Eq. (5). Amplitudes Now that the formalism II assures the systematic construction of observables that respect the symmetries of the angular distribution, we must focus on the cancellation of hadronic form factors. At leading At leading order in order 1/min b, 1/m α s for large E K b and α s, and at large > Λrecoil QCD (large (E K ), recoil), the transversity formamplitudes factors reduce A,R, to A,R ξ,ξ : and A,R can be written as: A,R = [ Nm B (1 ŝ) (C9 eff + C9 eff ) (C 1 + C 1 )+ˆm ] b ŝ (Ceff 7 + C7 eff ) ξ (E K ) A,R = [ Nm B (1 ŝ) (C9 eff C9 eff ) (C 1 C 1)+ ˆm ] b ŝ (Ceff 7 C7 eff ) ξ (E K ) A,R = Nm [ ] B(1 ŝ) (C9 ˆm K ŝ eff C9 eff ) (C 1 C 1 )+ˆm b(c7 eff C7 eff ) ξ (E K ) (9) where ŝ = q /m B,ˆm i = m i /m B,andtermsofO(ˆm K ) have been neglected. The normalization is Can given build by form factor independent observables using ratios of bilinear amplitude combinations N = [JHEP V tb Vts 131(13)48] β l G F α q λ 1/ Descotes-Genon, et al. e.g: (1) 3 1 π 5 m 3 B with λ =[q (m B +m P 5 K ) ][q Re(A (m B m K ) A ]. AR Therefore, AR ) at first order, we have n ξ ( A and n,n ξ. This establishes + A a clear R )( A guideline + A in R the + A construction + A R ) of clean observables, as ratios of quantities in Eq. (7) where the ξ, cancel [Form Factor Independent (FFI) K.A. Petridis (IC) B K µ + µ UK flavour workshop 13 6 / 1

7 An experimentalist s view Experimentalists fit the angular distribution of B K µ + µ decays in a particular basis, to extract observables Theorists use these observables to extract Wilson coefficients The observable basis is not set in stone Goal: Build any observable basis and the correlations, from a single fit to the angular distribution Fitting for the amplitudes will offer a complete description of the decay Fitting for amplitudes improves stability of fit relative to fitting for observables (complex relations between observables) Some important considerations 1 Need to account for symmetries as they will exhibit themselves as degenerate regions in the fit Account for the q dependence of the amplitudes K.A. Petridis (IC) B K µ + µ UK flavour workshop 13 7 / 1

8 Symmetries of the angular distribution... [JHEP1(1)56] Egede et al. Symmetry: Transformations of the A j s that leave the J i s and hence the differential decay distribution invariant. Number of degrees of freedom of J i s ( experimental ) and of A j s ( theoretical ) must match. Account for dependencies between J i s (n d ) and symmetries of A j s (n s ) with n J n d = n A n s. 6 = 1 real A j parameters, 8 independent J i s (in the limit of m l = and no scalar operators). Thus 4 symmetry transformations. K.A. Petridis (IC) B K µ + µ UK flavour workshop 13 8 / 1

9 A bit more on symmetries... A.1 Symmetries of the massless distribution [JHEP1(1)56] Egede et al. In this section we review the symmetry formalism for the massless angular distribut as presented ly in Ref. [7]. The six complex amplitudes present in this case can be arranged into three comp Define the basis: vectors: ( A ) ( ) ( ) n = A, n A R = A A symmetry of the angular distribution will therefore, n A R be= a unitary. transformation( acting in the same way on n A R, n and n, that is: n i Un i.suchasymmetryhasf independent All the coefficients parameters, J i can and be expressed can be written in terms as: Continuous transformations: of the products n i n j: [ ][ ][ ] J 1s = 3 ( n + n ) e iφ cos θ sin θ cosh i θ sinh, J 1c = n, J s = 1 ( i θ n n + n ) i = Un i = n, 4 e iφ i. ( R sin θ cos θ sinh i θ cosh 4 i θ J c = n, J 3 = 1 ( Of course, φ n n ), J 4 = 1 Re(n n,r are other global parametrizations phase changes are possible, of left and butright we prefer handed to keep amplitudes, this one to ) θmake, easy contact J 5 = with the generalization to the massive case and the notation Re(n n ), J 6s =Re(n n ), J 7 = introduced and θ are helicity+handedness transformations. Ref. [7]. The matrix U defines the four symmetries of the massless angular Im(n distributi n ), two global phase transformations (φ and φ R ), a rotation θ among the real and imagin components J 8 = of 1 the Im(n amplitudes n ), independently J 9 = Im(n and n another ), rotation J 6c =. θ that mixes real ( a imaginary components of the transversity amplitudes. A.1.1 Solution to the massless distribution 4 K.A. Petridis (IC) B K µ + µ UK flavour workshop 13 9 / 1

10 ...and a bit more... [JHEP1(1)56] Egede et al. Implement symmetry as a constraint on a set of amplitudes in order to have a well defined minimum in the angular fit Can choose to fix any 4 components as long as: 1 Solutions for φ,r,θ, θ exist, Transformed amplitudes can be parametrised by smooth function in q NB: The form and symmetries of the amplitudes in the large recoil region, give rise to 8 observables, 6 of which are form factor independent and which contain information on form factors.[jhep 14(1)14] Descotes-Genon et al. K.A. Petridis (IC) B K µ + µ UK flavour workshop 13 1 / 1

11 Parametrising amplitudes in q Focus on 1 < q < 6 GeV region mainly due to: 1 Potential resonant di-muon structures below 1 and above 6 GeV complicate situation Set of symmetries only apply only in m µ limit Squinting at.o. amplitude expression can see a general parametrisation: A i α i + β i q + γ i /q 3 parameters per real amplitude component gives 4 amplitude parameters (accounting for constraints) per B flavour K.A. Petridis (IC) B K µ + µ UK flavour workshop / 1

12 Fitting for the amplitudes Original attempt in [JHEP1(1)56] Egede et al. Amplitudes parametrised as nd order polynomials in q Choice of A =,Im(A R ) =,Im(A ) = leads to: azl_im 1 transformed q (GeV) Solution was to fit for.5 < q < 6 GeV. Propose a (better/physics) choice: A R =,Im(A ) =,Im(A R ) = K.A. Petridis (IC) B K µ + µ UK flavour workshop 13 1 / 1

13 apl_im transformed from system apl_re.3 transformed from system..1 atl_im.. transformed from system atl_re. transformed from system q (GeV) q (GeV) q (GeV) q (GeV) azl_im transformed from system azl_re.5 transformed from system apr_im.8.6 transformed from system apr_re.8 transformed from system q (GeV) q (GeV) q (GeV) q (GeV) atr_im transformed from system q (GeV) atr_re transformed from system q (GeV) azr_im.5 transformed from system q (GeV) transformed, Smooth behaviour of transformed amplitudes Also holds for wide range of new physics scenarios azr_re.8 transformed from system q (GeV) K.A. Petridis (IC) B K µ + µ UK flavour workshop / 1

14 Fit procedure Generate and fit toy datasets using rough estimates of signal and background yields in the 3 fb 1 HCb dataset. Signal generated using EOS central value amplitude predictions in the SM. [JHEP17(1)98] Bobeth, van Dyk et al. Use α, β, γ parametrisation to fit it back. Effects not yet accounted for: 1 S-wave in Kπ system. Detector bias of angular distribution. Results should be treated as a proof of principle. Events / ( ) B mass(mev/c ) K.A. Petridis (IC) B K µ + µ UK flavour workshop / 1

15 Results of amplitudes Results from ensembles of toy-experiments Stable fit behaviour, well defined error matrix Discrete degeneracies of amplitudes in full angular fit due to form of angular coefficients 5 Degeneracies not present in J i s 1 (bilinear combinations of 15 amplitudes) )> <Re(A <Re(A )> K.A. Petridis (IC) B K µ + µ UK flavour workshop / 1

16 s From amplitudes build the Js 3 fb 1 toy data Jc Js P5 prime.1.9 Experiment ± σ Experiment ± 1σ Experiment median Theory central value (SM EOS) c4) q (GeV/c4)...15 metrisation faithfully reproduces q dependence B! K µ+ µ q (GeV/c4) 6 J5 t. al. q (GeV/c4) - J6s -1.5 HCb week 4 17 / q (GeV/c4) K.A. Petridis (IC) B K µ+ µ 4 6 q (GeV/c4) UK flavour workshop / 1

17 From Js build observables 3 fb 1 toy data P P 5 prime Experiment ± σ Experiment ± 1σ Experiment median Theory central value (SM EOS) Experiment ± σ Experiment ± 1σ Experiment median Theory central value (SM EOS) 4 6 q (GeV /c 4 ) q (GeV /c 4 ) EOS [JHEP17(1)98] Bobeth, van Dyk et al. Can build any observable, e.g form factor independent P and P 5 [JHEP 131(13)48] Descotes-Genon et al. Choice of Amplitude q parametrisation faithfully reproduces q dependence of observables K.A. Petridis (IC) B K µ + µ UK flavour workshop / 1

18 From Js build observables 3 fb 1 toy data P prime P Experiment ± σ ( C =-1.5) 9 Experiment ± 1σ ( C =-1.5) 9 Experiment median ( C 9 =-1.5) Theory ± 1σ (SM EOS*) 4 6 q (GeV /c 4 ) -1 - Experiment ± σ ( C =-1.5) 9 Experiment ± 1σ ( C =-1.5) 9 Experiment median ( C 9 =-1.5) Theory ± 1σ (SM EOS*) 4 6 q (GeV /c 4 ) (*) EOS prediction with 15% uncertainty on sub-leading corrections [JHEP8(1)3] Beaujean et al. Full q shape information increases significance Combine observables to maximise sensitivity p-value to SM can be calculated using various observable bases. Correlations need to be taken into account. K.A. Petridis (IC) B K µ + µ UK flavour workshop / 1

19 Other q regions This study focused on 1 < q < 6 GeV region A lot of interest for low recoil and q < 1 GeV Separate treatment required. Ongoing effort in HCb Potential light and c c resonances Different binning required, single bin preferable for theory predictions? Could use resonances to extract additional information K.A. Petridis (IC) B K µ + µ UK flavour workshop / 1

20 Information transfer Accurate interpretation of measurements require correlations of observables, precise confidence intervals This information is available to experimentalists through the likelihood/dataset Ongoing discussion within HCb how best to provide results to theory community Input required from interested theory groups Is parametrisation of amplitudes satisfactory? Are observables preferable...etc... Tools like EOS allow experimentalists to extract Wilson coefficients Care needs to be taken on treatment of theory uncertainties A general consensus on form of prior, size, correlations etc, would be useful K.A. Petridis (IC) B K µ + µ UK flavour workshop 13 / 1

21 Conclusions Recent HCb measurements have revealed interesting phenomena in the sector of electroweak penguin transitions A lot of work within HCb to provide a comprehensive set of measurements in B K ( ) µ + µ and related decays Full angular analysis including a q parametrisation seems possible with the current dataset (1 < q < 6 GeV ) A lot of work required. Ongoing effort on measurements in rest of q region. Input from theory community is vital K.A. Petridis (IC) B K µ + µ UK flavour workshop 13 1 / 1