Four body charm amplitude analysis at LHCb T. Evans University of - PowerPoint PPT Presentation

Four body charm amplitude analysis at LHCb T. Evans University of Oxford Institute of Physics APP and HEPP Annual Conference 2017 April 11, 2017 1/13 1 / 13

Motivation ◮ Amplitude analysis of Cabibbo favoured decay D 0 → K − π + π + π − and suppressed decay D 0 → K + π − π − π + . ◮ Interference between these amplitudes in B → DX (see left hand example) can give access to γ → but need to know relative strong phase between amplitudes. ◮ Relative strong phase between DCS and CF varies across phase-space of D decay - can benefit from “local-knowledge”. ◮ Also of interest for Charm CPV, mixing. ◮ Amplitudes models are also a rich laboratory of strong and weak effects → very interesting in their own right. 2/13 2 / 13

Selection × 3 × 3 10 10 ) ) 160 2 2 Candidates / (1.20 MeV/c Candidates / (0.12 MeV/c 60 140 LHCb LHCb 50 Preliminary 120 Preliminary 40 100 ◮ Reconstruct B → D ∗ ( 2010 ) + [ D 0 π + ] µ X . 80 30 ◮ Charge of π + , µ relative to D 0 daughters gives 60 20 40 D 0 flavour. 10 20 0 0 exp exp ◮ Extremely clean ( ≈ 50 % purity after initial 2 2 N N N-N N-N 0 0 selection) source of D 0 decays. − − 2 2 1850 1900 140 145 150 π π π π π π π π π π 2 2 m(K ) [MeV/c ] m(K ) - m(K ) [MeV/c ] S ◮ Select offline with Boosted Decision Tree and PID on the kaon. ) ) 250 2 2 Candidates / (1.20 MeV/c Candidates / (0.12 MeV/c 600 LHCb LHCb ◮ Get ≈ 3000 WS events @ 80% purity, 900,000 500 200 Preliminary Preliminary RS events @ > 99 . 9 % purity. 400 150 300 ◮ Plot m ( K πππ ) and m ( K ππππ s ) − m ( K πππ ) for 100 200 both decay modes. 50 100 ◮ Flat(ish) efficiency over phase-space - correct 0 0 exp exp 2 2 N N N-N N-N 0 0 using simulated events. − − 2 2 1850 1900 140 145 150 π π π π π π π π π π 2 2 m(K ) [MeV/c ] m(K ) - m(K ) [MeV/c ] S 3/13 3 / 13

The Isobar Model The “traditional” isobar model. π + K − K ∗ (892) 0 π + a 1 (1260) + ρ (770) 0 π + D 0 D 0 π + π − ρ (770) 0 K − π − Extending to many bodies. 4/13 4 / 13

The Isobar Model (II) ◮ Divide multi-body process into chain of K − quasi-independent two body processes. ◮ Express two body strong FSI with dynamical T K ∗ ( p 2 functions T ( s ) . For example, Breit-Wigners, K ∗ ) L µ ( p K ∗ , q K ∗ ) K-matrices, Model Independent Partial Wave etc. π + ◮ Describe spin with covariant tensor formalism. D 0 T ρ ( p 2 ρ ) L µ ( p ρ , q ρ ) π + ◮ Main free parameters in the fit are complex S : g µν j µ 1 j ν coupling parameters between channels. 2 D q β D j µ P : ε µναβ p α 1 j ν 2 ◮ In pictured case, there is an independent coupling µν j µ 1 j ν D : L 2 2 for three different orbital configurations (S,P,D) ⇒ D 0 → VV system has 6 free parameters. π − = 5/13 5 / 13

The Isobar Model (III) ◮ Cascade processes are a bit more complicated as have to consider FSI between an isobar and a stable state. Consider the case of the a 1 ( 1260 ) : ◮ In the isobar model, express this in the width of the a 1 : π + T a 1 ( s ) − 1 = m 2 − s − i Γ( s ) m , a ) S µν ( p a , q a ) j ν T a 1 ( p 2 ρ T ρ ( p 2 ρ ) L µ ( p ρ , q ρ ) where π + � d x |M πππ ( x ) | 2 . Γ( s ) = D 0 ◮ The integral is over the three pion final state. L µ ( p D , q D ) j µ a 1 π − The matrix element coherently sums the K − different isobars that the a 1 couples to. ◮ Need to compute this numerically. Dependence of the width on couplings to isobars tends to be quite weak, so the problem is tractable. ◮ Simplified model of quite a complicated system - but works quite well. 6/13 6 / 13

Amplitude Fitting ◮ Maximum unbinned likelihood fit of both datasets using AmpGen package. ◮ Exploits natural tree structure of the isobar model. Example : D0{K*(892)bar0{K-,pi+},rho(770)0{pi+,pi-}} generates the tree shown on slide 4. ◮ Tree is used to generate the source code to evaluate the amplitude at runtime. ◮ Extremely fast, ≈ 10 6 amplitude evaluations /core/s. ◮ Can also pack up model, save the code for later : ◮ Plugin to MC generators (Gauss plugin = LbAmpGen) ◮ Distribute model code with documentation. ◮ Uses OpenMP for parallel evaluation of amplitudes, normalisation integrals. ◮ Correct for phase space acceptance variations by performing normalisation integrals for PDFs using MC events that have been propagated through full LHCb simulation. ◮ Integration events are generated according to an approximate model fitted by neglecting efficiency effects. ◮ Spin calculations verified against qft++, lineshapes against Laura++. 7/13 7 / 13

RS Results (I) × × ◮ Phase-space is five dimensional : 3 3 10 10 ) 45 ) 4 4 LHCb 30 LHCb /c /c show fit by projecting onto invariant 2 2 40 Unofficial Unofficial Events / (0.02 GeV Events / (0.02 GeV 25 mass squared s to show resonant 35 Signal 30 contributions. From top left to Combinatorial 20 25 bottom right : 15 20 ◮ K ∗ ( 892 ) 0 15 10 ◮ ρ ( 770 ) 0 10 5 5 ◮ K 1 ( 1270 ) − 0 0 ◮ a 1 ( 1260 ) − ( ≈ 30 % of the total 0.5 1 1.5 2 2.5 0.5 1 1.5 - π π π + 2 4 + - 2 4 s(K ) [GeV /c ] s( ) [GeV /c ] rate) × × 3 3 10 10 ) ) 22 22 4 4 ◮ χ 2 /ν ≈ 1 . 3 for ≈ ν ≈ 30 , 000, ≈ 60 LHCb LHCb /c /c 20 2 20 2 Unofficial Unofficial Events / (0.03 GeV Events / (0.02 GeV 18 18 free parameters. 16 16 14 14 ◮ Compare with BES III result with 12 12 10 10 ≈ 16 , 000 events. 8 8 6 6 ◮ LHCb model has comparable with 4 4 quality with comparable number of 2 2 0 0 free parameters with ≈ 60 × the 1 2 3 0.5 1 1.5 2 - π π π π π + - 2 4 + + - 2 4 s(K ) [GeV /c ] s( ) [GeV /c ] number of events. 8/13 8 / 13

RS Results (II) R ( g ) I ( g ) Fit Fraction K ∗ ( 892 ) 0 ρ ( 770 ) 0 5 . 734 ± 0 . 057 0 . 184 ± 0 . 001 − 0 . 079 ± 0 . 001 ◮ “Fit Fraction” defined for component i as : K ∗ ( 892 ) 0 ρ ( 770 ) 0 � L = 1 � 4 . 508 ± 0 . 038 0 . 062 ± 0 . 002 − 0 . 309 ± 0 . 002 K ∗ ( 892 ) 0 ρ ( 770 ) 0 � L = 2 � 9 . 459 ± 0 . 063 1 . 000 ± 0 . 000 0 . 000 ± 0 . 000 d x | g i A i | 2 � F i = , (1) � � d x g i g ∗ j A i A ∗ ij j for coupling parameters g . ◮ Becomes branching ratio in limit of narrow K − resonances - measure of the importance of a component in the model. T K ∗ ( p 2 K ∗ ) L µ ( p K ∗ , q K ∗ ) ◮ Curious pattern in the D 0 → VV π + D 0 polarisation L = 2 contribution greater T ρ ( p 2 ρ ) L µ ( p ρ , q ρ ) π + than the lower orbitals. S : g µν j µ 1 j ν 2 P : ε µναβ p α D q β D j µ 1 j ν ◮ Same pattern is seen in D 0 → ρρ . 2 µν j µ D : L 2 1 j ν 2 π − 9/13 9 / 13

WS Results (I) ◮ K 1 ( 1270 ) − coupling and shape parameters fixed from RS. ) ) 250 ◮ χ 2 /ν ≈ 1 . 7 ( ν ≈ 250, 12 free 4 4 /c LHCb /c LHCb 450 2 Unofficial 2 Unofficial Events / (0.05 GeV Events / (0.03 GeV 400 parameters) . 200 Signal 350 Combinatorial ◮ Dominated by K ∗ ( 892 ) 0 , mostly from 300 150 Mistag 250 cascades of the axial resonances 200 100 150 K 1 ( 1270 ) − / K 1 ( 1400 ) − . 100 50 a 1 / K 1 50 0 0 0.5 1 1.5 2 2.5 0.5 1 1.5 + π - 2 4 π - π 2 4 + s(K ) [GeV /c ] s( ) [GeV /c ] W ) 220 ) 4 4 200 /c LHCb /c LHCb 200 2 Unofficial 2 Unofficial Events / (0.05 GeV Events / (0.04 GeV 180 180 160 160 140 140 120 120 100 100 80 80 ◮ Connected to the a 1 ( 1260 ) − 60 60 40 40 contribution for the favoured mode. 20 20 0 0 ◮ Key features are strongly constrained 1 2 3 0.5 1 1.5 2 + π - π 2 4 π - π - π 2 4 + + s(K ) [GeV /c ] s( ) [GeV /c ] - but there are lots of possible parameterisations of equivalent fit quality .. 10/13 10 / 13

Coherence Factor and Binning Coherence factor defined as : ◮ Models can calculate coherence factor up to a global phase offset. Look at the prediction with � d x A K − 3 π ( x ) A K + 3 π ( x ) R K 3 π e − i δ K 3 π = . an ensemble of WS models with similar fit quality: �� �� d x |A K − 3 π ( x ) | 2 d x |A K + 3 π ( x ) | 2 # Models 25 LHCb Unofficial Appears in calculations for B → D h , D mixing. Can measure directly using a combination of charm 20 threshold data @ CLEO and LHCb mixing results : 15 10 5 0 0.5 0.55 0.6 0.65 0.7 0.75 R π K3 ◮ Models are consistent with the direct measurement. 11/13 11 / 13

Local coherence ) π LHCb K3 Unofficial δ 0 sin( Can also defined coherence factor in ‘bins‘: Binning algorithm π − K3 0.2 ◮ Divide data into “voxels” of equal R population using a decision tree. ◮ Group voxels by relative strong phase − 0.4 between two modes, so we get N bins of equal population where the phase − doesn’t vary much in each bin. 0.6 ◮ Measure R and δ in bins from one model for the ensemble of models. − 0.8 Phase description is rather stable w.r.t. model choice → − 0.5 0 δ R cos( ) π π K3 K3 12/13 12 / 13

Conclusions ◮ Developed amplitude models of D 0 → K − π + π + π − , D 0 → K + π − π − π + . ◮ Analysis is currently under review by collaboration. ◮ Models provide rather stable predictions for phase variations → can reliably use this to optimise future model independent measurements. 13/13 13 / 13