Evidence for CP violation in B + → K ∗ (892 ) + π 0 from a Dalitz plot analysis of B + → K 0 S π + π 0 decays
aa r X i v : . [ h e p - e x ] J a n B A B AR -PUB-14/011SLAC-PUB-16186 Evidence for CP violation in B + → K ∗ (892) + π from a Dalitz plot analysis of B + → K S π + π decays J. P. Lees, V. Poireau, and V. Tisserand
Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP),Universit´e de Savoie, CNRS/IN2P3, F-74941 Annecy-Le-Vieux, France
E. Grauges
Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain
A. Palano ab INFN Sezione di Bari a ; Dipartimento di Fisica, Universit`a di Bari b , I-70126 Bari, Italy G. Eigen and B. Stugu
University of Bergen, Institute of Physics, N-5007 Bergen, Norway
D. N. Brown, L. T. Kerth, Yu. G. Kolomensky, M. J. Lee, and G. Lynch
Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA
H. Koch and T. Schroeder
Ruhr Universit¨at Bochum, Institut f¨ur Experimentalphysik 1, D-44780 Bochum, Germany
C. Hearty, T. S. Mattison, J. A. McKenna, and R. Y. So
University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1
A. Khan
Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom
V. E. Blinov abc , A. R. Buzykaev a , V. P. Druzhinin ab , V. B. Golubev ab , E. A. Kravchenko ab ,A. P. Onuchin abc , S. I. Serednyakov ab , Yu. I. Skovpen ab , E. P. Solodov ab , and K. Yu. Todyshev ab Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090 a ,Novosibirsk State University, Novosibirsk 630090 b ,Novosibirsk State Technical University, Novosibirsk 630092 c , Russia A. J. Lankford
University of California at Irvine, Irvine, California 92697, USA
B. Dey, J. W. Gary, and O. Long
University of California at Riverside, Riverside, California 92521, USA
M. Franco Sevilla, T. M. Hong, D. Kovalskyi, J. D. Richman, and C. A. West
University of California at Santa Barbara, Santa Barbara, California 93106, USA
A. M. Eisner, W. S. Lockman, W. Panduro Vazquez, B. A. Schumm, and A. Seiden
University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA
D. S. Chao, C. H. Cheng, B. Echenard, K. T. Flood, D. G. Hitlin,T. S. Miyashita, P. Ongmongkolkul, F. C. Porter, and M. R¨ohrken
California Institute of Technology, Pasadena, California 91125, USA
R. Andreassen, Z. Huard, B. T. Meadows, B. G. Pushpawela, M. D. Sokoloff, and L. Sun
University of Cincinnati, Cincinnati, Ohio 45221, USA
P. C. Bloom, W. T. Ford, A. Gaz, J. G. Smith, and S. R. Wagner
University of Colorado, Boulder, Colorado 80309, USA
R. Ayad ∗ and W. H. Toki Colorado State University, Fort Collins, Colorado 80523, USA
B. Spaan
Technische Universit¨at Dortmund, Fakult¨at Physik, D-44221 Dortmund, Germany
D. Bernard and M. Verderi
Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France
S. Playfer
University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
D. Bettoni a , C. Bozzi a , R. Calabrese ab , G. Cibinetto ab , E. Fioravanti ab ,I. Garzia ab , E. Luppi ab , L. Piemontese a , and V. Santoro a INFN Sezione di Ferrara a ; Dipartimento di Fisica e Scienze della Terra, Universit`a di Ferrara b , I-44122 Ferrara, Italy A. Calcaterra, R. de Sangro, G. Finocchiaro, S. Martellotti,P. Patteri, I. M. Peruzzi, † M. Piccolo, M. Rama, and A. Zallo
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
R. Contri ab , M. R. Monge ab , S. Passaggio a , and C. Patrignani ab INFN Sezione di Genova a ; Dipartimento di Fisica, Universit`a di Genova b , I-16146 Genova, Italy B. Bhuyan and V. Prasad
Indian Institute of Technology Guwahati, Guwahati, Assam, 781 039, India
A. Adametz and U. Uwer
Universit¨at Heidelberg, Physikalisches Institut, D-69120 Heidelberg, Germany
H. M. Lacker
Humboldt-Universit¨at zu Berlin, Institut f¨ur Physik, D-12489 Berlin, Germany
U. Mallik
University of Iowa, Iowa City, Iowa 52242, USA
C. Chen, J. Cochran, and S. Prell
Iowa State University, Ames, Iowa 50011-3160, USA
H. Ahmed
Physics Department, Jazan University, Jazan 22822, Kingdom of Saudia Arabia
A. V. Gritsan
Johns Hopkins University, Baltimore, Maryland 21218, USA
N. Arnaud, M. Davier, D. Derkach, G. Grosdidier, F. Le Diberder,A. M. Lutz, B. Malaescu, ‡ P. Roudeau, A. Stocchi, and G. Wormser
Laboratoire de l’Acc´el´erateur Lin´eaire, IN2P3/CNRS et Universit´e Paris-Sud 11,Centre Scientifique d’Orsay, F-91898 Orsay Cedex, France
D. J. Lange and D. M. Wright
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
J. P. Coleman, J. R. Fry, E. Gabathuler, D. E. Hutchcroft, D. J. Payne, and C. Touramanis
University of Liverpool, Liverpool L69 7ZE, United Kingdom
A. J. Bevan, F. Di Lodovico, and R. Sacco
Queen Mary, University of London, London, E1 4NS, United Kingdom
G. Cowan
University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom
D. N. Brown and C. L. Davis
University of Louisville, Louisville, Kentucky 40292, USA
A. G. Denig, M. Fritsch, W. Gradl, K. Griessinger, A. Hafner, and K. R. Schubert
Johannes Gutenberg-Universit¨at Mainz, Institut f¨ur Kernphysik, D-55099 Mainz, Germany
R. J. Barlow § and G. D. Lafferty University of Manchester, Manchester M13 9PL, United Kingdom
R. Cenci, B. Hamilton, A. Jawahery, and D. A. Roberts
University of Maryland, College Park, Maryland 20742, USA
R. Cowan
Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA
R. Cheaib, P. M. Patel, ¶ and S. H. Robertson McGill University, Montr´eal, Qu´ebec, Canada H3A 2T8
N. Neri a and F. Palombo ab INFN Sezione di Milano a ; Dipartimento di Fisica, Universit`a di Milano b , I-20133 Milano, Italy L. Cremaldi, R. Godang, ∗∗ and D. J. Summers University of Mississippi, University, Mississippi 38677, USA
M. Simard and P. Taras
Universit´e de Montr´eal, Physique des Particules, Montr´eal, Qu´ebec, Canada H3C 3J7
G. De Nardo ab , G. Onorato ab , and C. Sciacca ab INFN Sezione di Napoli a ; Dipartimento di Scienze Fisiche,Universit`a di Napoli Federico II b , I-80126 Napoli, Italy G. Raven
NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands
C. P. Jessop and J. M. LoSecco
University of Notre Dame, Notre Dame, Indiana 46556, USA
K. Honscheid and R. Kass
Ohio State University, Columbus, Ohio 43210, USA
M. Margoni ab , M. Morandin a , M. Posocco a , M. Rotondo a , G. Simi ab , F. Simonetto ab , and R. Stroili ab INFN Sezione di Padova a ; Dipartimento di Fisica, Universit`a di Padova b , I-35131 Padova, Italy S. Akar, E. Ben-Haim, M. Bomben, G. R. Bonneaud, H. Briand,G. Calderini, J. Chauveau, Ph. Leruste, G. Marchiori, and J. Ocariz
Laboratoire de Physique Nucl´eaire et de Hautes Energies,IN2P3/CNRS, Universit´e Pierre et Marie Curie-Paris6,Universit´e Denis Diderot-Paris7, F-75252 Paris, France
M. Biasini ab , E. Manoni a , and A. Rossi a INFN Sezione di Perugia a ; Dipartimento di Fisica, Universit`a di Perugia b , I-06123 Perugia, Italy C. Angelini ab , G. Batignani ab , S. Bettarini ab , M. Carpinelli ab , †† G. Casarosa ab , M. Chrzaszcz a ,F. Forti ab , M. A. Giorgi ab , A. Lusiani ac , B. Oberhof ab , E. Paoloni ab , G. Rizzo ab , and J. J. Walsh a INFN Sezione di Pisa a ; Dipartimento di Fisica, Universit`a di Pisa b ; Scuola Normale Superiore di Pisa c , I-56127 Pisa, Italy D. Lopes Pegna, J. Olsen, and A. J. S. Smith
Princeton University, Princeton, New Jersey 08544, USA
F. Anulli a , R. Faccini ab , F. Ferrarotto a , F. Ferroni ab , M. Gaspero ab , A. Pilloni ab , and G. Piredda a INFN Sezione di Roma a ; Dipartimento di Fisica,Universit`a di Roma La Sapienza b , I-00185 Roma, Italy C. B¨unger, S. Dittrich, O. Gr¨unberg, M. Hess, T. Leddig, C. Voß, and R. Waldi
Universit¨at Rostock, D-18051 Rostock, Germany
T. Adye, E. O. Olaiya, and F. F. Wilson
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom
S. Emery and G. Vasseur
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
D. Aston, D. J. Bard, C. Cartaro, M. R. Convery, J. Dorfan, G. P. Dubois-Felsmann, W. Dunwoodie, M. Ebert,R. C. Field, B. G. Fulsom, M. T. Graham, C. Hast, W. R. Innes, P. Kim, D. W. G. S. Leith, D. Lindemann,S. Luitz, V. Luth, H. L. Lynch, D. B. MacFarlane, D. R. Muller, H. Neal, M. Perl, ¶ T. Pulliam, B. N. Ratcliff,A. Roodman, R. H. Schindler, A. Snyder, D. Su, M. K. Sullivan, J. Va’vra, W. J. Wisniewski, and H. W. Wulsin
SLAC National Accelerator Laboratory, Stanford, California 94309 USA
M. V. Purohit and J. R. Wilson
University of South Carolina, Columbia, South Carolina 29208, USA
A. Randle-Conde and S. J. Sekula
Southern Methodist University, Dallas, Texas 75275, USA
M. Bellis, P. R. Burchat, and E. M. T. Puccio
Stanford University, Stanford, California 94305-4060, USA
M. S. Alam and J. A. Ernst
State University of New York, Albany, New York 12222, USA
R. Gorodeisky, N. Guttman, D. R. Peimer, and A. Soffer
Tel Aviv University, School of Physics and Astronomy, Tel Aviv, 69978, Israel
S. M. Spanier
University of Tennessee, Knoxville, Tennessee 37996, USA
J. L. Ritchie and R. F. Schwitters
University of Texas at Austin, Austin, Texas 78712, USA
J. M. Izen and X. C. Lou
University of Texas at Dallas, Richardson, Texas 75083, USA
F. Bianchi ab , F. De Mori ab , A. Filippi a , and D. Gamba ab INFN Sezione di Torino a ; Dipartimento di Fisica, Universit`a di Torino b , I-10125 Torino, Italy L. Lanceri ab and L. Vitale ab INFN Sezione di Trieste a ; Dipartimento di Fisica, Universit`a di Trieste b , I-34127 Trieste, Italy F. Martinez-Vidal, A. Oyanguren, and P. Villanueva-Perez
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
J. Albert, Sw. Banerjee, A. Beaulieu, F. U. Bernlochner, H. H. F. Choi, G. J. King, R. Kowalewski,M. J. Lewczuk, T. Lueck, I. M. Nugent, J. M. Roney, R. J. Sobie, and N. Tasneem
University of Victoria, Victoria, British Columbia, Canada V8W 3P6
T. J. Gershon, P. F. Harrison, and T. E. Latham
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
H. R. Band, S. Dasu, Y. Pan, R. Prepost, and S. L. Wu
University of Wisconsin, Madison, Wisconsin 53706, USA
We report a Dalitz plot analysis of charmless hadronic decays of charged B mesons to the final state K S π + π using the full B A B AR dataset of 470 . ± . BB events collected at the Υ (4 S ) reso-nance. We measure the overall branching fraction and CP asymmetry to be B (cid:0) B + → K π + π (cid:1) = (cid:0) . ± . ± . +8 . − . (cid:1) × − and A CP (cid:0) B + → K π + π (cid:1) = 0 . ± . ± . +0 . − . , where the un-certainties are statistical, systematic, and due to the signal model, respectively. This is the firstmeasurement of the branching fraction for B + → K π + π . We find first evidence of a CP asym-metry in B + → K ∗ (892) + π decays: A CP (cid:0) B + → K ∗ (892) + π (cid:1) = − . ± . ± . +0 . − . . Thesignificance of this asymmetry, including systematic and model uncertainties, is 3 . CP asymmetries for three other intermediatedecay modes. PACS numbers: 13.25.Hw, 12.15.Hh, 11.30.Er
I. INTRODUCTION
The Cabibbo-Kobayashi-Maskawa (CKM) mecha-nism [1, 2] for quark mixing describes all weak chargedcurrent transitions between quarks in terms of a unitaritymatrix with four parameters: three rotation angles andan irreducible phase. The unitarity of the CKM matrixis usually expressed as triangle relationships among itselements. The interference between tree-level and loop(“penguin”) amplitudes can give rise to direct CP vio-lation, which is sensitive to the angles of the UnitarityTriangle, denoted α , β , and γ . Measurements of the pa-rameters of the CKM matrix provide an important testof the Standard Model (SM) since any deviation fromunitarity or discrepancies between measurements of thesame parameter in different decay processes would im-ply a possible signature of new physics. Tree ampli-tudes in B → K ∗ π decays are sensitive to γ , which canbe extracted from interferences between the intermedi-ate states that populate the Kππ
Dalitz plane. How-ever, these amplitudes are Cabibbo-suppressed relativeto contributions carrying a different phase and involv-ing radiation of either a gluon (QCD penguin) or photon(electroweak penguin or EWP) from a loop. ∗ Now at: University of Tabuk, Tabuk 71491, Saudi Arabia † Also at: Universit`a di Perugia, Dipartimento di Fisica, I-06123Perugia, Italy ‡ Now at: Laboratoire de Physique Nucl´eaire et de Hautes Energies,IN2P3/CNRS, F-75252 Paris, France § Now at: University of Huddersfield, Huddersfield HD1 3DH, UK ¶ Deceased ∗∗ Now at: University of South Alabama, Mobile, Alabama 36688,USA †† Also at: Universit`a di Sassari, I-07100 Sassari, Italy
QCD penguin contributions can be eliminated by con-structing a linear combination of the weak decay am-plitudes for B + → K ∗ π to form a pure isospin I = state [3]: A = A (cid:0) K ∗ π + (cid:1) + √ A (cid:0) K ∗ + π (cid:1) . (1)Since all transitions from I = to I = states occur viaonly ∆ I = 1 operators, A is free from QCD contribu-tions. The weak phase of A is often denoted asΦ = −
12 Arg (cid:16) ¯ A /A (cid:17) , (2)where ¯ A is the CP conjugate of the amplitude in Eq. (1).The phase Φ in Eq. (2) is the CKM angle γ in theabsence of EWP contributions [4].Measurements of the rates and CP asymmetries in B → Kπ have generated considerable interest becauseof possible hints of new-physics contributions [5, 6]. Ofparticular interest is the difference, ∆ A CP , between the CP asymmetry in B + → K + π and the CP asymme-try in B → K + π − , which in the SM is expected to beconsistent with zero within the theoretical uncertaintiesassuming U-spin symmetry and in the absence of color-suppressed tree and electroweak amplitudes [7, 8]. Usingthe average values of A CP of K + π and K + π − decays [9],∆ A CP ( Kπ ) is∆ A CP ( Kπ ) = A CP (cid:0) K + π (cid:1) − A CP (cid:0) K + π − (cid:1) = 0 . ± . , (3)which differs from zero by 5 . B → K ∗ π and B → Kρ [11–13], for which the ra-tios of tree-to-penguin amplitudes are expected to be twoto three times larger than for B → Kπ decays. Hence, B → K ∗ π and B → Kρ decays could have considerablylarger CP asymmetries.In this article, we present the results from an ampli-tude analysis of B + → K S π + π decays. The inclusion ofcharge conjugate processes is implied throughout this ar-ticle, except when referring to CP asymmetries. This isthe first Dalitz plot analysis of this decay by B A B AR ; theonly previous B A B AR analysis of this decay was restrictedto measuring the branching fraction and CP asymmetryof B + → K ρ + [14]. An upper limit on the branchingfraction for B + → K π + π was set by the CLEO Col-laboration: B ( B + → K π + π ) < × − [15].Two contributions to the K S π + π final state arisefrom the resonant decays B + → K ∗ (892) π + and B + → K ∗ (892) + π . Although both the rate and CP asymmetries for B + → K ∗ (892) π + have been well mea-sured, with K ∗ → K + π − , by both the B A B AR [16] andBelle [17] Collaborations, the measurements of the rateand CP asymmetry for B + → K ∗ (892) + π [18] have sig-nificant statistical uncertainties and could benefit fromthe additional information provided by a full amplitudeanalysis. In Table I we review the existing measurementsof the rates and CP asymmetries in the B → K ∗ (892) π system. TABLE I: Average values of the branching fractions B and CP asymmetries A CP for B → K ∗ (892) π decays as deter-mined by the Heavy Flavor Averaging Group [9].Mode B (10 − ) A CP References K ∗ + π − . ± . − . ± .
06 [19–22] K ∗ + π . ± . − . ± .
24 [18] K ∗ π + . +0 . − . − . ± .
042 [16, 17] K ∗ π . ± . − . ± .
13 [19, 23]
This article is organised as follows. The isobar modelused to parameterize the complex amplitudes describingthe intermediate resonances contributing to the K S π + π final state is presented in Section II. A brief descrip-tion of the B A B AR detector and the dataset is given inSection III. The event reconstruction and selection arediscussed in detail in Section IV, the background studyin Section V, and a description of the extended maxi-mum likelihood fit in Section VI. The results are given inSection VII, and a study of the systematic uncertaintiesis presented in Section VIII. In Section IX, we providea summary and conclusion, discussing the results andcombining the branching fractions and CP asymmetriesfor the decays B + → K ∗ (892) π + , B + → K ∗ (1430) π + ,and B + → K ∗ (892) + π with previous B A B AR resultsobtained from the final states B + → K + π − π + and B + → K + π π . II. AMPLITUDE ANALYSIS FORMALISM
A number of intermediate states contribute to thedecay B + → K S π + π . Their individual contributionsare measured by performing a maximum likelihood fitto the distribution of events in the Dalitz plot formedfrom the two variables, m K S π + and m π + π . We use the Laura++ [24] software to perform this fit.The total signal amplitudes for the B + and the B − decays are given in the isobar formalism by [25, 26] A (cid:16) m K S π + , m π + π (cid:17) = X j c j F j (cid:16) m K S π + , m π + π (cid:17) , (4)¯ A (cid:16) m K S π − , m π − π (cid:17) = X j ¯ c j ¯ F j (cid:16) m K S π − , m π − π (cid:17) , (5) where c j is the complex coefficient for a given resonantdecay mode j contributing to the Dalitz plot. This com-plex coefficient contains the weak-interaction phase de-pendence that is measured relative to one of the con-tributing resonant channels. In this article we reportresults for the relative phases between each pair of am-plitudes.The function F j describes the dynamics of the decayamplitudes and is the product of a resonant lineshape( R j ), two Blatt-Weisskopf barrier factors [27] ( X L ), andan angular-dependent term ( T j,L ) [28]: F j = R j × X L ( | ~p | , | ~p | ) × X L ( | ~q | , | ~q | ) × T j,L ( ~p, ~q ) , (6)where L is the orbital angular momentum between the in-termediate resonance and the bachelor particle (the bach-elor particle is the daughter of the B decay that does notarise from the resonance), ~q is the momentum of one ofthe daughters of the resonance in the rest frame of theresonance, ~p is the momentum of the bachelor particle inthe rest frame of the resonance, and ~p and ~q are thevalues of ~p and ~q , respectively, at the nominal mass ofthe resonance. The Blatt-Weisskopf barrier factors aregiven by X L =0 ( | ~u | , | ~u | ) = 1 , (7) X L =1 ( | ~u | , | ~u | ) = r z z , (8) X L =2 ( | ~u | , | ~u | ) = s ( z − + 9 z ( z − + 9 z , (9)where z = ( | ~u | r BW ) , z = ( | ~u | r BW ) , ~u is either ~q or ~p ,and r BW = 4 . /c ) − is the meson radius parameter.The uncertainty in r BW , used for systematic variations,is ± /c ) − for the K ∗ resonances, and ranges from − . . /c ) − for the ρ (770) + [28]. The an-gular term depends on the spin of the resonance and isgiven by [29, 30] T j,L =0 = 1 , (10) T j,L =1 = − ~p.~q, (11) T j,L =2 = 43 h ~p.~q ) − ( | ~p | | ~q | ) i . (12) π ) + π S (K )p( π )q( + π )q(- S K + π ) π S (K )p( + π )q( S K)q(- π S )K π + π ( )p( S K)q( + π )q(- π FIG. 1: Schematic representation of the definitions of ~q and ~p used in this analysis for the (left) K S π + , (center) K S π , and(right) π + π resonances.TABLE II: Parameters of the Dalitz plot model for B + → K S π + π used in the nominal fit. The mass and widthof the ρ (770) + and their uncertainties are taken from the anal-yses by the ALEPH [31] and CMD2 [32] Collaborations. Allother parameters are taken from Ref. [28]. The resonanceshapes are a Gounaris-Sakurai (GS) function, a relativisticBreit-Wigner (RBW) function, or based on measurements bythe LASS Collaboration [33], with a the scattering length and r the effective range of the LASS parametrization.Resonance Lineshape ParametersResonance mass Width (cid:0) MeV /c (cid:1) ( MeV) ρ (770) + GS 775 . ± . . ± . K ∗ (892) + RBW 891 . ± . . ± . K ∗ (892) RBW 896 . ± . . ± . Kπ ) ∗ / +0 LASS 1412 ±
50 294 ± m cutoff = 1800 MeV /c [16] a = 2 . ± . /c ) − [16] r = 3 . ± . /c ) − [16] The choice of which resonance daughter is defined tocarry the momentum ~q is a matter of convention. How-ever, its definition is important when comparing mea-surements from different experiments. In Fig. 1, we il-lustrate the momentum definitions used for the K S π + , K S π , and π + π resonance combinations.Table II lists the resonances used to model the signal.We determine a nominal model from data by studyingchanges in the log likelihood values for the best fit whenomitting or adding a resonance to the fit model, as de-scribed in Section VI.For the K ∗ (892) and K ∗ (892) + resonances, we use arelativistic Breit-Wigner (RBW) lineshape [28]: R RBW j ( m ) = 1 m − m − im Γ( m ) , (13)where m is the two-body invariant mass and Γ( m ) isthe mass-dependent width. In general, for a resonance decaying to spin-0 particles, Γ( m ) can be expressed asΓ( m ) = Γ (cid:18) | ~q || ~q | (cid:19) L +1 (cid:16) m m (cid:17) X L ( | ~q | , | ~q | ) , (14)where m and Γ are the nominal mass and width of theresonance.The Gounaris-Sakurai (GS) parametrization [34] isused to describe the lineshape of the ρ resonance decayinginto two pions. The parametrization takes the form R GS j = 1 + Γ · d/m m − m + f ( m ) − im Γ( m ) , (15)where Γ( m ) is given by Eq. (14). Expressions for f ( m ),in terms of Γ and m , and the constant d can be foundin Ref. [34]. The parameters specifying the ρ lineshapeare taken from Refs. [31, 32], which provides lineshapeinformation derived from fits to e + e − annihilation and τ lepton decay data.For the J P = 0 + component of the Kπ spectrum, de-noted ( Kπ ) ∗ / +0 , we make use of the LASS parametriza-tion [33], which consists of a K ∗ resonant term togetherwith an effective-range, nonresonant component to de-scribe the slowly increasing phase as a function of the Kπ mass: R LASS j = m | ~q | cot δ B − i | ~q | + e iδ B m Γ m | ~q | ( m − m ) − im Γ | ~q | m m | ~q | , (16)where cot δ B = a | ~q | + r | ~q | . The values used for thescattering length a and the effective range r are givenin Table II. The effective-range component has a cutoffimposed at 1800 MeV /c [16]. Integrating separately theresonant term, the effective-range term, and the coher-ent sum, we find that the K ∗ (1430) and the K ∗ (1430) + resonances account for 88% of the sum, and the effec-tive range component 49%; the 37% excess is due to de-structive interference between the two terms. The LASSparametrization is the least-well-determined componentof the signal model; we discuss the impact of these un-certainties in Section VIII.The complex coefficients c j and ¯ c j in Eqs. (4,5)can be parametrized in different ways; we follow theparametrization used in Ref. [16] as it avoids a bias inthe measurement of amplitudes and phases when the res-onant components have small magnitudes: c j = ( x j + ∆ x j ) + i ( y j + ∆ y j ) , (17)¯ c j = ( x j − ∆ x j ) + i ( y j − ∆ y j ) , where x j ± ∆ x j and y j ± ∆ y j are the real and imagi-nary parts of the amplitudes. The quantities ∆ x j and∆ y j parametrize the CP violation in the decay. The CP asymmetry for a given intermediate state is given by A CP,j = | ¯ c j | − | c j | | ¯ c j | + | c j | (18)= − x j ∆ x j + y j ∆ y j ) x j + ∆ x j + y j + ∆ y j . (19)The results quoted for the resonances in the follow-ing analysis use fit fractions (FF j ) as phase-convention-independent quantities representing the fractional rate ofeach contribution in the Dalitz plot. The FF for mode j is defined asFF j = R R (cid:16) | c j F j | + (cid:12)(cid:12) ¯ c j ¯ F j (cid:12)(cid:12) (cid:17) dm Kπ dm ππ R R (cid:16) | A | + (cid:12)(cid:12) ¯ A (cid:12)(cid:12) (cid:17) dm Kπ dm ππ . (20)The sum of all the fit fractions does not necessarily yieldunity due to constructive and destructive interference, asquantified by the interference fit fractions given by [30]FF ij = R R Re (cid:2) c i c ∗ j F i F ∗ j (cid:3) dm Kπ dm ππ R R | P k c k F k | dm Kπ dm ππ . (21)The parameters x j , ∆ x j , y j , and ∆ y j are determinedin the fit, except for the reference amplitude. Fit frac-tions, relative phases, and asymmetries are derived fromthe fit parameters and their statistical uncertainties de-termined from pseudo experiments generated from the fitresults. III. THE B A B AR DETECTOR AND MCSIMULATION
The data used in the analysis were collected withthe B A B AR detector at the PEP-II asymmetric-energy e + e − collider at SLAC National Accelerator Laboratory.The sample consists of 429 fb − of integrated luminosityrecorded at the Υ (4 S ) resonance mass (“on-peak”) and45 fb − collected 40 MeV below the resonance mass (“off-peak”) [35]. The on-peak sample corresponds to the full B A B AR Υ (4 S ) dataset and contains 470 . ± . BB events [30]. A detailed description of the B A B AR de-tector is given in Refs. [36, 37]. Charged-particle tracksare measured by means of a five-layer double-sided sil-icon vertex tracker (SVT) and a 40-layer drift chamber(DCH), both positioned within a solenoid that providesa 1.5 T magnetic field. Charged-particle identificationis achieved by combining the information from a ring-imaging Cherenkov detector (DIRC) and specific ioniza-tion energy loss (d E/ d x ) measurements from the DCHand SVT. Photons are detected and their energies mea-sured in a CsI(Tl) electromagnetic calorimeter (EMC).Muon candidates are identified in the instrumented fluxreturn of the solenoid.We use Geant4 -based software to simulate the de-tector response and account for the varying beam and experimental conditions [38, 39]. The
EvtGen [40] and
Jetset7.4 [41] software packages are used to generate sig-nal and background Monte-Carlo (MC) event samples inorder to determine efficiencies and evaluate backgroundcontributions for different selection criteria.
IV. EVENT SELECTION
We reconstruct B + → K S π + π candidates from one π candidate, one K S candidate reconstructed from a pair ofoppositely charged pions, and a charged pion candidate.The π candidate is formed from a pair of neutral en-ergy clusters in the EMC with laboratory energies above0 .
05 GeV and lateral moments [42] between 0 .
01 and 0 . π tolie in the range 0 . < m γγ < .
16 GeV /c . The K S can-didate is required to have a π + π − invariant mass within15 MeV /c of the K S mass [28], and a proper decay timegreater than 0 . × − s. To reduce combinatorial back-ground, we also require that the K S candidates have avertex probability greater than 10 − and that the cosineof the angle between the K S momentum direction andthe K S flight direction (as determined by the interactionpoint and the K S vertex) be greater than 0 . π + candidate, we use information from the tracking sys-tems, the EMC, and the DIRC to select a charged trackconsistent with the pion hypothesis. We constrain the π + track and K S candidate to originate from a commonvertex.Signal events that are misreconstructed with the de-cay products of one or more daughters completely orpartially exchanged with other particles in the rest ofthe event have degraded kinematic resolution. We re-fer to these as “self-cross-feed” (SCF) events. This mis-reconstruction has a strong dependence on the energyof the particles concerned and is more frequent for low-energy particles, i.e. , for decays in the corners of theDalitz plot. Because of the presence of a π in the finalstate, there is a significant probability for signal events tobe misreconstructed due to low-energy photons from the π decay. Using a classification based on MC informa-tion, we find that in simulated events the SCF fractiondepends strongly on the resonant substructure of the sig-nal and ranges from 34% for B + → K ∗ (892) + π to 50%for B + → ρ (770) + K S . In events simulated uniformly inphase space, hereafter referred to as nonresonant MC,the SCF fraction varies from less than 10% in the centerof the Dalitz plot to almost 70% in the two corners ofthe Dalitz plot, where either the π or the π + has lowenergy. We describe how the SCF events are handled inSection VI.In order to suppress the dominant background, due tocontinuum e + e − → qq ( q = u, d, s, c ) events, we employa boosted decision tree (BDT) algorithm that combinesfour variables commonly used to discriminate jet-like qq events from the more spherical BB events in the e + e − center-of-mass (CM) frame. The first of these is the ratioof the second-to-zeroth order momentum-weighted Leg-endre polynomial moments, L L = P i ∈ ROE 12 (cid:0) θ i − (cid:1) p i P i ∈ ROE p i , (22)where the summations are over all tracks and neutralclusters in the event, excluding those that form the B candidate (the “rest of the event” or ROE); p i is the par-ticle momentum, and θ i is the angle between the particleand the thrust axis of the B candidate, hereafter alsoreferred to as the B . The three other variables enteringthe BDT are the absolute value of the cosine of the an-gle between the B direction and the collision axis, thezeroth-order momentum-weighted Legendre polynomialmoment, and the absolute value of the output of anotherBDT used for “flavor tagging”, i.e. , for distinguishing B from B decays using inclusive properties of the decay ofthe other B meson in the Υ (4 S ) → BB event [43]. Themomentum-weighted Legendre polynomial moments andthe cosine of the angle between the B direction and thebeam axis are calculated in the e + e − CM frame. TheBDT is trained on a sample of signal MC events and off-peak data. We apply a loose criterion on the BDT outputof BDT out > .
06, which retains approximately 70% ofthe signal while rejecting 92% of the qq background.In addition to BDT out , we use two kinematic variablesto distinguish the signal from the background: m ES = q E − p B , (23)∆ E = E ⋆B − √ s/ , (24)where E X = ( s/ p e + e − · p B ) /E e + e − , (25)and where √ s is the total e + e − CM energy, with( E e + e − , p e + e − ) and ( E B , p B ) the four-momenta of theinitial e + e − system and the B candidate, respectively,both measured in the lab frame, while the star indicatesthe e + e − CM frame. The signal m ES distribution for cor-rectly reconstructed events is approximately independentof their position in the K S π + π Dalitz plot and peaksnear the B mass with a resolution of about 3 . /c .We retain all candidates satisfying the following selec-tion criteria: 5 . < m ES < .
29 GeV /c and − . < ∆ E < . . < m ES < .
287 GeV /c and − . < ∆ E < .
15 GeV. We also use candidates in the sideband re-gion of m ES defined by 5 . < m ES < .
26 GeV /c and − . < ∆ E < .
15 GeV and subtract from distribu-tions for these events the BB background contributionspredicted by MC simulations. We then add these dis-tributions to the off-peak data distributions to increasethe statistical precision of our model of the Dalitz plotdistribution for continuum background. Each of the B candidates is refit to determine theDalitz plot variables. In these fits the K S π + π invari-ant mass is constrained to the world average value of the B mass [28] to improve position resolution within theDalitz plot.We find that 20% of the remaining events in nonreso-nant MC have two or more candidates. We choose thebest candidate in multiple-candidate events based on thehighest B -vertex probability. This procedure is found toselect a correctly reconstructed candidate more than 60%of the time and does not bias the fit variables.The reconstruction efficiency over the Dalitz plot ismodeled using a two-dimensional (2D) binned distri-bution based on a generated sample of approximately2 × simulated B + → K S π + π MC events, where theevents uniformly populate phase space. All selection cri-teria are applied except for those corresponding to a Kπ invariant-mass veto described below, which is taken intoaccount separately. The 2D histogram of reconstructedMC events is then divided by the 2D histogram of thegenerated MC events. In order to expand regions of phasespace with large efficiency variations, the Dalitz plot vari-ables are transformed into “square Dalitz plot” [44] coor-dinates. We obtain an average efficiency, for nonresonantMC events, of approximately 15%. In the likelihood fitwe use an event-by-event efficiency that depends on theDalitz plot position. V. BB BACKGROUNDS
In addition to continuum events, background arisesfrom non-signal BB events. A major source of BB background arises from B + → D (cid:0) → K S π (cid:1) π + de-cays. To suppress this background, we veto events with1 . < m K S π < .
924 GeV /c .The remaining BB backgrounds are studied using MCsimulations and classified based on the shape of the m ES ,∆ E , and Dalitz plot distributions. We identify nine cat-egories of BB backgrounds: categories 1, 2 and 3 includedifferent types of three- and four-body B decays involv-ing an intermediate D meson; categories 4 and 5 includecharmless four-body B decays to intermediate resonanceswhere a π in the final state is not reconstructed; cate-gories 6 and 7 include two-body B decays with a radi-ated photon misreconstructed as a π decay product orwhere the π arises from the other B decay; category 8includes charmless three-body B decays where a chargedpion is interchanged with a π meson from the other B ;and finally category 9 includes all other simulated BB background contributions. Within each category, eachof the m ES , ∆ E , BDT out , and Dalitz plot distributionsare formed by combining the contributions of all decaymodes in the category. The combinations are done bynormalizing the distributions for each decay mode to theexpected number of events in the recorded data sam-ple, which is estimated using reconstruction efficienciesdetermined from MC, the number of BB pairs in the0recorded data sample, and the branching fractions listedin Refs. [9, 28]. For each category, the histograms of m ES ,∆ E , BDT out , and the Dalitz plot variables are used asthe probability density functions (PDF) in the likelihoodfit to data to model the BB background. VI. THE MAXIMUM LIKELIHOOD FIT
The extended likelihood function is given by L = exp − X k N k ! × (26) N e Y i =1 "X k N k P ik (cid:16) m K S π + , m π + π , m ES , ∆ E, BDT out , q B (cid:17) , where N k is the number of candidates in each signal orbackground category k , N e is the total number of eventsin the data sample, and P ik (the PDF for category k andevent i ) is the product of the PDFs describing the Dalitzplot, m ES , ∆ E , and BDT out distributions, with q B thecharge of the B candidate.To avoid possible biases in the determination of thefit parameters [45], we use MC samples to study corre-lations between the fit variables and the Dalitz plot pa-rameters, m K S π + and m π + π . We find that for correctlyreconstructed signal candidates, the ∆ E distribution isstrongly dependent on m K S π + . This is mostly due toa dependence of the energy resolution of the B candi-date on the π momentum. For SCF signal candidates,both the m ES and ∆ E distributions depend on all threetwo-body invariant masses: m K S π + , m K S π , and m π + π .The m ES , ∆ E , and BDT out distributions for continuumand BB backgrounds have negligible correlations withthe Dalitz plot parameters.For correctly reconstructed signal candidates, the m ES and ∆ E PDFs are parameterized by a Cruijff function,which is given by (omitting normalization factor) f Cruijff ( x ) = exp " − ( x − m ) σ L,R + α L,R ( x − m ) , (27)where m gives the peak of the distribution and the asym-metric width of the distribution is given by σ L for x < m and σ R for x > m . The asymmetric modulation is sim-ilarly given by α L for x < m and α R for x > m . The∆ E PDF parameters are calculated on an event-by-eventbasis in terms of the K S π + invariant mass, as a linearfunction for m K S π + <
20 GeV /c and as a quadraticfunction for m K S π + >
20 GeV /c . These functions aredetermined by fitting the ∆ E distribution in large non-resonant MC samples. For the SCF signal, in order tofollow the rapid shape variations across the Dalitz plot ofthe m ES and ∆ E distributions, we divide the Dalitz plotinto several regions as illustrated in Fig. 2. Each letter in-dicates whether the dependence is on m π + π , m K S π + , or m K S π . The regions are chosen based on the distribution ) (GeV/c + π S K m0 5 10 15 20 25 ) ( G e V / c π + π m C1C2C3C4 D1A1 B1B2B3
FIG. 2: Diagram illustrating the division of the Dalitz plotinto different regions for the definition of the PDFs for self-crossfeed signal events. Each letter indicates whether the de-pendence is on m π + π (A), m K S π + (B), or m K S π (C). Theremaining region of the Dalitz plot (D1) is where we expectto find fewer SCF events, and where the shapes for m ES and∆ E are less dependent on their position in the Dalitz plot,further described in Table III. in the Dalitz plot of the SCF fraction and the mean differ-ence between the true and reconstructed position in theDalitz plot; we include more regions in areas of the Dalitzplot where these quantities are largest. We use m ES and∆ E PDFs specific to each region, as listed in Table III.Some of the PDFs used in the parametrization of theSCF include Cruijff functions, Chebychev polynomials,Gaussian functions, and two-piece Gaussian (BGauss)functions. A two-piece Gaussian function is an asymmet-ric Gaussian described by the following functional form(omitting normalization factor) f BGauss ( x ) = exp " − ( x − m ) σ L,R . (28)For the continuum background, we use an ARGUSfunction [46] to parameterize the m ES shape. The ∆ E distribution is described by a linear function, and theBDT out distribution by an exponential function. The m ES , ∆ E , and BDT out PDFs for BB backgrounds aredefined by the sum of the histograms from the MC simu-lations for decay modes in each background category, asdescribed in Section V.The continuum and BB background Dalitz plot distri-butions are included in the likelihood as two-dimensionalhistograms. For BB backgrounds, we use MC samples.For continuum background, we combine events from theoff-peak data and the m ES sideband in on-peak data, af-ter subtracting contributions from B decays, as describedin Section IV. For the 2D histograms, we use the squareDalitz plot coordinates. A linear interpolation betweenbin centers is applied.The free parameters in the fit are the yields for signal,1 TABLE III: List of PDFs used to describe the m ES and ∆ E self-crossfeed signal distributions in each of the regions of the B + → K S π + π Dalitz plot shown in Fig. 2. The abbrevi-ations correspond to the following functional forms: Cruijfffunction described in Eq. (27) (Cruijff), Chebychev polyno-mial (Cheb), Gaussian (Gauss), two-piece Gaussian describedin Eq. (28) (BGauss), and exponential (Exp).Dalitz plot region m ES PDF ∆ E PDF m π + π (A1) Cruijff Cruijff m K S π + (B1) Cheb+Gauss Exp+Sigmoid(B2) Cheb+Gauss linear+BGauss(B3) Cruijff Exp+Sigmoid m K S π (C1) Cheb+Gauss Cheb(C2) Cheb+Gauss Cheb(C3) Cruijff Cheb(C4) Cruijff CruijffCentral region (D1) Cruijff Cruijff continuum background, and BB background categories1 and 9. The yields for the remaining BB backgroundcategories are fixed to the estimated values. All the PDFparameters for the correctly reconstructed m ES and ∆ E PDFs, except for the tail parameters, are determined inthe fit. All SCF signal PDF parameters are fixed to val-ues obtained from fits to nonresonant MC events. Theendpoint of the ARGUS function is fixed to 5 .
289 GeV /c while the shape parameter is determined in the fit. Theslope for the linear function of the ∆ E PDF and the expo-nent for the exponential function of the BDT out
PDF forcontinuum background are similarly determined in thefit. The isobar coefficients, x and y in Eq. (18), for allbut one of the isobar components are fitted parametersin the fit and are measured relative to the fixed isobarcomponent. The coefficients for the reference isobar arefixed to x = 1 and y = 0. In total, the fit is performedwith 21 free parameters. TABLE IV: Fit fractions obtained from the fit to data wheneach additional isobar is added to the fit model one at a time.Additional isobar Fit fraction ρ (1450) + . ± . K ∗ (1430) . ± . K ∗ (1430) + . ± . K ∗ (1680) . ± . K ∗ (1680) + . ± . We determine a nominal signal Dalitz plot model basedon information from previous studies [16, 18–20], and onthe changes in the log likelihood in the fit to data whenresonances are added to, or removed from, the list shownin Table II. In these fits to the combined B + and B − datasamples, the CP coefficients ∆ x and ∆ y are fixed to zero.We do not find significant contributions in the fit whenadding the resonances ρ (1450) + , K ∗ (1430) , K ∗ (1430) + , K ∗ (1680) , or K ∗ (1680) + , one at a time to the defaultmodel. We observe that the fit fractions for these ad- ditional resonances, reported in Table IV, are consistentwith zero. The most statistically significant of these fitfractions is FF (cid:0) K ∗ (1430) π + (cid:1) = 0 . ± . /c , suggesting that a nonreso-nant component, in addition to that included in the LASSparametrization, is not necessary. We observe that if weadd a nonresonant component to the fit, the change in loglikelihood for the binned data and the fit projections forthe K S π + , K S π , and π + π invariant masses are con-sistent with the expected change due to the additionalfree parameters in the fit, and do not indicate any sta-tistically significant nonresonant component. We there-fore conclude that, with the current level of statisticalsensitivity, the base model, which includes the ρ (770) + , K ∗ (892) + , K ∗ (892) , ( Kπ ) ∗ , and ( Kπ ) ∗ +0 resonances,provides an adequate description of the data. VII. RESULTS
We apply the fit described in Section VI to the 31 876selected B + → K S π + π candidates. A first fit is per-formed on the combined B ± sample. We obtain yields of1014 ±
60 signal events, 24 381 ±
200 continuum events,2745 ± BB events in category 1, and 1768 ± BB events in category 9. The results of the fit are shownin Fig. 3. For the purpose of this figure, the contribu-tions of signal events are enhanced by applying the morerestrictive selection criteria listed in Table V. TABLE V: Selection criteria imposed to enhance the contri-butions of signal events for the results presented in Figs. 3and 4.Projection plot Selections m ES − . < ∆ E < .
05 GeVBDT out > . E m ES > .
27 GeV /c BDT out > . out m ES > .
27 GeV /c − . < ∆ E < .
05 GeV m K S π + , m K S π , m π + π m ES > .
27 GeV /c − . < ∆ E < .
05 GeVBDT out > . The branching fraction for B + → K π + π is deter-mined from the number of signal events, the efficiencyestimated from MC events, and the total number of BB events in data. We take into account differences betweenthe π reconstruction efficiency in data and MC events,determined from control samples with either τ leptonsor initial-state radiation, as a function of π momentum( ǫ data ǫ MC = 97 . π momentum). We cor-rect for small biases in the branching fraction, as deter-mined from MC pseudo experiments generated with the2 ) (GeV/c ES m ) E ve n t s / ( . M e V / c (a) E (GeV) ∆ -0.2 -0.1 0 0.1 E ve n t s / ( M e V ) (b) out BDT E ve n t s / ( . ) (c) ) (GeV/c + π S K m ) E ve n t s / ( M e V / c (d) ) (GeV/c π S K m ) E ve n t s / ( M e V / c (e) ) (GeV/c π + π m ) E ve n t s / ( M e V / c (f) FIG. 3: Combined B ± fit: Measured distributions and fit projections for B ± → K S π ± π candidates; (a) m ES , (b) ∆ E ,(c) BDT out , (d) m K S π + , (e) m K S π , and (f) m π + π . The points with error bars correspond to data, the solid (blue) curvesto the total fit result, the dashed (green) curves to the total background contribution, and the dotted (red) curves to thecontinuum background component. The dash-dotted curves represent the signal contribution. The projected distributions areobtained from statistically precise pseudo experiments generated using the fit results. For all distributions in each panel, thesignal-to-background ratio is increased by applying tighter selection requirement on m ES , ∆ E , and/or BDT out , listed Table V. same number of signal events and resonance composi-tion as found in the fit to data. We divide the partialbranching fraction of B + → K S ( → π + π − ) π + π by thebranching fraction for K S → π + π − , and multiply theresult by a factor of 2 to account for K L decay, to ob-tain the branching fraction result B (cid:0) B + → K π + π (cid:1) = (cid:0) . ± . ± . +8 . − . (cid:1) × − , where the first uncertaintyis statistical, the second is systematic, and the third isdue to assumptions made concerning the signal model. The latter two uncertainties are described in Section VIIIand the breakdown of the systematic uncertainties is de-tailed in Table XI.We measure amplitudes and phases relative to eachof the five two-body decays in the signal model to takeadvantage of the smaller uncertainty observed when mea-suring the relative phases of the two pairs of decays withsame-charge K ∗ resonances. Table VI lists the relativephase, φ , between each pair of two-body decays in the3 TABLE VI: Combined B ± fit: Relative phases, φ , for the isobar amplitudes as measured from five fits to data, where each ofthe five isobar amplitudes is in turn taken as the reference. All phases are quoted in degrees. The uncertainties are statisticalonly. Relative phase (degrees)Reference amplitude Resonant contribution K ∗ (892) π + K ∗ (892) + π ( Kπ ) ∗ π + ( Kπ ) ∗ +0 π ρ (770) + K S B + → K ∗ (892) π + − ±
43 174 ± − ± − ± B + → K ∗ (892) + π – 0 − ±
42 6 ± − ± B + → ( Kπ ) ∗ π + – – 0 96 ±
42 63 ± B + → ( Kπ ) ∗ +0 π – – – 0 − ± B + → ρ (770) + K S – – – – 0TABLE VII: Combined B ± fit: Results for the fit fractions FF j (diagonal terms) and interference terms FF ij in data for eachresonant contribution. The uncertainties are statistical only. FF j and FF ij Resonant contribution K ∗ (892) π + K ∗ (892) + π ( Kπ ) ∗ π + ( Kπ ) ∗ +0 π ρ (770) + K S B + → K ∗ (892) π + . ± .
03 0 . ± . ± × − . ± . − . ± . B + → K ∗ (892) + π – 0 . ± . − . ± .
007 ( − ± × − . ± . B + → ( Kπ ) ∗ π + – – 0 . ± .
05 (1 . ± . × − − . ± . B + → ( Kπ ) ∗ +0 π – – – 0 . ± . − . ± . B + → ρ (770) + K S – – – – 0 . ± . signal model and its uncertainty. The statistical uncer-tainty in the relative phase is smallest ( ≈ ◦ ) for theresonances that decay to the same-charge Kπ state. Thisis due to a larger overlap in the Dalitz plot between thesame-charge K ∗ resonances than occurs for other pairs ofresonances that only overlap in the corners of the Dalitzplot.Since the statistical uncertainties of the fit fractions donot depend on the reference mode, we quote in Table VIIonly the fit fractions from a fit relative to the K ∗ (892) π + amplitude. The fit fractions for the K ∗ (1430) π + and K ∗ (1430) + π modes are the product of the ( Kπ ) ∗ S-wave fit fraction, shown in Table VII, and the fractiondue to the resonant contribution in the LASS parametri-sation (88%). The off-diagonal fit fractions are smallcompared to the diagonal elements. We calculate thebranching fractions for the resonant contributions shownin Table VIII as the product of the total branching frac-tion and the fit fractions returned by the fit to data,including appropriate Clebsch-Gordan coefficients.To determine the overall CP asymmetry as well as the CP asymmetries for the contributing isobar components,we simultaneously fit the separate B + and B − data sam-ples. The overall A CP value is calculated from the inte-grals of the positive and negative signal Dalitz plot dis-tributions. The ∆ x and ∆ y parameters from Eq. (18) areallowed to vary in the fit for all components except thereference isobar, for which the ∆ y parameter is fixed tozero (the relative phase of the B + and B − Dalitz plotscannot be determined since they do not interfere). Toaccount for possible differences in the reconstruction andparticle identification efficiencies for B + and B − , the ef-ficiency map as a function of the Dalitz plot position is TABLE VIII: Measured branching fractions B from a fit tothe combined B ± data sample, and CP asymmetries A CP (Eq. (19)). The first uncertainty is statistical, the second issystematic, and the third is due to the signal model.Decay channel B (cid:0) − (cid:1) A CP K π + π . ± . ± . +8 . − . . ± . ± . +0 . − . K ∗ (892) π + . ± . ± . +0 . − . − . ± . ± . +0 . − . K ∗ (892) + π . ± . ± . +0 . − . − . ± . ± . +0 . − . K ∗ (1430) π + . ± . ± . +2 . − . . ± . ± . +0 . − . K ∗ (1430) + π . ± . ± . +0 . − . . ± . ± . +0 . − . ρ (770) + K . ± . ± . +0 . − . . ± . ± . +0 . − . determined separately for B + and B − . The asymmetryfor the continuum background is allowed to vary in thefit. The CP asymmetries of the BB backgrounds are ex-pected to be small and so are fixed to zero in the nominalfit. They are varied within reasonable ranges based onworld average experimental results [28] in order to deter-mine the associated systematic uncertainty.We find an overall CP asymmetry of A CP (cid:0) B + → K π + π (cid:1) = 0 . ± . ± . +0 . − . ,where the first uncertainty is statistical, the second issystematic, and the third is due to the signal model.This is consistent with zero CP asymmetry. Invariantmass projections for the fit to data allowing for direct CP violation are shown in Fig. 4.Table VIII shows the results for the branching frac-tions and CP asymmetries obtained from the fit to data.The first uncertainty is statistical, the second is system-4 ) (GeV/c + π S K m ) E ve n t s / ( M e V / c (a) ) (GeV/c - π S K m ) E ve n t s / ( M e V / c (b) ) (GeV/c π S K m ) E ve n t s / ( M e V / c (c) ) (GeV/c π S K m ) E ve n t s / ( M e V / c (d) ) (GeV/c π + π m ) E ve n t s / ( M e V / c (e) ) (GeV/c π - π m ) E ve n t s / ( M e V / c (f) FIG. 4: The CP fit: Measured distributions and fit projections for B + → K S π + π (left column) and B − → K S π − π (rightcolumn) candidates; (a) m K S π + , (b) m K S π − , (c) m K S π (from B + → K S π + π ), (d) m K S π (from B − → K S π − π ), (e) m π + π ,and (f) m π − π . The points with error bars correspond to data, the solid (blue) curves to the total fit result, the dashed (green)curves to the total background contribution, and the dotted (red) curves to the continuum background component. Thedash-dotted curves represent the signal contribution. The projected distributions are obtained from statistically precise pseudoexperiments generated using the fit results. For all distributions in each panel, the signal-to-background ratio is increased byapplying the tighter selection requirements on m ES , ∆ E , and/or BDT out , listed in Table V. atic, and the third is the uncertainty associated with thesignal model. We observe a significant asymmetry be-tween the m K S π + and m K S π − distributions in the regionof the K ∗ (892) + resonance; see Figs. 4(a) and (b). Wedetermine the statistical significance, S , of a non-zero CP asymmetry in B + → K ∗ (892) + π from the difference be-tween the best-fit value of the likelihood, L A CP , and the value when the CP asymmetry is fixed to zero, L : S = r − (cid:16) L / L A CP (cid:17) . (29)Using this method, we measure a statistical signifi-cance of 3 . A CP in B + → K ∗ (892) + π . We obtain a consistent result of3 . x ∆± x -2 -1 0 1 2 3 y ∆ ± y -4-2024 + y) B ∆ x, y+ ∆ (x+ - y) B ∆ x, y- ∆ (x- FIG. 5: CP parameters ( x ± ∆ x, y ± ∆ y ) obtained from the fitto data for B ± → K ∗± (892) π resonant decay including the 1and 2 standard deviation contours (solid and dashed curves).The contours are estimated by calculating the uncertaintyand correlation between the two CP parameters. The starsindicate the central values of the CP parameters and the crosssign the origin of the plot. by dividing the central value of the CP asymmetryby the statistical uncertainty, indicating that the log-likelihood function is close to parabolic. Figure 5 dis-plays the contours in the complex plane of the coef-ficients c = ( x + ∆ x, y + ∆ y ), defined in Eq. (4), for B + → K ∗ (892) + π decays, and of ¯ c = ( x − ∆ x, y − ∆ y ),defined in Eq. (5), for B − → K ∗ (892) − π decays. Forother resonances the CP asymmetry is within 2 standarddeviations of zero.We also express the complex isobar coefficients c and¯ c of Eq. (18) in terms of amplitudes and phases, c = A + e iφ + , (30)¯ c = A − e − iφ − . (31)Table IX presents the results, measured with respect tothe B ± → K ∗ (892) π ± reference amplitude. The statis-tical uncertainties of the separate B + and B − decay am-plitudes, A + and A − , vary between 0 . .
3. We thusobtain significant statistical precision for these terms.With respect to the phases, φ + and φ − , only the ( Kπ ) ∗ amplitude yields a statistically precise result. For theother amplitudes, the statistical uncertainty ranges be-tween 70 ◦ and 170 ◦ , and only the statistical uncertaintyis quoted. For the more precisely determined variables,systematic uncertainties are evaluated as well. For thephases of the B ± → ( Kπ ) ∗± π decays relative to the B ± → K ∗ (892) ± π amplitude, we obtain φ + (cid:0) ( Kπ ) ∗ +0 π (cid:1) − φ + (cid:0) K ∗ (892) + π (cid:1) = (cid:0) − ± ± +4 − (cid:1) ◦ , (32) φ − (cid:0) ( Kπ ) ∗− π (cid:1) − φ − (cid:0) K ∗ (892) − π (cid:1) = (cid:0) ± ± +17 − (cid:1) ◦ . VIII. SYSTEMATIC UNCERTAINTIES
We evaluate systematic uncertainties to account foreffects that could affect the branching fractions, phases,and asymmetries, by varying the fixed parameters. Thesystematic uncertainties described in this section aresummarized in Tables XI through XVI of Appendix A.The uncertainties associated with the branching frac-tions are listed in Table XI. To estimate the uncertaintyrelated to the modeling of the SCF PDFs, we implementa simpler model consisting of only four regions in theDalitz plot. The PDFs are redefined using MC events tomatch the distributions found in the newly defined re-gions. We then fit the data using the new SCF modeland take the uncertainties to be the change in the fit pa-rameters compared to those obtained from the nominalfit to data. All relative systematic uncertainties due tothe SCF m ES and ∆ E PDFs range from approximately1% to 4%, except for the relative systematic uncertaintyfor the B + → ρ (770) + K decay, which is 7 . B + → ρ (770) + K events are due to SCF.The uncertainties associated with the number of BB background events are evaluated by varying the estimateswithin their uncertainties, which are primarily due to un-certainties in the branching fractions. The uncertaintiesrelated to the BB background m ES , ∆ E , and BDT out PDFs are accounted for by varying the histogram bincontents according to their statistical uncertainties. Theuncertainty is then taken as the RMS of the distributionof the difference in the fit parameters. The uncertain-ties related to the limited statistical precision of the MCand data-sideband samples are similarly accounted for byvarying the results in the corresponding histogram binsby their uncertainties.The uncertainty in the BDT out histogram PDFs forcorrectly reconstructed and SCF signal events is deter-mined by varying the bin contents in accordance withthe observed data/MC difference. For correctly recon-structed signal events, the tails of the asymmetric Gaus-sian PDFs for m ES and ∆ E are fixed. To account for anassociated uncertainty, we allow the relevant parametersto vary in a fit to data and use the variation in the fitparameters to define the uncertainty.To validate the fitting procedure, 500 MC pseudo ex-periments are generated, using the PDFs with parametervalues found from the fit to data. Small fit biases arefound for some of the fit parameters and are included inthe systematic uncertainties.We also account for uncertainties in the following pa-rameters describing the signal model: the mass and widthof each resonance and the value of the Blatt-Weisskopfbarrier radius. The associated uncertainties are deter-mined by varying the parameters within their uncertain-ties (some of which are given in Table II) and refitting.The uncertainties in the branching fractions relatedto particle identification, tracking efficiency, and the to-tal number of BB events are 1.0%, 1.0%, and 0.6%, re-6 TABLE IX: Results for the relative phases φ obtained from the combined B ± fit, the CP amplitudes A + and A − , and the CP phases φ + and φ − obtained from the CP fit. All parameters are measured relative to the B ± → K ∗ (892) π ± referenceamplitude. The first uncertainty is statistical, the second is systematic, and the third is due to the signal model. Note that forthe CP phases of all contributions except for B ± → ( Kπ ) ∗ π ± , only statistical uncertainties are quoted.Isobar φ ( ◦ ) A + A − φ + ( ◦ ) φ − ( ◦ ) K ∗ (892) ± π − ± +48 +8 − − . ± . ± . +0 . − . . ± . ± . +0 . − . − ± − ± Kπ ) ∗ π ± ± ± +0 − . ± . ± . +0 . − . . ± . ± . +0 . − . ± ± +4 − ± ± +1 − ( Kπ ) ∗± π − ± +53 +5 − − . ± . ± . +0 . − . . ± . ± . +0 . − . ± − ± ρ (770) ± K − ± +55 +16 − − . ± . ± . +0 . − . . ± . ± . +0 . − . − ± − ± spectively. We estimate systematic uncertainties in thebranching fractions associated with the π and K S recon-struction efficiencies to be 1.0% and 1.1%, respectively.Uncertainties from all the above sources are added inquadrature to yield the total systematic uncertainties,which are listed in Table XI.We determine changes in the branching fractions,∆ B , when the signal model is varied. The system-atic uncertainties in the branching fractions due to the( Kπ ) ∗ / +0 parametrization are estimated by replacing theLASS model with another phenomenologically inspiredparametrization [47]. We take the differences in branch-ing fractions with respect to the nominal fit as the sys-tematic uncertainty. This is the largest contribution tothe uncertainty due to the model. Another uncertaintyreflects any changes in the fit parameters for the nomi-nal model when including components that are omittedin the nominal fit, such as the ρ (1450) + , K ∗ (1430) , and K ∗ (1430) + . Positive and negative variations are addedseparately in quadrature to obtain the systematic uncer-tainties due to the signal model, listed in Table XI.We determine systematic uncertainties in the phasesaveraged over B + and B − decays from the same sourcesas considered for the branching fractions. The variationsin the phases are measured relative to the K ∗ (892) π + amplitude. Since the differences between positive andnegative shifts in the phases, shown in Table XII, arelarge in some cases, we quote for those phase shifts asym-metric systematic uncertainties.Reconstruction and particle identification efficienciescancel to first order in the fit to CP asymmetries; there-fore the only uncertainties that are included for A CP arethose coming from the fit and signal model. In additionto this, we do not evaluate any of the uncertainties thatare found to be negligible for the branching fractions.An additional uncertainty for A CP arises from havingfixed the CP asymmetries for individual BB backgroundcomponents to the mean asymmetry averaged over allsuch components. We take the largest variation of eachbackground asymmetry as the corresponding uncertainty.The uncertainty related to the efficiency model is deter-mined by exchanging the efficiency maps for the positiveand negative Dalitz plots and refitting the data. We thentake the difference in CP asymmetry with respect to thenominal fit as the uncertainty.We list in Table XIII the systematic uncertainties asso-ciated with the signal CP asymmetries and the variations in the asymmetry due to changes in the signal composi-tion.We evaluate systematic uncertainties for the CP amplitudes and CP phases from the same sourcesas for the CP asymmetries. We list the varia-tions to the amplitudes A + in Table XIV and tothe amplitudes A − in Table XV, including the un-certainties due to changes to the signal model. Ta-ble XVI lists the systematic variations and model uncer-tainties for φ +( − ) (cid:16) ( Kπ ) ∗ π +( − ) − K ∗ (892) π +( − ) (cid:17) and φ +( − ) (cid:16) ( Kπ ) ∗ +( − )0 π − K ∗ (892) +( − ) π (cid:17) . IX. SUMMARY AND CONCLUSION
The measured branching fractions and CP asymme-tries are summarized in Table VIII, and the amplitudeand phase values in Table IX, including statistical, sys-tematic, and model uncertainties. We have measured forthe first time the branching fraction and CP asymme-try for the decay B + → K π + π . We obtain first evi-dence for direct CP violation in the intermediate decay B + → K ∗ (892) + π , with a total significance of 3 . A CP by the total uncertainty.In addition, we have measured the branching fractions, CP asymmetries, and relative CP -averaged phase valuesof the decays B + → K ∗ (892) π + , B + → K ∗ (892) + π , B + → K ∗ (1430) π + , B + → K ∗ (1430) + π , and B + → ρ (770) + K . The results for the branchingfractions and CP asymmetries for B + → K ∗ (892) π + are consistent with the previous measurement from B + → K + π − π + decays by the Belle and B A B AR Col-laborations and the results for B + → K ∗ (1430) π + are within two standard deviations from the previ-ous B A B AR measurement [16, 17]. The branchingfraction for B + → K ∗ (892) + π are consistent withthe previous measurements from the B A B AR Collab-oration for the decay mode B + → K + π π and theresult for A CP is within 2 standard deviations of theprevious measurement [18]. The branching fractionand A CP results for B + → ρ (770) + K supersede theprevious B A B AR measurements [14]. The CP asym-metries of B + → K ∗ (892) π + , B + → K ∗ (1430) π + ,and B + → ρ (770) + K are all consistent with zero,7 TABLE X: Combined measurements of branching fractions and CP asymmetries from B + → K S π + π (this analysis) and from B A B AR analyses of B + → K ∗ (892) π + and B + → K ∗ (1430) π + from B + → K + π − π + [16], and of B + → K ∗ (892) + π from B + → K + π π [18]. The first uncertainty is statistical and the second is systematic.Decay channel B (cid:0) − (cid:1) A CP K ∗ (892) π + . ± . ± . . ± . ± . K ∗ (892) + π . ± . ± . − . ± . ± . K ∗ (1430) π + . ± . ± . . ± . ± . as expected. We obtain the first measurementsof the branching fraction and CP asymmetry for B + → K ∗ (1430) + π , with a significance of 5 . CP asymmetries of B + → K ∗ (892) π + , B + → K ∗ (1430) π + , and B + → K ∗ (892) + π with theprevious B A B AR measurements. The statistical uncer-tainties and all systematic uncertainties for the CP asym-metries are uncorrelated between the measurements. Forthe branching fractions, we account for possible corre-lations when combining the systematic uncertainties. Ifthe systematic uncertainties are asymmetric, the aver-age systematic uncertainty is calculated from the largestlimit. The combined results from B A B AR for these decaymodes are presented in Table X.Using the world average value for direct CP violationin B → K ∗ (892) + π − [9] and the final B A B AR result fordirect CP violation in B + → K ∗ (892) + π , we calculate∆ A CP for the K ∗ π system to be∆ A CP ( K ∗ π ) = A CP (cid:0) K ∗ + π (cid:1) − A CP (cid:0) K ∗ + π − (cid:1) = − . ± . . (33)Thus the value of ∆ A CP in K ∗ π is found to be consis-tent with zero. The uncertainty in the ∆ A CP ( K ∗ π ) resultremains large, rendering the comparison to ∆ A CP ( Kπ ),given in Eq. (3), inconclusive at present and motivatingimproved determinations in future experiments. X. ACKNOWLEDGEMENTS
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Table XI lists the uncertainties in the branching frac-tions due to systematic effects, efficiency corrections, andchanges to the signal model. Table XII lists uncertaintiesin the relative phase values (for B + and B − decays com-bined) due to systematic effects and changes to the signalmodel. Tables XIII, XIV, XV, and XVI list the system-atic and signal model uncertainties for the CP asymme-tries, the amplitudes for the B + and B − Dalitz plots, A + and A − , respectively, and the corresponding phases φ + and φ − for the B + → K ∗ (1430) π + amplitude rela-tive to that for B + → K ∗ (892) π + , and the phase valuesfor the B + → ( Kπ ) ∗ +0 π amplitude relative to that for B + → K ∗ (892) + π .9 TABLE XI: Combined B ± fit: Systematic uncertainties for the branching fraction measurements, including uncertainties dueto the signal model. Relative Variations of branching fraction (%)Source Resonant contribution Inclusive K ∗ (892) K ∗ (892) + K ∗ (1430) K ∗ (1430) + ρ (770) + Correctly reconstructed m ES and ∆ E PDF (fixed parameters) 0 . . . . . . out PDFs 3 . . . . . . m ES and ∆ E PDF models 3 . . . . . . . . . . . . BB background m ES , ∆ E and BDT out PDFs 0 . . . . . . BB background yields 0 . . . . . . . . . . . . . . . . . . K ∗ (892) mass and width 0 . . . . . . K ∗ (1430) mass and width 3 . . . . . . ρ (770) + mass and width < . . . . . . . . . . . . . . . . . . . . . . . . K S efficiency 1 . . . . . . . . . . . . . . . . . . N BB . . . . . . . . . . . . B (cid:0) − (cid:1) ( Kπ ) ∗ /( Kπ ) ∗ +0 parametrization +8 . − . − . − . ρ (1450) + +2 . . − . . − . − . K ∗ (1430) and K ∗ (1430) + +1 . − . . − . − . − . K ∗ (1680) and K ∗ (1680) + +1 . − . − . . − . − . . . . . . . − ) − . − . − . − . − . − . TABLE XII: Combined B ± fit: Systematic uncertainties due to the fit model, fixed shapes in the parametrization, and changesto the signal model for the relative phases (in degrees) measured relative to the K ∗ (892) amplitude.Systematic Variations ( ◦ )Systematic Resonant contribution K ∗ (892) + π K ∗ (1430) π + K ∗ (1430) + π ρ (770) + K S Self crossfeed PDFs and mapping 7 . . . . out PDFs 1 . . . . BB background yields 1 . . . . m ES and ∆ E PDF 0 . . . . . . . . BB background m ES , ∆ E , BDT out PDFs 0 . . . . . . . . . . . . K ∗ (892) mass 1 . . . . K ∗ (892) width 0 . . . < . K ∗ (1430) mass +43 −
33 +5 . − . −
37 +46 − K ∗ (1430) width 5 . . . . ρ (770) + mass 0 . . . . ρ (770) + width 0 . . . . +15 − . − . − . − . Total +48 −
36 +11 −
11 +53 −
40 +55 − Changes due to signal model( Kπ ) ∗ /( Kπ ) ∗ +0 parametrization − . − . ρ (1450) + − . − . − . − . K ∗ (1430) +7 . − . . . K ∗ (1680) − . − . − . . . . . . − ) − . − . − . − . TABLE XIII: Contributions to the uncertainties in the CP asymmetries for the overall and resonant isobar contributions,including uncertainties due to changes to the signal model. Systematic Variations of A CP (%)Systematic Resonant contribution Inclusive K ∗ (892) π + K ∗ (892) + π K ∗ (1430) π + K ∗ (1430) + π ρ (770) + K S Self crossfeed PDFs and mapping 2 . . . . . . out PDFs 0 . . . . . . BB background asymmetries 2 . . . . . . . . . . . . BB background m ES , ∆ E , BDT out PDFs 0 . . . . . . . . . . . . . . . . . . K ∗ (892) mass and width 0 . . . . . . K ∗ (1430) mass and width 1 . . . . . . ρ (770) + mass and width < . . . . . . < . . . . . . . . . . . . Kπ ) ∗ /( Kπ ) ∗ +0 parametrization − . . . − . ρ (1450) + +3 . . − . − . − . − . K ∗ (1430) − . . − . − . − . . K ∗ (1680) − . . . . − . . . . . . . . − ) − . − . − . − . − . − . CP amplitude, A + , including uncertainties due to changes to the signal model. In the fits, theamplitudes are measured relative to the K ∗ (892) amplitude. Variation of A + Systematic Resonant contribution K ∗ (892) + π ( Kπ ) ∗ π + ( Kπ ) ∗ +0 π ρ (770) + K S Self crossfeed PDFs and mapping 0 .
02 0 .
02 0 .
04 0 . out PDFs 0 .
01 0 .
03 0 .
02 0 . BB background asymmetries 0 .
01 0 .
03 0 .
07 0 . .
02 0 .
06 0 .
04 0 . BB background m ES , ∆ E , BDT out PDFs < .
01 0 .
01 0 . < . .
01 0 . < .
01 0 . .
01 0 .
03 0 .
02 0 . K ∗ (892) mass 0 .
01 0 .
01 0 . < . K ∗ (892) width < .
01 0 .
01 0 . < . Kπ ) ∗ mass 0 .
02 0 .
02 0 .
09 0 . Kπ ) ∗ width 0 .
02 0 .
06 0 .
01 0 . ρ (770) + mass < . < . < . < . ρ (770) + width < . < . < . < . .
02 0 .
03 0 .
02 0 . .
05 0 .
11 0 .
13 0 . Kπ ) ∗ /( Kπ ) ∗ +0 parametrization − . −− −− − . ρ (1450) + < . − . − . − . K ∗ (1430) 0 . − . − . − . K ∗ (1680) − .
02 0 . − . − . .
05 +0 .
07 +0 .
00 +0 . − ) − . − . − . − . TABLE XV: Variations in the CP amplitude, A − , including uncertainties due to changes to the signal model. In the fits, theamplitudes are measured relative to the K ∗ (892) amplitude. Variation of A − Systematic Resonant contribution K ∗ (892) − π ( Kπ ) ∗ π − ( Kπ ) ∗− π ρ (770) − K S Self crossfeed PDFs and mapping < .
01 0 .
05 0 .
04 0 . out PDFs 0 .
02 0 .
02 0 .
02 0 . BB background asymmetries 0 .
01 0 .
01 0 .
06 0 . .
02 0 .
05 0 .
04 0 . BB background m ES , ∆ E , BDT out PDFs < . < . < . < . . < .
01 0 .
02 0 . . < .
01 0 .
02 0 . K ∗ (892) mass < .
01 0 .
01 0 . < . K ∗ (892) width < .
01 0 .
01 0 .
01 0 . Kπ ) ∗ mass 0 .
01 0 .
08 0 .
08 0 . Kπ ) ∗ width 0 .
01 0 .
07 0 .
05 0 . ρ (770) + mass < . < . < . < . ρ (770) + width < . < . < . < . .
01 0 .
01 0 .
05 0 . .
05 0 .
13 0 .
14 0 . Kπ ) ∗ /( Kπ ) ∗ +0 parametrization − . −− −− − . ρ (1450) + .
04 0 .
12 0 .
22 0 . K ∗ (1430) 0 .
05 0 .
07 0 .
05 0 . K ∗ (1680) − . − . − . − . .
06 +0 .
14 +0 .
22 +0 . − ) − . − . − . − . CP phase values φ ± (in degrees) measured for the ( Kπ ) ∗ π ± amplitude relative to the K ∗ (892) π ± amplitude, and for the ( Kπ ) ∗± π amplitude relative to the K ∗ (892) ± π amplitude.Systematic Absolute variations of CP phase values( Kπ ) ∗ π ± − K ∗ (892) π ± ( Kπ ) ∗± π − K ∗ (892) ± π φ + φ − φ + φ − Self crossfeed PDFs and mapping 0 . . . . out PDFs 0 . . . . BB background asymmetries 1 . . . . . . . . BB background m ES , ∆ E , BDT out PDFs 0 . . . . . . . . . . . . K ∗ (892) mass 0 . . . . K ∗ (892) width 0 . . . . Kπ ) ∗ mass 6 . . . . Kπ ) ∗ width 4 . . . . ρ (770) + mass 0 . . < . . ρ (770) + width 0 . . . . . . . . . . . . ρ (1450) + +3 . − . − . . K ∗ (1430) − . . . − . K ∗ (1680) +1 . . . . . . . . − ) − ..
22 +0 . − ) − . − . − . − . CP phase values φ ± (in degrees) measured for the ( Kπ ) ∗ π ± amplitude relative to the K ∗ (892) π ± amplitude, and for the ( Kπ ) ∗± π amplitude relative to the K ∗ (892) ± π amplitude.Systematic Absolute variations of CP phase values( Kπ ) ∗ π ± − K ∗ (892) π ± ( Kπ ) ∗± π − K ∗ (892) ± π φ + φ − φ + φ − Self crossfeed PDFs and mapping 0 . . . . out PDFs 0 . . . . BB background asymmetries 1 . . . . . . . . BB background m ES , ∆ E , BDT out PDFs 0 . . . . . . . . . . . . K ∗ (892) mass 0 . . . . K ∗ (892) width 0 . . . . Kπ ) ∗ mass 6 . . . . Kπ ) ∗ width 4 . . . . ρ (770) + mass 0 . . < . . ρ (770) + width 0 . . . . . . . . . . . . ρ (1450) + +3 . − . − . . K ∗ (1430) − . . . − . K ∗ (1680) +1 . . . . . . . . − ) − .. − ..
22 +0 . − ) − . − . − . − . CP phase values φ ± (in degrees) measured for the ( Kπ ) ∗ π ± amplitude relative to the K ∗ (892) π ± amplitude, and for the ( Kπ ) ∗± π amplitude relative to the K ∗ (892) ± π amplitude.Systematic Absolute variations of CP phase values( Kπ ) ∗ π ± − K ∗ (892) π ± ( Kπ ) ∗± π − K ∗ (892) ± π φ + φ − φ + φ − Self crossfeed PDFs and mapping 0 . . . . out PDFs 0 . . . . BB background asymmetries 1 . . . . . . . . BB background m ES , ∆ E , BDT out PDFs 0 . . . . . . . . . . . . K ∗ (892) mass 0 . . . . K ∗ (892) width 0 . . . . Kπ ) ∗ mass 6 . . . . Kπ ) ∗ width 4 . . . . ρ (770) + mass 0 . . < . . ρ (770) + width 0 . . . . . . . . . . . . ρ (1450) + +3 . − . − . . K ∗ (1430) − . . . − . K ∗ (1680) +1 . . . . . . . . − ) − .. − .. − ..
22 +0 . − ) − . − . − . − . CP phase values φ ± (in degrees) measured for the ( Kπ ) ∗ π ± amplitude relative to the K ∗ (892) π ± amplitude, and for the ( Kπ ) ∗± π amplitude relative to the K ∗ (892) ± π amplitude.Systematic Absolute variations of CP phase values( Kπ ) ∗ π ± − K ∗ (892) π ± ( Kπ ) ∗± π − K ∗ (892) ± π φ + φ − φ + φ − Self crossfeed PDFs and mapping 0 . . . . out PDFs 0 . . . . BB background asymmetries 1 . . . . . . . . BB background m ES , ∆ E , BDT out PDFs 0 . . . . . . . . . . . . K ∗ (892) mass 0 . . . . K ∗ (892) width 0 . . . . Kπ ) ∗ mass 6 . . . . Kπ ) ∗ width 4 . . . . ρ (770) + mass 0 . . < . . ρ (770) + width 0 . . . . . . . . . . . . ρ (1450) + +3 . − . − . . K ∗ (1430) − . . . − . K ∗ (1680) +1 . . . . . . . . − ) − .. − .. − .. − ..