Final state interactions in B+- to K+ K- K+- decays
Agnieszka Furman, Robert Kaminski, Leonard Lesniak, Piotr Zenczykowski
aa r X i v : . [ h e p - ph ] M a y Final state interactions in B ± → K + K − K ± decays A. Furman a , R. Kami´nski b , L. Le´sniak b, ∗ , P. ˙Zenczykowski b a ul. Bronowicka 85 /
26, 30-091 Krak´ow, Poland b Henryk Niewodnicza´nski Institute of Nuclear Physics, Polish Academy of Sciences,PL 31-342 Krak´ow, Poland
Abstract
Charged B decays to three charged kaons are analysed in the framework of the QCD factorization approach. The strong final state K + K − interactions are described using the kaon scalar and vector form factors. The scalar non-strange and strange form factors atlow K + K − e ff ective masses are constrained by chiral perturbation theory and satisfy the two-body unitarity conditions. The latterstem from the properties of the meson-meson amplitudes which describe all possible S -wave transitions between three coupledchannels consisting of two kaons, two pions and four pions. The vector form factors are fitted to the data on the electromagnetickaon interactions. The model results are compared with the Belle and BaBar data. Away from φ (1020) resonance, in the S -wavedominated K + K − mass spectra, a possibility for a large CP asymmetry is identified. Keywords : charmless mesonic B decays, QCD factorization,final state interactions, CP violation
1. Introduction
Recently, charmless three-body decays of B mesons havebeen intensively studied both experimentally and theoretically.On the experimental side, Dalitz plot analyses of the charged B decays were performed by Belle [1] and BaBar [2] collab-orations. Likewise, several theoretical studies involving the B ± → K + K − K ± decays have been published [3], [4] and [5].Since charged kaons interact strongly, their long distance in-teractions in the final states have to be well understood if oneaims at extracting weak decay amplitudes from the B to KKK decays. In this Letter we go beyond an isobar model parameter-ization of the B decay amplitudes and introduce additional the-oretical constraints on the S -wave two-body K + K − interactionamplitudes, which follow, in particular, from unitarity. In orderto satisfy unitarity in two-body interactions we construct scalarstrange and non-strange form factors which enter into the S -wave parts of the decay amplitudes. These amplitudes are cal-culated in the framework of the QCD factorization approach. Inthe construction of form factors we use experimental informa-tion on the K + K − interactions coming from experiments otherthan B decays, for example from K + K − production processesin hadronic collisions or from e + e − reactions. We apply alsosome low-energy constraints coming from the chiral perturba-tion theory. Preliminary results of our analysis concerning the B ± → K + K − K ± reactions can be found in Ref. [6].In Section 2 we formulate the theoretical model of the B + and B − decay amplitudes. Presentation of results and their compar-ison with the experimental data are given in Sec. 3. Our con-clusions are presented in Sec. 4. ∗ Corresponding author
Email address:
[email protected] (L. Le´sniak) B ± → K + K − K ± decay amplitudes Inspection of the Dalitz plots of the Belle [1] and BaBar [2]experiments reveals an accumulation of events for the K + K − e ff ective masses below 1.8 GeV. Indeed, several mesonic reso-nances which can decay into the K + K − pairs exist in this range[7]. Among them there are scalar and vector resonances whichare formed via the S - and P -wave final state interactions. In thefirst approximation one can neglect their interaction with thethird kaon. This justifies using the QCD quasi-two-body fac-torization approach for the limited range of the e ff ective K + K − masses (see, for example Ref. [8]). The B − → K + K − K − ampli-tude is then expressed in terms of the following matrix elementof the weak e ff ective Hamiltonian H : h K − ( p ) K + ( p ) K − ( p ) | H | B − i = A − S + A − P , (1)where the S -wave part is A − S = G F √ n − r χ f K ( M B − s ) F B → ( K + K − ) S ( m K ) y Γ n ∗ ( s ) + B m b − m s ( M B − m K ) F BK ( s ) v Γ s ∗ ( s ) o , (2)the P -wave part is A − P = G F √ n f K f ρ A B ρ ( m K ) yF K + K − u ( s ) − F BK ( s ) h w u F K + K − u ( s ) + w d F K + K − d ( s ) + w s F K + K − s ( s ) io −→ p · −→ p (3)and the interacting kaons are taken to be kaons 2 and 3. Fur-thermore, s is the square of the K + ( p ) K − ( p ) e ff ective mass m ≡ m K + K − , while −→ p and −→ p are the kaon 1 and kaon 2 mo-menta in the center of mass system of the kaons 2 and 3. The Preprint submitted to Elsevier October 18, 2018 calar product of the kaon momenta can be written in terms ofthe helicity angle Θ H : −→ p · −→ p = −|−→ p ||−→ p | cos Θ H . (4)In these equations G F is the Fermi coupling constant, f K = . f ρ = .
220 GeV are the kaon and the ρ me-son decay constants, M B , m K , m b = . m s = . m u = .
004 GeV and m d = .
004 GeV are the masses of the B meson, kaon, b -quark, strange quark, down- and up-quarks,respectively.The functions Γ n and Γ s , present in the S -wave amplitudein Eq. (2), are the kaon non-strange and strange scalar formfactors. The vector form factors F K + K − q (for q = u , d and s ),introduced in Eq. (3), are defined through matrix elements < K + ( p ) K − ( p ) | ¯ q γ µ q | > = ( p − p ) µ F K + K − q ( s ) , (5)where | > is the vacuum state. The K + K − pair in the S -waveis then denoted by R S ≡ ( K + K − ) S . Similarly R P ≡ ( K + K − ) P stands for the P -state. Furthermore F B → ( K + K − ) S in Eq. (2) is theform factor of the transition from the B meson to the K + K − pair in the S -state, χ is the constant related to the decay ofthe ( K + K − ) S state into two kaons, and B = m π / ( m u + m d ),where m π is the pion mass. We take F B → ( K + K − ) S ( m K ) = . χ to the data. Functions F BK ( s ) and F BK ( s )are the B → K scalar and vector transition form factors and A B ρ ( m K ) = .
37 [8] is the B → ρ transition form factor. In ourapproximation, the ratio A B ρ / f ρ represents a general factor re-lated to the transition from B − to any ( K + K − ) P state and then itsdecay into the final K + K − pair. For the case of the ρ meson thiscoupling to the pair of kaons is e ff ectively realized only abovethe K + K − threshold.The weak decay amplitudes depend on QCD factorizationcoe ffi cients a pj and on the products Λ u = V ub V ∗ us , Λ c = V cb V ∗ cs ,where V i j are the CKM quark-mixing matrix elements. In or-der to describe B decay into mesons M and M we followRef. [8] and calculate the coe ffi cients a pj ( M M ) at the next-to-leading order in the strong coupling constant at the renoma-lization scale equal to m b /
2. Here the M meson has a com-mon spectator quark with the decaying B meson. In the caseof the B − → K + K − K − decays, M or M can be either kaon K − , or systems R S , R P . We take into account one-loop vertexand penguin corrections to a pj ( M M ) but neglect those due tohard scattering or the annihilation since they are expected tobe generally suppressed. In the QCD factorization approachthey receive logarithmically divergent contributions due to softgluon interaction which are “unavoidably model dependent”(see Ref. [8]). We treat such soft interactions by introducing theform factors constrained by data on meson-meson interactions,taken from analyses of reactions other than the B decays. Un-der these conditions we have a pj ( R S M ) = a pj ( R P M ), with theircommon value denoted below by a pj ( R S , P M ) ≡ a pjy . We alsouse the abbreviations: a pjw ≡ a pj ( K − R P ) and a pjv ≡ a pj ( K − R S ).The values of coe ffi cients a pj ( M M ) are given in Table 1. In terms of the quantities introduced above one defines: y = Λ u h a y + a u y + a u y − ( a u y + a u y ) r K χ i +Λ c h a c y + a c y − ( a c y + a c y ) r K χ i , (6)where r K χ = m K ( m b + m u )( m u + m s ) , (7) w u = Λ u ( a w + a w + a w + a w + a w ) + Λ c ( a w + a w + a w + a w ) , (8) w d = Λ u h a w + a w −
12 ( a w + a w ) i +Λ c h a w + a w −
12 ( a w + a w ) i , (9) w s = Λ u h a w + a u w + a w −
12 ( a w + a w + a u w ) i +Λ c h a w + a c w + a w −
12 ( a w + a w + a c w ) i , (10)and v = Λ u ( − a u v + a u v ) + Λ c ( − a c v + a c v ) . (11)One can notice that in the expressions for the decay ampli-tudes there are no transitions to the K + K − states of spin 2 orhigher. This results from the application of the factorizationapproach in which matrix elements to spin states higher thanone vanish. The contribution of f (1270) with its rather smallbranching fraction to K ¯ K (4.6 %) is thus not included in thisstudy.Since two identical charged kaons appear in the final state ofthe B − → K + K − K − decay, the amplitude of Eq. (1) has to besymmetrized A − sym = √ h h K − ( p ) K + ( p ) K − ( p ) | H | B − i + h K − ( p ) K + ( p ) K − ( p ) | H | B − i i . (12)The symmetrized amplitude for the B + → K + K − K + reactionreads A + sym = A − sym ( Λ u → Λ ∗ u , Λ c → Λ ∗ c , B − → B + ) . (13)The final state kaon-kaon S -wave interactions are dynami-cally coupled with systems consisting of two and four pions.Thus a system of three coupled channels: ππ, ¯ KK and 4 π (e ff ec-tive (2 π )(2 π ) or σσ , ρρ etc.), labelled by j = , , , is consid-ered in the construction of scalar form factors Γ n and Γ s . Herewe use an approach initiated in [4] and recently developed in[10] for the B ± → π + π − π ± decays. A set of the 3x3 transi-tion amplitudes T , describing all possible transitions betweenthe three channels, is taken from a unitary model of Ref. [11](solution A ). We introduce two kinds of production functions R n , sj , labeled by n (non-strange) or by s (strange): R n , sj ( E ) = α n , sj + τ n , sj E + ω n , sj E + cE , j = , , , (14)2here α n , sj , τ n , sj , ω n , sj and c are constant parameters, while E rep-resents the total energy and is related to the center of mass mo-menta k j = q E − m j , with m = m π , m = m K , m = s ≡ E ≡ m K + K − . The three scalar form factors, writ-ten in the compact row matrix form Γ n , s ∗ , are given by Γ n , s ∗ = R n , s + TGR n , s , (15)where R n , s are rows of the production functions and G is thematrix of the Green’s functions multiplied by the convergencefactors F j ( p ) = ( k j + κ ) / ( p + κ ). These factors, which reduceto unity on shell ( p = k j ), make finite the relevant integrals overthe intermediate momenta p . The parameter κ will be fitted tothe data of the BaBar [2] and Belle [1] Collaborations.For both the non-strange and strange form factors we alsoconstrain their low energy behaviour using the chiral perturba-tion model of Refs. [12, 13]. At low s values one writes thefollowing expansion: Γ n , sj ( s ) (cid:27) d n , sj + f n , sj s , j = , , , (16)with real coe ffi cients d n , sj and f n , sj . Explicit formulae for the setof non-strange form factors, in particular for the Γ n presented inEq. (2), are given in Eqs. (24-35) of Ref. [10]. For the strangeform factors we have d s = √ m π f (cid:16) L r − L r (cid:17) − m π π f + log m η µ , (17) f s = √ L r f − π f + log m K µ + m π π m η f , (18)and d s = + L r − L r ) f (cid:16) m π + m K (cid:17) − L r f m K + L r f m K + m η π f log m η µ + m K π f + log m η µ , (19) f s = L r f + L r f − m K π f m η − π f + log m η µ − π f + log m K µ . (20)In these equations m η is the η meson mass, µ is the scaleof the dimensional regularization and f = f π / √
2. Using f = . L rk , k = , , ,
8, given in Table X of Ref. [14], we obtain the non-strange-sector parameters: d n = . f n = . − , d n = . f n = . − and their strange-sectorcounterparts: d s = − . f s = . − , d s = . f s = . − . For the form factors related to thethird channel at low energies we make the simplest assumptions d n = d s = f n = f s =
0, as in Ref. [20].The coe ffi cients α n , sj , τ n , sj and ω n , sj are constrained by the val-ues of the form factors at low energies. They are calculated using the low energy expansion of Eq. (15) and are listed inTable 2. The parameter c , which controls the high energy be-haviour of R , is fixed while fitting the data.Our scalar form factors satisfy the following unitarity condi-tions: Im Γ ∗ = T † D Γ ∗ , (21)where D is the diagonal matrix of the kinematical coe ffi cientswhich are proportional to the channel momenta k j in the centerof mass frame: D i j = − k j √ s π δ i j θ ( √ s − m j ) , i , j = , , . (22)Presence of the resonances in the K + K − e ff ective mass distri-butions (see Refs. [2, 1]) is a direct manifestation of the K + K − final state interactions. The most prominent resonance in the P -wave is φ (1020). In 2005 Bruch, Khodjamirian and K¨uhn [15]described the electromagnetic form factors for charged and neu-tral kaons in terms of additive contributions from eight vectormesons: ρ ≡ ρ (770), ρ ′ ≡ ρ (1450), ρ ′′ ≡ ρ (1700), ω ≡ ω (782), ω ′ ≡ ω (1420), ω ′′ ≡ ω (1650), φ ≡ φ (1020) and φ ′ ≡ φ (1680).Using quark model assumptions and isospin symmetry as inRef. [15] one can deduce the following expressions for the three P -wave form factors F K + K − q defined in Eq. (5): F K + K − u =
12 ( c ρ BW ρ + c ρ ′ BW ρ ′ + c ρ ′′ BW ρ ′′ + c ω BW ω + c ω ′ BW ω ′ + c ω ′′ BW ω ′′ ) , (23) F K + K − d =
12 ( − c ρ BW ρ − c ρ ′ BW ρ ′ − c ρ ′′ BW ρ ′′ + c ω BW ω + c ω ′ BW ω ′ + c ω ′′ BW ω ′′ ) , (24) F K + K − s = − c φ BW φ − c φ ′ BW φ ′ . (25)In the above equations BW i , i = m i and width Γ i as BW i ( s ) = m i m i − s − i √ s Γ i ( s ) , (26)and c i are the constants given in Table 2 of Ref. [15] for theconstrained fit.The B to K transition form factors have been parametrizedaccording to Ref. [16]: F BK ( s ) = r − ss , (27)where r = . s = .
46 GeV , and F BK ( s ) = r − sm + r (1 − sm ) , (28)where r = . r = .
173 and m = .
41 GeV.3 able 1:
Leading order (LO) and next-to-leading order (NLO) coe ffi cients a piy , a piv and a piw entering into Eqs. (6-11). The NLOcoe ffi cients are the sum of the LO coe ffi cients plus next-to-leading order vertex and penguin corrections. The superscript p isomitted for i =
1, 2, 3, 5, 7 and 9, the penguin corrections being zero for these cases. a piy a piv a piw LO NLO LO NLO LO NLO a .
039 1 . + i . a . − . − i . a .
004 0 . − i . a u − . − . − i . − . − . − i . a c − . − . − i . − . − . − i . a − . − . − i . a u − . − . − i . − . − . − i . a c − . − . − i . − . − . − i . a . . + i . a u . . + i . . . + i . a c . . + i . . . + i . a − . − . − i . a u − . . + i . − . . + i . a c − . . + i . − . . + i . Table 2:
Parameters of production functions R ni ( E ) and R si ( E ) defined in Eq. (14) for κ = α ni τ ni (GeV − ) ω ni (GeV − ) α si τ si (GeV − ) ω si (GeV − )1 0 . − . . . − . . . . . . − . . . . . . + . .
3. Results
Partial wave analysis of the decay amplitudes helps in the in-vestigation of the density distributions in the Dalitz diagrams.In Eqs. (2,3) we have defined the S - and P - wave amplitudes towhich the double di ff erential B − → K − K + K − branching frac-tion Br is related through the symmetrized amplitude A − sym ofEq. (12): d Br − dm d cos Θ H = Γ B m |−→ p ||−→ p | π ) M B (cid:12)(cid:12)(cid:12) A − sym ( m , Θ H ) (cid:12)(cid:12)(cid:12) . (29)Here Γ B is the total width of the B − meson and the kaon mo-menta are: |−→ p | = q m − m K , (30) |−→ p | = m qh M B − ( m + m K ) i h M B − ( m − m K ) i . (31)The helicity angle Θ H is kinematically related to the e ff ectivemass m of the K − K + system:cos θ H = |−→ p ||−→ p | " m − (cid:16) M B − m + m K (cid:17) . (32)Due to the symmetry of the Dalitz plot density under the ex-change of the kaons K − and K − , one can define the e ff ectivemass m distribution integrated over the m masses larger than m : dBr − dm = Z Θ g d Br − dm d cos Θ H d cos Θ H , (33) where cos Θ g corresponds to the value of cos Θ H in Eq. (32)with m = m . The helicity angle distribution dBr − / d cos Θ H can be obtained from Eq. (29) by integration over the specificrange of the e ff ective mass m .Our aim is to describe the data of the Belle [1] and BaBar [2]Collaborations in one common fit. The data chosen by usinclude the total branching fraction for the decay B ± → [ φ (1020) K ± , φ (1020) → K + K − ], the averaged e ff ective massdistributions dBr ± / dm for m smaller than 1.8 GeV, and theaveraged helicity angle distribution dBr ± / d cos Θ H for m < .
05 GeV. The distributions of the B ± → K + K − K ± events areobtained from the published data by subtraction of the back-ground components. The total number of data points for tenplots from both collaborations is equal to 175. The theoreticaldistributions are normalized to the total number of experimen-tal events corresponding to each data set. In our fit we usedthe averaged B ± → [ φ (1020) K ± , φ (1020) → K + K − ] branch-ing fraction equal to (4 . ± . · − [7]. There are fourfitted parameters: χ, κ, c and N P . The first three parametersare related to the S -wave decay amplitudes and the fourth one, N P , is the common P -wave normalization constant by whichthe amplitudes A − P and A + P are multiplied. We have performedthe fit to the 176 data points obtaining the total value of χ equal to 343 and the following parameters: χ = (6 . ± . − , κ = (3 . ± .
20) GeV, c = (0 . ± . − and N P = . ± . N P = B ± → [ φ (1020) K ± , φ (1020) → K + K − ] branching fractionequal to 3.73 · − which is within one standard deviation from4 .4 0.6 0.8 1 1.2 1.4 1.6 1.8m K + K - (GeV)0102030 | Γ || Γ | Figure 1:
Moduli of kaon scalar non-strange and strange formfactors (solid lines) obtained in our fit. The dashed and dottedlines represent the variation of their moduli when parameter κ varies within its error band.the experimental value of (4 . ± . · − . One sees that theabsolute normalization of the P wave is very close to 1 whichmeans that the decay amplitudes calculated in our model areadequate.Our value of χ parametrizes a large range of K + K − e ff ec-tive mass up to 1.8 GeV and not just the region of f (980).Therefore it cannot be directly compared with the value given inRef. [4]. In addition, the estimate of χ given in Eq. (18) of [4]involves the coupling constant of f (980) to ππ while here wehave coupling to KK . Using g f ¯ KK / g f ππ = . χ ≈ . − .The κ parameter was not used in Ref. [4] whereonly the on-shell contributions to the form factors were takeninto account. The value of κ = .
51 GeV − is reasonably largerthan the typical KK mass considered. We have also done ananalogous fit to the data using the three P -wave form factorsbased on the parameterization of Ref. [5] obtaining similar val-ues of parameters as those written above, however with a higher χ value of 354.Fig. 1 shows the moduli of scalar form factors Γ n ∗ ( s ) and Γ s ∗ ( s ) which determine the functional dependence of the S -wave amplitudes on the K + K − e ff ective mass. There are twoprominent maxima of both form factors, one related to the f (980) resonance and the second one forming cusps due tothe opening of the third channel at 1400 MeV (in the presentmodel responsible e ff ectively for the production of four pions).Presence of f (980) leads to the threshold enhancement of the S -wave amplitude. This e ff ect can be directly studied in highstatistics experiment with a very good e ff ective K + K − mass res-olution of about 1 MeV and should be seen only a few MeVabove the threshold.In Fig. 2 the K + K − e ff ective mass distributions are shown fortwo mass ranges and for the data from the BaBar Collaboration.At low m K + K − the spectrum is influenced by the P -wave ampli-tude and dominated by the φ (1020) resonance. Above 1.05 GeVthe S -wave amplitude is much more important than the P -waveone. According to our analysis which uses the approach of Refs Figure 2:
The K + K − e ff ective mass distributions from the fitto BaBar experimental data [2] in the φ (1020) range (a) andbetween 1.05 GeV and 1.8 GeV (b). Theoretical results areshown as solid line in (a) and as histogram in (b).[11, 21], the experimental maximum near 1.5 GeV can be at-tributed to the f (1400 − A . Werecall that in Ref. [21] the coupling constant of the f (1400)decay to ¯ KK is much smaller than the corresponding couplingto ππ . Let us notice that the model distribution depends on thesharp 4 π threshold located at 2 · m = . K + K − e ff ective mass spec-tra and found that the quality of their description is similar tothat shown in Fig. 2 for the BaBar data. Fig. 3 shows a moredetailed comparison of the m K + K − theoretical distributions withthe Belle data [1], with events grouped in five ranges of m which is the other combination of the K + K − e ff ective masses.One observes an overall general agreement of theoretical his-tograms with experiment, with some surplus of experimentalevents in Fig. 3e for the case of the highest slice of m (largerthan 20 GeV ) where our model is not fully applicable due tothe proximity of the Dalitz plot edge.Finally, in Fig. 4 we present the helicity angle distributionin the K + K − mass range dominated by the φ (1020) resonance.Without the S -wave component of the decay amplitude the dis-5 igure 3: The K + K − e ff ective mass distributions from the fitto Belle experimental data [1] (a) for m < , (b) for 5GeV < m <
10 GeV , (c) for 10 GeV < m <
15 GeV ,(d) for 15 GeV < m <
20 GeV and (e) for 20 GeV < m .Theoretical results are shown as histograms.tribution should be symmetric with respect to cos Θ H = ff ect which distorts thedistribution. This is a direct evidence of a non-zero part of the S -wave present even under the huge peak of the φ (1020) reso-nance. A theoretical integration of the S -wave contribution tothe spectrum in the m K + K − range from threshold till 1.05 GeVleads to about 12% relative branching fraction. It correspondsto the average branching fraction of 4 . · − which is in agree-ment with the experimental upper bound of 2 . · − found inRef. [1]. This agrees also with the BaBar estimate (9 ± S -wave fraction in the region of masses between 1.013 and1.027 GeV [2]. In the range of the K + K − e ff ective mass from1200 to 1800 MeV, which might be relevant for the X (1550)discussed in Ref. [2], the CP averaged branching fraction cor-responding to the S − wave is equal to 4 . · − which is largerthan the total contribution of the Φ (1020) resonance.We have also studied the CP violation e ff ects comparing themagnitudes of the decay amplitudes of the B − and B + decays.While the moduli of the P -wave amplitudes for these chargeconjugated decays are rather similar, the S -wave amplitudesbehave di ff erently indicating an important CP violation e ff ectwhich depends on the m K + K − range. For the S -wave parameters Figure 4:
Helicity angle distribution for the Belle data [1] in the K + K − e ff ective mass up to 1.05 GeV. The dashed line representsthe S -wave contribution of our model, the dotted line - that ofthe P -wave, the dot-dashed - that of the interference term andthe solid line corresponds to the sum of these contributions.written above, starting from the K + K − threshold up to about 1.4GeV, the modulus of the B + S -wave amplitude is larger than thecorresponding modulus of the B − amplitude. Then, above 1.4GeV, the B − moduli become larger than the B + ones. Definingthe CP asymmetry as A CP ( m ) = (cid:16) dBr − dm − dBr + dm (cid:17) / (cid:16) dBr − dm + dBr + dm (cid:17) , (34)one gets very large asymmetries if one takes into account solelythe contribution of the S -wave. For example, A SCP (1 GeV) = − . A SCP (1 .
020 GeV) = − . A SCP (1 .
25 GeV) = − . A SCP (1 .
50 GeV) = + .
59 (here the superscript S standsfor the S - wave asymmetry). When the P -wave is includedthen the CP asymmetry is reduced to: A CP (1 GeV) = − . A CP (1 .
020 GeV) = + . A CP (1 .
25 GeV) = − .
85 and A CP (1 .
50 GeV ) = + . φ resonance, where the P -waveamplitude dominates, and an inversion of the A CP sign above1.4 GeV. Due to cancellations between the ranges of the nega-tive and positive asymmetries the resulting CP asymmetry av-eraged over the m K + K − range from threshold up to 1.8 GeV israther small, equal to -0.05. The averaged branching fractionfor the same mass range equals to 9 . · − . It is worthwhileto add that the S -wave gives to it the dominant contribution of5 . · − .6 . Conclusions We have studied final state interactions between kaons in the B ± → K + K − K ± decays. An overall general agreement withthe Belle and BaBar data has been obtained. Our formalismis based on the QCD factorization supplemented with the in-clusion of the long distance K + K − interactions. The latter aretaken into account through the functional dependence of thescalar and vector form factors on the e ff ective K + K − masses.A unitary model is constructed for the scalar non-strange andstrange form factors in which three scalar resonances f (600), f (980) and f (1400 − f (980) leads to the threshold enhancement ofthe S -wave K + K − amplitude. The K + K − structure seen near1.5 GeV can be attributed to the third scalar resonance. Apotentially large CP asymmetry is obtained in the mass spec-trum dominated by the S -wave. It originates from violent phasevariations of the two kaon scalar form factors which a ff ect the K + K − e ff ective mass dependence of the S − wave decay ampli-tudes. In general one can best study this e ff ect away from the φ (1020) peak. We have shown, however, that even under the φ maximum one observes nonnegligible helicity angle asymme-try. This e ff ect originates from the interference between the S -and P - waves.Our approach presented here for the B ± → K + K − K ± de-cays can be extended to study the B → K + K − K S reactionsfor which results of the time-dependent Dalitz analyses havebeen recently published by the Babar [17] and Belle [18] Col-laborations . For further studies of the charged B decays newexperimental data with better statistics are needed. Such dataalready exist! For example, the Belle Collaboration has nowfive times larger data sample than that used in their publication[1] analysed by us here. Future results from LHCb and fromsuper-B factories would also be very useful. Acknowledgments
This work has been supported in part by the Polish Ministryof Science and Higher Education (grant No. N N202 248135).
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