A complete model for D + → K − π + π+ S-wave
aa r X i v : . [ h e p - ph ] D ec August 16, 2018
A complete model for D + → K − π + π + s-wave Patr´ıcia Camargo Magalh˜aes and Manoel Roberto Robilotta on behalf of Institute of Physics, University of S˜ao Paulo, Brazil
The D + → K − π + π + decay have been observed in LHCb collaborationwith thousands of events per second from Kπ threshold. The theoreticaltreatment of this decay includes a rich dynamic behaviour that mix weakand strong interactions in a non trivial way.In the present work we show recently progress on D + → K − π + π + decay within a effective hadronic formalism. The weak sector followed asimilar formalism from heavy meson ChPT[1], where quark c is an ex-ternal object of SU(3) Goldstone bosons sector and FSI description isbased in three-body model from[2]. The results concerning the vectorweak transition revealed a very interesting new scenario in s-wave phase-shift: agrees well with data in the elastic region, and it starts from − ≈ − ρ meson propagator, and a rescattering model tothe final state interaction. Therefore, this work represent a progress in theunderstanding of D + → K − π + π + and could be applied to other decays.PRESENTED AT The 6 th International Workshop on Charm Physics(CHARM 2013)Manchester, UK, 31 August – 4 September, 2013 This work was supported by FAPESP, Brazilian founding agency.
Introduction
The S -wave K − π + sub-amplitude in the decay D + → K − π + π + , denoted by [ K − π + ] D + has been extracted from data by the E791[3] and FOCUS[4] collaborations. A remark-able feature of the results is a significant phase shift deviation between [ K − π + ] D + andelastic K − π + LASS data [5], which was considered to be a puzzle until recently. How-ever, this is a clearly indication that the dynamical relationship between both typesof processes is not simple and has motivated an effort by our group aimed at under-standing the origins of this problem. A schematic calculation was presented in [2] andrecent progress can be found in [6]. Here we briefly summarize the main issues andprogress. The programme is implemented by means of effective lagrangians, whichincorporate the symmetries of QCD, where weak and electromagnetic interactionsare included as external sources. The inclusion of heavy mesons can be performed bysuitable adaptations of the light sector [1].
The reaction D + → K − π + π + involves two distinct structures: weak vertex andfinal state interactions (FSI). The first one concerns the primary quark transition c → s W + , which occurs in the presence of the light quark condensate of the QCDvacuum and is dressed into hadrons. The second class of processes corresponds tothree-body FSI, associated with the strong propagation of the state produced in theweak vertex to the detector. These ideas are summarized in Fig.1. A W = + + W T T . . . + W T = +
Figure 1: Diagrammatic representation of the heavy meson decay into
Kππ , startingfrom the weak amplitude (red) and including hadronic final state interactions. weak vertex
For description of the weak vertex in D + → K − π + π + , we concentrate on the colourallowed class of diagrams proposed by Chau[7], which gives rise to the hadronic am-plitudes shown in Fig.2. Processes on the top involve an axial weak current, whereasthe bottom diagram is based on a vector current. The blob in the diagrams summa-rizes several hadronic processes which contribute to form factors. In the absence of1 KD ππ c s c s − KD ππ c s c s Figure 2: Mechanisms for hadronization; quarks u and d are not specified.form factors, the weak vertex entering Fig. 1 is given by the diagrams shown in Fig.3, where processes a and b involve the axial current and c contains a vector current.The inclusion of form factors can be made either by using phenomenological input or (a) (b) (c) += + W Figure 3: Topologies for the weak vertex: the dotted line is a scalar resonance andthe wavy line is a W + , which is contracted to a point in calculations.by allowing the intermediate propagation of ( cs ) states, as shown in Fig. 4. D+K − K − D+ K − D+ D * + s = µµ + µ ( v.1 ) ( v.2 ) V Figure 4: Vector form factor. final state interactions
When final state interactions are added to the processes, one finds three families ofcolor-allowed amplitudes denoted respectively by A a , A b , and A c . The class of FSIsconsidered is based on a succession of elastic two-body interactions, which bring the Kπ phase into the problem. The Kπ amplitude is derived by means of chiral effectivelagrangians, based on leading order contact terms [8] and supplemented by resonances[9], which allow for a wider energy range. 2 First results
In a previous publication [2], we evaluated the contributions of the three topologies(Fig. 3) up to second order in a perturbative series to the S -wave [ K − π + ] D + . Withthe purpose of taming the calculation we made some simplifying assumptions: theweak amplitudes of Fig. 3 were taken to be constants, isospin 3 / P waves werenot included in the Kπ amplitude and couplings to either vector mesons or inelasticchannels were neglected. Results for the phase are displayed in Fig. 5. Leading order P ha s e Aa = AbAcAc - 148 o Figure 5: Phase for A a , A b , A c at leading order and A c shifted by − comparedwith FOCUS [4](triangle) and E791[3](circle) data, together with elastic Kπ resultsfrom LASS[5](diamond).contributions from the axial weak current, represented by A a and A b , obey Watson’stheorem and fall on top of elastic Kπ data. The curve for the vector A c amplitudehas a different shape and, if shifted by − , can describe well FOCUS data[4], upto the region of the peak.The main lesson to be drawn from our first approach to this problem is that, forsome yet unknown reason, the amplitude which begins with a vector weak current,represented by diagram ( c ) of Fig. 3, seems to be favoured by data. This amplitudereceives no contribution at tree level, since the W + emitted by the charmed quarkdecays into a π + π pair. Therefore, the leading term in this kind of process necessarilyinvolves loops and the corresponding imaginary components.3 Vector vertex
A limitation of our first study [2] was that all weak vertices were described by mo-mentum independent functions. Those results are now improved by considering theproper P -wave structure of the weak vertex, corrections associated with form factorsand contributions from intermediate ρ mesons. The ρ is introduced by means of stan- π _K π + π + W ρ Ds*
K−T
Figure 6: Leading vector contribution.dard vector meson dominance, using the formalism given in Ref.[9]. The D → W K vertex may contain ( sc ) intermediate states and can be obtained either by means ofheavy-meson effective lagrangians [1] and the diagrams of Fig. 4, or by using phe-nomenological information parametrized in terms of nearest pole dominance [10]. Ourbasic interaction for this process became the diagram in Fig. 6. New predictions for -90090180270 pha s e s h i ft S - w a v e no rho no ff -90090180270 pha s e s h i ft S - w a v e fat rho Figure 7: Predictions for the phase, compared with FOCUS results[4].the phase are shown in the red curve of Fig. 7. Form factors, as expected, becomemore important at higher energies, as indicated by “no form factors” curve. The “norho” is obtained by taking the limit m ρ → ∞ in the calculation and tends to thatlabelled A c in Fig. 5.The most prominent feature of the full phase is that it now hasa negative value at threshold, showing that contributions from light intermediate res-onances are important. So far, the rho has just been treated as a point-like particle,4owever its width, associated with two-pion intermediate states, is a new source ofcomplex amplitudes and was included in Fig. 7(right) with the label “fat rho”, whichcompared with the point-like ρ show the relevance of including off-shell effects in thisvertex. We have presented results for the decay D + → K − π + π + and shown that final stateinteractions are visible in data. It is already clear that hadronic processes occurringbetween the primary weak decay and asymptotic propagation to the detector do playa key role in shaping experimental results. Although derived from a single instance,the patterns of hadronic interpolation are quite general and it is fair to assume thatthis conclusion can be extended to other processes. ACKNOWLEDGEMENTS
PCM would like to thank the organizers for the nice conference and for the opportu-nity to present a talk. PCM was supported by FAPESP, process 09/50634-0.
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