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a r X i v : h e p - e x / 0 6 1 2 0 3 2 v 1 1

3 D e c 2 0 0 6

A Non-parametric Approach to Measuring theK + Amplitudes in D + K K + + Decay

The FOCUS Collaboration1

J. M. Link a P. M. Yager a J. C. Anjos b I. Bediaga b

C. Castromonte b A. A. Machado b J. Magnin b A. Massafferri b

J. M. de Miranda b I. M. Pepe b E. Polycarpo b A. C. dos Reis b

S. Carrillo c E. Casimiro c E. Cuautle c A. Sanchez-Hern andez c

C. Uribe c F. V azquez c L. Agostino d L. Cinquini d

J. P. Cumalat d V. Frisullo d B. OReilly d I. Segoni d

K. Stenson d J. N. Butler e H. W. K. Cheung e G. Chiodini e

I. Gainese

P. H. Garbinciuse

L. A. Garrene

E. Gottschalke

P. H. Kasper e A. E. Kreymer e R. Kutschke e M. Wang e

L. Benussi f S. Bianco f F. L. Fabbri f A. Zallo f M. Reyes g

C. Cawleld h D. Y. Kim h A. Rahimi h J. Wiss h R. Gardner i

A. Kryemadhi i Y. S. Chung j J. S. Kang j B. R. Ko j

J. W. Kwak j K. B. Lee j K. Cho k H. Park k G. Alimonti

S. Barberis M. Boschini A. Cerutti P. DAngelo

M. DiCorato P. Dini L. Edera S. Erba P. Inzani

F. Leveraro S. Malvezzi D. Menasce M. Mezzadri

L. Moroni D. Pedrini C. Pontoglio F. Prelz M. Rovere S. Sala T. F. Davenport III m V. Arena n G. Boca n

G. Bonomi n G. Gianini n G. Liguori n D. Lopes Pegna n

M. M. Merlo n D. Pantea n S. P. Ratti n C. Riccardi n P. Vitulo n

C. Gobel o J. Otalora o H. Hernandez p A. M. Lopez p

H. Mendez p A. Paris p J. Quinones p J. E. Ramirez p Y. Zhang p

J. R. Wilson q T. Handler r R. Mitchell r D. Engh s M. Hosack s

W. E. Johns s E. Luiggi s M. Nehring s P. D. Sheldon s

E. W. Vaanderings

M. Websters

M. Sheaff t

a University of California, Davis, CA 95616 b Centro Brasileiro de Pesquisas Fsicas, Rio de Janeiro, RJ, Brazil

cCINVESTAV, 07000 Mexico City, DF, Mexico

Preprint submitted to Elsevier Preprint 15 December 2006
http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1http://arxiv.org/abs/hep-ex/0612032v1


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d University of Colorado, Boulder, CO 80309 eFermi National Accelerator Laboratory, Batavia, IL 60510

f Laboratori Nazionali di Frascati dellINFN, Frascati, Italy I-00044gUniversity of Guanajuato, 37150 Leon, Guanajuato, Mexico

h University of Illinois, Urbana-Champaign, IL 61801iIndiana University, Bloomington, IN 47405

j

Korea University, Seoul, Korea 136-701kKyungpook National University, Taegu, Korea 702-701INFN and University of Milano, Milano, Italy

m University of North Carolina, Asheville, NC 28804n Dipartimento di Fisica Nucleare e Teorica and INFN, Pavia, Italy

oPontifcia Universidade Cat olica, Rio de Janeiro, RJ, Brazil p University of Puerto Rico, Mayaguez, PR 00681

qUniversity of South Carolina, Columbia, SC 29208 rUniversity of Tennessee, Knoxville, TN 37996

sVanderbilt University, Nashville, TN 37235 t University of Wisconsin, Madison, WI 53706

Abstract

Using a large sample of D + K K + + decays collected by the FOCUS photo-production experiment at Fermilab, we present the rst non-parametric analysisof the K + amplitudes in D + K K + + decay. The technique is similar tothe technique used for our non-parametric measurements of the D + K

0e+

form factors. Although these results are in rough agreement with those of E687,we observe a wider S-wave contribution for the K 00 (1430) contribution than the

standard, PDG [1] Breit-Wigner parameterization. We have some weaker evidencefor the existence of a new, D-wave component at low values of the K + mass.

1 Introduction

This paper describes a non-parametric measurement of the K + amplitudes

present in the decay D+

K

K +

+

. Charm decay Dalitz plots are tra-ditionally t using the isobar model where the amplitude is represented bya sum of known Breit-Wigner resonances multiplied by complex amplitudesalong with a possible non-resonant term [2]. Often this approach gives a goodqualitative representation of the observed populations in the Dalitz plot. Theisobar approach, however, does not automatically incorporate some important

1 See http://www-focus.fnal.gov/authors.html for additional author information.

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theoretical constraints. If all nal state interactions are dominated by the two-body resonant system, with negligible contribution from the third body, thentwo-body unitarity is violated by the isobar model. These potential unitarityviolations are particularly severe for the case of broad, overlapping resonances.

An alternative, K-matrix formalism spearheaded by the FOCUS collaboration[6] for charm meson decays, is designed to automatically satisfy unitarity but

it is a difficult analysis that requires considerable input from low energy scat-tering experiments on the form and location of K-matrix poles. In particularthe FOCUS analysis of the D + , D+s +

+ nal state used the K-matrixdescription for just the broad dipion states, but the narrow P-wave states wereincorporated as isobar contributions.

This work takes a considerably different approach. We directly measure theK + spin amplitudes as a function of mK + mass by projecting thembased on the decay angular distribution. Our projective weighting technique,described in Section 3, is very similar to that used to make non-parametricmeasurements [5] of the q2 dependence of the helicity amplitudes in D + K + + . In our earlier work, each helicity amplitude is projected by makinga weighted histogram of q2 using special weights designed to block all otheramplitude contributions. The projective weighting technique is therefore anintrinsically one dimensional analysis. The D + K K + + nal state inprinciple is inuenced by three amplitudes: K + , K K + , and even K + + .We have chosen the D + K K + + nal state as a rst test case since theE687 isobar analysis [3] concluded that the observed D + K K + + Dalitzplot could be adequately described by just three resonant contributions: + ,K + K

0, and K + K 00 (1430). Although + is an important contribution, the

is a very narrow resonance that can be substantially removed by placing a

lower cut on the mK + K mass (such as mK + K > 1050 MeV/c2

). Because thereis no overlap of the band with the K 0 band and most of the kinematically

allowed K 00 (1430) region, there is a relatively small loss of information from

the anti- cut. The technique used to correct for residual contamination isdescribed in Section 5. It is important to establish to what extent undiscoveredcontributions in the K + K channel such as f 0(980)+ could inuence theseresults. 2 For example, the f 0(980)+ contribution was found [3] to be a majorcontributing channel in the related D +s K + K

+ decay. Uncertainty in theamplitudes describing the K K + channel will form the major systematic errorin this analysis. Throughout this paper, unless explicitly stated otherwise, thecharge conjugate state is implied when a decay mode of a specic charge isstated.

2 Evidence for a small a0(980) contribution based on an analysis of FOCUS datawas reported in a conference proceeding [4].

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2 Experimental and analysis details

The data for this paper were collected in the Wideband photoproduction ex-periment FOCUS during the Fermilab 19961997 xed-target run. In FOCUS,a forward multi-particle spectrometer is used to measure the interactions of high energy photons in a segmented BeO target. The FOCUS detector is a

large aperture, xed-target spectrometer with excellent track resolution andparticle identication. Most of the FOCUS experiment and analysis techniqueshave been described previously [7,8,9,10]. To obtain the signal displayed in g-ure 1, we required that the K K + + tracks formed a vertex with a condencelevel in excess of 10%, and the K K + + vertex was outside of our BeO targetand all other spectrometer material by at least 3 . Other tracks along withinformation from the K K + + secondary vertex were used for form a pri-mary vertex and the separation between the primary and secondary vertexwas required to exceed 7 . The kaon hypothesis was favored over the pionhypothesis by 3 units of log likelihood in our Cerenkov system for the K andK + candidates, and the + candidate response was consistent with that for

a pion. Finally, we required that the secondary vertex was well isolated. Werequired that no primary vertex track was consistent with the secondary ver-tex with a condence level exceeding 0.5 % , and no other track in the eventwas consistent with the secondary vertex with a condence level exceeding1 10 4. In this analysis we eliminated most of the D + + contributionby imposing an anti- cut that required mK + K > 1050 MeV/c 2.

Fig. 1. The K K + + invariant mass spectra for all of the cuts used in this analy-sis. The D + K K + + signal and sideband regions are shown with vertical lines.The peak to the right of the D + K K + + peak is due to D +s K

K + + .The D +s peak appears over a broad, ramped background due to misidentiedD + K + + decays.

We obtained a signal of 6400 D + K K + + events prior to the anti- cutand 4200 events after the anti- cut.

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3 Projection Weighting Technique

The projector method used in this analysis is nearly identical to that used todetermine the q2 dependence of the helicity form factor in D + K + + decay [5]. To apply the projector method, we must assume that after we imposethe anti- cut, the residual effects of the K K + amplitude contributions aresmall enough that we can reliably correct for them using the bias techniquedescribed in Section 5. In the absence of K K + resonances, we can write thedecay amplitude in terms of mK + = m and the decay angle (which is theangle between the K and the K + in the K + rest frame).

A =s,p,d

A (m) d00(cos )

Here d00(cos ) are the Wigner d-matrices that describe the amplitude fora K + system of angular momentum to simultaneously have 0 angularmomentum along its ( D + frame) helicity axis and the K + decay axis. Forsimplicity, we illustrate the technique assuming only P-wave and S-wave con-tributions although contributions up to and including D-wave are included inthe analysis. In the absence of D-wave or higher amplitudes, the decay inten-sity consists of three terms which depend on the two complex functions: S (m)and P (m) of m mK + .

I (m, cos ) = |A|2 = | S (m) + P (m)cos |2 =

|S (m)|2 + 2Re {S (m) P (m)} cos + |P (m)|2 cos2

For notational simplicity we will write the direct terms as SS (m) | S (m)|2

and P P (m) | P (m)|2 and the interference term as SP (m) Re {S (m) P (m)}:

I (m, cos ) = SS (m) + 2 SP (m)cos + P P (m)cos2 (1)

Our approach is to divide cos into twenty evenly spaced angular bins. Leti D = i n1 i n2 ... i n20 be a vector whose 20 components give the population

in data for each of the 20 cos bins. Here i species the ith mK + bin. Ourgoal is to represent the i D vector as a sum over the expected populationsfor each of the three partial waves. We will call these i m vectors. For thissimplied case, there are three such vectors computed for each mK + bin:{i m } = ( i mSS , i mSP , i mP P ). Each i m is generated using a phase space andfull detector simulation for D + K K + + decay with one amplitude turnedon, and all other amplitudes shut off. Hence the i mP P vector is computedassuming an intensity of cos 2 for each mK + bin. The i m vectors incorporatethe underlying angular distribution as well as all acceptance and cut efficiency

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effects. In particular, the anti- cut creates substantial inefficiencies at lowcos and high mK + , whereas the other cuts have a reasonably uniformacceptance in cos .

We use a weighting technique to t the bin populations in the data to theform: i D = F SS (m i ) i mSS + F SP (m i ) i mSP + F P P (m i ) i mP P . The termF SP (m i ), for example, is proportional to SP (m i ) along with the overall ac-ceptance and phase space for an SP interference term generated in a givenm i bin. Multiplying the i D data vector by each i m produces a componentequation:

i mSS i Di mSP i Di mP P i D

=

mSS mSS mSS mSP mSS mP P

mSP mSS mSP mSP mSP mP P

mP P mSS mP P mSP mP P mP P

F SS (m i )

F SP (m i )

F P P (m i )

The formal solution to this is:

F SS (m i )

F SP (m i )

F P P (m i )

=




1i mSS D ii mSP D ii mP P D i

This solution can be written as F SS (m i ) = i P SS D i , F SP (m i ) = i P SP D i ,and F P P (m i ) = i P P P D i , where the projection vectors are given by:

P SS P SP P P P

=




1i mSS i mSP i mP P

(2)

We can modify the projector weights to take into account cut efficiency, ac-ceptance and phase space corrections required to convert say F SP (m i ) intoSP (m i ) as discussed in Reference [5]. This procedure scales the i P SP , forexample, into a modied weight i P SP which includes all acceptance, efficiencycorrections and phase space effects.

The various projector dot products are implemented through a weighting tech-nique. For example, if we are trying to extract the 2 S (mK + ) P (mK + )interference in the i th mK + bin, we need to construct the dot product:

i P SP i D = i P SP 1

i n1 + i P SP 2

i n2 + i P SP 20

i n20

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We can do this by making a weighted histogram of mK + where all eventsthat are reconstructed in the rst cos bin are weighted by i P SP 1 ; all events

that are reconstructed in the second cos bin are weighted by i P SP 2 etc.

4 Amplitude ambiguity

Table 1 shows the cos dependences for each of the 3 direct and 3 interferenceterms for the S (mK + ), P (mK + ), and D(mK + ) amplitudes.Table 1The angular distributions for amplitudes up to and including D-wave. Here cos is the angle between the K + and K in the K + rest frame. These terms areproducts of the Wigner d-matrices d00(cos ).

SS 1 PP cos2

SP cos PD cos 3cos2 1 / 2

SD 3cos2 1 / 2 DD 3cos2 1 2 / 4

The problem is that these six terms are not independent. The relationshipbetween them is given in eq. (3).

i mSD =3 i mP P i mSS

2(3)

Hence one cannot make independent projectors of SS, PP, and SD, and mustchoose two out of these three or the inverse matrix of eq. (2) will becomesingular. Since it is known [3] that the K + spectrum in D + K K + +

is dominated by the PP contribution ( K 0) and SS contribution [ K

00 (1430)],

we made the choice of dropping the SD interference term and choosing SS,PP, DD, SP, and PD. As a result, the ve projectors we use can only partiallyblock any potential SD term. As long as there are no amplitude contributionsbeyond D-wave, one can show that a potential SD interference contributionwill contaminate both the SS and PP spectra but not the two interferenceterms: SP or PD.

5 The Bias Correction

Figure 2 compares the ve reconstructed spectra with the input spectra ac-cording to a simulation of the E687 model [3]. The discrepancies between the

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Fig. 2. The rst points (crosses) with error bars show the reconstructed spectrausing the method described in Section 3. The closely adjacent points (diamonds)are the actual mK + spectra used in the simulation based on the model of Refer-ence [3]. These two Monte Carlo spectra are normalized near the peak bins of theprominent K

0 present in the PP spectrum. The plots are: (a) S 2 (mK + ) directterm, (b) 2 S (mK + ) P (mK + ) interference term, (c) P 2 (mK + ) direct term,(d) 2 P (mK + ) D (mK + ) interference term and (e) D 2 (mK + ) direct term.

reconstructed and input spectra are due to residual contamination in the tailbeyond our 1050 MeV/c 2 anti- cut. The discrepancies are most prominent atthe high end of the mK + distribution (where the tail is the largest), butare relatively small on the scale of our statistical error elsewhere.

In order to correct for the tail that extends past mK + K = 1050 MeV/c 2, wetake the difference between the simulated input and reconstructed spectra asa bias that we subtract from the data after normalizing the bias by the ratioof the K

0 peak area of the PP spectra in the data to that in the simulation. 3

Varying the K K + amplitudes relative to the E687 model [3] will changethe bias correction. Uncertainty in these amplitudes is our major source of systematic error.

3 The normalization is actually done by nding a ratio that minimizes the 2between the data and MC prediction at the peak and one bin on either side of the peak.

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6 Systematic Errors

The major source of systematic error in this analysis is due to possible uncer-tainty in the K K + system amplitudes. In Section 5 we discussed the methodused to correct the amplitude spectrum in K + for residual K K + contribu-tions past our anti- cut of mK + K > 1050 MeV/c 2. This method, however,depends on our model for the K K + channel. The nominal results assumethe amplitude measured in the E687 [3] analysis, but we have consideredvariations in the result due to differences in the parameters as well as po-tential contributions from the f 0(980). Although the D +s K

K + + hasan 11% contribution from f 0(980)+ , there was no evidence for an f 0(980)+contribution in D + K K + + in the E687 analysis.

We have considered possible f 0(980) contributions or parameter variantsconsistent with our data in four areas: (a) the fraction of D + K K + +that appears as + : (b) the shape of the cos K + + distribution in the vicinityof the : (c) agreement with the shape of the complete mK + K spectrum and(d) agreement with the observed populations in two dimensional bins of theD + K K + + Dalitz plot. Here cos K + + is the cosine of the angle betweenthe K + and the + tracks in the K + K rest frame. In each case, the defaultmodel with a wide K

00 (1430) and no f 0(980) amplitude was among the best

models matching our data according to these tests. Essentially the same setof potential t variants were selected by the four tests.

We found that a potential f 0(980) when inserted with a phase of / 2 relativeto the K

0(890) satised our consistency criteria even with a t fraction aslarge as 3.3%. Such large f 0(980) amplitudes would be more inconsistent withthe data if they came in with a relative phase of zero. We construct the

systematic error as the r.m.s. spread of all acceptable t variants with f 0(980)amplitudes up to 40% of the K

0 amplitude that both matched the observedbinned Dalitz plot populations to within 10 of the best model and matchedthe observed cos K + + distribution in the vicinity of the with a condencelevel exceeding 1 10 4. Our default model matched the cos K + + distributionwith a condence level of 18%. Table 2 in Section 7 compares the sizes of systematic and statistical errors for our ve amplitude products. Generallythe systematic error was found to be smaller than the statistical error.

A variety of other checks of the results were made during the course of thiswork. We generated a Monte Carlo using the mK + amplitudes summarizedin Table 2 and compared the simulated to the observed cos distributions asa function of mK + as well as the simulated and observed mK + K , mK + ,and mK + + global mass projections. Agreement was good. We also comparedthe t results obtained by analyzing the data in different ways. For example,we analyzed the data by constructing only three rather than ve projectorsand found consistent results with the S 2 (mK + ) , P 2 (mK + ), and the2 S (mK + ) P (mK + ) interference terms. We raised the anti- cut from

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mK + K > 1050 MeV/c 2 to mK + K > 1100 MeV/c 2 and found consistentresults even though the errors in the high mass bins went up dramaticallyunless the D-wave projectors were excluded.

7 Results

Figure 3 and Table 2 show the results of this analysis. The relative correlationbetween the 5 amplitudes is typically less than 40% except at the highestmK + bins where they are as large as 65%. The comparison plot is basedon the E687 analysis [3] but with a much wider K

00 (1430) represented as a

Breit Wigner resonance with a pole at m0 = 1412 MeV/c 2 and a width of = 500 MeV/c 2 whereas the standard PDG [1] K

00 (1430) parameters are

m0 = 1414 MeV/c 2 and = 290 MeV /c 2. In order to roughly reproducethe results in gure 3 in plot (a), we also increased the magnitude of theK

00 (1430) amplitude by 40% relative to that obtained from the E687 analysis.

Hence the curves in g. 3 are drawn using a model with a 41.9% K

0

0 (1430) tfraction compared to the 37 3.5% t fraction quoted in Reference [3]. Theuse of a wider K

00 (1430) with a larger amplitude increased the level of the SP

interference in the model by 45% to a level in approximate agreement withour data as shown in plot(b).

8 Summary and Discussion

We presented a non-parametric amplitude analysis of the K

+

system inD + K K + + decay using the technique described in Reference [5]. There isno need to assume specic Breit-Wigner resonances, forms for mass dependentwidths, hadronic form factors, or Zemach momentum factors. As described inSection 3, a set of ve weights were generated using Monte Carlo simulationsthat were designed to project the various amplitude contributions. The veamplitude contributions appear in just ve weighted histograms in the mK +mass. Because this is an essentially one dimensional technique, we chose theD + K K + + nal state as a test case. According to an older, traditionalDalitz analysis done in E687 [3] the D + K K + + nal state is particu-larly simple consisting of just + , K

0K + , and K 00 (1430)K + . Hence only one

narrow resonance, the , should contribute to the K K + channel that can besignicantly reduced through an anti- cut such as mK + K > 1050 MeV/c 2.This leaves an amplitude that depends primarily on the K + mass. Anotherattractive feature of the D + K K + + nal state is that there should bea strong S-wave component in the K K + channel that E687 [3] modeled asthe K

00 (1430). Not that much is known about including broad S-wave reso-

nances in charm Dalitz analyzes, and we indeed nd considerable discrepan-

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Fig. 3. This gure shows the amplitudes measured in this analysis including system-atic errors in 73 MeV /c 2 bins. The default model described in the text are the curves.The data has been normalized such that the P 2 (mK + ) value at the K

0 mass is 1in plot (d). The plots are: (a) S 2 (mK + ) direct term, (b) 2 S (mK + ) P (mK + )interference term, (c) P 2 (mK + ) direct term, (d) 2 P (mK + ) D (mK + ) in-terference term and (e) D 2 (mK + ) direct term.

cies between our non-parametric description of the S-wave, K + amplitudein D + K K + + and the standard, PDG K

00 (1430) parameterization used

by E687 [3].

The other possibly signicant difference in these results compared to the E687analysis involves the glitch in the rst three bins of the S 2 (mK + ) spec-trum. We have observed this effect in all variants of the t we made in thecourse of understanding systematic errors. The same bins are also present asa glitch in the 2 P (mK + ) D(mK + ) interference term which is otherwiseconsistent with zero. If these are deemed to be signicant, one explanationwould be the presence of a small D-wave component at low K + masseswhich could account for both glitches through the mechanism discussed inSection 4 while being small enough to escape notice in the direct D 2 (mK + )terms where it is second order in the amplitude. Section 4 argues there is anambiguity in the cos distributions if one includes waves up to and includingD-wave which cannot be eliminated by this or any other method. Hence an2 S (mK + ) D(mK + ) interference term will also affect the S 2 (mK + )

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9 Acknowledgments

We wish to acknowledge the assistance of the staffs of Fermi National Ac-celerator Laboratory, the INFN of Italy, and the physics departments of thecollaborating institutions. This research was supported in part by the U. S.National Science Foundation, the U. S. Department of Energy, the Italian Isti-tuto Nazionale di Fisica Nucleare and Ministero dellUniversit` a e della RicercaScientica e Tecnologica, the Brazilian Conselho Nacional de DesenvolvimentoCientco e Tecnologico, CONACyT-Mexico, the Korean Ministry of Educa-tion, and the Korean Science and Engineering Foundation.

References

[1] W.-M. Yao et al., J. Phys. G 33 , 1 (2006)

[2] Dalitz Plot Analysis Formalism by D. Asner, in W.-M. Yao et al., J. Phys. G33 , 1 (2006)

[3] Rodney Greene thesis, University of Illinois at Champaign-Urbana (1995) ; P.LFrabetti et al., Phys. Lett. B 351 , 591 (1995).

[4] Sandra Malvezzi, D-Meson Dalitz Fit from FOCUS, Proc. of the 7th Conferenceon Intersection Between Particle and Nuclear Physics (CIPANP 2000), QuebecCity, 22-28 May 2000, AIP Conf. Proc. 549 : 569-574 (2002)

[5] FOCUS Collab., J.M. Link et al., Phys. Lett. B 633 (2006) 183 ; CLEO Collab.,Phys. Rev. D 74 , (2006) 052001.

[6] FOCUS Collab., J.M. Link et al., Phys. Lett. B 585 (2004) 200.

[7] E687 Collab., P. L. Frabetti et al., Nucl. Instrum. Methods A 320 (1992) 519.

[8] FOCUS Collab., J. M. Link et al., Nucl. Instrum. Methods A 484 (2002) 270.



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