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Note on helicity amplitudes in D V semileptonic decays → Svjetlana Fajfer1,2,∗ and Jernej Kamenik1,† 1J. Stefan Institute, Jamova 39, P. O. Box 3000, 1001 Ljubljana, Slovenia 2Department of Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia (Dated: February 2, 2008) Motivated bythe recent extraction of thehelicity amplitudes for theD+ K¯∗0µνµ decay, done by the FOCUS collaboration, we determine helicity amplitudes for the D+ → K¯∗0lν , D+ ρ0lν l l and D+ φlν semileptonic decays using the knowledge of the relevant for→m factors. The→vector s → l and axial form factors for D Vlν decays are parameterized byincluding contributions of charm l → meson resonances and using the HQET and SCET limits. In the case that the vector form factor receives contributions from two poles while axial form factors are dominated by a single pole for D+ K¯∗0lν, we obtain better agreement with the experimental result then when all of them are l → dominated by single poles. 6 0 PACSnumbers: 13.20.Fc,13.20.-v,12.39.Hg,12.39.Fe 0 2 Recently the FOCUS [1] collaboration has presented Next we follow the analysis of Ref. [2], where the F n + firstnon-parametricdeterminationofhelicityamplitudes formfactorinH P transitionsisgivenasasumoftwo a J inthesemileptonicdecayD+ K¯∗0µ+ν. Thismeasure- pole contribution→s,while theF0 formfactoris writtenas 4 mentallowsformoredetaileda→nalysisoftheD V form a single pole, based on form factor dispersion properties → factors, especially it enables the studying of the shapes as well as known HQET [6] and SCET [7, 8, 9] scaling 1 of the form factors. limits near zero and maximum recoilmomentum respec- v We have recently proposed a generalization of the tively. Utilizing the same approach we have proposed 8 Bećirević-Kaidalov(BK)formfactorparameterization[2] a general parametrization of the heavy to light vector 2 0 forthesemileptonicH V formfactorsbasedonHQET formfactors,whichalsotakesintoaccountalltheknown 1 and SCET scaling pred→ictions [3]. Furthermore we have scalingandresonancepropertiesoftheformfactors. The 0 calculated the D P and D V form factors shapes detailsoftheanalysisareoutlinedinRef.[3]andweonly 6 within a model w→hich combine→s properties of the heavy givetheresultsforthe derivedformfactorparameteriza- 0 meson chiral Lagrangian by taking into account known tions: h/ andpredictedcharmresonancesandutilizingthegeneral c′ (1 a) V(q2) = H − , p form factor parameterizations [3, 4]. (1 x)(1 ax) - − − p In this note we determine helicity amplitudes for the c′ (1 a) e D → V semileptonic decays and compare our model A1(q2) = ξ H1 −b′x , h predictions for the shapes of the form factors with the − c′′(1 a′) iv: nDe+w exKp¯e∗r0iµm+eνntdalecraeys.ults coming from FOCUS for the A0(q2) = (1−Hy)(1−−a′y), X → c′′′ The standard decomposition of the current matrix el- A (q2) = H , r ements relevant to semileptonic decays between a heavy 2 (1 b′x)(1 b′′x) a − − pseudoscalar meson state H(pH) with momentum pH (1) | i and a light vector meson V(p ,ǫ ) with momentum pV and polarization vector|ǫV Vis inV tierms of four form wa′h)]e/r(emc′H′′ m= )[(imsfiHxe+dbmyVt)hξec′Hre(l1ati−onab)e+twe2emnVthc′He′(fo1rm− factors V, A0, A1 and A2, functions of the exchanged factors Hat−q2 =V 0 while ξ =m2 /(m +m )2 is the pro- momentumsquaredq2 =(p p )2 [5]. HereV denotes H H V the vector form factor andHis−expVected to be dominated portionality factor between A1 and V from the SCET relation. Variables x = q2/m2 and y = q2/m2 en- by vector meson resonance exchange, the axial A and H∗ H 1 sure, that the V and A form factors are dominated by A form factors are expected to be dominated by ax- 0 2 the physical 1− and 0− resonance poles, while a and a′ ial resonances, while A denotes the pseudoscalar form 0 measure the contributions of higher states, parameter- factor and is expected to be dominated by pseudoscalar ized by additional effective poles. On the other hand meson resonance exchange [5]. In order that the ma- b′ in A and A measures the contribution of resonant trix elements are finite at q2 = 0, the form factors must 1 2 states with spin-parity assignment 1+ which are param- alsosatisfy the wellknownrelationA (0)+A (0)(m + 0 1 H eterized by the effective pole at m2 = m2 /b′ while mV)/2mV A2(0)(mH mV)/2mV =0. He′∗ff H∗ − − the scaling properties and form factor relations require an additional effective pole for the A form factor. At 2 the end we have parameterized the four H V vector ∗Electronicaddress:[email protected] form factors in terms of the six parameters c→′H, a, a′, b′, †Electronicaddress:[email protected] c′′ and b′′. H 2 We determine the above parameters via heavy meson compare them with the experimental results of FOCUS, chiral theory (HMχT) calculation of the form factors scaledbyanoverallfactordeterminedbytheleastsquare near q2 = (m m )2. We use the leading order fit of our model predictions, on FIGs. 1, 2 and 3. The max H − V heavy meson chiral Lagrangian in which we include ad- scale factor is common to all form factors. ditional charm meson states. The details of this frame- work are given in [3] and [4]. We first calculate values H+2Hq2L@GeV2D of the form factors in the small recoil region. The pres- 4 enceofcharmmesonresonancesinourLagrangianaffects the values of the form factors at q2 and induces sat- max 3 uration of the second poles in the parameterizations of the F (q2), V(q2) and A (q2) form factors by the next + 0 radial excitations of D∗ and D mesons respectively. 2 (s) (s) UsingHQETparameterizationofthecurrentmatrixele- ments[3],whichisespeciallysuitableforHMχT calcula- 1 tions of the form factors near zero recoil, we are able to extract consistently the contributions of individual resonances from our Lagrangian to the various D V 0 form factors. We use physical pole masses of ex→cited 0.2 0.4 0.6 0.8 state charmed mesons in the extrapolation, giving for q2@GeV2D the pole parameters a =m2 /m2 , a′ =m2 /m2 and H∗ H′∗ H H′ b′ = m2 /m2 . Although in the general parameteriza- FIG.1: Ourmodelpredictions(doublepoleinsolidlineand H∗ HA singlepoleindashedline)fortheq2dependenceofthehelicity tion of the form factors the extra poles in V and A 0,1,2 amplitudeH2(q2)incomparisonwithscaledFOCUSdataon parameterize all the neglected higher resonances beyond + D+ K¯∗0 semileptonic decay. thegroundstateheavymesonspindoublets(0−,1−),we → are saturating those by a single nearest resonance. The single pole q2 behavior of the A (q2) form factor is explained by the presence of a sing1le 1+ state relevant to H-2Hq2L@GeV2D eachdecay,whileinA (q2)inadditiontothesestatesone 2 10 mightalsoaccountfortheirnextradialexcitations. How- ever,duetothelackofdataontheirpresenceweassume their masses being much higher than the first 1+ states 8 and we neglect their effects, setting effectively b′′ =0. ThevaluesoftheunknownHMχTparametersappear- 6 ing in D Vlν decay amplitudes [3] are determined by l → fitting the model predictions to knownexperimentalval- ues of branching ratios and partial decay width ratios. 4 In order to compare our model predictions with recent experimental analysis performedby FOCUS collab- 0.2 0.4 0.6 0.8 oration, following [10] we introduce helicity amplitudes q2@GeV2D H : +,−,0 FIG.2: Ourmodelpredictions(doublepoleinsolidlineand 2m p~ (y) H±(y) = +(mH +mV)A1(m2Hy)∓ mHH|+VmV |V(m2Hy) saimngplleitpuodleeHin2d(aqs2h)eidnlcinoem)pfoarritshoenqw2itdhepsecnaldeednFceOoCfUthSedhaetliaciotny − H (y) = + mH +mV [m2 (1 y) m2]A (m2 y) D+ K¯∗0 semileptonic decay. 0 2mHmV√y H − − V 1 H → 2mH|p~V(y)| A (m2 y) (2) In addition to the two pole contributions we calculate −mV(mH +mV)√y 2 H helicity amplitudes in the case when all the form factors exhibit single pole behavior. Putting contributions of where y =q2/m2 and the three-momentum of the light higher charm resonances to be zero we fit the remaining H vector meson is given by: model parameters to existing branching ratios and partial decay ratios. We obtain the values for the following [m2 (1 y)+m2]2 parameter combinations as explained in [3]: p~ (y)2 = H − V m2. (3) | V | 4m2 − V H α˜µ˜ = 0 Becauseofthearbitrarynormalizationoftheformfac- α′ζ = 0.180 GeV3/2 tors in [1], we fit our model predictions for a common − α′µ = 0.00273 GeV1/2 overallscaleinordertocomparetheresults. Weplotthe − q2 dependence of the predicted helicity amplitudes and α = 0.203 GeV1/2 (4) 1 − 3 H02Hq2L@GeV2D H2Hq2L@GeV2D i 60 H- 20 H-HsingleL 50 H+ H+HsingleL 15 40 H0 H HsingleL 0 30 10 20 5 10 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 q2@GeV2D q2@GeV2D FIG.3: Ourmodelpredictions(doublepoleinsolid lineand FIG. 4: Our model predictions for the q2 dependence of singlepoleindashedline)fortheq2dependenceofthehelicity thehelicityamplitudesH2(q2)fortheD+ ρ0 semileptonic amplitudeH2(q2)incomparisonwithscaledFOCUSdataon decay. Double pole predicitions are rendere→d in thick (black) 0 D+ K¯∗0 semileptonic decay. lines while single pole predictions are rendered in thin (red) → lines: H−(solidlines),H0(dashedlines)andH+(dot-dashed lines). AsshownonFIGs.1and2theexperimentaldataforH ± do not favor such a parametrization, while in the case H2Hq2L@GeV2D i of H helicity amplitude there is almost no difference 0 H- since the H0 helicity amplitude is defined via the A1,2 20 H-HsingleL form factors, which are in our approach both effectively H+ dominated by a single pole. The agreement between the H+HsingleL 15 FOCUS results and our model predictions for the q2 de- H0 pendence of the helicity amplitudes is good, although as H0HsingleL 10 noted already in [1], the uncertainties of the data points are still rather large. On FIGs. 4 and 5 we present helicity amplitudes for the D+ ρ0lν and D+ φlν 5 → l s → l decays. Both decay modes are most promissing for the futureexperimentalstudies. Wemakepredictionsforthe 0.2 0.4 0.6 0.8 shapes of helicity amplitudes for both cases: where two q2@GeV2D poles contribute to the vector form factor and a single poleto the axialformfactors,andthesecondcasewhere FIG. 5: Our model predictions for the q2 dependence of all form factors exhibit single pole behavior. the helicity amplitudes H2(q2) for the D+ φ semileptonic In principle one can apply the above procedure to the i s → decay. Double pole predictions are rendered in thick (black) B ρlνl semileptonicdecays. However,duetothemuch lines while single pole predictions are rendered in thin (red) → broader leptons invariant mass dependence in this case, lines: H−(solidlines),H0(dashedlines)andH+(dot-dashed our procedure is much more sensitive to the values of lines). the form factors at q2 0. In addition, the semileptonic ≈ decayratesinourmodelfitarenumericallydominatedby thelongitudinalhelicityamplitudeH whichhasabroad D+ φlν decays. In all three cases that we have con- 0 s → l 1/pq2 pole [11]. This is true especially for D V but sidered we used two approaches: one with a two poles → to minor extent also for B V transitions. Since our shape for the vector form factor and single pole for the → model parameters are determined at q2 , this gives a axialformfactors,andsecondlytheusuallyassumedsin- max poorhandleonthedominatingeffectsintheoveralldecay gle pole behavior of all three relevant form factors. Our rate. Thus,accuratedeterminationofthemagnitudeand study indicates that the two pole shape for the V(q2) shape of the H helicity amplitude near q2 = 0 would form factor in D+ K¯∗0 transition is favored over the 0 → contribute much to clarifying this issue. single pole shape, when compared to the FOCUS result. We can summarize: we have investigated the predictions of the general H V form factor parametrization combined with HMχT→calculation for the D+ K¯∗0 Acknowledgments → semileptonic helicity amplitudes, recently determined by the FOCUS collaboration. In addition we have deter- We are thankful to D. Kim and J. Wiss from the FO- mined the helicity amplitudes for the D+ ρ0lν and CUScollaborationforsendingustheirdataandforhelp- l → 4 ing us understand it. This work is supported in part by ogy of the Republic of Slovenia. the Ministry of Higher Education, Science and Technol- [1] J. M. Link et al. (FOCUS) (2005), hep-ex/0509027. J. C. Raynal, Phys. Rev. D60, 014001 (1999), hep- [2] D.Becirevic and A.B. Kaidalov, Phys.Lett. B478, 417 ph/9812358. (2000), hep-ph/9904490. [8] M. Beneke and T. Feldmann, Nucl. Phys. B592, 3 [3] S. Fajfer and J. Kamenik, Phys. Rev. D72, 034029 (2001), hep-ph/0008255. (2005), hep-ph/0506051. [9] D. Ebert, R. N. Faustov, and V. O. Galkin, Phys. Rev. [4] S. 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