CLNS 94/1306 CLEO 94–24 January 13, 1995

Form Factor Ratio Measurement in Λ_{c}^{+} → Λe^{+}ν_{e}

G. Crawford,^{1} C. M. Daubenmier,^{1} R. Fulton,^{1} D. Fujino,^{1} K.K. Gan,^{1} K. Honscheid,^{1} H. Kagan,^{1} R. Kass,^{1} J. Lee,^{1} M. Sung,^{1} C. White,^{1} A. Wolf,^{1} M.M. Zoeller,^{1} F. Butler,^{2} X. Fu,^{2} B. Nemati,^{2} W.R. Ross,^{2} P. Skubic,^{2} M. Wood,^{2} M. Bishai,^{3} J. Fast,^{3} E. Gerndt,^{3}

J.W. Hinson,^{3} R.L. McIlwain,^{3} T. Miao,^{3} D.H. Miller,^{3} M. Modesitt,^{3} D. Payne,^{3} E.I. Shibata,^{3} I.P.J. Shipsey,^{3} P.N. Wang,^{3} M. Battle,^{4} J. Ernst,^{4} L. Gibbons,^{4} Y. Kwon,^{4} S. Roberts,^{4} E.H. Thorndike,^{4} C.H. Wang,^{4} J. Dominick,^{5} M. Lambrecht,^{5} S. Sanghera,^{5} V. Shelkov,^{5} T. Skwarnicki,^{5} R. Stroynowski,^{5} I. Volobouev,^{5} G. Wei,^{5} P. Zadorozhny,^{5} M. Artuso,^{6} M. Gao,^{6} M. Goldberg,^{6} D. He,^{6} N. Horwitz,^{6} G.C. Moneti,^{6} R. Mountain,^{6} F. Muheim,^{6} Y. Mukhin,^{6} S. Playfer,^{6} Y. Rozen,^{6} S. Stone,^{6} X. Xing,^{6} G. Zhu,^{6} J. Bartelt,^{7} S.E. Csorna,^{7} Z. Egyed,^{7} V. Jain,^{7} D. Gibaut,^{8} K. Kinoshita,^{8} P. Pomianowski,^{8} B. Barish,^{9}

M. Chadha,^{9} S. Chan,^{9} D.F. Cowen,^{9} G. Eigen,^{9} J.S. Miller,^{9} C. O’Grady,^{9} J. Urheim,^{9} A.J. Weinstein,^{9} M. Athanas,^{10} W. Brower,^{10} G. Masek,^{10} H.P. Paar,^{10} J. Gronberg,^{11}

C.M. Korte,^{11} R. Kutschke,^{11} S. Menary,^{11} R.J. Morrison,^{11} S. Nakanishi,^{11} H.N. Nelson,^{11}

T.K. Nelson,^{11} C. Qiao,^{11} J.D. Richman,^{11} A. Ryd,^{11} D. Sperka,^{11} H. Tajima,^{11} M.S. Witherell,^{11} M. Procario,^{12} R. Balest,^{13} K. Cho,^{13} W.T. Ford,^{13} D.R. Johnson,^{13} K. Lingel,^{13} M. Lohner,^{13} P. Rankin,^{13} J.G. Smith,^{13} J.P. Alexander,^{14} C. Bebek,^{14}

K. Berkelman,^{14} K. Bloom,^{14} T.E. Browder,^{14}^{∗} D.G. Cassel,^{14} H.A. Cho,^{14} D.M. Coffman,^{14}

D.S. Crowcroft,^{14} P.S. Drell,^{14} D.J. Dumas,^{14} R. Ehrlich,^{14} P. Gaidarev,^{14} M. Garcia-Sciveres,^{14} B. Geiser,^{14} B. Gittelman,^{14} S.W. Gray,^{14} D.L. Hartill,^{14} B.K. Heltsley,^{14} S. Henderson,^{14} C.D. Jones,^{14} S.L. Jones,^{14} J. Kandaswamy,^{14}

N. Katayama,^{14} P.C. Kim,^{14} D.L. Kreinick,^{14} G.S. Ludwig,^{14} J. Masui,^{14} J. Mevissen,^{14} N.B. Mistry,^{14} C.R. Ng,^{14} E. Nordberg,^{14} J.R. Patterson,^{14} D. Peterson,^{14} D. Riley,^{14} S. Salman,^{14} M. Sapper,^{14} F. Würthwein,^{14} P. Avery,^{15} A. Freyberger,^{15} J. Rodriguez,^{15} S. Yang,^{15} J. Yelton,^{15} D. Cinabro,^{16} T. Liu,^{16} M. Saulnier,^{16} R. Wilson,^{16} H. Yamamoto,^{16}

T. Bergfeld,^{17} B.I. Eisenstein,^{17} G. Gollin,^{17} B. Ong,^{17} M. Palmer,^{17} M. Selen,^{17} J. J. Thaler,^{17} K.W. Edwards,^{18} M. Ogg,^{18} A. Bellerive,^{19} D.I. Britton,^{19} E.R.F. Hyatt,^{19} D.B. MacFarlane,^{19} P.M. Patel,^{19} B. Spaan,^{19} A.J. Sadoff,^{20} R. Ammar,^{21} P. Baringer,^{21} A. Bean,^{21} D. Besson,^{21} D. Coppage,^{21} N. Copty,^{21} R. Davis,^{21} N. Hancock,^{21} M. Kelly,^{21}

S. Kotov,^{21} I. Kravchenko,^{21} N. Kwak,^{21} H. Lam,^{21} Y. Kubota,^{22} M. Lattery,^{22} M. Momayezi,^{22} J.K. Nelson,^{22} S. Patton,^{22} R. Poling,^{22} V. Savinov,^{22} S. Schrenk,^{22} R. Wang,^{22} M.S. Alam,^{23} I.J. Kim,^{23} Z. Ling,^{23} A.H. Mahmood,^{23} J.J. O’Neill,^{23}

H. Severini,^{23} C.R. Sun,^{23} and F. Wappler^{23}

(CLEO Collaboration)

1Ohio State University, Columbus, Ohio, 43210 2University of Oklahoma, Norman, Oklahoma 73019 3Purdue University, West Lafayette, Indiana 47907 4University of Rochester, Rochester, New York 14627 5Southern Methodist University, Dallas, Texas 75275 6Syracuse University, Syracuse, New York 13244 7Vanderbilt University, Nashville, Tennessee 37235

8Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 24061 9California Institute of Technology, Pasadena, California 91125 10University of California, San Diego, La Jolla, California 92093 11University of California, Santa Barbara, California 93106 12Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213 13University of Colorado, Boulder, Colorado 80309-0390 14Cornell University, Ithaca, New York 14853 15University of Florida, Gainesville, Florida 32611 16Harvard University, Cambridge, Massachusetts 02138 17University of Illinois, Champaign-Urbana, Illinois, 61801

18Carleton University, Ottawa, Ontario K1S 5B6 and the Institute of Particle Physics, Canada 19McGill University, Montréal, Québec H3A 2T8 and the Institute of Particle Physics, Canada

20Ithaca College, Ithaca, New York 14850 21University of Kansas, Lawrence, Kansas 66045 22University of Minnesota, Minneapolis, Minnesota 55455 23State University of New York at Albany, Albany, New York 12222

Abstract

The angular distributions of the decay Λ_{c}^{+} → Λe^{+}ν_{e} have been studied using the CLEO II detector. By performing a 3-dimensional maximum likelihood fit, the form factor ratio, R = f_{2}/f_{1}, is determined to be −0.25 ± 0.14 ± 0.08. The decay asymmetry parameter of the Λ_{c} averaged over q^{2} is calculated to be α_{Λ_{c}} = −0.82_{−0.06}^{+0.09} _{−0.03}^{+0.06}.

∗Permanent address: University of Hawaii at Manoa

Charm semileptonic decays allow a determination of the magnitude of the form factors which parameterize the hadronic current because the Cabibbo-Kobayashi-Maskawa (CKM) matrix element |V_{cs}| is known from unitarity [1]. Within Heavy Quark Effective Theory (HQET) [2], Λ-type baryons are more straightforward to treat than mesons as they consist of a heavy quark and a spin and isospin zero light diquark. This simplicity allows for more reliable predictions concerning heavy quark to light quark transitions [3] [4] than is the case for mesons. A measurement of the form factors in Λ_{c}^{+} → Λe^{+}ν_{e} will help the future determination of the matrix element V_{ub} using Λ_{b} decays since HQET relates the form factors in Λ_{c} decay to those governing Λ_{b} semileptonic decays.

In the limit of negligible lepton mass, the semileptonic decay of a charmed baryon (1/2^{+} → 1/2^{+}) is usually parameterized in terms of four form factors: two axial form factors F_{1}^{A} and F_{2}^{A}, and two vector form factors F_{1}^{V} and F_{2}^{V}. These form factors are functions of q^{2}, the invariant mass squared of the virtual W. In the zero lepton mass approximation, the decay may be described in terms of helicity amplitudes H_{λ_{Λ}λ_{W}} = H_{λ_{Λ}λ_{W}}^{V} + H_{λ_{Λ}λ_{W}}^{A}, where λ_{Λ} and λ_{W} are the helicities of the Λ and W. The helicity amplitudes are related to the form factors by [4]:

\sqrt{q^{2}}H_{\frac{1}{2} 0}^{V} = \sqrt{Q_{−}}[(M_{Λ_{c}} + M_{Λ})F_{1}^{V} − q^{2}F_{2}^{V}]
H_{\frac{1}{2} 1}^{V} = \sqrt{2Q_{−}}[−F_{1}^{V} + (M_{Λ_{c}} + M_{Λ})F_{2}^{V}]
\sqrt{q^{2}}H_{\frac{1}{2} 0}^{A} = \sqrt{Q_{+}}[(M_{Λ_{c}} − M_{Λ})F_{1}^{A} + q^{2}F_{2}^{A}]
H_{\frac{1}{2} 1}^{A} = \sqrt{2Q_{+}}[−F_{1}^{A} − (M_{Λ_{c}} − M_{Λ})F_{2}^{A}]  (1)

where Q_{±} = (M_{Λ_{c}} ± M_{Λ})^{2} − q^{2}. The remaining helicity amplitudes can be obtained using the parity relations: H_{−λ_{Λ}−λ_{W}}^{V (A)} = +(−)H_{λ_{Λ}λ_{W}}^{V (A)} . In terms of the helicity amplitudes, the decay angular distribution can be written as [4]:

Γ_{S} = \frac{dΓ}{dq^{2}d cos Θ_{W} d cos Θ_{Λ}} = B(Λ → pπ)\frac{1}{2} \frac{G_{F}^{2}}{(2π)^{4}} |V_{cs}|^{2} \frac{q^{2}P}{24M_{Λ_{c}}^{2}}

×\{\frac{3}{8}(1 − cos Θ_{W})^{2}|H_{\frac{1}{2} 1}|^{2}(1 + α_{Λ} cos Θ_{Λ}) + \frac{3}{8}(1 + cos Θ_{W})^{2}|H_{− \frac{1}{2} −1}|^{2}(1 − α_{Λ} cos Θ_{Λ})

+\frac{3}{4} sin^{2} Θ_{W} [|H_{\frac{1}{2} 0}|^{2}(1 + α_{Λ} cos Θ_{Λ}) + |H_{− \frac{1}{2} 0}|^{2}(1 − α_{Λ} cos Θ_{Λ})]\}  (2)

where G_{F} is the Fermi coupling constant, V_{cs} is the CKM matrix element, P is the Λ momentum in the Λ_{c} rest frame, Θ_{Λ} is the angle between the momentum vector of the proton in the Λ rest frame and the Λ momentum in the Λ_{c} rest frame, Θ_{W} is the angle between the momentum vector of the positron in the W rest frame and the Λ momentum in the Λ_{c} rest frame, and α_{Λ} is the Λ → pπ decay asymmetry parameter measured to be 0.64 ± 0.01 [1].

Within the framework of HQET the heavy flavor and spin symmetries imply relations among the form factors which reduce their number to one when the decay involves only heavy quarks. In this paper we follow reference [4] where it is argued that the HQET formalism may also work for Λ_{c}^{+} → Λe^{+}ν_{e}. Treating the c quark as a heavy quark, two independent form factors f_{1} and f_{2} are required to describe the hadronic current. The relationship between these form factors and the standard form factors are: F_{1}^{V} (q^{2}) = −F_{1}^{A}(q^{2}) = f_{1}(q^{2})+\frac{M_{Λ}}{M_{Λc}} f_{2}(q^{2}) and F_{2}^{V} (q^{2}) = −F_{2}^{A}(q^{2}) = \frac{1}{M_{Λc}} f_{2}(q^{2}). In general f_{2} is expected to be negative and smaller in magnitude than f_{1}. If the strange quark is treated as heavy, f_{2} is zero.

In order to extract the form factor ratio R = f_{2}/f_{1} from a fit to Γ_{S} an assumption must be made about the q^{2} dependence of the form factors. We follow the model of Körner and Krämer (KK) [4] who use the dipole form, f(q^{2}) = f(q_{max}^{2})(1 − q_{max}^{2}/m_{ff}^{2})^{2}/(1 − q^{2}/m_{ff}^{2})^{2}, where the pole mass is chosen to be m_{ff} = 2.11GeV/c^{2} . With the current statistical power, the angular distributions are not sensitive to the q^{2} dependence of the form factors.

The data sample used in this study was collected with the CLEO II detector [5] at the Cornell Electron Storage Ring (CESR). The integrated luminosity consists of 3.0 fb^{−1} taken at and just below the Υ(4S) resonance, corresponding to approximately 4 million e^{+}e^{−} → c\overline{c} events. We search for the decay Λ_{c}^{+} → Λe^{+}ν_{e} in e^{+}e^{−} → cc̄ events by detecting a Λe^{+} pair with invariant mass in the range m_{Λ} < m_{Λe^{+}} < m_{Λ_{c}} [6]. The minimum allowed momentum for positron candidates is 0.7 GeV/c. The Λ is reconstructed through its decay to pπ. The dE/dx measurement of the proton candidate is required to be consistent with the expected value. We require the momentum of the pπ pair to be greater than 0.8 GeV/c in order to reduce combinatorial background. These Λ candidates are then combined with positrons, and the sum of Λ and e^{+} momentum, p_{Λe^{+}}, is required to be greater than 1.4 GeV/c. This cut reduces our dependence on the shape of the Λ_{c} fragmentation function at low momentum, which is poorly known.

The number of events passing these requirements is 1076, of which 137±12 are consistent with fake Λ background, 120 ± 36 are consistent with fake positron background and 106 ± 21 with Ξ_{c} → Ξe^{+}ν_{e}, Ξ → Λ(π) feedthrough. Details of the background estimates can be found in a previous paper [7]. The sidebands of the pπ invariant mass distribution are used to estimate the fake Λ background in this study [8]. The fake positron background is estimated using events containing a Λ candidate, by multiplying the number of tracks satisfying our kinematic criteria which are not positively identified as a positron with the probability for a hadron to be identified as positron. The background from Ξ_{c} → Ξe^{+}ν_{e} decays is estimated using the result of a previous analysis [9]. With the selection criteria described above, we find no evidence for Λe^{+}ν_{e} final states in which there are additional Λ_{c}^{+} decay products [7]. In Fig. 1, we show the m_{Λe^{+}} distribution after Λ sideband subtraction and comparisons between the data and Monte Carlo (MC) distributions.

Calculating kinematic variables requires knowledge of the Λ_{c} momentum which is unknown due to the undetected neutrino. The direction of the Λ_{c} is approximated by the thrust axis of the event. The magnitude of the Λ_{c} momentum is then obtained by solving the equation P⃗_{Λ_{c}}^{2} = (P⃗_{Λ} + P⃗_{e} + P⃗_{ν_{e}})^{2}. We determine the best solution by MC simulation following the procedure described in detail in [7]. After the Λ_{c} momentum is estimated, the three kinematic variables are obtained by working in the Λ_{c} center-of-mass frame. The resolutions on q^{2}/q_{max}^{2}, cos Θ_{Λ} and cos Θ_{W} determined by MC are 0.25, 0.25 and 0.2 respectively.

Using q^{2}/q_{max}^{2}, cos Θ_{Λ} and cos Θ_{W}, we perform a 3-dimensional unbinned maximum likelihood fit in a manner similar to reference [10]. This technique enables a multi-dimensional likelihood fit to be performed to variables modified by experimental acceptance and resolution, and is necessary for this analysis due to the substantial smearing of the kinematic variables. The essence of the method is to determine the probability density function by using the population of appropriately weighted MC events in the three dimensional kinematic space. This is accomplished by generating one high statistics sample of MC events with a known value of the form factor ratio R and corresponding known values of the three kinematic variables q^{2}/q_{max}^{2}, cos Θ_{Λ}, and cos Θ_{W} for each event. The generated events are then processed through the full detector simulation, offline analysis programs and selection criteria. Using the generated kinematic variables, the accepted MC events are weighted by the ratio of the decay distribution for the trial values of R to that of the generated distribution. The accepted MC events are now, therefore, distributed according to the probability density corresponding to the trial values of R. By such weighting, a likelihood may be evaluated for each data event for different values of the form factor ratio, and a fit performed. The probability for each event is determined by sampling this distribution using a search volume around each data point. The volume size is chosen so that the systematic effect from finite search volumes is small and the required number of MC events is not prohibitively high.

The Λ_{c}^{+} → Λe^{+}ν_{e} sample has signal to background in the approximate ratio 2:1. Background is incorporated into the fitting technique by constructing the log-likelihood function lnL = ∑_{i=1}^{N} ln(P_{S}Γ_{S} + P_{B}Γ_{B}), where N is the number of events in the signal region, P_{S} and P_{B} are the probabilities that events in this region are signal and background respectively, Γ_{S} is the signal shape modeled from signal MC and Γ_{B} is the background shape modeled by the background samples. Fake positron background is modeled by the fake positron data sample. Feedthrough background from Ξ_{c} → Ξe^{+}ν_{e} is modeled by the MC sample which is generated according to the HQET consistent KK model. Fake Λ background is modeled using data events in the sidebands of the pπ invariant mass distribution.

Using the above method we find R = −0.25 ± 0.14 where the error is statistical. The confidence level of the fit is determined to be 15% using the method of nearest neighbors [11]. Fig. 2 and Fig. 3 show the projections of the q^{2}/q_{max}^{2}, cos Θ_{Λ} and cos Θ_{W} distributions for the data and for the fit. Correlations between kinematic variables for the data and fit are shown in Fig. 4.

We have considered the following sources of systematic error and give our estimate of their magnitude in parentheses. The statistical error in the MC sample is estimated by dividing the MC into three equal parts and repeating the fit (0.02). The error due to the search volume size is determined by varying the size of the search volume (0.04). The error due to background normalization and shape are determined by varying the estimated number of the background events in the signal region by one standard deviation and by using different background samples (0.03). The error associated with the ratio of form factors for the decay of Ξ_{c} → Ξe^{+}ν_{e} is determined by varying this ratio and repeating the fit (0.04). As this measurement is insensitive to the production polarization, P, of the Λ_{c} no systematic error has been included from this source [12]. The sensitivity to the q^{2} dependence is also investigated by using functional forms other than the dipole form and by using a modified dipole form which gives mildly different q^{2} dependences for f_{1} and f_{2}. The variation of the value of R derived from these different fits is found to be small and no systematic error has been included. The error associated with the uncertainty in the Λ_{c} fragmentation function is estimated by varying this function (0.01). The error associated with MC modeling of slow pions from Λ decay is obtained by varying this efficiency according to our understanding of the CLEO II detector (0.01). To estimate the error associated with the technique of incorporating background into the fit, two other methods of computing the log-likelihood ratio are used: direct background subtraction and a hybrid of the method outlined earlier and direct subtraction. The difference between the central values found by these three methods (0.03) is taken as the systematic error. Adding all sources of systematic error in quadrature we find R = −0.25 ± 0.14 ± 0.08.

Using the value of R obtained here and the HQET consistent KK model, the mean value of the decay asymmetry parameter of Λ_{c}^{+} → Λe^{+}ν_{e} is calculated to be α_{Λ_{c}} = −0.82_{−0.06}^{+0.09} _{−0.03}^{+0.06}. The average value of q^{2} in the KK model (R = −0.25) is 0.7 (GeV/c^{2})^{2}. This is consistent with our previous measurement of α_{Λ_{c}} determined from the cos Θ_{Λ} distribution [7] but has a smaller error since it is obtained using more experimental information and additional data.

In conclusion, using a 3-dimensional maximum likelihood fit the angular distributions of Λ_{c}^{+} → Λe^{+}ν_{e} have been studied and the form factor ratio, R = f_{2}/f_{1}, is found to be −0.25 ± 0.14 ± 0.08. At the current level of experimental sensitivity the hadronic matrix element in the semileptonic decay of the Λ_{c} baryon is adequately described by two form factors as predicted by HQET.

We gratefully acknowledge the effort of the CESR staff in providing us with excellent luminosity and running conditions. This work was supported by the National Science Foundation, the U.S. Dept. of Energy, the Heisenberg Foundation, the SSC Fellowship program of TNRLC, Natural Sciences and Engineering Research Council of Canada, and the A.P. Sloan Foundation.

REFERENCES

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[6] Throughout this paper charge conjugate modes are implicitly included.

[7] T. Bergfeld et al. (CLEO), Phys. Lett. B323 219 (1994).

[8] In ref [7] the fake Λ background was allowed for by fitting the pπ invariant mass. The Λ sideband method used in this work gives a consistent estimate of the fake Λ background.

[9] J. Alexander et al. (CLEO), CLNS 94 /1288 (submitted to Phys. Rev. Lett. (1994)).

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[12] Parity conservation in the electromagnetic and strong interaction requires Λ_{c} production polarization to be normal to the production plane, and charge conjugation invariance requires the sign of the polarization to be the same for particle and anti-particle. Since α_{Λ_{c}} is CP odd, (α_{Λ_{c}}P) is CP odd so that averaged over charge conjugate states the net polarization is zero.

FIGURES

FIG. 1. Invariant Λe^{+} mass for right sign combinations. The points with error bars are data after subtraction of the contribution of fake lambdas estimated using the pπ invariant mass sidebands. The dotted line and the open circles show the fake positron and the feedthrough backgrounds respectively. The dashed line shows the MC prediction for Λ_{c}^{+} → Λe^{+}ν_{e} normalized to the data after subtraction of the backgrounds. The solid line shows the sum of the MC prediction and the backgrounds.

FIG. 2. Projections of the data (points with error bars) and the fit (solid histogram) for q^{2}/q_{max}^{2}, cos Θ_{Λ} and cos Θ_{W}. The dashed lines show the background distributions.

FIG. 3. Projections of the data (points with error bars) and the fit (solid histogram) onto cos Θ_{Λ} and cos Θ_{W} for different q^{2}/q_{max}^{2} regions. The dashed lines show the background distributions.

FIG. 4. Correlations between q^{2}/q_{max}^{2}, cos Θ_{Λ} and cos Θ_{W} for data (left) and the fit (right). The size of the box in the plots is proportional to the number of events.