Measurement of the branching ratio of the decay D 0 → π − µ + ν relative to D 0 → K − µ + ν

We present a new measurement of the branching ratio of the Cabibbo suppressed decay D 0 → π − µ + ν relative to the Cabibbo favored decay D 0 → K − µ + ν and an improved measurement of the ratio . Our results are 0 . 074 ± 0 . 008 ± 0 . 007 for the branching ratio and 0 . 85 ± 0 . 04 ± 0 . 04 ± 0 . 01 for the form factor ratio, respectively.


Introduction
Semileptonic decays provide the advantage of having factorizable weak currents in the Hamiltonian which allows for a clean theoretical description.The hadronic current can be described in terms of two form factors, f + (q 2 ) and f − (q 2 ) which are functions only of the lepton-neutrino invariant mass squared, q 2 .Assuming a pole dominance parameterization of the form factors, we present a parametric analysis of the pseudoscalar semileptonic decays D 0 → π − µ + ν and D 0 → K − µ + ν from the FOCUS experiment.This paper concentrates on the relative branching ratio and the form factor ratio of the Cabibbo suppressed decay relative to the Cabibbo favored mode.
Since the efficiency tends to have a non-negligible q 2 dependence (see Figure 3), we allow the pole masses and the ratio f − (0)/f + (0) to vary freely in the fit.The results and description of the detailed analysis of the pole masses and f − (0)/f + (0) are included in another paper [1] along with a non-parametric analysis of the high statistics decay D 0 → K − µ + ν .
We report a new measurement for the branching ratio Γ(π − µ + ν)/Γ(K − µ + ν) in agreement with recent results from the CLEO collaboration [2,3].These results indicate a lower value for this branching ratio than the one reported in the PDG [4].We also report a new measurement of the form factor ratio with greatly improved errors with respect to existing measurements and compare it to recent theoretical predictions from an unquenched Lattice QCD calculation [5,6] .

Data Selection
This analysis is based on data collected by the FOCUS experiment during the 1996-97 fixed target run at Fermilab.FOCUS is a photoproduction experiment which collected a large sample of charm decays produced in the interactions of a photon beam [7] with an average energy of ∼ 180 GeV on a BeO segmented target.The FOCUS spectrometer [8,9,10,11] is equipped with a 16 plane silicon strip vertex detector; 4 planes are interleaved with the targets and 12 planes are located downstream of the target area.Momentum analysis is accomplished by two magnets with opposite polarities and 5 multiwire proportional chambers.Three multi-cell threshold Čerenkov counters provide charged particle identification.A muon counter located at the end of the spectrometer is responsible for muon identification.
We reconstruct the semileptonic decays D 0 → π − µ + ν and D 0 → K − µ + ν requiring a D * + -tag where the D * + is reconstructed in the D 0 π + final state. 1henever possible we apply identical selection criteria to both decay modes to reduce systematic effects.As the decay D 0 → π − µ + ν has more background and less statistics, the selection cuts have been optimized for this mode.The signal and normalization samples are selected requiring two opposite charged tracks form a good vertex with a confidence level greater than 1%.One of the two tracks from the D 0 decay vertex must be identified as a muon from the inner muon detector with a confidence level greater than 1% and must have momentum greater than 10 GeV/c.To suppress pion and kaon in-flight decays, this track is required to have a consistent momentum when measured in the first and second magnets.The other track must satisfy a Čerenkov re-quirement based on the value of the negative log-likelihood W for a given hypothesis: in the D 0 → π − µ + ν mode, the pion must be favored with respect to the kaon hypothesis by at least 3 units of likelihoods (W (K) − W (π) > 3); in the case of K − µ + ν the kaon must be favored over the pion hypothesis by 3 units of likelihoods (W (π) − W (K) > 3).To reduce non-charm background, the candidate hadron must have a momentum greater than 14 GeV/c.The primary vertex is found after excluding the candidate tracks from the D 0 decay vertex; the remaining tracks are used to form candidate vertices.Of these vertices we choose the one with highest multiplicity and we break ambiguities by picking the most upstream vertex as the primary vertex.This vertex is required to be isolated from other tracks in the silicon strip vertex detector by requiring that the confidence level of any another track not used in the determination of the primary or decay vertex be less than 1%.For each hadron-lepton combination that satisfies the above requirements, another track coming from the primary vertex must be found as the candidate "soft" pion from the D * + → D 0 π + s .The π + s candidate must not have the pion hypothesis strongly disfavored over all other particle hypotheses from the Čerenkov system (min{W (e), W (K), W (p)} − W (π) > −6).It must also have a momentum greater than 2.5 GeV/c.To suppress backgrounds from decays where a final state particle is lost (usually π 0 ), such as K − π + π 0 , K − π + π 0 π 0 , ρ − µ + ν and Kπµ + ν2 , we place a lower cut on the hadron-lepton invariant mass (visible mass) of 1.0 GeV/c 2 .Contamination from D 0 → K − π + is eliminated by requiring the visible mass to be less than 1.7 GeV/c 2 .Since the neutrino is not reconstructed, the resultant smearing effects on the resolution play an important factor in this analysis.Rather than using the standard neutrino closure resulting in a two-fold ambiguity on the D 0 momentum, we take advantage of the D * + -tag by boosting the final state particles in the hadron-lepton center of mass frame.By constraining the K − µ + ν (π − µ + ν ) mass to the D 0 mass and the K − µ + ν π + s (π − µ + ν π + s ) mass to the D * + mass, we are able to determine the angle between the neutrino3 and the π + s direction.We then sample the azimuthal angle and choose the one that gives the direction of the D 0 most consistent with pointing to the primary vertex.

Analysis
The fit to the data is designed to constrain the background in the π − µ + ν sample and to supply information about the pole mass and form factors. plish these goals we perform fits on two-dimensional distributions where the free parameters are the signal and background yields.All the fits are binned maximum likelihood fit where the likelihood is defined as: where f ij (n ij ) is the number of expected (observed) events in the bin ij.First, a fit of q 2 and D * + − D 0 mass difference is performed to establish the amount of non-peaking background (Fig. 1). 4 We next place a mass cut on the D * + − D 0 mass difference of less than 0.154 GeV/c 2 to reduce the background and to obtain more reliable results for parameters such as pole masses and form factors.A fit is then made to the two-dimensional distribution q 2 vs. cos θ ℓ (where cos θ ℓ is defined as the cosine of the angle between the neutrino direction and the D 0 direction in the rest frame of the lepton-neutrino system).
The fit is first performed on the K − µ + ν sample and the results from this fit are used to set the background from K − µ + ν and Kπµ + ν in the π − µ + ν sample.
In the fit to the K − µ + ν distribution we make use of the recent vector to pseudoscalar branching ratio measurement Γ(D + → Kπµ + ν)/Γ(D + → K 0 µ + ν) = 0.63 ± 0.05 [12] in the form of a penalty term added to the log-likelihood as shown in Eq. 2: where we assume isospin invariance to relate D + and D 0 decays.The likelihood L is constructed using the expected number of events is each ij bin of the twodimensional distribution given by: where in Eqs. 2 and 3 the fit parameters Y α are the fitted yields, S α are the normalized shapes obtained from Monte Carlo and ǫ the reconstruction efficiency.We define the cc component as the background obtained from a high statistics charm-charmbar Monte Carlo sample after removing the modes handled specifically in Eq. 3 (and 5).
In a similar way we fit the π − µ + ν distribution.We use the branching ratio Γ(D 0 → ρ − µ + ν)/Γ(D 0 → Kπµ + ν) = 0.086 ± 0.0105 to constrain the background in the fit as shown in Eq. 4: The expected number of events in each two-dimensional bin used to construct the likelihood is: where Y 0 K − µ + ν and Y 0 K − π 0 µ + ν in Eq. 5 are fixed to the results obtained from the fit to the K − µ + ν data (Eq.3).The symbol (X → Y ) means that a hadron X is misidentified as Y .
To measure pole masses and the form factor ratio η ≡ f K − (0)/f K + (0) we apply an event-by-event weighting procedure [14].This is achieved by re-weighting each Monte Carlo event according to the ratio of the probability that the event was generated with a pole mass M ′ pole and a form factor ratio η ′ relative to the probability that the event was generated with the default values M D * s (M D * for πµν ) and η 0 . 6The relative efficiencies of the decays D 0 → π − µ + ν and D 0 → K − µ + ν are defined as the ratio of the reconstructed and generated Monte Carlo events.At each fit iteration these efficiencies change as a function of the pole masses and η values.
The weight W i for an event with q 2 = q 2 i is given by the equation: where the intensity is: and the normalization is determined by: Events Fig. 2. Fit projections for π − µ + ν and D 0 → K − µ + ν.The fit is performed on a two-dimensional distribution of q 2 and cos θ ℓ .In the D 0 → π − µ + ν, the peaking background contribution is defined as the sum of the contributions from D 0 → K − µ + ν, ρ − µ + ν and Kπµ + ν.
The form factor f + (M pole ; q 2 ) is assumed to have the following q 2 dependence: and g(η) can be written in terms of three kinematic coefficients A, B and C:7 From the fit to the π − µ + ν (K − µ + ν) distributions (Fig. 2) we find 288 ± 29 2 Fig. 3. Reconstruction efficiency as a function of the q 2 for π − µ + ν (top) and π − µ + ν relative to the Cabibbo allowed decay D 0 → K − µ + ν to be: From the same fits we find M π = 1.91 +0.30 −0.15 and M K = 1.93 +0.05 −0.04 for the π − µ + ν and the K − µ + ν pole masses respectively.We also measure the ra- A detailed description of the pole mass results has been included in Ref. [1].
Using the yields from the fit it is possible to obtain the ratio of the form factors f π + (0)/f K + (0).In order to do this we compute a numerical integration of the differential decay rate modulated by the reconstruction efficiency as a function of the q 2 [15].This efficiency is found by sampling the q 2 Monte Carlo distribution and dividing the reconstructed events by the generated events in each bin.The resultant distribution is then fit to a third degree polynomial (Fig. 3) which is used in the computation of the integral.We quote the result: Applying the unitarity constraints on the CKM matrix elements [4] we use the value | V cd Vcs | 2 = 0.051 ± 0.001 in Eq. 12 and measure the ratio f π + (0)/f K + (0) to be:

Systematic Studies
Several studies have been performed to search for possible systematic uncertainties.The fitting procedure was tested on a Monte Carlo set whose size is roughly 20 times the FOCUS data set and we verified that the fit returned the input values used in our simulation.
We checked for possible biases as well as the accuracy of our statistical error by performing a fit on fluctuated data distributions multiple times and comparing the mean and width of the distribution of the fit results to our measurement.
We found that we have to add a 0.005 contribution to the systematic error to compensate for K − µ + ν and Kπµ + ν contributions that were not allowed to float in the π − µ + ν fit.We also performed an analogous study using the fit function as the parent distribution to establish how well our fit function described the data.We compared the likelihood obtained from our measurement to the distribution of the likelihoods from the fluctuated fit function.
We found good agreement indicating that our fit function well represents the data.
We investigated the stability of our results by changing a variety of selection criteria: the significance of separation between the primary and secondary vertex, muon identification, track momenta, visible mass cut, and Čerenkov identification.We found no significant change in our results and assign a systematic uncertainty of 0.003 on the branching ratio due to cut variations.This number is found by computing the variance of this set of results.
We further investigated fit variations by using a different approach in which we fit the q 2 and D * + − D 0 mass difference.Rather than fitting the K − µ + ν distribution first, this fit was performed simultaneously on the π − µ + ν and K − µ + ν samples.The results are nearly identical to the results obtained from the fit to q 2 and cos θ ℓ .Other fit variations include changing the bin size.By computing the variance of these a priori likely results, we assigned a systematic uncertainty of 0.004 from fit variations.
Since the Monte Carlo is used to determine the amount of K − µ + ν background in the π − µ + ν sample, we are sensitive to the simulated misidentification rate.
Using | V cd Vcs | 2 = 0.051 ±0.001 from unitarity constraints, we find the form factor ratio to be: = 0.85 ± 0.04 (stat.)± 0.04 (sys.) ± 0.01 (CKM) (16) where the last error (CKM) corresponds to the uncertainty on the ratio |V cd /V cs |.We compare our results to the measurement reported by the CLEO collaboration in Ref. [2] where they report the branching ratio of D 0 → π − e + ν relative to D 0 → K − e + ν to be 0.082 ± 0.006 ± 0.005 and the form factor ratio f π + (0) f K + (0) = 0.86 ± 0.07 +0.06 −0.04 ± 0.01.We also compare our branching ratio result to the recent measurement from absolute branching ratios for D 0 → π − e + ν and D 0 → K − e + ν from CLEO-c [3] where they report a relative branching ratio of 0.070 ± 0.007 ± 0.003.Our results are consistent with both of these new measurements.Further, we report an improved measurement of in good agreement with SU(3) breaking expected in recent lattice QCD calculations where they quote a form factor ratio value of 0.85 ± 0.05 [5] and 0.86 ± 0.05 ± 0.11 [6].

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events.Correcting for the relative Monte Carlo efficiency we find the branching ratio for the Cabibbo suppressed decay D 0 →