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The search for the fundamental constituents of matter has been pursued and studied since the dawn of civilization. As early as the fourth century BCE, Democritus, expanding the teachings of Leucippus, proposed small, indivisible entities called atoms, interacting with each other to form the Universe. Democritus was convinced of this by observing the environment around him. He observed, for example, how a collection of tiny grains of sand can make out smooth beaches. Today, following the lead set by Democritus more than 2500 years ago, at the heart of particle physics is the hypothesis that everything we can observe in the Universe is made of a small number of fundamental particles interacting with each other. In contrast to Democritus, for the last hundred years we have been able to perform experiments that probe deeper and deeper into matter in the search for the fundamental particles of nature. Today's knowledge is encapsulated in the Standard Model of particle physics, a model describing the fundamental particles and their interactions. It is within this model that the work in this thesis is presented. This work attempts to add to the understanding of the Standard Model by measuring the relative branching fraction of the charmed semileptonic decay D{sup +} {yields} {rho}{sup 0}{mu}{sup +}{nu} with respect to D{sup +} {yields} {bar K}*{sup 0} {mu}{sup +}{nu}. Many theoretical models that describe hadronic interactions predict the value of this relative branching fraction, but only a handful of experiments have been able to measure it with any precision. By making a precise measurement of this relative branching fraction theorists can distinguish between viable models as well as refine existing ones. In this thesis we presented the measurement of the branching fraction ratio of the Cabibbo suppressed semileptonic decay mode D{sup +} {yields} {rho}{sup 0}{mu}{sup +}{nu} with respect to the Cabibbo favored mode D{sup +} {yields} {bar K}*{sup 0} {mu}{sup +}{nu} using data collected by the FOCUS collaboration. We used a binned maximum log-likelihood fit that included all known semileptonic backgrounds as well as combinatorial and muonmisidentification backgrounds to extract the yields for both the signal and normalization modes. We reconstructed 320 {+-} 44 D{sup +} {yields} {rho}{sup 0}{mu}{sup +}{nu} events and 11372 {+-} 161 D{sup +} {yields} K{sup -}{pi}{sup +}{mu}{sup +}{nu} events. Taking into account the non-resonant contribution to the D{sup +} {yields} K{sup -}{pi}{sup +}{mu}{sup +}{nu} yield due to a s-wave interference first measured by FOCUS the branching fraction ratio is: {Lambda}(D{sup +} {yields} {rho}{sup 0}{mu}{sup +}{nu}) = 0.0412 {+-} 0.0057 {+-} 0.0040 (VII.1) where the first error is statistical and the second error is the systematic uncertainty. This represents a substantial improvement over the previous world average. More importantly, the new world average for {Lambda}(D{sup +}{yields}{rho}{sup 0}{mu}{sup +}{nu})/{Lambda}(D{sup +}{yields}{bar K}*{sup 0}{mu}{sup +}{nu}) along with the improved measurements in the electronic mode can be used to discriminate among different theoretical approaches that aim to understand the hadronic current involved in the charm to light quark decay process. The average of the electronic and muonic modes indicate that predictions for the partial decay width {Lambda}(D{sup +} {yields} {rho}{sup 0}{ell}{sup +}{nu}) and the ratio {Lambda}(D{sup +}{yields}{rho}{sup 0}{ell}{sup +}{nu})/{Lambda}(D{sup +}{yields}{bar K}*{sup 0}{ell}{sup +}{nu}) based on Sum Rules are too low. Using the same data used to extract {Lambda}(D{sup +}{yields}{rho}{sup 0}{mu}{sup +}{nu})/{Lambda}(D{sup +}{yields}{bar K}*{sup 0}{mu}{sup +}{nu}) we studied the feasibility of measuring the form factors for the D{sup +} {yields} {rho}{sup 0}{mu}{sup +}{nu} decay. We found that the need to further reduce the combinatorial and muon misidentification backgrounds left us with a much smaller sample of 52 {+-} 12 D{sup +} {yields} {rho}{sup 0}{mu}{sup +}{mu} events; not enough to make a statistically significant measurement of the form factors

Analysis of the charmed semileptonic decay D<sup>+</sup>→ ρ<sup>0</sup> μ<sup>+</sup> v