Measurement of the pseudoscalar decay constantfDsusingDs+→τ+ν,τ+→ρ+ν¯decays

Analyzing 600 pb{sup -1} of e{sup +}e{sup -} collisions at 4170 MeV center-of-mass energy with the CLEO-c detector, we measure the branching fraction B(D{sub s}{sup +}{yields}{tau}{sup +}{nu})=(5.52{+-}0.57{+-}0.21)% using the {tau}{sup +}{yields}{rho}{sup +}{nu} decay mode. Combining with other CLEO measurements of B(D{sub s}{sup +}{yields}{tau}{sup +}{nu}) we determine the pseudoscalar decay constant f{sub D{sub s}}=(259.7{+-}7.8{+-}3.4) MeV consistent with the value obtained from our D{sub s}{sup +}{yields}{mu}{sup +}{nu} measurement of (257.6{+-}10.3{+-}4.3) MeV. Combining these measurements we find a value of f{sub D{sub s}}=(259.0{+-}6.2{+-}3.0) MeV, that differs from the most accurate prediction based on unquenched lattice gauge theory of (241{+-}3) MeV by 2.4 standard deviations. We also present the first measurements of B(D{sub s}{sup +}{yields}K{sup 0}{pi}{sup +}{pi}{sup 0})=(1.00{+-}0.18{+-}0.04)%, and B(D{sub s}{sup +}{yields}{pi}{sup +}{pi}{sup 0}{pi}{sup 0})=(0.65{+-}0.13{+-}0.03)%, and measure a new value for B(D{sub s}{sup +}{yields}{eta}{rho}{sup +})=(8.9{+-}0.6{+-}0.5)%.