Observation of D+ → f0 (500)e+ νe and Improved Measurements of D → ρe+ νe

Using a data sample corresponding to an integrated luminosity of 2.93 fb(-1) recorded by the BESIII detector at a center-of-mass energy of 3.773 GeV, we present an analysis of the decays D-0 -> pi(-) pi(0) e(vertical bar) nu(e) and D-vertical bar -> pi(-) pi(vertical bar) e(vertical bar) nu(e). By performing a partial wave analysis, the pi(vertical bar) pi(-) S-wave contribution to D-vertical bar -> pi(-) pi(vertical bar) e(vertical bar) nu(e) is observed to be (25.7 +/- 1.6 +/- 1.1)% with a statistical significance greater than 10 sigma, besides the dominant P-wave contribution. This is the first observation of the S-wave contribution. We measure the branching fractions B(D0 -> rho(-) e(+) nu(e)) = (1.445 +/- 0.058 +/- 0.039) x 10(-3), B(D+ -> rho(0) e(+) nu(e)) = (1.860 +/- 0.070 +/- 0.061) x 10(-3), and B(D+ -> f(0)(500)e(+) nu(e),f(0)(500) -> pi(+) pi(-)) = (6.30 +/- 0.43 +/- 0.32) x 10(-4). An upper limit of B(D+ -> f(0)(980)e(+) nu(e),f(0)(980)-> pi(+) pi(-)) < 2.8 x 10(-5) is set at the 90% confidence level. We also obtain the hadronic form factor ratios of D -> rho e(+) nu(e) at q(2) = 0 assuming the single-pole dominance parametrization: r(V) ={[V(0)]/[A(1)(0)]}=1.695 +/- 0.083 +/- 0.051, r(2) = {[A(2)(0)]/[A(1)(0)]} = 0.845 +/- 0.056 +/- 0.039.