Using
2.93 fb−1 of
e+e− collision data collected with the BESIII detector at the center-of-mass energy of 3.773 GeV, we investigate the semileptonic decays
D+→π+π−ℓ+νℓ (
ℓ=e and
μ). The
D+→f0(500)μ+νμ decay is observed for the first time. By analyzing simultaneously the differential decay rates of
D+→f0(500)μ+νμ and
D+→f0(500)e+νe in different
ℓ+νℓ four-momentum transfer intervals, the product of the relevant hadronic form factor
f+f0(0) and the magnitude of the
c→d Cabibbo-Kobayashi-Maskawa matrix element
|Vcd| is determined to be
f+f0(0)|Vcd|=0.143±0.014stat±0.011syst for the first time. With the input of
|Vcd| from the global fit in the standard model, we determine
f+f0(0)=0.63±0.06stat±0.05syst. The absolute branching fractions of
D+→f0(500)(π+π−)μ+νμ and
D+→ρ(π+π−)0μ+νμ are determined as
(0.72±0.13stat±0.08syst)×10−3 and
(1.64±0.13stat±0.10syst)×10−3. Combining these results with those of previous BESIII measurements on their semielectronic counterparts from the same data sample, we test lepton flavor universality by measuring the branching fraction ratios
BD+→ρ0μ+νμ/BD+→ρ0e+νe=0.88±0.10 and
BD+→f0(500)μ+νμ/BD+→f0(500)e+νe=1.14±0.26, which are compatible with the standard model expectation. Published by the American Physical Society 2024