Context: Quantifying the gas content inside the dust gaps of transition disks is important to establish their origin. Aims: We seek to constrain the surface density of warm gas in the disk of HD 139614, a Herbig Ae star with a transition disk exhibiting a dust gap from 2.3 to 6 AU. Methods: We have obtained ESO/VLT CRIRES high-resolution spectra of CO ro-vibrational emission. We derived constraints on the disks structure by modeling the line-profiles, the spectroastrometric signal, and the rotational diagrams using flat Keplerian disk models. Results: We detected v=1-0 12CO, 2-1 12CO, 1-0 13CO, 1-0 C18O, and 1-0 C17O ro-vibrational lines. 12CO v=1-0 lines have an average width of 14 km/s, Tgas of 450 K and an emitting region from 1 to 15 AU. 13CO and C18O lines are on average 70 and 100 K colder, 1 and 4 km/s narrower, and are dominated by emission at R>6 AU. The 12CO v=1-0 line-profile indicates that if there is a gap in the gas it must be narrower than 2 AU. We find that a drop in the gas surface density (delta_gas) at R<5-6 AU is required to simultaneously reproduce the line-profiles and rotational diagrams of the three CO isotopologs. Delta_gas can range from 10^-2 to 10^-4 depending on the gas-to-dust ratio of the outer disk. We find that at 1<R<6 AU the gas surface density profile is flat or increases with radius. We derive a gas column density at 1<R<6 AU of NH=3x10^19 - 10^21 cm^-2. We find a 5sigma upper limit on NCO at R<1 AU of 5x10^15 cm^-2 (NH<5x10^19 cm^-2). Conclusions: The dust gap in the disk of HD 139614 has gas. The gas surface density in the disk at R<6 AU is significantly lower than the surface density expected from HD 139614s accretion rate assuming a viscous alpha-disk model. The gas density drop, the non-negative density gradient of the gas inside 6 AU, and the absence of a wide (>2 AU) gas gap suggest the presence of an embedded <2 MJ planet at around 4 AU.
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