Measuring the Heating and Cooling of the Interstellar Medium at High
  redshift: PAH and [C II] Observations of the Same Star Forming Galaxies at
  z~2
  Star formation depends critically on cooling mechanisms in the interstellar
medium (ISM); however, thermal properties of gas in galaxies at the peak epoch
of star formation (z ~ 2) remain poorly understood. A limiting factor in
understanding the multiphase ISM is the lack of multiple tracers detected in
the same galaxies, such as Polycyclic Aromatic Hydrocarbon (PAH) emission, a
tracer of a critical photoelectric heating mechanism in interstellar gas, and
[C II] 158{\mu}m fine-structure emission, a principal coolant. We present ALMA
Band 9 observations targeting [C II] in six z ~ 2 star-forming galaxies with
strong Spitzer IRS detections of PAH emission. All six galaxies are detected in
dust continuum and marginally resolved. We compare the properties of PAH and [C
II] emission, and constrain their relationship as a function of total infrared
luminosity (LIR) and IR surface density. [C II] emission is detected in one
galaxy at high signal-to-noise (34{\sigma}), and we place a secure upper limit
on a second source. The rest of our sample are not detected in [C II] likely
due to redshift uncertainties and narrow ALMA bandpass windows. Our results are
consistent with the deficit in [C II]/LIR and PAH/LIR observed in the
literature. However, the ratio of [C II] to PAH emission at z ~ 2 is possibly
much lower than what is observed in nearby dusty star-forming galaxies. This
could be the result of enhanced cooling via [O I] at high-z, hotter gas and
dust temperatures, and/or a reduction in the photoelectric efficiency, in which
the coupling between interstellar radiation and gas heating is diminished.
