ON THE DISTANCE OF THE GLOBULAR CLUSTER M4 (NGC 6121) USING RR LYRAE STARS. II. MID-INFRARED PERIOD–LUMINOSITY RELATIONS

New mid-infrared (MIR) period–luminosity (PL) relations are presented for RR Lyræ variables in the globular cluster M4 (NGC 6121). Accurate photometry was obtained for 37 RR Lyræ variables using observations from the Infrared Array Camera on board the Spitzer Space Telescope. The dispersion of M4's PL relations is 0.056, and the uncertainty in the slope is 0.11 mag. Additionally, we established calibrated PL relations at 3.6 and 4.5 μm using published Hubble Space Telescope geometric parallaxes of five Galactic RR Lyræ stars. The resulting band-averaged distance modulus for M4 is $\mu =11.399\pm 0.007(\mathrm{stat})$ ± $0.080(\mathrm{syst})$ ± $0.015(\mathrm{cal})\pm 0.020(\mathrm{ext})$. The systematic uncertainty will be greatly reduced when parallaxes of more stars become available from the GAIA mission. Optical and infrared period–color (PC) relations are also presented, and the lack of an MIR PC relation suggests that RR Lyræ stars are not affected by CO absorption in the 4.5 μm band.