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On the distance of the globular cluster M4 (NGC 6121) using RR Lyrae stars: II. Mid-infrared period-luminosity relations

New mid-infrared period-luminosity (PL) relations are presented for \rrl{} variables in the globular cluster M4 (NGC 6121). Accurate photometry was obtained for 37 \rrl{} variables using observations from the Infrared Array Camera onboard 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, weestablished calibrated PL relations at 3.6 and 4.5~\micron{} using published Hubble Space Telescopegeometric parallaxes of five Galactic \rrl{} stars. The resulting band-averaged distance modulus for M4 is $ μ= 11.399 \pm 0.007 \textrm{(stat)} \pm 0.080 \textrm{(syst)} \pm 0.015 \textrm{(cal)} \pm 0.020 \textrm{(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 a MIR PC relation suggests that \rrl{} stars are not affected by CO absorptionin the 4.5~\micron{} band.