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The extinction law in the inner $3\times3$ deg$^2$ of the Milky Way and the red clump absolute magnitude in the inner bar-bulge

The extinction law from $0.9$ to $8$ microns in the inner $3\times3$ deg$^2$ of the Milky Way is measured using data from VISTA Variables in the Via Lactea, GLIMPSE and WISE. Absolute extinction ratios are found by requiring that the observed red clump density peaks at the GRAVITY collaboration distance to the Galactic centre. When combined with selective extinction ratios measured from the bulge giant colour-colour diagrams, we find an extinction law of $A_Z:A_Y:A_J:A_H:A_{K_s}:A_{W1}:A_{[3.6]}:A_{[4.5]}:A_{W2}:A_{[5.8]}:A_{[8.0]} =7.19(0.30):5.11(0.20):3.23(0.11):1.77(0.04):1:0.54(0.02):0.46(0.03):0.34(0.03):0.32(0.03):0.24(0.04):0.28(0.03)$ valid for low extinctions where non-linearities are unimportant. These results imply an extinction law from the Rayleigh Jeans colour excess method (RJCE) of $A_{K_s}=0.677(H-[4.5]-0.188)$. We find little evidence for significant selective extinction ratio variation over the inspected region (around $5\%$). Assuming the absolute extinction ratios do not vary across the inspected region gives an independent measurement of the absolute $K_s$ magnitude of the red clump at the Galactic Centre of $(-1.61\pm0.07)\,\mathrm{mag}$. This is very similar to the value measured for solar neighbourhood red clump stars giving confidence in the use of red clump stars as standard candles across the Galaxy. As part of our analysis, we inspect the completeness of PSF photometry from the VVV survey using artificial star tests, finding $90\%$ completeness at $K_s\approx16 \,(17)$ in high (low) density regions and good agreement with the number counts with respect to the GALACTICNUCLEUS and DECAPS catalogues over small regions of the survey.