The AISI Type 4140 alloy steel can certainly be welded to mild steel with a local preheat of 400 deg F. I am presuming certain things here by your statement of "mild steel". I would know for certain your "mild steel" base material composition, prior to welding, to assure your carbon content is less than 0.35%. The carbon equivalent for 4140 is approximately 0.75, meaning this base metal requires a preheat and subsequent stress relief (post weld heat treatment).
I am not sure about your choice of GTAW wire so my suggestion would be to use SFA 5.28 ER80S-B2. This filler metal would better match the 4140 and is very compatible for joining to carbon steel.
I presume the air hardening 4140 is normalized, with an average hardness of 300 BHN. I could not determine if your method of attachment will be by fillet weld or a partial penetration groove weld. Can you provide information as to your weld joint configuration?
After the completion of welding, you could either perform a local post weld heat treatment or if the component is small enough to handle, place it in an oven. The recommended post weld heat treatment temperature is 1075 deg F for 30 minutes at temperature to avoid softening the 4140 base material. The 1075 deg F PWHT will have little affect on the "mild steel" base material. Follow a gradual heat up and cool down ramp rate at less than 600 deg F per hour to avoid distortion.