Euclidean partition function for a relativistic particle

Question

Consider the Euclidean partition function for a relativistic particle. $Z=\int DX e^{-M\int d\tau \sqrt{-\dot{X}^{2}}}$ evaluated for all worldlines with a specific boundary condition. Now , This sum includes all worldlines which involve tachyonic "spacelike" paths , timelike and lightlike. But for a spacelike trajectory , this leads to minus sign under the square root and so the argument of the exponential becomes pure imaginary. So , the path integral becomes a sum of real decaying exponentials and complex exponentials. What are the consequences of this  ? What happens if we use $\sqrt{sgn(\dot{X}^{2})\dot{X}^{2}}$ instead where $sgn$ is the sign function ?

Comments

As you said, it is a sum, not a single integral over a weird path. An in this sum there is a cancelation of paths contributions which are "far" from the classical path. No need in modifying this sum; otherwise you will add weird contributions.