Constraints on long range force from perihelion precession of planets in a gauged $L_e-L_{\mu,\tau}$ scenario

The standard model particles can be gauged in an anomaly free way by three possible gauge symmetries namely ${L_e-L_\mu}$, ${L_e-L_\tau}$, and ${L_\mu-L_\tau}$. Of these, ${L_e-L_\mu}$ and ${L_e-L_\tau}$ forces can mediate between the Sun and the planets and change the perihelion precession of planetary orbits. It is well known that a deviation from the $1/r^2$ Newtonian force can give rise to a perihelion advancement in the planetary orbit, for instance, as in the well known case of Einstein's gravity which was tested from the observation of the perihelion advancement of the Mercury. We consider the long range Yukawa potential which arises between the Sun and the planets if the mass of the gauge boson is $M_{Z^{\prime}}\leq \mathcal{O}(10^{-19})\rm {eV}$. We derive the formula of perihelion advancement for Yukawa type fifth force due to the mediation of such $U(1)_{L_e-L_{\mu,\tau}}$ gauge bosons. The perihelion advancement for Yukawa potential is proportional to the square of the semi major axis of the orbit for small $M_{Z^{\prime}}$, unlike GR, where it is largest for the nearest planet. However for higher values of $M_{Z^{\prime}}$, an exponential suppression of the perihelion advancement occurs. We take the observational limits for all planets for which the perihelion advancement is measured and we obtain the upper bound on the gauge boson coupling $g$ for all the planets. The Mars gives the stronger bound on $g$ for the mass range $\leq 10^{-19}\rm{eV}$ and we obtain the exclusion plot. This mass range of gauge boson can be a possible candidate of fuzzy dark matter whose effect can therefore be observed in the precession measurement of the planetary orbits.