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 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 Yukawa potential of ${L_e-L_{\mu,\tau}}$ force 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 for the perihelion advancement for such Yukawa type fifth force. We find that perihelion advancement is proportional to the square of the semi major axis of the orbit for the Yukawa potential, unlike GR, where it is largest for the nearest planet. We take the observational limits of all planets for which the perihelion advancement is measured and we obtain the gauge boson coupling $g$ in the range $10^{-18}$ to $10^{-16}$ for the mass range $10^{-22}$eV to $10^{-18}$eV. 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.


I. INTRODUCTION
It is well known that deviation from the inverse square law force between the Sun and the planets results in the perihelion precession of the planetary orbits around the Sun. One of the most prominent example is the case of the Einstein's general relativity (GR) which predicts a deviation from Newtonian 1/r 2 gravity. In fact, one of the famous classical tests of GR was to explain the perihelion advancement of the Mercury. There was a mismatch of about 43 arc seconds per century from the observation [1] which could not be explained from Newtonian mechanics by considering all non-relativistic effects such as perturbations from the other Solar System bodies, oblateness of the Sun, etc. GR explains the discrepancy with a prediction of contribution of 42.9799 /Julian century [2]. However there is an uncertainty in the GR prediction which is about 10 −3 arc seconds per century [2][3][4][5] for the Mercury orbit. The current most accurate detection of perihelion precession of Mercury is done by MESSENGER mission [3]. In the near future, more accurate results will come from BepiColombo mission [6]. Other planets also experience such perihelion shift, although the shifts are small since they are at larger distance from the Sun. Their orbital time periods are larger and could not be observed accurately [7,8].
The uncertainty in GR prediction opens up the possibility to explore the existence of Yukawa type potential between the Sun and the planets leading to the fifth force which is a deviation from the inverse-square law. Massless or ultralight scalar, pseudoscalar or vector particles can mediate such fifth force between the Sun and the planets. Many recent papers constrain the fifth force originated from either scalar-tensor theories of gravity [9][10][11] or the dark matter components [11][12][13]. Fifth forces due to ultra light axions was studied in [14]. The unparticle long range force from perihelion precession of Mercury was studied in [15]. Perihelion precession of planets can also constrain the fifth force of dark matter [5]. In this paper, we consider the Yukawa type potential which arises in a gauged L e −L µ,τ scenario and we calculate the perihelion shift of planets (Mercury, Venus, Earth, Mars, Jupiter, and Saturn) due to coupling of the ultralight vector gauge bosons with the electron current of the macroscopic objects along with the GR effect.
In standard model, we can construct three gauge symmetries L e − L µ , L e − L τ , L µ − L τ in an anomaly free way and they can be gauged [16][17][18][19]. L e − L µ and L e − L τ [20][21][22][23] long range forces can be probed in a neutrino oscillation experiment. L µ − L τ long range force cannot be probed in neutrino oscillation experiment because Earth and Sun do not contain any muon charge. Recently in [24], L µ − L τ long range force was probed from the orbital period decay of neutron star-neutron star and neutron star-white dwarf binary systems since they contain large muon charge. However, as the Sun and the planets contain lots of electrons and the number of electrons is approximately equal to the number of baryons, we can probe L e − L µ,τ long range force from the Solar System.
The number of electrons in i'th macroscopic object (Sun or planet) is given by The paper is organised as follows. In section II, we give a detail calculation of the perihelion precession of planets due to such fifth force in the background of the Schwarzschild geometry around the Sun. In section III, we obtain constraints on the L e − L µ,τ gauge coupling and the mass of the gauge boson for planets Mercury, Venus, Earth, Mars, Jupiter, and Saturn and we obtain the exclusion plot of g versus M Z for all the planets mentioned before. In section IV, we summarize our results. We use the natural system of units throughout the paper.

TYPE OF POTENTIAL IN THE CURVED SPACETIME BACKGROUND
The dynamics of a Sun-planet system in presence of a Schwarzschild background and a non gravitational Yukawa type L e − L µ,τ long range force is given by the following action: where . denotes the derivative with respect to the proper time τ , M p is the mass of the planet, g is the coupling constant which couples the classical current J µ = qẋ µ of the planet with the L e − L µ,τ gauge field A µ and q is the charge due to the presence of electrons in the planet. Varying the action Eq. (2.1) we obtain the equation of motion of the planet as In Appendix A, we show the detailed calculation of Eq. (2.2). For the static case A µ = {V (r), 0, 0, 0}, where V (r) is the potential leading to long range L e − L µ,τ Yukawa type of force.
Γ α µν denotes the Christoffel symbol for the background spacetime. For the Sun-Planet system, the background is a Schwarzschild spacetime around the Sun. The christoffel symbols for the metric Eq. (2.3) are given in Appendix B. The Schwarzschild metric outside the Sun is given by where M is the mass of Sun. Here we assume the Newton's gravitational constant G = 1.
Hence, to obtain the solution for temporal part of the Eqs. (2.2), we writë t + 2M where E is the constant of motion. E is identified as the total energy of the system per unit mass.
Similarly, the φ part of Eq. (2.2) isφ After integration we getφ where L is the angular momentum of the system which is a constant of motion.
The radial part of Eq.
The potential V (r) is generated due to the presence of electrons in the Sun and it is given as . Note that we keep only the Yukawa term in the form of V (r) as we are interested in the leading order contribution only (see Appendix C). Hence, from Eq. (2.10) we Where we have neglected g 4 term because the coupling is small and its contribution will be negligible. Here Q = N 1 is the number of electrons in the Sun and q = N 2 is the number of electrons in the planet. For planar motion, L x = L y = 0, and θ = π/2. Since the orbit of the planet is stable, the total energy of the system is negative. So E = −|E|. Hence, we write Eq. (2.12) as The first term on the right hand side of Eq. (2.13) represents the kinetic energy part, the second term is the centrifugal potential part, and the fourth term is the usual Newtonian potential. Due to general relativistic M L 2 r 3 term, there is an advancement of perihelion motion of a planet. The last term arises due to exchange of a U (1) Le−Lµ,τ gauge bosons between electrons of a planet and the Sun. Here, M Z is the mass of the gauge boson. M Z is constrained from the range of the potential which is basically the distance between the planet and the Sun. Usingṙ = L r 2 dr dφ , we write Eq. (2.13) as d dφ (2.14) Applying d dφ on both sides and using the reciprocal coordinate u = 1 r we obtain from Eq. (2.14) Now expanding Eq. (2.15) upto the leading order of M Z we get where M = M + g 2 N 1 N 2 |E|/4πM p .
We assume that u = u 0 (φ) + ∆u(φ), where, u 0 (φ) is the solution of Newton's theory with the effective mass M and ∆u(φ) is the solution due to general relativistic correction and Yukawa potential. Thus we write The solution of Eq. (2.18) is where e is the eccentricity of the planetary orbit. The equation of motion for ∆u(φ) is The solution of Eq. (2.20) is The ∆u increases linearly with φ and contributes to the perihelion precession of planets. Therefore we identify only the related terms in Eq. (2.21) and rewrite ∆u as or, (1−e 2 )(1+e) . Under φ → φ + 2π, u is not same. Hence, the planet does not follow the previous orbit. So the motion of the planet is not periodic. The change in azimuthal angle after one precession is The semi major axis and the orbital angular momentum are related by a = L 2 M (1−e 2 ) . Using this expression in Eq. (2.25) we get . (2.26) In natural system of units Eq. (2.26) is The total energy of the system per unit mass is (2.28) In Appendix D, we discuss Eq. (2.28) in detail. The energy due to gravity is much larger than the energy due to long range Yukawa type force. The last term of Eq. (2.27) indicates that long range force, which arises due to U (1) Le−Lµ,τ gauge boson exchange between the electrons of composite objects, contributes to the perihelion advance of planets within the permissible limit. System. For Mercury planet, we write   We can write from the fifth force constraint Suppose S 1 = M p −g µνẋ µẋν dτ . For this action, the Lagrangian is Hence, the equation of motion is d dτ or, Multiplying g ρσ we have, or, where, Γ ρ µν = 1 2 g ρσ (∂ ν g µσ + ∂ µ g νσ − ∂ σ g µν ) is called the Christoffel symbol. We can choose τ in such a way that dL dτ = 0. This is called affine parametrization. So, or, δS 2 = gq ∂A µ ∂x ν δx ν dx µ + gq A µ d(δx µ ).
Using equation by parts and using the fact that the total derivative term will not contribute to the integration, we can write or, Since µ and ν are dummy indices, we interchange µ and ν in the first term. Hence, we can write Imposing the fact δS 1 + δS 2 = 0 and using Eq. (A4), Eq. (A7) and Eq. (A12) we can writë which matches with Eq. (2.2).