Constraints on long range force from perihelion precession of planets in a gauged L e − L μ,τ scenario
CConstraints on long range force from perihelion precession of planetsin a gauged L e − L µ,τ scenario Tanmay Kumar Poddar,
1, 2, ∗ Subhendra Mohanty, † and Soumya Jana
3, 4, ‡ Theoretical Physics Division, Physical Research Laboratory, Ahmedabad - 380009, India Discipline of Physics, Indian Institute of Technology, Gandhinagar - 382355, India D´epartement de Physique Th´eorique, Universit´e de Gen`eve,24 quai Ernest Ansermet, 1211Gen`eve 4, Switzerland Department of Physics, Sitananda College, Nandigram, 721631, India
Abstract
The standard model particles can be gauged in an anomaly free way by three possible gauge symmetriesnamely L e − L µ , L e − L τ , and L µ − L τ . Of these, L e − L µ and L e − L τ forces can mediate between theSun and the planets and change the perihelion precession of planetary orbits. It is well known that adeviation from the 1 /r 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 ofthe perihelion advancement of the Mercury. We consider the long range Yukawa potential which arisesbetween the Sun and the planets if the mass of the gauge boson is M Z (cid:48) ≤ O (10 − )eV. We derive theformula of perihelion advancement for Yukawa type fifth force due to the mediation of such U (1) L e − L µ,τ gauge bosons. The perihelion advancement for Yukawa potential is proportional to the square of the semimajor axis of the orbit for small M Z (cid:48) , unlike GR, where it is largest for the nearest planet. Howeverfor higher values of M Z (cid:48) , an exponential suppression of the perihelion advancement occurs. We take theobservational limits for all planets for which the perihelion advancement is measured and we obtain theupper bound on the gauge boson coupling g for all the planets. The Mars gives the stronger bound on g for the mass range ≤ − eV and we obtain the exclusion plot. This mass range of gauge boson canbe a possible candidate of fuzzy dark matter whose effect can therefore be observed in the precessionmeasurement of the planetary orbits. ∗ Email Address: [email protected] † Email Address: [email protected] ‡ Email Address: [email protected] a r X i v : . [ h e p - ph ] J u l . INTRODUCTION It is well known that a deviation from the inverse square law force between the Sun and theplanets results in the perihelion precession of the planetary orbits around the Sun. One of the mostprominent example is the case of the Einstein’s general relativity (GR) which predicts a deviationfrom Newtonian 1 /r gravity. In fact, one of the famous classical tests of GR was to explain theperihelion advancement of the Mercury. There was a mismatch of about 43 arc seconds per centuryfrom the observation [1] which could not be explained from Newtonian mechanics by considering allnon-relativistic effects such as perturbations from the other Solar System bodies, oblateness of theSun, etc. GR explains the discrepancy with a prediction of contribution of 42 . (cid:48)(cid:48) / Julian century[2]. However there is an uncertainty in the GR prediction which is about 10 − arc seconds percentury [2–5] for the Mercury orbit. The current most accurate detection of perihelion precessionof Mercury is done by MESSENGER mission [3]. In the near future, more accurate results willcome from BepiColombo mission [6]. Other planets also experience such perihelion shift, althoughthe shifts are small since they are at larger distance from the Sun [7, 8].The uncertainty in GR prediction opens up the possibility to explore the existence of Yukawatype potential between the Sun and the planets leading to the fifth force which is a deviation fromthe inverse-square law. Massless or ultralight scalar, pseudoscalar or vector particles can mediatesuch fifth force between the Sun and the planets. Many recent papers constrain the fifth forceoriginated from either scalar-tensor theories of gravity [9–11] or the dark matter components [11–13]. Fifth forces due to ultra light axions was studied in [14]. Ultra light scalar particles can alsobe probed from the coupling of electron in long range force effects in torsion balance experiment[15]. They can also be probed from superradiance phenomena [16, 17]. The unparticle long rangeforce from perihelion precession of Mercury was studied in [18]. Perihelion precession of planetscan also constrain the fifth force of dark matter [5]. In this paper, we consider the Yukawa typepotential 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 gaugebosons 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 ananomaly free way and they can be gauged [20–23]. L e − L µ and L e − L τ [24–27] long range forces2an be probed in a neutrino oscillation experiment. L µ − L τ long range force cannot be probed inneutrino oscillation experiment because Earth and Sun do not contain any muon charge. However,if there is an inevitable Z − Z (cid:48) mixing, then L µ − L τ force can be probed [28]. Recently in [29, 30], L µ − L τ long range force was probed from the orbital period decay of neutron star-neutron starand neutron star-white dwarf binary systems since they contain large muon charge. However, asthe Sun and the planets contain lots of electrons and the number of electrons is approximatelyequal 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 N i = M i /m n , where M i is the mass of the i’th object and m n is the mass of nucleon which is roughly 1GeV. L e − L µ,τ gauge boson is mediated between the classical electron current sources: Sun and planet as shownin FIG.1. This causes a fifth force between the planet and the Sun along with the gravitationalforce and contributes to the perihelion shift of the planets. The Yukawa type of potential in sucha scenario is V ( r ) (cid:39) g πr e − M Z (cid:48) r , where g is the constant of coupling between the electron and thegauge boson and M Z (cid:48) is the mass of the gauge boson. M Z (cid:48) is restricted by the distance betweenthe Sun and the planet which gives the strongest bound on gauge boson mass M Z (cid:48) < − eV.Therefore, the lower bound of the range of this force is given by λ = M Z (cid:48) > Km. L e − L µ,τ long range force can also be probed from MICROSCOPE experiment [31–33]. In this mass rangethe vector gauge boson can also be a candidate for fuzzy dark matter (FDM), although FDM isusually referred to as ultralight scalars [34, 35].The paper is organised as follows. In section II, we give a detail calculation of the perihelionprecession of planets due to such fifth force in the background of the Schwarzschild geometryaround the Sun. In section III, we obtain constraints on the L e − L µ,τ gauge coupling and themass of the gauge boson for planets Mercury, Venus, Earth, Mars, Jupiter, and Saturn and weobtain the exclusion plot of g versus M Z (cid:48) for all the planets mentioned before. In section IV, wesummarize our results. We use the natural system of units throughout the paper.3IG. 1: Mediation of L e − L µ,τ vector gauge bosons between planet and Sun. II. PERIHELION PRECESSION OF PLANETS DUE TO LONG RANGE YUKAWATYPE OF POTENTIAL IN THE SCHWARZSCHILD SPACETIME BACKGROUND
The dynamics of a Sun-planet system in presence of a Schwarzschild background and a nongravitational Yukawa type L e − L µ,τ long range force is given by the following action: S = − M p (cid:90) (cid:113) − g µν ˙ x µ ˙ x ν dτ − g (cid:90) A µ J µ dτ, (2.1)where “˙” (overdot) denotes the derivative with respect to the proper time τ , g µν is the metrictensor for the background spacetime, M p is the mass of the planet, g is the coupling constant whichcouples the classical current J µ = q ˙ x µ of the planet with the L e − L µ,τ gauge field A µ due to theSun, and q is the total charge due to the presence of electrons in the planet. Varying the actionEq. (2.1), we obtain the equation of motion of the planet as¨ x α + Γ αµν ˙ x µ ˙ x ν = gqM p g αµ ( ∂ µ A ν − ∂ ν A µ ) ˙ x ν . (2.2)In Appendix A, we show the detailed calculation of Eq. (2.2). For the static case A µ = { V ( r ) , , , } , where V ( r ) is the potential leading to a long range L e − L µ,τ Yukawa type force.Γ αµν denotes the Christoffel symbol for the background spacetime. For the Sun-Planet system, the4ackground is a Schwarzschild spacetime outside the Sun and it is described by the line element ds = − (cid:16) − Mr (cid:17) dt + (cid:16) − Mr (cid:17) − dr + r dθ + r sin θdφ , (2.3)where M is the mass of the Sun. The Christoffel symbols for the metric Eq. (2.3) are given inAppendix B.Hence, to obtain the solution for temporal part of the Eq. (2.2), we write¨ t + 2 Mr (cid:16) − Mr (cid:17) ˙ r ˙ t = gqM p (cid:16) − Mr (cid:17) dVdr ˙ r. (2.4)Integrating Eq. (2.4) once, we get ˙ t = (cid:16) E + gqVM p (cid:17)(cid:16) − Mr (cid:17) , (2.5)where E is the constant of motion. E is interpreted as the total energy per unit rest mass for atimelike geodesic relative to a static observer at infinity.Similarly, the φ part of Eq. (2.2) is ¨ φ + 2 r ˙ r ˙ φ = 0 . (2.6)After integration we get ˙ φ = Lr , (2.7)where L is the angular momentum of the system per unit mass, which is also a constant of motion.The radial part of Eq. (2.2) is¨ r − M ˙ r r (cid:16) − Mr (cid:17) + M (cid:16) − Mr (cid:17) r ˙ t − r (cid:16) − Mr (cid:17) ˙ φ = gqM p (cid:16) − Mr (cid:17) dVdr ˙ t. (2.8)Using Eqs. (2.5) and (2.7) in Eq. (2.8), we obtain¨ r + Mr (cid:16) − Mr (cid:17) (cid:16)(cid:16) E + gqVM p (cid:17) − ˙ r (cid:17) − L r (cid:16) − Mr (cid:17) = gqM p (cid:16) E + gqVM p (cid:17) dVdr . (2.9)Again, for a timelike particle g µν ˙ x µ ˙ x ν = − (cid:16) E + gqVM p (cid:17) −
12 = ˙ r L r − M L r − Mr . (2.10)5sing Eq. (2.10) in Eq. (2.9), we get¨ r + 3 M L r + Mr − L r = gqM p (cid:16) E + gqVM p (cid:17) dVdr . (2.11)We can also obtain Eq. (2.11) by directly differentiating Eq. (2.10).The potential V ( r ) is generated due to the presence of electrons in the Sun and it is given as V ( r ) (cid:39) gQ πr e − M Z (cid:48) r + O ( MR ), where R is the radius of the Sun. Note that we keep only the Yukawaterm in the form of V ( r ) as we are interested in the leading order contribution only (see AppendixC). Hence, from Eq. (2.10) we write E −
12 = ˙ r L r − M L r − Mr − g N N E πM p r e − M Z (cid:48) r , (2.12)where we have neglected O ( g ) term because the coupling is small and its contribution will benegligible. Here Q = N is the number of electrons in the Sun and q = N is the number ofelectrons in the planet. For planar motion, L x = L y = 0, and θ = π/
2. The orbit of theplanet is stable when
E <
1. In the presence of gravitational potential and fifth force E = E (cid:39) − M a + g Qq πM p (cid:18) u + u − e − MZ (cid:48) /u + − u u − e − MZ (cid:48) /u − u − u − (cid:19) which is explained in Appendix D.The first term on the right hand side of Eq. (2.12) represents the kinetic energy part, the secondterm is the centrifugal potential part, and the fourth term is the usual Newtonian potential. Dueto general relativistic ML r term, there is an advancement of perihelion motion of a planet. Thelast term arises due to exchange of a U (1) L e − L µ,τ gauge bosons between electrons of a planet andthe Sun. Here, M Z (cid:48) is the mass of the gauge boson. M Z (cid:48) is constrained from the range of thepotential which is basically the distance between the planet and the Sun. Using ˙ r = Lr drdφ , we writeEq. (2.12) as (cid:104) ddφ (cid:16) r (cid:17)(cid:105) + 1 r = E − L + 2 Mr + 2 ML r + g N N E πL rM p e − M Z (cid:48) r . (2.13)Applying ddφ on both sides and using the reciprocal coordinate u = r we obtain from Eq. (2.13) d udφ + u = ML + 3 M u + g N N πL M p e − MZ (cid:48) u + g N N EM Z (cid:48) πL M p u e − MZ (cid:48) u . (2.14)As E appears as a multiplication factor in Eq. (2.14), we take E ≈ M Z (cid:48) , we get d udφ + u = ML + 3 M u + g N N πL M p − g N N M Z (cid:48) πL M p u , (2.15)6here for non circular orbit ddφ (cid:16) r (cid:17) (cid:54) = 0. The first term on the right hand side of Eq. (2.15) isthe usual term which comes in Newton’s theory. The second term is the general relativistic termwhich is a perturbation of Newton’s theory. The last two terms arise due to the presence of longrange Yukawa type potential in the theory.We write Eq. (2.15) as d udφ + u = M (cid:48) L + 3 M u − g N N M Z (cid:48) πL M p u , (2.16)where M (cid:48) = M + g N N / πM p .We assume that u = u ( φ ) + ∆ u ( φ ), where, u ( φ ) is the solution of Newton’s theory with theeffective mass M (cid:48) and ∆ u ( φ ) is the solution due to general relativistic correction and Yukawapotential. Thus we write d u dφ + u = M (cid:48) L . (2.17)The solution of Eq. (2.17) is u = M (cid:48) L (1 + e cos φ ) , (2.18)where e is the eccentricity of the planetary orbit. The equation of motion for ∆ u ( φ ) is d ∆ udφ + ∆ u = 3 M M (cid:48) L (1 + e cos φ + 2 e cos φ ) − g N N M Z (cid:48) L πL M p M (cid:48) (1 + e cos φ + 2 e cos φ ) . (2.19)The solution of Eq. (2.19) is∆ u = 3 M M (cid:48) L (cid:104) e − e φ + eφ sin φ (cid:105) − g N N M Z (cid:48) L πL M p M (cid:48) (cid:104) − cos φe (1 + e cos φ ) +sin φ (1 − e )(1 + e cos φ ) − e (1 − e ) / sin φ cos − (cid:16) e + cos φ e cos φ (cid:17)(cid:105) . (2.20)When ∆ u increases linearly with φ , it contributes to the perihelion precession of planets. Therefore,we identify only the related terms in Eq. (2.20), neglect all other terms, and rewrite ∆ u as∆ u = 3 M M (cid:48) L eφ sin φ + g N N M Z (cid:48) L πM p M (cid:48) e (1 − e )(1 + e ) φ sin φ, (2.21)where we used cos − (cid:16) e +cos φ e cos φ (cid:17) (cid:39) √ − e e φ + O ( φ ).Using Eqs. (2.18) and (2.21), we get the total solution as u = M (cid:48) L (1 + e cos φ ) + 3 M M (cid:48) L eφ sin φ + g N N M Z (cid:48) L πM p M (cid:48) e (1 − e )(1 + e ) φ sin φ, (2.22)7r, u = M (cid:48) L [1 + e cos φ (1 − α )] , (2.23)where, α = 3 M M (cid:48) L + g N N M Z (cid:48) L πM p M (cid:48) − e )(1 + e ) . (2.24)Under φ → φ + 2 π , u is not same. Hence, the planet does not follow the previous orbit. So themotion of the planet is not periodic. The change in azimuthal angle after one precession is∆ φ = 2 π − α − π ≈ πα. (2.25)The semi major axis and the orbital angular momentum are related by a = L M (cid:48) (1 − e ) . Using thisexpression in Eq. (2.25) we get∆ φ = 6 πMa (1 − e ) + g N N M Z (cid:48) a (1 − e )4 M p M (cid:48) (1 + e ) . (2.26)In natural system of units Eq. (2.26) is∆ φ = 6 πGMa (1 − e ) + g N N M Z (cid:48) a (1 − e )4 M p ( GM + g N N πM p )(1 + e ) . (2.27)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) L e − L µ,τ gaugeboson exchange between the electrons of composite objects, contributes to the perihelion advanceof planets within the permissible limit. III. CONSTRAINTS ON U (1) L e − L µ,τ GAUGE COUPLING FOR PLANETS IN SOLARSYSTEM
The contribution of the gauge boson must be within the excess of perihelion advance from theGR prediction, i.e; (∆ φ ) obs − (∆ φ ) GR ≥ (∆ φ ) L e − L µ,τ . The first term in the right hand side ofEq. (2.27) is (∆ φ ) GR and the second term is (∆ φ ) L e − L µ,τ . Putting the observed and GR values for(∆ φ ), we can constrain the U (1) L e − L µ,τ gauge coupling constants for all the planets in our SolarSystem. For Mercury planet, we write g N N M Z (cid:48) a (1 − e )4 M p ( GM + g N N πM p )(1 + e ) (cid:16) century T (cid:17) < . × − arcsecond / century , (3.1)8here 3 × − arcsecond/century is the uncertainty in the perihelion advancement from its GRprediction and put upper bound on the gauge coupling. T = 88 days is the orbital time period ofMercury. Similarly, we can put upper bounds on g for other planets. In this section, we constrainthe U (1) L e − L µ,τ gauge coupling from the observed perihelion advancement of the planets in theSolar System. We consider six planets: Mercury, Venus, Earth, Mars, Jupiter, and Saturn. Here,we take the mass of the Sun as M = 10 GeV. Using Eqs. (2.27), we put an upper bound on g fromthe uncertainty of their perihelion advance. In TABLE I, we obtain the upper bound on massesof the gauge bosons which are mediated between the Sun and the planets and, in TABLE II, weshow the constraints on the gauge coupling constants from the uncertainties [19, 36] of perihelionadvance.TABLE I: Summary of the masses, eccentricities [37] of the orbits, perihelion distances from theSun and upper bounds on gauge boson mass M Z (cid:48) which are mediated between the planets andSun in our Solar System. Planet Mass M p (GeV) Eccentricity (e) Perihelion distance a (AU) Mass of gauge boson M Z (cid:48) (eV)Mercury 1 . × .
206 0 . ≤ . × − Venus 2 . × .
007 0 . ≤ . × − Earth 3 . × .
017 0 . ≤ . × − Mars 3 . × .
093 1 . ≤ . × − Jupiter 1 . × .
048 4 . ≤ . × − Saturn 3 . × .
056 9 . ≤ . × − We can write from the fifth force constraint g N N πGM M p < . (3.2)This gives the upper bound on g as g < . × − for all the planets. In FIG.2 we show thevalues of gauge coupling of the planets corresponding to the planet-Sun distance. For U (1) L e − L µ,τ vector gauge bosons exchange between the planet and the Sun, the mass of the gauge boson is M Z (cid:48) ≤ O (10 − ) eV . In FIG.3, we obtain the exclusion plots of gauge boson electron couplingfor the six planets by numerically solving Eqs. (2.14). There is an extra multiplicative factor9ABLE II: Summary of the uncertainties in the perihelion advance in arcseconds per centuryand upper bounds on gauge boson-electron coupling g for the values of M Z (cid:48) discussed in TABLEIfor planets in our Solar System. Planet Uncertainty in perihelion advance (as/cy) g from perihelion advanceMercury 3 . × − ≤ . × − Venus 1 . × − ≤ . × − Earth 1 . × − ≤ . × − Mars 3 . × − ≤ . × − Jupiter 2 . × − ≤ . × − Saturn 4 . × − ≤ . × − Torsion Balance Mars Earth MercurySaturn VenusJupiter × - × - × - × - × - × - × - - - - - - M Z ' ( in eV ) g FIG. 2: Values of the gauge coupling of each planets corresponding to the Sun-planet distanceobtained from TABLEII. Violet dot is for Jupiter planet, blue dot is for Mercury planet, blackdot is for Venus, cyan dot is for Saturn, green dot is for Earth and yellow dot is for Mars. Theyellow shaded region is excluded from the torsion balance experimentsexp[ − M (cid:48) Z L M (cid:48) ] in the expression of α if we solve Eqs. (2.14) numerically in order to incorporatethe exponential suppresion due to higher values of M Z (cid:48) . The regions above the coloured lines10 ercuryVenusEarthMarsJupiterSaturn - - - - - - - - - - - - - - - M Z ' ( in eV ) g FIG. 3: Plot of coupling constant g vs the mass of the gauge bosons M (cid:48) Z for all the planets.Violet line is for Jupiter planet, red line is for Mercury planet, black line is for Venus, cyan line isfor Saturn, green line is for Earth and yellow line is for Mars.corresponding to every planets are excluded. Eqs. (2.27) suggests that the perihelion shift dueto the mediation of L e − L µ,τ gauge bosons is proportional to the square of the semi major axis.This is completely opposite from the standard GR result where the perihelion shift is inverselyproportional to a for small M Z (cid:48) . However, for higher values of M Z (cid:48) , the exponential suppressionstarts dominating. So the contribution of the gauge boson mediation for perihelion shift is largerfor outer planets. However it also depends on the available uncertainties for perihelion precessionof the planets and other parameters like orbital time period and eccentricity. From TABLE II,we obtain the stronger bound on the gauge boson coupling is g ≤ O (10 − ). From FIG.3 it isclear that the Mars gives the strongest bound among all the planets considered. As we go to thelower mass region, the exponential term in the potential will become less effective and the Yukawapotential effectively becomes Coulomb potential at M Z (cid:48) →
0. Thus it will be degenerate with1 /r -Newtonian force and will not contribute to the perihelion precession of planets at all. So aswe go to the lower mass ( < − eV ) region, we get weaker bound on g . On the other hand, forhigher mass region ( > − eV ) the long range force theory breaks down and, thus we can not go11rbitrarily for higher masses. IV. DISCUSSIONS
Since the Sun and the planets contain a significant number of electrons, long range Yukawa typefifth force can be mediated between the electrons of Sun and planet in a gauged L e − L µ,τ scenario.Also there can be the dipole radiation of the gauge bosson for the planeraty orbits. Following ourprevious work [29] on compact binary systems in a gauged L µ − L τ scenario, the energy loss dueto dipole radiation is proportional to the fourth power of the orbital frequency. For planet-Sunbinary system, the orbital frequency is smaller than the orbital frequency of the compact binarysystems. Hence, the contribution due to dipole radiation for the planetary systems is smaller andits effect will be neglected for planetary motion.This ultralight vector gauge bosons mediated between the Sun and the planets can contributeto the perihelion shift in addition to the GR prediction. From the perihelion shift calculation inpresence of a long range Yukawa type potential, we obtain an upper bound on the gauge coupling g ≤ O (10 − ) in a gauged L e − L µ,τ scenario. The mass of the gauge bosons is constrained bythe distance between the Sun and the planet which gives M Z (cid:48) ≤ O (10 − )eV. The electron-gaugeboson coupling obtained from perihelion shift measurement is six order of magnitude more stringentthan our fifth force constraint Eq. (3.2). From Eq. (2.27) we conclude that, while the precession ofperihelion due to GR is largely contributed by the planets close to Sun, the contribution of vectorgauge bosons in perihelion precession is dominated by the outer planets.The bound on coupling g that we have obtained is not only as good as the torsion balance [38]or the neutrino oscillation experiment [25], but also our results possess additional importance forthe following reasons:( a ) Our analysis of the perihelion precession is sensitive to the magnitude of the potential andthe nature of the potential, i.e. the deviation from the inverse square law.( b ) In our analysis, we are probing larger distance (upto the planet Saturn) compare to the earthSun distance.( c ) Since the perihelion shift depends on the value of uncertainty in GR prediction, the future12epiColombo mission [6] can give more accurate result and the bound on coupling willbecome even more stronger.Moreover, we emphasize the novel physics behind the work which suggests that we can studythe gauge boson electron coupling in a gauged L e − L µ,τ scenario by planetary observations andwe can constrain the arising long range force from perihelion precession of planets. These gaugebosons ( M Z (cid:48) ≤ − eV) can be a possible candidate of fuzzy dark matter and can be probed fromprecession measurement of planetary orbits. ACKNOWLEDGMENTS
SJ was supported by the Swiss Government Excellence Scholarship 2019 (Postdoctoral) forforeign researchers offered via the Federal Commission for Scholarships (FCS) for Foreign Students.
Appendix A: Equation of motion of a planet in presence of a Schwarzschild backgroundand a non gravitational Yukawa type of potential
The action which describes the motion of a planet in Schwarzschild background and a nongravitational long range Yukawa type of potential is given by Eq. (2.1).Suppose S = M p (cid:82) (cid:112) − g µν ˙ x µ ˙ x ν dτ . For this action, the Lagrangian is L = M p (cid:114) g µν dx µ dτ dx ν dτ . (A1)Hence, the equation of motion is ddτ (cid:16) ∂ L ∂ (cid:0) ∂x σ ∂τ (cid:1) (cid:17) − ∂ L ∂x σ = 0 , (A2)or, 1 L d L dτ g µσ dx µ dτ = g µσ d x µ dτ + ∂ α g µσ dx α dτ dx µ dτ − ∂ σ g µν dx µ dτ dx ν dτ . (A3)Multiplying g ρσ we have, d x ρ dτ + g ρσ ∂ ν g µσ dx ν dτ dx µ dτ − g ρσ ∂ σ g µν dx µ dτ dx ν dτ = 1 L d L dτ dx ρ dτ , (A4)13r, d x ρ dτ + 12 g ρσ ( ∂ ν g µσ + ∂ µ g νσ − ∂ σ g µν ) dx µ dτ dx ν dτ = 1 L (cid:16) d L dτ (cid:17) dx ρ dτ , (A5)or, d x ρ dτ + Γ ρµν dx µ dτ dx ν dτ = 1 L d L dτ dx ρ dτ , (A6)where, Γ ρµν = g ρσ ( ∂ ν g µσ + ∂ µ g νσ − ∂ σ g µν ) is called the Christoffel symbol. We can choose τ insuch a way that d L dτ = 0. This is called affine parametrization. So, d x ρ dτ + Γ ρµν dx µ dτ dx ν dτ = 0 . (A7)Suppose S = gq (cid:82) A µ dx µ dτ dτ = gq (cid:82) A µ dx µ . Hence, δS = gq (cid:90) δA µ dx µ + gq (cid:90) A µ δ ( dx µ ) , (A8)or, δS = gq (cid:90) ∂A µ ∂x ν δx ν dx µ + gq (cid:90) A µ d ( δx µ ) . (A9)Using integration by parts and using the fact that the total derivative term will not contribute tothe integration, we can write δS = gq (cid:90) ∂A µ ∂x ν δx ν dx µ − gq (cid:90) dA µ δx µ . (A10)or, δS = gq (cid:90) ∂A µ ∂x ν δx ν dx µ − gq (cid:90) ∂A µ ∂x ν dx ν δx µ . (A11)Since µ and ν are dummy indices, we interchange µ and ν in the first term. Hence, we can write δS = gq (cid:90) ( ∂ µ A ν − ∂ ν A µ ) dx ν δx µ = gq (cid:90) ( ∂ µ A ν − ∂ ν A µ ) dx ν dτ δx µ dτ. (A12)Imposing the fact δS + δS = 0 and using Eq. (A4), Eq. (A7) and Eq. (A12) we can write¨ x ρ + Γ ρµν ˙ x µ ˙ x ν = gqM p g ρµ ( ∂ µ A ν − ∂ ν A µ ) ˙ x ν , (A13)which matches with Eq. (2.2). 14 ppendix B: Christoffel symbols for the Schwarzschiild metric The christoffel symbols for the Schwarzschiild metric defined in Eq. (2.3) areΓ trt = Mr (cid:16) − Mr (cid:17) , Γ rtt = Mr (cid:16) − Mr (cid:17) , Γ rrr = − Mr (cid:16) − Mr (cid:17) , Γ rθθ = − r (cid:16) − Mr (cid:17) Γ rφφ = − r sin θ (cid:16) − Mr (cid:17) , Γ θrθ = 1 r , Γ θφφ = − sin θ cos θ, Γ φφr = 1 r , Γ φθφ = cot θ. (B1) Appendix C: Equation of motion for the vector field A µ The vector field A µ satisfies the Klein-Gordon equation (cid:3) A µ = M Z (cid:48) A µ . (C1)Now, for the static case, A µ = { V ( r ) , , , } . Hence, (cid:3) V ( r ) = M Z (cid:48) V ( r ) . (C2)In the background of the Schwarzschild spacetime, Eq. (C2) becomes (cid:16) − Mr (cid:17) d Vdr + 2 r (cid:16) − Mr (cid:17) dVdr = M Z (cid:48) V ( r ) . (C3)So, in the Schwarzschild background, V ( r ) will not satisfy the Klein-Gordon equation. So weexpand V ( r ) in a perturbation series where the perturbation parameter is MR , and the leadingorder term is the Yukawa term. Let, V ( r ) = V ( r ) + MR V ( r ) + O (cid:16) MR (cid:17) , (C4)where V ( r ) = c e − M (cid:48) Z r r , c = g N N π , (C5)such that d V dr + 2 r dV dr = M Z (cid:48) V . (C6)Inserting Eq. (C4) in Eq. (C3), we get the equation for V ( r )1 R d V dr + 2 rR dV dr = M Z (cid:48) V R + 2 r d V dr + 2 r dV dr . (C7)15et, V ( r ) = χ ( r ) e − M (cid:48) Z r r . (C8)Now, Eq. (C7) becomes 1 R d χdr − R M (cid:48) Z dχdr = 2 c (cid:16) M Z (cid:48) r + 1 r + M (cid:48) Z r (cid:17) . (C9)Integrating Eq. (C9) once we get dχdr − M (cid:48) Z χ = 2 cR (cid:104) M Z (cid:48) ln( M (cid:48) Z r ) − r − M (cid:48) Z r (cid:105) + k R, (C10)where k is the integration constant. Eq. (C10) can be written as ddr (cid:16) e − M (cid:48) Z r χ (cid:17) = 2 cRe − M (cid:48) Z r (cid:104) M Z (cid:48) ln( M (cid:48) Z r ) − r − M (cid:48) Z r (cid:105) + k Re − M (cid:48) Z r . (C11)From Eq. (C11), we can write e − M (cid:48) Z r χ ( r ) = 2 cR (cid:104) M Z (cid:48) (cid:90) r ∞ e − M (cid:48) Z x ln( M (cid:48) Z x ) dx − (cid:90) r ∞ e − M (cid:48) Z x x dx − (cid:90) r ∞ M (cid:48) Z e − M (cid:48) Z x x dx (cid:105) − k R M (cid:48) Z e − M (cid:48) Z r + k , (C12)where k is an integration constant. Doing integration by parts, Eq. (C12) becomes χ ( r ) = cR (cid:104) − M (cid:48) Z ln( M (cid:48) Z r ) + 1 r + M (cid:48) Z e M (cid:48) Z r E i ( − M (cid:48) Z r ) (cid:105) − k R M (cid:48) Z + k e M (cid:48) Z r , (C13)where E i ( x ) is a special function called the exponential integral function which is defined as E i ( x ) = − (cid:90) ∞− x e − t t dt. (C14)We chose k = 0 as e M (cid:48) Z r diverges. We also chose k = 0 as we are looking for particular integral.Hence, from Eq. (C13) we get V ( r ) = cRe − M (cid:48) Z r r (cid:104) r − M (cid:48) Z ln( M (cid:48) Z r ) + M (cid:48) Z e M (cid:48) Z r E i ( − M (cid:48) Z r ) (cid:105) . (C15)So the total solution of the potential is V ( r ) = ce − M (cid:48) Z r r (cid:104) Mr { − M (cid:48) Z r ln( M (cid:48) Z r ) + M (cid:48) Z re M (cid:48) Z r E i ( − M (cid:48) Z r ) } (cid:105) + O (cid:16) M R (cid:17) . (C16)We take the leading order term which is the Yukawa term in our calculation. The higher orderterms are comparatively small. 16 ppendix D: Total energy of the binary system due to gravity and long range Yukawatype potential For Newtonian gravity, we can write E − L = − a (1 − e ) , ML = 2 a (1 − e ) . (D1)Dividing the above two expression, we obtain E − M = − a , (D2)or, E (cid:39) (cid:114) − Ma ≈ − M a . (D3)In presence of long range Yukawa potential, we obtain E from the condition dudφ = 0 at u = u + =1 /a (1 + e ) (aphelion) and u = u − = 1 /a (1 − e ) (perihelion), E (cid:39) − M a + g Qq πM p (cid:18) u + u − e − M Z (cid:48) /u + − u u − e − M Z (cid:48) /u − u − u − (cid:19) (D4)where 1 in the right hand side is the rest energy per unit mass in the Minkowski background. Thesecond term is ≈ − and the third Yukawa term is smaller than the Newtonian term. [1] I. Shapiro, ”Solar system tests of general relativity: recent results and presentplans” , Proceedings ofthe 12th International Conference on General Relativity and Gravitation, University of Colorado atBoulder, Cambridge University Press, Cambridge, 313-330, 1990.[2] R. S. Park et al, The Astronomical Journal , 121 (2017).[3] A. Genova et al, Nature Communication , 289 (2018).[4] L. Iorio, Planetary and Space Science 55, 1290 (2007).[5] B. Sun, Z. Cao, and L. Shao, Phys. Rev. D , 084030 (2019).[6] C. M. Will, Phys. Rev. Lett. 120, 191101 (2018).[7] A. Biswas, K. R. S. Mani, Cent. Eur. J. Phys. 6(3)(2008) 754-758.[8] L. Iorio, The Astronomical Journal, 137:36153618, 2009 March.
9] T. Liu, X. Zhang, and W. Zhao, Phys. Lett. B , 286-293 (2018).[10] Soumya Jana, Subhendra Mohanty., Constraints on f(R) theories of gravity from GW170817Phys.Rev.D99,(2019)no4 ,044056.[11] S. Alexander, E. McDonough, R. Sims, and N. Yunes, Class. Quant. Grav. , 235012 (2018).[12] D. Croon, A. E. Nelson, C. Sun, D. G. E. Walker, and Z.-Z. Xianyu, ApJ Lett. :L2 (5pp), 2018.[13] J. Kopp, R. Laha, T. Opferkuch, and W. Shepherd, arXiv:1807.02527.[14] T. K. Poddar, S. Mohanty, S. Jana, Phys. Rev. D , 083007 (2020).[15] K. S. Babu, G. Chauhan, P. S. B. Dev, arXiv:1912.13488.[16] M. Baryakhtar, R. Lasenby, M. Teo, Phys. Rev. D 96, 035019.[17] H. Davoudiasl, P. B. Denton, PhysRevLett.123.021102.[18] S. Das, S. Mohanty, K. Rao, Phys. Rev. D , 076001 (2008).[19] C. M. Will, arXiv:1805.10523.[20] R.Foot, Mod. Phys. Lett. A 6 , 527 (1991).[21] X.-G. He, G.C.Joshi, H. Lew and R.R. Volkas, Phys. Rev
D 44 , 2118 (1991).[22] R. Foot, X.-G. He, H. Lew, and R. R. Volkas Phys. Rev. D , 4571 (1994).[23] G. Dutta, A.S. Joshipura, and K. B. Vijaykumar, Phys. Rev. D 50 , 2109 (1994).[24] J. A. Grifols, E. Masso, Phys.Lett. B579 (2004) 123-126.[25] A. S. Joshipura, S. Mohanty, Phys. Lett. B584 (2004) 103-108.[26] A. Bandyopadhyay, A. Dighe, A. S. Joshipura, Phys. Rev. D 75, 093005 (2007).[27] M. Bustamante, S.K.Agarwalla, Phys. Rev. Lett. 122, 061103 (2019).[28] J. Heeck, W. Rodejohann, J.Phys.G38:085005,2011.[29] T. K. Poddar, S. Mohanty, S. Jana, Phys. Rev. D 100, 123023 (2019).[30] J. A. Dror, R. Laha, T. Opferkuch, arXiv:1909.12845.[31] P. Touboul et.al, Phys. Rev. Lett. 119, 231101 (2017).[32] P. Fayet, Phys. Rev. D 97, 055039 (2018).[33] P. Fayet, Phys. Rev. D 99, 055043 (2019).[34] W. Hu, R. Barkana, and A. Gruzinov, Phys. Rev. Lett. , 1158 (2000).[35] L. Hui, J. P. Ostriker, S. Tremaine, E. Witten, Phys. Rev. D , 043541 (2017).[36] E. V. Pitjeva, N. V. Pitjev, MNRAS , 34313437 (2013).
37] https://solarsystem.nasa.gov/planets/mercury/by-the-numbers/[38] T A Wagner, S Schlamminger, J. H. Gundlach and E. G.Adelberger, Class. Quant. Grav., vol. 29, p.184002, 2012, 1207.2442.37] https://solarsystem.nasa.gov/planets/mercury/by-the-numbers/[38] T A Wagner, S Schlamminger, J. H. Gundlach and E. G.Adelberger, Class. Quant. Grav., vol. 29, p.184002, 2012, 1207.2442.